We're talking about theoretic possibilities, because a lot of it hasn't been studied.
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The smallest theoretic mass for a star is around 0.07 to 0.08 solar masses.
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"We define housing as something other than healthcare but that's purely a theoretic definition," Bamberger said.
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And it was an amazing surprise—why should this topological tool prove a graph theoretic thing?
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Prudence is not a "theoretic" virtue, not a virtue only concerned with the truth of something.
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" ON THE "MODERN MONETARY THEORY" DEBATED IN THE U.S. "I don't think there is enough theoretic background to this idea.
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The theoretic risk of torsion was present in my case because the cyst was lopsided and more than seven centimeters in diameter.
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Instead, the goal is to prevent "information theoretic brain death," whereby the "stuff" that's needed to restore a person's personality and memories remains intact.
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He accuses quantum information theorists of confusing different kinds of entropy—the thermodynamic and information-theoretic kinds—and using the latter in domains where it doesn't apply.
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We dig into the company's approach, which is based on a methodology developed at MIT called Systems Theoretic Process Analysis (STPA) as the foundation for Ike's product development.
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This information theoretic criterion of death is obviously much more difficult to meet than current legal or medical definitions, hence the belief that cryopreserved patients are not actually dead.
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The general idea is that, while a person may be considered legally dead, they may not have attained information theoretic death, whereby integral brain structures have been irrevocably destroyed.
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A major theoretic justification of abortion in this and other countries came from Margaret Sanger's eugenic theories, from the fear of the increase of babies born to black or other supposedly inferior women.
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Under the right circumstances, conservatives can be important for the development of good liberal law However, the game-theoretic scholarship on the institutional Supreme Court is not uniformly so optimistic about judicial decision-making.
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I could just trust the game-theoretic and mathematical properties of the platform and then I can program new things and there's been, the early things, Bitcoin is a great example, I can program money.
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I recently spoke to Randol Aikin, the head of systems engineering at self-driving trucks startup Ike Robotics, about the company's approach, which is based on a methodology developed at MIT called Systems Theoretic Process Analysis.
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"(China's) measures are unlikely to see full implementation and are designed to tilt game-theoretic odds in favor of a compromise," said Karl Schamotta, director of global product & market strategy at Cambridge Global Payments in Toronto.
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She says that her research into bubbles of nothing aims, in part, at establishing what the implications are for string-theoretic descriptions of the universe, given that spacetime decay via a bubble of nothing is very unlikely.
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"Our result shows that quantum information processing really does provide benefits — without having to rely on unproven complexity-theoretic conjectures," said Robert König, a co-author of the paper, in a statement from the Technical University of Munich, where he teaches.
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From 1879 to 19703 the philosopher-mathematicians Gottlob Frege, Bertrand Russell, Kurt Gödel, Alonzo Church and Alan Turing invented symbolic logic, helped establish the set-theoretic foundations of mathematics, and gave us the formal theory of computation that ushered in the digital age.
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Less painful and rigorous, and hence more promising, is the langsec initiative: The Language-theoretic approach (LANGSEC) regards the Internet insecurity epidemic as a consequence of ad hoc programming of input handling at all layers of network stacks, and in other kinds of software stacks.
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This explanation builds, for example, on the intuition of a game-theoretic model by Jeffrey Lax and Charles Cameron (23): A forceful judge who disagrees with another judge's views incentivizes that second judge to clarify and hone her argument, especially if she is trying to win him over.
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In these cases, so-called information theoretic brain death—a term coined by cryoncist and futurist Ralph Merkle in 1994—will have been achieved if the cancer, for example, destroys critical brain functions that future technologies, no matter how advanced, will not be able to recover and restore.
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"Seeking Truth," the party's leading theoretic journal, recently celebrated the fact that the "People's Leader" had handled the disaster with unflappable confidence, proving himself to be not only "the guiding light of China and the backbone of 1.4 billion Chinese," but also a "calming balm" for a world whose nerves had been jangled by the outbreak.
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While his work has inspired discussion of accelerated industrialization, a broad cultural gesture toward the digital, and all things that make New York City modern, now Marianne Preger-Simon's memoir Dancing with Merce Cunningham rescues the conversation about her and fellow artists' lived experiences from weighty, theoretic explanations of Cunningham that attempt to talk over their dancing.
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On the one hand, you could see this as simply assuming a new costume for a new career stage, playing another role, especially since it can be traced to the rollout of her fourth album, "Cheek to Cheek," a compilation of jazz standards made in collaboration with Tony Bennett, or even as yet another example of the way fashion co-opts its own theoretic antitheses, be they ripped jeans or down jackets, and adapts them into its own version of the same.
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There are three widely-used approaches to the semantics of Datalog programs: model-theoretic, fixed-point, and proof-theoretic.
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With model-theoretic means, a stronger notion is obtained: an extension T_2 of a theory T_1 is model-theoretically conservative if T_1 \subseteq T_2 and every model of T_1 can be expanded to a model of T_2. Each model-theoretic conservative extension also is a (proof-theoretic) conservative extension in the above sense. The model theoretic notion has the advantage over the proof theoretic one that it does not depend so much on the language at hand; on the other hand, it is usually harder to establish model theoretic conservativity.
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The number theoretic Hilbert transform is an extension () of the discrete Hilbert transform to integers modulo an appropriate prime number. In this it follows the generalization of discrete Fourier transform to number theoretic transforms. The number theoretic Hilbert transform can be used to generate sets of orthogonal discrete sequences().
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Therefore, some of the more difficult machinery used in set-theoretic forcing can be eliminated or substantially simplified when defining forcing in recursion theory. But while the machinery may be somewhat different, recursion-theoretic and set-theoretic forcing are properly regarded as an application of the same technique to different classes of formulas.
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Recent work by Hamkins proposes a more flexible alternative: a set-theoretic multiverse allowing free passage between set- theoretic universes that satisfy the continuum hypothesis and other universes that do not.
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He was accused of heresy because of his interest in philosophy. In one case Al- Amidi defended philosophical doctrine against the criticism of well known Ash’ari theologian Fakhr al-Din al Razi. He also had interest in pre-theoretic belief, creating A Treatise on the Division of Theoretical Scholarship, to explain the difference between pre-theoretic and theoretic belief.
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It is possible to interpret Gentzen's proof in game-theoretic terms .
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The following contains the basic principles of decision-theoretic rough sets.
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Several set-theoretic definitions of the ordered pair are given below.
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Girard's paradox is the type-theoretic analogue of Russell's paradox in set theory.
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Group-theoretic Algorithms for Matrix Multiplication. . Proceedings of the 46th Annual Symposium on Foundations of Computer Science, 23–25 October 2005, Pittsburgh, PA, IEEE Computer Society, pp. 379–388.Henry Cohn, Chris Umans. A Group-theoretic Approach to Fast Matrix Multiplication. .
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Lattice-theoretic characterizations of this type also exist for solvable groups and perfect groups .
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It also helped him to formulate a theoretic explanation of his own supersensual vision.
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285–330 .Martin Shubik (1987). A Game-Theoretic Approach to Political Economy. MIT Press. Description.
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Contract-theoretic screening models have been successfully tested in laboratory experiments and using field data.
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As such the measure theoretic formalization of the concept also serves as a unifying discipline.
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In contrast to the generative-enumerative (proof- theoretic) approach to syntax assumed by transformational grammar, arc pair grammar takes a model-theoretic approach. In arc pair grammar, linguistic laws and language-specific rules of grammar are formalized as axiomatic logical statements. Sentences of a language, understood as structures of a certain type, follow the set of linguistic laws and language-specific statements. This reduces grammaticality to the logically satisfiable notion of model-theoretic satisfaction.
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Then the (scheme-theoretic) intersection of Lagrangian submanifolds of M carries a canonical symmetric obstruction theory.
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Sound diffusers have been based on number-theoretic concepts such as primitive roots and quadratic residues.
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Pre-theoretic belief is a term used in philosophical arguments for and against libertarianism and determinism.
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P. Herbert, J.T. Yao, Game-theoretic Rough Sets, Fundamenta Informaticae, , 108 (3–4): pp. 267–286, 2011.J.T. Yao, J.P. Herbert, A Game- Theoretic Perspective on Rough Set Analysis, 2008 International Forum on Knowledge Technology (IFKT'08), Chongqing, Journal of Chongqing University of Posts and Telecommunications, Vol.
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Several approximations have been published (see for example the results section below), assuming set-theoretic assumptions (such as the existence of large cardinals or variations of the generalized continuum hypothesis), or model-theoretic assumptions (such as amalgamation or tameness). As of 2014, the original conjecture remains open.
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An alternative "abstract nonsense" proof of the splitting lemma may be formulated entirely in category theoretic terms.
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Antokoletz, Elliott (2014). A History of Twentieth-Century Music in a Theoretic-Analytical Context, p.166. Routledge. .
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It is a group-theoretic analogue of the Jacobi identity for the ring-theoretic commutator (see next section). N.B., the above definition of the conjugate of by is used by some group theorists. Many other group theorists define the conjugate of by as . This is often written {}^x a.
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Finally, both the realist criticisms here described ignore new possible explanations, like the game-theoretic one discussed below .
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S-matrices are not substitutes for a field-theoretic treatment, but rather, complement the end results of such.
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In complexity-theoretic cryptography, security against adaptive chosen-ciphertext attacks is commonly modeled using ciphertext indistinguishability (IND-CCA2).
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Nengo is built upon two theoretic underpinnings, the Neural Engineering Framework (NEF) and the Semantic Pointer Architecture (SPA).
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In mathematics, categorification is the process of replacing set-theoretic theorems by category-theoretic analogues. Categorification, when done successfully, replaces sets by categories, functions with functors, and equations by natural isomorphisms of functors satisfying additional properties. The term was coined by Louis Crane. The reverse of categorification is the process of decategorification.
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In particular, kernel pairs can be used to interpret kernels in monoid theory or ring theory in category-theoretic terms.
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Cartier duality is a scheme-theoretic analogue of Pontryagin duality taking finite commutative group schemes to finite commutative group schemes.
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The game-theoretic rough set model determine and interprets the required thresholds by utilizing a game-theoretic environment for analyzing strategic situations between cooperative or conflicting decision making criteria. The essential idea is to implement a game for investigating how the probabilistic thresholds may change in order to improve the rough set based decision making.N. Azam, J. T. Yao, Analyzing Uncertainties of Probabilistic Rough Set Regions with Game-theoretic Rough Sets, International Journal of Approximate Reasoning, Vol. 55, No.1, pp 142-155, 2014.Y. Zhang, Optimizing Gini Coefficient of Probabilistic Rough Set Regions using Game- Theoretic Rough Sets, Proceeding of 26th Annual IEEE Canadian Conference on Electrical and Computer Engineering (CCECE'13), Regina, Canada, May 5–8, 2013, pp 699–702J.
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In proof theory, ordinal analysis assigns ordinals (often large countable ordinals) to mathematical theories as a measure of their strength. If theories have the same proof-theoretic ordinal they are often equiconsistent, and if one theory has a larger proof-theoretic ordinal than another it can often prove the consistency of the second theory.
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In mathematics, Ψ0(Ωω) is a large countable ordinal that is used to measure the proof-theoretic strength of some mathematical systems. In particular, it is the proof theoretic ordinal of the subsystem \Pi_1^1-CA0 of second-order arithmetic; this is one of the "big five" subsystems studied in reverse mathematics (Simpson 1999).
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To understand Nielsen equivalence of non-minimal generating sets, module theoretic investigations have been useful, as in . Continuing in these lines, a K-theoretic formulation of the obstruction to Nielsen equivalence was described in and . These show an important connection between the Whitehead group of the group ring and the Nielsen equivalence classes of generators.
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Examples of proof-theoretic formalizations of non-monotonic reasoning, which revealed some undesirable or paradoxical properties or did not capture the desired intuitive comprehensions, that have been successfully (consistent with respective intuitive comprehensions and with no paradoxical properties, that is) formalized by model-theoretic means include first-order circumscription, closed-world assumption, and autoepistemic logic.
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Another possibility is to use Monte Carlo (MC) algorithms and to sample the full partition function integral in field-theoretic formulation. The resulting procedure is then called a polymer field-theoretic simulation. In a recent work, however, Baeurle demonstrated that MC sampling in conjunction with the basic field- theoretic representation is impracticable due to the so-called numerical sign problem (Baeurle 2002). The difficulty is related to the complex and oscillatory nature of the resulting distribution function, which causes a bad statistical convergence of the ensemble averages of the desired thermodynamic and structural quantities.
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A logical graph is a special type of graph-theoretic structure in any one of several systems of graphical syntax that Charles Sanders Peirce developed for logic. In his papers on qualitative logic, entitative graphs, and existential graphs, Peirce developed several versions of a graphical formalism, or a graph-theoretic formal language, designed to be interpreted for logic. In the century since Peirce initiated this line of development, a variety of formal systems have branched out from what is abstractly the same formal base of graph-theoretic structures.
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A-not-A operator lowering must satisfy four conditions: # The A-not-A operator targets the closest MWd that is the X′-theoretic head that it c-commands. # Closeness of the head is qualified by: (i) The closest head is a X′-theoretic head of the maximal which is immediately dominated by the maximal projection of the A-not-A operator. (ii) The target must have overt phonological realization. # There is not any non-X′-theoretic head or SWd intervening between the A-not-A operator and its target.
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The Blahut–Arimoto algorithm, is often used to refer to a class of algorithms for computing numerically either the information theoretic capacity of a channel, or the rate-distortion function of a source. They are iterative algorithms that eventually converge to the optimal solution of the convex optimization problem that is associated with these information theoretic concepts.
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Mathematical models can take many forms, including but not limited to dynamical systems, statistical models, differential equations, or game theoretic models. These and other types of models can overlap, with a given model involving a variety of abstract structures. A more comprehensive type of mathematical modelDI Spivak, RE Kent. "Ologs: a category-theoretic approach to knowledge representation" (2011).
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Utility-Theoretic Indexing developed by Cooper and Maron is a theory of indexing based on utility theory. To reflect the value for documents that is expected by the users, index terms are assigned to documents. Also, Utility-Theoretic Indexing is related an "event space" in the statistical word. There are several basic spaces Ω in the Information Retrieval.
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Also, they are exactly ideals in the ring-theoretic sense on the Boolean ring formed by the powerset of the underlying set.
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A systematic study of webs was started by Blaschke in the 1930s. He extended the same group-theoretic approach to web geometry.
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In: P. Kolman, J. Kratochvíl (eds), Graph-Theoretic Concepts in Computer Science. WG 2011. Lecture Notes Comp. Sci. 6986, Springer, 191−202.
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Fortunately, the deviations from equilibrium do not cause much damage to the social welfare - the final welfare is close to the theoretic optimum.
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Most number-theoretic functions definable using recursion on a single variable are primitive recursive. Basic examples include the addition and truncated subtraction functions.
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In the theoretic of Plotinus, nous produces nature through intellectual mediation, thus the intellectualizing gods are followed by a triad of psychic gods.
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Forcing in recursion theory is a modification of Paul Cohen's original set- theoretic technique of forcing to deal with the effective concerns in recursion theory. Conceptually the two techniques are quite similar: in both one attempts to build generic objects (intuitively objects that are somehow 'typical') by meeting dense sets. Both techniques are described as a relation (customarily denoted \Vdash) between 'conditions' and sentences. However, where set-theoretic forcing is usually interested in creating objects that meet every dense set of conditions in the ground model, recursion-theoretic forcing only aims to meet dense sets that are arithmetically or hyperarithmetically definable.
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Any regular local ring is a complete intersection ring, but not conversely. A ring R is a set-theoretic complete intersection if the reduced ring associated to R, i.e., the one obtained by dividing out all nilpotent elements, is a complete intersection. As of 2017, it is in general unknown, whether curves in three-dimensional space are set- theoretic complete intersections.
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Tait (2005) gives a game-theoretic interpretation of Gentzen's method. Gentzen's consistency proof initiated the program of ordinal analysis in proof theory. In this program, formal theories of arithmetic or set theory are assigned ordinal numbers that measure the consistency strength of the theories. A theory will be unable to prove the consistency of another theory with a higher proof theoretic ordinal.
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"Contrasting applications of logic in natural language syntactic description." In Petr Hájek, Luis Valdés- Villanueva, and Dag Westerståhl (eds.), Logic, Methodology and Philosophy of Science: Proceedings of the Twelfth International Congress, 481-503. #Pullum, Geoffrey K. (2007) "The evolution of model-theoretic frameworks in linguistics." In the proceedings of the Model-Theoretic Syntax at 10 workshop at ESSLLI 2007, Trinity College, Dublin.
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Recent developments in proof theory include the study of proof mining by Ulrich Kohlenbach and the study of proof-theoretic ordinals by Michael Rathjen.
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An Information-theoretic security technique known as physical layer encryption ensures that a wireless communication link is provably secure with communications and coding techniques.
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In geometric measure theory an approximate tangent space is a measure theoretic generalization of the concept of a tangent space for a differentiable manifold.
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Every comparability graph is perfect: this is essentially just Mirsky's theorem, restated in graph-theoretic terms . By the perfect graph theorem of , the complement of any perfect graph is also perfect. Therefore, the complement of any comparability graph is perfect; this is essentially just Dilworth's theorem itself, restated in graph-theoretic terms . Thus, the complementation property of perfect graphs can provide an alternative proof of Dilworth's theorem.
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A field-theoretic simulation is a numerical strategy to calculate structure and physical properties of a many-particle system within the framework of a statistical field theory, like e.g. a polymer field theory. A convenient possibility is to use Monte Carlo (MC) algorithms, to sample the full partition function integral expressed in field-theoretic representation. The procedure is then called the auxiliary field Monte Carlo method.
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The QED vacuum is the field-theoretic vacuum of quantum electrodynamics. It is the lowest energy state (the ground state) of the electromagnetic field when the fields are quantized. When Planck's constant is hypothetically allowed to approach zero, QED vacuum is converted to classical vacuum, which is to say, the vacuum of classical electromagnetism. Another field-theoretic vacuum is the QCD vacuum of the Standard Model.
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In metalogic, 'syntax' has to do with formal languages or formal systems without regard to any interpretation of them, whereas, 'semantics' has to do with interpretations of formal languages. The term 'syntactic' has a slightly wider scope than 'proof- theoretic', since it may be applied to properties of formal languages without any deductive systems, as well as to formal systems. 'Semantic' is synonymous with 'model-theoretic'.
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For example, Shannon's entropy in the context of an information diagram must be taken as a signed measure. (See the article Information theory and measure theory for more information.). Information diagrams have also been applied to specific problems such as for displaying the information theoretic similarity between sets of ontological terms . Venn diagram of information theoretic measures for three variables x, y, and z.
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Using this formula and certain number theoretic and Galois-cohomological estimates, Armand Borel and Gopal Prasad proved several finiteness theorems about arithmetic groups, [6]. The volume formula, together with number-theoretic and Bruhat-Tits theoretic considerations led to a classification, by Gopal Prasad and Sai-Kee Yeung, of fake projective planes (in the theory of smooth projective complex surfaces) into 28 non-empty classes [21] (see also [22] and [23]). This classification, together with computations by Donald Cartwright and Tim Steger, has led to a complete list of fake projective planes. This list consists of exactly 50 fake projective planes, up to isometry (distributed among the 28 classes).
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In model theory, a weakly o-minimal structure is a model theoretic structure whose definable sets in the domain are just finite unions of convex sets.
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Lastly, using a game-theoretic model, it has been argued that sometimes it is easier to reach an agreement if the initial property rights are unclear.
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Rationality and Society provides an forum in which theoretic developments, empirical research, and policy analysis that are relevant to the rational action paradigm can be shared.
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The goal here is to hide all functions of the plaintext rather than all information about it. # Quantum cryptography is largely part of information-theoretic cryptography.
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Moshe Y. Vardi. An Automata-Theoretic Approach to Linear Temporal Logic. Proceedings of the 8th Banff Higher Order Workshop (Banff'94). Lecture Notes in Computer Science, vol.
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Strategic epidemiology is a branch of economic epidemiology that adopts an explicitly game theoretic approach to analyzing the interplay between individual behavior and population wide disease dynamics.
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His current interest is applying systems control theoretic approach to turbulence control. John Kim served as Co-Editor-in-Chief of Physics of Fluids from 1998-2015.
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100 Horsemen and the empty city: A game theoretic examination of deception in Chinese military legend. Journal of Peace Research, 2011; 48 (2): 217 DOI: 10.1177/0022343310396265.
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Lattice-theoretic (algebraic) treatments of mereotopology as contact algebras have been applied to separate the topological from the mereological structure, see Stell (2000), Düntsch and Winter (2004).
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Models and theoretic equations need to take into account "social, economic, cultural, geographical, and even climatological variables" in order to accurately reflect the statistics of the time.
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Wald greatly impressed Basu. Wald had developed a decision-theoretic foundations for statistics in which Bayesian statistics was a central part, because of Wald's theorem characterising admissible decision rules as Bayesian decision rules (or limits of Bayesian decision rules). Wald also showed the power of using measure-theoretic probability theory in statistics. He married Kalyani Ray in 1952 and subsequently had two children, Monimala (Moni) Basu and Shantanu Basu.
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James Douglas Montgomery (born April 13, 1963) is professor of sociology and economics at the University of Wisconsin–Madison. He received his Ph.D. in economics from Massachusetts Institute of Technology. He has applied game- theoretic models and non-monotonic logic to present formal analysis and description of social theories and sociological phenomena. He was the recipient of James Coleman Award (1999) for his paper “Toward a Role-Theoretic Conception of Embeddedness”.
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In this section, the central concepts and definitions of domain theory will be introduced. The above intuition of domains being information orderings will be emphasized to motivate the mathematical formalization of the theory. The precise formal definitions are to be found in the dedicated articles for each concept. A list of general order-theoretic definitions, which include domain theoretic notions as well can be found in the order theory glossary.
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A well-working machinery of intersecting algebraic cycles and requires more than taking just the set-theoretic intersection of the cycles in question. If the two cycles are in "good position" then the intersection product, denoted , should consist of the set-theoretic intersection of the two subvarieties. However cycles may be in bad position, e.g. two parallel lines in the plane, or a plane containing a line (intersecting in 3-space).
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Robert Kleinberg is known for his research work on group theoretic algorithms for matrix multiplication, online learning, network coding and greedy embedding, social networks and algorithmic game theory.
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As Armstrong observes, any family of sets of this type forms an antimatroid. Armstrong also provides a lattice-theoretic characterization of the antimatroids that this construction can form.
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He has stated plans to expand these to many languages and is experimenting with other poetic forms."Graph theoretic" Poetic Forms . websters-online-dictionary.org. Icon Group International, Inc.
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Other important information theoretic quantities include Rényi entropy (a generalization of entropy), differential entropy (a generalization of quantities of information to continuous distributions), and the conditional mutual information.
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The function type in programming languages does not correspond to the space of all set-theoretic functions. Given the countably infinite type of natural numbers as the domain and the booleans as range, then there are an uncountably infinite number (2ℵ0 = c) of set-theoretic functions between them. Clearly this space of functions is larger than the number of functions that can be defined in any programming language, as there exist only countably many programs (a program being a finite sequence of a finite number of symbols) and one of the set-theoretic functions effectively solves the halting problem. Denotational semantics concerns itself with finding more appropriate models (called domains) to model programming language concepts such as function types.
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In particular, HOL with Henkin semantics has all the model-theoretic properties of first-order logic, and has a complete, sound, effective proof system inherited from first-order logic.
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Wolsey has made seminal contributions in duality theory for integer programming, submodular optimization, the group-theoretic approach and polyhedral analysis of fixed-charge network flow and production planning models.
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Computational number theory, also known as algorithmic number theory, is the study of algorithms for performing number theoretic computations. The best known problem in the field is integer factorization.
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Alfred Tarski developed the basics of model theory. Beginning in 1935, a group of prominent mathematicians collaborated under the pseudonym Nicolas Bourbaki to publish Éléments de mathématique, a series of encyclopedic mathematics texts. These texts, written in an austere and axiomatic style, emphasized rigorous presentation and set- theoretic foundations. Terminology coined by these texts, such as the words bijection, injection, and surjection, and the set-theoretic foundations the texts employed, were widely adopted throughout mathematics.
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The concept of an incidence structure is very simple and has arisen in several disciplines, each introducing its own vocabulary and specifying the types of questions that are typically asked about these structures. Incidence structures use a geometric terminology, but in graph theoretic terms they are called hypergraphs and in design theoretic terms they are called block designs. They are also known as a set system or family of sets in a general context.
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Frances E. Allen and John Cocke. Graph theoretic constructs for program control flow analysis. Technical Report IBM Res. Rep. RC 3923, IBM T.J. Watson Research Center, Yorktown Heights, NY, 1972.
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Julia Elisenda (Eli) Grigsby is an American mathematician who works as a professor at Boston College. Her research concerns low-dimensional topology, including knot theory and category-theoretic knot invariants.
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An Information-theoretic Model for Adaptive Side-channel Attacks. In Proceedings of the 14th ACM Conference on Computer and Communications Security (CCS ’07). ACM, New York, NY, USA, 286–296..
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Another information- theoretic metric is Variation of information, which is roughly a symmetrization of conditional entropy. It is a metric on the set of partitions of a discrete probability space.
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In Kripke semantics where the semantic values of formulae are sets of possible worlds, negation can be taken to mean set-theoretic complementation (see also possible world semantics for more).
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In game-theoretic terms this makes pledging to build the public good a dominant strategy: the best move is to pledge to the contract regardless of the actions of others.
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In algebraic geometry, the Enriques–Babbage theorem states that a canonical curve is either a set-theoretic intersection of quadrics, or trigonal, or a plane quintic. It was proved by and .
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Chapter two contains Lewis's response to several arguments, many set theoretic in nature, that attempt to show that modal realism flounders because of the quantity of worlds that must be postulated.
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Caryn Linda Navy (born July 5, 1953) is an American mathematician and computer scientist. Blind since childhood, she is chiefly known for her work in set- theoretic topology and Braille technology.
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This is the order-theoretic dual to the notion of cofinal subset. Note that cofinal and coinitial subsets are both dense in the sense of appropriate (right- or left-) order topology.
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Milgrom made several fundamental contributions to game theory in the 1980s and 1990s on topics including the game-theoretic analysis of reputation formation, repeated games, supermodular games and learning in games.
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Naturally, we need to have at least a fraction R of the transmitted symbols to be correct in order to recover the message. This is an information-theoretic lower bound on the number of correct symbols required to perform decoding and with list decoding, we can potentially achieve this information-theoretic limit. However, to realize this potential, we need explicit codes (codes that can be constructed in polynomial time) and efficient algorithms to perform encoding and decoding.
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In applications of the TBDE to QED, the two particles interact by way of four-vector potentials derived from the field theoretic electromagnetic interactions between the two particles. In applications to QCD, the two particles interact by way of four-vector potentials and Lorentz invariant scalar interactions, derived in part from the field theoretic chromomagnetic interactions between the quarks and in part by phenomenological considerations. As with the Breit equation a sixteen-component spinor Ψ is used.
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Ann Arbor, MI: University of Michigan Press. # Gates, Scott and Brian D. Humes. 1997. Games, Information, and Politics: Applying Game Theoretic Models to Political Science. Ann Arbor, MI: University of Michigan Press.
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1103 - 1114, 2017., two-level game modelV. Nanduri, T. K. Das and P. Rocha, "Generation capacity expansion in energy markets using a two-level game- theoretic model," IEEE trans. power sys, vol.
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It was used by André Weil to give a representation-theoretic interpretation of theta functions, and is important in the theory of modular forms of half-integral weight and the theta correspondence.
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Variational free energy is an information theoretic functional and is distinct from thermodynamic (Helmholtz) free energy.Evans, D. J. (2003). A non-equilibrium free energy theorem for deterministic systems. Molecular Physics , 101, 15551–4.
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Rosický, J. "Injectivity and accessible categories." Cubo Matem. Educ 4 (2002): 201-211. Grothendieck continued the development of the theory for homotopy-theoretic purposes in his (still partly unpublished) 1991 manuscript Les dérivateurs.
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Whereas taking a more geomorphic research approach tends to derive patterns via theoretic knowledge and detailed measurements of multiple factors. In turn, this uses smaller sample sizes than that of large replication studies.
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However, this need not always be the case, as non-symmetric monoidal categories can be encountered in category-theoretic formulations of linguistics; roughly speaking, this is because word-order in natural language matters.
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A probabilistic and decision theoretic extension of affect control theoryHoey, Alhothali, and Schroeder (2013). generalizes the original theory in order to allow for uncertainty about identities, changing identities, and explicit non-affective goals.
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To make the field-theoretic methodology amenable for computation, Baeurle proposed to shift the contour of integration of the partition function integral through the homogeneous mean field (MF) solution using Cauchy's integral theorem, which provides its so-called mean-field representation. This strategy was previously successfully employed in field- theoretic electronic structure calculations (Rom 1997, Baer 1998). Baeurle could demonstrate that this technique provides a significant acceleration of the statistical convergence of the ensemble averages in the MC sampling procedure (Baeurle 2002).
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In order theory, a different notion of a tree is used: an order-theoretic tree is a partially ordered set with one minimal element in which each element has a well-ordered set of predecessors. Every tree in descriptive set theory is also an order-theoretic tree, using a partial ordering in which two sequences T and U are ordered by T if and only if T is a proper prefix of U. The empty sequence is the unique minimal element, and each element has a finite and well-ordered set of predecessors (the set of all of its prefixes). An order-theoretic tree may be represented by an isomorphic tree of sequences if and only if each of its elements has finite height (that is, a finite set of predecessors).
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The history and credit for the fundamental theorem is complicated by the fact that it was proven when group theory was not well- established, and thus early forms, while essentially the modern result and proof, are often stated for a specific case. Briefly, an early form of the finite case was proven in , the finite case was proven in , and stated in group-theoretic terms in . The finitely presented case is solved by Smith normal form, and hence frequently credited to , though the finitely generated case is sometimes instead credited to ; details follow. Group theorist László Fuchs states: The fundamental theorem for finite abelian groups was proven by Leopold Kronecker in , using a group-theoretic proof, though without stating it in group-theoretic terms; a modern presentation of Kronecker's proof is given in , 5.2.
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The field of ordinal analysis was formed when Gerhard Gentzen in 1934 used cut elimination to prove, in modern terms, that the proof-theoretic ordinal of Peano arithmetic is ε0. See Gentzen's consistency proof.
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Eisenträger appears in the documentary film Julia Robinson and Hilbert's Tenth Problem (2008). In 2017, she became a Fellow of the American Mathematical Society "for contributions to computational number theory and number-theoretic undecidability".
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That article also discusses how we may rephrase the above definition in terms of the existence of suitable Galois connections between related posets — an approach of special interest for category theoretic investigations of the concept.
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It is sometimes a computationally intensive method. The theoretic basis for the Kolmogorov complexity approach was laid by Bennett, Gacs, Li, Vitanyi, and Zurek (1998) by proposing the information distance.Bennett,C.H., Gacs,P., Li,M.
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In mathematics and theoretical computer science, entropy compression is an information theoretic method for proving that a random process terminates, originally used by Robin Moser to prove an algorithmic version of the Lovász local lemma...
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Dobrinci is the birthplace of writer Jovan Subotić (1817–1886). It is also the birthplace of Slavko Vorkapić-Vorki (1894–1976), a famous film theoretic and professor. Famous Nadja Blagojevic was born and raised here.
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The decision-theoretic approach to statistical inference was reinvigorated by Abraham Wald and his successors, and makes extensive use of scientific computing, analysis, and optimization; for the design of experiments, statisticians use algebra and combinatorics.
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These strategies of pattern recognition are useful in purely group theoretic context, as well as for applications in algebraic number theory concerning Galois groups of higher p-class fields and Hilbert p-class field towers.
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From 2–6 April, Stepanović completed the theoretic part of his exam in twelve subjects. He was released of doing practical part of exam, because he received very good grade on headquarters’ journeys and missions.
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They are named after the Swiss mathematician Julius Richard Büchi, who invented them in 1962. Büchi automata are often used in model checking as an automata-theoretic version of a formula in linear temporal logic.
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In theoretical physics, Hamiltonian field theory is the field-theoretic analogue to classical Hamiltonian mechanics. It is a formalism in classical field theory alongside Lagrangian field theory. It also has applications in quantum field theory.
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It has since been extended both to weaker type systems such as the untyped lambda calculus using a domain theoretic approach, and to richer type systems such as several variants of Martin-Löf type theory.
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A logical graph is a special type of diagrammatic structure in any one of several systems of graphical syntax that Charles Sanders Peirce developed for logic. In his papers on qualitative logic, entitative graphs, and existential graphs, Peirce developed several versions of a graphical formalism, or a graph-theoretic formal language, designed to be interpreted for logic. In the century since Peirce initiated this line of development, a variety of formal systems have branched out from what is abstractly the same formal base of graph-theoretic structures.
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Elaborating further on basic field-theoretic notions, it can be shown that two finite fields with the same order are isomorphic. It is thus customary to speak of the finite field with elements, denoted by or .
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This turns out not to be a full differential operator in the usual sense but has many of the desired properties. There are a number of approaches to defining the Laplacian: probabilistic, analytical or measure theoretic.
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In abstract algebra, Morita equivalence is a relationship defined between rings that preserves many ring-theoretic properties. It is named after Japanese mathematician Kiiti Morita who defined equivalence and a similar notion of duality in 1958.
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A History of Twentieth-Century Music in a Theoretic-Analytical Context, p.166. Routledge. . "[Riegger and Becker] were grouped with Ives, Ruggles, and Cowell as the 'American Five'." Ruggles's music is published by Theodore Presser Company.
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The family of all subsets of a set S, ordered by set inclusion, forms a lattice in which the meet is represented by the set-theoretic intersection and the join is represented by the set-theoretic union; a lattice formed in this way is called a Boolean lattice. The lattice-theoretic version of Frankl's conjecture is that in any finite lattice there exists an element x that is not the join of any two smaller elements, and such that the number of elements greater than or equal to x totals at most half the lattice, with equality only if the lattice is a Boolean lattice. As shows, this statement about lattices is equivalent to the Frankl conjecture for union-closed sets: each lattice can be translated into a union-closed set family, and each union-closed set family can be translated into a lattice, such that the truth of the Frankl conjecture for the translated object implies the truth of the conjecture for the original object. This lattice-theoretic version of the conjecture is known to be true for several natural subclasses of lattices; ; but remains open in the general case.
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Proof-theoretic formalization of a non-monotonic logic begins with adoption of certain non-monotonic rules of inference, and then prescribes contexts in which these non-monotonic rules may be applied in admissible deductions. This typically is accomplished by means of fixed-point equations that relate the sets of premises and the sets of their non-monotonic conclusions. Default logic and autoepistemic logic are the most common examples of non-monotonic logics that have been formalized that way.. Model-theoretic formalization of a non-monotonic logic begins with restriction of the semantics of a suitable monotonic logic to some special models, for instance, to minimal models, and then derives the set of non- monotonic rules of inference, possibly with some restrictions in which contexts these rules may be applied, so that the resulting deductive system is sound and complete with respect to the restricted semantics. Unlike some proof-theoretic formalizations that suffered from well-known paradoxes and were often hard to evaluate with respect of their consistency with the intuitions they were supposed to capture, model-theoretic formalizations were paradox-free and left little, if any, room for confusion about what non- monotonic patterns of reasoning they covered.
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A list of his publications, complete through 2000, appears in the 1999 volume of History and Philosophy of Logic, which also includes the expository article by M. Scanlan and S. Shapiro "The Work of John Corcoran: An Appreciation". Other articles about his work include "Corcoran the Mathematician" by S. Shapiro, "Corcoran the Philosopher" by J. M. Sagüillo, and "Corcoran in Spanish" by C. Martínez-Vidal; all appear in a 2007 volume published by the University of Santiago de Compostela Press. Corcoran's work in the 1990s on information- theoretic logic is discussed by José M. Sagüillo in the article "Methodological Practice and Complementary Concepts of Logical Consequence: Tarski's Model-Theoretic Consequence and Corcoran's Information-Theoretic Consequence" (History and Philosophy of Logic volume 30, 2009, 21-48), which received the 2009 Ivor Grattan-Guinness Award for the History and Philosophy of Logic (informaworld.com).
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Gábor Bódy (30 August 1946 – 24 October 1985) was a Hungarian film director, screenwriter, theoretic, and occasional actor. A pioneer of experimental filmmaking and film language, Bódy is one of the most important figures of Hungarian cinema.
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Dr. Yener is interested in fundamental performance limits of networked systems, communications and information theory. The applications of these fields include but not limited to information theoretic physical layer security, energy harvesting communication networks, and caching systems.
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The Data processing inequality is an information theoretic concept which states that the information content of a signal cannot be increased via a local physical operation. This can be expressed concisely as 'post-processing cannot increase information'.
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3, pp. 1851–1895 • Martin Shubik, 1987. A Game-Theoretic Approach to Political Economy, Part II. MIT Press. Description. A third aspect is oriented to public policy related to economic regulation,Richard Schmalensee and Robert Willig, eds.
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General topology is the branch of topology dealing with the basic set-theoretic definitions and constructions used in topology.Munkres, James R. Topology. Vol. 2. Upper Saddle River: Prentice Hall, 2000.Adams, Colin Conrad, and Robert David Franzosa.
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The algebra of sets defines the properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations. Any family of sets that is closed under the set- theoretic operations forms a Boolean algebra with the join operator being union, the meet operator being intersection, the complement operator being set complement, the bottom being and the top being the universe set under consideration.
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An important related notion is that of a succinct data structure, which uses space roughly equal to the information- theoretic minimum, which is a worst-case notion of the space needed to represent the data. In contrast, the size of a compressed data structure depends upon the particular data being represented. When the data are compressible, as is often the case in practice for natural language text, the compressed data structure can occupy space very close to the information- theoretic minimum, and significantly less space than most compression schemes.
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The subsequent year Mendelsohn was barred from competition as at that time the winning university set the examination for the next year and its students were barred from competition. Mendelsohn completed his Ph.D. dissertation in 1941. It was titled "A Group-Theoretic Characterization of the General Projective Collineation Group", and summarized in the Proceedings of the National Academy of Sciences in 1944.N. Mendelsohn, A Group-Theoretic Characterization of the General Projective Collineation Group, Proc Natl Acad Sci U S A. 1944 September 15; 30(9): 279–283.
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More recently, new information-theoretic estimators have been developed in an attempt to reduce this problem,Lukacs, P. M., Burnham, K. P. & Anderson, D. R. (2010) "Model selection bias and Freedman's paradox." Annals of the Institute of Statistical Mathematics, 62(1), 117-125 in addition to the accompanying issue of model selection bias,Burnham, K. P., & Anderson, D. R. (2002). Model Selection and Multimodel Inference: A Practical-Theoretic Approach, 2nd ed. Springer-Verlag. whereby estimators of predictor variables that have a weak relationship with the response variable are biased.
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He also gave one of the earliest approaches to higher dimensional bosonization of fermionic field theories (with Fidel Schaposnik) as well as two-dimensional Fermi surfaces (with his then student Antonio Castro-Neto) and has later applied them to important problems in condensed matter. Some of his other important works in recent times that have not already been mentioned above include a graph-theoretic lattice discretization scheme for Chern-Simons theories and its applications to condensed matter problems, and novel field theoretic approaches to describe fractional topological insulators.
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Search-theoretic models, on the other hand, are based on explicit descriptions of specialization, the pattern of meetings, and the information structure. Kiyotaki and Wright (1989) was the first attempt to use a search-theoretic model to endogenously determine which commodities would become media of exchange, i.e. commodity money. Later, Kiyotaki and Wright (1991) constructed an alternative search-based model to prove that fiat money can be valued as a medium of exchange even if it has a rate of return that is inferior to other available assets.
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One disadvantage of the scheme-theoretic definition is that a scheme over k cannot have an L-valued point if L is not an extension of k. For example, the rational point (1,1,1) is a solution to the equation x1 + ix2 - (1+i)x3 but the corresponding Q[i]-variety V has no Spec(Q)-valued point. The two definitions of field of definition are also discrepant, e.g. the (scheme- theoretic) minimal field of definition of V is Q, while in the first definition it would have been Q[i].
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It is often the case that the evolution function can be understood to compose the elements of a group, in which case the group- theoretic orbits of the group action are the same thing as the dynamical orbits.
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He is the author of over 180 academic papers. His notable accomplishments include the introduction of interactive proof systems,. the introduction of the term Las Vegas algorithm,. and the introduction of group theoretic methods in graph isomorphism testing.
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Tsur did a PhD degree's in application of game theoretic models to law, another one in Economic Analysis of Law from New York University. Moreover, she was a post- doctoral fellow at Yale Law School's Information Society Project.
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Bezhanishvili et al. (2010) Finally, let be set-theoretic inclusion on the set of prime filters of and let . Then is a Priestley space. Moreover, is a lattice isomorphism from onto the lattice of all clopen up-sets of .
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Notable works include the 1969 automata-theoretic approach by Büchi and Landweber, and the works by Manna and Waldinger (c. 1980). The development of modern high- level programming languages can also be understood as a form of program synthesis.
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As in many other game theoretic experiments, scholars have investigated the effect of increasing the stakes. As with other games, for instance the ultimatum game, as the stakes increase the play approaches (but does not reach) Nash equilibrium play.
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Marcello Pelillo from the University of Venice, Venezia Mestre, Italy was named Fellow of the Institute of Electrical and Electronics Engineers (IEEE) in 2013 for contributions to graph-theoretic and optimization-based approaches in pattern recognition and computer vision.
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Directed completeness relates in various ways to other completeness notions such as chain completeness. Directed completeness alone is quite a basic property that occurs often in other order-theoretic investigations, using for instance algebraic posets and the Scott topology.
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Scientists of the faculty are involved in both theoretic and applied fields. #Faculty of Humanitarian and social sciences was founded in 1996 (after re-organization of the Historical-Philological faculty). It has more than 2,500 students and 12 departments.
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Gentzen's result introduced the ideas of cut elimination and proof-theoretic ordinals, which became key tools in proof theory. Gödel (1958) gave a different consistency proof, which reduces the consistency of classical arithmetic to that of intuitionistic arithmetic in higher types.
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Pálfy and Pudlák. Congruence lattices of finite algebras and intervals in subgroup lattices of finite groups. Algebra Universalis 11(1), 22–27 (1980). DOI For an overview of the group theoretic approach to the problem, see Pálfy (1993)Péter Pál Pálfy.
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Subject courses in Research are now offered in all grades. 9th graders are offered basic theoretic principles in research and Statistics. Students may specialize either in Biotechnology or Microbiology. Seniors are given advanced applied principles in research and Thesis Writing.
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The measure-theoretic version of differentiation under the integral sign also applies to summation (finite or infinite) by interpreting summation as counting measure. An example of an application is the fact that power series are differentiable in their radius of convergence.
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A right Noetherian, right self-injective ring is called a quasi-Frobenius ring, and is two-sided Artinian and two-sided injective, . An important module theoretic property of quasi-Frobenius rings is that the projective modules are exactly the injective modules.
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Euler's advice is vague; see Euler et al., pp. 3, 26. John Baez even suggests a category-theoretic method involving multiply pointed sets and the quantum harmonic oscillator. Baez, John C. Euler's Proof That 1 + 2 + 3 + ... = −1/12 (PDF). math.ucr.
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In addition, order theory does not restrict itself to the various classes of ordering relations, but also considers appropriate functions between them. A simple example of an order theoretic property for functions comes from analysis where monotone functions are frequently found.
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Since the two definitions are equivalent, lattice theory draws on both order theory and universal algebra. Semilattices include lattices, which in turn include Heyting and Boolean algebras. These "lattice-like" structures all admit order-theoretic as well as algebraic descriptions.
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David L. Childs is a computer scientist noted for his work on his Extended Set Theoretic approach to data base management and cited by Edgar F. Codd in his key paper "A Relational Model of Data for Large Shared Data Banks".
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The State and Revolution (1917) is a book by Vladimir Lenin describing the role of the State in society, the necessity of proletarian revolution, and the theoretic inadequacies of social democracy in achieving revolution to establish the dictatorship of the proletariat.
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But Paul Zweifel uses a group-theoretic approach to analyse different sets, concluding especially that a set of twenty divisions of the octave is another viable option for retaining certain properties associated with the conventional "diatonic" selections from twelve pitch classes.
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For two-player finite zero-sum games, the different game theoretic solution concepts of Nash equilibrium, minimax, and maximin all give the same solution. If the players are allowed to play a mixed strategy, the game always has an equilibrium.
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As a result, and following proposals according to which asymptotic symmetries could explain the microscopic origin of black hole entropy, BMS symmetry and its extensions as well as its gauge-theoretic cousins are subjects of active research as of May 2020.
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However, it is well known that MC sampling in conjunction with the basic field-theoretic representation of the partition function integral, directly obtained via the Hubbard-Stratonovich transformation, is impracticable, due to the so-called numerical sign problem (Baeurle 2002, Fredrickson 2002). The difficulty is related to the complex and oscillatory nature of the resulting distribution function, which causes a bad statistical convergence of the ensemble averages of the desired structural and thermodynamic quantities. In such cases special analytical and numerical techniques are required to accelerate the statistical convergence of the field-theoretic simulation (Baeurle 2003, Baeurle 2003a, Baeurle 2004).
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New gauge-theoretic problems arise out of superstring theory models. In such models the universe is 10 dimensional consisting of four dimensions of regular spacetime and a 6-dimensional Calabi–Yau manifold. In such theories the fields which act on strings live on bundles over these higher dimensional spaces, and one is interested in gauge-theoretic problems relating to them. For example, the limit of the natural field theories in superstring theory as the string radius approaches zero (the so-called large volume limit) on a Calabi–Yau 6-fold is given by Hermitian Yang–Mills equations on this manifold.
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The proof of Gödel's incompleteness theorem just sketched is proof-theoretic (also called syntactic) in that it shows that if certain proofs exist (a proof of or its negation) then they can be manipulated to produce a proof of a contradiction. This makes no appeal to whether is "true", only to whether it is provable. Truth is a model-theoretic, or semantic, concept, and is not equivalent to provability except in special cases. By analyzing the situation of the above proof in more detail, it is possible to obtain a conclusion about the truth of in the standard model ℕ of natural numbers.
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The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic addition and multiplication are associative and commutative, so are set union and intersection; just as the arithmetic relation "less than or equal" is reflexive, antisymmetric and transitive, so is the set relation of "subset". It is the algebra of the set-theoretic operations of union, intersection and complementation, and the relations of equality and inclusion. For a basic introduction to sets see the article on sets, for a fuller account see naive set theory, and for a full rigorous axiomatic treatment see axiomatic set theory.
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More generally, properties of sets which describe their being computationally weak (when used as a Turing oracle) are referred to under the umbrella term lowness properties. By the Low basis theorem of Jockusch and Soare, any nonempty \Pi^0_1 class in 2^\omega contains a set of low degree. This implies that, although low sets are computationally weak, they can still accomplish such feats as computing a completion of Peano Arithmetic. In practice, this allows a restriction on the computational power of objects needed for recursion theoretic constructions: for example, those used in the analyzing the proof- theoretic strength of Ramsey's theorem.
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Contract theory is concerned with providing incentives in situations in which some variables cannot be observed by all parties. Hence, contract theory is difficult to test in the field: If the researcher could verify the relevant variables, then the contractual parties could contract on these variables, hence any interesting contract-theoretic problem would disappear. Yet, in laboratory experiments it is possible to directly test contract-theoretic models. For instance, researchers have experimentally studied moral hazard theory, adverse selection theory, exclusive contracting, deferred compensation, the hold-up problem, flexible versus rigid contracts, and models with endogenous information structures.
