Music "hypnotizes and causes man to abandon the expressible," he writes.
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The title of Forrest Gander's latest book of poems, " Be With " (New Directions), is a blurted command welling up from yearnings not quite expressible in language.
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Then in the mid-1890s Henri Poincaré showed that actually there couldn't be any constants of the motion that were expressible as any analytic functions of the positions, velocities and mass ratios.
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In 1887, though, Heinrich Bruns showed that there couldn't be any such constants of the motion, at least expressible as algebraic functions of the standard {x,y,z} position and velocity coordinates of the three bodies.
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Scruton is not suggesting that in those cases, some numinous entity — the image — is created; he is suggesting that a different way of seeing the lines and fields is available to us, a way of seeing that exposes us to a world beyond the one expressible by any purely physical description of paint.
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Both type of spheroidal harmonics are expressible in terms of Legendre functions.
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Still, algorithms like classification, filter kernels and general convolutions, histograms, and Discrete Fourier Transform are expressible.
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Similar formulae are known for cubic and quartic equations, but do not exist in general for degree 5 and higher. (see in particular p. 273 for concrete examples) Abstract properties of Galois groups associated with polynomials (in particular their solvability) give a criterion for polynomials that have all their solutions expressible by radicals, i.e., solutions expressible using solely addition, multiplication, and roots similar to the formula above.
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The first significant result in this area, Fagin's theorem (1974) established that NP is precisely the set of languages expressible by sentences of existential second-order logic.
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2000 (two thousand) is a natural number following 1999 and preceding 2001. Two thousand is the highest number expressible using only two unmodified characters in Roman numerals (MM).
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With the advent of a universal currency as an intermediary, these systems became reconcilable, as everything tended to become expressible in a single quantifiable metric: its monetary cost.
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Conversely, the restriction to the elements of order coprime to p of each ordinary irreducible character is uniquely expressible as a non-negative integer combination of irreducible Brauer characters.
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Overprinted genes are particularly common features of the genomic organization of viruses, likely to greatly increase the number of potential expressible genes from a small set of viral genetic information.
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Unlike palindromes, it is also font dependent. The concept of strobogrammatic numbers is not neatly expressible algebraically, the way that the concept of repunits is, or even the concept of palindromic numbers.
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The fact that the musician meant to pluck it at a mathematically expressible point. However, if the mathematical proportion between the points on the string were to be broken, the sound would become unsettled.
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The tool use in tool-assisted speedrunning is therefore different from the sort of state manipulation that tools like Gameshark provide, since such manipulation would not be expressible as a sequence of timed inputs.
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In the case of Boolean algebra x = y can also be translated as (x \land y) \lor ( eg x \land eg y), but this translation is incorrect intuitionistically. In both Boolean and Heyting algebra, inequality x \le y can be used in place of equality. The equality x = y is expressible as a pair of inequalities x \le y and y \le x. Conversely the inequality x \le y is expressible as the equality x \land y = x, or as x \lor y = y.
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Projective determinacy is the assertion that every two-player perfect information game with moves being integers, game length ω and projective payoff set is determined, that is one of the players has a winning strategy. (The first player wins the game if the play belongs to the payoff set; otherwise, the second player wins.) A set is projective iff (as a predicate) it is expressible by a formula in the language of second-order arithmetic, allowing real numbers as parameters, so projective determinacy is expressible as a schema in the language of Z2. Many natural propositions expressible in the language of second-order arithmetic are independent of Z2 and even ZFC but are provable from projective determinacy. Examples include coanalytic perfect subset property, measurability and the property of Baire for \Sigma^1_2 sets, \Pi^1_3 uniformization, etc.
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These constructions can be applied to all topological spaces, and so singular homology can be expressed in terms of category theory, where homology is expressible as a functor from the category of topological spaces to the category of graded abelian groups.
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Its most common use is in Abraham Robinson's nonstandard analysis of the hyperreal numbers, where the transfer principle states that any sentence expressible in a certain formal language that is true of real numbers is also true of hyperreal numbers.
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An overlapping gene is a gene whose expressible nucleotide sequence partially overlaps with the expressible nucleotide sequence of another gene. In this way, a nucleotide sequence may make a contribution to the function of one or more gene products. Overprinting refers to a type of overlap in which all or part of the sequence of one gene is read in an alternate reading frame from another gene at the same locus. Overprinting has been hypothesized as a mechanism for de novo emergence of new genes from existing sequences, either older genes or previously non-coding regions of the genome.
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The primary solution angles in the form (cos,sin) on the unit circle are at multiples of 30 and 45 degrees. Exact algebraic expressions for trigonometric values are sometimes useful, mainly for simplifying solutions into radical forms which allow further simplification. All trigonometric numbers – sines or cosines of rational multiples of 360° – are algebraic numbers (solutions of polynomial equations with integer coefficients); moreover they may be expressed in terms of radicals of complex numbers; but not all of these are expressible in terms of real radicals. When they are, they are expressible more specifically in terms of square roots.
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For the polynomial , the lone real root is algebraic, but not expressible in terms of radicals. The other four roots are complex numbers. Van der Waerdenvan der Waerden, Modern Algebra (1949 English edn.), Vol. 1, Section 61, p.191 cites the polynomial .
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720 is expressible as the product of consecutive integers in two different ways: , and . There are 49 solutions to the equation φ(x) = 720, more than any integer below it, making 720 a highly totient number. 720 is a 241-gonal number.
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Quine further made a distinction between the ontological commitments of a theory (what the theory says exists) and the ideological commitments of a theory (those concepts, logical or non-logical, that are expressible within the theory)."Ontological Commitment". Stanford Encyclopedia of Philosophy.
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The legality of this mapping depends on the nature of the correspondence between G and S. Two popular ways to model this correspondence exist: Global as View or GAV and Local as View or LAV. Figure 3: Illustration of tuple space of the GAV and LAV mappings. In GAV, the system is constrained to the set of tuples mapped by the mediators while the set of tuples expressible over the sources may be much larger and richer. In LAV, the system is constrained to the set of tuples in the sources while the set of tuples expressible over the global schema can be much larger.
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This sampling mode is not expressible in J:a:b notation. "4:2:1" is an obsolete term from a previous notational scheme, and very few software or hardware codecs use it. Cb horizontal resolution is half that of Cr (and a quarter of the horizontal resolution of Y).
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This is a useful approximation as the total energy consists of contributions only from the kinetic energy and exchange- correlation energy, and that the wavefunction is expressible in terms of planewaves. In particular, for a constant density ρ, the exchange energy density is proportional to ρ⅓.
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All constraints are expressible in the same language. A database can be considered a structure in realization of the database language. The states of a created conceptual schema are transformed into an explicit mapping, the database schema. This describes how real-world entities are modeled in the database.
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In topology, a topological space is said to be resolvable if it is expressible as the union of two disjoint dense subsets. For instance, the real numbers form a resolvable topological space because the rationals and irrationals are disjoint dense subsets. A topological space that is not resolvable is termed irresolvable.
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As Quine puts it, > the general adoption of class variables of quantification ushers in a theory > whose laws were not in general expressible in the antecedent levels of > logic. The price paid for this increased power is ontological: objects of a > special and abstract kind, viz. classes, are now presupposed.
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W. Galway wrote a computer programme to determine odd integers not expressible as . Galway verified that there are only eighteen numbers less than not representable in the form . Based on Galway's computations, Ken Ono and K. Soundararajan formulated the following conjecture: :The odd positive integers which are not of the form x2 \+ are: .
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In classical mechanics a system may be defined as holonomic if all constraints of the system are holonomic. For a constraint to be holonomic it must be expressible as a function: : f(x_1,\ x_2,\ x_3,\ \ldots,\ x_N,\ t)=0, \, i.e. a holonomic constraint depends only on the coordinates x_j\,\\! and time t\,\\!.
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The prime decomposition of the number 2450 is given by 2450 = 257. Of the primes occurring in this decomposition, 2, 5, and 7, only 7 is congruent to 3 modulo 4. Its exponent in the decomposition, 2, is even. Therefore, the theorem states that it is expressible as the sum of two squares.
