Most of us are much more mathematical than we think.
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For more mathematical breakdowns of 15-year-old rap songs, follow Trey Smith on Twitter.
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Nate Silver at FiveThirtyEight has a more mathematical and data-oriented analysis of all that.
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The models have gotten more accurate over time with the help of more mathematical theories and observation.
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"I'd love to make more mathematical animation tools, where artists and creators can explore the spaces created by parameters," she said.
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To satisfy critics they built more mathematical models, which aimed to show how decisions by rational, forward-looking people could, in aggregate, cause downturns.
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While staying in a hotel at a conference, he found himself seized in the middle of the night by the idea of making programming more mathematical.
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Crunching the numbers for a worldwide layer of clouds, which might be just a couple hundred feet tall, simply requires more mathematical brawn than any supercomputer can deliver.
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The question is will the Valley change the Pentagon, as people like McCord suggest it will, or is the relationship destined to be much more mathematical than all that?
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Which is, one side of the brain is the creative, imaginative side and then the other side of the brain is the logical, more mathematical, putting pieces together side. Yeah.
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Yet, unlike Spears and Gomez who use curation to help express a moment in time or help tell their story, Beyoncé's approach is, in some ways, more mathematical and assertive.
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More and more mathematical journals were published and, by the end of the century, the development of the World Wide Web led to online publishing.
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A more mathematical version of pangenesis, one which dropped much of Darwin's Lamarckian holdovers, was developed as the "biometrical" school of heredity by Darwin's cousin, Francis Galton.
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Nomograph, also called Alignment Chart, calculating chart with scales that contain values of three or more mathematical variables, widely used in engineering, industry, and the natural and physical sciences.
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To avoid complications arising from the presence of singularities, usually one requires the vector field to be nonvanishing. If we add more mathematical structure, our congruence may acquire new significance.
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The term gnurd (spelled with the "g") was in use at the Massachusetts Institute of Technology (MIT) by 1965.More Mathematical People (D.J. Albers, J.L. Alexanderson and C. Reid), p. 105 (1990).
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Trajectories correspond to the unfolding of biological pathways and transients of the equations to short-term biological events. For a more mathematical discussion, see the articles on nonlinearity, dynamical systems, bifurcation theory, and chaos theory.
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Finite action-set learning automata (FALA) are a class of learning automata for which the number of possible actions is finite or, in more mathematical terms, for which the size of the action-set is finite.
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The Church–Turing thesis conjectures that there is no effective model of computing that can compute more mathematical functions than a Turing machine. Computer scientists have imagined many varieties of hypercomputers, models of computation that go beyond Turing computability.
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Her name may have a more mathematical origin, as the DX Goods locomotive No. 183 was given the name Theorem. The first locomotives built were given tenders, but with the introduction of water scoops for refilling en route they were replaced with tenders.
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By revealing such a direct connection between CNN and multigrid method, this opens up a new door to the design and study of deep learning models from a more mathematical viewpoint and in particular the rich mathematical techniques developed for multigrid method can be applied in the study of deep learning.
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Statistical mechanics forms a separate field, which includes the theory of phase transitions. It relies upon the Hamiltonian mechanics (or its quantum version) and it is closely related with the more mathematical ergodic theory and some parts of probability theory. There are increasing interactions between combinatorics and physics, in particular statistical physics.
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An artistic representation of a Turing machine. Turing machines are used to model general computing devices. Theoretical computer science (TCS) is a subset of general computer science and mathematics that focuses on more mathematical topics of computing, and includes the theory of computation. It is difficult to circumscribe the theoretical areas precisely.
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Direct-reading ferrography is a more mathematical approach to ferrography. Essentially, the buildup on the glass slide is measured by shining a light across the slide. The blockage of the light by the buildup of particles is then used, over time, to calculate an average. An increase in blockage indicates higher amounts of machine wear.
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In general, more mathematical tools have been developed to test the fit of statistical models than models involving differential equations. Tools from nonparametric statistics can sometimes be used to evaluate how well the data fit a known distribution or to come up with a general model that makes only minimal assumptions about the model's mathematical form.
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Cram et al (2003) pp. 243–291. He also studied the first games that babies and children play, and wrote a more mathematical section "On the rebounding of tennis balls".Cram et al (2003) pp. 234–235. As with his biological works, the Book of Games is organised on the empirical principles of observation, description, and classification.
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Quantitative uses of the terms uncertainty and risk are fairly consistent from fields such as probability theory, actuarial science, and information theory. Some also create new terms without substantially changing the definitions of uncertainty or risk. For example, surprisal is a variation on uncertainty sometimes used in information theory. But outside of the more mathematical uses of the term, usage may vary widely.