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Information- theoretically secure cryptosystems have been used for the most sensitive governmental communications, such as diplomatic cables and high-level military communications, because of the great efforts enemy governments expend toward breaking them. There are a variety of cryptographic tasks for which information-theoretic security is a meaningful and useful requirement. A few of these are: # Secret sharing schemes such as Shamir's are information- theoretically secure (and also perfectly secure) in that having less than the requisite number of shares of the secret provides no information about the secret. # More generally, secure multiparty computation protocols often have information-theoretic security.
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Michael Paul Wellman (born March 27, 1961) is the Richard H. Orenstein Division Chair of Computer Science and Engineering and Lynn A. Conway Collegiate Professor of Computer Science and Engineering at the University of Michigan, Ann Arbor. Wellman received a PhD from the Massachusetts Institute of Technology in 1988 for his work in qualitative probabilistic reasoning and decision-theoretic planning. From 1988 to 1992, Wellman conducted research in these areas at the USAF's Wright Laboratory. For the past 25 years, his research has focused on computational market mechanisms and game-theoretic reasoning methods, with applications in electronic commerce, finance, and cyber-security.
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The theory has also found applications in other areas, including statistical inference, natural language processing, cryptography, neurobiology, human vision, the evolution and function of molecular codes (bioinformatics), model selection in statistics,Burnham, K. P. and Anderson D. R. (2002) Model Selection and Multimodel Inference: A Practical Information- Theoretic Approach, Second Edition (Springer Science, New York) . thermal physics, quantum computing, linguistics, plagiarism detection, pattern recognition, and anomaly detection. Important sub-fields of information theory include source coding, algorithmic complexity theory, algorithmic information theory, and information-theoretic security. Applications of fundamental topics of information theory include lossless data compression (e.g.
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Therefore, the VP of (21b) counts as given. Schwarzschild assumes an optimality theoretic grammar. Accent placement is determined by a set of violable, hierarchically ranked constraints as shown in (24): (24) ::a. GIVENness: A constituent that is not F-marked is given. ::b.
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This method produces an explicit phylogenic network having an underlying tree with additional contact edges. Characters can be borrowed but evolve without homoplasy. To produce such networks, a graph-theoretic algorithm Nakhleh et al. Perfect Phylogenic networks, Language 81 (2005) has been used.
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Any vector space can be made into a unital associative algebra, called functional-theoretic algebra, by defining products in terms of two linear functionals. In general, it is a non-commutative algebra. It becomes commutative when the two functionals are the same.
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Hana, Jiri. Czech Clitics in Higher Order Grammar. Diss. The Ohio State University, 2007. It can be viewed simultaneously as generative-enumerative (like categorial grammar and principles and parameters) or model theoretic (like head-driven phrase structure grammar or lexical functional grammar).
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Most times, especially in amateur football, the goalkeeper is often forced to guess. Game theoretic research shows that both the penalty taker and also the goalkeeper must randomize their strategies in precise ways to avoid having the opponent take advantage of their predictability.
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In number theory, an arithmetic, arithmetical, or number-theoretic function is for most authorsNiven & Zuckerman, 4.2.Nagell, I.9.Bateman & Diamond, 2.1. any function f(n) whose domain is the positive integers and whose range is a subset of the complex numbers.
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In all other cases, it is true. All of the following are disjunctions: : A \lor B : eg A \lor B : A \lor eg B \lor eg C \lor D \lor eg E. The corresponding operation in set theory is the set-theoretic union.
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In geometry, the link of a vertex of a 2-dimensional simplicial complex is a graph that encodes information about the local structure of the complex at the vertex. It is a graph-theoretic analog to a sphere centered at a point.
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Here he studied the Riemann-Roch theorem. He was able to combine Riemann's function theoretic approach with the Italian geometric approach and with the Weierstrass arithmetical approach. His arithmetic setting of this result led eventually to the modern abstract theory of algebraic functions.
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The team semantics for dependence logic is a variant of Wilfrid Hodges' compositional semantics for IF logic.Hodges 1997Väänänen 2007, §3.2 There exist equivalent game-theoretic semantics for dependence logic, both in terms of imperfect information games and in terms of perfect information games.
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Graph-Based Generation of Referring Expressions. Computational Linguistics 23:53-72 present a graph-theoretic model of definite NP generation with many nice properties. In recent years a shared-task event has compared different algorithms for definite NP generation, using the TUNA corpus.
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Many common notions from mathematics (e.g. surjective, injective, free object, basis, finite representation, isomorphism) are definable purely in category theoretic terms (cf. monomorphism, epimorphism). Category theory has been suggested as a foundation for mathematics on par with set theory and type theory (cf. topos).
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Zech logarithms are used to implement addition in finite fields when elements are represented as powers of a generator \alpha. Zech logarithms are named after Julius Zech, and are also called Jacobi logarithms, after Carl G. J. Jacobi who used them for number theoretic investigations.
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20, No. 3, pp 291–298, 2008. Y. Zhang, J.T. Yao, Rule Measures Tradeoff Using Game-theoretic Rough Sets, Proceedings of the International Conference on Brian Informatics (BI'12), Macau, China, Dec 4–7, 2012, Lecture Notes in Computer Science 7670, pp 348–359.
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Several cryptographic methods rely on its hardness, see Applications. An efficient algorithm for the quadratic residuosity problem immediately implies efficient algorithms for other number theoretic problems, such as deciding whether a composite N of unknown factorization is the product of 2 or 3 primes.
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A decision-theoretic justification of the use of Bayesian inference (and hence of Bayesian probabilities) was given by Abraham Wald, who proved that every admissible statistical procedure is either a Bayesian procedure or a limit of Bayesian procedures. Conversely, every Bayesian procedure is admissible.
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John Wiley & Sons. :Description: Exposition of statistical hypothesis testing using the statistical decision theory of Abraham Wald, with some use of measure-theoretic probability. :Importance: Made Wald's ideas accessible. Collected and organized many results of statistical theory that were scattered throughout journal articles, civilizing statistics.
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Various model-theoretic ideas are related to quantifier elimination, and there are various equivalent conditions. Every first-order theory with quantifier elimination is model complete. Conversely, a model- complete theory, whose theory of universal consequences has the amalgamation property, has quantifier elimination.Hodges, Wilfrid (1993).
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If one considers sequences of measurable functions, then several modes of convergence that depend on measure-theoretic, rather than solely topological properties, arise. This includes pointwise convergence almost-everywhere, convergence in p-mean and convergence in measure. These are of particular interest in probability theory.
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Under the measure-theoretic definition of a probability space, the probability of an outcome need not even be defined. In particular, the set of events on which probability is defined may be some σ-algebra on S and not necessarily the full power set.
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Game semantics (, translated as dialogical logic) is an approach to formal semantics that grounds the concepts of truth or validity on game-theoretic concepts, such as the existence of a winning strategy for a player, somewhat resembling Socratic dialogues or medieval theory of Obligationes.
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Both, the algebraic, group theoretic access to the principalization problem by Hilbert- Artin-Furtwängler and the arithmetic, cohomological access by Hilbert- Herbrand-Iwasawa are also presented in detail in the two bibles of capitulation by J.-F. Jaulent 1988 and by K. Miyake 1989.
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The free market equilibrium in such an environment is generally not considered Pareto efficient. This is an important welfare-theoretic justification for macroprudential regulation that may require the introduction of targeted policy tools. Roland McKean was the first to distinguish technological and pecuniary effects.
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In 1917, Bernstein suggested the first axiomatic foundation of probability theory, based on the underlying algebraic structure. It was later superseded by the measure-theoretic approach of Kolmogorov. In the 1920s, he introduced a method for proving limit theorems for sums of dependent random variables.
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There is no coproduct in Met. The forgetful functor Met → Set assigns to each metric space the underlying set of its points, and assigns to each metric map the underlying set-theoretic function. This functor is faithful, and therefore Met is a concrete category.
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In mathematics, the Riemann–Roch theorem for surfaces describes the dimension of linear systems on an algebraic surface. The classical form of it was first given by , after preliminary versions of it were found by and . The sheaf- theoretic version is due to Hirzebruch.
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Under the measure- theoretic definition of a probability space, the probability of an elementary event need not even be defined. In particular, the set of events on which probability is defined may be some σ-algebra on S and not necessarily the full power set.
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Information-theoretic security is a cryptosystem whose security derives purely from information theory; the system cannot be broken even if the adversary has unlimited computing power. The cryptosystem is considered cryptoanalytically unbreakable if the adversary does not have enough information to break the encryption.
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For some, however, the Game of Life had more philosophical connotations. It developed a cult following through the 1970s and beyond; current developments have gone so far as to create theoretic emulations of computer systems within the confines of a Game of Life board.
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This class of hyperbolic surfaces is further subdivided into subclasses according to whether function spaces other than the negative subharmonic functions are degenerate, e.g. Riemann surfaces on which all bounded holomorphic functions are constant, or on which all bounded harmonic functions are constant, or on which all positive harmonic functions are constant, etc. To avoid confusion, call the classification based on metrics of constant curvature the geometric classification, and the one based on degeneracy of function spaces the function-theoretic classification. For example, the Riemann surface consisting of "all complex numbers but 0 and 1" is parabolic in the function-theoretic classification but it is hyperbolic in the geometric classification.
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Although pre-twentieth-century naturalists such as Charles Darwin made game- theoretic kinds of statements, the use of game-theoretic analysis in biology began with Ronald Fisher's studies of animal behavior during the 1930s. This work predates the name "game theory", but it shares many important features with this field. The developments in economics were later applied to biology largely by John Maynard Smith in his 1982 book Evolution and the Theory of Games. In addition to being used to describe, predict, and explain behavior, game theory has also been used to develop theories of ethical or normative behavior and to prescribe such behavior.
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Chromatic sums and sum coloring were introduced by Supowit in 1987 using non-graph-theoretic terminology, and first studied in graph theoretic terms by Ewa Kubicka (independently of Supowit) in her 1989 doctoral thesis. Obtaining the chromatic sum may require using more distinct labels than the chromatic number of the graph, and even when the chromatic number of a graph is bounded, the number of distinct labels needed to obtain the optimal chromatic sum may be arbitrarily large. Computing the chromatic sum is NP-hard. However it may be computed in linear time for trees and pseudotrees, and in polynomial time for outerplanar graphs.
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As Professor, he was a Visiting Scholar in the Logic Group at the University of Buffalo and in the Logic and Methodology Group at the University of California at Berkeley. He is the winner of the first Ivor Grattan-Guinness Best Paper Award for History and Philosophy of Logic. The award winning paper is ‘Methodological Practice and Complementary Concepts of Logical Consequence: Tarski’s Model-Theoretic Consequence and Corcoran’s Information-Theoretic Consequence’, published in the journal History and Philosophy of Logic in the first issue of 2009, pp. 21–48. Candidates are authors of contributions published in History and Philosophy of Logic in the past year.
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Multi-agent (multi-agent simulation', equilibrium, game theoretic) models simulate the operation of a system of heterogeneous agents (generating units, companies) interacting with each other, and build the price process by matching the demand and supply in the market. This class includes cost-based models (or production-cost models, PCM), equilibrium or game theoretic approaches (like the Nash-Cournot framework, supply function equilibrium - SFE, strategic production-cost models - SPCM) and agent-based models. Multi-agent models generally focus on qualitative issues rather than quantitative results. They may provide insights as to whether or not prices will be above marginal costs, and how this might influence the players’ outcomes.
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Abraham Robinson's nonstandard analysis does not need any axioms beyond Zermelo–Fraenkel set theory (ZFC) (as shown explicitly by Wilhelmus Luxemburg's ultrapower construction of the hyperreals), while its variant by Edward Nelson, known as internal set theory, is similarly a conservative extension of ZFC.This is shown in Edward Nelson's AMS 1977 paper in an appendix written by William Powell. It provides an assurance that the newness of nonstandard analysis is entirely as a strategy of proof, not in range of results. Further, model theoretic nonstandard analysis, for example based on superstructures, which is now a commonly used approach, does not need any new set-theoretic axioms beyond those of ZFC.
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The late 1960s saw Childs working on the CONCOMP project for Research in Conversational Use of Computers under project director Franklin H. Westervelt. Childs proposed the Extended Set Theoretic approach to data base management in 1968 in his paper Feasibility of a set-theoretical data structure based on a reconstituted definition of relation. The MICRO Relational Database Management System which was implemented in 1970 was based on Childs' Set-Theoretic Data Structure (STDS) work. In 1970 Codd in his key paper on relational databases, A Relational Model of Data for Large Shared Data Banks, cited one of Childs' 1968 papers as part of the basis for the work.
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D.L. Childs, Vice President of Set Theoretic Information Systems (STIS) Corporation, provided continuing guidance in the use of Set-Theoretic Data Structure (STDS) data access software for MICRO. Funding came from the Office of Manpower Administration within the U.S. Department of Labor. MICRO was first used for the study of large social science data bases referred to as micro data; hence the name. Organizations such as the US Department of Labor, the US Environmental Protection Agency, and researchers from the University of Alberta, the University of Michigan, Wayne State University, the University of Newcastle upon Tyne, and Durham University used MICRO to manage very large scale databases until 1998.
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By treating fields of sets on pre-orders as a category in its own right this deep connection can be formulated as a category theoretic duality that generalizes Stone representation without topology. R. Goldblatt had shown that with restrictions to appropriate homomorphisms such a duality can be formulated for arbitrary modal algebras and modal frames. Naturman showed that in the case of interior algebras this duality applies to more general topomorphisms and can be factored via a category theoretic functor through the duality with topological fields of sets. The latter represent the Lindenbaum–Tarski algebra using sets of points satisfying sentences of the S4 theory in the topological semantics.
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The sector is home to more than fifty kindergartens, school and public high schools as well as the Hyperion Private University. The most prestigious high schools in the sector are Matei Basarab National College, situated in Downtown Bucharest and Alexandru Ioan Cuza Theoretic Lyceum, situated in Titan.
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Other promising field- theoretic simulation techniques have been developed recently, but they either still lack the proof of correct statistical convergence, like e.g. the Complex Langevin method (Ganesan 2001), and/or still need to prove their effectiveness on systems, where multiple saddle points are important (Moreira 2003).
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In order-theoretic mathematics, the deviation of a poset is an ordinal number measuring the complexity of a partially ordered set. The deviation of a poset is used to define the Krull dimension of a module over a ring as the deviation of its poset of submodules.
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The constants and the design parameters are linked by several logical and biochemical constraints (e.g., encoded automata theoretic variables must not be recognized as splicing junctions). The input of the automaton are molecular markers given by single stranded DNA (ssDNA) molecules. These markers are signalling aberrant (e.g.
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This chapter is focused on the function field case; the Riemann-Roch theorem is stated and proved in measure-theoretic language, with the canonical class defined as the class of divisors of non-trivial characters of the adele ring which are trivial on the embedded field.
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This many-to-one situation arises in, for example, the uplink of cellular networks. Other channel types exist such as the many- to-many situation. These (information theoretic) channels are also referred to as network links, many of which will be simultaneously active at any given time.
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Geometry & Topology. vol. 9 (2005), pp. 1835-1880 and others. Grushko decomposition theorem is a group-theoretic analog of the Kneser prime decomposition theorem for 3-manifolds which says that a closed 3-manifold can be uniquely decomposed as a connected sum of irreducible 3-manifolds.
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This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry. For simplicity, a reference to the base scheme is often omitted; i.e.
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Category theory deals with morphisms instead of functions. Morphisms are arrows from one object to another. The domain of any morphism is the object from which an arrow starts. In this context, many set theoretic ideas about domains must be abandoned—or at least formulated more abstractly.
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Eisner is the author of the book Stability of Operators and Operator Semigroups (Operator Theory: Advances and Applications, Vol. 209, Birkhäuser, 2010). She is a coauthor of Operator Theoretic Aspects of Ergodic Theory (with Bálint Farkas, Markus Haase, Rainer Nagel, Graduate Texts in Mathematics 272, Springer, 2015).
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Categorical quantum mechanics can also be seen as a type theoretic form of quantum logic that, in contrast to traditional quantum logic, supports formal deductive reasoning.R. Duncan (2006) Types for Quantum Computing, DPhil. thesis. University of Oxford. There exists software that supports and automates this reasoning.
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Torrence Douglas Parsons (1941-1987) was an American mathematician. He worked mainly in graph theory, and is known for introducing a graph-theoretic view of pursuit-evasion problems (Parsons 1976, 1978). He obtained his Ph.D. from Princeton University in 1966 under the supervision of Albert W. Tucker.
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37, pp. 156–189, 1988. that introduced zero-knowledge arguments, as well as a security model using information-theoretic private-channels, and also first formalized the concept of a commitment scheme. 1991, with Torben Pedersen, he demonstrated a well-cited zero-knowledge proof of a DDH tuple.
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The unused hashes do not need to be included in the signature if a cryptographic accumulator is used instead of a hash list. However if the accumulator is based on number-theoretic assumptions this probably defeats the benefit of employing Lamport signatures, e.g. quantum computing resistance.
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It is a time and resource-consuming strategy, affecting performance. The scope is known. It cannot be successful if not supported by other strategies. Claude Shannon's theorems show that if the encryption key is smaller than the secured information, the information-theoretic security can not be achieved.
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Helium's first ionization energy is −24.587387936(25) eV. This value was derived by experiment. The theoretic value of Helium atom's second ionization energy is −54.41776311(2) eV. The total ground state energy of the helium atom is −79.005151042(40) eV, or −2.90338583(13) Atomic units a.u.
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Universal logic is the field of logic that studies the common features of all logical systems, aiming to be to logic what universal algebra is to algebra. A number of approaches to universal logic have been proposed since the twentieth century, using model theoretic, and categorical approaches.
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Mary Ellen Rudin (December 7, 1924 – March 18, 2013) was an American mathematician known for her work in set-theoretic topology. In 2013, Elsevier established the Mary Ellen Rudin Young Researcher Award which is awarded annually to a young researcher, mainly in fields adjacent to general topology.
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The membership of an element of an intersection set in set theory is defined in terms of a logical conjunction: x ∈ A ∩ B if and only if (x ∈ A) ∧ (x ∈ B). Through this correspondence, set- theoretic intersection shares several properties with logical conjunction, such as associativity, commutativity and idempotence.
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The term psychic apparatus (also psychical apparatus, mental apparatus) denotes a central, theoretic construct of Freudian metapsychology, wherein an implicit intake and processing of information takes place, and thereby acts on said information in pursuit of pleasure by way of resolving tension through the reactional discharge of “instinctual impulses”.
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An important special case of an ideal is constituted by those ideals whose set-theoretic complements are filters, i.e. ideals in the inverse order. Such ideals are called prime ideals. Also note that, since we require ideals and filters to be non-empty, every prime ideal is necessarily proper.
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Only rewriting the sentence, or placing appropriate punctuation can resolve a syntactic ambiguity.Critical Thinking, 10th ed., Ch 3, Moore, Brooke N. and Parker, Richard. McGraw-Hill, 2012 For the notion of, and theoretic results about, syntactic ambiguity in artificial, formal languages (such as computer programming languages), see Ambiguous grammar.
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Comments on Knives And Beer Bar Trick: Amazing Balance The Discordian "mandala", containing five Borromean rings configurations Some knot-theoretic links contain multiple Borromean rings configurations; one five-loop link of this type is used as a symbol in Discordianism, based on a depiction in the Principia Discordia.
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Computational immunology began over 90 years ago with the theoretic modeling of malaria epidemiology. At that time, the emphasis was on the use of mathematics to guide the study of disease transmission. Since then, the field has expanded to cover all other aspects of immune system processes and diseases.
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The second definition clarified the meaning of the topological entropy: for a system given by an iterated function, the topological entropy represents the exponential growth rate of the number of distinguishable orbits of the iterates. An important variational principle relates the notions of topological and measure-theoretic entropy.
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His research interests centered on group theoretic methods in physics.Asım Barut: A Personal Tribute by Jonathan P. Dowling, Foundations of Physics, Vol . 28, No. 3, 1998. His books include Theory of the Scattering Matrix, Electrodynamics and Classical Theory of Fields and Particles and Representations of Noncompact Groups and Applications.
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In mathematics, the Bachmann–Howard ordinal (or Howard ordinal) is a large countable ordinal. It is the proof-theoretic ordinal of several mathematical theories, such as Kripke–Platek set theory (with the axiom of infinity) and the system CZF of constructive set theory. It was introduced by and .
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Applications of group theory abound. Almost all structures in abstract algebra are special cases of groups. Rings, for example, can be viewed as abelian groups (corresponding to addition) together with a second operation (corresponding to multiplication). Therefore, group theoretic arguments underlie large parts of the theory of those entities.
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Scientific projects at the Center for Social Complexity focus on investigating social systems and processes on multiple scales: groups, organizations, economies, societies, regions, international systems. Researchers use a variety of interdisciplinary tools, including multi-agent systems and agent-based models (including the MASON toolkit in Java), cellular automata and other social simulation methods, network and graph-theoretic models, GIS (geographic information systems), events data analysis, complexity-theoretic models and other advanced computational methods. The Center houses a specialized simulation environment (the Simulatorium), where faculty, postdoctoral researchers, and graduate research assistants collaborate in a variety of projects. Conflict and cooperation, emergent economic systems, network dynamics, and long-term societal adaptation to environmental change are among the current lines of investigation.
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They may be provable, even if they cannot all be derived from a single consistent set of axioms."Platonism in the Philosophy of Mathematics", (Stanford Encyclopedia of Philosophy) Set-theoretic realism (also set-theoretic Platonism)Ivor Grattan-Guinness (ed.), Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, Routledge, 2002, p. 681. a position defended by Penelope Maddy, is the view that set theory is about a single universe of sets.Naturalism in the Philosophy of Mathematics (Stanford Encyclopedia of Philosophy) This position (which is also known as naturalized Platonism because it is a naturalized version of mathematical Platonism) has been criticized by Mark Balaguer on the basis of Paul Benacerraf's epistemological problem.
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There are three important themes in the categorical approach to logic: ;Categorical semantics: Categorical logic introduces the notion of structure valued in a category C with the classical model theoretic notion of a structure appearing in the particular case where C is the category of sets and functions. This notion has proven useful when the set-theoretic notion of a model lacks generality and/or is inconvenient. R.A.G. Seely's modeling of various impredicative theories, such as system F is an example of the usefulness of categorical semantics. :It was found that the connectives of pre-categorical logic were more clearly understood using the concept of adjoint functor, and that the quantifiers were also best understood using adjoint functors.
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Timsort has been Python's standard sorting algorithm since version 2.3. It is also used to sort arrays of non-primitive type in Java SE 7, on the Android platform, in GNU Octave, on V8, Swift, and Rust. It uses techniques from Peter McIlroy's 1993 paper "Optimistic Sorting and Information Theoretic Complexity".
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This article discusses a few related problems of this type. The unifying theme is that each problem has an operator-theoretic characterization which gives a corresponding parametrization of solutions. More specifically, finding self-adjoint extensions, with various requirements, of symmetric operators is equivalent to finding unitary extensions of suitable partial isometries.
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In algebraic geometry, a log structure provides an abstract context to study semistable schemes, and in particular the notion of logarithmic differential form and the related Hodge-theoretic concepts. This idea has applications in the theory of moduli spaces, in deformation theory and Fontaine's p-adic Hodge theory, among others.
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Considering subgroups, and in particular maximal subgroups, of semigroups often allows one to apply group-theoretic techniques in semigroup theory. There is a one-to-one correspondence between idempotent elements of a semigroup and maximal subgroups of the semigroup: each idempotent element is the identity element of a unique maximal subgroup.
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We take the functional theoretic algebra C[0, 1] of curves. For each loop γ at 1, and each positive integer n, we define a curve \gamma_n called n-curve. The n-curves are interesting in two ways. #Their f-products, sums and differences give rise to many beautiful curves.
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Both GF itself and the GF Resource Grammar Library are open-source. Typologically, GF is a functional programming language. Mathematically, it is a type-theoretic formal system (a logical framework to be precise) based on Martin-Löf's intuitionistic type theory, with additional judgments tailored specifically to the domain of linguistics.
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He also proposed reconstructions of ancient Greek music, and prepared, on historical-theoretic principles, settings for musical performance of Homer, Sappho, Pindar, and the Persai (The Persians) of Aeschylus. Principal works include: Trio (1960) fl, hn, pf. Perspectives (1964) hn. Quintet 1964 cl, bn, tp, db, org. Antifonia (1965) 2tp,2tb.
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In mathematics, a Prüfer domain is a type of commutative ring that generalizes Dedekind domains in a non-Noetherian context. These rings possess the nice ideal and module theoretic properties of Dedekind domains, but usually only for finitely generated modules. Prüfer domains are named after the German mathematician Heinz Prüfer.
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An information theoretic analysis using a simplified but useful model shows that in asexual reproduction, the information gain per generation of a species is limited to 1 bit per generation, while in sexual reproduction, the information gain is bounded by where G is the size of the genome in bits.
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Periodization of History: A theoretic-mathematical analysis. In: History & Mathematics. Moscow: KomKniga/URSS. P.10-38. . As steam power was the technology standing behind industrial society, so information technology is seen as the catalyst for the changes in work organisation, societal structure and politics occurring in the late 20th century.
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As an economic tool market concentration is useful because it reflects the degree of competition in the market. Tirole (1988, p. 247) notes that: :Bain's (1956) original concern with market concentration was based on an intuitive relationship between high concentration and collusion. There are game theoretic models of market interaction (e.g.
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There is no evidence which suggests that zero-sum thinking is an enduring feature of human psychology. Game-theoretic situations rarely apply to instances of individual behaviour. This is demonstrated by the ordinary response to the prisoner's dilemma. Zero- sum thinking is the result of both proximate and ultimate causes.
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The Hofmann voltameter is an apparatus for electrolyzing water, invented by August Wilhelm von Hofmann in 1866.von Hofmann, A. W. Introduction to Modern Chemistry: Experimental and Theoretic; Embodying Twelve Lectures Delivered in the Royal College of Chemistry, London. Walton and Maberly, London, 1866. It consists of three joined upright cylinders, usually glass.
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It allows complex analytic methods to be used in algebraic geometry, and algebraic- geometric methods in complex analysis and field-theoretic methods to be used in both. This is characteristic of a much wider class of problems in algebraic geometry. See also algebraic geometry and analytic geometry for a more general theory.
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Bhaduri's stories are the instances of double session, where the existing civil and political society and the stereotypical literary critiques are at a time attacked with subtle wit. Bhaduri declared the death of model-theoretic formularized criticism. Most of his writings portray the lifestyle of the people of Bengal and eastern Bihar.
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However, game theoretic models suggest that if individuals are able to migrate between groups (which is common in small-scale societies), differences between groups should be difficult to maintain.Henrich, J. Cultural group selection, coevolutionary processes and large-scale cooperation. Journal of Economic Behavior & Organization. Volume 53, Issue 1, January 2004, 3–35.
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His students continued to employ Neyman's entropy technique to achieve a better understanding of repeated games under complexity constraints. Neyman's information theoretic approach opened new research areas beyond bounded complexity. A classic example is the communication game he introduced jointly with Olivier Gossner and Penelope Hernandez.Gossner, O., Hernandez, P., and Neyman, A. (2006).
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Lazarus also published writings on homeopathy and Christianity, in which he advocates spiritual and physical treatments that appear to be precursors to modern day New-Age spirituality.Lazarus, M. E., Trinity and Incarnation. New York: Fowler and Wells, 1851.Lazarus, M. E., Homoeopathy : a theoretic demonstration, with social applications, Making of America Series.
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To clear up the chronology of the beginning of each respective stage he proposes the three production revolutions: the Agrarian or Neolithic Revolution; the Industrial Revolution, and the Information-Scientific RevolutionGrinin L. Production Revolutions and Periodization of History: A Comparative and Theoretic- mathematical Approach. Social Evolution & History. Vol. 6, num, 2, 2007.
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In 2006G. Japaridze, "Introduction to cirquent calculus and abstract resource semantics". Journal of Logic and Computation 16 (2006), pages 489-532. Japaridze conceived cirquent calculus as a proof-theoretic approach that manipulates graph-style constructs, termed cirquents, instead of the more traditional and less general tree-like constructs such as formulas or sequents.
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One of the ways in which the production of a nominative–accusative case marking system can be explained is from an Optimality Theoretic perspective. Case marking is said to fulfill two functions, or constraints: an identifying function and a distinguishing function.de Hoop, Helen and Malchukov, Andrej L. (2008) "Case-marking strategies". Linguistic Inquiry.
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Levi first made a name for himself with his first book, Gambling with Truth. In the text Levi offered a decision theoretic reconstruction of epistemology with a close-eye towards the classical pragmatist philosophers like William James and Charles Sanders Peirce. Levi was known for his work in belief revision and imprecise probability.
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Minimalist grammars are a class of formal grammars that aim to provide a more rigorous, usually proof-theoretic, formalization of Chomskyan Minimalist program than is normally provided in the mainstream Minimalist literature. A variety of particular formalizations exist, most of them developed by Edward Stabler, Alain Lecomte, Christian Retoré, or combinations thereof.
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The subsequent notion of "automorphic representation" has proved of great technical value for dealing with the case that G is an algebraic group, treated as an adelic algebraic group. As a result, an entire philosophy, the Langlands program has developed around the relation between representation and number theoretic properties of automorphic forms..
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He received an honorary degree from the University of Vienna in 2006. In 1953, he married fellow mathematician Mary Ellen Estill, known for her work in set-theoretic topology. The two resided in Madison, Wisconsin, in the eponymous Walter Rudin House, a home designed by architect Frank Lloyd Wright. They had four children.
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Mathematics of Control, Signals, and Systems is a peer-reviewed scientific journal that covers research concerned with mathematically rigorous system theoretic aspects of control and signal processing. The journal was founded by Eduardo Sontag and Bradley Dickinson in 1988. The editors-in-chief are Lars Gruene, Eduardo Sontag, and Jan H. van Schuppen.
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Myers' principal research interests are collective behavior and social movements. His most recent work focuses on racial rioting in the 1960s and 1970s, deterministic and stochastic models of diffusion for collective violence, mathematical models of collective action, media coverage of protests, demonstrations, and riots, and game theoretic analyses of small group negotiation.
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This usage dates from a historical period where classes and sets were not distinguished as they are in modern set-theoretic terminology. Many discussions of "classes" in the 19th century and earlier are really referring to sets, or perhaps rather take place without considering that certain classes can fail to be sets.
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Using a collection of automation programs called "Eve", Parker has applied his techniques within his dictionary project to digital poetry; he reports posting over 1.3 million poems, aspiring to reach one poem for each word found in the English language.An Introduction to "graph theoretic poetry" . websters-online- dictionary.org. Icon Group International, Inc.
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In proof theory, a structural rule is an inference rule that does not refer to any logical connective, but instead operates on the judgment or sequents directly. Structural rules often mimic intended meta-theoretic properties of the logic. Logics that deny one or more of the structural rules are classified as substructural logics.
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In mathematics, abstract nonsense, general abstract nonsense, generalized abstract nonsense, and general nonsense are terms used by mathematicians to describe abstract methods related to category theory and homological algebra. More generally, “abstract nonsense” may refer to a proof that relies on category-theoretic methods, or even to the study of category theory itself.
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Fraïssé used the back-and- forth method to determine whether two model-theoretic structures were elementarily equivalent. This method of determining elementary equivalence was later formulated as the Ehrenfeucht–Fraïssé game. Fraïssé worked primarily in relation theory. Another of his important works was the Fraïssé construction of a Fraïssé limit of finite structures.
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"Computability Logic" is a proper noun referring to a research programme initiated by Giorgi Japaridze in 2003. Its ambition is to redevelop logic from a game-theoretic semantics. Such a semantics sees games as formal equivalents of interactive computational problems, and their "truth" as existence of algorithmic winning strategies. See Computability logic.
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During that period Karpaty played in final of the Ukrainian Cup and played against Swedish Helsingborgs IF in the UEFA Cup. Eventually Karimov returned to coach for the Karpaty sports school where he worked as a theoretic. The Ukrainian football defender and native of Lviv Ihor Karimov is a son of Andriy Karimov.
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Turning his attention to game theory, Ho in 1965 published the paper Differential Games and Optimal Pursuit-Evasion Strategies, that proved the optimality of a proportional guidance scheme. His paper Nonzero Sum Differential Games was an influential game-theoretic study in systems and control, following earlier work by Samuel Karlin, for example.
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In other words, there would be efficient quantum algorithms that perform tasks that do not have efficient probabilistic algorithms. This would not however invalidate the original Church–Turing thesis, since a quantum computer can always be simulated by a Turing machine, but it would invalidate the classical complexity-theoretic Church–Turing thesis for efficiency reasons. Consequently, the quantum complexity-theoretic Church–Turing thesis states: "A quantum Turing machine can efficiently simulate any realistic model of computation." Eugene Eberbach and Peter Wegner claim that the Church–Turing thesis is sometimes interpreted too broadly, stating "the broader assertion that algorithms precisely capture what can be computed is invalid".. They claim that forms of computation not captured by the thesis are relevant today, terms which they call super-Turing computation.
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In measure-theoretic analysis and related branches of mathematics, Lebesgue–Stieltjes integration generalizes Riemann–Stieltjes and Lebesgue integration, preserving the many advantages of the former in a more general measure-theoretic framework. The Lebesgue–Stieltjes integral is the ordinary Lebesgue integral with respect to a measure known as the Lebesgue–Stieltjes measure, which may be associated to any function of bounded variation on the real line. The Lebesgue–Stieltjes measure is a regular Borel measure, and conversely every regular Borel measure on the real line is of this kind. Lebesgue–Stieltjes integrals, named for Henri Leon Lebesgue and Thomas Joannes Stieltjes, are also known as Lebesgue–Radon integrals or just Radon integrals, after Johann Radon, to whom much of the theory is due.
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The Greeks distinguished theoretic > from problematic analysis. A theoretic analysis is of the following kind. To > prove that A is B, assume first that A is B. If so, then, since B is C and C > is D and D is E, therefore A is E. If this be known a falsity, A is not B. > But if this be a known truth and all the intermediate propositions be > convertible, then the reverse process, A is E, E is D, D is C, C is B, > therefore A is B, constitutes a synthetic proof of the original theorem. > Problematic analysis is applied in all cases where it is proposed to > construct a figure which is assumed to satisfy a given condition.
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One such was given in 2009 by Voevodsky, another in 2010 by van den Berg and Garner. A general solution, building on Voevodsky's construction, was eventually given by Lumsdaine and Warren in 2014. At the PSSL86 in 200786th edition of the Peripatetic Seminar on Sheaves and Logic, Henri Poincaré University, September 8-9 2007 Awodey gave a talk titled "Homotopy type theory" (this was the first public usage of that term, which was coined by AwodeyPreliminary list of PSSL86 participants). Awodey and Warren summarized their results in the paper "Homotopy theoretic models of identity types", which was posted on the ArXiv preprint server in 2007 and published in 2009; a more detailed version appeared in Warren's thesis "Homotopy theoretic aspects of constructive type theory" in 2008.
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The economics of scientific knowledge is an approach to understanding science which is predicated on the need to understand scientific knowledge creation and dissemination in economic terms. The approach has been developed as a contrast to the sociology of scientific knowledge, which places scientists in their social context and examines their behavior using social theory. The economics of scientific knowledge typically involves thinking of scientists as having economic interests with these being thought of as utility maximisation and science as being a market process. Modelling strategies might use any of a variety of approaches including the neoclassical, game theoretic, behavioural (bounded rationality) information theoretic and transaction costs. Boumans and Davis (2010) mention Dasgupta and David (1994) as being an interesting early example of work in this area.
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MICRO runs under the Michigan Terminal System (MTS), the interactive time-sharing system developed at the University of Michigan that runs on IBM System/360 Model 67, System/370, and compatible mainframe computers."Chapter 6: MICRO" in Introduction to database management systems on MTS, Rick Rilio, User Guide Series, Computing Center, University of Michigan, March 1986, pages 147-189 MICRO provides a query language, a database directory, and a data dictionary to create an interface between the user and the very efficient proprietary Set-Theoretic Data Structure (STDS) software developed by the Set-Theoretic Information Systems Corporation (STIS) of Ann Arbor, Michigan. The lower level routines from STIS treat the data bases as sets and perform set operations on them, e.g., union, intersection, restrictions, etc.
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In her interdisciplinary paper "The Internal Description of a Causal Set: What the Universe Looks Like from the Inside", Markopoulou instantiates some abstract terms from mathematical category theory to develop straightforward models of space-time. It proposes simple quantum models of space-time based on category-theoretic notions of a topos and its subobject classifier (which has a Heyting algebra structure, but not necessarily a Boolean algebra structure). For example, hard-to-picture category-theoretic "presheaves" from topos theory become easy-to-picture "evolving (or varying) sets" in her discussions of quantum spacetime. The diagrams in Markopoulou's papers (including hand-drawn diagrams in one of the earlier versions of "The Internal Description of a Causal Set") are straightforward presentations of possible models of space-time.
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Timothy Avelin Roughgarden is an American computer scientist and a Professor of Computer Science at Columbia University. Roughgarden's work deals primarily with game theoretic questions in computer science. Roughgarden received his Ph.D. at Cornell University in 2002, under the supervision of Éva Tardos. He earned his postdoc from University of California, Berkeley in 2004.
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Contract-theoretic principal–agent models have been applied in various fields, including financial contracting, regulation, public procurement, monopolistic price- discrimination, job design, internal labor markets, team production, and many others. From the cybernetics point of view, the Cultural Agency Theory arose in order to better understand the socio-cultural nature of organisations and their behaviours.
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Dual affine planes can be viewed as a point residue of a projective plane, a 1-design, and, more classically, as a tactical configuration. Since they are not pairwise balanced designs (PBDs), they have not been studied extensively from the design-theoretic viewpoint. However, tactical configurations are central topics in geometry, especially finite geometry.
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Her dissertation, The Finite Primitive Collineation Groups which contain Homologies of Period Two, concerned the group-theoretic properties of collineations, geometric transformations preserving straight lines; she also published this material in three journal papers. J. A. Todd, who supervised her research work, observed that "the detailed results contained in her papers" were "of permanent value".
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The terms "abuse of language" and "abuse of notation" depend on context. Writing "f: A → B" for a partial function from A to B is almost always an abuse of notation, but not in a category theoretic context, where f can be seen as a morphism in the category of sets and partial functions.
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The theorem has a natural interpretation in the theory of finite Markov chains (where it is the matrix-theoretic equivalent of the convergence of an irreducible finite Markov chain to its stationary distribution, formulated in terms of the transition matrix of the chain; see, for example, the article on the subshift of finite type).
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The term information- theoretic security is often used interchangeably with the term unconditional security. The latter term can also refer to systems that do not rely on unproven computational hardness assumptions. Today, such systems are essentially the same as those that are information-theoretically secure. Nevertheless, it does not always have to be that way.
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The version discussed here was developed independently by Daniel Shanks in 1973, who explained: > My tardiness in learning of these historical references was because I had > lent Volume 1 of Dickson's History to a friend and it was never > returned.Daniel Shanks. Five Number-theoretic Algorithms. Proceedings of the > Second Manitoba Conference on Numerical Mathematics.
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In their 2017 article published in the Journal of International Economics, the authors examined WTO disputes filed by the United States between 1995 and 2014. They developed a theoretic model to explain the regularity with which incumbent presidential candidates filed trade disputes involving industries in swing states in the year prior to presidential elections.
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Honora Sneyd, through her early contact with members of the Lunar Society, had always taken a keen interest in science, an attribute that drew the intention of Richard Edgeworth who considered himself an inventor. Following their marriage, she worked on his projects with him and in his words, "became an excellent theoretic mechanick" herself.
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SMT solvers are useful both for verification, proving the correctness of programs, software testing based on symbolic execution, and for synthesis, generating program fragments by searching over the space of possible programs. Outside of software verification, SMT solvers have also been used for modelling theoretic scenarios, including modelling actor beliefs in nuclear arms control .
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Information fluctuation complexity is an information-theoretic quantity defined as the fluctuation of information about entropy. It is derivable from fluctuations in the predominance of order and chaos in a dynamic system and has been used as a measure of complexity in many diverse fields. It was introduced in a 1993 paper by Bates and Shepard.
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Hence NF and related theories usually employ Quine's set-theoretic definition of the ordered pair, which yields a type-level ordered pair. Holmes (1998) takes the ordered pair and its left and right projections as primitive. Fortunately, whether the ordered pair is type-level by definition or by assumption (i.e., taken as primitive) usually does not matter.
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The sum total of fingers displayed is either odd or even. If the result is odd, then the person who called odds is the victor, and can decide the issue as they see fit. Often, the participants continue to shoot for a best two out of three. From a game-theoretic perspective, the game is equivalent to matching pennies.
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Note this quantity is not same as Holevo's chi quantity or coherent information( each of them plays important role in quantum information theory). The information theoretic meaning of Ohya's quantum mutual information is still obscure He also proposed 'Information Dynamics' and 'Adaptive Dynamics', which he applied to the study of Chaos theory, quantum information and biosciences as related fields.
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Hearn proved that Kōnane is PSPACE-complete with respect to the dimensions of the board, by a reduction from Constraint Logic. There have been some positive results for restricted configurations. Ernst derives Combinatorial-Game-Theoretic values for several interesting positions. Chan and Tsai analyze the 1 × n game, but even this version of the game is not yet solved.
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His research helped establish the link between artificial intelligence and decision science. As an example, he coined the concept of bounded optimality, a decision-theoretic approach to bounded rationality. The influences of bounded optimality extend beyond computer science into cognitive science and psychology. He studied the use of probability and utility to guide automated reasoning for decision making.
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These and other aspects of the Majumdar–Papapetrou metric have attracted considerable attention on the classical side, as well as in the work and applications from the perspective of string theory. In particular, the mass equal to charge aspect of these models was used extensively in certain string theoretic considerations connected to black hole entropy and related issues.
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Analytic number theory studies numbers, often integers or rationals, by taking advantage of the fact that they can be regarded as complex numbers, in which analytic methods can be used. This is done by encoding number-theoretic information in complex-valued functions. For example, the Riemann zeta function is related to the distribution of prime numbers.
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The state of the world's animal genetic resources for food and agriculture. Barbara Rischkowsky and Dafydd Pilling. Commission on Genetic Resources for Food and Agriculture. 2007 It was shown by set-theoretic means that for the term breed an infinite number of different definitions, which more or less meet the common requirements found in literature, can be given.
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Two to four movements were common, with contrasting tempos (slow-fast-slow-fast). In the 18th century England, the word 'voluntary' and 'fuge' were interchangeable. These English style 'fuges' (or fugue) do not follow the strict theoretic form of German-style fugues. They are more related to the 'fugues' written by Italian composers of the time.
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The positions of the nodes in this equilibrium are used to generate a drawing of the graph. For forces defined from springs whose ideal length is proportional to the graph-theoretic distance, stress majorization gives a very well-behaved (i.e., monotonically convergent). and mathematically elegant way to minimise these differences and, hence, find a good layout for the graph.
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Intraocular lens power calculation formulas fall into two major categories: regression formulas and theoretical formulas. Regression formulas are now obsolete and modern theoretic formulas are used instead. The regression formulas are empiric formulas generated by averaging large numbers of postoperative clinical results (i.e. from retrospective computer analysis of data obtained from a great many patients who have undergone surgery).
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This is why NBG is finitely axiomatizable. Classes are also used for other constructions, for handling the set-theoretic paradoxes, and for stating the axiom of global choice, which is stronger than ZFC's axiom of choice. John von Neumann introduced classes into set theory in 1925. The primitive notions of his theory were function and argument.
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If the device ends in an accepting state, the device is said to accept the sequence of symbols. A family of acceptors is a set of acceptors with the same type of internal store. The study of AFA is part of AFL (abstract families of languages) theory.Seymour Ginsburg, Algebraic and automata theoretic properties of formal languages, North-Holland, 1975, .
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The polyhedral graph formed as the Schlegel diagram of a regular dodecahedron. Schlegel diagram of truncated icosidodecahedral graph In geometric graph theory, a branch of mathematics, a polyhedral graph is the undirected graph formed from the vertices and edges of a convex polyhedron. Alternatively, in purely graph-theoretic terms, the polyhedral graphs are the 3-vertex-connected planar graphs.
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In 1998, z-Tree was developed by Urs Fischbacher. z-Tree is the first and the most cited software tool for experimental economics. z-Tree allows the definition of game rules in z-Tree-language for game-theoretic experiments with human subjects. It also allows definition of computer players, which participate in a play with human subjects.
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Insights from related work have been applied over the past two decades. At the LAP 2004 - Conference, Kalle Lyytinen discussed the academic/theoretic success of LAP.The Struggle with the Language in the IT - Why is LAP not in the Mainstream? LAP 2004 - Conference Yet, these LAP successes have not found entry into the wider stream of applications.
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Game Description Language, or GDL, is a logic programming language designed by Michael Genesereth as part of the General Game Playing Project at Stanford University, California. GDL describes the state of a game as a series of facts, and the game mechanics as logical rules. GDL is hereby one of alternative representations for game theoretic problems.
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Zoltán "Zoli" Tibor Balogh (December 7, 1953 – June 19, 2002) was a Hungarian- born mathematician, specializing in set-theoretic topology. His father, Tibor Balogh, was also a mathematician. His best-known work concerned solutions to problems involving normality of products, most notably the first ZFC constructionZ. Balogh, A small Dowker space in ZFC, Proc. Amer. Math. Soc.
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The desire to understand hard optimization problems from the perspective of approximability is motivated by the discovery of surprising mathematical connections and broadly applicable techniques to design algorithms for hard optimization problems. One well-known example of the former is the Goemans–Williamson algorithm for maximum cut, which solves a graph theoretic problem using high dimensional geometry.
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While at Oxford Farrar published his major work, Science in Theology, Sermons before the University of Oxford, in 1859, followred by A Critical History of Free Thought, the Bampton Lectures in 1862. In the former he sought "to bring some of the discoveries and methods of the physical and moral sciences to bear upon theoretic questions of theology".
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In mathematical optimization, the network simplex algorithm is a graph theoretic specialization of the simplex algorithm. The algorithm is usually formulated in terms of a minimum-cost flow problem. The network simplex method works very well in practice, typically 200 to 300 times faster than the simplex method applied to general linear program of same dimensions.
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In the mathematical theory of decisions, decision-theoretic rough sets (DTRS) is a probabilistic extension of rough set classification. First created in 1990 by Dr. Yiyu Yao, the extension makes use of loss functions to derive \textstyle \alpha and \textstyle \beta region parameters. Like rough sets, the lower and upper approximations of a set are used.
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Combinatorial number theory deals with number theoretic problems which involve combinatorial ideas in their formulations or solutions. Paul Erdős is the main founder of this branch of number theory. Typical topics include covering system, zero-sum problems, various restricted sumsets, and arithmetic progressions in a set of integers. Algebraic or analytic methods are powerful in this field.
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This approximation ratio is close to best possible, as under standard complexity-theoretic assumptions a ratio of (1-\epsilon)\log n cannot be achieved in polynomial time for any \epsilon>0 . The latter hardness of approximation still holds for instances restricted to subcubic graphs , and even to bipartite subcubic graphs as shown in Hartung's PhD thesis .
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David Baron similarly presents a game-theoretic model of media behaviour, suggesting that mass media outlets only hire journalists whose writing is aligned with their political positions. This engages false consensus bias, as beliefs are determined to be common due to the surrounding of aligned views. This effectively heightens the political bias within media representation of information.