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This is the easier part of the theorem, and follows immediately from the observation that all squares are congruent to 0 or 1 modulo 4. Since the Diophantus identity implies that the product of two integers each of which can be written as the sum of two squares is itself expressible as the sum of two squares, by applying Fermat's theorem to the prime factorization of any positive integer n, we see that if all the prime factors of n congruent to 3 modulo 4 occur to an even exponent, then n is expressible as a sum of two squares. The converse also holds.For a proof of the converse see for instance 20.1, Theorems 367 and 368, in: G.H. Hardy and E.M. Wright.
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Gestus is an acting technique developed by the German theatre practitioner Bertolt Brecht. It carries the sense of a combination of physical gestures and "gist" or attitude. It is a means by which "an attitude or single aspect of an attitude" is revealed, insofar as it is "expressible in words or actions."Willett (1964, 42).
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See the footnote at the end of Soare: 1996. showing that a general solution to the ' is impossible, assuming that the intuitive notion of "effectively calculable" is captured by the functions computable by a Turing machine (or equivalently, by those expressible in the lambda calculus). This assumption is now known as the Church–Turing thesis.
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Incidence and prevalence are terms commonly used in describing disease epidemiology. Incidence is newly diagnosed cases, which can be expressed as new cases per year per million people. Prevalence is the number of cases alive, expressible as existing cases per million people during a period of time."Incidence and Prevalence" Advanced Renal Education Program (Accessed 17 October 2017).
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Some models of microscope use automatic electronic models for reciprocity failure compensation, generally of a form for correct time, Tc, expressible as a power law of metered time, Tm, that is, Tc=(Tm)p, for times in seconds. Typical values of p are 1.25 to 1.45, but some are low as 1.1 and high as 1.8.
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A commemorative plaque now appears at the site of the Hardy–Ramanujan incident, at 2 Colinette Road in Putney. The same expression defines 1729 as the first in the sequence of "Fermat near misses" defined, in reference to Fermat's Last Theorem, as numbers of the form 1 + z3 which are also expressible as the sum of two other cubes.
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Although this model is very robust, no practical circuits are possible due to the lack of expressible conditionals in DI circuits. Instead the Quasi-Delay-Insensitive model is the smallest compromise model yet capable of generating useful computing circuits. For this reason circuits are often incorrectly referred to as Delay- Insensitive when they are Quasi Delay-Insensitive.
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275 For rigid elastic spheres, \Omega(T) is independent of T and very close to 1. More complex interaction laws introduce a weak temperature dependence. The precise nature of the dependence is not always easy to discern, however, as \Omega(T) is defined as a multi- dimensional integral which may not be expressible in terms of elementary functions.
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In a programming language where function parameters are statically typed, a function may be defined as a partial function because the language's type system cannot express the exact domain of the function, so the programmer instead gives it the smallest domain which is expressible as a type and contains the domain of definition of the function.
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Albert Girard was the first to make the observation, describing all positive integer numbers (not necessarily primes) expressible as the sum of two squares of positive integers; this was published in 1625.Simon Stevin. l'Arithmétique de Simon Stevin de Bruges, annotated by Albert Girard, Leyde 1625, p. 622.L. E. Dickson, History of the Theory of Numbers, Vol.
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Establishing whether a constraint satisfaction problem on a finite domain has solutions is an NP complete problem in general. This is an easy consequence of a number of other NP complete problems being expressible as constraint satisfaction problems. Such other problems include propositional satisfiability and three-colorability. Tractability can be obtained by considering specific classes of constraint satisfaction problems.
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Highly significant for that application is whether each character of G is a non-negative integer combination of characters induced from linear characters of subgroups. In general, this is not the case. In fact, by a theorem of Taketa, if all characters of G are so expressible, then G must be a solvable group (although solvability alone does not guarantee such expressions- for example, the solvable group SL(2,3) has an irreducible complex character of degree 2 which is not expressible as a non-negative integer combination of characters induced from linear characters of subgroups). An ingredient of the proof of Brauer's induction theorem is that when G is a finite nilpotent group, every complex irreducible character of G is induced from a linear character of some subgroup.
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A special attribute was that the lyrics were not monolingual. Besides English, they were written also in German, French and Turkish. Turkish had an extra role due to this language obtained to be exotic in alternative music back then. 2005/2006 αlabay realized that several things one cannot think or feel in German or English were better expressible in Turkish.
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A Rights Expression Language or REL is a machine-processable language used to express intellectual property rights (such as copyright) and other terms and conditions for use over content. RELs can be used as standalone expressions (i.e. metadata usable for search, compatibility tracking) or within a DRM system. RELs are expressible in a machine-language (such as XML, RDF , RDF Schema, and JSON).
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There is an increased chance for women over the age of 35 to have multiple births. IVF is a common genetic and ethical topic. Through IVF individuals can produce offspring successfully when natural procreation is not viable. However, in vitro can become genetically specific and allow for the selection of particular genes or expressible traits to be dominantly present in the formed embryo.
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Although it is fairly easy to get an intuitive grasp of this idea, it is nevertheless desirable to have some more definite, mathematically expressible definition. Such a definition was first given by Gödel at Princeton in 1934 ... . These functions are described as "general recursive" by Gödel ... . Another definition of effective calculability has been given by Church ... who identifies it with λ-definability.
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Although its equations of motion can be linearized, a bike is a nonlinear system. The variable(s) to be solved for cannot be written as a linear sum of independent components, i.e. its behavior is not expressible as a sum of the behaviors of its descriptors. Generally, nonlinear systems are difficult to solve and are much less understandable than linear systems.
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For example, the symmetric groups is not solvable for . Consequently, as can be shown, the zeros of the following polynomials are not expressible by sums, products, and radicals. For the latter polynomial, this fact is known as the Abel–Ruffini theorem: : (and ), : (where is regarded as a polynomial in , for some indeterminates , is any field, and ). The tensor product of fields is not usually a field.
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As in the Gaussian units, the Heaviside–Lorentz units (HLU in this article) use the length–mass–time dimensions. This means that all of the electric and magnetic units are expressible in terms of the base units of length, time and mass. Coulomb's equation, used to define charge in these systems, is in the Gaussian system, and in the HLU. The unit of charge then connects to .
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Thus expressing complexities in terms of \omega provide a more realistic complexity, since it remains valid whichever algorithm is chosen for matrix computation. Problems that have the same asymptotic complexity as matrix multiplication include determinant, matrix inversion, Gaussian elimination (see next section). Problems with complexity that is expressible in terms of \omega include characteristic polynomial, eigenvalues (but not eigenvectors), Hermite normal form, and Smith normal form.
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Corrective glasses are used to cure visual dyslexia A writing system is a type of symbolic system used to represent elements or statements expressible in language. The orthography of a language specifies the correct way of using a specific writing system to write the language. Where more than one writing system is used for a language, for example for Kurdish, there can be more than one orthography.
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Conjunctive grammars are a class of formal grammars studied in formal language theory. They extend the basic type of grammars, the context-free grammars, with a conjunction operation. Besides explicit conjunction, conjunctive grammars allow implicit disjunction represented by multiple rules for a single nonterminal symbol, which is the only logical connective expressible in context-free grammars. Conjunction can be used, in particular, to specify intersection of languages.
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The intellectual purpose of Newspeak is to make Ingsoc-approved thoughts the only expressible thoughts. As constructed, Newspeak's vocabulary communicates the exact expression of sense and meaning that a member of the Party could wish to express. It excludes secondary denotations and connotations. The linguistic simplification of Oldspeak into Newspeak was realised with neologisms, the elimination of ideologically undesirable words, and the elimination of the politically unorthodox meanings of words.
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That is, all facts related to the process of understanding must be expressible in terms of natural facts. If this is not true, i.e. there are facts which cannot be expressed as natural facts, science would have no means of investigating them. In this vein, Roderick Chisholm argues that there are epistemic principles (or facts) which are necessary to knowledge acquisition, but may not be, themselves, natural facts.
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Fagin's theorem is a result in descriptive complexity theory that states that the set of all properties expressible in existential second-order logic is precisely the complexity class NP. It is remarkable since it is a characterization of the class NP that does not invoke a model of computation such as a Turing machine. The theorem was proven by Ronald Fagin in 1974 (strictly, in 1973 in his doctoral thesis).
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Not every polynomial time graph property can be modeled by a formula in a logic that uses only fixed points and counting. However, for some special classes of graphs the polynomial time properties are the same as the properties expressible in fixed point logic with counting. These include random graphs, interval graphs, and (through a logical expression of the graph structure theorem) every class of graphs characterized by forbidden minors.
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Before collapse, the wave function may be any square-integrable function. This function is expressible as a linear combination of the eigenstates of any observable. Observables represent classical dynamical variables, and when one is measured by a classical observer, the wave function is projected onto a random eigenstate of that observable. The observer simultaneously measures the classical value of that observable to be the eigenvalue of the final state.
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While they can be added to other words, they by themselves are just a name. When these elements are added together, Socrates says that a 'complex' is formed (202b). The primary elements are 'unaccountable and unknowable, but perceivable' while the complexes are 'knowable and expressible' and so can be objects of 'true judgement' (202b). He concludes his dream by agreeing with Theaetetus that knowledge is 'true judgement with an account' (202c).
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For 1366×768 pixel Wide XGA panels the nearest resolution expressible in the EDID standard timing descriptor syntax is 1360×765 pixels, typically leading to 3 pixel thin black bars. Specifying 1368 pixels as the screen width would yield an unnatural screen height of 769.5 pixels. Many Wide XGA panels do not advertise their native resolution in the standard timing descriptors, instead offering only a resolution of 1280×768.
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That is, all facts related to the process of understanding must be expressible in terms of natural facts. If this is not true, i.e. there are facts which cannot be expressed as natural facts, science would have no means of investigating them. In this vein, Roderick Chisholm argues that there are epistemic principles (or facts) which are necessary to knowledge acquisition, but may not be, themselves, natural facts.
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The equal-tempered intervals are black; the Pythagorean intervals are green. Below is a list of intervals expressible in terms of a prime limit (see Terminology), completed by a choice of intervals in various equal subdivisions of the octave or of other intervals. For commonly encountered harmonic or melodic intervals between pairs of notes in contemporary Western music theory, without consideration of the way in which they are tuned, see Interval (music) § Main intervals.
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The P = NP problem can be restated in terms of expressible certain classes of logical statements, as a result of work in descriptive complexity. Consider all languages of finite structures with a fixed signature including a linear order relation. Then, all such languages in P can be expressed in first-order logic with the addition of a suitable least fixed-point combinator. Effectively, this, in combination with the order, allows the definition of recursive functions.
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An R-algebra S is called finitely generated (as an algebra) if there are finitely many elements s1, ..., sn such that any element of s is expressible as a polynomial in the si. Equivalently, S is isomorphic to :R[T1, ..., Tn] / I. A much stronger condition is that S is finitely generated as an R-module, which means that any s can be expressed as a R-linear combination of some finite set s1, ..., sn.
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Alistair MacLeod, (July 20, 1936 - April 20, 2014) was a Canadian novelist, short story writer and academic. His powerful and moving stories vividly evoke the beauty of Cape Breton Island's rugged landscape and the resilient character of many of its inhabitants, the descendants of Scottish immigrants, who are haunted by ancestral memories and who struggle to reconcile the past and the present.Joan Thomas. "Alistair MacLeod's expressible island." The Globe and Mail, April 15, 2000, p.
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Boolean grammars, introduced by , are a class of formal grammars studied in formal language theory. They extend the basic type of grammars, the context- free grammars, with conjunction and negation operations. Besides these explicit operations, Boolean grammars allow implicit disjunction represented by multiple rules for a single nonterminal symbol, which is the only logical connective expressible in context-free grammars. Conjunction and negation can be used, in particular, to specify intersection and complement of languages.
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Arthur Cayley and Felix Klein found an application of the cross-ratio to non- Euclidean geometry. Given a nonsingular conic C in the real projective plane, its stabilizer GC in the projective group acts transitively on the points in the interior of C. However, there is an invariant for the action of GC on pairs of points. In fact, every such invariant is expressible as a function of the appropriate cross ratio.
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Attitudinals are a set of cmavo which allow the speakers to express their emotional state or source of knowledge, or the present stage of discourse. In natural languages, attitudes are expressed using interjections, but also by the tone of voice when speaking, and (very imperfectly) by punctuation when writing; in Lojban, such information are extensively expressible in words. And the meanings are to be understood separately from the main predicate. :.iu (love) :.
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Using Solèr's theorem, the field K over which the vector space is defined can be proven, with additional hypotheses, to be either the real numbers, complex numbers, or the quaternions, as is needed for Gleason's theorem to hold. By invoking Gleason's theorem, the form of a probability function on lattice elements can be restricted. Assuming that the mapping from lattice elements to probabilities is noncontextual, Gleason's theorem establishes that it must be expressible with the Born rule.
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This is true regardless of whether the Hilbert space is finite-dimensional or not. Geometrically, when the state is not expressible as a convex combination of other states, it is a pure state. The family of mixed states is a convex set and a state is pure if it is an extremal point of that set. It follows from the spectral theorem for compact self-adjoint operators that every mixed state is a countable convex combination of pure states.
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This remix in Lessig's eyes is exemplary of the power this type of expression holds - to not tell but show. Using preexisting images is vital to the art form because the production of meaning draws heavily on cultural reference an image or sound brings with it. > Their meaning comes not from the content of what they say; it comes from the > reference, which is expressible only if it is the original that gets > used.Lessig, Lawrence. 2008.
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Much of the material in this section is about the limitations of language in expressing levels of significance beyond that which can be effectively captured by words, so that, seemingly, what we need to understand may only be expressible in a "language" that we do not know! That paradox is explored in a page entitled Language and the reconstruction of reality.Human potential and development project – Comments: language and the reconstruction of reality. Union of International Associations.
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In mathematics, Euler's idoneal numbers (also called suitable numbers or convenient numbers) are the positive integers D such that any integer expressible in only one way as x2 ± Dy2 (where x2 is relatively prime to Dy2) is a prime power or twice a prime power. In particular, a number that has two distinct representations as a sum of two squares is composite. Every idoneal number generates a set containing infinitely many primes and missing infinitely many other primes.
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Chen's theorem proves that for all sufficiently large n, 2n=p+q where p is prime and q is either prime or semiprime.A semiprime is a natural number that is the product of two prime factors. Montgomery and Vaughan showed that the exceptional set (even numbers not expressible as the sum of two primes) was of density zero. The best current bound on the exceptional set is E(x) < x^{0.72} (for large enough x) due to Pintz.
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A notable property is the closure of the set of all G-functions not only under differentiation but also under indefinite integration. In combination with a functional equation that allows to liberate from a G-function G(z) any factor zρ that is a constant power of its argument z, the closure implies that whenever a function is expressible as a G-function of a constant multiple of some constant power of the function argument, f(x) = G(cxγ), the derivative and the antiderivative of this function are expressible so too. The wide coverage of special functions also lends power to uses of Meijer's G-function other than the representation and manipulation of derivatives and antiderivatives. For example, the definite integral over the positive real axis of any function g(x) that can be written as a product G1(cxγ)·G2(dxδ) of two G-functions with rational γ/δ equals just another G-function, and generalizations of integral transforms like the Hankel transform and the Laplace transform and their inverses result when suitable G-function pairs are employed as transform kernels.
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Subshifts of finite type are identical to free (non-interacting) one- dimensional Potts models (n-letter generalizations of Ising models), with certain nearest-neighbor configurations excluded. Interacting Ising models are defined as subshifts together with a continuous function of the configuration space (continuous with respect to the product topology, defined below); the partition function and Hamiltonian are explicitly expressible in terms of this function. Subshifts may be quantized in a certain way, leading to the idea of the quantum finite automata.
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There are other ordinal notations capable of capturing ordinals well past \varepsilon_0, but because there are only countably many strings over any finite alphabet, for any given ordinal notation there will be ordinals below \omega_1 (the first uncountable ordinal) that are not expressible. Such ordinals are known as large countable ordinals. The operations of addition, multiplication and exponentiation are all examples of primitive recursive ordinal functions, and more general primitive recursive ordinal functions can be used to describe larger ordinals.
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However, its probability density function is not expressible in terms of elementary functions; rather, the probability density function is expressed in terms of hypergeometric functions. The Holtsmark distribution has applications in plasma physics and astrophysics. In 1919, Norwegian physicist J. Holtsmark proposed the distribution as a model for the fluctuating fields in plasma due to chaotic motion of charged particles. It is also applicable to other types of Coulomb forces, in particular to modeling of gravitating bodies, and thus is important in astrophysics.