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A convenient notation for theoretic scheduling problems was introduced by Ronald Graham, Eugene Lawler, Jan Karel Lenstra and Alexander Rinnooy Kan in. It consists of three fields: α, β and γ. Each field may be a comma separated list of words. The α field describes the machine environment, β the job characteristics and constraints, and γ the objective function.
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She is primarily known for her work on syllable structure within an optimality theoretic framework. Her work has addressed the phonology of the Japanese language, including the rendaku phenomenon. Her work has been published in Linguistic Inquiry amongst other peer-reviewed research journals in linguistics. She often collaborates with her UCSC colleague and husband Armin Mester.
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This section gives a measure-theoretic proof of the theorem. There is also a functional-analytic proof, using Hilbert space methods, that was first given by von Neumann. For finite measures and , the idea is to consider functions with . The supremum of all such functions, along with the monotone convergence theorem, then furnishes the Radon–Nikodym derivative.
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In 2002 he retired from RIMS as professor emeritus and then became a professor at Chūō University. Ihara has done important research on geometric and number theoretic applications of Galois theory. In the 1960s he introduced the eponymous Ihara zeta function.Ihara: On discrete subgroups of the two by two projective linear group over p-adic fields.
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Hofmann voltameter A Hofmann voltameter is an apparatus for electrolysing water, invented by August Wilhelm von Hofmann (1818–1892)von Hofmann, A. W. Introduction to Modern Chemistry: Experimental and Theoretic; Embodying Twelve Lectures Delivered in the Royal College of Chemistry, London. Walton and Maberly, London, 1866. in 1866. It consists of three joined upright cylinders, usually glass.
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Penelope Maddy (born 4 July 1950) is an American philosopher. She is a UCI Distinguished Professor of Logic and Philosophy of Science and of Mathematics at the University of California, Irvine. She is well known for her influential work in the philosophy of mathematics, where she has worked on mathematical realism (especially set-theoretic realism) and mathematical naturalism.
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Order theory is a branch of mathematics which investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and provides basic definitions. A list of order-theoretic terms can be found in the order theory glossary.
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T. Derham, S. Doughty, C. Baker, K. Woodbridge, "Ambiguity Functions for Spatially Coherent and Incoherent Multistatic Radar," IEEE Trans. Aerospace and Electronic Systems (in press). If not, the optimal detector performs incoherent summation of received signals which gives diversity gain. Such systems are sometimes described as MIMO radars due to the information theoretic similarities to MIMO communication systems.
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Information-theoretic death is a term of art used in cryonics to define death in a way that is permanent and independent of any future medical advances, no matter how distant or improbable that may be. Because detailed reading or restoration of information-storing brain structures is well beyond current technology, the term lacks practical importance in medicine.
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Francez's current research focuses on proof-theoretic semantics for logic and natural language. He has also carried out work in formal semantics of natural language, type-logical grammar, computational linguistics, unification-based grammar formalisms (LFG, HPSG). In the past he was interested in semantics of programming languages, program verification, concurrent and distributed programming and logic programming.
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When Wittgenstein visited Vienna, Carnap would meet with him. He (with Hahn and Neurath) wrote the 1929 manifesto of the Circle, and (with Hans Reichenbach) initiated the philosophy journal Erkenntnis. In February 1930 Alfred Tarski lectured in Vienna, and during November 1930 Carnap visited Warsaw. On these occasions he learned much about Tarski's model theoretic method of semantics.
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Most eusocial insect societies have haplodiploid sexual determination, which means that workers are unusually closely related. This explanation of insect eusociality has however been challenged by a few highly noted evolutionary game theorists (Nowak and Wilson) who have published a controversial alternative game theoretic explanation based on a sequential development and group selection effects proposed for these insect species.
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Contract-theoretic screening models have been pioneered by Roger Myerson and Eric Maskin. They have been extended in various directions, e.g. it has been shown that in the context of patent licensing optimal screening contracts may actually yield too much trade compared to the first-best solution. Applications of screening models include regulation, public procurement, and monopolistic price discrimination.
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There are many named subsystems of second-order arithmetic. A subscript 0 in the name of a subsystem indicates that it includes only a restricted portion of the full second-order induction scheme (Friedman 1976). Such a restriction lowers the proof-theoretic strength of the system significantly. For example, the system ACA0 described below is equiconsistent with Peano arithmetic.
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The Jacobson radical of a ring has numerous internal characterizations, including a few definitions that successfully extend the notion to rings without unity. The radical of a module extends the definition of the Jacobson radical to include modules. The Jacobson radical plays a prominent role in many ring and module theoretic results, such as Nakayama's lemma.
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Tor Nørretranders' mother is Yvonne Levy (1920-) and his father was Bjarne Nørretranders (1922-1986). Tor Nørretranders graduated at "Det frie gymnasium" in 1973 and reached a cand.techn.soc-degree from Roskilde University (Roskilde) in 1982, specialized in environment planning and its scientific theoretic basis. He lives north of Copenhagen with his wife Rikke Ulk and three children.
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On a Riemannian manifold, one may define a -dimensional Hausdorff measure for any (integer or real), which may be integrated over -dimensional subsets of the manifold. A function times this Hausdorff measure can then be integrated over -dimensional subsets, providing a measure-theoretic analog to integration of -forms. The -dimensional Hausdorff measure yields a density, as above.
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The image segmentation problem is concerned with partitioning an image into multiple regions according to some homogeneity criterion. This article is primarily concerned with graph theoretic approaches to image segmentation applying graph partitioning via minimum cut or maximum cut. Segmentation-based object categorization can be viewed as a specific case of spectral clustering applied to image segmentation.
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In addition to the thermodynamic perspective of entropy, the tools of information theory can be used to provide an information perspective of entropy. In particular, it is possible to derive the Sackur–Tetrode equation in information-theoretic terms. The overall entropy is represented as the sum of four individual entropies, i.e., four distinct sources of missing information.
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From the computer science point of view, however, the resulting pospaces have a severe drawback. Because partial orders are by definition antisymmetric, their only directed loops i.e. directed paths which end where they start, are the constant loops. Inspired by smooth manifolds, L. Fajstrup, E. Goubault, and M. Raussen use the sheaf-theoretic approach to define local pospaces.
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In model theory, a branch of mathematical logic, U-rank is one measure of the complexity of a (complete) type, in the context of stable theories. As usual, higher U-rank indicates less restriction, and the existence of a U-rank for all types over all sets is equivalent to an important model-theoretic condition: in this case, superstability.
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Access structures are used in the study of security system where multiple parties need to work together to obtain a resource. Groups of parties that are granted access are called qualified. In set theoretic terms they are referred to as qualified sets. In turn, the set of all such qualified sets is called the access structure of the system.
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A twisted cubic curve, the subject of Atiyah's first paper Atiyah's early papers on algebraic geometry (and some general papers) are reprinted in the first volume of his collected works. As an undergraduate Atiyah was interested in classical projective geometry, and wrote his first paper: a short note on twisted cubics. He started research under W. V. D. Hodge and won the Smith's prize for 1954 for a sheaf-theoretic approach to ruled surfaces, which encouraged Atiyah to continue in mathematics, rather than switch to his other interests—architecture and archaeology. His PhD thesis with Hodge was on a sheaf-theoretic approach to Solomon Lefschetz's theory of integrals of the second kind on algebraic varieties, and resulted in an invitation to visit the Institute for Advanced Study in Princeton for a year.
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Cohen- Macaulay schemes have a special relation with intersection theory. Precisely, let X be a smooth varietysmoothness here is somehow extraneous and is used in part to make sense of a proper component. and V, W closed subschemes of pure dimension. Let Z be a proper component of the scheme-theoretic intersection V \times_X W, that is, an irreducible component of expected dimension.
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Gasarch received his doctorate in computer science from Harvard in 1985, advised by Harry R. Lewis. His thesis was titled Recursion-Theoretic Techniques in Complexity Theory and Combinatorics. He was hired into a tenure track professorial job at the University of Maryland in the Fall of 1985. He was promoted to Associate Professor with Tenure in 1991, and to Full Professor in 1998.
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A subgroupoid is a subcategory that is itself a groupoid. A groupoid morphism is simply a functor between two (category- theoretic) groupoids. The category whose objects are groupoids and whose morphisms are groupoid morphisms is called the groupoid category, or the category of groupoids, denoted Grpd. It is useful that this category is, like the category of small categories, Cartesian closed.
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See also pages 188, 250., gives a very brief proof of the cut-elimination theorem. a result with far-reaching meta-theoretic consequences, including consistency. Gentzen further demonstrated the power and flexibility of this technique a few years later, applying a cut- elimination argument to give a (transfinite) proof of the consistency of Peano arithmetic, in surprising response to Gödel's incompleteness theorems.
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A diagram of a link farm. Each circle represents a website, and each arrow represents a pair of hyperlinks between two websites. On the World Wide Web, a link farm is any group of websites that all hyperlink to other sites in the group for the purpose of increasing SEO rankings. In graph theoretic terms, a link farm is a clique.
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From a game-theoretic point of view, "chicken" and "hawk–dove" are identical; the different names stem from parallel development of the basic principles in different research areas.Osborne and Rubenstein (1994) p. 30. The game has also been used to describe the mutual assured destruction of nuclear warfare, especially the sort of brinkmanship involved in the Cuban Missile Crisis.Russell (1959) p. 30.
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4, 1963, pp. 765-767 (in Russian). provide C^k continuity for the functions exactly defining the set-theoretic operations (min/max functions are a particular case). Because of this property, the result of any supported operation can be treated as the input for a subsequent operation; thus very complex models can be created in this way from a single functional expression.
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One issue for the Everett interpretation is the role that probability plays on this account. The Everettian account is completely deterministic, whereas probability seems to play an ineliminable role in quantum mechanics.David Wallace, 'The Emergent Multiverse', pp. 113–117 Contemporary Everettians have argued that one can get an account of probability that follows the Born Rule through certain decision- theoretic proofs.
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This novel proof-theoretic approach was later successfully used to "tame" various fragments of computability logic,G. Japaridze, "The taming of recurrences in computability logic through cirquent calculus, Part I". Archive for Mathematical Logic 52 (2013), pages 173-212.G. Japaridze, "The taming of recurrences in computability logic through cirquent calculus, Part II". Archive for Mathematical Logic 52 (2013), pages 213-259.
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However, in many circumstances the opposite categories have no inherent meaning, which makes duality an additional, separate concept. A category that is equivalent to its dual is called self-dual. An example of self-dual category is the category of Hilbert spaces. Many category-theoretic notions come in pairs in the sense that they correspond to each other while considering the opposite category.
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With more complex expressions one can capture all kinds of nameless yet potentially meaningful relations and operations on computational problems, such as, for instance, "Turing-reducing the problem of semideciding r to the problem of many-one reducing q to p". Imposing time or space restrictions on the work of the machine, one further gets complexity-theoretic counterparts of such relations and operations.
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The DC-4s had a theoretic capacity of 44, but SAS chose to install 28 seats to increase comfort. The first flight with the SAS livery was flown on 17 September, and included a large delegation from SAS's management. Prior to this, SAS had established an American subsidiary, SAS Inc, which was able to sell as many tickets as those sold in Scandinavia.
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To any irreducible algebraic variety is associated its function field. The points of an algebraic variety correspond to valuation rings contained in the function field and containing the coordinate ring. The study of algebraic geometry makes heavy use of commutative algebra to study geometric concepts in terms of ring-theoretic properties. Birational geometry studies maps between the subrings of the function field.
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Schmeidler has made many other contributions, ranging from conceptual issues in implementation theory, to mathematical results in measure theory. But his most influential contribution is probably in decision theory. Schmeidler was the first to propose a general-purpose, axiomatically- based decision theoretic model that deviated from the Bayesian dictum, according to which any uncertainty can and should be quantified by probabilities.
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763–770, 2005. It has also been extended to other applications. The algorithm was initially published by Leo Grady as a conference paperLeo Grady, Gareth Funka-Lea: Multi-Label Image Segmentation for Medical Applications Based on Graph-Theoretic Electrical Potentials, Proc. of the 8th ECCV Workshop on Computer Vision Approaches to Medical Image Analysis and Mathematical Methods in Biomedical Image Analysis, pp.
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Joan Helene Hambidge (born 11 September 1956 in Aliwal North, South Africa) (the English surname notwithstanding), is an Afrikaans poet, literary theorist and academic. She is a prolific poet in Afrikaans, controversial as a public figure and critic and notorious for her out-of-the-closet style of writing. Her theoretic contributions deal mainly with Roland Barthes, deconstruction, postmodernism, psychoanalysis and metaphysics.
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Takeuchi's work, however, was in Japanese and was not widely known outside Japan for many years. AICc was originally proposed for linear regression (only) by . That instigated the work of , and several further papers by the same authors, which extended the situations in which AICc could be applied. The first general exposition of the information-theoretic approach was the volume by .
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She has also co-authored two monographs with Justin Brown: Flag Varieties: An Interplay of Geometry, Combinatorics, and Representation Theory (Texts and Readings in Mathematics 53, Hindustan Book Agency, 2009) and The Grassmannian Variety: Geometric and Representation-Theoretic Aspects (Developments in Mathematics 42, Springer, 2015). In 2012 she was selected as one of the inaugural fellows of the American Mathematical Society.
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1991 (Wye, Kent, UK): H. Isil Bozma, Yale University, New Haven, CT, USA. H.I. Bozma, J.S. Duncan: Model-based recognition of multiple deformable objects using a game-theoretic framework. 1993 (Flagstaff, AZ, USA): Jeffrey A. Fessler, University of Michigan, Ann Arbor, MI, USA. J.A. Fessler: Tomographic reconstruction using information-weighted spline smoothing. 1995 (Brest, France): Maurits K. Konings, University Hospital, Utrecht, The Netherlands.
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Randomized algorithms are algorithms that employ a degree of randomness as part of their logic. These algorithms can be used to give good average-case results (complexity-wise) to problems which are hard to solve deterministically, or display poor worst-case complexity. An algorithmic game theoretic approach can help explain why in the average case randomized algorithms may work better than deterministic algorithms.
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There are three general strategies for constructing families of expander graphs.see, e.g., The first strategy is algebraic and group-theoretic, the second strategy is analytic and uses additive combinatorics, and the third strategy is combinatorial and uses the zig-zag and related graph products. Noga Alon showed that certain graphs constructed from finite geometries are the sparsest examples of highly expanding graphs.
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Several set-theoretic principles about determinacy stronger than Borel determinacy are studied in descriptive set theory. They are closely related to large cardinal axioms. The axiom of projective determinacy states that all projective subsets of a Polish space are determined. It is known to be unprovable in ZFC but relatively consistent with it and implied by certain large cardinal axioms.
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More generally, hypergeometric series can be generalized to describe the symmetries of any symmetric space; in particular, hypergeometric series can be developed for any Lie group.N. Vilenkin, Special Functions and the Theory of Group Representations, Am. Math. Soc. Transl.,vol. 22, (1968).J. D. Talman, Special Functions, A Group Theoretic Approach, (based on lectures by E.P. Wigner), W. A. Benjamin, New York (1968).
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The problem of spoofing in the hands-off case can be solved using two fundamental information-theoretic properties of quantum physics: # A single quantum in an unknown state cannot be cloned.W.K. Wootters and W.H. Zurek. "A single quantum cannot be cloned". Nature, 299: 802–803, 1982 # When a quantum state is measured most of the information it contains is destroyed.
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Although first theoretic publications about copyright in Switzerland date back to 1738,Rehbinder, p. 32; mentioning the dissertation of Johann Rudolf Thurneisen, Dissertatio juridica inauguralis de recursione librorum furtiva, zu Teutsch dem unerlaubten Büchernachdruck, Basel, 1738. Thurneisen already suggested an international treaty by which countries should protect each others copyrights based on reciprocity. the topic remained unregulated by law until the 19th century.
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The phrasing of "Satisfaction" is a good example of syncopation. It is derived here from its theoretic unsyncopated form, a repeated trochee (¯ ˘ ¯ ˘). A backbeat transformation is applied to "I" and "can't", and then a before-the-beat transformation is applied to "can't" and "no". 1 & 2 & 3 & 4 & 1 & 2 & 3 & 4 & Repeated trochee: ¯ ˘ ¯ ˘ I can't get no – o Backbeat trans.
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For an order theoretic example, let be some set, and let and both be the power set of , ordered by inclusion. Pick a fixed subset of . Then the maps and , where , and , form a monotone Galois connection, with being the lower adjoint. A similar Galois connection whose lower adjoint is given by the meet (infimum) operation can be found in any Heyting algebra.
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In operator algebra, the Koecher–Vinberg theorem is a reconstruction theorem for real Jordan algebras. It was proved independently by Max Koecher in 1957 and Ernest Vinberg in 1961. It provides a one-to-one correspondence between formally real Jordan algebras and so-called domains of positivity. Thus it links operator algebraic and convex order theoretic views on state spaces of physical systems.
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The theoretic bases of Scoptophilia were developed by the psychoanalyst Otto Fenichel, in special reference to the process and stages of psychological identification.Otto Fenichel, The Scoptophilic Instinct and Identification (1953) That in developing a personal identity, "a child, who is looking for libidinous purposes . . . wants to look at an object in order [for it] to ‘feel along with him’."Fenichel, Otto.
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Automatic sequences were introduced by Büchi in 1960, although his paper took a more logico-theoretic approach to the matter and did not use the terminology found in this article. The notion of automatic sequences was further studied by Cobham in 1972, who called these sequences "uniform tag sequences". The term "automatic sequence" first appeared in a paper of Deshouillers.
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It should be mentioned that, though called "tensor product", this is not a tensor product of graphs in the above sense; actually it is the category-theoretic product in the category of graphs and graph homomorphisms. However it is actually the Kronecker tensor product of the adjacency matrices of the graphs. Compare also the section Tensor product of linear maps above.
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In engineering, mathematics, physics, and biology Shannon's theory is used more literally and is referred to as Shannon theory, or information theory. This means that outside of the social sciences, fewer people refer to a "Shannon–Weaver" model than to Shannon's information theory; some may consider it a misinterpretation to attribute the information theoretic channel logic to Weaver as well.
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But we can do this for systems far beyond Peano's axioms. For example, the proof-theoretic strength of Kripke–Platek set theory is the Bachmann–Howard ordinal, and, in fact, merely adding to Peano's axioms the axioms that state the well-ordering of all ordinals below the Bachmann–Howard ordinal is sufficient to obtain all arithmetical consequences of Kripke–Platek set theory.
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Its benefit is that by re-expressing problems in terms of multilinear algebra, there is a clear and well-defined "best solution": the constraints the solution exerts are exactly those you need in practice. In general there is no need to invoke any ad hoc construction, geometric idea, or recourse to co- ordinate systems. In the category-theoretic jargon, everything is entirely natural.
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STEIM is a foundation, financially supported by the Dutch ministry of Culture. It invites international artists in residence of different musical and artistic styles and scenes. Aside from offering support in theoretic and practical development of contemporary musical instruments, STEIM also hosts in-house concerts, exhibitions and workshops. The work in progress of supported artists is presented in open studio events.
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He was recognized in the 1980s for his contribution to world peace through nuclear conflict restraint via his game theoretic models of psychological conflict resolution. He won the Lenz International Peace Research Prize in 1976. Professor Rapoport was also a member of the editorial board of the Journal of Environmental Peace published by the International Innovation Projects at the University of Toronto.
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In a twin prime pair (p, p + 2) with p > 5, p is always a strong prime, since 3 must divide p − 2, which cannot be prime. It is possible for a prime to be a strong prime both in the cryptographic sense and the number theoretic sense. For the sake of illustration, 439351292910452432574786963588089477522344331 is a strong prime in the number theoretic sense because the arithmetic mean of its two neighboring primes is 62 less. Without the aid of a computer, this number would be a strong prime in the cryptographic sense because 439351292910452432574786963588089477522344330 has the large prime factor 1747822896920092227343 (and in turn the number one less than that has the large prime factor 1683837087591611009), 439351292910452432574786963588089477522344332 has the large prime factor 864608136454559457049 (and in turn the number one less than that has the large prime factor 105646155480762397).
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The semantic view of theories is a position in the philosophy of science that holds that a scientific theory can be identified with a collection of models. The semantic view of theories was originally proposed by Patrick Suppes in “A Comparison of the Meaning and Uses of Models in Mathematics and the Empirical Sciences”Suppes, P. (1960), “A Comparison of the Meaning and Uses of Models in Mathematics and the Empirical Sciences,” Synthese 12: 287–301. as a reaction against the received view of theories popular among the logical positivists. Many varieties of the semantic view propose identifying theories with a class of set-theoretic models in the Tarskian sense,Suppes, P. (1960) and da Costa, Newton C. A., and Steven French (1990), “The Model-Theoretic Approach in the Philosophy of Science”, Philosophy of Science 57: 248–265.
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If the channel from Alice to Bob is statistically better than the channel from Alice to Eve, it had been shown that secure communication is possible. That is intuitive, but Wyner measured the secrecy in information theoretic terms defining secrecy capacity, which essentially is the rate at which Alice can transmit secret information to Bob. Shortly afterward, Imre Csiszár and Körner showed that secret communication was possible even if Eve had a statistically better channel to Alice than Bob did. The basic idea of the information theoretic approach to securely transmit confidential messages (without using an encryption key) to a legitimate receiver is to use the inherent randomness of the physical medium (including noises and channel fluctuations due to fading) and exploit the difference between the channel to a legitimate receiver and the channel to an eavesdropper to benefit the legitimate receiver.
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The Three Against Two Relationship as the Foundation of Timelines in West African Musics Urbana, IL: University of Illinois. UnlockingClave.com. 3:2 is the generative or theoretic form of sub-Saharan rhythmic principles. Agawu succinctly states: "[The] resultant [3:2] rhythm holds the key to understanding … there is no independence here, because 2 and 3 belong to a single Gestalt."Agawu, Kofi (2003: 92).
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History of human thought spans across the history of humanity. It covers the history of philosophy, history of science and history of political thought, among others. The discipline studying it is called intellectual history. Merlin Donald has claimed that human thought has progressed through three historic stages: the episodic, the mimetic, and the mythic stages, before reaching the current stage of theoretic thinking or culture.
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The first clear definition of an abstract field is due to .. See also . In particular, Heinrich Martin Weber's notion included the field Fp. Giuseppe Veronese (1891) studied the field of formal power series, which led to introduce the field of p-adic numbers. synthesized the knowledge of abstract field theory accumulated so far. He axiomatically studied the properties of fields and defined many important field-theoretic concepts.
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In mathematics and computer science, the gradations of applicable meaning of a fuzzy concept are described in terms of quantitative relationships defined by logical operators. Such an approach is sometimes called "degree-theoretic semantics" by logicians and philosophers,Roy T. Cook, A dictionary of philosophical logic. Edinburgh University Press, 2009, p. 84. but the more usual term is fuzzy logic or many-valued logic.
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Victor Ufnarovski and Bo Åhlander have detailed the function's connection to famous number-theoretic conjectures like the twin prime conjecture, the prime triples conjecture, and Goldbach's conjecture. For example, Goldbach's conjecture would imply, for each k > 1 the existence of an n so that D(n) = 2k. The twin prime conjecture would imply that there are infinitely many k for which D^2(k) = 1.
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Some of this criticism is intense: see the introduction by Willard Quine preceding Mathematical logic as based on the theory of types in . See also in the introduction to his Axiomatization of Set Theory in Zermelo's set-theoretic response was his 1908 Investigations in the foundations of set theory I – the first axiomatic set theory; here too the notion of "propositional function" plays a role.
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The Nash bargaining solution, however, only deals with the simplest structure of bargaining. It is not dynamic (failing to deal with how pareto outcomes are achieved). Instead, for situations where the structure of the bargaining game is important, a more mainstream game- theoretic approach is useful. This can allow players' preferences over time and risk to be incorporated into the solution of bargaining games.
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Informally distributed, 1969. Notes, December 1969, Oxford Univ. Introduced November 1969, Dana Scott's untyped set theoretic model constructed a proper topology for any λ-calculus model whose function space is limited to continuous functions. The result of a Scott continuous λ-calculus topology is a function space built upon a programming semantic allowing fixed point combinatorics, such as the Y combinator, and data types.
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Ring theory is the branch of mathematics in which rings are studied: that is, structures supporting both an addition and a multiplication operation. This is a glossary of some terms of the subject. For the items in commutative algebra (the theory of commutative rings), see glossary of commutative algebra. For ring-theoretic concepts in the language of modules, see also Glossary of module theory.
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The word "clique", in its graph-theoretic usage, arose from the work of , who used complete subgraphs to model cliques (groups of people who all know each other) in social networks. The same definition was used by in an article using less technical terms. Both works deal with uncovering cliques in a social network using matrices. For continued efforts to model social cliques graph-theoretically, see e.g.
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IL provides an axiomatically formulated theory of language, which currently covers, in particular, phonology, morphology, syntax, semantics, and language variability. IL is a non- generative and non-transformational approach in linguistics: it assumes neither "deep structures" nor transformational relations between sequentially ordered structures. Rather, IL conceives linguistic entities as interrelated, "multidimensional" objects, which are typically modelled as set-theoretic constructs.Nolda, Andreas; and Oliver Teuber (2011).
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In game theory, it is often called a simplex plot.Karl Tuyls, "An evolutionary game-theoretic analysis of poker strategies", Entertainment Computing January 2009 , p. 9 Ternary plots are tools for analyzing compositional data in the three- dimensional case. Approximate colours of Ag–Au–Cu alloys in jewellery making In a ternary plot, the values of the three variables , , and must sum to some constant, .
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Mathematical Methods of Statistics :Author: Harald Cramér :Publication data: Princeton Mathematical Series, vol. 9. Princeton University Press, Princeton, N. J., 1946. xvi+575 pp. (A first version was published by Almqvist & Wiksell in Uppsala, Sweden, but had little circulation because of World War II.) :Description: Carefully written and extensive account of measure-theoretic probability for statisticians, along with careful mathematical treatment of classical statistics.
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In mathematics, the Feferman–Schütte ordinal Γ0 is a large countable ordinal. It is the proof-theoretic ordinal of several mathematical theories, such as arithmetical transfinite recursion. It is named after Solomon Feferman and Kurt Schütte. It is sometimes said to be the first impredicative ordinal,Kurt Schütte, Proof theory, Grundlehren der Mathematischen Wissenschaften, Band 225, Springer-Verlag, Berlin, Heidelberg, New York, 1977, xii + 302 pp.
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This can be considered the central postulate of musical set theory. In practice, set-theoretic musical analysis often consists in the identification of non-obvious transpositional or inversional relationships between sets found in a piece. Some authors consider the operations of complementation and multiplication as well. The complement of set X is the set consisting of all the pitch classes not contained in X .
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28, No. 11 minimum cut,Z. Wu and R. Leahy (1993): "An optimal graph theoretic approach to data clustering: Theory and its application to image segmentation", IEEE Transactions on Pattern Analysis and Machine Intelligence, pp. 1101–1113, Vol. 15, No. 11 isoperimetric partitioning,Leo Grady and Eric L. Schwartz (2006): "Isoperimetric Graph Partitioning for Image Segmentation" , IEEE Transactions on Pattern Analysis and Machine Intelligence, pp.
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The model-theoretic properties of HOL with standard semantics are also more complex than those of first-order logic. For example, the Löwenheim number of second-order logic is already larger than the first measurable cardinal, if such a cardinal exists.Menachem Magidor and Jouko Väänänen. "On Löwenheim- Skolem-Tarski numbers for extensions of first order logic", Report No. 15 (2009/2010) of the Mittag-Leffler Institute.
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Sartre notes that his contemporary Marxists maintained a focus on "analysis" but criticizes this analysis as a superficial study focused on verifying Marxist absolutes ("eternal knowledge") instead of gaining an understanding of historical perspective, as Marx himself did.Sartre, 27. Sartre turns his criticism on to other methods of investigation. He says that "American Sociology" has too much "theoretic uncertainty" while the once promising psychoanalysis has stagnated.
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An alternative to Tarskian (model theoretic) semantics is proposed for some uses where "the truth conditions for quantified formuli are given purely in terms of truth with no appeal to domains of interpretation". This has come to be called "truth-value semantics". Marcus shows that the claim that such a semantics leads to contradictions is false. Such a semantics may be of interest for mathematics e.g.
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The membership of an element of a union set in set theory is defined in terms of a logical disjunction: x ∈ A ∪ B if and only if (x ∈ A) ∨ (x ∈ B). Because of this, logical disjunction satisfies many of the same identities as set- theoretic union, such as associativity, commutativity, distributivity, and de Morgan's laws, identifying logical conjunction with set intersection, logical negation with set complement.
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In both cases the intersection should be a point, because, again, if one cycle is moved, this would be the intersection. The intersection of two cycles and is called proper if the codimension of the (set-theoretic) intersection is the sum of the codimensions of and , respectively, i.e. the "expected" value. Therefore, the concept of moving cycles using appropriate equivalence relations on algebraic cycles is used.
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Non-linear behaviour does manifest itself, yet mostly on pathological inputs. Thus the complexity theoretic proofs by and came as a surprise to the research community. HM is preferably used for functional languages. It was first implemented as part of the type system of the programming language ML. Since then, HM has been extended in various ways, most notably with type class constraints like those in Haskell.
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Hemiola can be used to describe the ratio of the lengths of two strings as three-to-two (3:2), that together sound a perfect fifth. The early Pythagoreans, such as Hippasus and Philolaus, used this term in a music-theoretic context to mean a perfect fifth.Andrew Barker, Greek Musical Writings: [vol. 2] Harmonic and Acoustic Theory (Cambridge: Cambridge University Press, 1989): 31, 37–38.
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A magnet levitating above a high-temperature superconductor. Today some physicists are working to understand high-temperature superconductivity using the AdS/CFT correspondence.Merali 2011 Over the decades, experimental condensed matter physicists have discovered a number of exotic states of matter, including superconductors and superfluids. These states are described using the formalism of quantum field theory, but some phenomena are difficult to explain using standard field theoretic techniques.
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In order to understand the nature of memristor function, some knowledge of fundamental circuit theoretic concepts is useful, starting with the concept of device modelling. Engineers and scientists seldom analyze a physical system in its original form. Instead, they construct a model which approximates the behaviour of the system. By analyzing the behaviour of the model, they hope to predict the behaviour of the actual system.
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Though New Public Administration brought public administration closer to political science, it was criticized as anti-theoretic and anti-management. Robert T. Golembiewski describes it as radicalism in words and status quo in skills and technologies. Further, it must be counted as only a cruel reminder of the gap in the field between aspiration and performance. Golembiewski considers it as a temporary and transitional phenomena.
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His research interests span federal, state, and local finance, and various aspects of the economics of education and econometric methodology. His current research includes the design of least regressive national consumption tax systems, the effects of safety net structures on the labor force participation of adults, the relationship between demography and the fisc, and the game theoretic characterization of the overloading of trucks on the public roads.
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Elwyn Berlekamp, listing at the Department of Mathematics, University of California, Berkeley. Berlekamp was the inventor of an algorithm to factor polynomials, and was one of the inventors of the Berlekamp–Welch algorithm and the Berlekamp–Massey algorithms, which are used to implement Reed–Solomon error correction. Berlekamp had also been active in money management. In 1986, he began information-theoretic studies of commodity and financial futures.
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Subsequent authors have greatly extended Dehn's algorithm and applied it to a wide range of group theoretic decision problems. It was shown by Pyotr Novikov in 1955 that there exists a finitely presented group G such that the word problem for G is undecidable. It follows immediately that the uniform word problem is also undecidable. A different proof was obtained by William Boone in 1958.
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Andreone and Giacoma (1989) speculated that newt migration into the ponds increases after rainy days, since after rainfall, newt activity is not limited by humidity. Higher altitudes, where temperatures begin to decrease, have a direct effect on the size of T. carnifex.(Ficetola, G. F., et al. 2010. Ecogeographical variation of body size in the newt Triturus carnifex: comparing the hypotheses using an information-theoretic approach.
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These two gradations must be compatible, and there is often disagreement as to how they should be regarded. See Deligne's discussion of this difficulty. Still greater generalizations are possible to Lie algebras over a class of braided monoidal categories equipped with a coproduct and some notion of a gradation compatible with the braiding in the category. For hints in this direction, see Lie algebra#Category theoretic definition.
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The notion of D-spaces was introduced by Eric Karel van Douwen and E.A. Michael. It first appeared in a 1979 paper by van Douwen and Washek Frantisek Pfeffer in the Pacific Journal of Mathematics. Whether every Lindelöf and regular topological space is a D-space is known as the D-space problem. This problem is among twenty of the most important problems of set theoretic topology.
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The philosophy of information (PI) is a branch of philosophy that studies topics relevant to computer science, information science and information technology. It includes: # the critical investigation of the conceptual nature and basic principles of information, including its dynamics, utilisation and sciences # the elaboration and application of information-theoretic and computational methodologies to philosophical problems.Luciano Floridi, "What is the Philosophy of Information?" , Metaphilosophy, 2002, (33), 1/2.
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"Radically elementary probability theory" of Edward Nelson combines the discrete and the continuous theory through the infinitesimal approach. The model-theoretical approach of nonstandard analysis together with Loeb measure theory allows one to define Brownian motion as a hyperfinite random walk, obviating the need for cumbersome measure-theoretic developments. Jerome Keisler used this classical approach of nonstandard analysis to characterize general stochastic processes as hyperfinite ones.
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Generalized relative entropy (\epsilon-relative entropy) is a measure of dissimilarity between two quantum states. It is a "one-shot" analogue of quantum relative entropy and shares many properties of the latter quantity. In the study of quantum information theory, we typically assume that information processing tasks are repeated multiple times, independently. The corresponding information-theoretic notions are therefore defined in the asymptotic limit.
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This rule may also more generally be applied to any graph. In graph-theoretic terms, each move is made to the adjacent vertex with the least degree. Although the Hamiltonian path problem is NP-hard in general, on many graphs that occur in practice this heuristic is able to successfully locate a solution in linear time. The knight's tour is such a special case.
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In the above form, the functional to be extended must already be bounded by a sublinear function. In some applications, this might close to begging the question. However, in locally convex spaces, any continuous functional is already bounded by the norm, which is sublinear. One thus hasIn category-theoretic terms, the field is an injective object in the category of locally convex vector spaces.
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In knot theory, the Milnor conjecture says that the slice genus of the (p, q) torus knot is :(p-1)(q-1)/2. It is in a similar vein to the Thom conjecture. It was first proved by gauge theoretic methods by Peter Kronheimer and Tomasz Mrowka.. Jacob Rasmussen later gave a purely combinatorial proof using Khovanov homology, by means of the s-invariant..
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Information theoretic concepts apply to cryptography and cryptanalysis. Turing's information unit, the ban, was used in the Ultra project, breaking the German Enigma machine code and hastening the end of World War II in Europe. Shannon himself defined an important concept now called the unicity distance. Based on the redundancy of the plaintext, it attempts to give a minimum amount of ciphertext necessary to ensure unique decipherability.
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The failures in the reduction of mathematics to pure logic imply that scientific knowledge can at best be defined with the aid of less certain set-theoretic notions. Even if set theory's lacking the certainty of pure logic is deemed acceptable, the usefulness of constructing an encoding of scientific knowledge as logic and set theory is undermined by the inability to construct a useful translation from logic and set-theory back to scientific knowledge. If no translation between scientific knowledge and the logical structures can be constructed that works both ways, then the properties of the purely logical and set-theoretic constructions do not usefully inform understanding of scientific knowledge. On Quine's account, attempts to pursue the traditional project of finding the meanings and truths of science philosophically have failed on their own terms and failed to offer any advantage over the more direct methods of psychology.
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About 1818 Danish scholar Ferdinand Degen displayed the Degen's eight-square identity, which was later connected with norms of elements of the octonion algebra: :Historically, the first non- associative algebra, the Cayley numbers ... arose in the context of the number-theoretic problem of quadratic forms permitting composition…this number-theoretic question can be transformed into one concerning certain algebraic systems, the composition algebras... In 1919 Leonard Dickson advanced the study of the Hurwitz problem with a survey of efforts to that date, and by exhibiting the method of doubling the quaternions to obtain Cayley numbers. He introduced a new imaginary unit , and for quaternions and writes a Cayley number . Denoting the quaternion conjugate by , the product of two Cayley numbers is :(q + Qe)(r + Re) = (qr - R'Q) + (Rq + Q r')e . The conjugate of a Cayley number is , and the quadratic form is , obtained by multiplying the number by its conjugate.
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In the mathematical field of group theory, an Artin transfer is a certain homomorphism from an arbitrary finite or infinite group to the commutator quotient group of a subgroup of finite index. Originally, such mappings arose as group theoretic counterparts of class extension homomorphisms of abelian extensions of algebraic number fields by applying Artin's reciprocity maps to ideal class groups and analyzing the resulting homomorphisms between quotients of Galois groups. However, independently of number theoretic applications, a partial order on the kernels and targets of Artin transfers has recently turned out to be compatible with parent-descendant relations between finite p-groups (with a prime number p), which can be visualized in descendant trees. Therefore, Artin transfers provide a valuable tool for the classification of finite p-groups and for searching and identifying particular groups in descendant trees by looking for patterns defined by the kernels and targets of Artin transfers.
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In mathematics, specifically in order theory and functional analysis, the order topology of an ordered vector space (X, ≤) is the finest locally convex topological vector space (TVS) topology on X for which every order interval is bounded, where an order interval in X is a set of the form [a, b] := { z ∈ X : a ≤ z and z ≤ b } where a and b belong to X. The order topology is an important topology that is used frequently in the theory of ordered topological vector spaces because the topology stems directly from the algebraic and order theoretic properties of (X, ≤), rather than from some topology that X starts out having. This allows for establishing intimate connections between this topology and the algebraic and order theoretic properties of (X, ≤). For many ordered topological vector spaces that occur in analysis, their topologies are identical to the order topology.
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A planar graph and its dual. Every cycle in the blue graph is a minimal cut in the red graph, and vice versa, so the two graphs are algebraic duals and have dual graphic matroids. In mathematics, Whitney's planarity criterion is a matroid-theoretic characterization of planar graphs, named after Hassler Whitney.. It states that a graph G is planar if and only if its graphic matroid is also cographic (that is, it is the dual matroid of another graphic matroid). In purely graph-theoretic terms, this criterion can be stated as follows: There must be another (dual) graph G'=(V',E') and a bijective correspondence between the edges E' and the edges E of the original graph G, such that a subset T of E forms a spanning tree of G if and only if the edges corresponding to the complementary subset E-T form a spanning tree of G'.
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The failures in the reduction of mathematics to pure logic imply that scientific knowledge can at best be defined with the aid of less certain set-theoretic notions. Even if set theory's lacking the certainty of pure logic is deemed acceptable, the usefulness of constructing an encoding of scientific knowledge as logic and set theory is undermined by the inability to construct a useful translation from logic and set-theory back to scientific knowledge. If no translation between scientific knowledge and the logical structures can be constructed that works both ways, then the properties of the purely logical and set-theoretic constructions do not usefully inform understanding of scientific knowledge. On Quine's account, attempts to pursue the traditional project of finding the meanings and truths of science philosophically have failed on their own terms and failed to offer any advantage over the more direct methods of psychology.
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The right hand expression could specify a new pattern to transform the left hand side into. For example, transform a set theoretic data type into code using an Ada set library. The initial purpose for transformation rules was to refine a high level logical specification into well designed code for a specific hardware and software platform. This was inspired by early work on theorem proving and automatic programming.
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They have been used recently for truckload transportation, bus routes, industrial procurement, and in the allocation of radio spectrum for wireless communications. In recent years, procurement teams have applied reverse combinatorial auctions in the procurement of goods and services. This application is often referred to as sourcing optimization. Although they allow bidders to be more expressive, combinatorial auctions present both computational and game-theoretic challenges compared to traditional auctions.
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Mo Abdi Mohammad Abdi (Persian:محمد عبدی, Tehran, Iran, 1974) is a writer, film critic, and art researcher. He has written articles for more than 40 publications and journals in Persian in Iran and abroad. From 1999 - 2000, Abdi served as editor-in-chief of "Seventh Art" magazine, a theoretic magazine on film and cinema. In 1998, Abdi published "Film Criticism in Iran", an analytical, historical study on film criticism in Iran.
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Kleene and Rosser were able to show that both systems are able to characterize and enumerate their provably total, definable number-theoretic functions, which enabled them to construct a term that essentially replicates the Richard paradox in formal language. Curry later managed to identify the crucial ingredients of the calculi that allowed the construction of this paradox, and used this to construct a much simpler paradox, now known as Curry's paradox.
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In these applications, ƒ is expected to behave randomly; Flajolet and Odlyzko. study the graph-theoretic properties of the functional graphs arising from randomly chosen mappings. In particular, a form of the birthday paradox implies that, in a random functional graph with n vertices, the path starting from a randomly selected vertex will typically loop back on itself to form a cycle within O() steps. Konyagin et al.
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J. 30 1967 121–127 . Recently these problems got some attention as they can help to generate random points on smooth manifolds (in particular, unit sphere) with prescribed probability density function. The problem of finding the polarization constant is connected to the problem of energy minimization. In particular, for connections with the Thomson problem, see Farkas, Bálint; Révész, Szilárd Gy. "Potential theoretic approach to rendezvous numbers". Monatsh. Math.
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Most common applications of parity game solving. Despite its interesting complexity theoretic status, parity game solving can be seen as the algorithmic backend to problems in automated verification and controller synthesis. The model-checking problem for the modal μ-calculus for instance is known to be equivalent to parity game solving. Also, decision problems like validity or satisfiability for modal logics can be reduced to parity game solving.
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From a complexity theoretic perspective, the Wallace tree algorithm puts multiplication in the class NC1. The downside of the Wallace tree, compared to naive addition of partial products, is its much higher gate count. These computations only consider gate delays and don't deal with wire delays, which can also be very substantial. The Wallace tree can be also represented by a tree of 3/2 or 4/2 adders.
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In economics and philosophy, scholars have applied game theory to help in the understanding of good or proper behavior. Game-theoretic arguments of this type can be found as far back as Plato. An alternative version of game theory, called chemical game theory, represents the player's choices as metaphorical chemical reactant molecules called “knowlecules”. Chemical game theory then calculates the outcomes as equilibrium solutions to a system of chemical reactions.
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Specification languages are generally not directly executed. They are meant to describe the what, not the how. Indeed, it is considered as an error if a requirement specification is cluttered with unnecessary implementation detail. A common fundamental assumption of many specification approaches is that programs are modelled as algebraic or model-theoretic structures that include a collection of sets of data values together with functions over those sets.
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Alternatively, the model can be portrayed in game theoretic terms as initially a game with multiple Nash equilibria, with government having the capability of affecting the payoffs to switch to a game with just one equilibrium. Although it is possible for the national government to increase a country's welfare in the model through export subsidies, the policy is of beggar thy neighbor type.Cohen and Lipson, p. 22.Baldwin, p. 69.
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Ruthild Winkler-Oswatitsch (born 1941, also known as Ruthild Oswatitsch Eigen) is an Austrian biochemist associated with the Max Planck Institute for Biophysical Chemistry in Germany, and known for two books she coauthored with Nobel prize winner Manfred Eigen. Her research has concerned fast biochemical reactions, game-theoretic models for molecular evolution, and the use of sequence analysis of DNA and RNA in studying the early history of biological evolution.
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A densely defined linear operator T from one topological vector space, X, to another one, Y, is a linear operator that is defined on a dense linear subspace dom(T) of X and takes values in Y, written T : dom(T) ⊆ X → Y. Sometimes this is abbreviated as T : X → Y when the context makes it clear that X might not be the set- theoretic domain of T.
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Two descriptions and two definitions of the catena unit are now given. :Catena (everyday description) :Any single word or any combination of words that are linked together by dependencies. :Catena (graph-theoretic description) :In terms of graph theory, any syntactic tree or connected subgraph of a tree is a catena. Any individual element (word or morph) or combination of elements linked together in the vertical dimension is a catena.
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Three channel types or connection situations in wireless networks. A wireless network can be seen as a collection of (information theoretic) channels sharing space and some common frequency band. Each channel consists of a set of transmitters trying to send data to a set of receivers. The simplest channel is the point-to-point channel which involves a single transmitter aiming at sending data to a single receiver.
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The first of these generalizes chip-firing from Laplacian matrices of graphs to M-matrices, connecting this generalization to root systems and representation theory. The second considers chip-firing on abstract simplicial complexes instead of graphs. The third uses chip-firing to study graph-theoretic analogues of divisor theory and the Riemann–Roch theorem. And the fourth applies methods from commutative algebra to the study of chip-firing.
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Ambient on the spacetime there is also a B-field or Kalb–Ramond field B (not to be confused with the B in B-model), which is the string theoretic equivalent of the classical background electromagnetic field (hence the use of B, which commonly denotes the magnetic field strength).Freed, D.S. and Witten, E., 1999. Anomalies in string theory with $ D $-branes. Asian Journal of Mathematics, 3(4), pp.819-852.
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André Chapuis has argued that the reasoning agents use in rational choice exhibits an interdependence characteristic of circular concepts.Chapuis (2003) Revision theory can be adapted to model other sorts of phenomena. For example, vagueness has been analyzed in revision-theoretic terms by Conrad Asmus.Asmus (2013) To model a vague predicate on this approach, one specifies pairs of similar objects and which objects are non- borderline cases, and so are unrevisable.
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The dilaton made its first appearance in Kaluza–Klein theory, a five-dimensional theory that combined gravitation and electromagnetism. It appears in string theory. However, it has become central to the lower-dimensional many-bodied gravity problem based on the field theoretic approach of Roman Jackiw. The impetus arose from the fact that complete analytical solutions for the metric of a covariant N-body system have proven elusive in general relativity.
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More generally, for every d and k > d there exists a number m(d,k) such that every set of m(d,k) points in general position has a subset of k points that form the vertices of a neighborly polytope., Ex. 7.3.6, p. 126. This result follows by applying a Ramsey-theoretic argument similar to Szekeres's original proof together with Perles's result on the case k = d + 2.
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Bernard Neumann and Hanna Neumann produced their study of varieties of groups, groups defined by group theoretic equations rather than polynomial ones. Continuous groups also had explosive growth in the 1900-1940 period. Topological groups began to be studied as such. There were many great achievements in continuous groups: Cartan's classification of semisimple Lie algebras, Hermann Weyl's theory of representations of compact groups, Alfréd Haar's work in the locally compact case.
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In graph-theoretic mathematics, a biregular graph. or semiregular bipartite graph. is a bipartite graph G=(U,V,E) for which every two vertices on the same side of the given bipartition have the same degree as each other. If the degree of the vertices in U is x and the degree of the vertices in V is y, then the graph is said to be (x,y)-biregular.
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This is a vector space over K, which has the multiplicity m_i of the root as a dimension. This definition of intersection multiplicity, which is essentially due to Jean-Pierre Serre in his book Local Algebra, works only for the set theoretic components (also called isolated components) of the intersection, not for the embedded components. Theories have been developed for handling the embedded case (see Intersection theory for details).
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In graph theoretic terms, the states are Eulerian orientations of an underlying 4-regular undirected graph. The partition function also counts the number of nowhere-zero 3-flows. For two-dimensional models, the lattice is taken to be the square lattice. For more realistic models, one can use a three-dimensional lattice appropriate to the material being considered; for example, the hexagonal ice lattice is used to analyse ice.