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The sine function is periodic with period 2\pi, since :\sin(x + 2\pi) = \sin x for all values of x. This function repeats on intervals of length 2\pi (see the graph to the right). Everyday examples are seen when the variable is time; for instance the hands of a clock or the phases of the moon show periodic behaviour. Periodic motion is motion in which the position(s) of the system are expressible as periodic functions, all with the same period.
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However, the number 2 raised to any positive integer power must be even (because it is divisible by 2) and the number 3 raised to any positive integer power must be odd (since none of its prime factors will be 2). Clearly, an integer cannot be both odd and even at the same time: we have a contradiction. The only assumption we made was that log2 3 is rational (and so expressible as a quotient of integers m/n with n ≠ 0).
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Several states in the United States have adopted learning trajectories from the Common Core State Standards Initiative's guidelines for mathematics education. Aside from sequencing the learning of fractions and operations with fractions, the document provides the following definition of a fraction: "A number expressible in the form where a is a whole number and b is a positive whole number. (The word fraction in these standards always refers to a non- negative number.)" The document itself also refers to negative fractions.
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A primary focus of her work was violence against women and children. In a series of letters published in 1994 as Two Women Talking: Correspondence 1985-1987, Wallace and poet Erin Mouré discuss feminist theory. Mouré defends the language philosophers (particularly Wittgenstein) who demonstrate that our speech, and the concepts expressible in language, governs our knowledge and actions. However, Wallace disagreed that language-centred writing rescues women from the patriarchy, claiming that it can be easily co-opted by patriarchs.
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Many computer programs for molecular viewing are compatible with this format, including Jmol. Closely related is mmCIF, macromolecular CIF, which is intended as an alternative to the Protein Data Bank (PDB) format. It is now the default format used by the Protein Data Bank. Also closely related is Crystallographic Information Framework, a broader system of exchange protocols based on data dictionaries and relational rules expressible in different machine-readable manifestations, including, but not restricted to, Crystallographic Information File and XML.
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Such an element a is called a generator or a primitive element of the group. In additive notation, the requirement for an element to be primitive is that each element of the group can be written as : ..., −a−a, −a, 0, a, a+a, ... In the groups Z/nZ introduced above, the element 1 is primitive, so these groups are cyclic. Indeed, each element is expressible as a sum all of whose terms are 1. Any cyclic group with n elements is isomorphic to this group.
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In mathematical logic, an elementary definition is a definition that can be made using only finitary first-order logic, and in particular without reference to set theory or using extensions such as plural quantification. Elementary definitions are of particular interest because they admit a complete proof apparatus while still being expressive enough to support most everyday mathematics (via the addition of elementarily-expressible axioms such as Zermelo–Fraenkel set theory (ZFC)). Saying that a definition is elementary is a weaker condition than saying it is algebraic.
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One way of implementing iterators is to use a restricted form of coroutine, known as a generator. By contrast with a subroutine, a generator coroutine can yield values to its caller multiple times, instead of returning just once. Most iterators are naturally expressible as generators, but because generators preserve their local state between invocations, they're particularly well-suited for complicated, stateful iterators, such as tree traversers. There are subtle differences and distinctions in the use of the terms "generator" and "iterator", which vary between authors and languages.
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A replacement package called Blaze attempts to overcome this limitation. Algorithms that are not expressible as a vectorized operation will typically run slowly because they must be implemented in "pure Python", while vectorization may increase memory complexity of some operations from constant to linear, because temporary arrays must be created that are as large as the inputs. Runtime compilation of numerical code has been implemented by several groups to avoid these problems; open source solutions that interoperate with NumPy include `scipy.weave`, numexpr and Numba.
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Written transcriptions of rhythms tend to be imprecise. Usually only the basic idea of the rhythm is transcribed but the real feeling that it carries can't be easily put down on paper. This is due to the nature of the West African music—the different types of swing (at least four of them) that are not easily expressible with western notation. For this reason the written material for advanced players is still scarce if not unavailable, while the general and informational literature are readily obtained.
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The use of templates as a metaprogramming technique requires two distinct operations: a template must be defined, and a defined template must be instantiated. The template definition describes the generic form of the generated source code, and the instantiation causes a specific set of source code to be generated from the generic form in the template. Template metaprogramming is Turing-complete, meaning that any computation expressible by a computer program can be computed, in some form, by a template metaprogram. Templates are different from macros.
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See, for example, A new explicit formula in the additive theory of primes with applications I. The explicit formula for the Goldbach and Generalized Twin Prime Problems by Janos Pintz. Chen Jingrun showed in 1973 using the methods of sieve theory that every sufficiently large even number can be written as the sum of either two primes, or a prime and a semiprime (the product of two primes). See Chen's theorem for further information. In 1975, Hugh Montgomery and Robert Charles Vaughan showed that "most" even numbers are expressible as the sum of two primes.
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A counting quantifier is a mathematical term for a quantifier of the form "there exists at least k elements that satisfy property X". In first-order logic with equality, counting quantifiers can be defined in terms of ordinary quantifiers, so in this context they are a notational shorthand. However, they are interesting in the context of logics such as two-variable logic with counting that restrict the number of variables in formulas. Also, generalized counting quantifiers that say "there exists infinitely many" are not expressible using a finite number of formulas in first-order logic.
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Unlike more traditional techniques of quantum field theory, conformal bootstrap does not use the Lagrangian of the theory. Instead, it operates with the general axiomatic parameters, such as the scaling dimensions of the local operators and their operator product expansion coefficients. A key axiom is that the product of local operators must be expressible as a sum over local operators (thus turning the product into an algebra); the sum must have a non-zero radius of convergence. This leads to decompositions of correlation functions into structure constants and conformal blocks.
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For the avoidance of ambiguity, zero will always be a valid possible constituent of "sums of two squares", so for example every square of an integer is trivially expressible as the sum of two squares by setting one of them to be zero. 1\. The product of two numbers, each of which is a sum of two squares, is itself a sum of two squares. ::This is a well-known property, based on the identity :::(a^2+b^2)(p^2+q^2) = (ap+bq)^2 + (aq-bp)^2 ::due to Diophantus. 2\.
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This is one of a number of characterizations of the exponential function; others involve series or differential equations. From any of these definitions it can be shown that the exponential function obeys the basic exponentiation identity, :\exp(x + y) = \exp x \cdot \exp y which justifies the notation for . The derivative (rate of change) of the exponential function is the exponential function itself. More generally, a function with a rate of change proportional to the function itself (rather than equal to it) is expressible in terms of the exponential function.
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Genetics 183: 503-517 Since different tissues require different genes to be expressed, reciprocal silencing can occur between tissues. Importantly, while the pattern of gene expression is the same as in the population case, the genetic means by which this pattern is achieved are very different. While silencing mutations are thought to be the main source of reciprocal silencing at the population level, at the tissue level only epigenetic factors are in play, since expressible copies of both homeologous loci must exist in all cells in an individual if different tissues express different homeologs.
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A central focus of database theory is on understanding the complexity and power of query languages and their connection to logic. Starting from relational algebra and first-order logic (which are equivalent by Codd's theorem) and the insight that important queries such as graph reachability are not expressible in this language, more powerful language based on logic programming and fixpoint logic such as datalog were studied. Another focus was on the foundations of query optimization and data integration. Here most work studied conjunctive queries, which admit query optimization even under constraints using the chase algorithm.
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An entangled system is defined to be one whose quantum state cannot be factored as a product of states of its local constituents; that is to say, they are not individual particles but are an inseparable whole. In entanglement, one constituent cannot be fully described without considering the other(s). The state of a composite system is always expressible as a sum, or superposition, of products of states of local constituents; it is entangled if this sum necessarily has more than one term. Quantum systems can become entangled through various types of interactions.
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A classic proof by contradiction from mathematics is the proof that the square root of 2 is irrational. If it were rational, it would be expressible as a fraction a/b in lowest terms, where a and b are integers, at least one of which is odd. But if a/b = , then a2 = 2b2. Therefore, a2 must be even, and because the square of an odd number is odd, that in turn implies that a is itself even — which means that b must be odd because a/b is in lowest terms.