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In other words, every instance of a problem in the complexity class #P can be reduced to an instance of the #SAT problem. This is an important result because many difficult counting problems arise in Enumerative Combinatorics, Statistical physics, Network Reliability, and Artificial intelligence without any known formula. If a problem is shown to be hard, then it provides a complexity theoretic explanation for the lack of nice looking formulas.
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David Preiss FRS (born 1947) is a professor of mathematics at the University of WarwickMathematics Staff and Postgraduates, University of Warwick Institute of Mathematics. Revised 2 March 2012. Retrieved 26 October 2015. and the winner of the 2008 London Mathematical Society Pólya Prize for his 1987 result on Geometry of Measures, where he solved the remaining problem in the geometric theoretic structure of sets and measures in Euclidean space.
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In mathematics, a setoid (X, ~) is a set (or type) X equipped with an equivalence relation ~. A Setoid may also be called E-set, Bishop set, or extensional set.Alexandre Buisse, Peter Dybjer, "The Interpretation of Intuitionistic Type Theory in Locally Cartesian Closed Categories - an Intuitionistic Perspective", Electronic Notes in Theoretical Computer Science 218 (2008) 21–32. Setoids are studied especially in proof theory and in type- theoretic foundations of mathematics.
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The decomposition theorem, a far- reaching extension of the hard Lefschetz theorem decomposition, requires the usage of perverse sheaves. Hodge modules are, roughly speaking, a Hodge- theoretic refinement of perverse sheaves. The geometric Satake equivalence identifies equivariant perverse sheaves on the affine Grassmannian Gr_G with representations of the Langlands dual group of a reductive group G - see . A proof of the Weil conjectures using perverse sheaves is given in .
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Mathematical models can take many forms, including dynamical systems, statistical models, differential equations, or game theoretic models. These and other types of models can overlap, with a given model involving a variety of abstract structures. In general, mathematical models may include logical models. In many cases, the quality of a scientific field depends on how well the mathematical models developed on the theoretical side agree with results of repeatable experiments.
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From a category-theoretic point of view, the fundamental group is a functor :{Pointed algebraic varieties} -> {Profinite groups}. The inverse Galois problem asks what groups can arise as fundamental groups (or Galois groups of field extensions). Anabelian geometry, for example Grothendieck's section conjecture, seeks to identify classes of varieties which are determined by their fundamental groups. studies higher étale homotopy groups by means of the étale homotopy type of a scheme.
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Selecting the minimum length description of the available data as the best model observes the principle identified as Occam's razor. Prior to the advent of computer programming, generating such descriptions was the intellectual labor of scientific theorists. It was far less formal than it has become in the computer age. If two scientists had a theoretic disagreement, they rarely could formally apply Occam's razor to choose between their theories.
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Work has been done on several models of physical systems with similar characteristics, which are described in detail in the main publication on this model. There are ongoing attempts to extend this model in various ways, such as van Enk's model. The toy model has also been analyzed from the viewpoint of categorical quantum mechanics. Currently, there is work being done to reproduce quantum formalism from information-theoretic axioms.
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Canons of epistemic value used to identify degrees of explanatory coherence are themselves justified by appeal to natural teleology. (Judgement and Justification 1988.) In defense of his view, Lycan critically assesses major competitors (especially reliabilism) and other views (e.g., epistemic minimalism). Meaning in natural language (Logical Form in Natural Language 1984), including the meaning of indicative conditionals (Real Conditionals 2001), is explained by Lycan in truth- theoretic terms.
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In the closed loop quantum control, the feedback may be entirely dynamical (that is, the plant and controller form a single dynamical system and the controller with the two influencing each other through direct interaction). This is named Coherent Control. Alternatively, the feedback may be entirely information theoretic insofar as the controller gains information about the plant due to measurement of the plant. This is measurement-based control.
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One examines and criticizes the existing theory. One tries to pin-point the faults in it and then tries to remove them. The difficulty here is to remove the faults without destroying the very great successes of the existing theory." Abdus Salam remarked in 1972, "Field-theoretic infinities first encountered in Lorentz's computation of electron have persisted in classical electrodynamics for seventy and in quantum electrodynamics for some thirty-five years.
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They used an agglomerative algorithm and did not penalize for merging dissimilar nodes. 3\. Fred and Jain: They proposed to use a single linkage algorithm to combine multiple runs of the k-means algorithm. 4\. Dana Cristofor and Dan Simovici: They observed the connection between clustering aggregation and clustering of categorical data. They proposed information theoretic distance measures, and they propose genetic algorithms for finding the best aggregation solution. 5\.
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Carol Saunders Wood (born February 9, 1945, in Pennington Gap, Virginia)Candidate biography, Trustee election, American Mathematical Society, Notices of the AMS 53 (8): 930, September 2006. is a retired American mathematician, the Edward Burr Van Vleck Professor of Mathematics, Emerita, at Wesleyan University.Mathematics and Computer Science faculty listing, Wesleyan, retrieved January 2, 2015. Her research concerns mathematical logic and model-theoretic algebra,Curriculum vitae, retrieved January 2, 2015.
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Bauer earned his Abitur in 1942 and served in the Wehrmacht during World War II, from 1943 to 1945. From 1946 to 1950, he studied mathematics and theoretical physics at Ludwig-Maximilians-Universität in Munich. Bauer received his Doctor of Philosophy (Ph.D.) under the supervision of Fritz Bopp for his thesis Gruppentheoretische Untersuchungen zur Theorie der Spinwellengleichungen ("Group-theoretic investigations of the theory of spin wave equations") in 1952.
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Landau's fourth problem asked whether there are infinitely many primes which are of the form p=n^2+1 for integer n. (The list of known primes of this form is .) The existence of infinitely many such primes would follow as a consequence of other number-theoretic conjectures such as the Bunyakovsky conjecture and Bateman–Horn conjecture. , this problem is open. One example of near-square primes are Fermat primes.
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Structural models are a recent alternative to econometric estimates of the triangle under an estimated money demand curve. Cooley and Hansen (1989) calibrate a cash-in-advance version of a business cycle model. They find that the welfare cost of 10 percent inflation is about 0.4 percent of GNP. Craig and Rocheteau (2008) argue that a search-theoretic framework is necessary for appropriately measuring the welfare cost of inflation.
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If X is an affine algebraic variety, then the set of all regular functions on X forms a ring called the coordinate ring of X. For a projective variety, there is an analogous ring called the homogeneous coordinate ring. Those rings are essentially the same things as varieties: they correspond in essentially a unique way. This may be seen via either Hilbert's Nullstellensatz or scheme-theoretic constructions (i.e., Spec and Proj).
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Author-level metrics are citation metrics that measure the bibliometric impact of individual authors, researchers, academics, and scholars. Many metrics have been developed that take into account varying numbers of factors (from only considering total number of citations, to looking at their distribution across papers or journals using statistical or graph-theoretic principles). The main motivation for these quantitative comparisons between researchers is to allocate resources (e.g. funding, academic appointments).
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Version 1.0, 1.1 and 2.1 have been released so far. Version 1.0 is the first implementation of the CKKS scheme without bootstrapping. In the second version, the bootstrapping algorithm was attached so that users are able to address large-scale homomorphic computations. In Version 2.1, currently the latest version, the multiplication of ring elements in R_q was accelerated by utilizing fast Fourier transform (FFT)-optimized number theoretic transform (NTT) implementation.
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In the mathematical field of set theory, an ideal is a partially ordered collection of sets that are considered to be "small" or "negligible". Every subset of an element of the ideal must also be in the ideal (this codifies the idea that an ideal is a notion of smallness), and the union of any two elements of the ideal must also be in the ideal. More formally, given a set X, an ideal I on X is a nonempty subset of the powerset of X, such that: # \emptyset \in I, # if A \in I and B \subseteq A, then B\in I, and # if A,B \in I, then A\cup B \in I. Some authors add a fourth condition that X itself is not in I; ideals with this extra property are called proper ideals. Ideals in the set-theoretic sense are exactly ideals in the order-theoretic sense, where the relevant order is set inclusion.
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Model-theoretic grammars, also known as constraint-based grammars, contrast with generative grammars in the way they define sets of sentences: they state constraints on syntactic structure rather than providing operations for generating syntactic objects. A generative grammar provides a set of operations such as rewriting, insertion, deletion, movement, or combination, and is interpreted as a definition of the set of all and only the objects that these operations are capable of producing through iterative application. A model-theoretic grammar simply states a set of conditions that an object must meet, and can be regarded as defining the set of all and only the structures of a certain sort that satisfy all of the constraints. The approach applies the mathematical techniques of model theory to the task of syntactic description: a grammar is a theory in the logician's sense (a consistent set of statements) and the well-formed structures are the models that satisfy the theory.
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In the 1970s, concepts from MPT found their way into the field of regional science. In a series of seminal works, Michael Conroy modeled the labor force in the economy using portfolio- theoretic methods to examine growth and variability in the labor force. This was followed by a long literature on the relationship between economic growth and volatility. More recently, modern portfolio theory has been used to model the self-concept in social psychology.
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In 1974 he graduated B.Sc. cum laude in Mathematics in Physics from the Hebrew University. In 1980 he graduated M.Sc. cum laude in Theoretical Physics at Tel Aviv University; his M.Sc. thesis topic was "Spallation nuclear reactions in the galactic cosmic rays". In 1985 he completed Ph.D. studies in the Hebrew University, publishing the thesis "Reduced Dynamical Description: An Information Theoretic Approach". He did his post-doctoral studies at MIT in 1985-1986.
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There have been various attempts to provide decision-theoretic explanations of Ellsberg's observation. Since the probabilistic information available to the decision-maker is incomplete, these attempts sometimes focus on quantifying the non-probabilistic ambiguity which the decision-maker faces – see Knightian uncertainty. That is, these alternative approaches sometimes suppose that the agent formulates a subjective (though not necessarily Bayesian) probability for possible outcomes. One such attempt is based on info-gap decision theory.
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In measure-theoretic probability theory, the density function is defined as the Radon–Nikodym derivative of the probability distribution relative to a common dominating measure. The likelihood function is that density interpreted as a function of the parameter (possibly a vector), rather than the possible outcomes. This provides a likelihood function for any statistical model with all distributions, whether discrete, absolutely continuous, a mixture or something else. (Likelihoods will be comparable, e.g.
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Statistical field theory attempts to extend the field-theoretic paradigm toward many-body systems and statistical mechanics. As above, it can be approached by the usual infinite number of degrees of freedom argument. Much like statistical mechanics has some overlap between quantum and classical mechanics, statistical field theory has links to both quantum and classical field theories, especially the former with which it shares many methods. One important example is mean field theory.
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The following are proposals for demonstrating quantum computational supremacy using current technology, often called NISQ devices. Such proposals include (1) a well-defined computational problem, (2) a quantum algorithm to solve this problem, (3) a comparison best-case classical algorithm to solve the problem, and (4) a complexity-theoretic argument that, under a reasonable assumption, no classical algorithm can perform significantly better than current algorithms (so the quantum algorithm still provides a superpolynomial speedup).
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The th roots of unity form under multiplication a cyclic group of order , and in fact these groups comprise all of the finite subgroups of the multiplicative group of the complex number field. A generator for this cyclic group is a primitive th root of unity. The th roots of unity form an irreducible representation of any cyclic group of order . The orthogonality relationship also follows from group-theoretic principles as described in character group.
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The fundamental objects of interest in gauge theory are connections on vector bundles and principal bundles. In this section we briefly recall these constructions, and refer to the main articles on them for details. The structures described here are standard within the differential geometry literature, and an introduction to the topic from a gauge-theoretic perspective can be found in the book of Donaldson and Peter Kronheimer.Donaldson, S.K., Donaldson, S.K. and Kronheimer, P.B., 1990.
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When reasoning about the meta-theoretic properties of a deductive system in a proof assistant, it is sometimes desirable to limit oneself to first-order representations and to have the ability to name or rename assumptions. The locally nameless approach uses a mixed representation of variables--De Bruijn indices for bound variables and names for free variables--that is able to benefit from the α-canonical form of De Bruijn indexed terms when appropriate.
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The star chromatic number of Dyck graph is 4, while the chromatic number is 2. In graph-theoretic mathematics, a star coloring of a graph G is a (proper) vertex coloring in which every path on four vertices uses at least three distinct colors. Equivalently, in a star coloring, the induced subgraphs formed by the vertices of any two colors has connected components that are star graphs. Star coloring has been introduced by .
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Achieving information theoretic security requires the assumption that there are multiple non-cooperating servers, each having a copy of the database. Without this assumption, any information-theoretically secure PIR protocol requires an amount of communication that is at least the size of the database n. Multi-server PIR protocols tolerant of non-responsive or malicious/colluding servers are called robust or Byzantine robust respectively. These issues were first considered by Beimel and Stahl (2002).
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Universal algebra defines a notion of kernel for homomorphisms between two algebraic structures of the same kind. This concept of kernel measures how far the given homomorphism is from being injective. There is some overlap between this algebraic notion and the categorical notion of kernel since both generalize the situation of groups and modules mentioned above. In general, however, the universal-algebraic notion of kernel is more like the category-theoretic concept of kernel pair.
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The twisted cubic (green) is a set-theoretic complete intersection, but not a complete intersection. By Krull's principal ideal theorem, a foundational result in the dimension theory of rings, the dimension of :R = k[T1, ..., Tr] / (f1, ..., fn) is at least r − n. A ring R is called a complete intersection ring if it can be presented in a way that attains this minimal bound. This notion is also mostly studied for local rings.
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The two main models of bounded complexity, automaton size and recall capacity, continued to pose intriguing open problems in the following decades. A major breakthrough was achieved when Neyman and his Ph.D. student Daijiro Okada proposed a new approach to these problems, based on information theoretic techniques, introducing the notion of strategic entropy.Neyman, A. and Okada, D. (1999). "Strategic entropy and complexity in repeated games." Games and Economic Behavior, 29(1), 191–223.
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The Basic Logic Dialect (BLD) adds features to the Core dialect that are not directly available such as: logic functions, equality in the then-part and named arguments. RIF BLD corresponds to positive datalogs, that is, logic programs without functions or negations. RIF-BLD has a model-theoretic semantics. The frame syntax of RIF BLD is based on F-logic, but RIF BLD doesn't have the non- monotonic reasoning features of F-logic.
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One can then define functions from that type by induction on the way the elements of the type are generated. Induction-recursion generalizes this situation since one can simultaneously define the type and the function, because the rules for generating elements of the type are allowed to refer to the function. Induction-recursion can be used to define large types including various universe constructions. It increases the proof-theoretic strength of type theory substantially.
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There are also models that combine hidden action and hidden information. Since there is no data on unobservable variables, the contract-theoretic moral hazard model is difficult to test directly, but there have been some successful indirect tests with field data. Direct tests of moral hazard theory are feasible in laboratory settings, using the tools of experimental economics. In such a setup, Hoppe and Schmitz (2018) have corroborated central insights of moral hazard theory.
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In very abstract terms, general position is a discussion of generic properties of a configuration space; in this context one means properties that hold on the generic point of a configuration space, or equivalently on a Zariski-open set. This notion coincides with the measure theoretic notion of generic, meaning almost everywhere on the configuration space, or equivalently that points chosen at random will almost surely (with probability 1) be in general position.
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An augmented transition network or ATN is a type of graph theoretic structure used in the operational definition of formal languages, used especially in parsing relatively complex natural languages, and having wide application in artificial intelligence. An ATN can, theoretically, analyze the structure of any sentence, however complicated. ATN are modified transition networks and an extension of RTNs. ATNs build on the idea of using finite state machines (Markov model) to parse sentences.
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Intensional Logics and Logical Structure of Theories. Metsniereba, Tbilisi, 1988, pages 16-48 (Russian). Japaridze proved the arithmetical completeness of this system, as well as its inherent incompleteness with respect to Kripke frames. GLP has been extensively studied by various authors during the subsequent three decades, especially after Lev Beklemishev, in 2004,L. Beklemishev, "Provability algebras and proof-theoretic ordinals, I". Annals of Pure and Applied Logic 128 (2004), pages 103-123.
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Knot theory can be used to determine if a molecule is chiral (has a "handedness") or not. Chemical compounds of different handedness can have drastically differing properties, thalidomide being a notable example of this. More generally, knot theoretic methods have been used in studying topoisomers, topologically different arrangements of the same chemical formula. The closely related theory of tangles have been effectively used in studying the action of certain enzymes on DNA.
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The task of the simulator is to act as a wrapper around the idealised protocol to make it appear like the cryptographic protocol. The simulation succeeds with respect to an information theoretic, respectively computationally bounded adversary if the output of the simulator is statistically close to, respectively computationally indistinguishable from the output of the cryptographic protocol. A two-party computation protocol is secure, if for all adversaries there exists a successful simulator.
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The primary motivation for Lorenzen and Kuno Lorenz was to find a game-theoretic (their term was dialogical, in German ) semantics for intuitionistic logic. Andreas BlassAndreas R. Blass was the first to point out connections between game semantics and linear logic. This line was further developed by Samson Abramsky, Radhakrishnan Jagadeesan, Pasquale Malacaria and independently Martin Hyland and Luke Ong, who placed special emphasis on compositionality, i.e. the definition of strategies inductively on the syntax.
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The trend of her ministry was in the direction of the practical and spiritual, rather than the theoretic. The motive of her ministry was to add something to the helpful forces of the world. The secret of her success was hard work, making no account of difficulties. The methods and means of her progress included the habit of learning from experience and from passing events, taking great lessons for life from humble sources.
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Aside from algebraic spaces, no straightforward generalization is possible for stacks. The complication already appears in the orbifold case (Kawasaki's Riemann–Roch). The equivariant Riemann–Roch theorem for finite groups is equivalent in many situations to the Riemann–Roch theorem for quotient stacks by finite groups. One of the significant applications of the theorem is that it allows one to define a virtual fundamental class in terms of the K-theoretic virtual fundamental class.
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In mathematics, computer science and especially graph theory, a distance matrix is a square matrix (two-dimensional array) containing the distances, taken pairwise, between the elements of a set. Depending upon the application involved, the distance being used to define this matrix may or may not be a metric. If there are elements, this matrix will have size . In graph-theoretic applications the elements are more often referred to as points, nodes or vertices.
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Set-Theoretic Foundations. A knob on a radio does not take on an uncountably infinite number of possible values -- it takes a finite number of possible values fully limited by the mechanical, physical, nature of the knob itself. There exists no one-to-one mapping between the continuous mathematics used for engineering applications and the physical product(s) produced by the engineering. Indeed, this is one of the core open problems within Philosophy of Mathematics.
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A common theme of this work is the adoption of a sign-theoretic perspective on issues of artificial intelligence and knowledge representation. Many of its applications lie in the field of human-computer interaction (HCI) and fundamental devices of recognition. One part of this field, known as algebraic semiotics, combines aspects of algebraic specification and social semiotics, and has been applied to user interface design and to the representation of mathematical proofs.
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When Topology Meets Chemistry: A Topological Look At Molecular Chirality is a book in chemical graph theory on the graph-theoretic analysis of chirality in molecular structures. It was written by Erica Flapan, based on a series of lectures she gave in 1996 at the Institut Henri Poincaré, and was published in 2000 by the Cambridge University Press and Mathematical Association of America as the first volume in their shared Outlooks book series.
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In 1923, Wintringham joined the recently formed Communist Party of Great Britain. In 1925, he was one of the twelve CPGB officials imprisoned for seditious libel and incitement to mutiny. In 1930, he helped to found the Communist newspaper, the Daily Worker, and was one of the few named writers to publish articles in it. In writing for the Communist party's theoretic journal Labour Monthly, he established himself as the party's military expert.
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By erasing and adding ink in the proper places, the writer can convey just as much information as if the paper were clean, even though the reader does not know where the dirt was. In this analogy, the paper is the channel, the dirt is interference, the writer is the transmitter, and the reader is the receiver. Note that DPC at the encoder is an information- theoretic dual of Wyner-Ziv coding at the decoder.
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The proteomic networks contain many biomarkers that are proxies for development and illustrate the potential clinical application of this technology as a way to monitor normal and abnormal fetal development. An information theoretic framework has also been introduced for biomarker discovery, integrating biofluid and tissue information. This new approach takes advantage of functional synergy between certain biofluids and tissues with the potential for clinically significant findings not possible if tissues and biofluids were considered individually.
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A recurring problem with attempts to ground mathematics in mereology is how to build up the theory of relations while abstaining from set-theoretic definitions of the ordered pair. Martin argued that Eberle's (1970) theory of relational individuals solved this problem. Topological notions of boundaries and connection can be married to mereology, resulting in mereotopology; see Casati and Varzi (1999: chpts. 4,5). Whitehead's 1929 Process and Reality contains a good deal of informal mereotopology.
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While traditionally load balancing strategies have been designed to change consumers' consumption patterns to make demand more uniform, developments in energy storage and individual renewable energy generation have provided opportunities to devise balanced power grids without affecting consumers' behavior. Typically, storing energy during off-peak times eases high demand supply during peak hours. Dynamic game-theoretic frameworks have proved particularly efficient at storage scheduling by optimizing energy cost using their Nash equilibrium.
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In collaboration with the mathematician Jürgen Jost, he proposed an analysis of gene expressions based on information theoryScherrer, Klaus & Jürgen Jost (2007), " The gene and the genon concept : A functional and information-theoretic analysis", Mol Syst Biol 3, 87-98. Based on biochemical and molecular-biological methods, pushed to the their technological limits, Scherrer's early results were obtained before the advent of modern DNA technology and were thus met, at the time, with apprehension.
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Hardy & Wright include in their definition the requirement that an arithmetical function "expresses some arithmetical property of n".Hardy & Wright, intro. to Ch. XVI An example of an arithmetic function is the divisor function whose value at a positive integer n is equal to the number of divisors of n. There is a larger class of number-theoretic functions that do not fit the above definition, for example, the prime- counting functions.
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For example, integers can be represented in binary notation, and graphs can be encoded directly via their adjacency matrices, or by encoding their adjacency lists in binary. Even though some proofs of complexity-theoretic theorems regularly assume some concrete choice of input encoding, one tries to keep the discussion abstract enough to be independent of the choice of encoding. This can be achieved by ensuring that different representations can be transformed into each other efficiently.
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Every order theoretic definition has its dual: it is the notion one obtains by applying the definition to the inverse order. Since all concepts are symmetric, this operation preserves the theorems of partial orders. For a given mathematical result, one can just invert the order and replace all definitions by their duals and one obtains another valid theorem. This is important and useful, since one obtains two theorems for the price of one.
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As already mentioned, the methods and formalisms of universal algebra are an important tool for many order theoretic considerations. Beside formalizing orders in terms of algebraic structures that satisfy certain identities, one can also establish other connections to algebra. An example is given by the correspondence between Boolean algebras and Boolean rings. Other issues are concerned with the existence of free constructions, such as free lattices based on a given set of generators.
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He described his views as contrary to those held by contemporary authorities in Leeds, Newcastle, Edinburgh, and Cambridge. These administrations, he argued, promoted only scientific and theoretic instruction at the expense of practical work. Webb described his prime objective as reinforcing theoretical knowledge with the practical experience gained from daily instruction on farms, as embodied in the college motto "Scientia et Labore". He also stressed the importance of educating people, irrespective of their age.
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It is named after the ancient Greek mathematician Euclid, who first described it in his Elements (c. 300 BC). It is an example of an algorithm, a step-by-step procedure for performing a calculation according to well-defined rules, and is one of the oldest algorithms in common use. It can be used to reduce fractions to their simplest form, and is a part of many other number-theoretic and cryptographic calculations.
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Problems that require some privacy in the data (typically cryptographic problems) can use randomization to ensure that privacy. In fact, the only provably secure cryptographic system (the one-time pad) has its security relying totally on the randomness of the key data supplied to the system. The field of cryptography utilizes the fact that certain number-theoretic functions are randomly self-reducible. This includes probabilistic encryption and cryptographically strong pseudorandom number generation.
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A study led by the European Food Safety Authority (EFSA) also found viral genes in transgenic plants. Transgenic carrots have been used to produce the drug Taliglucerase alfa which is used to treat Gaucher's disease. In the laboratory, transgenic plants have been modified to increase photosynthesis (currently about 2% at most plants versus the theoretic potential of 9–10%).NWT magazine, April 2011 This is possible by changing the rubisco enzyme (i.e.
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MDL applies in machine learning when algorithms (machines) generate descriptions. Learning occurs when an algorithm generates a shorter description of the same data set. The theoretic minimum description length of a data set, called its Kolmogorov complexity, cannot, however, be computed. That is to say, even if by random chance an algorithm generates the shortest program of all that outputs the data set, an automated theorem prover cannot prove there is no shorter such program.
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The extra effort in gift- exchange games is modeled to be a negative payoff if not compensated by salary. The IKEA effect of own extra work is not considered in the payoff structure of this game. Therefore, this model rather fits labor conditions, which are less meaningful for the employees. Like in trust games, game- theoretic solution for rational players predicts that employees’ effort will be minimum for one-shot and finitely repeated interactions.
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In anonymity networks (e.g., Tor, Crowds, Mixmaster, I2P, etc.), it is important to be able to measure quantitatively the guarantee that is given to the system. The degree of anonymity d is a device that was proposed at the 2002 Privacy Enhancing Technology (PET) conference. There were two papers that put forth the idea of using entropy as the basis for formally measuring anonymity: "Towards an Information Theoretic Metric for Anonymity", and "Towards Measuring Anonymity".
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Calvin attended the University of Maryland from 1999 to 2003 where she earned bachelor's degrees in computer science and mathematics. She then attended Stanford University where she earned her master's degree and PhD in management science and engineering. While earning her PhD, Calvin worked at the US Energy Information Administration for two years as an international energy analyst. She completed her thesis, titled "Participation in international environmental agreements : a game-theoretic study" in 2008.
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Nikolai Nikolaevich Luzin (also spelled Lusin; ; 9 December 1883 – 28 January 1950) was a Soviet/Russian mathematician known for his work in descriptive set theory and aspects of mathematical analysis with strong connections to point- set topology. He was the eponym of Luzitania, a loose group of young Moscow mathematicians of the first half of the 1920s. They adopted his set-theoretic orientation, and went on to apply it in other areas of mathematics.
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50 As the group's main theoretic writer during its existence, Jeantet sought to steer the group towards a socialist economic position, arguing in 1942 in favour of a "national and socialist revolution" similar to that associated with Strasserism. This was despite the fact that Jeantet was fully aware of La Cagoule being funded by wealthy industrialists such as Jacques Lemaigre-Dubreuil and Louis Renault, all of whom despised the concept of socialism.
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If samples from a joint distribution are available, a Bayesian approach can be used to estimate the mutual information of that distribution. The first work to do this, which also showed how to do Bayesian estimation of many other information-theoretic properties besides mutual information, was . Subsequent researchers have rederived and extended this analysis. See for a recent paper based on a prior specifically tailored to estimation of mutual information per se.
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A complication is that this multivariate mutual information (as well as the interaction information) can be positive, negative, or zero, which makes this quantity difficult to interpret intuitively. In fact, for n random variables, there are 2^n-1 degrees of freedom for how they might be correlated in an information- theoretic sense, corresponding to each non-empty subset of these variables. These degrees of freedom are bounded by the various inequalities in information theory.
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Richard is a distinguished university professor of Economics at the University of Pittsburgh. He also holds an appointment in the department of statistics. Prior to coming to Pittsburgh in 1991, he held teaching and research positions at the University of Louvain, University of Chicago, London School of Economics, University of London, and Duke University. His research interests include econometric modeling, time series, Bayesian methods, and empirical game theoretic models of auctions and collusion.
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This is the fundamental theorem of drama theory. Until a resolution meeting these conditions is arrived at, the characters are under emotional pressure to rationalize re-definitions of the game that they will play. Re-definitions inspired by new dilemmas then follow each other until eventually, with or without a resolution, characters become players in the game they have defined for themselves. In game-theoretic terms, this is a game with a focal point – i.e.
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At that time he was already a well-known authoritative theoretic of Slavophilia. Together with another notable slavophile, Ivan Aksakov, he participated in the preparation of the project of emperor's manifest that should have reanimated Zemsky Sobor. This project made Konstantin Pobedonostsev furious, which immediately forced the resignations of both Golohvastov and the minister. Since a young age, Golohvastov studied the history of the Russian people, especially the period of the Time of Troubles.
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From a module-theoretic point of view this was integrated into the Cartan–Eilenberg theory of homological algebra in the early 1950s. The application in algebraic number theory to class field theory provided theorems valid for general Galois extensions (not just abelian extensions). The cohomological part of class field theory was axiomatized as the theory of class formations. In turn, this led to the notion of Galois cohomology and étale cohomology (which builds on it) .
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Social therapy is an activity-theoretic practice developed outside of academia at the East Side Institute for Group and Short Term Psychotherapy in New York. Its primary methodologists are cofounders of the East Side Institute, Fred Newman and Lois Holzman. In evolution since the late 1970s, the social therapeutic approach to human development and learning is informed by a variety of intellectual traditions especially the works of Karl Marx, Lev Vygotsky and Ludwig Wittgenstein.
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In mathematics, a rectifiable set is a set that is smooth in a certain measure-theoretic sense. It is an extension of the idea of a rectifiable curve to higher dimensions; loosely speaking, a rectifiable set is a rigorous formulation of a piece-wise smooth set. As such, it has many of the desirable properties of smooth manifolds, including tangent spaces that are defined almost everywhere. Rectifiable sets are the underlying object of study in geometric measure theory.
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In proof theory, a branch of mathematical logic, elementary function arithmetic (EFA), also called elementary arithmetic and exponential function arithmetic, is the system of arithmetic with the usual elementary properties of 0, 1, +, ×, xy, together with induction for formulas with bounded quantifiers. EFA is a very weak logical system, whose proof theoretic ordinal is ω3, but still seems able to prove much of ordinary mathematics that can be stated in the language of first-order arithmetic.
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An intelligent agent is intrinsically motivated to act if the information content alone, of the experience resulting from the action, is the motivating factor. Information content in this context is measured in the information- theoretic sense of quantifying uncertainty. A typical intrinsic motivation is to search for unusual, surprising situations (exploration), in contrast to a typical extrinsic motivation such as the search for food (homeostasis). Extrinsic motivations are typically described in artificial intelligence as task-dependent or goal-directed.
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Ouedeyer and Kaplan have made a substantial contribution to the study of intrinsic motivation. They define intrinsic motivation based on Berlyne's theory, and divide approaches to the implementation of intrinsic motivation into three categories that broadly follow the roots in psychology: "knowledge- based models", "competence-based models" and "morphological models". Knowledge-based models are further subdivided into "information-theoretic" and "predictive". Baldassare and Mirolli present a similar typology, differentiating knowledge-based models between prediction-based and novelty- based.
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Any field has an algebraic closure, which is moreover unique up to (non-unique) isomorphism. It is commonly referred to as the algebraic closure and denoted . For example, the algebraic closure of is called the field of algebraic numbers. The field is usually rather implicit since its construction requires the ultrafilter lemma, a set-theoretic axiom that is weaker than the axiom of choice.. Mathoverflow post In this regard, the algebraic closure of , is exceptionally simple.
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William Alvin Howard (born 1926) is a proof theorist best known for his work demonstrating formal similarity between intuitionistic logic and the simply typed lambda calculus that has come to be known as the Curry–Howard correspondence. He has also been active in the theory of proof-theoretic ordinals. He earned his Ph.D. at the University of Chicago in 1956 for a dissertation entitled "k-fold recursion and well-ordering". He was a student of Saunders Mac Lane.
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In addition to the theme, the New Critics also looked for paradox, ambiguity, irony, and tension to help establish the single best and most unified interpretation of the text. Although the New Criticism is no longer a dominant theoretical model in American universities, some of its methods (like close reading) are still fundamental tools of literary criticism, underpinning a number of subsequent theoretic approaches to literature including poststructuralism, deconstruction theory, New Testament narrative criticism, and reader-response theory.
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G.J. Chaitin, Register Allocation and Spilling via Graph Coloring, US Patent 4,571,678 (1986) [cited from Register Allocation on the Intel® Itanium® Architecture, p.155] He was formerly a researcher at IBM's Thomas J. Watson Research Center in New York and remains an emeritus researcher. He has written more than 10 books that have been translated to about 15 languages. He is today interested in questions of metabiology and information-theoretic formalizations of the theory of evolution.
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The first example of such a use comes from the work of the physicist Gustav Kirchhoff, who published in 1845 his Kirchhoff's circuit laws for calculating the voltage and current in electric circuits. The introduction of probabilistic methods in graph theory, especially in the study of Erdős and Rényi of the asymptotic probability of graph connectivity, gave rise to yet another branch, known as random graph theory, which has been a fruitful source of graph-theoretic results.
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In 1918 Dvorniković arrives in Zagreb where he works in a Musical school. During 1919 he begins lecturing at the University of Zagreb with the theme "Philosophy and Science." In 1925 he becomes a regular professor on the board of directors for the "theoretic and practical philosophy and for the history of philosophy." He is the fourth professor of philosophy in Croatia in half a century; his predecessors being Franjo Marković, Đuro Arnold and Albert Bazala.
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Callas received her musical education in Athens. Initially, her mother tried to enroll her at the prestigious Athens Conservatoire, without success. At the audition, her voice, still untrained, failed to impress, while the conservatoire's director refused to accept her without her satisfying the theoretic prerequisites (solfege). In the summer of 1937, her mother visited Maria Trivella at the younger Greek National Conservatoire, asking her to take Mary, as she was then called, as a student for a modest fee.
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For this reason, the theory presents itself as a "clinical anthropology". The theoretic model developed by Gagnepain and his research group at Rennes has inspired the work of professors and researchers in a number of European countries and in the United States in a wide variety of disciplinary fields, among them linguistics, literature, psychology, art history, archeology, psychoanalysis, theology. Its aim is deliberately trans- disciplinary - or, as Gagnepain humorously puts it, the theory of mediation cultivates "in-discipline".
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A few were written by Willard Quine and Burton Dreben. The Source Book did much to advance the view that modern logic begins with, and builds on, the Begriffsschrift. Grattan-Guinness (2000) argues that this perspective on the history of logic is mistaken, because Frege employed an idiosyncratic notation and was far less read than, say, Peano. Ironically, van Heijenoort (1967a) is often cited by those who prefer the alternative model theoretic stance on logic and mathematics.
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Ivan Turgenev, who first popularized the term nihilism in his 1862 novel Fathers and Sons (portrait by Ilya Repin) Nihilism came into conflict with Orthodox religious authorities, as well as with the Tsarist autocracy. Young radicals began calling themselves nihilists in university protests, innocuous youthful rebellions, and ever-escalating revolutionary activities, which included widespread arson. The theoretic side of nihilism was somewhat distinct from this violent expression however. Nevertheless, nihilism was widely castigated by conservative publicists and government authorities.
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Robert Michael "Mike" Canjar"Dissertation: Model-Theoretic Properties of Countable Ultraproducts without the Continuum Hypothesis" by Robert Michael Kanjar, Ph.D., University of Michigan, 1982 (September 9, 1953 - May 7, 2012) was a Professor in the Department of Mathematics and Computer Science at University of Detroit Mercy (UDM). He started there in 1995, and served as department Chairman from 1995–2002. He was promoted to Full Professor in 2001. He previously taught at several universities, including the University of Baltimore.
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Many operations such as set-theoretic, blending, offsetting, projection, non-linear deformations, metamorphosis, sweeping, hypertexturing, and others, have been formulated for this representation in such a manner that they yield continuous real-valued functions as output, thus guaranteeing the closure property of the representation. R-functions originally introduced in V.L. Rvachev's "On the analytical description of some geometric objects",V.L. Rvachev, "On the analytical description of some geometric objects", Reports of Ukrainian Academy of Sciences, vol. 153, no.
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Meakins received her Bachelor of Arts (Honours) and Master of Arts at the University of Queensland. She completed her master's thesis, Lashings of Tongue: A Relevance Theoretic Account of Impoliteness, in 2001. Meakins earned her Ph.D. from the University of Melbourne in 2008 for her work with the Aboriginal Child Language Project. Rachel Nordlinger was main supervisor for Meakins' dissertation, Case-marking in contact: the development and function of case morphology in Gurindji Kriol, an Australian mixed language.
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In model theory, interpretation of a structure M in another structure N (typically of a different signature) is a technical notion that approximates the idea of representing M inside N. For example every reduct or definitional expansion of a structure N has an interpretation in N. Many model-theoretic properties are preserved under interpretability. For example if the theory of N is stable and M is interpretable in N, then the theory of M is also stable.
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The MIP SCOC architecture includes powerful ALFU (Algorithm Level Functional units) like chain matrix adders, multipliers, sorters, multiple operand adders and graph theoretic units like Depth-First-Search, Breadth-First-Search. This introduces a higher level of abstraction through the algorithm-level instructions (ALISA). A single ALISA is equivalent to multiple parallel VLIW. The MIP SCOC architecture includes an on-chip compiler (Compiler-On-Silicon) to generate the required instructions to feed the ALFUs of the MIP node.
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A hypergame has the same rules as a super game except that I may name any somewhat finite game on the first move. The hypergame is closely related to the "hypergame paradox" a self-referential, set-theoretic paradox like Russell's paradox and Cantor's paradox. The hypergame paradox arises from trying to answer the question "Is a hypergame somewhat finite?" The paradox, as Zwicker note, satisfies conditions 1- 4 making it somewhat finite in the same way a supergame was.
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It was published by Erdős and Rado in 1956. Rado's theorem is another Ramsey-theoretic result concerning systems of linear equations, proved by Rado in his thesis. The Milner–Rado paradox, also in set theory, states the existence of a partition of an ordinal into subsets of small order-type; it was published by Rado and E. C. Milner in 1965. The Erdős–Ko–Rado theorem can be described either in terms of set systems or hypergraphs.
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Over the decades, experimental condensed matter physicists have discovered a number of exotic states of matter, including superconductors and superfluids. These states are described using the formalism of quantum field theory, but some phenomena are difficult to explain using standard field theoretic techniques. Some condensed matter theorists including Subir Sachdev hope that the AdS/CFT correspondence will make it possible to describe these systems in the language of string theory and learn more about their behavior.Merali 2011, p.
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The field focuses on modeling phenomena in cognitive science that have resisted traditional techniques or where traditional models seem to have reached a barrier (e.g., human memory), and modeling preferences in decision theory that seem paradoxical from a traditional rational point of view (e.g., preference reversals). Since the use of a quantum-theoretic framework is for modeling purposes, the identification of quantum structures in cognitive phenomena does not presuppose the existence of microscopic quantum processes in the human brain.
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Maxentius was jealous of Constantine's power, and on October 28, 306, he persuaded a cohort of imperial guardsmen to declare him Augustus. Uncomfortable with sole leadership, Maxentius sent a set of imperial robes to Maximian and saluted him as "Augustus for the second time", offering him theoretic equal rule but less actual power and a lower rank.Barnes, Constantine and Eusebius, 30–32. Galerius refused to recognize Maxentius and sent Severus with an army to Rome to depose him.
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The ring-theoretic approach can be further generalized to the semidirect sum of Lie algebras. For geometry, there is also a crossed product for group actions on a topological space; unfortunately, it is in general non-commutative even if the group is abelian. In this context, the semidirect product is the space of orbits of the group action. The latter approach has been championed by Alain Connes as a substitute for approaches by conventional topological techniques; c.f.
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Drew Fudenberg (born March 2, 1957 in New York City) is the Paul A. Samuelson Professor of Economics at MIT. His extensive research spans many aspects of game theory, including equilibrium theory, learning in games, evolutionary game theory, and many applications to other fields. Fudenberg was also one of the first to apply game theoretic analysis in industrial organization, bargaining theory, and contract theory. He has also authored papers on repeated games, reputation effects, and behavioral economics.
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These concepts can even assist with in number- theoretic questions solely concerned with integers. For example, prime ideals in the ring of integers of quadratic number fields can be used in proving quadratic reciprocity, a statement that concerns the existence of square roots modulo integer prime numbers. Early attempts to prove Fermat's Last Theorem led to Kummer's introduction of regular primes, integer prime numbers connected with the failure of unique factorization in the cyclotomic integers., Section I.7, p.
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The graph coloring game is a mathematical game related to graph theory. Coloring game problems arose as game-theoretic versions of well-known graph coloring problems. In a coloring game, two players use a given set of colors to construct a coloring of a graph, following specific rules depending on the game we consider. One player tries to successfully complete the coloring of the graph, when the other one tries to prevent him from achieving it.
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The hyperreals can be developed either axiomatically or by more constructively oriented methods. The essence of the axiomatic approach is to assert (1) the existence of at least one infinitesimal number, and (2) the validity of the transfer principle. In the following subsection we give a detailed outline of a more constructive approach. This method allows one to construct the hyperreals if given a set- theoretic object called an ultrafilter, but the ultrafilter itself cannot be explicitly constructed.
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The development of these algorithms led to the method of iterative compression, a more general tool for many other parameterized algorithms. The parameterized algorithms known for these problems take nearly-linear time for any fixed value of k. Alternatively, with polynomial dependence on the graph size, the dependence on k can be made as small as 2.3146^k. In contrast, the analogous problem for directed graphs does not admit a fixed-parameter tractable algorithm under standard complexity- theoretic assumptions.
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The development of the fundamentals of model theory (such as the compactness theorem) rely on the axiom of choice, or more exactly the Boolean prime ideal theorem. Other results in model theory depend on set-theoretic axioms beyond the standard ZFC framework. For example, if the Continuum Hypothesis holds then every countable model has an ultrapower which is saturated (in its own cardinality). Similarly, if the Generalized Continuum Hypothesis holds then every model has a saturated elementary extension.
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A wide class of cases has been proved by Benson Farb and Mark Kisin; these equations are on a locally symmetric variety X subject to some group-theoretic conditions. This work is based on the previous results of Katz for Picard–Fuchs equations (in the contemporary sense of the Gauss–Manin connection), as amplified in the Tannakian direction by André. It also applies a version of superrigidity particular to arithmetic groups. Other progress has been by arithmetic methods.
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Phillips did not himself state there was any relationship between employment and inflation; this notion was a trivial deduction from his statistical findings. Samuelson and Solow made the connection explicit and subsequently Milton Friedman and Edmund Phelps put the theoretical structure in place. In so doing, Friedman was to successfully predict the imminent collapse of Phillips' a-theoretic correlation. While there is a short run tradeoff between unemployment and inflation, it has not been observed in the long run.
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In the game of Go, by contrast, a ply is the normal unit of counting moves; so for example to say that a game is 250 moves long is to imply 250 plies. The word "ply" used as a synonym for "layer" goes back to the 15th century.Online Etymology Dictionary, "ply" (cited 24 April 2011) Arthur Samuel first used the term in its game-theoretic sense in his seminal paper on machine learning in checkers in 1959,A.
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Slicing the Truth: On the Computability Theoretic and Reverse Mathematical Analysis of Combinatorial Principles is a book on reverse mathematics in combinatorics, the study of the axioms needed to prove combinatorial theorems. It was written by Denis R. Hirschfeldt, based on a course given by Hirschfeldt at the National University of Singapore in 2010, and published in 2014 by World Scientific, as volume 28 of the Lecture Notes Series of the Institute for Mathematical Sciences, National University of Singapore.
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The hyperreals can be developed either axiomatically or by more constructively oriented methods. The essence of the axiomatic approach is to assert (1) the existence of at least one infinitesimal number, and (2) the validity of the transfer principle. In the following subsection we give a detailed outline of a more constructive approach. This method allows one to construct the hyperreals if given a set-theoretic object called an ultrafilter, but the ultrafilter itself cannot be explicitly constructed.
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In more formal graph-theoretic terms, the problem asks whether the complete bipartite graph K3,3 is planar.. Bóna introduces the puzzle (in the form of three houses to be connected to three wells) on p. 275, and writes on p. 277 that it "is equivalent to the problem of drawing K3,3 on a plane surface without crossings". This graph is often referred to as the utility graph in reference to the problem;Utility Graph from mathworld.wolfram.
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In mathematics, an iterated binary operation is an extension of a binary operation on a set S to a function on finite sequences of elements of S through repeated application. Common examples include the extension of the addition operation to the summation operation, and the extension of the multiplication operation to the product operation. Other operations, e.g., the set theoretic operations union and intersection, are also often iterated, but the iterations are not given separate names.
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Freund was one of the originators of two-component duality which gave the original impetus to what then developed into string theory. He pioneered the modern unification of physics through the introduction of extra dimensions of space and found mechanisms by which the extra dimensions curl up. Freund made significant contributions, to the theory of magnetic monopoles, to supersymmetry and supergravity, to number-theoretic aspects of string theory, as well as to the phenomenology of hadrons.
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Later, the Neoplatonist Iamblichus changed the role of the "One", effectively altering the role of the Demiurge as second cause or dyad, which was one of the reasons that Iamblichus and his teacher Porphyry came into conflict. The figure of the Demiurge emerges in the theoretic of Iamblichus, which conjoins the transcendent, incommunicable “One,” or Source. Here, at the summit of this system, the Source and Demiurge (material realm) coexist via the process of henosis.See Theurgy, Iamblichus and henosis .
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In his review, Simoncelli notes "cortical neurons tend to have lower firing rates and may use a different form of code as compared to retinal neurons". Cortical Neurons may also have the ability to encode information over longer periods of time than their retinal counterparts. Experiments done in the auditory system have confirmed that redundancy is decreased. Difficult to test: Estimation of information- theoretic quantities requires enormous amounts of data, and is thus impractical for experimental verification.
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Dependence logic is a logic of imperfect information, like branching quantifier logic or independence-friendly logic: in other words, its game theoretic semantics can be obtained from that of first-order logic by restricting the availability of information to the players, thus allowing for non-linearly ordered patterns of dependence and independence between variables. However, dependence logic differs from these logics in that it separates the notions of dependence and independence from the notion of quantification.
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In graph theory, two graphs G and G' are homeomorphic if there is a graph isomorphism from some subdivision of G to some subdivision of G'. If the edges of a graph are thought of as lines drawn from one vertex to another (as they are usually depicted in illustrations), then two graphs are homeomorphic to each other in the graph-theoretic sense precisely if they are homeomorphic in the sense in which the term is used in topology.
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The term "algebraic combinatorics" was introduced in the late 1970s.Algebraic Combinatorics by Eiichi Bannai Through the early or mid-1990s, typical combinatorial objects of interest in algebraic combinatorics either admitted a lot of symmetries (association schemes, strongly regular graphs, posets with a group action) or possessed a rich algebraic structure, frequently of representation theoretic origin (symmetric functions, Young tableaux). This period is reflected in the area 05E, Algebraic combinatorics, of the AMS Mathematics Subject Classification, introduced in 1991.
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An order-theoretic analog to the intersection graphs are the inclusion orders. In the same way that an intersection representation of a graph labels every vertex with a set so that vertices are adjacent if and only if their sets have nonempty intersection, so an inclusion representation f of a poset labels every element with a set so that for any x and y in the poset, x ≤ y if and only if f(x) ⊆ f(y).
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Anton Buslov was born and raised in Voronezh. In February 2003 he successfully passed entry exams to the MEPhI Graduate School of Physics. In February 2006 he obtained Specialist degree with the Department of Experimental Nuclear Physics and Theoretical Physics and Cosmophysics. In May 2006 he entered the Post-Graduate School under the Department of Experimental and Theoretic Physics, and started working on his thesis, "Control System and Data Computing in Solar Research Project 'Koronas-Foton'".
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A traditional axiomatic foundation of mathematics is set theory, in which all mathematical objects are ultimately represented by sets (including functions, which map between sets). More recent work in category theory allows this foundation to be generalized using topoi; each topos completely defines its own mathematical framework. The category of sets forms a familiar topos, and working within this topos is equivalent to using traditional set-theoretic mathematics. But one could instead choose to work with many alternative topoi.