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Impossibility theorems are usually expressible as negative existential propositions, or universal propositions in logic (see universal quantification for more). Perhaps one of the oldest proofs of impossibility is that of the irrationality of square root of 2, which shows that it is impossible to express the square root of 2 as a ratio of integers. Another famous proof of impossibility was the 1882 proof of Ferdinand von Lindemann, showing that the ancient problem of squaring the circle cannot be solved, because the number is transcendental (i.e., non-algebraic) and only a subset of the algebraic numbers can be constructed by compass and straightedge.
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The informal definition of an algebraic function provides a number of clues about their properties. To gain an intuitive understanding, it may be helpful to regard algebraic functions as functions which can be formed by the usual algebraic operations: addition, multiplication, division, and taking an nth root. This is something of an oversimplification; because of the fundamental theorem of Galois theory, algebraic functions need not be expressible by radicals. First, note that any polynomial function y = p(x) is an algebraic function, since it is simply the solution y to the equation : y-p(x) = 0.
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In abstract algebra, a module is indecomposable if it is non-zero and cannot be written as a direct sum of two non-zero submodules. Jacobson (2009), p. 111. Indecomposable is a weaker notion than simple module (which is also sometimes called irreducible module): simple means "no proper submodule" N < M, while indecomposable "not expressible as N \oplus P = M". A direct sum of indecomposables is called completely decomposable; this is weaker than being semisimple, which is a direct sum of simple modules. A direct sum decomposition of a module into indecomposable modules is called an indecomposable decomposition.
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More strongly, if the exponential time hypothesis is true, then clique-finding and first-order model checking would necessarily take time proportional to a power of whose exponent is proportional to . On restricted classes of graphs, model checking of first-order sentences can be much more efficient. In particular, every graph property expressible as a first-order sentence can be tested in linear time for the graphs of bounded expansion. These are the graphs in which all shallow minors are sparse graphs, with a ratio of edges to vertices bounded by a function of the depth of the minor.
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This is related to Tarski's indefinability theorem. The example of ZFC illustrates the importance of distinguishing the metamathematics of a formal system from the statements of the formal system itself. The property D(φ) that a formula φ of ZFC defines a unique real number is not itself expressible by ZFC, but must be considered as part of the metatheory used to formalize ZFC. From this viewpoint, Richard's paradox results from treating a construction of the metatheory (the enumeration of all statements in the original system that define real numbers) as if that construction could be performed in the original system.
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But it is not syntactically complete, since there are sentences expressible in the language of first order logic that can be neither proved nor disproved from the axioms of logic alone. In a mere system of logic it would be absurd to expect syntactic completeness. But in a system of mathematics, thinkers such as Hilbert had believed that it is just a matter of time to find such an axiomatization that would allow one to either prove or disprove (by proving its negation) each and every mathematical formula. A formal system might be syntactically incomplete by design, as logics generally are.
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If Ladner languages were not expressible in this way, the set of all constraint languages could be divided exactly into those defining polynomial-time and those defining NP-complete problems; that is, this set would exhibit a dichotomy. Partial results are known for some sets of constraint languages. The best known such result is Schaefer's dichotomy theorem, which proves the existence of a dichotomy in the set of constraint languages on a binary domain. More precisely, it proves that a relation restriction on a binary domain is tractable if all its relations belong to one of six classes, and is NP-complete otherwise.
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Nithyananda's speeches have gathered widespread mixed reception in the Indian press, having broadcast a range of pseudoscientific claims, including claims of delaying Sun to rise for 40 minutes, his ability to make cows speak in Tamil and Sanskrit, and explaining Einstein's mass–energy equivalence wrong. He has claimed to have discovered over 400 esoteric powers expressible by humans and alleges having initiated his disciples into 60 such powers including kundalini and third-eye awakening. He has since asserted he would open the third eye, for anyone, free of charge by 2021. He also claimed that person would be able to see through smog and walls.
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Java 8 supports lambda expressions as a replacement for some anonymous classes. In C#, anonymous classes are not necessary, because closures and lambdas are fully supported. Libraries and language extensions for immutable data structures are being developed to aid programming in the functional style in C#. Many object- oriented design patterns are expressible in functional programming terms: for example, the strategy pattern simply dictates use of a higher-order function, and the visitor pattern roughly corresponds to a catamorphism, or fold. Similarly, the idea of immutable data from functional programming is often included in imperative programming languages, for example the tuple in Python, which is an immutable array.
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Rule 4: Dynamic online catalog based on the relational model: :The data base description is represented at the logical level in the same way as ordinary data, so that authorized users can apply the same relational language to its interrogation as they apply to the regular data. Rule 5: The comprehensive data sublanguage rule: :A relational system may support several languages and various modes of terminal use (for example, the fill-in-the-blanks mode). However, there must be at least one language whose statements are expressible, per some well- defined syntax, as character strings and that is comprehensive in supporting all of the following items: ::#Data definition. ::#View definition.
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The fear of nuclear war with the Soviet Union, along with less expressible qualms about radioactive fallout from America's own atomic tests, energized many of the era's genre films. Science fiction, horror, and various hybrids of the two were now of central economic importance to the low-budget end of the business. Most down-market films of the type—like many of those produced by William Alland at Universal (such as Creature from the Black Lagoon (1954)) and Sam Katzman at Columbia (including It Came from Beneath the Sea (1955))—provided little more than thrills, though their special effects could be impressive.Kinnard (1988), pp. 67–73.
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Foundationalism is a theory of epistemology which states that there are certain fundamental principles which are the basis for all other knowledge. In the case of ethics, foundationalists hold that certain fundamental moral rules are their own justification. Walter Sinnott-Armstrong explains: > The deepest challenge in moral epistemology, as in general epistemology, is > raised by a skeptical regress argument: Someone is justified in believing > something only if the believer has a reason that is expressible in an > inference with premises that the believer is already justified in believing. > This requires a chain of inferences that must continue infinitely, close > into a circle, or stop arbitrarily.
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Macros are capable of conditional control over compilation based on predetermined criteria, but cannot instantiate new types, recurse, or perform type evaluation and in effect are limited to pre-compilation text-substitution and text- inclusion/exclusion. In other words, macros can control compilation flow based on pre-defined symbols but cannot, unlike templates, independently instantiate new symbols. Templates are a tool for static polymorphism (see below) and generic programming. In addition, templates are a compile time mechanism in C++ that is Turing-complete, meaning that any computation expressible by a computer program can be computed, in some form, by a template metaprogram prior to runtime.
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Fagin's theorem is the oldest result of descriptive complexity theory, a branch of computational complexity theory that characterizes complexity classes in terms of logic-based descriptions of their problems rather than by the behavior of algorithms for solving those problems. The theorem states that the set of all properties expressible in existential second-order logic is precisely the complexity class NP. It was proven by Ronald Fagin in 1973 in his doctoral thesis, and appears in his 1974 paper. The arity required by the second-order formula was improved (in one direction) in Lynch's theorem, and several results of Grandjean have provided tighter bounds on nondeterministic random-access machines.
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Second-order arithmetic can also be seen as a weak version of set theory in which every element is either a natural number or a set of natural numbers. Although it is much weaker than Zermelo–Fraenkel set theory, second-order arithmetic can prove essentially all of the results of classical mathematics expressible in its language. A subsystem of second-order arithmetic is a theory in the language of second-order arithmetic each axiom of which is a theorem of full second-order arithmetic (Z2). Such subsystems are essential to reverse mathematics, a research program investigating how much of classical mathematics can be derived in certain weak subsystems of varying strength.
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A square matrix A that is equal to its transpose, that is, A = A, is a symmetric matrix. If instead, A is equal to the negative of its transpose, that is, A = −A, then A is a skew-symmetric matrix. In complex matrices, symmetry is often replaced by the concept of Hermitian matrices, which satisfy A = A, where the star or asterisk denotes the conjugate transpose of the matrix, that is, the transpose of the complex conjugate of A. By the spectral theorem, real symmetric matrices and complex Hermitian matrices have an eigenbasis; that is, every vector is expressible as a linear combination of eigenvectors. In both cases, all eigenvalues are real.