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In the United States, a master program in statistics requires courses in probability, mathematical statistics, and applied statistics (e.g., design of experiments, survey sampling, etc.). For a doctoral degree in statistics, it has been traditional that students complete a course in measure-theoretic probability as well as courses in mathematical statistics. Such courses require a good course in real analysis, covering the proofs of the theory of calculus and topics like the uniform convergence of functions.
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The reason for this discrepancy is that the scheme-theoretic definitions only keep track of the polynomial set up to change of basis. In this example, one way to avoid these problems is to use the Q-variety Spec(Q[x1,x2,x3]/(x12+ x22+ 2x32- 2x1x3 - 2x2x3)), whose associated Q[i]-algebraic set is the union of the Q[i]-variety Spec(Q[i][x1,x2,x3]/(x1 + ix2 - (1+i)x3)) and its complex conjugate.
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Such functions are called bijections. The inverse of an injection that is not a bijection (that is, not a surjection), is only a partial function on , which means that for some , is undefined. If a function is invertible, then both it and its inverse function are bijections. Another convention is used in the definition of functions, referred to as the "set- theoretic" or "graph" definition using ordered pairs, which makes the codomain and image of the function the same.
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At its core, mathematical logic deals with mathematical concepts expressed using formal logical systems. These systems, though they differ in many details, share the common property of considering only expressions in a fixed formal language. The systems of propositional logic and first-order logic are the most widely studied today, because of their applicability to foundations of mathematics and because of their desirable proof-theoretic properties.Ferreirós (2001) surveys the rise of first-order logic over other formal logics in the early 20th century.
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Historically the most popular construction of a gerbe is a category-theoretic model featured in Giraud's theory of gerbes, which are roughly sheaves of groupoids over M. In 1994 Murray introduced bundle gerbes, which are geometric realizations of 1-gerbes. For many purposes these are more suitable for calculations than Giraud's realization, because their construction is entirely within the framework of classical geometry. In fact, as their name suggests, they are fiber bundles. This notion was extended to higher gerbes the following year.
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Goguen's research with Thatcher, Wagner and Wright (also in the 1970s) was one of the earliest works to formalise the algebraic basis for data abstraction.V. S. Alagar, "Specification of Software Systems", Springer, pp. 216 (1999). . In the early 1990s Goguen and Rod Burstall developed the theory of institutions, a category-theoretic description of logical systems in computer science.J. A. Goguen and R. M. Burstall, "Institutions: Abstract Model Theory for Specification and Programming", Journal of the ACM 39: 95–146 (1992).
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We can prove by several means either proof theoretic (algebraic steps); or semantic (truth table, method of analytic tableaux, Venn diagram, Veitch diagram / Karnaugh map) known in propositional calculus that :\left(A \land \left( A \rightarrow B\right)\right) \rightarrow B holds. Since i-j \mid m, by the transitivity property of the divisibility relation, p \mid i-j \rightarrow p \mid m. Thus (as equality axioms postulate identity to be a congruence relation ) :p \mid m can be proven.
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It is called a strongly minimal set if this is true even in all elementary extensions. A strongly minimal set, equipped with the closure operator given by algebraic closure in the model-theoretic sense, is an infinite matroid, or pregeometry. A model of a strongly minimal theory is determined up to isomorphism by its dimension as a matroid. Totally categorical theories are controlled by a strongly minimal set; this fact explains (and is used in the proof of) Morley's theorem.
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The paper develops a game-theoretic model that addresses the effects of a luxury tax on competitive balance, team profits, and social welfare. This model has half the teams above a certain tax threshold, and the other half below. The teams above would pay taxable balance from their excess amount, and it would be redistributed to the teams below. This research proved that the small teams could have a larger salary than before, and the larger teams would not be affected as much.
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The Stefan Bergman Prize in mathematics was initiated by Bergman's wife in her will, in memory of her husband's work. The American Mathematical Society supports the prize and selects the committee of judges.Other Prizes and Awards Supported by the AMS The prize is awarded for: # the theory of the kernel function and its applications in real and complex analysis; or # function-theoretic methods in the theory of partial differential equations of elliptic type with a special attention to Bergman's and related operator methods.
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The biggest technical change after 1950 has been the development of sieve methods, particularly in multiplicative problems. These are combinatorial in nature, and quite varied. The extremal branch of combinatorial theory has in return been greatly influenced by the value placed in analytic number theory on quantitative upper and lower bounds. Another recent development is probabilistic number theory, which uses methods from probability theory to estimate the distribution of number theoretic functions, such as how many prime divisors a number has.
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Galois' theory was notoriously difficult for his contemporaries to understand, especially to the level where they could expand on it. For example, in his 1846 commentary, Liouville completely missed the group-theoretic core of Galois' method. Joseph Alfred Serret who attended some of Liouville's talks, included Galois' theory in his 1866 (third edition) of his textbook Cours d'algèbre supérieure. Serret's pupil, Camille Jordan, had an even better understanding reflected in his 1870 book Traité des substitutions et des équations algébriques.
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Sentence structure is conceived of as existing in two dimensions. Combinations organized along the horizontal dimension (in terms of precedence) are called strings, whereas combinations organized along the vertical dimension (in terms of dominance) are catenae. In terms of a cartesian coordinate system, strings exist along the x-axis, and catenae along the y-axis. :Catena (informal graph- theoretic definition) :Any single word or any combination of words that are continuous in the vertical dimension, that is, with respect to dominance (y-axis).
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Emil Artin established the Artin reciprocity law in a series of papers (1924; 1927; 1930). This law is a general theorem in number theory that forms a central part of global class field theory. The term "reciprocity law" refers to a long line of more concrete number theoretic statements which it generalized, from the quadratic reciprocity law and the reciprocity laws of Eisenstein and Kummer to Hilbert's product formula for the norm symbol. Artin's result provided a partial solution to Hilbert's ninth problem.
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To make the methodology amenable for computation, Baeurle proposed to shift the contour of integration of the partition function integral through the homogeneous MF solution using Cauchy's integral theorem, providing its so- called mean-field representation. This strategy was previously successfully employed by Baer et al. in field-theoretic electronic structure calculations (Baer 1998). Baeurle could demonstrate that this technique provides a significant acceleration of the statistical convergence of the ensemble averages in the MC sampling procedure (Baeurle 2002, Baeurle 2002a).
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Boyd earned his Ph.D. from MIT in 1970. Boyd's doctoral thesis, directed by Richard Cartwright, was titled A Recursion-Theoretic Characterization of the Ramified Analytical Hierarchy. Boyd taught for most of his career at Cornell University, though he also taught briefly at Harvard University, the University of Michigan, Ann Arbor, and the University of California, Berkeley. He has also been a visiting professor at the University of Canterbury in Christchurch, New Zealand, and the University of Melbourne in Melbourne, Victoria, Australia.
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Among his other important ideas is the notion of local conjunction of constraints - the idea that two constraints can combine into a single constraint that is violated only when both of its conjuncts are violated. Local conjunction has been applied to the analysis of various "super-additive" effects in Optimality Theory. With Bruce Tesar (Rutgers University), Smolensky has also contributed significantly to the study of the learnability of Optimality Theoretic grammars. He is a member of the Center for Language and Speech Processing.
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The broadcast and retransmission strategies used by the algorithm were already described in the literature."Exploiting Distributed Spatial Diversity in Networks," J. N. Laneman, G. Wornell; Analyzes some information-theoretic cooperative diversity schemes, but the radios use special techniques to share spectrum. ExOR adapts the time-slot scheme to a longer time-scale that can be implemented in software using commodity radios."Selection Diversity Forwarding in a Multihop Packet Radio Network with Fading Channels and Capture," P. Larsson, SIGMOBIL Mob. Comm. Rev.
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Several independent efforts to give a formal characterization of effective calculability led to a variety of proposed definitions (general recursion, Turing machines, λ-calculus) that later were shown to be equivalent. The notion captured by these definitions is known as recursive or effective computability. The Church–Turing thesis states that the two notions coincide: any number- theoretic function that is effectively calculable is recursively computable. As this is not a mathematical statement, it cannot be proven by a mathematical proof.
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The \Delta-lemma states that every uncountable collection of finite sets contains an uncountable \Delta- system. The \Delta-lemma is a combinatorial set-theoretic tool used in proofs to impose an upper bound on the size of a collection of pairwise incompatible elements in a forcing poset. It may for example be used as one of the ingredients in a proof showing that it is consistent with Zermelo–Fraenkel set theory that the continuum hypothesis does not hold. It was introduced by .
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System memory addresses for both the PPE and SPE are expressed as 64-bit values for a theoretic address range of 264 bytes (16 exabytes or 16,777,216 terabytes). In practice, not all of these bits are implemented in hardware. Local store addresses internal to the SPU (Synergistic Processor Unit) processor are expressed as a 32-bit word. In documentation relating to Cell a word is always taken to mean 32 bits, a doubleword means 64 bits, and a quadword means 128 bits.
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In statistical physics, the BBGKY hierarchy (Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy, sometimes called Bogoliubov hierarchy) is a set of equations describing the dynamics of a system of a large number of interacting particles. The equation for an s-particle distribution function (probability density function) in the BBGKY hierarchy includes the (s + 1)-particle distribution function, thus forming a coupled chain of equations. This formal theoretic result is named after Nikolay Bogolyubov, Max Born, Herbert S. Green, John Gamble Kirkwood, and Jacques Yvon.
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In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms ('arrows') called 'weak equivalences', 'fibrations' and 'cofibrations'. These abstract from a conventional homotopy category of topological spaces or of chain complexes (derived category theory), via the acyclic model theorem. The concept was introduced by . In recent decades, the language of model categories has been used in some parts of algebraic K-theory and algebraic geometry, where homotopy-theoretic approaches led to deep results.
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Duality of polytopes and order-theoretic duality are both involutions: the dual polytope of the dual polytope of any polytope is the original polytope, and reversing all order-relations twice returns to the original order. Choosing a different center of polarity leads to geometrically different dual polytopes, but all have the same combinatorial structure. A planar graph in blue, and its dual graph in red. From any three- dimensional polyhedron, one can form a planar graph, the graph of its vertices and edges.
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In mathematics, Paley graphs are dense undirected graphs constructed from the members of a suitable finite field by connecting pairs of elements that differ by a quadratic residue. The Paley graphs form an infinite family of conference graphs, which yield an infinite family of symmetric conference matrices. Paley graphs allow graph-theoretic tools to be applied to the number theory of quadratic residues, and have interesting properties that make them useful in graph theory more generally. Paley graphs are named after Raymond Paley.
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He analyzes the concept of "medical theory" in line with the so-called structuralist view of theories to represent their structure and content, according to Patrick Suppes and Joseph D. Sneed's approach, as set-theoretic predicates. This enables him to show that a theory in medicine cannot be confirmed, supported, disconfirmed, verified or falsified simply because it is merely a conceptual structure and no epistemic entity to be true, probable or false. It does not make any empirical claims about the world.
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In mathematics, a dyadic compactum is a Hausdorff topological space that is the continuous image of a product of discrete two-point sets, and a dyadic space is a topological space with a compactification which is a dyadic compactum. However, many authors use the term dyadic space with the same meaning as dyadic compactum above. T. C. Przymusinski, Products of normal spaces, Ch. XVIII In K. Kunen and J.E. Vaughan (eds) Handbook of Set-Theoretic Topology. North-Holland, Amsterdam, 1984, p. 794.
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Kalai's work on large games showed that the equilibria of Bayesian games with many players are structurally robust, thus large games escape major pitfalls in game-theoretic modeling. Kalai is also known for seminal collaborative research on flow games and totally balanced games; strategic complexity and its implications in economics and political systems; arbitration, strategic delegation and commitments; extensions of Arrow’s Impossibility Theorem in social choice; competitive service speed in queues; and on rational strategic polarization in group decision making.
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Their scheme allows these trees to be encoded in a number of bits that is close to the information-theoretic lower bound (the base-2 logarithm of the Wedderburn–Etherington number) while still allowing constant-time navigation operations within the tree.. use unordered binary trees, and the fact that the Wedderburn–Etherington numbers are significantly smaller than the numbers that count ordered binary trees, to significantly reduce the number of terms in a series representation of the solution to certain differential equations..
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A 1:10 scale model of on display at the Vasa Museum in Stockholm. The richly decorated stern is a sign of the great prestige that contemporary warship enjoyed as symbols of power and wealth. Vasa syndrome is inspired by the disastrous sinking of the Swedish warship on its short maiden voyage in 1628. Vasa was one of the earliest examples of a warship with two full gun decks, and was built when the theoretic principles of shipbuilding were still poorly understood.
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In 1906 he started teaching at a high school in Graz, at the same time collaborating with Adalbert Meingast and working as Meinong's assistant at the university. He also maintained close contacts with the Graz Psychological Institute, founded by Meinong. In 1912, he wrote his habilitation thesis entitled Gegenstandstheoretische Grundlagen der Logik und Logistik (Object-theoretic Foundations for Logics and Logistics) at Graz with Meinong as supervisor. From 1915 to 1918 he served as an officer in the Austro-Hungarian Army.
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Description-theoretic approaches are theories of donkey pronouns in which definite descriptions play an important role. They were pioneered by Gareth Evans's E-type approach, which holds that donkey pronouns can be understood as referring terms whose reference is fixed by description. Later authors have attributed an even larger role to definite descriptions, to the point of arguing that donkey pronouns have the semantics, and even the syntax, of definite descriptions. Approaches of the latter kind are usually called D-type.
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In mathematics, Solèr's theorem is a result concerning certain infinite- dimensional vector spaces. It states that any orthomodular form that has an infinite orthonormal sequence is a Hilbert space over the real numbers, complex numbers or quaternions. Originally proved by Maria Pia Solèr, the result is significant for quantum logic and the foundations of quantum mechanics. In particular, Solèr's theorem helps to fill a gap in the effort to use Gleason's theorem to rederive quantum mechanics from information-theoretic postulates.
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Late Rorty and Jürgen Habermas are closer to Continental thought. Neopragmatist thinkers who are more loyal to classical pragmatism include Sidney Hook and Susan Haack (known for the theory of foundherentism). Many pragmatist ideas (especially those of Peirce) find a natural expression in the decision-theoretic reconstruction of epistemology pursued in the work of Isaac Levi. Nicholas Rescher advocates his version of methodological pragmatism, based on construing pragmatic efficacy not as a replacement for truths but as a means to its evidentiation.
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The traditional explanation for this effect is that it is an extension of the anchoring bias, as studied by Tversky and Kahneman. The initial "anchor" is the .5 probability given when there are two choices without any other evidence, and people fail to adjust sufficiently far away. However, a recent study suggests that the belief revising conservatism can be explained by an information-theoretic generative mechanism and that assumes a noisy conversion of objective evidence (observation) into subjective estimates (judgment).
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Notable accomplishments here include a proof of Langlands' conjecture on the discrete series, along with a later proof (joint with Michael Atiyah) constructing all such discrete series representations on spaces of harmonic spinors. Schmid along with his student Henryk Hecht proved Blattner's conjecture in 1975. In the 1970s he described the singularities of the Griffith's period map by applying Lie-theoretic methods to problems in algebraic geometry. Schmid has been very involved in K–12 mathematics education both nationally and internationally.
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C. Naturman observed that these spaces were the Alexandrov-discrete spaces and extended the result to a category-theoretic duality between the category of Alexandrov-discrete spaces and (open) continuous maps, and the category of preorders and (bounded) monotone maps, providing the preorder characterizations as well as the interior and closure algebraic characterizations. A systematic investigation of these spaces from the point of view of general topology which had been neglected since the original paper by Alexandrov was taken up by F.G. Arenas.
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An important advantage of this string theory at that time was also that the unphysical tachyon of the bosonic string theory was eliminated. This was an early appearance of the ideas of supersymmetry which were being developed independently at that time by several groups. A few years later, Neveu, working in Princeton with David Gross, developed the Gross–Neveu model.A quantum-field-theoretic model of Dirac fermions with a four-fermion interaction vertex and unitary symmetry in one spatial dimension.
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Within mathematics, Gauld works in set-theoretic topology, with emphasis on applications to non- metrisable manifolds and topological properties of manifolds close to metrisability. Gauld has authored two monographs and over 70 research papers. Gauld was born on 28 June 1942 in Inglewood and grew up there. He was educated at Wanganui Technical College, Inglewood High School and New Plymouth Boys’ High School, and later obtained his BSc and MSc degrees with first-class honours in mathematics from the University of Auckland.
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The lack of electrical brain activity may not be enough to consider someone scientifically dead. Therefore, the concept of information-theoretic death has been suggested as a better means of defining when true death occurs, though the concept has few practical applications outside the field of cryonics. There have been some scientific attempts to bring dead organisms back to life, but with limited success. In science fiction scenarios where such technology is readily available, real death is distinguished from reversible death.
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In aphasia, the inherent neurological damage is frequently assumed to be a loss of implicit linguistic competence that has damaged or wiped out neural centers or pathways that are necessary for maintenance of the language rules and representations needed to communicate. The measurement of implicit language competence, although apparently necessary and satisfying for theoretic linguistics, is complexly interwoven with performance factors. Transience, stimulability, and variability in aphasia language use provide evidence for an access deficit model that supports performance loss.LaPointe, Leonard L. (2008).
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This was provided in two steps; the first one was done in 1970 (Publ. Math. de l'IHÉS) by Peter Donovan and Max Karoubi; the second one in 1988 by Jonathan Rosenberg in Continuous- Trace Algebras from the Bundle Theoretic Point of View. In physics, it has been conjectured to classify D-branes, Ramond-Ramond field strengths and in some cases even spinors in type II string theory. For more information on twisted K-theory in string theory, see K-theory (physics).
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One of the approaches to representations of finite groups is through module theory. Representations of a group G are replaced by modules over its group algebra K[G] (to be precise, there is an isomorphism of categories between K[G]-Mod and RepG, the category of representations of G). Irreducible representations correspond to simple modules. In the module-theoretic language, Maschke's theorem asks: is an arbitrary module semisimple? In this context, the theorem can be reformulated as follows: :Maschke's Theorem.
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In the third chapter titled "An Elementary Linguistic Theory", Chomsky tries to determine what sort of device or model gives an adequate account of a given set of "grammatical" sentences. Chomsky hypothesizes that this device has to be finite instead of infinite. He then considers finite state grammar, a communication theoretic modelSpecifically, the model proposed in which treats language as a Markov process. Then in the fourth chapter titled "Phrase Structure", he discusses phrase structure grammar, a model based on immediate constituent analysis.
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He was presented as a prominent authority on all the areas of aviation. Jordanoff eventually became the largest American publisher and editor of specialized military manuals. In the 1940s Jordanoff was assigned the task by the United States Department of Defense to prepare instruction manuals for military aircraft, submarines and aircraft carriers, on topics like land support with radio equipment, air meteorology, theoretic and flight preparation of the pilots for diurnal and nocturnal piloting. These were for crew use, inspection, maintenance and repair.
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A pseudo algebraically closed field (in short PAC) K is a field satisfying the following geometric property. Each absolutely irreducible algebraic variety V defined over K has a K-rational point. Over PAC fields there is a firm link between arithmetic properties of the field and group theoretic properties of its absolute Galois group. A nice theorem in this spirit connects Hilbertian fields with ω-free fields (K is ω-free if any embedding problem for K is properly solvable). Theorem.
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An encryption protocol with information-theoretic security does not depend for its effectiveness on unproven assumptions about computational hardness. Such a protocol is not vulnerable to future developments in computer power such as quantum computing. An example of an information-theoretically secure cryptosystem is the one-time pad. The concept of information-theoretically secure communication was introduced in 1949 by American mathematician Claude Shannon, the inventor of information theory, who used it to prove that the one-time pad system was secure.
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In other words, the plaintext message is independent of the transmitted ciphertext if we do not have access to the key. It has been proved that any cipher with the perfect secrecy property must use keys with effectively the same requirements as one- time pad keys. It is common for a cryptosystem to leak some information but nevertheless maintain its security properties even against an adversary that has unlimited computational resources. Such a cryptosystem would have information-theoretic but not perfect security.
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In particular, his work implies that if a finitely generated group G is elementarily equivalent to a word- hyperbolic group then G is word-hyperbolic as well. Sela also proved that the first-order theory of a finitely generated free group is stable in the model- theoretic sense, providing a brand-new and qualitatively different source of examples for the stability theory. An alternative solution for the Tarski conjecture has been presented by Olga Kharlampovich and Alexei Myasnikov.O. Kharlampovich, and A. Myasnikov.
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The theory of unique bid auctions has been the subject of mathematical investigation. In a 2007 paper Bruss, Louchard and Ward proposed a technique for calculating game-theoretic probabilistic optimal strategies for unique bid auctions, given a small set of extra assumptions about the nature of the auction. Another paper by Raviv and Virag in the same year made theoretical predictions and compared their results to the results of real-world unique bid auctions. Another paper by Rapoport et al.
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Formally, an implicit data structure is one with constant space overhead (above the information-theoretic lower bound). Historically, defined an implicit data structure (and algorithms acting on one) as one "in which structural information is implicit in the way data are stored, rather than explicit in pointers." They are somewhat vague in the definition, defining it most strictly as a single array, with only the size retained (a single number of overhead),"Thus, only a simple array is needed for the data.", p.
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There is also an alternative graph-theoretic description of the same lattice: the independent sets of any bipartite graph may be given a partial order in which one independent set is less than another if they differ by removing elements from one side of the bipartition and adding elements to the other side of the bipartition; with this order, the independent sets form a distributive lattice,. and applying this construction to a path graph results in the lattice associated with the Fibonacci cube.
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Berger has worked on the decision theoretic bases of Bayesian inference, including advances on the Stein phenomenon during and after his thesis. He has also greatly contributed to advances in the so-called objective Bayes approach where prior distributions are constructed from the structure of the sampling distributions and/or of frequentist properties. He is also recognized for his analysis of the opposition between Bayesian and frequentist visions on testing statistical hypotheses, with criticisms of the use of p-values and critical levels.
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Kamp's 1971 paper on "now" (published in Theoria) was the first employment of double- indexing in model theoretic semantics. His doctoral committee included Richard Montague as chairman, Chen Chung Chang, Alonzo Church, David Kaplan, Yiannis N. Moschovakis, and Jordan Howard Sobel. Kamp became a corresponding member of the Royal Netherlands Academy of Arts and Sciences in 1997. He was awarded the Jean Nicod Prize in 1996 and was elected a Fellow of the American Academy of Arts & Sciences in 2015.
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His research is in theoretical condensed matter physics, although he is also known for his earlier work in theoretical particle physics. In 2009, Shankar was awarded the Julius Edgar Lilienfeld Prize from the American Physical Society for "innovative applications of field theoretic techniques to quantum condensed matter systems". After three years at the Harvard Society of Fellows, he joined the Yale physics department, which he chaired between 2001-2007. He is a fellow of the American Academy of Arts and Sciences.
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The study of complete subgraphs in mathematics predates the "clique" terminology. For instance, complete subgraphs make an early appearance in the mathematical literature in the graph-theoretic reformulation of Ramsey theory by . But the term "clique" and the problem of algorithmically listing cliques both come from the social sciences, where complete subgraphs are used to model social cliques, groups of people who all know each other. used graphs to model social networks, and adapted the social science terminology to graph theory.
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From a security-theoretic point of view, modes of operation must provide what is known as semantic security. Informally, it means that given some ciphertext under an unknown key one cannot practically derive any information from the ciphertext (other than the length of the message) over what one would have known without seeing the ciphertext. It has been shown that all of the modes discussed above, with the exception of the ECB mode, provide this property under so-called chosen plaintext attacks.
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Cohn et al. put methods such as the Strassen and Coppersmith–Winograd algorithms in an entirely different group- theoretic context, by utilising triples of subsets of finite groups which satisfy a disjointness property called the triple product property (TPP). They show that if families of wreath products of Abelian groups with symmetric groups realise families of subset triples with a simultaneous version of the TPP, then there are matrix multiplication algorithms with essentially quadratic complexity.Henry Cohn, Robert Kleinberg, Balázs Szegedy, and Chris Umans.
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In set theory, the core model is a definable inner model of the universe of all sets. Even though set theorists refer to "the core model", it is not a uniquely identified mathematical object. Rather, it is a class of inner models that under the right set-theoretic assumptions have very special properties, most notably covering properties. Intuitively, the core model is "the largest canonical inner model there is" (Ernest Schimmerling and John R. Steel) and is typically associated with a large cardinal notion.
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An order theoretic meet-semilattice gives rise to a binary operation ∧ such that is an algebraic meet-semilattice. Conversely, the meet-semilattice gives rise to a binary relation ≤ that partially orders S in the following way: for all elements x and y in S, x ≤ y if and only if x = x ∧ y. The relation ≤ introduced in this way defines a partial ordering from which the binary operation ∧ may be recovered. Conversely, the order induced by the algebraically defined semilattice coincides with that induced by ≤.
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It was shown to be true in dimensions at most 6 by . However, for higher dimensions it is false, as was shown in dimensions at least 10 by and in dimensions at least 8 by . These disproofs used a reformulation of the problem in terms of the clique number of certain graphs now known as Keller graphs. Although this graph-theoretic version of the conjecture was resolved for all dimensions by 2002, Keller's original cube- tiling conjecture remained open in dimension 7 until 2019.
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From 2002 until 2008, Maurer also served on the board of Tamedia AG.Tamedia AG Medienmitteilung 23. April 2008: Jahresabschluss 2007 Maurer is editor of the Journal of Cryptology and used to be editor-in-chief. In 2008, Maurer was named a Fellow of the International Association for Cryptologic Research "for fundamental contributions to information-theoretic cryptography, service to the IACR, and sustained educational leadership in cryptology." In 2015, he was named a Fellow of the Association for Computing Machinery "for contributions to cryptography and information security.".
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Cognitive Computation, 1(1), 50–63. Abundant evidence indicates that consciously perceived inputs elicit widespread brain activation, as compared with inputs that do not reach consciousness. The dynamic core hypothesis (DCH) proposes that consciousness arises from neural dynamics in the thalamocortical system, as measured by the quantity neural complexity (CN). CN is an information- theoretic measure; the CN value is high if each subset of a neural system can take on many different states, and if these states make a difference to the rest of the system.
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Graph theory is also used to study molecules in chemistry and physics. In condensed matter physics, the three-dimensional structure of complicated simulated atomic structures can be studied quantitatively by gathering statistics on graph-theoretic properties related to the topology of the atoms. Also, "the Feynman graphs and rules of calculation summarize quantum field theory in a form in close contact with the experimental numbers one wants to understand." In chemistry a graph makes a natural model for a molecule, where vertices represent atoms and edges bonds.
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Categorical logic is now a well-defined field based on type theory for intuitionistic logics, with applications in functional programming and domain theory, where a cartesian closed category is taken as a non-syntactic description of a lambda calculus. At the very least, category theoretic language clarifies what exactly these related areas have in common (in some abstract sense). Category theory has been applied in other fields as well. For example, John Baez has shown a link between Feynman diagrams in physics and monoidal categories.
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Several properties of λ-terms that are difficult to state and prove using the traditional notation are easily expressed in the De Bruijn notation. For example, in a type-theoretic setting, one can easily compute the canonical class of types for a term in a typing context, and restate the type checking problem to one of verifying that the checked type is a member of this class. De Bruijn notation has also been shown to be useful in calculi for explicit substitution in pure type systems..
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As such, he proposed a theory of the three stages for the smooth transition of China's economic system: liberalization, marketization and privatization. Echoing with China's reform and opening-up initiative, in 1993 Tian co-edited A Series of Popular Economics Books for Institutional Transition in China (14 volumes), which was awarded China's National Book Award in 1994. In 2014, Tian published the book China’s Reform: History, Logic and Future. It offers a review of China's marketization endeavor with theoretic insights and practical policy implications.
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A game-theoretic auction model is a mathematical game represented by a set of players, a set of actions (strategies) available to each player, and a payoff vector corresponding to each combination of strategies. Generally, the players are the buyer(s) and the seller(s). The action set of each player is a set of bid functions or reservation prices (reserves). Each bid function maps the player's value (in the case of a buyer) or cost (in the case of a seller) to a bid price.
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" A similar conclusion was reached by Corner et al., who after investigating the psychological mechanism of the slippery slope argument say, "Despite their philosophical notoriety, SSAs are used (and seem to be accepted) in a wide variety of practical contexts. The experimental evidence reported in this paper suggests that in some circumstances, their practical acceptability can be justified, not just because the decision-theoretic framework renders them subjectively rational, but also because it is demonstrated how, objectively, the slippery slopes they claim do in fact exist.
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There are many widely used formulas having terms involving natural-number exponents that require to be evaluated to . For example, regarding as an empty product assigns it the value , even when . Alternatively, the combinatorial interpretation of is the number of empty tuples of elements from a set with elements; there is exactly one empty tuple, even if . Equivalently, the set-theoretic interpretation of is the number of functions from the empty set to the empty set; there is exactly one such function, the empty function.
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3G MIMO describes MIMO techniques which have been considered as 3G standard techniques. MIMO, as the state of the art of intelligent antenna (IA), improves the performance of radio systems by embedding electronics intelligence into the spatial processing unit. Spatial processing includes spatial precoding at the transmitter and spatial postcoding at the receiver, which are dual each other from information signal processing theoretic point of view. Intelligent antenna is technology which represents smart antenna, multiple antenna (MIMO), self-tracking directional antenna, cooperative virtual antenna and so on.
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Unlike the social democratic orthodoxy of the Second International, she did not regard organisation as a product of scientific-theoretic insight to historical imperatives, but as product of the working classes' struggles:The Political Leader of the German Working Classes. Collected Works. Vol. 2. p. 280. > Social democracy is simply the embodiment of the modern proletariat's class > struggle, a struggle which is driven by a consciousness of its own historic > consequences. The masses are in reality their own leaders, dialectically > creating their own development process.
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In joint work with Dennis Gaitsgory, he used his non-abelian Poincaré duality in an algebraic-geometric setting, to prove the Siegel mass formula for function fields. Lurie was one of the inaugural winners of the Breakthrough Prize in Mathematics in 2014, "for his work on the foundations of higher category theory and derived algebraic geometry; for the classification of fully extended topological quantum field theories; and for providing a moduli-theoretic interpretation of elliptic cohomology." Lurie was also awarded a MacArthur "Genius Grant" Fellowship in 2014.
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Some believe that Georg Cantor's set theory was not actually implicated in the set-theoretic paradoxes (see Frápolli 1991). One difficulty in determining this with certainty is that Cantor did not provide an axiomatization of his system. By 1899, Cantor was aware of some of the paradoxes following from unrestricted interpretation of his theory, for instance Cantor's paradoxLetter from Cantor to David Hilbert on September 26, 1897, p. 388. and the Burali-Forti paradox,Letter from Cantor to Richard Dedekind on August 3, 1899, p. 408.
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A cycle double cover of the Petersen graph, corresponding to its embedding on the projective plane as a hemi-dodecahedron. In graph-theoretic mathematics, a cycle double cover is a collection of cycles in an undirected graph that together include each edge of the graph exactly twice. For instance, for any polyhedral graph, the faces of a convex polyhedron that represents the graph provide a double cover of the graph: each edge belongs to exactly two faces. It is an unsolved problem, posed by George Szekeres.
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Category utility is a measure of "category goodness" defined in and . It attempts to maximize both the probability that two objects in the same category have attribute values in common, and the probability that objects from different categories have different attribute values. It was intended to supersede more limited measures of category goodness such as "cue validity" (; ) and "collocation index" . It provides a normative information-theoretic measure of the predictive advantage gained by the observer who possesses knowledge of the given category structure (i.e.
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A rigorous construction of localization of categories, avoiding these set-theoretic issues, was one of the initial reasons for the development of the theory of model categories: a model category M is a category in which there are three classes of maps; one of these classes is the class of weak equivalences. The homotopy category Ho(M) is then the localization with respect to the weak equivalences. The axioms of a model category ensure that this localization can be defined without set-theoretical difficulties.
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Based on previously described measures, we want to recognize nodes that are the most important in disease spreading. Approaches based only on centralities, that focus on individual features of nodes, may not be good idea. Nodes in the red square, individually cannot stop disease spreading, but considering them as a group, we clearly see that they can stop disease if it has started in nodes v_1, v_4, and v_5. Game-theoretic centralities try to consult described problems and opportunities, using tools from game-theory.
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In algebra, the free product (coproduct) of a family of associative algebras A_i, i \in I over a commutative ring R is the associative algebra over R that is, roughly, defined by the generators and the relations of the A_i's. The free product of two algebras A, B is denoted by A ∗ B. The notion is a ring- theoretic analog of a free product of groups. In the category of commutative R-algebras, the free product of two algebras (in that category) is their tensor product.
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The large amount of data collected by Hampton on the combination of two concepts can be modeled in a specific quantum-theoretic framework in Fock space where the observed deviations from classical set (fuzzy set) theory, the above-mentioned over- and under- extension of membership weights, are explained in terms of contextual interactions, superposition, interference, entanglement and emergence. And, more, a cognitive test on a specific concept combination has been performed which directly reveals, through the violation of Bell's inequalities, quantum entanglement between the component concepts.
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She married Manuel Blum, then a student at the Massachusetts Institute of Technology, and transferred in 1961 to Simmons College, a private women's liberal arts college in Boston. Simmons did not have a strong mathematics program but she was eventually able to take Isadore Singer's mathematics classes at MIT, graduating from Simmons with a B.S. in mathematics in 1963. She received her Ph.D. in mathematics from the Massachusetts Institute of Technology in 1968. Her dissertation, Generalized Algebraic Theories: A Model Theoretic Approach, was supervised by Gerald Sacks.
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Primarily a particle physicist, his works encompass studies in the standard model of particle physics. Some of his pioneering works are particularly focused on the topic of massive neutrinos, whose theoretical basis requires physics beyond the standard model. His original contributions to the field of neutrino physics include calculation of Feynman diagrams for Majorana fermions., determination of properties of fermions in matters and magnetic fields, and a quantum field theoretic reworking of Lincoln Wolfenstein's formula predicting the properties of neutrino oscillation in presence of matter.
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The place of the disruptive innovations and innovative entrepreneurs in traditional economic theory (which describes many efficiency-based ratios assuming uniform outputs) presents theoretic quandaries. Baumol contributed greatly to this area of economic theory. The 2006 Annual Meetings of the American Economic Association held a special session in his name, and honoring his many years of work in the field of entrepreneurship and innovation, where 12 papers on entrepreneurship were presented. The Baumol Research Centre for Entrepreneurship Studies at Zhejiang Gongshang University is named after William Baumol.
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It includes an English presentation of the work of Takeuchi. The volume led to far greater use of AIC, and it now has more than 48,000 citations on Google Scholar. Akaike called his approach an "entropy maximization principle", because the approach is founded on the concept of entropy in information theory. Indeed, minimizing AIC in a statistical model is effectively equivalent to maximizing entropy in a thermodynamic system; in other words, the information-theoretic approach in statistics is essentially applying the Second Law of Thermodynamics.
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The primitive recursive functions are among the number-theoretic functions, which are functions from the natural numbers (nonnegative integers) {0, 1, 2, ...} to the natural numbers. These functions take n arguments for some natural number n and are called n-ary. The basic primitive recursive functions are given by these axioms: More complex primitive recursive functions can be obtained by applying the operations given by these axioms: Example. We take f(x) as the S(x) defined above. This f is a 1-ary primitive recursive function.
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Using estimates on Kloosterman sums he was able to derive estimates for Fourier coefficients of modular forms in cases where Pierre Deligne's proof of the Weil conjectures was not applicable. It was later translated by Jacquet to a representation theoretic framework. Let be a reductive group over a number field F and H\subset G be a subgroup. While the usual trace formula studies the harmonic analysis on G, the relative trace formula a tool for studying the harmonic analysis on the symmetric space .
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Both the precedence and release time constraints in the standard notation for theoretic scheduling problems may be modeled by antimatroids. use antimatroids to generalize a greedy algorithm of Eugene Lawler for optimally solving single-processor scheduling problems with precedence constraints in which the goal is to minimize the maximum penalty incurred by the late scheduling of a task. use antimatroids to model the ordering of events in discrete event simulation systems. uses antimatroids to model progress towards a goal in artificial intelligence planning problems.
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Take, for example, . In 1952, Davenport still had to specify that he meant The Higher Arithmetic. Hardy and Wright wrote in the introduction to An Introduction to the Theory of Numbers (1938): "We proposed at one time to change [the title] to An introduction to arithmetic, a more novel and in some ways a more appropriate title; but it was pointed out that this might lead to misunderstandings about the content of the book." In particular, arithmetical is preferred as an adjective to number-theoretic.
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The s-cobordism theorem states for a closed connected oriented manifold M of dimension n > 4 that an h-cobordism W between M and another manifold N is trivial over M if and only if the Whitehead torsion of the inclusion M\hookrightarrow W vanishes. Moreover, for any element in the Whitehead group there exists an h-cobordism W over M whose Whitehead torsion is the considered element. The proofs use handle decompositions. There exists a homotopy theoretic analogue of the s-cobordism theorem.
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An important special case is a monoid action or act, in which the semigroup is a monoid and the identity element of the monoid acts as the identity transformation of a set. From a category theoretic point of view, a monoid is a category with one object, and an act is a functor from that category to the category of sets. This immediately provides a generalization to monoid acts on objects in categories other than the category of sets. Another important special case is a transformation semigroup.
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Binding Theory refers to 3 different theoretic principles that regulate DP's (Determiner Phrase). In consideration of the following definitions of the principles, the local domain refers to the closest XP with a subject. If a DP(1) is bound, this means it is c-commanded and co-indexed by a DP(2) that is sister to the XP dominating over DP (1) .To contrast, if it is free, then the DP in question must not be c-commanded and co-indexed by another DP.
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Every contractible space is simply connected. ;Coproduct topology: If {Xi} is a collection of spaces and X is the (set-theoretic) disjoint union of {Xi}, then the coproduct topology (or disjoint union topology, topological sum of the Xi) on X is the finest topology for which all the injection maps are continuous. ;Cosmic space: A continuous image of some separable metric space. ;Countable chain condition: A space X satisfies the countable chain condition if every family of non-empty, pairswise disjoint open sets is countable.
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The typical diagram of the definition of a universal morphism. In category theory, a branch of mathematics, a universal property is an important property which is satisfied by a universal morphism (see Formal Definition). Universal morphisms can also be thought of more abstractly as initial or terminal objects of a comma category (see Connection with Comma Categories). Universal properties occur almost everywhere in mathematics, and hence the precise category theoretic concept helps point out similarities between different branches of mathematics, some of which may even seem unrelated.
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Debraj RayInformation for this entry has been compiled from Ray's webpage, Ray's cv available on his webpage, as well as a short biography in his recent book, A Game-Theoretic Perspective on Coalition Formation (2007; Oxford University Press). (born 3 September 1957) is an Indian-American economist whose focus is development economics and game theory. Ray is currently Julius Silver Professor in the Faculty of Arts and Science, and Professor of Economics at New York University. He is Co-Editor of the American Economic Review.
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People's Daily Online'. January 25, 2011. they called for the creation of research, teaching and treatment hospitals that combine expertise in genetic engineering, theoretic and experimental physics, American, Chinese and stem cell medicine and the comparative analysis of the cost, effectiveness and painfulness of each of those treatment modalities alone or in combination to make 21st century health care globally more effective and less costly and painful. They have proposed that the prototype of such research, educational and clinical hospitals be created in Jilin Province, China.
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A drama unfolds through episodes in which characters interact. The episode is a period of preplay communication between characters who, after communicating, act as players in a game that is constructed through the dialogue between them. The action that follows the episode is the playing out of this game; it sets up the next episode. Most drama-theoretic terminology is derived from a theatrical model applied to real life interactions; thus, an episode goes through phases of scene-setting, build-up, climax and decision.
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Ravdonikas was the first scholar who interpreted Onega objects, symbolic figures that probably date back to 5000 BCE, in the context of their astronomical orientation. His finding was that the figures fix a complete Lunar 18.6-year's cycle and that their complex was a lunar calendar. In 1980 Ravdonikas restored a Peter Kinzing's music automaton from the State Hermitage collection. Writing his report of the restoration works led Feliks to the idea of putting together a theoretic inquiry in what he later named 'spatial musical symbolism'.
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Bushkovitch argues that the theoretic lack of limitations on the power of the tsar is irrelevant and instead claims that the "crucial question" is where real power lay. In his view, this can only be shown by the political narrative of events.D.L. Ransel, The Politics of Catherinian Russia: The Panin Party (New Haven 1975); Bushkovitch, Peter the Great: The Struggle for Power, 29. Bushkovitch placed the balance of power between the tsar, the individual boyars and the tsar’s favourites at the centre of political decision-making.
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Kuhn poker is an extremely simplified form of poker developed by Harold W. Kuhn as a simple model zero-sum two-player imperfect-information game, amenable to a complete game-theoretic analysis. In Kuhn poker, the deck includes only three playing cards, for example a King, Queen, and Jack. One card is dealt to each player, which may place bets similarly to a standard poker. If both players bet or both players pass, the player with the higher card wins, otherwise, the betting player wins.
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Kalmar defined what are known as elementary functions, number-theoretic functions (i.e. those based on the natural numbers) built up from the notions of composition and variables, the constants 0 and 1, repeated addition + of the constants, proper subtraction ∸, bounded summation and bounded product (Kleene 1952:526). Elimination of the bounded product from this list yields the subelementary or lower elementary functions. By use of the abstract computational model called a register machine Schwichtenberg provides a demonstration that "all elementary functions are computable and totally defined" (Schwichtenberg 58).
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In literary theoretic approach, narrative is being narrowly defined as fiction-writing mode in which the narrator is communicating directly to the reader. Until the late 19th century, literary criticism as an academic exercise dealt solely with poetry (including epic poems like the Iliad and Paradise Lost, and poetic drama like Shakespeare). Most poems did not have a narrator distinct from the author. But novels, lending a number of voices to several characters in addition to narrator's, created a possibility of narrator's views differing significantly from the author's views.
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Winsberg wrote his doctoral dissertation on the use of computer simulation to study complex physical systems. Over the next several years, he published a number of articles on computer simulation, including their implications for understanding the nature of scientific theories and their application, scientific realism, the role of fiction in science, and the nature of inter-theoretic reduction. His work on computer simulation has been called "pioneering," "groundbreaking," and "trailblazing." He also contributed to the literature on the role of the thermodynamics for understanding the arrow of time.
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In group theory, two subgroups Γ1 and Γ2 of a group G are said to be commensurable if the intersection Γ1 ∩ Γ2 is of finite index in both Γ1 and Γ2. Example: Let a and b be nonzero real numbers. Then the subgroup of the real numbers R generated by a is commensurable with the subgroup generated by b if and only if the real numbers a and b are commensurable, in the sense that a/b is rational. Thus the group-theoretic notion of commensurability generalizes the concept for real numbers.
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One might wonder which ring-theoretic property of A=k[x_1,\ldots,x_n] causes the Hilbert syzygy theorem to hold. It turns out that this is regularity, which is an algebraic formulation of the fact that affine -space is a variety without singularities. In fact the following generalization holds: Let A be a Noetherian ring. Then A has finite global dimension if and only if A is regular and the Krull dimension of A is finite; in that case the global dimension of A is equal to the Krull dimension.
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Pearce promotes replacing the pain/pleasure axis with a robot-like response to noxious stimuliSee Vanity Fair interview with Pearce or with gradients of bliss,See Life in the Far North – An information-theoretic perspective on Heaven through genetic engineering and other technical scientific advances. Hedonistic psychology,Kahneman, D., E. Diener and N. Schwartz (eds.) Well-being: The Foundations of Hedonistic Psychology, Russell Sage Foundation, 1999 affective science, and affective neuroscience are some of the emerging scientific fields that could in the coming years focus their attention on the phenomenon of suffering.
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In the frequentist interpretation, probabilities are discussed only when dealing with well- defined random experiments (or random samples). Neyman's derivation of confidence intervals embraced the measure theoretic axioms of probability published by Kolmogorov a few years previously and referenced the subjective (Bayesian) probability definitions of Jeffreys published earlier in the decade. Neyman defined frequentist probability (under the name classical) and stated the need for randomness in the repeated samples or trials. He accepted in principle the possibility of multiple competing theories of probability while expressing several specific reservations about the existing alternative probability interpretation.
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The scalars (and hence the vectors, matrices and tensors) can be real or complex as both are fields in the abstract-algebraic/ring-theoretic sense. In a general setting, classical fields are described by sections of fiber bundles and their dynamics is formulated in the terms of jet manifolds (covariant classical field theory).Giachetta, G., Mangiarotti, L., Sardanashvily, G. (2009) Advanced Classical Field Theory. Singapore: World Scientific, () In modern physics, the most often studied fields are those that model the four fundamental forces which one day may lead to the Unified Field Theory.
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Haddad's Thermodynamics: A Dynamical Systems Approach, Princeton, NJ: Princeton University Press, 2005, develops a novel and unique system-theoretic framework of thermodynamics. Thermodynamics is one of the bedrock disciplines of physics and engineering, yet its foundation has been lacking rigor and clarity as very eloquently pointed out by the American mathematician and natural philosopher Clifford Truesdell. Over the years, researchers from the systems and control community have acknowledged the need for developing a solid foundation for thermodynamics. Haddad's book brings together a vast range of ideas and tools to construct a powerful framework for thermodynamics.
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Gregory John Chaitin ( ; born 25 June 1947) is an Argentine-American mathematician and computer scientist. Beginning in the late 1960s, Chaitin made contributions to algorithmic information theory and metamathematics, in particular a computer-theoretic result equivalent to Gödel's incompleteness theorem.Review of Meta Math!: The Quest for Omega, By Gregory Chaitin SIAM News, Volume 39, Number 1, January/February 2006 He is considered to be one of the founders of what is today known as algorithmic (Solomonoff-Kolmogorov- Chaitin, Kolmogorov or program-size) complexity together with Andrei Kolmogorov and Ray Solomonoff.
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Complexity economics has a complex relation to previous work in economics and other sciences, and to contemporary economics. Complexity- theoretic thinking to understand economic problems has been present since their inception as academic disciplines. Research has shown that no two separate micro-events are completely isolated,Albert-László Barabási "Unfolding the science behind the idea of six degrees of separation" and there is a relationship that forms a macroeconomic structure. However, the relationship is not always in one direction; there is a reciprocal influence when feedback is in operation.
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If a(F) and b(F) are identically distributed for all search algorithms a and b, then F has an NFL distribution. This condition holds if and only if F and F o j are identically distributed for all j in J. In other words, there is no free lunch for search algorithms if and only if the distribution of objective functions is invariant under permutation of the solution space.The "only if" part was first published by Set-theoretic NFL theorems have recently been generalized to arbitrary cardinality X and Y.
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Solving a parity game played on a finite graph means deciding, for a given starting position, which of the two players has a winning strategy. It has been shown that this problem is in NP and Co-NP, more precisely UP and co-UP, as well as in QP (quasipolynomial time). It remains an open question whether this decision problem is solvable in PTime. Given that parity games are history- free determined, solving a given parity game is equivalent to solving the following simple looking graph-theoretic problem.
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André Weil's famous Weil conjectures proposed that certain properties of equations with integral coefficients should be understood as geometric properties of the algebraic variety that they define. His conjectures postulated that there should be a cohomology theory of algebraic varieties that gives number-theoretic information about their defining equations. This cohomology theory was known as the "Weil cohomology", but using the tools he had available, Weil was unable to construct it. In the early 1960s, Alexander Grothendieck introduced étale maps into algebraic geometry as algebraic analogues of local analytic isomorphisms in analytic geometry.