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Hamiltonicity can be expressed in MSO2 by the existence of a set of edges that forms a connected 2-regular graph on all the vertices, with connectivity expressed as above and 2-regularity expressed as the incidence of two but not three distinct edges at each vertex. However, Hamiltonicity is not expressible in MSO1, because MSO1 is not capable of distinguishing complete bipartite graphs with equal numbers of vertices on each side of the bipartition (which are Hamiltonian) from unbalanced complete bipartite graphs (which are not).; , Corollary 7.24, pp. 126–127. Although not part of the definition of MSO2, orientations of undirected graphs can be represented by a technique involving Trémaux trees.
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The ' stands for phonogram, the legal term used in most English-speaking countries to refer to works known in U.S. copyright law as "sound recordings".Statement of Marybeth Peters, United States Register of Copyrights, before the Subcommittee on Courts, the Internet, and Intellectual Property, Committee on the Judiciary (July 31, 2007). A sound recording has a separate copyright that is distinct from that of the underlying work (usually a musical work, expressible in musical notation and written lyrics), if any. The sound recording copyright notice extends to a copyright for just the sound itself and will not apply to any other rendition or version, even if performed by the same artist(s).
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Thus the first-order theory of real numbers and sets of real numbers has many models, some of which are countable. The second-order theory of the real numbers has only one model, however. This follows from the classical theorem that there is only one Archimedean complete ordered field, along with the fact that all the axioms of an Archimedean complete ordered field are expressible in second-order logic. This shows that the second-order theory of the real numbers cannot be reduced to a first-order theory, in the sense that the second-order theory of the real numbers has only one model but the corresponding first-order theory has many models.
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Some graphics card drivers have historically coped poorly with the EDID, using only its standard timing descriptors rather than its Detailed Timing Descriptors (DTDs). Even in cases where the DTDs were read, the drivers are/were still often limited by the standard timing descriptor limitation that the horizontal/vertical resolutions must be evenly divisible by 8. This means that many graphics cards cannot express the native resolutions of the most common wide screen flat panel displays and liquid crystal display televisions. The number of vertical pixels is calculated from the horizontal resolution and the selected aspect ratio. To be fully expressible, the size of wide screen display must thus be a multiple of 16×9 pixels.
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Ghinnawas (literally "little songs") are short, two line emotional lyric poems written by the Bedouins of Egypt, in a fashion similar to haiku, but similar in content to the American blues.Veiled Sentiments: Honor and Poetry in a Bedouin Society, by Lila Abu-Lugodh, University of California Press, Berkeley, 1986 Ghinnawas typically talk of deep, personal feelings and are often an outlet for personal emotions which might not be otherwise expressible in Bedouin society. Ghinnawas may also be sung. Lila Abu Lughod - the Arab American anthropologist, who studied the Awlad Ali Bedouins in Northern Egypt in the late 1970s, and collected over 450 ghinnawas, has published the most comprehensive work on ghinnawas to date.
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Consider fitting a line: for each data point the product of the vertical and horizontal residuals equals twice the area of the triangle formed by the residual lines and the fitted line. We choose the line which minimizes the sum of these areas. Nobel laureate Paul Samuelson proved in 1942 that, in two dimensions, it is the only line expressible solely in terms of the ratios of standard deviations and the correlation coefficient which (1) fits the correct equation when the observations fall on a straight line, (2) exhibits scale invariance, and (3) exhibits invariance under interchange of variables. This solution has been rediscovered in different disciplines and is variously known as standardised major axis (Ricker 1975, Warton et al.
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Owen Flanagan argues that Jackson's thought experiment "is easy to defeat". He grants that "Mary knows everything about color vision that can be expressed in the vocabularies of a complete physics, chemistry, and neuroscience," and then distinguishes between "metaphysical physicalism" and "linguistic physicalism": > Metaphysical physicalism simply asserts that what there is, and all there > is, is physical stuff and its relations. Linguistic physicalism is the > thesis that everything physical can be expressed or captured in the > languages of the basic sciences…Linguistic physicalism is stronger than > metaphysical physicalism and less plausible. Flanagan argues that, while Mary has all the facts that are expressible in "explicitly physical language", she can only be said to have all the facts if one accepts linguistic physicalism.
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6 An example of a non-regular open set is the set U = ∪ in R with its normal topology, since 1 is in the interior of the closure of U, but not in U. The regular open subsets of a space form a complete Boolean algebra. ; Relatively compact: A subset Y of a space X is relatively compact in X if the closure of Y in X is compact. ; Residual: If X is a space and A is a subset of X, then A is residual in X if the complement of A is meagre in X. Also called comeagre or comeager. ; Resolvable: A topological space is called resolvable if it is expressible as the union of two disjoint dense subsets.
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Relational completeness clearly does not imply that any interesting database query can be expressed in relationally complete languages. Well-known examples of inexpressible queries include simple aggregations (counting tuples, or summing up values occurring in tuples, which are operations expressible in SQL but not in relational algebra) and computing the transitive closure of a graph given by its binary edge relation (see also expressive power). Codd's theorem also doesn't consider SQL nulls and the three-valued logic they entail; the logical treatment of nulls remains mired in controversy.For recent work extending Codd's theorem in this direction see Additionally, SQL allows duplicate rows (has multiset semantics.) Nevertheless, relational completeness constitutes an important yardstick by which the expressive power of query languages can be compared.
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As long as the signature contains at least one predicate or function in addition to the distinguished order relation, so that the amount of space taken to store such finite structures is actually polynomial in the number of elements in the structure, this precisely characterizes P. Similarly, NP is the set of languages expressible in existential second-order logic—that is, second-order logic restricted to exclude universal quantification over relations, functions, and subsets. The languages in the polynomial hierarchy, PH, correspond to all of second-order logic. Thus, the question "is P a proper subset of NP" can be reformulated as "is existential second-order logic able to describe languages (of finite linearly ordered structures with nontrivial signature) that first-order logic with least fixed point cannot?".Elvira Mayordomo.
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If a number which can be written as a sum of two squares is divisible by a number which is not a sum of two squares, then the quotient has a factor which is not a sum of two squares. (This is Euler's second Proposition). ::Suppose q is a number not expressible as a sum of two squares, which divides a^2+b^2. Write the quotient, factored into its (possibly repeated) prime factors, as p_1p_2\cdots p_n so that a^2+b^2 = q p_1p_2\cdots p_n. If all factors p_i can be written as sums of two squares, then we can divide a^2+b^2 successively by p_1, p_2, etc., and applying step (2.) above we deduce that each successive, smaller, quotient is a sum of two squares.
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The tree consists of all finite sequences of elements of A, and the children of a particular node σ of the tree are exactly the sequences that extend σ by one element. Thus if A = { 0, 1 }, the first level of the tree consists of the sequences ⟨ 0 ⟩ and ⟨ 1 ⟩; the second level consists of the four sequences ⟨ 0, 0 ⟩, ⟨ 0, 1 ⟩, ⟨ 1, 0 ⟩, ⟨ 1, 1 ⟩; and so on. For each of the finite sequences σ in the tree, the set of all elements of Aω that begin with σ is a basic open set in the topology on A. The open sets of Aω are precisely the sets expressible as unions of these basic open sets. The closed sets, as usual, are those whose complement is open.
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However, as David Seetapun originally proved, the version of the theorem for graphs is weaker than ACA0, and it turns out to be inequivalent to any one of the big five subsystems. The version for uniform hypergraphs of fixed order greater than two is equivalent to ACA0, and the version of the theorem stated for all numbers of colors and all orders of hypergraphs simultaneously is stronger than ACA0. Chapter seven discusses conservative extensions of theories, in which the statements of a powerful theory (such as one of the forms of second-order arithmetic) that are both provable in that theory and expressible in a weaker theory (such as Peano arithmetic) are only the ones that are already provably in the weaker theory. Chapter eight summarizes the results so far in diagrammatic form.
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The concept of the form of value shows how, with the development of commodity trade, anything with a utility for people can be transformed into an abstract value, objectively expressible as a sum of money; but, also, how this transformation changes the organization of labour to maximize its value-creating capacity, how it changes social interactions and the very way people are aware of their interactions. However, the quantification of objects and the manipulation of quantities ineluctably leads to distortions (reifications) of their qualitative properties. For the sake of obtaining a measure of magnitude, it is frequently assumed that objects are quantifiable, but in the process of quantification, various qualitative aspects are conveniently ignored or abstracted away from.Viktor Mayer- Schönberger and Thomas Ramge, Reinventing capitalism in the age of big data.
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One notable exception is the journal Capital & Class, which published a translation by Mike Roth and Wal Suchting of Marx's original text on the value-form as it appears in the first edition of Capital, Volume I. See "The value-form", in: Capital and Class, No.4 Spring 1978, pp. 130–150. Two other journals referring to the value-form discussion are Thesis Eleven and Telos. In the reified perception of the political economists and the vulgar Marxists, products have value because they are expressible in money-prices, but Marx argues that in reality it is just the other way round: because commodities have value, i.e. because they are all products with a replacement cost of social labour,By "social labour" is meant "cooperative labour to produce things which are used by others".
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London: Longmans, Green and Co.; [etc.]. pp. 65–69 One of the simplest applications of these theorems was to perfect the theory of the Leyden phial, a result which (if we except the peculiar action of the insulating solid medium, since discovered by Faraday) we owe to his genius. He has also shown how an infinite number of forms of conductors may be invented, so that the distribution of electricity in equilibrium on each may be expressible in finite algebraic terms – an immense stride in the science, when we consider that the distribution of electricity on a single spherical conductor, an uninfluenced ellipsoidal conductor, and two spheres mutually influencing one another, were the only cases solved by Poisson, and indeed the only cases conceived to be solvable by mathematical writers.Baynes, T. S. (1888).
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Logically, many theorems are of the form of an indicative conditional: if A, then B. Such a theorem does not assert B—only that B is a necessary consequence of A. In this case, A is called the hypothesis of the theorem ("hypothesis" here means something very different from a conjecture), and B the conclusion of the theorem. Alternatively, A and B can be also termed the antecedent and the consequent, respectively. The theorem "If n is an even natural number, then n/2 is a natural number" is a typical example in which the hypothesis is "n is an even natural number", and the conclusion is "n/2 is also a natural number". In order for a theorem be proved, it must be in principle expressible as a precise, formal statement.
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In an infinite distributed lag model, an infinite number of lag weights need to be estimated; clearly this can be done only if some structure is assumed for the relation between the various lag weights, with the entire infinitude of them expressible in terms of a finite number of assumed underlying parameters. In a finite distributed lag model, the parameters could be directly estimated by ordinary least squares (assuming the number of data points sufficiently exceeds the number of lag weights); nevertheless, such estimation may give very imprecise results due to extreme multicollinearity among the various lagged values of the independent variable, so again it may be necessary to assume some structure for the relation between the various lag weights. The concept of distributed lag models easily generalizes to the context of more than one right-side explanatory variable.
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The term promise was coined by Liskov and Shrira, although they referred to the pipelining mechanism by the name call-stream, which is now rarely used. Both the design described in Liskov and Shrira's paper, and the implementation of promise pipelining in Xanadu, had the limit that promise values were not first-class: an argument to, or the value returned by a call or send could not directly be a promise (so the example of promise pipelining given earlier, which uses a promise for the result of one send as an argument to another, would not have been directly expressible in the call-stream design or in the Xanadu implementation). It seems that promises and call-streams were never implemented in any public release of Argus, the programming language used in the Liskov and Shrira paper. Argus development stopped around 1988.
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The granularity-related inconsistency of means (GRIM) test is a simple statistical test used to identify inconsistencies in the analysis of data sets. The test relies on the fact that, given a dataset containing N integer values, the arithmetic mean (commonly called simply the average) is restricted to a few possible values: it must always be expressible as a fraction with an integer numerator and a denominator N. If the reported mean does not fit this description, there must be an error somewhere; the preferred term for such errors is "inconsistencies", to emphasise that their origin is, on first discovery, typically unknown. GRIM inconsistencies can result from inadvertent data-entry or typographical errors or from scientific fraud. The GRIM test is most useful in fields such as psychology where researchers typically use small groups and measurements are often integers.
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With this contribution of von Neumann, the axiomatic system of the theory of sets avoided the contradictions of earlier systems and became usable as a foundation for mathematics, despite the lack of a proof of its consistency. The next question was whether it provided definitive answers to all mathematical questions that could be posed in it, or whether it might be improved by adding stronger axioms that could be used to prove a broader class of theorems. A strongly negative answer to whether it was definitive arrived in September 1930 at the historic Second Conference on the Epistemology of the Exact Sciences of Königsberg, in which Kurt Gödel announced his first theorem of incompleteness: the usual axiomatic systems are incomplete, in the sense that they cannot prove every truth expressible in their language. Moreover, every consistent extension of these systems necessarily remains incomplete.
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There are numerous known examples of sixth powers that can be expressed as the sum of seven other sixth powers, but no examples are yet known of a sixth power expressible as the sum of just six sixth powers.Quoted in This makes it unique among the powers with exponent k = 1, 2, ... , 8, the others of which can each be expressed as the sum of k other k-th powers, and some of which (in violation of Euler's sum of powers conjecture) can be expressed as a sum of even fewer k-th powers. In connection with Waring's problem, every sufficiently large integer can be represented as a sum of at most 24 sixth powers of integers. There are infinitely many different nontrivial solutions to the Diophantine equation :a^6+b^6+c^6=d^6+e^6+f^6.
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All values of the sines, cosines, and tangents of angles at 3° increments are expressible in terms of square roots, using identities – the half-angle identity, the double-angle identity, and the angle addition/subtraction identity – and using values for 0°, 30°, 36°, and 45°. For an angle of an integer number of degrees that is not a multiple of 3° ( radians), the values of sine, cosine, and tangent cannot be expressed in terms of real radicals. According to Niven's theorem, the only rational values of the sine function for which the argument is a rational number of degrees are 0, , 1, −, and −1. According to Baker's theorem, if the value of a sine, a cosine or a tangent is algebraic, then the angle is either a rational number of degrees or a transcendental number of degrees.
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An alternative characterization of PSPACE is the set of problems decidable by an alternating Turing machine in polynomial time, sometimes called APTIME or just AP.Arora & Barak (2009) p.100 A logical characterization of PSPACE from descriptive complexity theory is that it is the set of problems expressible in second- order logic with the addition of a transitive closure operator. A full transitive closure is not needed; a commutative transitive closure and even weaker forms suffice. It is the addition of this operator that (possibly) distinguishes PSPACE from PH. A major result of complexity theory is that PSPACE can be characterized as all the languages recognizable by a particular interactive proof system, the one defining the class IP. In this system, there is an all-powerful prover trying to convince a randomized polynomial-time verifier that a string is in the language.
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Fourth, this article only deals with trigonometric function values when the expression in radicals is in real radicals – roots of real numbers. Many other trigonometric function values are expressible in, for example, cube roots of complex numbers that cannot be rewritten in terms of roots of real numbers. For example, the trigonometric function values of any angle that is one-third of an angle θ considered in this article can be expressed in cube roots and square roots by using the cubic equation formula to solve :4\cos^3 \frac \theta 3 - 3\cos \frac \theta 3 = \cos\theta, but in general the solution for the cosine of the one-third angle involves the cube root of a complex number (giving casus irreducibilis). In practice, all values of sines, cosines, and tangents not found in this article are approximated using the techniques described at Trigonometric tables.
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Matiyasevich proved that there is no algorithm that, given a multivariate polynomial p(x1, x2,...,xk) with integer coefficients, determines whether there is an integer solution to the equation p = 0. Because polynomials with integer coefficients, and integers themselves, are directly expressible in the language of arithmetic, if a multivariate integer polynomial equation p = 0 does have a solution in the integers then any sufficiently strong system of arithmetic T will prove this. Moreover, if the system T is ω-consistent, then it will never prove that a particular polynomial equation has a solution when in fact there is no solution in the integers. Thus, if T were complete and ω-consistent, it would be possible to determine algorithmically whether a polynomial equation has a solution by merely enumerating proofs of T until either "p has a solution" or "p has no solution" is found, in contradiction to Matiyasevich's theorem.
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Fifty-one is a pentagonal number as well as a centered pentagonal number and an 18-gonal number and a Perrin number. It is also the 6th Motzkin number, telling the number of ways to draw non-intersecting chords between any six points on a circle's boundary, no matter where the points may be located on the boundary. Since the greatest prime factor of 512 \+ 1 = 2602 is 1301, which is substantially more than 51 twice, 51 is a Størmer number. There are 51 different cyclic Gilbreath permutations on 10 elements, and therefore there are 51 different real periodic points of order 10 on the Mandelbrot set.. Since 51 is the product of the distinct Fermat primes 3 and 17, a regular polygon with 51 sides is constructible with compass and straightedge, the angle is constructible, and the number cos is expressible in terms of square roots.