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As a method of applied mathematics, game theory has been used to study a wide variety of human and animal behaviors. It was initially developed in economics to understand a large collection of economic behaviors, including behaviors of firms, markets, and consumers. The first use of game-theoretic analysis was by Antoine Augustin Cournot in 1838 with his solution of the Cournot duopoly. The use of game theory in the social sciences has expanded, and game theory has been applied to political, sociological, and psychological behaviors as well.
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The application of game theory to political science is focused in the overlapping areas of fair division, political economy, public choice, war bargaining, positive political theory, and social choice theory. In each of these areas, researchers have developed game-theoretic models in which the players are often voters, states, special interest groups, and politicians. Early examples of game theory applied to political science are provided by Anthony Downs. In his 1957 book An Economic Theory of Democracy, he applies the Hotelling firm location model to the political process.
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2003), which identifies cross- rotation peaks consistent with non-crystallographic symmetry, was used in the structure determination of the enzyme dihydrofolate reductase-thymidylate synthase (DHFR-TS) from Cryptosporidium hominis, an important advancement in Cryptosporidium biology. He has designed many algorithms and computational protocols to extract structural information from NMR data, and used that information to compute structures of globular proteins and symmetric homo- oligomers. A distinct feature of his algorithms is that they use less data, and provide complexity-theoretic guarantees on time and space (See, e.g., B. R. Donald and J. Martin.
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The same applies if the game length is unknown but has a known upper limit. Unlike the standard prisoner's dilemma, in the iterated prisoner's dilemma the defection strategy is counter-intuitive and fails badly to predict the behavior of human players. Within standard economic theory, though, this is the only correct answer. The superrational strategy in the iterated prisoner's dilemma with fixed N is to cooperate against a superrational opponent, and in the limit of large N, experimental results on strategies agree with the superrational version, not the game-theoretic rational one.
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When both and are matrices, the trace of the (ring-theoretic) commutator of and vanishes: , because and is linear. One can state this as "the trace is a map of Lie algebras from operators to scalars", as the commutator of scalars is trivial (it is an Abelian Lie algebra). In particular, using similarity invariance, it follows that the identity matrix is never similar to the commutator of any pair of matrices. Conversely, any square matrix with zero trace is a linear combinations of the commutators of pairs of matrices.
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Model theory has a different scope that encompasses more arbitrary theories, including foundational structures such as models of set theory. From the model-theoretic point of view, structures are the objects used to define the semantics of first-order logic. For a given theory in model theory, a structure is called a model if it satisfies the defining axioms of that theory, although it is sometimes disambiguated as a semantic model when one discusses the notion in the more general setting of mathematical models. Logicians sometimes refer to structures as interpretations.
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Tarski's student, Vaught, has ranked Tarski as one of the four greatest logicians of all time -- along with Aristotle, Gottlob Frege, and Kurt Gödel. However, Tarski often expressed great admiration for Charles Sanders Peirce, particularly for his pioneering work in the logic of relations. Tarski produced axioms for logical consequence and worked on deductive systems, the algebra of logic, and the theory of definability. His semantic methods, which culminated in the model theory he and a number of his Berkeley students developed in the 1950s and 60s, radically transformed Hilbert's proof-theoretic metamathematics.
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In the 1960s, Abraham Robinson used model-theoretic techniques to develop calculus and analysis based on infinitesimals, a problem that first had been proposed by Leibniz. In proof theory, the relationship between classical mathematics and intuitionistic mathematics was clarified via tools such as the realizability method invented by Georg Kreisel and Gödel's Dialectica interpretation. This work inspired the contemporary area of proof mining. The Curry–Howard correspondence emerged as a deep analogy between logic and computation, including a correspondence between systems of natural deduction and typed lambda calculi used in computer science.
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The payoff of each player under a combination of strategies is the expected utility (or expected profit) of that player under that combination of strategies. Game-theoretic models of auctions and strategic bidding generally fall into either of the following two categories. In a private values model, each participant (bidder) assumes that each of the competing bidders obtains a random private value from a probability distribution. In a common value model, the participants have equal valuations of the item, but they do not have perfectly accurate information about this valuation.
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In the category of unital rings, there are no kernels in the category-theoretic sense; indeed, this category does not even have zero morphisms. Nevertheless, there is still a notion of kernel studied in ring theory that corresponds to kernels in the category of non-unital rings. In the category of pointed topological spaces, if f : X → Y is a continuous pointed map, then the preimage of the distinguished point, K, is a subspace of X. The inclusion map of K into X is the categorical kernel of f.
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Each Euler curve divides the plane into two regions or "zones": the interior, which symbolically represents the elements of the set, and the exterior, which represents all elements that are not members of the set. The sizes or shapes of the curves are not important; the significance of the diagram is in how they overlap. The spatial relationships between the regions bounded by each curve (overlap, containment or neither) corresponds to set- theoretic relationships (intersection, subset and disjointness). Curves whose interior zones do not intersect represent disjoint sets.
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In some common situations may be a set of subsets of in which case incidence will be containment ( if and only if is a member of ). Incidence structures of this type are called set-theoretic. This is not always the case, for example, if is a set of vectors and a set of square matrices, we may define vector is an eigenvector of matrix }. This example also shows that while the geometric language of points and lines is used, the object types need not be these geometric objects.
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Cournot also introduced the concept of best response dynamics in his analysis of the stability of equilibrium. Cournot did not use the idea in any other applications, however, or define it generally. The modern game-theoretic concept of Nash equilibrium is instead defined in terms of mixed strategies, where players choose a probability distribution over possible actions (rather than choosing a deterministic action to be played with certainty). The concept of a mixed-strategy equilibrium was introduced by John von Neumann and Oskar Morgenstern in their 1944 book The Theory of Games and Economic Behavior.
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The discipline of Interpreting Studies is often referred to as the sister of Translation Studies. This is due to the similarities between the two disciplines, consisting in the transfer of ideas from one language into another. Indeed, interpreting as an activity was long seen as a specialized form of translation, before scientifically founded Interpreting Studies emancipated gradually from Translation Studies in the second half of the 20th century. While they were strongly oriented towards the theoretic framework of Translation Studies, Interpreting Studies have always been concentrating on the practical and pedagogical aspect of the activity.
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Another commonly used algorithm for finding communities is the Girvan–Newman algorithm. This algorithm identifies edges in a network that lie between communities and then removes them, leaving behind just the communities themselves. The identification is performed by employing the graph-theoretic measure betweenness centrality, which assigns a number to each edge which is large if the edge lies "between" many pairs of nodes. The Girvan–Newman algorithm returns results of reasonable quality and is popular because it has been implemented in a number of standard software packages.
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In mathematics, specifically ring theory, a principal ideal is an ideal I in a ring R that is generated by a single element a of R through multiplication by every element of R. The term also has another, similar meaning in order theory, where it refers to an (order) ideal in a poset P generated by a single element x \in P, which is to say the set of all elements less than or equal to x in P. The remainder of this article addresses the ring-theoretic concept.
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Homomorphisms are also used in the study of formal languagesSeymour Ginsburg, Algebraic and automata theoretic properties of formal languages, North-Holland, 1975, , and are often briefly referred to as morphisms.T. Harju, J. Karhumӓki, Morphisms in Handbook of Formal Languages, Volume I, edited by G. Rozenberg, A. Salomaa, Springer, 1997, . Given alphabets Σ1 and Σ2, a function such that for all u and v in Σ1∗ is called a homomorphism on Σ1∗.The ∗ denotes the Kleene star operation, while Σ∗ denotes the set of words formed from the alphabet Σ, including the empty word.
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Kantor originally named his system Organismic Psychology, but around the time of the publication of the first volume of his Principles of Psychology (Kantor, 1924), he had already renamed it to interbehavioral psychology. Interbehaviorism as developed by Kantor was characterized as "field-theoretic, not lineal-mechanistic, self-actional, or mediational; a system that is naturalistic, not dualistic; and a system that is comprehensive, not narrowly focused." (Midgley & Morris, 2006). At the University of Chicago, Kantor was heavily influenced by the pragmatism and functionalism of Dewey (who had retired earlier from the University), Angell, and Mead.
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Crossover is a particular manifestation of binding, which is one of the most explored and discussed areas of theoretical syntax. The factors that determine when the coreferential reading is possible have been extensively debated. Simple linear order plays a role, but the other key factor might be c-command as associated (primarily) with government and binding, or it might be o-command as associated with head-driven phrase structure grammar. One such analysis (scope-theoretic) of the determining factors for a coreferential reading is outlined by Ruys (2000).
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Here we see that we drop metacompactness from Traylor's theorem, but at the cost of a set- theoretic assumption. Another example of this is Fleissner's theorem that the axiom of constructibility implies that locally compact, normal Moore spaces are metrizable. On the other hand, under the Continuum hypothesis (CH) and also under Martin's Axiom and not CH, there are several examples of non- metrizable normal Moore spaces. Nyikos proved that, under the so-called PMEA (Product Measure Extension Axiom), which needs a large cardinal, all normal Moore spaces are metrizable.
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Sommer was born in 1949 in Haifa, Israel, the son of Jewish parents. His mother had fled Bessarabia (now Moldavia) in 1943 and his father had emigrated from Würzburg to Palestine in 1935. After his parents separated, Sommer moved with his mother and her new partner to Vienna, where he adopted the name of his stepfather, studying mathematics at the University of Vienna and was awarded a doctorate at the age of 21 in 1971 with the title "Limit theorems on the entropy of number-theoretic transformations." Sommer's PhD supervisor was Fritz Schweiger.
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In information theoretic cryptography physical assumptions appear, which do not rely on any hardness assumptions, but merely assume a limit on some other resource. In classical cryptography, the bounded-storage model introduced by Ueli Maurer assumes that the adversary can only store a certain number of classical bits. Protocols are known that do (in principle) allow the secure implementation of any cryptographic task as long as the adversary's storage is small. Very intuitively, security becomes possible under this assumption since the adversary has to make a choice which information to keep.
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The Kripke–Platek set theory with urelements (KPU) is an axiom system for set theory with urelements, based on the traditional (urelement-free) Kripke–Platek set theory. It is considerably weaker than the (relatively) familiar system ZFU. The purpose of allowing urelements is to allow large or high-complexity objects (such as the set of all reals) to be included in the theory's transitive models without disrupting the usual well-ordering and recursion-theoretic properties of the constructible universe; KP is so weak that this is hard to do by traditional means.
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NBG is finitely axiomatizable, while ZFC and MK are not. A key theorem of NBG is the class existence theorem, which states that for every formula whose quantifiers range only over sets, there is a class consisting of the sets satisfying the formula. This class is built by mirroring the step-by-step construction of the formula with classes. Since all set-theoretic formulas are constructed from two kinds of atomic formulas (membership and equality) and finitely many logical symbols, only finitely many axioms are needed to build the classes satisfying them.
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The economics of marriage includes the economic analysis of household formation and break up, of production and distribution decisions within the household. It is closely related to the law and economics of marriages and households. Grossbard-Shechtman (1999a and 1999b) identifies three approaches to the subject: the Marxist approach (Friedrich Engels (1884) and Himmelweit and Mohun (1977)), the neo-classical approach (Gary Becker (1974)) and the game theoretic approaches (Marilyn Manser, Murray Brown, Marjorie McElroy and Mary Jane Horney).Grossbard-Shechtman, Shoshana (1999)a "Marriage" in Encyclopedia of Political Economy, edited by Phillip O'Hara.
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Anderson’s nonstandard construction of Brownian motion is a single object which, when viewed from a nonstandard perspective, has all the formal properties of a discrete random walk; however, when viewed from a measure-theoretic perspective, it is a standard Brownian motion. This permits a pathwise definition of the Itô Integral and pathwise solutions of stochastic differential equations. Anderson’s contributions to mathematical economics are primarily within General Equilibrium Theory. Some of this work uses nonstandard analysis, but much of it provides simple elementary treatments that generalize work that had originally been done using sophisticated mathematical machinery.
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The Schlegel diagram of a convex polyhedron represents its vertices and edges as points and line segments in the Euclidean plane, forming a subdivision of an outer convex polygon into smaller convex polygons (a convex drawing of the graph of the polyhedron). It has no crossings, so every polyhedral graph is also a planar graph. Additionally, by Balinski's theorem, it is a 3-vertex-connected graph. According to Steinitz's theorem, these two graph-theoretic properties are enough to completely characterize the polyhedral graphs: they are exactly the 3-vertex-connected planar graphs.
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In quantum mechanics, explicit descriptions of the representations of SO(3) are very important for calculations, and almost all the work has been done using Euler angles. In the early history of quantum mechanics, when physicists and chemists had a sharply negative reaction towards abstract group theoretic methods (called the Gruppenpest), reliance on Euler angles was also essential for basic theoretical work. Many mobile computing devices contain accelerometers which can determine these devices' Euler angles with respect to the earth's gravitational attraction. These are used in applications such as games, bubble level simulations, and kaleidoscopes.
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Lattice theoretic information about the lattice of subgroups can sometimes be used to infer information about the original group, an idea that goes back to the work of . For instance, as Ore proved, a group is locally cyclic if and only if its lattice of subgroups is distributive. If additionally the lattice satisfies the ascending chain condition, then the group is cyclic. The groups whose lattice of subgroups is a complemented lattice are called complemented groups , and the groups whose lattice of subgroups are modular lattices are called Iwasawa groups or modular groups .
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Samuel Standfield Wagstaff Jr. (born 21 February 1945) is an American mathematician and computer scientist, whose research interests are in the areas of cryptography, parallel computation, and analysis of algorithms, especially number theoretic algorithms. He is currently a professor of computer science and mathematics at Purdue UniversityPurdue University: Department of Computer Science: Faculty: Samuel Wagstaff Jr. who coordinates the Cunningham project, a project to factor numbers of the form bn ± 1, since 1983. He has authored/coauthored over 50 research papers and three books.Selected Publications of Sam Wagstaff He has an Erdős number of 1.
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The methods of noncommutative algebraic geometry are analogs of the methods of commutative algebraic geometry, but frequently the foundations are different. Local behavior in commutative algebraic geometry is captured by commutative algebra and especially the study of local rings. These do not have a ring-theoretic analogue in the noncommutative setting; though in a categorical setup one can talk about stacks of local categories of quasicoherent sheaves over noncommutative spectra. Global properties such as those arising from homological algebra and K-theory more frequently carry over to the noncommutative setting.
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In mathematics and logic, a higher-order logic is a form of predicate logic that is distinguished from first-order logic by additional quantifiers and, sometimes, stronger semantics. Higher-order logics with their standard semantics are more expressive, but their model-theoretic properties are less well-behaved than those of first-order logic. The term "higher-order logic", abbreviated as HOL, is commonly used to mean higher-order simple predicate logic. Here "simple" indicates that the underlying type theory is the theory of simple types, also called the simple theory of types (see Type theory).
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In anthropology, anthropopoiesis is the self-building process of social man and of a whole culture, particularly referred to what concerns modifications of socialized body. The concept found applications mainly in French and Italian contemporary literatures. In accordance with theoretic background which supports the idea, man is an unfinished being or better his behaviour is not strongly prefixed by genetic heritage. Human beings become fully finished only by means of culture acquisition. Anthropopoiesis is both anthropogenesis (man “reborn” as social creature) and manufacturing of ”mankind patterns and fictions”.
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Rojas has received numerous awards for his research work and publications. In 2002, he co-wrote an article with Kirby D. Schroeder titled "A Game Theoretic Model of Sexually Transmitted Disease Epidemics" and this won the 2003 Outstanding Graduate Student Paper Award by the ASA in Mathematical Sociology. His book Party in the Streets was selected as a Choice Top 25 Outstanding Academic Book for 2015 by the American Library Association and also received the APSA Political Organizations and Parties Section’s Leon Epstein Outstanding Book Award in 2016.
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In his 1995 book, Symmetric Bends: How to Join Two Lengths of Cord, Miles presents a knot theoretic analysis of 60 symmetric bends. The Vice Versa Bend appears as number 19 in this sequence. Miles attributes the knot to Asher and describes it as a 'pure lanyard bend' in which "two ends of equal status emerge from the knot in each of two opposite directions". Budworth, a founding member of the International Guild of Knot Tyers, includes the Vice Versa Bend in his 2000 book The Book of Practical Knots.
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This property is shared only by the diamond crystal (the strong isotropy should not be confused with the edge-transitivity or the notion of symmetric graph; for instance, the primitive cubic lattice is a symmetric graph, but not strongly isotropic). The K4 crystal and the diamond crystal as networks in space are examples of “standard realizations”, the notion introduced by Sunada and Motoko Kotani as a graph-theoretic version of Albanese maps (Abel-Jacobi maps) in algebraic geometry. For his work, see also Isospectral, Reinhardt domain, Ihara zeta function, Ramanujan graph, quantum ergodicity, quantum walk.
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Any satisficing problem can be formulated as an optimization problem. To see that this is so, let the objective function of the optimization problem be the indicator function of the constraints pertaining to the satisficing problem. Thus, if our concern is to identify a worst-case scenario pertaining to a constraint, this can be done via a suitable Maximin/Minimax worst-case analysis of the indicator function of the constraint. This means that the generic decision theoretic models can handle outcomes that are induced by constraint satisficing requirements rather than by say payoff maximization.
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The intersection type discipline was pioneered by Mario Coppo, Mariangiola Dezani-Ciancaglini, Patrick Sallé, and Garrel Pottinger. The underlying motivation was to study semantic properties (such as normalization) of the λ-calculus by means of type theory. While the initial work by Coppo and Dezani established a type theoretic characterization of strong normalization for the λI-calculus, Pottinger extended this characterization to the λK-calculus. In addition, Sallé contributed the notion of the universal type \omega that can be assigned to any λ-term, thereby corresponding to the empty intersection.
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In that case, the quotient :A/K(L) is isomorphic to the dual abelian variety Â. This construction of  extends to any field K of characteristic zero.Mumford, Abelian Varieties, pp.74-80 In terms of this definition, the Poincaré bundle, a universal line bundle can be defined on :A × Â. The construction when K has characteristic p uses scheme theory. The definition of K(L) has to be in terms of a group scheme that is a scheme- theoretic stabilizer, and the quotient taken is now a quotient by a subgroup scheme.
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Haikala (1956) claims that cobweb theorem is a theorem of deceiving farmers, thus seeing cobweb theorem as a kind of rational or rather, consistent, expectations model with a game-theoretic feature. This makes sense when considering the argument of Hans-Peter Martin and Harald Schumann. The truth-value of a prediction is one measure in differentiating between non-deceiving and deceiving models. In Martin and Schumann's context, a claim that anti-Keynesian policies lead to a greater welfare of the majority of mankind should be analyzed in terms of truth.
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See Katsuhito Iwai is often cited for his formulation of a bootstrap theory of money. Everybody uses money as money, Iwai writes, because everybody else uses money as money. He has offered proof, in his search- theoretic model of decentralized exchanges, that, to sustain itself as an equilibrium, the monetary system requires no "real" conditions. Iwai is credited with constructing mathematical models of Schumpeterian evolutionary processes that describe how large numbers of firms interact with one another, by competing to innovate, trying to imitate and struggling to grow.
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First-price sealed-bid auctions are when a single bid is made by all bidding parties and the single highest bidder wins, and pays what they bid. The main difference between this and English auctions is that bids are not openly viewable or announced as opposed to the competitive nature which is generated by public bids. From the game-theoretic point of view, the first-price sealed-bid auction is strategically equivalent to the Dutch auction; that is, in both auctions the players will be using the same bidding strategies.
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Polytopes may exist in any general number of dimensions n as an n-dimensional polytope or n-polytope. Flat sides mean that the sides of a (k+1)-polytope consist of k-polytopes that may have (k−1)-polytopes in common. For example, a two- dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. Some theories further generalize the idea to include such objects as unbounded apeirotopes and tessellations, decompositions or tilings of curved manifolds including spherical polyhedra, and set-theoretic abstract polytopes.
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In mathematics, a Meyer set or almost lattice is a set relatively dense X of points in the Euclidean plane or a higher-dimensional Euclidean space such that its Minkowski difference with itself is uniformly discrete. Meyer sets have several equivalent characterizations; they are named after Yves Meyer, who introduced and studied them in the context of diophantine approximation. Nowadays Meyer sets are best known as mathematical model for quasicrystals. However, Meyer's work precedes the discovery of quasicrystals by more than a decade and was entirely motivated by number theoretic questions. ..
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These conflicts of interest make bargaining a necessary fact of household life and create a household environment that is not universally governed by altruism. These conflicts of interest have the potential to create a spectrum of intra-household dynamics, ranging from a non-cooperative to a cooperative household (which is directly reflective of game theoretic bargaining models). In the non-cooperative model, each household member acts in order to maximize his or her own utility; in the cooperative model, households act as a unit to "maximize the welfare of their members" (described above as altruism).
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Arnold Zellner (January 2, 1927 – August 11, 2010) was an American economist and statistician specializing in the fields of Bayesian probability and econometrics. Zellner contributed pioneering work in the field of Bayesian analysis and econometric modeling. In Bayesian analysis, Zellner not only provided many applications of it but also a new information-theoretic derivation of rules that are 100% efficient information processing rules — this class includes Bayes's theorem. In econometric modeling, he, in association with Franz Palm, developed the structural time-series approach for constructing new models and for checking the adequacy of old models.
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The field is at the intersection of mathematics, statistics, computer science, physics, neurobiology, and electrical engineering. Its impact has been crucial to the success of the Voyager missions to deep space, the invention of the compact disc, the feasibility of mobile phones, the development of the Internet, the study of linguistics and of human perception, the understanding of black holes, and numerous other fields. Important sub-fields of information theory are source coding, channel coding, algorithmic complexity theory, algorithmic information theory, information-theoretic security, and measures of information.
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The Shannon information is closely related to information theoretic entropy, which is the expected value of the self-information of a random variable, quantifying how surprising the random variable is "on average." This is the average amount of self-information an observer would expect to gain about a random variable when measuring it.Jones, D.S., Elementary Information Theory, Vol., Clarendon Press, Oxford pp 11-15 1979 The information content can be expressed in various units of information, of which the most common is the "bit" (sometimes also called the "shannon"), as explained below.
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An information-theoretic framework for biomarker discovery, integrating biofluid and tissue information, has been introduced; this approach takes advantage of functional synergy between certain biofluids and tissues, with the potential for clinically significant findings (not possible if tissues and biofluids were considered separately). By conceptualizing tissue biofluids as information channels, significant biofluid proxies were identified and then used for guided development of clinical diagnostics. Candidate biomarkers were then predicted, based on information-transfer criteria across the tissue-biofluid channels. Significant biofluid-tissue relationships can be used to prioritize the clinical validation of biomarkers.
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This problem was formulated in 1891 by Édouard Lucas and independently, a few years earlier, by Peter Guthrie Tait in connection with knot theory.. For a number of couples equal to 3, 4, 5, ... the number of seating arrangements is :12, 96, 3120, 115200, 5836320, 382072320, 31488549120, ... . Mathematicians have developed formulas and recurrence equations for computing these numbers and related sequences of numbers. Along with their applications to etiquette and knot theory, these numbers also have a graph theoretic interpretation: they count the numbers of matchings and Hamiltonian cycles in certain families of graphs.
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Crown graphs with six, eight, and ten vertices. The outer cycle of each graph forms a Hamiltonian cycle; the eight and ten-vertex graphs also have other Hamiltonian cycles. Solutions to the ménage problem may be interpreted in graph-theoretic terms, as directed Hamiltonian cycles in crown graphs. A crown graph is formed by removing a perfect matching from a complete bipartite graph Kn,n; it has 2n vertices of two colors, and each vertex of one color is connected to all but one of the vertices of the other color.
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In a totally ordered set, the terms maximal element and greatest element coincide, which is why both terms are used interchangeably in fields like analysis where only total orders are considered. This observation applies not only to totally ordered subsets of any poset, but also to their order theoretic generalization via directed sets. In a directed set, every pair of elements (particularly pairs of incomparable elements) has a common upper bound within the set. If a directed set has a maximal element, it is also its greatest element,Let m \in D be maximal.
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It also offers LTE- Advanced in limited locations on the 1800 MHz band, whereas the majority of KPN's 4G network operates in the 8/900 MHz bands, which will allow theoretic download speeds up to 200Mbit/sec. The 3G network is to be shut down in January 2022. In September 2019, KPN announced that Dominique Leroy will succeed Maximo Ibarra as CEO and Chairman of the Board of Management, with effect from December 1, 2019. Several weeks later in the same month, KPN announced withdrawal of appointment of Dominique Leroy as CEO.
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In earlier work, several computer scientists had advanced using category theory to provide semantics for the lambda calculus. Moggi's key insight was that a real-world program is not just a function from values to other values, but rather a transformation that forms computations on those values. When formalized in category-theoretic terms, this leads to the conclusion that monads are the structure to represent these computations. Several others popularized and built on this idea, including Philip Wadler and Simon Peyton Jones, both of whom were involved in the specification of Haskell.
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Wright, together with co-author Nobuhiro Kiyotaki, pioneered the use of search theory in monetary economics. The application of search theory to macroeconomics would later be known as matching theory. Search-theoretic models of monetary exchange are based on explicit descriptions of the frictions that make money essential, which contrasts with earlier reduced form approaches to money in macroeconomics, such as putting money in the utility function or imposing cash-in-advance constraints. These earlier ways of modeling money's role did not show explicitly how it helps overcome informational, spatial, or temporal frictions.
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Technicians prepare a body for cryopreservation in 1985. Cryonics (from Greek κρύος 'kryos-' meaning 'icy cold') is the low-temperature preservation of animals and humans who cannot be sustained by contemporary medicine, with the hope that healing and resuscitation may be possible in the future. Cryopreservation of people or large animals is not reversible with current technology. The stated rationale for cryonics is that people who are considered dead by current legal or medical definitions may not necessarily be dead according to the more stringent information-theoretic definition of death.
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An order-theoretic lattice gives rise to the two binary operations ∨ and ∧. Since the commutative, associative and absorption laws can easily be verified for these operations, they make into a lattice in the algebraic sense. The converse is also true. Given an algebraically defined lattice , one can define a partial order ≤ on L by setting : if , or : if , for all elements a and b from L. The laws of absorption ensure that both definitions are equivalent: a = a ∧ b implies b = b ∨ (b ∧ a) = (a ∧ b) ∨ b = a ∨ b and dually for the other direction.
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When lattices with more structure are considered, the morphisms should "respect" the extra structure, too. In particular, a bounded-lattice homomorphism (usually called just "lattice homomorphism") f between two bounded lattices L and M should also have the following property: : f(0L) = 0M , and : f(1L) = 1M . In the order-theoretic formulation, these conditions just state that a homomorphism of lattices is a function preserving binary meets and joins. For bounded lattices, preservation of least and greatest elements is just preservation of join and meet of the empty set.
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An illustration of the selection principle S1(A,B) In mathematics, a selection principle is a rule asserting the possibility of obtaining mathematically significant objects by selecting elements from given sequences of sets. The theory of selection principles studies these principles and their relations to other mathematical properties. Selection principles mainly describe covering properties, measure- and category-theoretic properties, and local properties in topological spaces, especially function spaces. Often, the characterization of a mathematical property using a selection principle is a nontrivial task leading to new insights on the characterized property.
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In a 2012 article in Psychological Bulletin it is suggested the subadditivity effect can be explained by an information- theoretic generative mechanism that assumes a noisy conversion of objective evidence (observation) into subjective estimates (judgment). This explanation is different than support theory, proposed as an explanation by Tversky and Koehler, which requires additional assumptions. Since mental noise is a sufficient explanation that is much simpler and straightforward than any explanation involving heuristics or behavior, Occam's razor would argue in its favor as the underlying generative mechanism (it is the hypotheses which makes the fewest assumptions).
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Their global dimension coincides with the Krull dimension, whose definition is module-theoretic. When the ring A is noncommutative, one initially has to consider two versions of this notion, right global dimension that arises from consideration of the right A-modules, and left global dimension that arises from consideration of the left A-modules. For an arbitrary ring A the right and left global dimensions may differ. However, if A is a Noetherian ring, both of these dimensions turn out to be equal to weak global dimension, whose definition is left-right symmetric.
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Commutative diagram for the set product X1×X2. A category-theoretic product A × B in a category of sets represents the set of ordered pairs, with the first element coming from A and the second coming from B. In this context the characteristic property above is a consequence of the universal property of the product and the fact that elements of a set X can be identified with morphisms from 1 (a one element set) to X. While different objects may have the universal property, they are all naturally isomorphic.
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In category theory, a small set is one in a fixed universe of sets (as the word universe is used in mathematics in general). Thus, the category of small sets is the category of all sets one cares to consider. This is used when one does not wish to bother with set-theoretic concerns of what is and what is not considered a set, which concerns would arise if one tried to speak of the category of "all sets". A category C is called small if both the collection of objects and arrows are sets.
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In 2012, the results of the full dataset (collected 2000-2010) of RICE (RICE/NARC) were published and the RICE (RICE/NARC) experiment was described as "presently at the end of useful data-taking." No ultra-high energy (UHE) neutrinos were detected; this is in accordance with theoretic expectation. The radio Cherenkov technique of detecting neutrinos is continued by RICE's successor experiment, Askaryan Radio Array (ARA), to which RICE hardware (and some of the researchers) was transferred. ARA is also deployed at the South Pole Station under ice.
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Shermer still cycles actively, and participated in the Furnace Creek 508 in October 2011, a qualifying race for RAAM, finishing second in the four man team category. Shermer has written on the subject of pervasive doping in competitive cycling and a game theoretic view of the dynamics driving the problem in several sports. He wrote specifically about r-EPO doping, which he saw as both widespread and well known within the sport, which was later shown to be instrumental in the doping scandal surrounding Lance Armstrong in 2010.
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Jacques Lacan was an intellectual who defended obscurantism to a degree. To his students' complaint about the deliberate obscurity of his lectures, he replied: "The less you understand, the better you listen." In the 1973 seminar Encore, he said that his Écrits (Writings) were not to be understood, but would effect a meaning in the reader, like that induced by mystical texts. The obscurity is not in his writing style, but in the repeated allusions to Hegel, derived from Alexandre Kojève's lectures on Hegel, and similar theoretic divergences.
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Severini works in quantum information science and complex systems. Together with Adan Cabello and Andreas Winter, he defined a graph-theoretic framework for studying quantum contextuality, and together with Tomasz Konopka, Fotini Markopoulou, and Lee Smolin, he introduced a random graph model of spacetime called quantum graphity.Roberto Mangabeira Unger, Lee Smolin, The Singular Universe and the Reality of Time: A Proposal in Natural Philosophy, Cambridge University Press (2014).Shyam Wuppuluri and Giancarlo Ghirardi (Eds.), Space, Time and the Limits of Human Understanding (Foreword by John Stachel and Afterword by Noam Chomsky), Springer (2017).
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Chemical graph theory concerns the graph-theoretic structure of molecules and other clusters of atoms. Both the Errera graph itself and its dual graph are relevant in this context. Atoms of metals such as gold can form clusters in which a central atom is surrounded by twelve more atoms, in the pattern of an icosahedron. Another, larger, type of cluster can be formed by coalescing two of these icosahedral clusters, so that the central atom of each cluster becomes one of the boundary atoms for the other cluster.
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To assume that only one of the relata of a relation could cause that relation is as silly as assuming that the female (or the male) member of a marriage causes the marriage. Nor is the proper relation between postulation and intuition "identity", as can easily been seen using "blue". Concept-by-postulation blue is not identical with concept-by-intuition blue, but is just one among many relata that go to form this complex secondary quality. Neither identity nor causality is the proper relation between sensed blue and theoretic blue.
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These are the graphs K such that a product G × H has a homomorphism to K only when one of G or H also does. Identifying multiplicative graphs lies at the heart of Hedetniemi's conjecture.. Graph homomorphisms also form a category, with graphs as objects and homomorphisms as arrows. The initial object is the empty graph, while the terminal object is the graph with one vertex and one loop at that vertex. The tensor product of graphs is the category-theoretic product and the exponential graph is the exponential object for this category.
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A strangulated graph, formed by using clique-sums to glue together a maximal planar graph (yellow) and two chordal graphs (red and blue). The red chordal graph can in turn be decomposed into clique-sums of four maximal planar graphs (two edges and two triangles). In graph theoretic mathematics, a strangulated graph is a graph in which deleting the edges of any induced cycle of length greater than three would disconnect the remaining graph. That is, they are the graphs in which every peripheral cycle is a triangle.
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Otto Rank behind Sigmund Freud, and other psychoanalysts (1922). In classical Freudian psychology the super-ego, "the heir to the Oedipus complex", is formed as the infant boy internalizes the familial rules of his father. In contrast, in the early 1920s, using the term "pre-Oedipal", Otto Rank proposed that a boy's powerful mother was the source of the super-ego, in the course of normal psychosexual development. Rank's theoretic conflict with Freud excluded him from the Freudian inner circle; nonetheless, he later developed the psychodynamic Object relations theory in 1925.
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From 1909, he studied under musical theoretic Boleslav Yavorsky, whom he periodically visited in Moscow and Kyiv over the next twelve years. Leontovych also became involved with theatrical music in Tulchyn and its community life by taking charge of a local organisation called Prosvita, meaning "enlightenment". This period in his career was among the most productive, as he created numerous choral arrangements. These included his famous Shchedryk, as well as Піють півні (The Roosters are Singing), ' (A Mother had One Daughter), ' (Little Dudka Player), ' (Oh, the Star has Risen), amongst others.
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Set-theoretic, algebraic and topological operations on multivalued maps (like union, composition, sum, convex hull, closure) usually preserve the type of continuity. But this should be taken with appropriate care since, for example, there exists a pair of lower hemicontinuous correspondences whose intersection is not lower hemicontinuous. This can be fixed upon strengthening continuity properties: if one of those lower hemicontinuous multifunctions has open graph then their intersection is again lower hemicontinuous. Crucial to set-valued analysis (in view of applications) are the investigation of single-valued selections and approximations to multivalued maps.
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In geometry, a dissection problem is the problem of partitioning a geometric figure (such as a polytope or ball) into smaller pieces that may be rearranged into a new figure of equal content. In this context, the partitioning is called simply a dissection (of one polytope into another). It is usually required that the dissection use only a finite number of pieces. Additionally, to avoid set-theoretic issues related to the Banach–Tarski paradox and Tarski's circle-squaring problem, the pieces are typically required to be well-behaved.
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In 2004 Kühn published a pair of papers in Combinatorica with her thesis advisor, Reinhard Diestel, concerning the cycle spaces of infinite graphs. In these graphs the appropriate generalizations of cycles and spanning trees hinge on a proper treatment of the ends of the graph. Reviewer R. Bruce Richter writes that "the results are extremely satisfactory, in the sense that standard theorems for finite graphs have perfect analogues" but that "there is nothing simple about any aspect of this work. It is a nice mix of graph-theoretic and topological ideas.".
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Frontpage of the publication: "Farbendruck und Farbenphotographie". Published 1908 in Leipzig. Emanuel Goldberg (; ; ) (born: 31 August 1881; died: 13 September 1970) was an Israeli physicist and inventor. He was born in Moscow and moved first to Germany and later to Israel. He described himself as “a chemist by learning, physicist by calling, and a mechanic by birth.” He contributed a wide range of theoretic and practical advances relating to light and media and was the founding head of Zeiss Ikon, the famous photographic products company in Dresden, Germany.
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Gal, Mash, Procaccia and Zick, based on their experience with the rent division application in the Spliddit website, note that envy-freeness alone is insufficient to guarantee the satisfaction of the participants. Therefore they build an algorithmic framework, based on linear programming, for calculating allocations that are both envy-free and optimize some criterion. Based on theoretic and experimental tests, they conclude that the maximin criterion - maximizing the minimum utility of an agent subject to envy-freeness - attains optimal results. Note that, since their solution is always EF, it might return negative prices.
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The origins of communication theory is linked to the development of information theory in the early 1920s. Limited information-theoretic ideas had been developed at Bell Labs, all implicitly assuming events of equal probability. Harry Nyquist's 1924 paper, Certain Factors Affecting Telegraph Speed, contains a theoretical section quantifying "intelligence" and the "line speed" at which it can be transmitted by a communication system. Ralph Hartley's 1928 paper, Transmission of Information, uses the word "information" as a measurable quantity, reflecting the receiver's ability to distinguish one sequence of symbols from any other.
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Sachdev developed the theory of magneto-thermoelectric transport in 'strange' metals: these are states of quantum matter with variable density without quasiparticle excitations. Such metals are found, most famously, near optimal doping in the hole-doped cuprates, but also appear in numerous other correlated electron compounds. For strange metals in which momentum is approximately conserved, a set of hydrodynamic equations were proposed in 2007, describing two-component transport with momentum drag component and a quantum-critical conductivity. This formulation was connected to the holography of charged black holes, memory functions, and new field-theoretic approaches.
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This use of > legal and enforcement presumptions based on economic presumptions arising > from economic theory and evidence and judicial experience has now spread > across all of antitrust. The PNB presumption for mergers also has evolved as > economic theory and evidence have advanced. Looking forward, merger > presumptions should be neither abandoned nor set in stone; instead, they > should be permitted to continue to evolve, based on new or additional > economic factors besides market shares and concentration.Steven C. Salop, > The Evolution and Vitality of Merger Presumptions: A Decision-Theoretic > Approach at 50.
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SN(d_1) is correctly interpreted as the present value, using the risk-free interest rate, of the expected asset price at expiration, given that the asset price at expiration is above the exercise price. For related discussion and graphical representation see section "Interpretation" under Datar–Mathews method for real option valuation. The equivalent martingale probability measure is also called the risk-neutral probability measure. Note that both of these are probabilities in a measure theoretic sense, and neither of these is the true probability of expiring in-the-money under the real probability measure.
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Information theory leads us to believe it is much more difficult to keep secrets than it might first appear. A brute force attack can break systems based on asymmetric key algorithms or on most commonly used methods of symmetric key algorithms (sometimes called secret key algorithms), such as block ciphers. The security of all such methods currently comes from the assumption that no known attack can break them in a practical amount of time. Information theoretic security refers to methods such as the one-time pad that are not vulnerable to such brute force attacks.
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In the early days of the development of K-triviality, attention was paid to separation of K-trivial sets and computable sets. Chaitin in his 1976 paper Gregory J. Chaitin (1976), "Information-Theoretic Characterizations of Recursive Infinite Strings", Theoretical Computer Science Volume 2, Issue 1, June 1976, Pages 45–48 mainly studied sets such that there exists b ∈ℕ with : \forall n C(A\upharpoonright n)\leq C(n)+b where C denotes the plain Kolmogorov complexity. These sets are known as C-trivial sets. Chaitin showed they coincide with the computable sets.
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Group theory has three main historical sources: number theory, the theory of algebraic equations, and geometry. The number- theoretic strand was begun by Leonhard Euler, and developed by Gauss's work on modular arithmetic and additive and multiplicative groups related to quadratic fields. Early results about permutation groups were obtained by Lagrange, Ruffini, and Abel in their quest for general solutions of polynomial equations of high degree. Évariste Galois coined the term "group" and established a connection, now known as Galois theory, between the nascent theory of groups and field theory.
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In collaboration with Andrei Rapinchuk, Prasad has studied Zariski-dense subgroups of semi-simple groups and proved the existence in such a subgroup of regular semi-simple elements with many desirable properties, [15], [16]. These elements have been used in the investigation of geometric and ergodic theoretic questions. Prasad and Rapinchuk introduced a new notion of "weak-commensurability" of arithmetic subgroups and determined "weak- commensurability classes" of arithmetic groups in a given semi-simple group. They used their results on weak-commensurability to obtain results on length-commensurable and isospectral arithmetic locally symmetric spaces, see [17], [18] and [19].
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Modal logics include additional modal operators, such as an operator which states that a particular formula is not only true, but necessarily true. Although modal logic is not often used to axiomatize mathematics, it has been used to study the properties of first-order provability (Solovay 1976) and set-theoretic forcing (Hamkins and Löwe 2007). Intuitionistic logic was developed by Heyting to study Brouwer's program of intuitionism, in which Brouwer himself avoided formalization. Intuitionistic logic specifically does not include the law of the excluded middle, which states that each sentence is either true or its negation is true.
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The returns of equity and fixed income markets as well as alpha generating strategies have a natural positive skew that manifests in a smoothed return histogram as a positive slope near zero. Fixed income strategies with a relatively constant positive return ("carry") also exhibit total return series with a naturally positive slope near zero. Cash investments such as 90-day T-Bills have large bias ratios, because they generally do not experience periodic negative returns. Consequently, the bias ratio is less reliable for the theoretic hedge fund that has an un-levered portfolio with a high cash balance.
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Efforts have been made within the field of artificial intelligence to perform and analyze the act of argumentation with computers. Argumentation has been used to provide a proof-theoretic semantics for non-monotonic logic, starting with the influential work of Dung (1995). Computational argumentation systems have found particular application in domains where formal logic and classical decision theory are unable to capture the richness of reasoning, domains such as law and medicine. In Elements of Argumentation, Philippe Besnard and Anthony Hunter show how classical logic-based techniques can be used to capture key elements of practical argumentation.
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His plan was to familiarize the child with the surface of the earth by going from the near to the distant, and from the concrete to the abstract, and this system at once overthrew theoretic geography, and initiated the modern practical and descriptive science. The immediate success of the work led Olney to give up teaching and devote himself to authorship. Leaving Hartford in 1833, he settled in Southington, Connecticut, until 1854, when he moved to Stratford. His text-books (1831–52) included other geographies, a series of readers, a Common School Arithmetic, and a History of the United States.
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As a result of these rules, equirecursive types contribute significantly more complexity to a type system than isorecursive types do. Algorithmic problems such as type checking and type inference are more difficult for equirecursive types as well. Since direct comparison does not make sense on an equirecursive type, they can be converted into a canonical form in O(n log n) time, which can easily be compared. Equirecursive types capture the form of self-referential (or mutually referential) type definitions seen in procedural and object- oriented programming languages, and also arise in type-theoretic semantics of objects and classes.
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But Fermat's crowning achievement was in the theory of numbers." Regarding Fermat's work in analysis, Isaac Newton wrote that his own early ideas about calculus came directly from "Fermat's way of drawing tangents." Of Fermat's number theoretic work, the 20th-century mathematician André Weil wrote that: "what we possess of his methods for dealing with curves of genus 1 is remarkably coherent; it is still the foundation for the modern theory of such curves. It naturally falls into two parts; the first one ... may conveniently be termed a method of ascent, in contrast with the descent which is rightly regarded as Fermat's own.
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The method tests the hypothesis that, given some initial setting and parameter values, a certain network structure will emerge as an equilibrium of this game. Since the number of nodes usually fixed, they can very rarely explain the properties of huge real-world networks; however, they are very useful to examine the network formation in smaller groups. Jackson and Wolinsky pioneered these types of models in a 1996 paper, which has since inspired several game- theoretic models. These models were further developed by Jackson and Watts, who put this approach to a dynamic setting to see how the network structure evolve over time.
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Scripta Mathematica was a quarterly journal published by Yeshiva University devoted to the philosophy, history, and expository treatment of mathematics.. It was said to be, at its time, "the only mathematical magazine in the world edited by specialists for laymen.". The journal was established in 1932 under the editorship of Jekuthiel Ginsburg, a professor of mathematics at Yeshiva University,. and its first issue appeared in 1933 at a subscription price of three dollars per year.. It ceased publication in 1973. Notable papers published in Scripta Mathematica included work by Nobelist Percy Williams Bridgman concerning the implications for physics of set-theoretic paradoxes,.
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Haddad's key work on impulsive and hybrid dynamical systems and control include. His book on Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control, Princeton, NJ: Princeton University Press, 2006, provides a highly detailed, general analysis and synthesis framework for impulsive and hybrid dynamical systems. In particular, this research monograph develops fundamental results on stability, dissipativity theory, energy-based hybrid control, optimal control, disturbance rejection control, and robust control for nonlinear impulsive and hybrid dynamical systems. The monograph is written from a system-theoretic point of view and provides a fundamental contribution to mathematical system theory and control system theory.
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This led to an increasing interest in gauge theory for its own sake, independent of its successes in fundamental physics. In 1994, Edward Witten and Nathan Seiberg invented gauge-theoretic techniques based on supersymmetry that enabled the calculation of certain topological invariants; (the Seiberg–Witten invariants). These contributions to mathematics from gauge theory have led to a renewed interest in this area. The importance of gauge theories in physics is exemplified in the tremendous success of the mathematical formalism in providing a unified framework to describe the quantum field theories of electromagnetism, the weak force and the strong force.
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Graph-theoretic methods, in various forms, have proven particularly useful in linguistics, since natural language often lends itself well to discrete structure. Traditionally, syntax and compositional semantics follow tree-based structures, whose expressive power lies in the principle of compositionality, modeled in a hierarchical graph. More contemporary approaches such as head- driven phrase structure grammar model the syntax of natural language using typed feature structures, which are directed acyclic graphs. Within lexical semantics, especially as applied to computers, modeling word meaning is easier when a given word is understood in terms of related words; semantic networks are therefore important in computational linguistics.
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Representations of groups are important because they allow many group-theoretic problems to be reduced to problems in linear algebra, which is well understood. They are also important in physics because, for example, they describe how the symmetry group of a physical system affects the solutions of equations describing that system. The term representation of a group is also used in a more general sense to mean any "description" of a group as a group of transformations of some mathematical object. More formally, a "representation" means a homomorphism from the group to the automorphism group of an object.
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Mathematically, Bass–Serre theory builds on exploiting and generalizing the properties of two older group-theoretic constructions: free product with amalgamation and HNN extension. However, unlike the traditional algebraic study of these two constructions, Bass–Serre theory uses the geometric language of covering theory and fundamental groups. Graphs of groups, which are the basic objects of Bass–Serre theory, can be viewed as one-dimensional versions of orbifolds. Apart from Serre's book, the basic treatment of Bass–Serre theory is available in the article of Bass, the article of G. Peter Scott and C. T. C. WallPeter Scott and Terry Wall.
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Kennedy's research at the University of Helsinki focuses on mathematical logic in the area of set-theoretic model theory and set theory. In the course of her mathematical work she also researches the history of mathematics and the foundations of mathematics. In this context she has sustained an extensive project to place the works of Kurt Gödel in its historical and foundational context. In 2017 she published her research on the interplay between the works of Alan Turing and that of Gödel, who in 1956 defined the P versus NP problem in a letter to John von Neumann.
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A game-theoretic explanation for democratic peace is that public and open debate in democracies sends clear and reliable information regarding their intentions to other states. In contrast, it is difficult to know the intentions of nondemocratic leaders, what effect concessions will have, and if promises will be kept. Thus there will be mistrust and unwillingness to make concessions if at least one of the parties in a dispute is a non-democracy. On the other hand, game theory predicts that two countries may still go to war even if their leaders are cognizant of the costs of fighting.
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Separately, game theory has played a role in online algorithms; in particular, the -server problem, which has in the past been referred to as games with moving costs and request- answer games. Yao's principle is a game-theoretic technique for proving lower bounds on the computational complexity of randomized algorithms, especially online algorithms. The emergence of the Internet has motivated the development of algorithms for finding equilibria in games, markets, computational auctions, peer-to-peer systems, and security and information markets. Algorithmic game theory and within it algorithmic mechanism design combine computational algorithm design and analysis of complex systems with economic theory.