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New York: Alfred A. Knopf. Carl Van Vechten, widely recognised as a patron of the Harlem Renaissance and also for his work as the literary executor of Gertrude Stein, stated himself ‘a great admirer of Frederick Buechner's A Long Day's Dying’, while also noting its impressiveness as a debut: ‘It is the book of a first novelist already arrived, most original, and filled with wit, nostalgia, and emotion.’Carl Van Vechten (see inside reverse cover for review): Buechner, Frederick (1950). A Long Day's Dying. New York: Alfred A. Knopf. The renowned composer, conductor, and author, Leonard Bernstein, also eulogised the novel, remarking: > I have rarely been so moved by a perception. Mr. Buechner shows a remarkable > insight into one of the least easily expressible tragedies of modern man; > the basic incapacity of persons really to communicate with one another. That > he has made this frustration manifest, in such a personal and magnetic way, > and at the age of twenty-three, constitutes a literary triumph.
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In computational complexity theory, SNP (from Strict NP) is a complexity class containing a limited subset of NP based on its logical characterization in terms of graph-theoretical properties. It forms the basis for the definition of the class MaxSNP of optimization problems. It is defined as the class of problems that are properties of relational structures (such as graphs) expressible by a second-order logic formula of the following form: : \exists S_1 \dots \exists S_\ell \, \forall v_1 \dots \forall v_m \,\phi(R_1,\dots,R_k,S_1,\dots,S_\ell,v_1,\dots,v_m), where R_1,\dots,R_k are relations of the structure (such as the adjacency relation, for a graph), S_1,\dots,S_\ell are unknown relations (sets of tuples of vertices), and \phi is a quantifier-free formula: any boolean combination of the relations. That is, only existential second-order quantification (over relations) is allowed and only universal first-order quantification (over vertices) is allowed.
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In the subfield of abstract algebra known as module theory, a right R module M is called a balanced module (or is said to have the double centralizer property) if every endomorphism of the abelian group M which commutes with all R-endomorphisms of M is given by multiplication by a ring element. Explicitly, for any additive endomorphism f, if fg = gf for every R endomorphism g, then there exists an r in R such that f(x) = xr for all x in M. In the case of non- balanced modules, there will be such an f that is not expressible this way. In the language of centralizers, a balanced module is one satisfying the conclusion of the double centralizer theorem, that is, the only endomorphisms of the group M commuting with all the R endomorphisms of M are the ones induced by right multiplication by ring elements. A ring is called balanced if every right R module is balanced.
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In differential topology, the transversality theorem, also known as the Thom transversality theorem after French mathematician René Thom, is a major result that describes the transverse intersection properties of a smooth family of smooth maps. It says that transversality is a generic property: any smooth map f\colon X\rightarrow Y, may be deformed by an arbitrary small amount into a map that is transverse to a given submanifold Z \subseteq Y. Together with the Pontryagin–Thom construction, it is the technical heart of cobordism theory, and the starting point for surgery theory. The finite-dimensional version of the transversality theorem is also a very useful tool for establishing the genericity of a property which is dependent on a finite number of real parameters and which is expressible using a system of nonlinear equations. This can be extended to an infinite-dimensional parametrization using the infinite-dimensional version of the transversality theorem.
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In 1901, Giovanni Giorgi proposed a new system of units that would remedy this state of affairs. Original manuscript with handwritten notes by Oliver Heaviside He noted that the mechanical practical units such as the joule and the watt are coherent not only in the QES system, but also in the meter-kilogram-second (MKS) system. It was of course known that just adopting the meter and the kilogram as base units—obtaining the three dimensional MKS system—would not solve the problem: while the watt and the joule would be coherent, this would not be so for the volt, the ampere, the ohm, and the rest of the practical units for electric and magnetic quantities (the only three-dimensional absolute system in which all practical units are coherent is the QES system). But Giorgi pointed out that the volt and the rest could be made coherent if one gave up on the idea that all physical quantities must be expressible in terms of dimensions of length, mass, and time, and admitted a fourth base dimension for electric quantities.
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Mixture densities are complicated densities expressible in terms of simpler densities (the mixture components), and are used both because they provide a good model for certain data sets (where different subsets of the data exhibit different characteristics and can best be modeled separately), and because they can be more mathematically tractable, because the individual mixture components can be more easily studied than the overall mixture density. Mixture densities can be used to model a statistical population with subpopulations, where the mixture components are the densities on the subpopulations, and the weights are the proportions of each subpopulation in the overall population. Mixture densities can also be used to model experimental error or contamination – one assumes that most of the samples measure the desired phenomenon, Parametric statistics that assume no error often fail on such mixture densities – for example, statistics that assume normality often fail disastrously in the presence of even a few outliers – and instead one uses robust statistics. In meta-analysis of separate studies, study heterogeneity causes distribution of results to be a mixture distribution, and leads to overdispersion of results relative to predicted error.
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A key result (obtained jointly with Arthur Fischer of the University of California at Santa Cruz) was to relate the reduced Hamiltonian for Einstein's equations to a topological invariant known as the Yamabe invariant (or sigma constant) for the spatial manifold and to show that the reduced Hamiltonian is monotonically decreasing along all solutions of the field equations (in the direction of cosmological expansion) and therefore evidently seeking to attain its infimum which in turn is expressible in terms of the sigma constant. A discussion of this and related work (with Lars Andersson of the University of Miami and Yvonne Choquet-Bruhat of the Université Paris VI) may be found in Moncrief's and Choquet-Bruhat's lectures at the Cargese summer school on 50 years of the Cauchy Problem in General Relativity. Moncrief's own research is mainly concerned with the global existence and asymptotic properties of cosmological solutions of Einstein's equations and especially the question of how these properties depend upon the topology of spacetime. He is also interested in how a study of the "Einstein flow" on various manifolds might shed light on open questions in 3-manifold topology itself.
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Leonard Eugene Dickson studied generalizations of Waring's problem for seventh powers, showing that every non-negative integer can be represented as a sum of at most 258 non-negative seventh powers. All but finitely many positive integers can be expressed more simply as the sum of at most 46 seventh powers. If negative powers are allowed, only 12 powers are required. The smallest number that can be represented in two different ways as a sum of four positive seventh powers is 2056364173794800. The smallest seventh power that can be represented as a sum of eight distinct seventh powers is: :102^7=12^7+35^7+53^7+58^7+64^7+83^7+85^7+90^7. The two known examples of a seventh power expressible as the sum of seven seventh powers are :568^7 = 127^7+ 258^7 + 266^7 + 413^7 + 430^7 + 439^7 + 525^7 (M. Dodrill, 1999); and : 626^7=625^7+309^7+258^7+255^7+158^7+148^7+91^7 (Maurice Blondot, 11/14/2000); any example with fewer terms in the sum would be a counterexample to Euler's sum of powers conjecture, which is currently only known to be false for the powers 4 and 5.
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Note that winsorizing is not equivalent to simply excluding data, which is a simpler procedure, called trimming or truncation, but is a method of censoring data. In a trimmed estimator, the extreme values are discarded; in a winsorized estimator, the extreme values are instead replaced by certain percentiles (the trimmed minimum and maximum). Thus a winsorized mean is not the same as a truncated mean. For instance, the 10% trimmed mean is the average of the 5th to 95th percentile of the data, while the 90% winsorized mean sets the bottom 5% to the 5th percentile, the top 5% to the 95th percentile, and then averages the data. In the previous example the trimmed mean would be obtained from the smaller set: :{92, 19, 101, 58, 91, 26, 78, 10, 13, 101, 86, 85, 15, 89, 89, 28, −5, 41} (N = 18, mean = 56.5) In this case, the winsorized mean can equivalently be expressed as a weighted average of the truncated mean and the 5th and 95th percentiles (for the 10% winsorized mean, 0.05 times the 5th percentile, 0.9 times the 10% trimmed mean, and 0.05 times the 95th percentile) though in general winsorized statistics need not be expressible in terms of the corresponding trimmed statistic.
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