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In mathematics, and especially differential geometry and mathematical physics, gauge theory is the general study of connections on vector bundles, principal bundles, and fibre bundles. Gauge theory in mathematics should not be confused with the closely related concept of a gauge theory in physics, which is a field theory which admits gauge symmetry. In mathematics theory means a mathematical theory, encapsulating the general study of a collection of concepts or phenomena, whereas in the physical sense a gauge theory is a physical model of some natural phenomenon. Gauge theory in mathematics is typically concerned with the study of gauge-theoretic equations.
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For cooperation to emerge between game theoretic rational players, the total number of rounds N must be unknown to the players. In this case "always defect" may no longer be a strictly dominant strategy, only a Nash equilibrium. Amongst results shown by Robert Aumann in a 1959 paper, rational players repeatedly interacting for indefinitely long games can sustain the cooperative outcome. According to a 2019 experimental study in the American Economic Review which tested what strategies real-life subjects used in iterated prisoners' dilemma situations with perfect monitoring, the majority of chosen strategies were always defect, tit-for-tat, and grim trigger.
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For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. The second is that if such a system is also capable of proving certain basic facts about the natural numbers, then the system cannot prove the consistency of the system itself. These two results are known as Gödel's incompleteness theorems, or simply Gödel's Theorem. Later in the decade, Gödel developed the concept of set-theoretic constructibility, as part of his proof that the axiom of choice and the continuum hypothesis are consistent with Zermelo–Fraenkel set theory.
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A computationally viable alternative to full analytic response to Kohn-Sham density functional theoretic (DFT) approach, which solves coupled-perturbed Kohn-Sham (CPKS) procedure in non-iteratively has been formulated by Sourav. In the above procedure, the derivative of KS matrix is obtained using finite field and then the density matrix derivative is obtained by single-step CPKS solution followed by analytic evaluation of properties. He has implemented this in deMON2K software and used for calculation of electric properties. Density functional response approach for the linear and non-linear electric properties of molecules K.B. Sophy and Sourav Pal (2003) J.Chem.Phys.
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It is easy to see that ker f is an equivalence relation on A, and in fact a congruence relation. Thus, it makes sense to speak of the quotient algebra A/(ker f). The first isomorphism theorem in general universal algebra states that this quotient algebra is naturally isomorphic to the image of f (which is a subalgebra of B). Note that the definition of kernel here (as in the monoid example) doesn't depend on the algebraic structure; it is a purely set-theoretic concept. For more on this general concept, outside of abstract algebra, see kernel of a function.
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For every number and every formula , where is a free variable, we define , a relation between two numbers and , such that it corresponds to the statement " is not the Gödel number of a proof of ". Here, can be understood as with its own Gödel number as its argument. Note that takes as an argument , the Gödel number of . In order to prove either , or , it is necessary to perform number-theoretic operations on that mirror the following steps: decode the number into the formula , replace all occurrences of in with the number , and then compute the Gödel number of the resulting formula .
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Toward the delineation of therapeutic change principles. American Psychologist, 35, 991-999. Principles of change are shared by all theoretic orientations of therapy, and include strategies such as: promoting client belief in the effectiveness of therapy, the formation and maintenance of a therapeutic alliance with the client, facilitating client awareness of the factors influencing their problems, and encouraging the client to engage in corrective experiences. Emphasizing ongoing reality testing in the client's life has been demonstrated to be among the principles of change that can be used to explain and account for the underlying effectiveness of therapeutic counseling techniques, regardless of theoretical ideals.
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Bayesian probability has produced a number of algorithms that are in common use in many advanced control systems, serving as state space estimators of some variables that are used in the controller. The Kalman filter and the Particle filter are two examples of popular Bayesian control components. The Bayesian approach to controller design often requires an important effort in deriving the so-called system model and measurement model, which are the mathematical relationships linking the state variables to the sensor measurements available in the controlled system. In this respect, it is very closely linked to the system- theoretic approach to control design.
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This method he applied in like manner to the Zohar, which he, far from all mysticism, considered as a rich source of speculative knowledge. This view referred only to the theoretic or intuitive, and not the practical, Kabbalah, the belief in which he considered as contradictory to sound reason. At the beginning of this book are printed the approbation of Rabbi Moses Münz and a eulogistic Hebrew poem of Rabbi Moses Kunitz. This work gave great offense to the Orthodox party, which thwarted the publication of a second edition, for which Chorin had prepared many corrections and additions.
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Brodie accepted post-doctorate offers first at the Stanford Linear Accelerator Center, and in 2001 at the Perimeter Institute for Theoretical Physics in Waterloo, Ontario, Canada. One of the first postdoctoral researchers at Perimeter institute, Brodie's work was notable for its breadth, ranging from non-perturbative effects in supersymmetric gauge theories to string theoretic descriptions of quantum Hall fluids and of inflationary cosmology.ALUMNI WINS PI'S JOHN BRODIE MEMORIAL AWARD During his short career, he published fifteen articles in peer-review journals, many of which have proven to be influential. In 2002, Brodie had a psychotic episode and was diagnosed with bipolar disorder.
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Currently, there is no commonly accepted evaluation framework or benchmark that would allow for a comparison of the models under a set of representative and common conditions. A game-theoretic approach in this direction has been proposed, where the configuration of a trust model is optimized assuming attackers with optimal attack strategies; this allows in a next step to compare the expected utility of different trust models. Similarly, a model-based analytical framework for predicting the effectiveness of reputation mechanisms against arbitrary attack models in arbitrary system models has been proposed for Peer-to-Peer systems.
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186–187 (text condensed). On 24 October 1994, Wiles submitted two manuscripts, "Modular elliptic curves and Fermat's Last Theorem" and "Ring theoretic properties of certain Hecke algebras", the second of which was co-authored with Taylor and proved that certain conditions were met that were needed to justify the corrected step in the main paper. The two papers were vetted and published as the entirety of the May 1995 issue of the Annals of Mathematics. These papers established the modularity theorem for semistable elliptic curves, the last step in proving Fermat's Last Theorem, 358 years after it was conjectured.
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Then the plane is partitioned into a collection of disjoint subregions. For example, each subregion may consist of the collection of all the locations of this plane that are closer to some point of the underlying point pattern than any other point of the point pattern. This mathematical structure is known as a Voronoi tessellation and may represent, for example, the association cells in a cellular network where users associate with the closest base station. Instead of placing a disk or a Voronoi cell on a point, one could place a cell defined from the information theoretic channels described above.
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He pioneered the modern unification of physics through the introduction of extra dimensions of space and found mechanisms by which the extra dimensions curl up. Professor Freund made significant contributions, to the theory of magnetic monopoles, to supersymmetry and supergravity, to number-theoretic aspects of string theory, as well as to the phenomenology of hadrons. The Archaeological Museum in Sofia Dyankov has a passion for antique and old historic and finance books, history of Bulgarian finance, etc. While in the Bulgarian government, Dyankov started a program for restoring cultural heritage sites from the Thracian, Roman and early Bulgarian history.
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He is the Horace White Professor of Physics, Emeritus, at Cornell University and a fellow of the American Physical Society. He joined the Hong Kong University of Science and Technology in 2011 and was the Director of HKUST Jockey Club Institute for Advanced Study during 2011-2016. Together with Gia Dvali, he suggested the idea of brane inflation in 1998 in which inflation arises because of the weak forces supersymmetry allows between identical branes. A variant of this proposal based on branes and antibranes was later put on concrete string theoretic grounds by Shamit Kachru and collaborators.
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The Taming of the Shrew, by C. R. Leslie The shrew – an unpleasant, ill- tempered woman characterised by scolding, nagging, and aggression – is a comedic, stock character in literature and folklore, both Western and Eastern. The theme is illustrated in Shakespeare's play The Taming of the Shrew. In critical-theoretic analysis, the figure represents insubordinate female behaviour in a marital system of polarised gender roles, that is male- dominated in a moral hierarchy. As a reference to actual women, rather than the stock character, shrew is considered old-fashioned, and the synonym scold (as a noun) is archaic.
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Games and Economic Behavior (GEB) is a journal of game theory published by Elsevier. Founded in 1989, the journal's stated objective is to communicate game-theoretic ideas across theory and applications.Webpage of Games and Economic Behavior It is considered to be the leading journal of game theory and one of the top journals in economics, and it is one of the two official journals of the Game Theory Society. Apart from game theory and economics, the research areas of the journal also include applications of game theory in political science, biology, computer science, mathematics and psychology.
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This results in odd bet sizing and a much different strategy than humans are used to seeing. Methods are being developed to at least approximate perfect poker strategy from the game theory perspective in the heads-up (two player) game, and increasingly good systems are being created for the multi-player game. Perfect strategy has multiple meanings in this context. From a game-theoretic optimal point of view, a perfect strategy is one that cannot expect to lose to any other player's strategy; however, optimal strategy can vary in the presence of sub-optimal players who have weaknesses that can be exploited.
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Throughout his career Scanlan solved some of the most important circuit theoretic challenges of the time. Initially he contributed to the theory of high frequency transistor amplifiers and oscillators, and that of tunnel diode amplifiers. After making a number of fundamental contributions to the synthesis of lumped networks, he turned to the synthesis of distributed circuits, then exercising the minds of the leading circuit theorists, and working with David Rhodes produced what remain the most important results in distributed circuit synthesis. These methods continue to be central to the design of microwave filters for the most challenging applications.
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The results are mixed. Some authors have found beta larger than alpha, which contradicts a central assumption made by Fehr and Schmidt (1999). Other authors have found that inequity aversion with Fehr and Schmidt's (1999) distribution of alphas and betas explains data of contract-theoretic experiments not better than standard theory; they also estimate average values of alpha that are much smaller than suggested by Fehr and Schmidt (1999). Moreover, Levitt and List (2007) have pointed out that laboratory experiments tend to exaggerate the importance of pro-social behaviors because the subjects in the laboratory know that they are being monitored.
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Gentzen's main work was on the foundations of mathematics, in proof theory, specifically natural deduction and the sequent calculus. His cut-elimination theorem is the cornerstone of proof-theoretic semantics, and some philosophical remarks in his "Investigations into Logical Deduction", together with Ludwig Wittgenstein's later work, constitute the starting point for inferential role semantics. One of Gentzen's papers had a second publication in the ideological Deutsche Mathematik that was founded by Ludwig Bieberbach who promoted "Aryan" mathematics.Dipl.Math. Walter Tydecks, Neuere Geschichte der Mathematik in Deutschland (in German) Gentzen proved the consistency of the Peano axioms in a paper published in 1936.
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Why should we trust our moral intuitions, no matter how strong they are, if we have a reasonable explanation of their origin that is compatible with their being entirely false? Joyce has developed and defended what has come to be known as an "evolutionary debunking argument," according to which the evolutionary origin of human moral thinking might give us cause to doubt our moral judgments. The conclusion of Joyce's debunking argument is not the error-theoretic view that all moral judgments are false (though this is a conclusion he argues for elsewhere), but the epistemological view that all moral judgments are unjustified.
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Lorenz developed (along with Paul Lorenzen) an approach to arithmetic and logic as dialogue games. In dialogical logic (game semantics), tree calculations (generally, of Gentzen type calculus) are written upside down, so that the initial assertion of a proponent stays above and is defended against an opponent as in a game. This is a linguistically more congenial approach to logic which is more suitable as a model for argumentation than the formal derivation in a calculus or truth tables. Lorenz presented for the first time a simple demonstration of Gentzen's consistency proof on this game-theoretic basis.
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Spectral graph theory emerged in the 1950s and 1960s. Besides graph theoretic research on the relationship between structural and spectral properties of graphs, another major source was research in quantum chemistry, but the connections between these two lines of work were not discovered until much later.Eigenspaces of Graphs, by Dragoš Cvetković, Peter Rowlinson, Slobodan Simić (1997) The 1980 monograph Spectra of GraphsDragoš M. Cvetković, Michael Doob, Horst Sachs, Spectra of Graphs (1980) by Cvetković, Doob, and Sachs summarised nearly all research to date in the area. In 1988 it was updated by the survey Recent Results in the Theory of Graph Spectra.
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The term measure here refers to the measure-theoretic approach to probability. Violations of unit measure have been reported in arguments about the outcomes of events R. Christensen and T. Reichert: "Unit measure violations in pattern recognition: ambiguity and irrelevancy" Pattern Recognition, 8, No. 4 1976.T. Oldberg and R. Christensen "Erratic measure" NDE for the Energy Industry 1995, American Society of Mechanical Engineers, New York, NY. under which events acquire "probabilities" that are not the probabilities of probability theory. In situations such as these the term "probability" serves as a false premise to the associated argument.
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Morita equivalence is a relationship defined between rings that preserves many ring-theoretic properties. It is named after Japanese mathematician Kiiti Morita who defined equivalence and a similar notion of duality in 1958. Two rings R and S (associative, with 1) are said to be (Morita) equivalent if there is an equivalence of the category of (left) modules over R, R-Mod, and the category of (left) modules over S, S-Mod. It can be shown that the left module categories R-Mod and S-Mod are equivalent if and only if the right module categories Mod-R and Mod-S are equivalent.
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There is no natural concept of distance (a metric) in an incidence structure. However, a combinatorial metric does exist in the corresponding incidence graph (Levi graph), namely the length of the shortest path between two vertices in this bipartite graph. The distance between two objects of an incidence structure – two points, two lines or a point and a line – can be defined to be the distance between the corresponding vertices in the incidence graph of the incidence structure. Another way to define a distance again uses a graph-theoretic notion in a related structure, this time the collinearity graph of the incidence structure.
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Traits which people tend to underestimate include juggling ability, the ability to ride a unicycle, the odds of living past 100 or of finding a U.S. twenty dollar bill on the ground in the next two weeks. Some have attempted to explain this cognitive bias in terms of the regression fallacy or of self-handicapping. In a 2012 article in Psychological Bulletin it is suggested the worse-than- average effect (as well as other cognitive biases) can be explained by a simple information-theoretic generative mechanism that assumes a noisy conversion of objective evidence (observation) into subjective estimates (judgment).
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Although it does not use complex analysis, it is in fact much more technical than the standard proof of PNT. One possible definition of an "elementary" proof is "one that can be carried out in first order Peano arithmetic." There are number-theoretic statements (for example, the Paris–Harrington theorem) provable using second order but not first order methods, but such theorems are rare to date. Erdős and Selberg's proof can certainly be formalized in Peano arithmetic, and in 1994, Charalambos Cornaros and Costas Dimitracopoulos proved that their proof can be formalized in a very weak fragment of PA, namely .
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Mikhail Khovanov and Lev Rozansky have since defined several other related cohomology theories whose Euler characteristics recover other classical invariants. Catharina Stroppel gave a representation theoretic interpretation of Khovanov homology by categorifying quantum group invariants. There is also growing interest from both knot theorists and scientists in understanding "physical" or geometric properties of knots and relating it to topological invariants and knot type. An old result in this direction is the Fary–Milnor theorem states that if the total curvature of a knot in \R^3 satisfies :\oint_K \kappa \,ds \leq 4\pi, where is the curvature at , then is an unknot.
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In type-theoretic foundations of mathematics, setoids may be used in a type theory that lacks quotient types to model general mathematical sets. For example, in Per Martin- Löf's intuitionistic type theory, there is no type of real numbers, only a type of regular Cauchy sequences of rational numbers. To do real analysis in Martin-Löf's framework, therefore, one must work with a setoid of real numbers, the type of regular Cauchy sequences equipped with the usual notion of equivalence. Predicates and functions of real numbers need to be defined for regular Cauchy sequences and proven to be compatible with the equivalence relation.
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A series-parallel partial order, shown as a Hasse diagram. In order-theoretic mathematics, a series-parallel partial order is a partially ordered set built up from smaller series-parallel partial orders by two simple composition operations... The series-parallel partial orders may be characterized as the N-free finite partial orders; they have order dimension at most two.. They include weak orders and the reachability relationship in directed trees and directed series-parallel graphs. The comparability graphs of series-parallel partial orders are cographs. Series-parallel partial orders have been applied in job shop scheduling,.
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In mathematics, the Haagerup property, named after Uffe Haagerup and also known as Gromov's a-T-menability, is a property of groups that is a strong negation of Kazhdan's property (T). Property (T) is considered a representation-theoretic form of rigidity, so the Haagerup property may be considered a form of strong nonrigidity; see below for details. The Haagerup property is interesting to many fields of mathematics, including harmonic analysis, representation theory, operator K-theory, and geometric group theory. Perhaps its most impressive consequence is that groups with the Haagerup Property satisfy the Baum–Connes conjecture and the related Novikov conjecture.
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Ligeti used technology including sine wave, white noise, and impulse generators, as well as filters. Having conceived of many various possible and artificial phonemes, created recordings of them, and grouped them into various categories and bins, Ligeti created a formula (and many tables) to determine the maximum length of each tape used (the louder the shorter), and then went through a process of grabbing randomly, without-looking, similar "phonemes" out of their bins, combining them into "texts", and then cutting these in half, down to "words".Elliott Antokoletz (2014). A History of Twentieth- Century Music in a Theoretic-Analytical Context, p.371. Routledge. .
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Among most subtle representatives of "pluritonic order" there were Mozart and Rossini; this stage he saw as the culmination and perfection of tonalité moderne. The romantic tonality of Berlioz and especially Wagner he related to "omnitonic order" with its "insatiable desire for modulation" . His prophetic vision of the omnitonic order (though he didn't approve it personally) as the way of further development of tonality was a remarkable innovation to historic and theoretic concepts of the 19th century . Tonalité ancienne Fetis described as tonality of ordre unitonique (establishing one key and remaining in that key for the duration of the piece).
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Another issue was the representation of the infinite dimensional irreducible components of a State Property System as the set of closed subspaces of one of three standard Hilbert spaces, real, complex or quaternionic. This was now concluded by introducing a new axiom called 'plane transitivity'. An interesting mathematical result proven was the fact that the category of State Property Systems is categorically equivalent with the category of closure spaces. The structure of State Property Systems was generalized to that of State Context Property Systems with the aim of providing a generalized quantum theoretic framework for the modeling of concepts and their dynamics.
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Grothendieck's construction of new cohomology theories, which use algebraic techniques to study topological objects, has influenced the development of algebraic number theory, algebraic topology, and representation theory. As part of this project, his creation of topos theory, a category-theoretic generalization of point-set topology, has influenced the fields of set theory and mathematical logic. The Weil conjectures were formulated in the later 1940s as a set of mathematical problems in arithmetic geometry. They describe properties of analytic invariants, called local zeta functions, of the number of points on an algebraic curve or variety of higher dimension.
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This global political organization would allow individuals to participate in an actual concrete form of freedom that gives voice to their concrete particularity and to the concrete particularity of their community. In this way a cosmopolitan global structure of States links back to the ultimate goal of Weil's theoretic philosophy, which is to explain the unity of action and discourse. Because, for Weil, it is in political organization that humanity's activity, as well as history itself, becomes meaningful, political organization is the backdrop against which moral action is possible and the concrete content of an individual's life can be articulated.
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One can more generally view the classification problem from a homotopy-theoretic point of view. There is a universal bundle for real line bundles, and a universal bundle for complex line bundles. According to general theory about classifying spaces, the heuristic is to look for contractible spaces on which there are group actions of the respective groups C2 and S1, that are free actions. Those spaces can serve as the universal principal bundles, and the quotients for the actions as the classifying spaces BG. In these cases we can find those explicitly, in the infinite-dimensional analogues of real and complex projective space.
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He also chaired the Theoretic Section of the Music School Programming Committee and the State Music Publishing Council. In his academic research he dealt with the history of the Polish Renaissance and Baroque music as well as musical ethnography. He studied the works of, among others Mikołaj Gomółka, Jan z Lublina and Jacek Różycki. He was the initiator of making copies of XV - XVIII-century music manuscripts, and discovered many unknown monuments of Polish music from that period. He prepared the 22 issue of the cycle of the Publishing House of Old Polish Music (1928-1951), and the individual works of former composers.
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A subtree for the idiom "tie the knot," meaning "marry." By adopting a theoretic architecture of grammar which does not separate syntactic, morphological, and semantic processes and by allowing terminals to represent sub-morphemic information, Nanosyntax is equipped to address various failings and areas of uncertainty in previous theories. One example that supports these tools of Nanosyntax is idioms, in which a single lexical item is represented using multiple words whose meaning cannot be determined cumulatively. Because terminals in Nanosyntax represent sub-morphemic information, a single morpheme is able to span several terminals, thus creating a subtree.
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In a sense, the controller assigns its index to PRO, which identifies the argument that is understood as the subject of the subordinate predicate. A (constituency-based) X-bar theoretic tree that is consistent with the standard GB-type analysis is given next:Numerous GB trees like the one here can be found in, for instance, Haegeman (1994). ::Syntactic tree The details of this tree are, again, not so important. What is important is that by positing the existence of the null subject PRO, the theoretical analysis of control constructions gains a useful tool that can help uncover important traits of control constructions.
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It is considered by some to be a better formulation of Hilbert's fifth problem, than the characterisation in the category of topological groups of the Lie groups often cited as a solution. In 1997, Dušan Repovš and Evgenij Ščepin proved the Hilbert-Smith conjecture for groups acting by Lipschitz maps on a Riemannian manifold using the covering, fractal and cohomological dimension theory. In 1999, Gaven Martin extended their dimension-theoretic argument to quasiconformal actions on a Riemannian manifold and gave applications concerning unique analytic continuation for Beltrami systems. In 2013, John Pardon proved the three-dimensional case of the Hilbert–Smith conjecture.
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Leinfellner long viewed game and decision theory as theoretical and methodological frameworks within which the social sciences could be integrated. This was the major motivation for the founding of the journal Theory and Decision and for confounding Theory and Decision Library. Later, Leinfellner would come to view evolutionary game theory as a theoretical framework for integrating biological and cultural evolution. Once placed in an evolutionary game-theoretic framework, it is possible to explain how societal cooperation evolves even though selfishness is favored at the individual level He viewed evolution as always at work but always producing surprises.
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Positive political theory (PPT) or explanatory political theory is the study of politics using formal methods such as social choice theory, game theory, and statistical analysis. In particular, social choice theoretic methods are often used to describe and (axiomatically) analyze the performance of rules or institutions. The outcomes of the rules or institutions described are then analyzed by game theory, where the individuals/parties/nations involved in a given interaction are modeled as rational agents playing a game, guided by self-interest. Based on this assumption, the outcome of the interactions can be predicted as an equilibrium of the game.
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For the standard tools of probability theory, such as joint and conditional probabilities, to work, it is necessary to use a σ-algebra, that is, a family closed under complementation and countable unions of its members. The most natural choice of σ-algebra is the Borel measurable set derived from unions and intersections of intervals. However, the larger class of Lebesgue measurable sets proves more useful in practice. In the general measure-theoretic description of probability spaces, an event may be defined as an element of a selected σ-algebra of subsets of the sample space.
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Perfect security is a special case of information-theoretic security. For an encryption algorithm, if there is ciphertext produced that uses it, no information about the plaintext is provided without knowledge of the key. If E is a perfectly secure encryption function, for any fixed message m, there must be, for each ciphertext c, at least one key k such that c = E_k(m). Mathematically, let m and c be the random variables representing the plaintext and ciphertext messages, respectively; then, we have that : I(m;c) = 0, where I(m;c) is the mutual information between m and c.
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Many researchers in axiomatic set theory have subscribed to what is known as set-theoretic Platonism, exemplified by Kurt Gödel. Several set theorists followed this approach and actively searched for axioms that may be considered as true for heuristic reasons and that would decide the continuum hypothesis. Many large cardinal axioms were studied, but the hypothesis always remained independent from them and it is now considered unlikely that CH can be resolved by a new large cardinal axiom. Other types of axioms were considered, but none of them has reached consensus on the continuum hypothesis yet.
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For more refined questions, the nature of the intersection has to be addressed more closely. The hypersurfaces may be required to satisfy a transversality condition (like their tangent spaces being in general position at intersection points). The intersection may be scheme-theoretic, in other words here the homogeneous ideal generated by the Fi(X0, ..., Xn) may be required to be the defining ideal of V, and not just have the correct radical. In commutative algebra, the complete intersection condition is translated into regular sequence terms, allowing the definition of local complete intersection, or after some localization an ideal has defining regular sequences.
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An agreement forest for two unrooted -trees and is a partition } of the taxon set satisfying the following conditions: # and are isomorphic for every and # the subtrees in and are vertex-disjoint subtrees of and , respectively. The set partition } is identified with the forest of restricted subtrees }, with either or (the choice of it begin irrelevant because of condition 1). Therefore, an agreement forest can either be seen as a partition of the taxon set or as a forest (in the classical graph-theoretic sense) of restricted subtrees. The size of an agreement forest is simply its number of components.
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SimRank is a general similarity measure, based on a simple and intuitive graph-theoretic model. SimRank is applicable in any domain with object-to- object relationships, that measures similarity of the structural context in which objects occur, based on their relationships with other objects. Effectively, SimRank is a measure that says "two objects are considered to be similar if they are referenced by similar objects." Although SimRank is widely adopted, it may output unreasonable similarity scores which are influenced by different factors, and can be solved in several ways, such as introducing an evidence weight factor,I.
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He is a member of the US National Academy of Sciences, and the American Academy of Arts and Sciences, an associate member of the faculty of Canada's Perimeter Institute for Theoretical Physics, and a distinguished professor of the Korea Institute for Advanced Study. Susskind is widely regarded as one of the fathers of string theory. He was the first to give a precise string-theoretic interpretation of the holographic principle in 1995 and the first to introduce the idea of the string theory landscape in 2003. Susskind was awarded the 1998 J. J. Sakurai Prize, and the 2018 Oskar Klein Medal.
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These tasks can be equally formulated from a phenomenological, hermeneutical, or grammatical point of view. This open hermeneutics is focused by Marc Jean- Bernard on both practical and theoretical dimensions: Epistemic Perspective The Greek locution Dialegein synthesizes a theoretic set of axiological investigations, corresponding to a unified epistemic methodology that embraces Cultural Hermeneutics, specifically Aesthetics and Ethics. His construction of ethical and aesthetical responsibility stands as a philosophical cantus firmus for the understanding of cultural polyphony. Intentionality and Cognition Firstly, Marc Jean-Bernard seeks to provide a philosophical account of cognitive sciences in the field of culture.
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You may want to know, in what integer residue class rings you have a primitive k-th root of unity. You need it for instance if you want to compute a Discrete Fourier Transform (more precisely a Number theoretic transform) of a k-dimensional integer vector. In order to perform the inverse transform, you also need to divide by k, that is, k shall also be a unit modulo n. A simple way to find such an n is to check for primitive k-th roots with respect to the moduli in the arithmetic progression k+1, 2k+1, 3k+1, \dots.
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In mathematics, more specifically in the theory of dynamical systems and probability theory, ergodicity is a property of a (discrete or continuous) dynamical system which expresses a form of irreducibility of the system, from a measure-theoretic viewpoint. It includes the ergodicity of stochastic processes; though the language used for the study of ergodic processes is usually more probabilist. The origin of the notion and the nomenclature lie in statistical physics, where L. Boltzmann formulated the ergodic hypothesis. An informal way to phrase it is that the average behaviour over time on a trajectory does not depend on the particular trajectory chosen.
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The Zariski topology in the set-theoretic sense is then replaced by a Zariski topology in the sense of Grothendieck topology. Grothendieck introduced Grothendieck topologies having in mind more exotic but geometrically finer and more sensitive examples than the crude Zariski topology, namely the étale topology, and the two flat Grothendieck topologies: fppf and fpqc. Nowadays some other examples have become prominent, including the Nisnevich topology. Sheaves can be furthermore generalized to stacks in the sense of Grothendieck, usually with some additional representability conditions, leading to Artin stacks and, even finer, Deligne–Mumford stacks, both often called algebraic stacks.
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SDIBT has more than 1,000 staff, including over 700 teachers, 100 professors and 209 associate professors; 144 doctorate degree holders (including those currently undertaking doctorate studies) and 355 master's degree holders. Six experts enjoy State council special pension and more than 20 people have obtained the title of “National Excellent Teachers”, “Young and middle-aged Provincial- ranking Experts with Outstanding Contribution”, “First Ten Young and Middle- aged Jurisconsults” and “Theoretic Talents in One-hundred-talent Program of Shandong Province”. SDITB receives students from across China. At present, over 18,600 students attend regular full- time courses in the day.
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Monks reading and copying the medical texts learnt a lot about human anatomy and methods of treatment, and then put their theoretic skills into practice at monastery hospital. By the 10–11th centuries Monte Cassino became the most famous cultural, educational, and medical center of Europe with great library in Medicine and other sciences. Many physicians came there for medical and other knowledge. That is why the first High Medical School in the world was soon opened in nearby Salerno which is considered today to have been the earliest Institution of Higher Education in Western Europe.
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In model theory, a branch of mathematical logic, the Hrushovski construction generalizes the Fraïssé limit by working with a notion of strong substructure \leq rather than \subseteq. It can be thought of as a kind of "model-theoretic forcing", where a (usually) stable structure is created, called the generic or rich Slides on Hrushovski construction from Frank Wagner model. The specifics of \leq determine various properties of the generic, with its geometric properties being of particular interest. It was initially used by Ehud Hrushovski to generate a stable structure with an "exotic" geometry, thereby refuting Zil'ber's Conjecture.
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Although higher-order logics are more expressive, allowing complete axiomatizations of structures such as the natural numbers, they do not satisfy analogues of the completeness and compactness theorems from first- order logic, and are thus less amenable to proof-theoretic analysis. Another type of logics are s that allow inductive definitions, like one writes for primitive recursive functions. One can formally define an extension of first- order logic -- a notion which encompasses all logics in this section because they behave like first-order logic in certain fundamental ways, but does not encompass all logics in general, e.g. it does not encompass intuitionistic, modal or fuzzy logic.
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It has an edge whenever can reach . That is, it has an edge for every related pair of distinct elements in the reachability relation of , and may therefore be thought of as a direct translation of the reachability relation into graph-theoretic terms. The same method of translating partial orders into DAGs works more generally: for every finite partially ordered set , the graph that has a vertex for each member of and an edge for each pair of elements related by is automatically a transitively closed DAG, and has as its reachability relation. In this way, every finite partially ordered set can be represented as the reachability relation of a DAG.
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In mathematics, specifically in group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted . This operation is the group-theoretic analogue of the Cartesian product of sets and is one of several important notions of direct product in mathematics. In the context of abelian groups, the direct product is sometimes referred to as the direct sum, and is denoted G \oplus H. Direct sums play an important role in the classification of abelian groups: according to the fundamental theorem of finite abelian groups, every finite abelian group can be expressed as the direct sum of cyclic groups.
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It was a substantial improvement of his academic situation when Hausdorff was appointed in 1921 to Bonn. Here he could develop a thematically wide-spanned teaching and always lecture on the latest research. He gave a particularly noteworthy lecture on probability theory (NL Hausdorff: Capsule 21: Fasz 64) in the summer semester 1923, in which he grounded this theory in measure-theoretic axiomatic theory, and this occurred ten years before A. N. Kolmogorov's "Basic concepts of probability theory" (reprinted in full in the collected works, Volume V). In Bonn, Hausdorff had Eduard Study, and later with Otto Toeplitz, outstanding mathematicians as well as colleagues and friends.
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As such, the conjecture is seen as providing some game-theoretic foundations for the usual assumption in general equilibrium theory of price taking agents. In particular, it means that in a "large" economy people act as if they were price takers, even though theoretically they have all the power to set prices and renegotiate their trades. Hence, the fictitious Walrasian auctioneer of general equilibrium, while strictly speaking completely unrealistic, can be seen as a "short-cut" to getting the right answer. Edgeworth himself did not quite prove this result—hence the term conjecture rather than theorem—although he did provide most of the necessary intuition and went some way towards it.
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In 1967 he moved to UC Irvine to become the dean of the Graduate School of Administration (now Paul Merage School of Business). During this career transition from early computing technologies to administration, he worked on applying decision theory and game theoretic techniques to organizational structure and business administration. He stayed at Irvine until his retirement in 1982. Outside of academia, Brown was a member of the SATCOM task group for the 'Interchange of Scientific and Technical Information in Machine Language (ISTIM)' established in 1969 by the President's Special Assistant for Science and Technology (precursor to the modern-day Office of Science and Technology Policy).
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This includes cryptographic systems such as Lamport signatures and the Merkle signature scheme and the newer XMSS and SPHINCS schemes. Hash based digital signatures were invented in the late 1970s by Ralph Merkle and have been studied ever since as an interesting alternative to number-theoretic digital signatures like RSA and DSA. Their primary drawback is that for any hash-based public key, there is a limit on the number of signatures that can be signed using the corresponding set of private keys. This fact had reduced interest in these signatures until interest was revived due to the desire for cryptography that was resistant to attack by quantum computers.
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That is, the image is the kernel of the cokernel, and the coimage is the cokernel of the kernel. Note that this notion of image may not correspond to the usual notion of image, or range, of a function, even assuming that the morphisms in the category are functions. For example, in the category of topological abelian groups, the image of a morphism actually corresponds to the inclusion of the closure of the range of the function. For this reason, people will often distinguish the meanings of the two terms in this context, using "image" for the abstract categorical concept and "range" for the elementary set-theoretic concept.
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There was some irony that in the pushing through of David Hilbert's long-range programme a natural home for intuitionistic logic's central ideas was found: Hilbert had detested the school of L. E. J. Brouwer. Existence as 'local' existence in the sheaf- theoretic sense, now going by the name of Kripke–Joyal semantics, is a good match. On the other hand Brouwer's long efforts on 'species', as he called the intuitionistic theory of reals, are presumably in some way subsumed and deprived of status beyond the historical. There is a theory of the real numbers in each topos, and so no one master intuitionist theory.
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"Can a ball be decomposed into a finite number of point sets and reassembled into two balls identical to the original?" The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in 3‑dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball. Indeed, the reassembly process involves only moving the pieces around and rotating them without changing their shape. However, the pieces themselves are not "solids" in the usual sense, but infinite scatterings of points.
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This later suggestion has been pursued by several philosophers since Lewis. Following game-theoretic account of conventions, Edna Ullmann-Margalit (1977) and Bicchieri (2006) have developed theories of social norms that define them as Nash equilibria that result from transforming a mixed-motive game into a coordination game. Game theory has also challenged philosophers to think in terms of interactive epistemology: what it means for a collective to have common beliefs or knowledge, and what are the consequences of this knowledge for the social outcomes resulting from the interactions of agents. Philosophers who have worked in this area include Bicchieri (1989, 1993), Skyrms (1990), and Stalnaker (1999).
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Kotarbiński first introduced reism in his work called Elements of the Theory of Knowledge, Formal Logic and Methodology of the Sciences and the emergent theory was developed independent of the ideas previously put forward by the German philosopher Franz Brentano. The latter's account reism is considered a metaphysical view of the mind. However, Kotarbiński's reism - as proposed - was not a ready theory of the world but a program with the aim of eliminating apparent terms (onomatoids) and partial successes. Kotarbiński's model adopted the formal logic of Lesniewski and his rejection of the classical set-theoretic conception of classes in favor of the mereological whole.
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In 1936, Tarski published Polish and German versions of a lecture he had given the preceding year at the International Congress of Scientific Philosophy in Paris. A new English translation of this paper, Tarski (2002), highlights the many differences between the German and Polish versions of the paper and corrects a number of mistranslations in Tarski (1983). This publication set out the modern model-theoretic definition of (semantic) logical consequence, or at least the basis for it. Whether Tarski's notion was entirely the modern one turns on whether he intended to admit models with varying domains (and in particular, models with domains of different cardinalities).
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The geometric Langlands correspondence is a relationship between abstract geometric objects associated to an algebraic curve such as the elliptic curves illustrated above. In mathematics, the classical Langlands correspondence is a collection of results and conjectures relating number theory to the branch of mathematics known as representation theory.Frenkel 2007 Formulated by Robert Langlands in the late 1960s, the Langlands correspondence is related to important conjectures in number theory such as the Taniyama–Shimura conjecture, which includes Fermat's last theorem as a special case. In spite of its importance in number theory, establishing the Langlands correspondence in the number theoretic context has proved extremely difficult.
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But by convention, the LR name stands for the form of parsing invented by Donald Knuth, and excludes the earlier, less powerful precedence methods (for example Operator-precedence parser). LR parsers can handle a larger range of languages and grammars than precedence parsers or top-down LL parsing.Language theoretic comparison of LL and LR grammars This is because the LR parser waits until it has seen an entire instance of some grammar pattern before committing to what it has found. An LL parser has to decide or guess what it is seeing much sooner, when it has only seen the leftmost input symbol of that pattern.
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Title page of the first edition of Disquisitiones Arithmeticae, one of the founding works of modern algebraic number theory. Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields. These properties, such as whether a ring admits unique factorization, the behavior of ideals, and the Galois groups of fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations.
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In the "chickie run" scene from the film Rebel Without a Cause, this happens when Buzz cannot escape from the car and dies in the crash. The opposite scenario occurs in Footloose where Ren McCormack is stuck in his tractor and hence wins the game as they cannot play "chicken". A similar event happens in two different games in the film The Heavenly Kid, when first Bobby, then later Lenny become stuck in their cars and drive off a cliff. The basic game-theoretic formulation of Chicken has no element of variable, potentially catastrophic, risk, and is also the contraction of a dynamic situation into a one-shot interaction.
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Firstly, the agent may be risk-averse, so there is a trade-off between providing the agent with incentives and insuring the agent. Secondly, the agent may be risk-neutral but wealth-constrained and so the agent cannot make a payment to the principal and there is a trade-off between providing incentives and minimizing the agent's limited-liability rent. Among the early contributors to the contract-theoretic literature on moral hazard were Oliver Hart and Sanford J. Grossman. In the meantime, the moral hazard model has been extended to the cases of multiple periods and multiple tasks, both with risk-averse and risk-neutral agents.
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As evidenced by Efimov and Ganbold in an earlier work (Efimov 1991), the procedure of tadpole renormalization can be employed very effectively to remove the divergences from the action of the basic field-theoretic representation of the partition function and leads to an alternative functional integral representation, called the Gaussian equivalent representation (GER). They showed that the procedure provides functional integrals with significantly ameliorated convergence properties for analytical perturbation calculations. In subsequent works Baeurle et al. developed effective low-cost approximation methods based on the tadpole renormalization procedure, which have shown to deliver useful results for prototypical polymer and PE solutions (Baeurle 2006a, Baeurle 2006b, Baeurle 2007a).
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The terminology of algebraic geometry changed drastically during the twentieth century, with the introduction of the general methods, initiated by David Hilbert and the Italian school of algebraic geometry in the beginning of the century, and later formalized by André Weil, Jean-Pierre Serre and Alexander Grothendieck. Much of the classical terminology, mainly based on case study, was simply abandoned, with the result that books and papers written before this time can be hard to read. This article lists some of this classical terminology, and describes some of the changes in conventions. translates many of the classical terms in algebraic geometry into scheme-theoretic terminology.
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Arunava Sen is President Elect of the Society for Social Choice and Welfare, Fellow of the Econometric Society and an Economic Theory Fellow. He has been awarded the Mahalanobis Memorial Medal of the Indian Econometric Society for his contribution to Economics. He is a recipient of the 2012 Infosys Prize in the Social Sciences category for his work on "game-theoretic analyses of mechanism design for implementing social choice rules, when individuals have diverse information and incentives". In 2017, he received the TWAS-Siwei Cheng Prize for his " theoretical work on the collective, strategic behavior of people trying to get what they want from rule-based institutions".
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She received a B.A. degree from Lady Shri Ram College for Women of the University of Delhi in 1992 and an M.A. degree in economics from Delhi School of Economics, also of the University of Delhi, in 1994. She further completed an M.A. degree at the University of Washington in 1996. She earned her Ph.D. in economics from Princeton University in 2001 after completing a doctoral dissertation titled "Three essays on international capital flows: a search theoretic approach," under the supervision of Ben Bernanke and Kenneth Rogoff. She was awarded the Princeton's Woodrow Wilson Fellowship Research Award while doing her doctoral research at Princeton.
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Usually, field equations are postulated (like the Einstein field equations and the Schrödinger equation, which underlies all quantum field equations) or obtained from the results of experiments (like Maxwell's equations). The extent of their validity is their extent to correctly predict and agree with experimental results. From a theoretical viewpoint, field equations can be formulated in the frameworks of Lagrangian field theory, Hamiltonian field theory, and field theoretic formulations of the principle of stationary action. Given a suitable Lagrangian or Hamiltonian density, a function of the fields in a given system, as well as their derivatives, the principle of stationary action will obtain the field equation.
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The institution is deemed to be the follow-on institution of the Academia de Bellas Artes de Santa Bárbara, which was closed in 1759. The institution was founded under the name Real Academia de las Tres Nobles Artes de San Carlos (Royal Academy of the three Noble Arts of Saint Charles) by decret of Charles III from February 14, 1768, according to the Real Academia de Bellas Artes de San Fernando in Madrid. The so-called "three noble arts" were painting, sculpture and architecture. Until 1910 the academic training was rather practical, before the course offer was increased by essential theoretic and practical knowledge.
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In fact, given that the geometrization conjecture is now settled, the only case needed to be proven for the virtually fibered conjecture is that of hyperbolic 3-manifolds. The original interest in the virtually fibered conjecture (as well as its weaker cousins, such as the virtually Haken conjecture) stemmed from the fact that any of these conjectures, combined with Thurston's hyperbolization theorem, would imply the geometrization conjecture. However, in practice all known attacks on the "virtual" conjecture take geometrization as a hypothesis, and rely on the geometric and group-theoretic properties of hyperbolic 3-manifolds. The virtually fibered conjecture was not actually conjectured by Thurston.
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It is now known as a conditional GAN or cGAN. An idea similar to GANs was used to model animal behavior by Li, Gauci and Gross in 2013. Adversarial machine learning has other uses besides generative modeling and can be applied to models other than neural networks. In control theory, adversarial learning based on neural networks was used in 2006 to train robust controllers in a game theoretic sense, by alternating the iterations between a minimizer policy, the controller, and a maximizer policy, the disturbance . In 2017, a GAN was used for image enhancement focusing on realistic textures rather than pixel-accuracy, producing a higher image quality at high magnification.
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Having established the concept of veto players, Tsebelis then applies this to social choice, following Anthony Downs' approach of continuous policy space with veto players concerned solely about proximity of choices to their ideal on a policy spectrum. Further he assumes that there is a status quo point (apparently analogous to a disagreement point in game theoretic bargaining analysis). He argues that the status quo will only change if it is weakly preferred by all veto players (since otherwise one of the players would veto the social choice). This is analogous to saying that the status quo will only change if the status quo is not Pareto efficient for veto players.
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Abbott was a professor at Michigan State University where she taught linguistics and philosophy from 1976 to 2006. Her main concentrations are semantics and pragmatics Her book Reference focuses on the issue of how far reference is and if it is a two-place or three-place relation. Abbott is also known for her other published works which include Natural Language Semantics, Language, Linguistics and Philosophy, Journal of Pragmatics, and Mind. She has also released a wide range of articles beginning in 1974 with an article titled Some Problems In Giving An Adequate Model- Theoretic Account of Cause to her most recent article, titled Some Remarks on Referentiality, in 2011.
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Second is mining more useful information and can get the corresponding information in test clusters and words clusters. This corresponding information can be used to describe the type of texts and words, at the same time, the result of words clustering can be also used to text mining and information retrieval. Several approaches have been proposed based on the information contents of the resulting blocks: matrix-based approaches such as SVD and BVD, and graph-based approaches. Information-theoretic algorithms iteratively assign each row to a cluster of documents and each column to a cluster of words such that the mutual information is maximized.
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In the mathematical theory of categories, a sketch is a category D, together with a set of cones intended to be limits and a set of cocones intended to be colimits. A model of the sketch in a category C is a functor :M:D\rightarrow C that takes each specified cone to a limit cone in C and each specified cocone to a colimit cocone in C. Morphisms of models are natural transformations. Sketches are a general way of specifying structures on the objects of a category, forming a category-theoretic analog to the logical concept of a theory and its models. They allow multisorted models and models in any category.
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A ground atom is true in an interpretation I if it is an element of I. A rule is true in an interpretation I if for each ground instance of that rule, if all the clauses in the body are true in I, then the head of the rule is also true in I. A model of a Datalog program P is an interpretation I of P which contains all the ground facts of P, and makes all of the rules of P true in I. Model- theoretic semantics state that the meaning of a Datalog program is its minimal model (equivalently, the intersection of all its models).
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It has also helped build complex computational models of populations to predict the outcome of the system over time and track and share information on an increasingly large number of species and organisms. Future endeavors are to reconstruct a now more complex tree of life. Christoph Adami, a professor at the Keck Graduate Institute made this point in Evolution of biological complexity: :To make a case for or against a trend in the evolution of complexity in biological evolution, complexity must be both rigorously defined and measurable. A recent information-theoretic (but intuitively evident) definition identifies genomic complexity with the amount of information a sequence stores about its environment.
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The foundations of relevance theory have been criticised because relevance, in the technical sense it is used there, cannot be measured, so it is not possible to say what exactly is meant by "relevant enough" and "the most relevant". Carston generally agrees with the relevance theoretic concept of implicature, but argues that Sperber and Wilson let implicatures do too much work. The mentioned embedding tests not only categorize utterances on the likes of the vodka bottle example as explicatures, but also loose use and metaphors: : If your steak is raw, you can send it back. : If Jane is your anchor in the storm, you should let her help you now.
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In this version of the puzzle, A, B, C and D take 5, 10, 20, and 25 minutes, respectively, to cross, and the time limit is 60 minutes. In all these variations, the structure and solution of the puzzle remain the same. In the case where there are an arbitrary number of people with arbitrary crossing times, and the capacity of the bridge remains equal to two people, the problem has been completely analyzed by graph-theoretic methods. Martin Erwig from Oregon State University has used a variation of the problem to argue for the usability of the Haskell programming language over Prolog for solving search problems.
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For these authors, the alternative reflects one of the clashes effectively existent in the Argentine politics of that time: between the illustrated classes, based on the principles of the theoretic right of the millenary European tradition; and the pragmatic provincial leaders, men of action rather than theory. Given the intellectual ambient of the moment, in which the ideologists of the French revolutionists had given place to the illumining positivism, it was natural that the thought of the first inclined for the defence of the liberal order, in which the abolition of the historical and traditional limits gave in for a new era of cooperation between people.
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One of the ways students interact with each other and with the instructors within fully online learning environments is via asynchronous discussion forums. However, student engagement in online discussion forums does not always take place automatically and there is a lack of clarity about the ideal role of the instructors in them. In their research, Dip Nandi and his colleges report on their research on the quality of discussion in fully online courses through analysis of discussion forum communication. They have conducted the research on two large fully online subjects for computing students over two consecutive semesters and used a grounded theoretic approach for data analysis.
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Upward planar drawings are particularly important for Hasse diagrams of partially ordered sets, as these diagrams are typically required to be drawn upwardly. In graph-theoretic terms, these correspond to the transitively reduced directed acyclic graphs; such a graph can be formed from the covering relation of a partial order, and the partial order itself forms the reachability relation in the graph. If a partially ordered set has one minimal element, has one maximal element, and has an upward planar drawing, then it must necessarily form a lattice, a set in which every pair of elements has a unique greatest lower bound and a unique least upper bound., 6.7.
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In mathematics, a Leinster group is a finite group whose order equals the sum of the orders of its proper normal subgroups. The Leinster groups are named after Tom Leinster, a mathematician at the University of Edinburgh, who wrote about them in a paper written in 1996 but not published until 2001. He called them "perfect groups", and later "immaculate groups", but they were renamed as the Leinster groups by , because "perfect group" already had a different meaning (a group that equals its commutator subgroup). Leinster groups give a group-theoretic way of analyzing the perfect numbers and of approaching the still-unsolved problem of the existence of odd perfect numbers.
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A publishing company has turned his work into a classic book for those not mathematically inclined, while opting for a title with more pizazz, "Friendship, As Easy as Pi." Charlie takes joy in the belief that this book will allow his thoughts to reach a much wider audience than before. By the episode "In Security," the published book appears with the title "The Attraction Equation" and a dapper photo on the back cover of him holding a sculpture of a stellated icosidodecahedron with bevelled edges. A decision theoretic approach to relationships is covered in the book. His proud father hands copies to friends and Larry sells signed copies on eBay.
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The primary achievement of ludics is the discovery of a relationship between two natural, but distinct notions of type, or proposition. The first view, which might be termed the proof-theoretic or Gentzen-style interpretation of propositions, says that the meaning of a proposition arises from its introduction and elimination rules. Focalization refines this viewpoint by distinguishing between positive propositions, whose meaning arises from their introduction rules, and negative propositions, whose meaning arises from their elimination rules. In focused calculi, it is possible to define positive connectives by giving only their introduction rules, with the shape of the elimination rules being forced by this choice.
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It is usual to distinguish two main kinds of theories about the semantics of donkey pronouns. The most classical proposals fall within the so-called description-theoretic approach, a label that is meant to encompass all the theories that treat the semantics of these pronouns as akin to, or derivative from, the semantics of definite descriptions. The second main family of proposals goes by the name dynamic theories, and they model donkey anaphora -and anaphora in general- on the assumption that the meaning of a sentence lies in its potential to change the context (understood as the information shared by the participants in a conversation).
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A set-theoretic representation (also known as a group action or permutation representation) of a group G on a set X is given by a function ρ from G to XX, the set of functions from X to X, such that for all g1, g2 in G and all x in X: :\rho(1)[x] = x :\rho(g_1 g_2)[x]=\rho(g_1)[\rho(g_2)[x . This condition and the axioms for a group imply that ρ(g) is a bijection (or permutation) for all g in G. Thus we may equivalently define a permutation representation to be a group homomorphism from G to the symmetric group SX of X.
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Gregory Lawrence Eyink is an American mathematical physicist at Johns Hopkins University. He received his bachelor’s degree in mathematics and philosophy (1981) and Doctor of Philosophy (1987) from Ohio State University. He now holds joint appointments in the departments of Physics and Astronomy, Mathematics, and Mechanical Engineering at Johns Hopkins. He was awarded the status of Fellow of the American Physical Society , after being nominated by their Topical Group on Statistical and Nonlinear Physics in 2003, for his work in nonequilibrium statistical mechanics, in particular on the foundation of transport laws in chaotic dynamical systems, on field-theoretic methods in statistical hydrodynamics and on singularities and dissipative anomalies in fluid turbulence.
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Gordon was also appointed as a non-tenured associate research professor in the institute. The MPC website stated: > The Michael Polanyi Center (MPC) is a cross-disciplinary research and > educational initiative focused on advancing the understanding of science. It > has a fourfold purpose: (1) to support and pursue research in the history > and conceptual foundations of the natural and social sciences; (2) to study > the impact of contemporary science on the humanities and the arts; (3) to be > an active participant in the growing dialogue between science and religion; > and (4) to pursue the mathematical development and empirical application of > design-theoretic concepts in the natural sciences.
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His review on the neutrino physics gives a lucid description of the solar neutrino problem which appeals to a wider class of physicists. Apart from his contributions to neutrino physics, Pal has done important work related to grand unification theory and statistical field theory (popularly called thermal field theory). Some of his works deal with the quantum field theoretic calculations of self-energies of photon and neutrino in a background of charged particles (as electrons, positrons or charged weak gauge bosons) in thermal equilibrium. Pal has also ventured into calculations where the background for various elementary particle processes includes a thermal bath of charged particles in presence of constant magnetic field.
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Harcourt's writings focus on punishment, social control, legal and political theory, and political economy from a critical, empirical, and social theoretic perspective. In 2012, he published, The Illusion of Free Markets: Punishment and the Myth of Natural Order which explored the relationship between laissez faire and mass incarceration. In Illusion of Order: The False Promise of Broken Windows Policing he challenged evidence for the broken windows theory and critiqued the assumptions of the policing strategy. In Language of the Gun, he develops a post-structuralist theory of social science, arguing that social scientists should embrace the ethical choices they make when they interpret data.
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Clifford's theorem has led to a branch of representation theory in its own right, now known as Clifford theory. This is particularly relevant to the representation theory of finite solvable groups, where normal subgroups usually abound. For more general finite groups, Clifford theory often allows representation-theoretic questions to be reduced to questions about groups that are close (in a sense which can be made precise) to being simple. found a more precise version of this result for the restriction of irreducible unitary representations of locally compact groups to closed normal subgroups in what has become known as the "Mackey machine" or "Mackey normal subgroup analysis".
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One of the main successes of the categorical quantum mechanics research program is that from seemingly weak abstract constraints on the compositional structure, it turned out to be possible to derive many quantum mechanical phenomena. In contrast to earlier axiomatic approaches, which aimed to reconstruct Hilbert space quantum theory from reasonable assumptions, this attitude of not aiming for a complete axiomatization may lead to new interesting models that describe quantum phenomena, which could be of use when crafting future theories.J. C. Baez, Quantum quandaries: a category-theoretic perspective. In: The Structural Foundations of Quantum Gravity, D. Rickles, S. French and J. T. Saatsi (Eds), pages 240–266.
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As the weight increases and the molecules become more complex, the number of possible compounds increases drastically. Thus, a program that is able to reduce this number of candidate solutions through the process of hypothesis formation is essential. New graph-theoretic algorithms were invented by Lederberg, Harold Brown, and others that generate all graphs with a specified set of nodes and connection-types (chemical atoms and bonds) -- with or without cycles. Moreover, the team was able to prove mathematically that the generator is complete, in that it produces all graphs with the specified nodes and edges, and that it is non-redundant, in that the output contains no equivalent graphs (e.g.
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In graph theory, a rooted tree is a directed graph in which every vertex except for a special root vertex has exactly one outgoing edge, and in which the path formed by following these edges from any vertex eventually leads to the root vertex. If T is a tree in the descriptive set theory sense, then it corresponds to a graph with one vertex for each sequence in T, and an outgoing edge from each nonempty sequence that connects it to the shorter sequence formed by removing its last element. This graph is a tree in the graph-theoretic sense. The root of the tree is the empty sequence.
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In philosophy and mathematical logic, mereology (from the Greek μέρος meros (root: μερε- mere-, "part") and the suffix -logy "study, discussion, science") is the study of parts and the wholes they form. Whereas set theory is founded on the membership relation between a set and its elements, mereology emphasizes the meronomic relation between entities, which—from a set-theoretic perspective—is closer to the concept of inclusion between sets. Mereology has been explored in various ways as applications of predicate logic to formal ontology, in each of which mereology is an important part. Each of these fields provides its own axiomatic definition of mereology.
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Firms face a kinked demand curve if, when one firm decreases its price, other firms are expected to follow suit in order to maintain sales; when one firm increases its price, however, its rivals are unlikely to follow, as they would lose the sales' gains that they would otherwise get by holding prices at the previous level. Kinked demand potentially fosters supra- competitive prices because any one firm would receive a reduced benefit from cutting price, as opposed to the benefits accruing under neoclassical theory and certain game theoretic models such as Bertrand competition. Collusion may occur also in auction markets, where independent firms coordinate their bids (bid rigging).
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Cauchy used an infinitesimal \alpha to write down a unit impulse, infinitely tall and narrow Dirac-type delta function \delta_\alpha satisfying \int F(x)\delta_\alpha(x) = F(0) in a number of articles in 1827, see Laugwitz (1989). Cauchy defined an infinitesimal in 1821 (Cours d'Analyse) in terms of a sequence tending to zero. Namely, such a null sequence becomes an infinitesimal in Cauchy's and Lazare Carnot's terminology. Modern set- theoretic approaches allow one to define infinitesimals via the ultrapower construction, where a null sequence becomes an infinitesimal in the sense of an equivalence class modulo a relation defined in terms of a suitable ultrafilter.
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It is renormalizable and asymptotically free. In this model phenomena such as dynamic bulk production and spontaneous symmetric breaking can be studied. With Roger Dashen and Brosl Hasslacher, he examined, among other things, quantum-field-theoretic models of extended hadrons and semiclassical approximations in quantum field theory which are reflected in the DHN method of the quantization of solitons. From 1972 to 1977 Neveu was at the Institute for Advanced Study while spending half of the time in Orsay. From 1974 to 1983 he was at the Laboratory for Theoretical Physics of the ENS and from 1983 to 1989 in the theory department at CERN.
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The application of these theories emerged in Kiyotaki and Wright (1993), when the authors developed a tractable model of the exchange process that captures the "double coincidence of wants problem" in a pure barter setup. In this model, the essential function of money is its role as a medium of exchange. The model can be used to address issues in monetary economics, such as the interaction between specialization and monetary exchange, and the possibility of equilibria with multiple fiat currencies. A shortcoming of search-theoretic models of money is that these models becomes intractable without very strong assumptions, and are therefore impractical for the analysis of monetary policy.
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The bulk of Grothendieck's published work is collected in the monumental, yet incomplete, Éléments de géométrie algébrique (EGA) and Séminaire de géométrie algébrique (SGA). The collection Fondements de la Géometrie Algébrique (FGA), which gathers together talks given in the Séminaire Bourbaki, also contains important material. Grothendieck's work includes the invention of the étale and l-adic cohomology theories, which explain an observation of André Weil's that there is a connection between the topological characteristics of a variety and its diophantine (number theoretic) properties. For example, the number of solutions of an equation over a finite field reflects the topological nature of its solutions over the complex numbers.
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The rules on pre- and postconditions are identical to those introduced by Bertrand Meyer in his 1988 book Object- Oriented Software Construction. Both Meyer, and later Pierre America, who was the first to use the term behavioral subtyping, gave proof-theoretic definitions of some behavioral subtyping notions, but their definitions did not take into account aliasing that may occur in programming languages that support references or pointers. Taking aliasing into account was the major improvement made by Liskov and Wing (1994), and a key ingredient is the history constraint. Under the definitions of Meyer and America, a MutablePoint would be a behavioral subtype of ImmutablePoint, whereas LSP forbids this.
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Source: NAS Etel Solingen (2018) - For providing the first systematic analysis in contemporary international relations connecting political economy, globalization, and nuclear choices on the one hand with domestic politics and nuclear behavior on the other. Her theoretical and empirical contributions have left an indelible impact on work within the academy and on broader public understanding of nuclear war. Scott D. Sagan (2015) - For his pioneering theoretical and empirical work addressing the risks of nuclear possession and deployment and the causes of nuclear proliferation. Robert Powell (2012) - For sophisticated game theoretic models of conflict that illuminate the heart of the strategic dilemmas of nuclear deterrence, including the importance of private information.
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In 1970, the University of Michigan began development of the MICRO Information Management System based on D.L. Childs' Set-Theoretic Data model. MICRO was used to manage very large data sets by the US Department of Labor, the U.S. Environmental Protection Agency, and researchers from the University of Alberta, the University of Michigan, and Wayne State University. It ran on IBM mainframe computers using the Michigan Terminal System.MICRO Information Management System (Version 5.0) Reference Manual, M.A. Kahn, D.L. Rumelhart, and B.L. Bronson, October 1977, Institute of Labor and Industrial Relations (ILIR), University of Michigan and Wayne State University The system remained in production until 1998.
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"Tutte's theorem is the basis for solutions to other computer graphics problems, such as morphing."Steven J. Gortle; Craig Gotsman; Dylan Thurston. "Discrete One-Forms on Meshes and Applications to 3D Mesh Parameterization", Computer Aided Geometric Design, 23(2006)83–112 Tutte was mainly responsible for developing the theory of enumeration of planar graphs, which has close links with chromatic and dichromatic polynomials. This work involved some highly innovative techniques of his own invention, requiring considerable manipulative dexterity in handling power series (whose coefficients count appropriate kinds of graphs) and the functions arising as their sums, as well as geometrical dexterity in extracting these power series from the graph-theoretic situation.
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Dirichlet earned the Habilitation and lectured in the 1827/28 year as a Privatdozent at Breslau. While in Breslau, Dirichlet continued his number theoretic research, publishing important contributions to the biquadratic reciprocity law which at the time was a focal point of Gauss's research. Alexander von Humboldt took advantage of these new results, which had also drawn enthusiastic praise from Friedrich Bessel, to arrange for him the desired transfer to Berlin. Given Dirichlet's young age (he was 23 years old at the time), Humboldt was only able to get him a trial position at the Prussian Military Academy in Berlin while remaining nominally employed by the University of Breslau.
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In the so- called behavioral system theoretic approach due to Willems (see people in systems and control), models considered do not directly define an input-output structure. In this framework systems are described by admissible trajectories of a collection of variables, some of which might be interpreted as inputs or outputs. A system is then defined to be controllable in this setting, if any past part of a behavior (trajectory of the external variables) can be concatenated with any future trajectory of the behavior in such a way that the concatenation is contained in the behavior, i.e. is part of the admissible system behavior.
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In algebraic number theory, the Hilbert class field E of a number field K is the maximal abelian unramified extension of K. Its degree over K equals the class number of K and the Galois group of E over K is canonically isomorphic to the ideal class group of K using Frobenius elements for prime ideals in K. In this context, the Hilbert class field of K is not just unramified at the finite places (the classical ideal theoretic interpretation) but also at the infinite places of K. That is, every real embedding of K extends to a real embedding of E (rather than to a complex embedding of E).
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In mathematics, a Lie bialgebra is the Lie-theoretic case of a bialgebra: it is a set with a Lie algebra and a Lie coalgebra structure which are compatible. It is a bialgebra where the comultiplication is skew-symmetric and satisfies a dual Jacobi identity, so that the dual vector space is a Lie algebra, whereas the comultiplication is a 1-cocycle, so that the multiplication and comultiplication are compatible. The cocycle condition implies that, in practice, one studies only classes of bialgebras that are cohomologous to a Lie bialgebra on a coboundary. They are also called Poisson- Hopf algebras, and are the Lie algebra of a Poisson–Lie group.
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Another important question is the existence of automorphisms in recursion-theoretic structures. One of these structures is that one of recursively enumerable sets under inclusion modulo finite difference; in this structure, A is below B if and only if the set difference B − A is finite. Maximal sets (as defined in the previous paragraph) have the property that they cannot be automorphic to non-maximal sets, that is, if there is an automorphism of the recursive enumerable sets under the structure just mentioned, then every maximal set is mapped to another maximal set. Soare (1974) showed that also the converse holds, that is, every two maximal sets are automorphic.
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Jerzy Bańczerowski (born on October 28, 1938 in Ożarów ) is a Polish professor of Philology. Bańczerowski is a member of the Committee on Oriental Studies of the Polish Academy of Sciences, and has served as director of the Institute of Linguistics (1977-1991, 1994-2008), and head of the Department of General and Comparative Linguistics at Adam Mickiewicz University (1969-). The main subjects of Bańczerowski's research are general and comparative linguistics, set theoretic axiomatization of linguistic theory, Finno-Ugric linguistics, and Asian languages. On his initiative, new specialties (in the field of philology) of studies were established at the Adam Mickiewicz University, including Finnish philology and ethnolinguistics.
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When we consider the labor, the power and > the powder required to mine and to quarry products in the aggregate so > enormously valuable, we can not fail to be impressed with the scantiness of > literature on the economics of rock excavation. A dozen years ago, when > called upon to estimate the cost of some open cut rock excavation, I was > astonished to find no text book that in the least served to guide me. I > subsequently learned also that most of the matter to be found in the few > books on blasting was either theoretic or too meagre to be of material > value. What was true then has unfortunately remained true.
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Christopher Cotton, an economist from the Queen's University, and Chang Liu, a graduate student, used game theory to model the bluffing strategies used in the Chinese military legends of Li Guang and his 100 horsemen (144 BC), and Zhuge Liang and the Empty City (228 AD). In the case of these military legends, the researchers found that bluffing arose naturally as the optimal strategy in each situation. The findings were published under the title 100 Horsemen and the empty city: A game theoretic examination of deception in Chinese military legend in the Journal of Peace Research in 2011.The full reference of the work is: C. Cotton; C. Liu.
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Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems of intuitionistic logic do not include the law of the excluded middle and double negation elimination, which are fundamental inference rules in classical logic. Formalized intuitionistic logic was originally developed by Arend Heyting to provide a formal basis for Brouwer's programme of intuitionism. From a proof-theoretic perspective, Heyting’s calculus is a restriction of classical logic in which the law of excluded middle and double negation elimination have been removed.
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There are four theoretic paradigms of cognitive dissonance, the mental stress people suffer when exposed to information that is inconsistent with their beliefs, ideals or values: Belief Disconfirmation, Induced Compliance, Free Choice, and Effort Justification, which respectively explain what happens after a person acts inconsistently, relative to his or her intellectual perspectives; what happens after a person makes decisions and what are the effects upon a person who has expended much effort to achieve a goal. Common to each paradigm of cognitive-dissonance theory is the tenet: People invested in a given perspective shall—when confronted with contrary evidence—expend great effort to justify retaining the challenged perspective.
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There are two distinct senses of the word "undecidable" in mathematics and computer science. The first of these is the proof-theoretic sense used in relation to Gödel's theorems, that of a statement being neither provable nor refutable in a specified deductive system. The second sense, which will not be discussed here, is used in relation to computability theory and applies not to statements but to decision problems, which are countably infinite sets of questions each requiring a yes or no answer. Such a problem is said to be undecidable if there is no computable function that correctly answers every question in the problem set (see undecidable problem).
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For many years Volodymyr Naumenko was a teacher of gymnasium teaching literature in the 2nd Kiev Gymnasium (1880), the State Female Gymnasium of St. Olga (1883), Collegiate of Pavel Galagan, Kiev Funduklei Gymnasium (1889), Vladimir Cadet Corps (1893). For his work he received multiple awards such as the Order of St Anna (IV, 1883), Order of St Stanislaus (II, 1886), Order of St Ana (II, 1893), Order of St Vladimir (IV, 1897). On February 28, 1898 Naumenko was granted a title of Merited Teacher. Volodymyr Naumenko entered history of Ukrainian Pedagogy not only as a brilliant practical teacher, but also as an experienced Methodist-innovator and theoretic.
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The AdS/CFT correspondence is closely related to another duality conjectured by Igor Klebanov and Alexander Markovich Polyakov in 2002.Klebanov and Polyakov 2002 This duality states that certain "higher spin gauge theories" on anti-de Sitter space are equivalent to conformal field theories with O(N) symmetry. Here the theory in the bulk is a type of gauge theory describing particles of arbitrarily high spin. It is similar to string theory, where the excited modes of vibrating strings correspond to particles with higher spin, and it may help to better understand the string theoretic versions of AdS/CFT and possibly even prove the correspondence.
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In her dissertation and postdoctoral research, Malliaris studied unstable model theory and its connection, via characteristic sequences, to graph theoretic concepts such as the Szemerédi regularity lemma. She is also known for two joint papers with Saharon Shelah connecting topology, set theory, and model theory. In this work, Malliaris and Shelah used Keisler's order, a construction from model theory, to prove the equality between two cardinal characteristics of the continuum, 𝖕 and 𝖙, which are greater than the smallest infinite cardinal and less than or equal to the cardinality of the continuum. This resolved a problem in set theory that had been open for fifty years.
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It is of importance, however, in the study of non-ω-models. The system consisting of ACA0 plus induction for all formulas is sometimes called ACA with no subscript. The system ACA0 is a conservative extension of first-order arithmetic (or first-order Peano axioms), defined as the basic axioms, plus the first-order induction axiom scheme (for all formulas φ involving no class variables at all, bound or otherwise), in the language of first-order arithmetic (which does not permit class variables at all). In particular it has the same proof-theoretic ordinal ε0 as first-order arithmetic, owing to the limited induction schema.
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As a successful theoretical framework today, quantum field theory emerged from the work of generations of theoretical physicists spanning much of the 20th century. Its development began in the 1920s with the description of interactions between light and electrons, culminating in the first quantum field theory—quantum electrodynamics. A major theoretical obstacle soon followed with the appearance and persistence of various infinities in perturbative calculations, a problem only resolved in the 1950s with the invention of the renormalization procedure. A second major barrier came with QFT's apparent inability to describe the weak and strong interactions, to the point where some theorists called for the abandonment of the field theoretic approach.
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In algebraic geometry, Chevalley's structure theorem states that a smooth connected algebraic group over a perfect field has a unique normal smooth connected affine algebraic subgroup such that the quotient is an abelian variety. It was proved by (though he had previously announced the result in 1953), , and . Chevalley's original proof, and the other early proofs by Barsotti and Rosenlicht, used the idea of mapping the algebraic group to its Albanese variety. The original proofs were based on Weil's book Foundations of algebraic geometry and are hard to follow for anyone unfamiliar with Weil's foundations, but later gave an exposition of Chevalley's proof in scheme- theoretic terminology.
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Most of the algorithmic strategies are implemented using modern programming languages, although some still implement strategies designed in spreadsheets. Increasingly, the algorithms used by large brokerages and asset managers are written to the FIX Protocol's Algorithmic Trading Definition Language (FIXatdl), which allows firms receiving orders to specify exactly how their electronic orders should be expressed. Orders built using FIXatdl can then be transmitted from traders' systems via the FIX Protocol.FIXatdl – An Emerging Standard, FIXGlobal, December 2009 Basic models can rely on as little as a linear regression, while more complex game-theoretic and pattern recognition or predictive models can also be used to initiate trading.
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Characters of irreducible representations encode many important properties of a group and can thus be used to study its structure. Character theory is an essential tool in the classification of finite simple groups. Close to half of the proof of the Feit–Thompson theorem involves intricate calculations with character values. Easier, but still essential, results that use character theory include Burnside's theorem (a purely group-theoretic proof of Burnside's theorem has since been found, but that proof came over half a century after Burnside's original proof), and a theorem of Richard Brauer and Michio Suzuki stating that a finite simple group cannot have a generalized quaternion group as its Sylow -subgroup.
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Charles Akemann and Nik Weaver showed in 2003 that the statement "there exists a counterexample to Naimark's problem which is generated by ℵ1, elements" is independent of ZFC. Miroslav Bačák and Petr Hájek proved in 2008 that the statement "every Asplund space of density character ω1 has a renorming with the Mazur intersection property" is independent of ZFC. The result is shown using Martin's maximum axiom, while Mar Jiménez and José Pedro Moreno (1997) had presented a counterexample assuming CH. As shown by Ilijas Farah and N. Christopher Phillips and Nik Weaver, the existence of outer automorphisms of the Calkin algebra depends on set theoretic assumptions beyond ZFC.
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In algebraic geometry, the Chow group of a stack is a generalization of the Chow group of a variety or scheme to stacks. For a quotient stack X = [Y/G], the Chow group of X is the same as the G-equivariant Chow group of Y. A key difference from the theory of Chow groups of a variety is that a cycle is allowed to carry non-trivial automorphisms and consequently intersection- theoretic operations must take this into account. For example, the degree of a 0-cycle on a stack need not be an integer but is a rational number (due to non-trivial stabilizers).
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At each point of the projective variety, all the polynomials in the set were required to equal zero. The complement of the zero set of a linear polynomial is an affine space, and an affine variety was the intersection of a projective variety with an affine space. André Weil saw that geometric reasoning could sometimes be applied in number-theoretic situations where the spaces in question might be discrete or even finite. In pursuit of this idea, Weil rewrote the foundations of algebraic geometry, both freeing algebraic geometry from its reliance on complex numbers and introducing abstract algebraic varieties which were not embedded in projective space.
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Rate distortion theory has been applied to choosing k called the "jump" method, which determines the number of clusters that maximizes efficiency while minimizing error by information-theoretic standards. The strategy of the algorithm is to generate a distortion curve for the input data by running a standard clustering algorithm such as k-means for all values of k between 1 and n, and computing the distortion (described below) of the resulting clustering. The distortion curve is then transformed by a negative power chosen based on the dimensionality of the data. Jumps in the resulting values then signify reasonable choices for k, with the largest jump representing the best choice.
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Moral realism is the view that there are mind-independent, and therefore objective, moral facts that moral judgments are in the business of describing. This combines a cognitivist view about moral judgments (that they are belief-like mental states in the business of describing the way the world is), a view about the existence of moral facts (that they do in fact exist), and a view about the nature of moral facts (that they are objective: independent of our cognizing them or our stance towards them). This contrasts with expressivist theories of moral judgment (e.g., Stevenson, Hare, Blackburn, Gibbard), error- theoretic/fictionalist denials of the existence of moral facts (e.g.
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This axiom is closely related to the von Neumann construction of the natural numbers in set theory, in which the successor of x is defined as x ∪ {x}. If x is a set, then it follows from the other axioms of set theory that this successor is also a uniquely defined set. Successors are used to define the usual set-theoretic encoding of the natural numbers. In this encoding, zero is the empty set: :0 = {}. The number 1 is the successor of 0: :1 = 0 ∪ {0} = {} ∪ {0} = {0} = . Likewise, 2 is the successor of 1: :2 = 1 ∪ {1} = {0} ∪ {1} = {0,1} = { {}, }, and so on: :3 = {0,1,2} = { {}, , {{}, } }; :4 = {0,1,2,3} = { {}, , { {}, }, { {}, , {{}, } } }.
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Many articles in The Fibonacci Quarterly deal directly with topics that are very closely related to Fibonacci numbers, such as Lucas numbers, the golden ratio, Zeckendorf representations, Binet forms, Fibonacci polynomials, and Chebyshev polynomials. However, many other topics, especially as related to recurrences, are also well represented. These include primes, pseudoprimes, graph colorings, Euler numbers, continued fractions, Stirling numbers, Pythagorean triples, Ramsey theory, Lucas-Bernoulli numbers, quadratic residues, higher-order recurrence sequences, nonlinear recurrence sequences, combinatorial proofs of number-theoretic identities, Diophantine equations, special matrices and determinants, the Collatz sequence, public-key crypto functions, elliptic curves, fractal dimension, hypergeometric functions, Fibonacci polytopes, geometry, graph theory, music, and art.
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Bubbio's main thesis is that there is a strong interrelation between the kenotic conception of sacrifice and the tradition of Kantian and post-Kantian idealism. In other words, this conception of sacrifice can be seen in the works of most of the thinkers of the post-Kantian tradition. Bubbio argues that the very possibility of a “realm of reason” made up by values and norms depends on the recognition of “the other” as another human being. Particularly he emphasizes the reciprocal connection of the Hegel's recognition-theoretic approach and his emphasis on kenotic sacrifice, both of which are evidence of his belonging to perspectivism.
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Stone duality provides a category theoretic duality between Boolean algebras and a class of topological spaces known as Boolean spaces. Building on nascent ideas of relational semantics (later formalized by Kripke) and a result of R. S. Pierce, Jónsson, Tarski and G. Hansoul extended Stone duality to Boolean algebras with operators by equipping Boolean spaces with relations that correspond to the operators via a power set construction. In the case of interior algebras the interior (or closure) operator corresponds to a pre-order on the Boolean space. Homomorphisms between interior algebras correspond to a class of continuous maps between the Boolean spaces known as pseudo-epimorphisms or p-morphisms for short.
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A 2008 re-evaluation of this algorithm showed it to be no faster than ordinary heapsort for integer keys, presumably because modern branch prediction nullifies the cost of the predictable comparisons which bottom-up heapsort manages to avoid. A further refinement does a binary search in the path to the selected leaf, and sorts in a worst case of comparisons, approaching the information-theoretic lower bound of comparisons. A variant which uses two extra bits per internal node (n−1 bits total for an n-element heap) to cache information about which child is greater (two bits are required to store three cases: left, right, and unknown) uses less than compares.
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For the group of affine transformations on the parameter space of the normal distribution, the right Haar measure is the Jeffreys prior measure. Unfortunately, even right Haar measures sometimes result in useless priors, which cannot be recommended for practical use, like other methods of constructing prior measures that avoid subjective information. Another use of Haar measure in statistics is in conditional inference, in which the sampling distribution of a statistic is conditioned on another statistic of the data. In invariant-theoretic conditional inference, the sampling distribution is conditioned on an invariant of the group of transformations (with respect to which the Haar measure is defined).
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Nina Claire Snaith is a British mathematician at the University of Bristol working in random matrix theory and quantum chaos. In 1998, she and her then adviser Jonathan Keating conjectured a value for the leading coefficient of the asymptotics of the moments of the Riemann zeta function. Keating and Snaith's guessed value for the constant was based on random-matrix theory, following a trend that started with Montgomery's pair correlation conjecture. Keating's and Snaith's work extended works by Conrey, Ghosh and Gonek, also conjectural, based on number theoretic heuristics; Conrey, Farmer, Keating, Rubinstein, and Snaith later conjectured the lower terms in the asymptotics of the moments.
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Prof. Agrawal obtained a PhD degree in Mechanical Engineering from Stanford University in 1990 with emphasis on robotics, dynamics, and control. He currently directs the Robotics and Rehabilitation Laboratory (ROAR) and Robotic Systems Engineering Laboratory (ROSE), which have an active group of PhD, MS, UG, and post-doctoral researchers at Columbia University. He joined Columbia University as a professor in 2013. Before joining Columbia, he was a faculty member at Ohio University and University of Delaware. Agrawal’s current and past research has focused on the design of intelligent machines using non-linear system theoretic principles, computational algorithms for planning and optimization, design of novel rehabilitation machines, and training algorithms for functional rehabilitation of neural impaired adults and children.
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More limited versions of constructivism limit themselves to natural numbers, number-theoretic functions, and sets of natural numbers (which can be used to represent real numbers, facilitating the study of mathematical analysis). A common idea is that a concrete means of computing the values of the function must be known before the function itself can be said to exist. In the early 20th century, Luitzen Egbertus Jan Brouwer founded intuitionism as a part of philosophy of mathematics . This philosophy, poorly understood at first, stated that in order for a mathematical statement to be true to a mathematician, that person must be able to intuit the statement, to not only believe its truth but understand the reason for its truth.
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The coin toss at the start of Super Bowl XLIII Coin tossing is a simple and unbiased way of settling a dispute or deciding between two or more arbitrary options. In a game theoretic analysis it provides even odds to both sides involved, requiring little effort and preventing the dispute from escalating into a struggle. It is used widely in sports and other games to decide arbitrary factors such as which side of the field a team will play from, or which side will attack or defend initially; these decisions may tend to favor one side, or may be neutral. Factors such as wind direction, the position of the sun, and other conditions may affect the decision.
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Condensed Matter Physics at the Racah Institute contains both a strong theoretical and an experimental effort. Most investigations are performed within the expansive field of many-body physics, with particular emphasis on nonequilibrium phenomena, the effects of decoherence and dissipation, the study of low-dimensional systems, and glassy systems, to name but a few subjects. Another direction of research includes statistical physics applied, for example, to reaction diffusion systems, especially in cases where fluctuations have an important effect. Within the realm of theory, the methods being employed range from various field theoretic methods, both exact and perturbative, to numerical methods and exact methods based on the theory of both classical and quantum integrability.
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Similarly, reduced the dimension in which a counterexample to the conjecture is known by finding a clique of size 28 in the Keller graph of dimension eight. Subsequently, showed that the Keller graph of dimension seven has a maximum clique of size 124 < 27. Because this is less than 27, the graph-theoretic version of Keller's conjecture is true in seven dimensions. However, the translation from cube tilings to graph theory can change the dimension of the problem, so this result doesn't settle the geometric version of the conjecture in seven dimensions. Finally, a 200-gigabyte computer- assisted proof in 2019 used Keller graphs to establish that the conjecture holds true in seven dimensions.
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His current research topics include Rise of China, Islamic State and Police Brutality. His previous research includes Applied Game Theoretic Analysis, International Relations Theory, International Political Economy, Formal Models of Bureaucracy and Economic Modeling. Gates holds a BA in political science and anthropology from the University of Minnesota (1980), an MA in political science from the University of Michigan (1983), an MSc in applied economics from the University of Minnesota (1985) and a PhD in political science from the University of Michigan (1989). He moved to Norway in 2003, was a visiting research fellow at PRIO from 1997 to 1999 and continued as program leader and research professor until 2002, when he was appointed director of the CSCW.
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Other formulations of the vertex algebra axioms include Borcherds's later work on singular commutative rings, algebras over certain operads on curves introduced by Huang, Kriz, and others, and D-module- theoretic objects called chiral algebras introduced by Alexander Beilinson and Vladimir Drinfeld. While related, these chiral algebras are not precisely the same as the objects with the same name that physicists use. Important basic examples of vertex operator algebras include lattice VOAs (modeling lattice conformal field theories), VOAs given by representations of affine Kac–Moody algebras (from the WZW model), the Virasoro VOAs (i.e., VOAs corresponding to representations of the Virasoro algebra) and the moonshine module V♮, which is distinguished by its monster symmetry.
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Logical positivists were generally committed to "Unified Science", and sought a common language or, in Neurath's phrase, a "universal slang" whereby all scientific propositions could be expressed.For a review of "unity of science" to, see Gregory Frost-Arnold, "The large-scale structure of logical empiricism: Unity of science and the rejection of metaphysics" . The adequacy of proposals or fragments of proposals for such a language was often asserted on the basis of various "reductions" or "explications" of the terms of one special science to the terms of another, putatively more fundamental. Sometimes these reductions consisted of set-theoretic manipulations of a few logically primitive concepts (as in Carnap's Logical Structure of the World, 1928).
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It was a possible question to ask, around 1957, for a purely category-theoretic characterisation of categories of sheaves of sets, the case of sheaves of abelian groups having been subsumed by Grothendieck's work (the Tôhoku paper). Such a definition of a topos was eventually given five years later, around 1962, by Grothendieck and Verdier (see Verdier's Nicolas Bourbaki seminar Analysis Situs). The characterisation was by means of categories 'with enough colimits', and applied to what is now called a Grothendieck topos. The theory was rounded out by establishing that a Grothendieck topos was a category of sheaves, where now the word sheaf had acquired an extended meaning, since it involved a Grothendieck topology.
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To get a more classical set theory one can look at toposes in which it is moreover a Boolean algebra, or specialising even further, at those with just two truth-values. In that book, the talk is about constructive mathematics; but in fact this can be read as foundational computer science (which is not mentioned). If one wants to discuss set-theoretic operations, such as the formation of the image (range) of a function, a topos is guaranteed to be able to express this, entirely constructively. It also produced a more accessible spin-off in pointless topology, where the locale concept isolates some insights found by treating topos as a significant development of topological space.
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A slight modification of the above game, and the related graph-theoretic problem, makes solving the game NP-hard. The modified game has the Rabin acceptance condition. Specifically, in the above bipartite graph scenario, the problem now is to determine if there is a choice function selecting a single out-going edge from each vertex of V0, such that the resulting subgraph has the property that in each cycle (and hence each strongly connected component) it is the case that there exists an i and a node with color 2i, and no node with color 2i + 1... Note that as opposed to parity games, this game is no longer symmetric with respect to players 0 and 1\.
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The term architrave has also been used in academic writing to mean the fundamental part of something (a speech, a thought or a reasoning), or the basis upon which an idea, reasoning, thought or philosophy is built. Examples: # "...the Mature Hegel – the Hegel of the Philosophy of Right – who becomes the architrave on which he (Honneth, ed.) constructs his social philosophy."Page: XIV, The Ethics of Democracy: A Contemporary Reading of Hegel's Philosophy of Right (Lucio Cortella, SUNY Press, 2015) #"to become the architrave of his theoretic construction"Pag. 281, Economics and institutions Contributions from the History of Economic thought (Pier Francesco Asso, Luca Fiorito, Italian Association for History and Economic Thought, Vol.
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With this in hand, the proof becomes easy: the (filter generated by the) image of an ultrafilter on the product space under any projection map is an ultrafilter on the factor space, which therefore converges, to at least one xi. One then shows that the original ultrafilter converges to x = (xi). In his textbook, Munkres gives a reworking of the Cartan–Bourbaki proof that does not explicitly use any filter-theoretic language or preliminaries. 4) Similarly, the Moore–Smith theory of convergence via nets, as supplemented by Kelley's notion of a universal net, leads to the criterion that a space is compact if and only if each universal net on the space converges.
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However, the spaces in which every convergent filter has a unique limit are precisely the Hausdorff spaces. In general we must select, for each element of the index set, an element of the nonempty set of limits of the projected ultrafilter base, and of course this uses AC. However, it also shows that the compactness of the product of compact Hausdorff spaces can be proved using (BPI), and in fact the converse also holds. Studying the strength of Tychonoff's theorem for various restricted classes of spaces is an active area in set-theoretic topology. The analogue of Tychonoff's theorem in pointless topology does not require any form of the axiom of choice.
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In cryptography, a private information retrieval (PIR) protocol is a protocol that allows a user to retrieve an item from a server in possession of a database without revealing which item is retrieved. PIR is a weaker version of 1-out-of-n oblivious transfer, where it is also required that the user should not get information about other database items. One trivial, but very inefficient way to achieve PIR is for the server to send an entire copy of the database to the user. In fact, this is the only possible protocol (in the classical or the quantum setting) that gives the user information theoretic privacy for their query in a single-server setting.
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A motivating example of spectral triple is given by the algebra of smooth functions on a compact spin manifold, acting on the Hilbert space of L2-spinors, accompanied by the Dirac operator associated to the spin structure. From the knowledge of these objects one is able to recover the original manifold as a metric space: the manifold as a topological space is recovered as the spectrum of the algebra, while the (absolute value of) Dirac operator retains the metric.A. Connes, Noncommutative Geometry, Academic Press, 1994 On the other hand, the phase part of the Dirac operator, in conjunction with the algebra of functions, gives a K-cycle which encodes index-theoretic information. The local index formulaA.
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A network is a graph with real numbers associated with each of its edges, and if the graph is a digraph, the result is a directed network. A flow graph is more general than a directed network, in that the edges may be associated with gains, branch gains or transmittances, or even functions of the Laplace operator s, in which case they are called transfer functions. There is a close relationship between graphs and matrices and between digraphs and matrices. "The algebraic theory of matrices can be brought to bear on graph theory to obtain results elegantly", and conversely, graph-theoretic approaches based upon flow graphs are used for the solution of linear algebraic equations.
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GTS is mutually interpretable with Peano arithmetic (thus it has the same proof-theoretic strength as PA); The most remarkable fact about ST (and hence GST), is that these tiny fragments of set theory give rise to such rich metamathematics. While ST is a small fragment of the well-known canonical set theories ZFC and NBG, ST interprets Robinson arithmetic (Q), so that ST inherits the nontrivial metamathematics of Q. For example, ST is essentially undecidable because Q is, and every consistent theory whose theorems include the ST axioms is also essentially undecidable.Burgess (2005), 2.2, p. 91. This includes GST and every axiomatic set theory worth thinking about, assuming these are consistent.
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The theory of mediation, which is the principal referent of the research group of the Interdisciplinary Laboratory for Language Research (L.I.R.L.), is a theoretic model developed at Rennes (France) since the 1960' by Professor Jean Gagnepain, linguist and epistemologist. This model, whose principles Jean Gagnepain has methodically set forth in his three volume study On Meaning (Du Vouloir Dire),Jean Gagnepain Jean, Du Vouloir Dire I, Bruxelles, De Boeck, 1990Jean Gagnepain Jean, Du Vouloir Dire II, Bruxelles, De Boeck, 1991 covers the whole field of the human sciences. One essential feature of the theory is that it seeks to find a kind of experimental verification of its theorems in the clinic of psychopathology.
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It can be defined in graph theoretic terms by choosing an arbitrary orientation of the graph, and defining an integral cycle of a graph G to be an assignment of integers to the edges of G (an element of the free abelian group over the edges) with the property that, at each vertex, the sum of the numbers assigned to incoming edges equals the sum of the numbers assigned to outgoing edges.. A member of H_1(G,\Z) or of H_1(G,\Z_k) (the cycle space modulo k) with the additional property that all of the numbers assigned to the edges are nonzero is called a nowhere-zero flow or a nowhere-zero k-flow..
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This resulted in a series of papers with Jere Behrman and the late Paul Taubman which are included in a book published by the University of Chicago Press in 1995. Pollak's move to the University of Washington in 1985 marked the beginning of his long and fruitful collaboration with Shelly Lundberg on bargaining in marriage and other family issues. The "separate spheres bargaining model," developed in their most widely cited article, provides a game theoretic analysis of bargaining in marriage. A closely related empirical paper (joint with Shelly Lundberg and Terence J. Wales) finds strong evidence that the fraction of household resources controlled by each spouse is an important determinant of allocation within marriage.
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Ordinal analysis concerns true, effective (recursive) theories that can interpret a sufficient portion of arithmetic to make statements about ordinal notations. The proof- theoretic ordinal of such a theory T is the smallest ordinal (necessarily recursive, see next section) that the theory cannot prove is well founded--the supremum of all ordinals \alpha for which there exists a notation o in Kleene's sense such that T proves that o is an ordinal notation. Equivalently, it is the supremum of all ordinals \alpha such that there exists a recursive relation R on \omega (the set of natural numbers) that well-orders it with ordinal \alpha and such that T proves transfinite induction of arithmetical statements for R.
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The homotopy category of a model category C is the localization of C with respect to the class of weak equivalences. This definition of homotopy category does not depend on the choice of fibrations and cofibrations. However, the classes of fibrations and cofibrations are useful in describing the homotopy category in a different way and in particular avoiding set-theoretic issues arising in general localizations of categories. More precisely, the "fundamental theorem of model categories" states that the homotopy category of C is equivalent to the category whose objects are the objects of C which are both fibrant and cofibrant, and whose morphisms are left homotopy classes of maps (equivalently, right homotopy classes of maps) as defined above.
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The utility graph K3,3 has no 2-page book embedding, but it can be drawn as shown in a 2-page book with only one crossing. Therefore, its 2-page book crossing number is 1. This 1-page embedding of the diamond graph has pagewidth 3, because the yellow ray crosses three edges. A book is a particular kind of topological space, also called a fan of half-planes... It consists of a single line , called the spine or back of the book, together with a collection of one or more half-planes, called the pages or leaves of the book,The "spine" and "pages" terminology is more standard in modern graph-theoretic approaches to the subject.
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He was a leader in critical behavior theory and developed methods for distilling testable predictions for critical exponents. In using field theoretic techniques in the study of condensed matter, Brezin helped further modern theories of magnetism and the quantum Hall effect. Brézin was elected a member of the French Academy of Sciences on 18 February 1991 and served as president of the academy in 2005-2006. He also is a foreign associate of the United States National Academy of Sciences (since 2003), a foreign honorary member of the American Academy of Arts and Sciences (since 2002), a foreign member of the Royal Society (since 2006) and a member of the Academia Europaea (since 2003).
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Algorithms for Molecular Biology, 2010 Since 2014 he has focused on the application and development of integer linear programming in computational biology. Gusfield is most well known for his book Algorithms on Strings, Trees and Sequences: Computer Science and Computational Biology, which provides a comprehensive presentation of the algorithmic foundations of molecular sequence analysis for computer scientists, and has been cited more than 6000 times. This book has helped to define and develop the intersection of computer science and computational biology. His second book in computational biology is on phylogenetic networks, which are graph- theoretic models of evolution that go beyond the classical tree model, to address biological processes such as hybridization, recombination, and horizontal gene transfer.
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By using this method, in n steps of communication, the sender can communicate up to 5n/2 messages, significantly more than the 2n that could be transmitted with the simpler one- digit code. The effective number of values that can be transmitted per unit time step is (5n/2)1/n = . In graph-theoretic terms, this means that the Shannon capacity of the 5-cycle is at least . As showed, this bound is tight: it is not possible to find a more complicated system of code words that allows even more different messages to be sent in the same amount of time, so the Shannon capacity of the 5-cycle is exactly .
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He provided the first axiomatic measure-theoretic description of coin-tossing, which was to influence the full axiomatization of probability by the Russian mathematician Andrey Kolmogorov a decade later. Steinhaus was also the first to offer precise definitions of what it means for two events to be "independent", as well as for what it means for a random variable to be "uniformly distributed". While in hiding during World War II, Steinhaus worked on the fair cake-cutting problem: how to divide a heterogeneous resource among several people with different preferences such that every person believes he received a proportional share. Steinhaus' work has initiated the modern research of the fair cake-cutting problem.
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Systematic musicologists who are oriented toward the humanities often make reference to fields such as aesthetics, philosophy, semiotics, hermeneutics, music criticism, Media studies, Cultural studies, gender studies, and (theoretic) sociology. Those who are oriented toward science tend to regard their discipline as empirical and data-oriented, and to borrow their methods and ways of thinking from psychology, acoustics, psychoacoustics, physiology, cognitive science, and (empirical) sociology. More recently emerged areas of research which at least partially are in the scope of systematic musicology comprise cognitive musicology, neuromusicology, biomusicology, and music cognition including embodied music cognition. As an academic discipline, systematic musicology is closely related to practically oriented disciplines such as music technology, music information retrieval, and musical robotics.
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K-theory was invented in the late 1950s by Alexander Grothendieck in his study of intersection theory on algebraic varieties. In the modern language, Grothendieck defined only K0, the zeroth K-group, but even this single group has plenty of applications, such as the Grothendieck–Riemann–Roch theorem. Intersection theory is still a motivating force in the development of (higher) algebraic K-theory through its links with motivic cohomology and specifically Chow groups. The subject also includes classical number-theoretic topics like quadratic reciprocity and embeddings of number fields into the real numbers and complex numbers, as well as more modern concerns like the construction of higher regulators and special values of L-functions.
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Computable model theory is a branch of model theory which deals with questions of computability as they apply to model-theoretical structures. Computable model theory introduces the ideas of computable and decidable models and theories and one of the basic problems is discovering whether or not computable or decidable models fulfilling certain model-theoretic conditions can be shown to exist. Computable model theory was developed almost simultaneously by mathematicians in the West, primarily located in the United States and Australia, and Soviet Russia during the middle of the 20th century. Because of the Cold War there was little communication between these two groups and so a number of important results were discovered independently.
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By forgetting the face structure, any polyhedron gives rise to a graph, called its skeleton, with corresponding vertices and edges. Such figures have a long history: Leonardo da Vinci devised frame models of the regular solids, which he drew for Pacioli's book Divina Proportione, and similar wire-frame polyhedra appear in M.C. Escher's print Stars. Coxeter's analysis of Stars is on pp. 61–62. One highlight of this approach is Steinitz's theorem, which gives a purely graph- theoretic characterization of the skeletons of convex polyhedra: it states that the skeleton of every convex polyhedron is a 3-connected planar graph, and every 3-connected planar graph is the skeleton of some convex polyhedron.
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Z. Eckstein and I. Zilcha "The Effects of Compulsory Schooling on Growth, Income Distribution and Welfare ", Journal of Public Economics, 54, 1994, 339-359. Duration to First Job and the Return to Schooling: Estimates from a Search-Matching Model The article investigates the properties of joint distribution of the duration to the first post-schooling full-time job and of the accepted wage for that job within a search-matching- bargaining theoretic model. The article was co-written with prof. Kenneth I. Wolpin of University of Pennsylvania. Z. Eckstein and K.I. Wolpin "Duration to First Job and the Return to Schooling: Estimates from a Search-Matching Model ", Review of Economic Studies, 1995, 62, 263-286.
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