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Carothers developed a series of mathematic equations to describe the behavior of step-growth polymerization systems which are still known as the Carothers equations today. Collaborating with Paul Flory, a physical chemist, they developed theories that describe more mathematical aspects of step-growth polymerization including kinetics, stoichiometry, and molecular weight distribution etc. Carothers is also well known for his invention of Nylon.
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The operational calculus generally is typified by two symbols, the operator p, and the unit function 1. The operator in its use probably is more mathematical than physical, the unit function more physical than mathematical. The operator p in the Heaviside calculus initially is to represent the time differentiator . Further, it is desired this operator bear the reciprocal relation such that p denotes the operation of integration.
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Efficiency is the amount of divisions ordered from a task. In more mathematical or scientific terms, it is a measure of the extent to which input is well used for an intended task or function (output). It often specifically comprises the capability of a specific application of effort to produce a specific outcome with a minimum amount or quantity of waste, expense, or unnecessary effort. Sickles, R., and Zelenyuk, V. (2019).
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A record can be viewed as the computer analog of a mathematical tuple, although a tuple may or may not be considered a record, and vice versa, depending on conventions and the specific programming language. In the same vein, a record type can be viewed as the computer language analog of the Cartesian product of two or more mathematical sets, or the implementation of an abstract product type in a specific language.
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Pedro Nunes (1502 – 1578) was a famed mathematician of the Portuguese Renaissance. The first university of Portugal was founded in 1290 as a Studium Generale in Lisbon. It was focused on the arts and humanities, but also included a medical school since its foundation. During the 16th century, in the Age of Discovery, a more mathematical educational approach flourished in this university with the creation of specialized courses and classes in the field.
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In natural philosophy and the philosophy of science, medieval philosophers were mainly influenced by Aristotle. However, from the fourteenth century onward, the increasing use of mathematical reasoning in natural philosophy prepared the way for the rise of science in the early modern period. The more mathematical reasoning techniques of William Heytesbury and William of Ockham are indicative of this trend. Other contributors to natural philosophy are Albert of Saxony, John Buridan, and Nicholas of Autrecourt.
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After the war Irwin embarked on a number of long-term collaborative studies, often for official committees. These were applied studies but he continued to work on more mathematical problems, e.g. he produced a series of papers on the Generalized Waring Distribution. Oscar Irwin received many honours and served as president of the Royal Statistical Society in 1962–64; he had played an important part in the affairs of the society for many years.
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In 1861, Roch went to work at the University of Göttingen, studying under Wilhelm Eduard Weber, but also attending lectures by Bernhard Riemann. After three terms in Göttingen, Roch went to the University of Berlin, where he met Leopold Kronecker, Karl Weierstrass and others. In 1862 he was awarded a magister artium degree from Leipzig and later a doctorate for his work on electromagnetism. From this point his work took a more mathematical slant.
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The TI-89 Titanium also has a USB On-The-Go port, for connectivity to other TI-89 Titanium calculators, or to a computer (to store programs or update the operating system). The TI-89 Titanium also features some pre-loaded applications, such as "CellSheet", a spreadsheet program also offered with other TI calculators. The Titanium has a slightly updated CAS, which adds a few more mathematical functions, most notably implicit differentiation.
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The Algebra FX series was the successor of the CFX-9970G, the first Casio calculator with computer algebra system (CAS) released in 1998. The computer algebra system in the Algebra FX series had been largely improved from the previous model so that more mathematical functions were added. The Casio's CAS was mainly developed by its R&D; team and Professor John Kenelly of Clemson University. The calculators were sold to different parts of the world.
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However, he was attracted to the more mathematical discipline of physical chemistry, which he thought more challenging, and in the end this is what he specialized in. He graduated in 1950. Panish enrolled in graduate school at Michigan State University, majoring in physical chemistry and minoring in organic chemistry. His master's thesis involved a "series of measurements of the electric dipole behavior of some organic compounds," and he did not consider it very challenging.
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A rare snippet of Baker at work was included in the 1968 documentary film Music, which also featured the Beatles working on Hey Jude in the studio. A jazz pianist, he brought a sense of rhythm to the Workshop which some of the other more mathematical composers lacked. His work included many signature tunes for BBC television and radio. He was also particularly interested in combining recorded electronic music with live musicians.
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FST values are also used to understand dispersing behaviors by calculating an FST value for each sex. The sex that disperses more displays a lower FST value, which measures levels of inbreeding between the subpopulation and the total population. Additionally, assignment tests can be utilized to quantify the number of individuals of a certain sex dispersing to other populations. A more mathematical approach to quantifying sex-biased dispersal on the molecular level is the use of spatial autocorrelation.
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The left is used to manipulate the operating system (moving screens and the like) and the right delivers commands to the application. The absence of Num lock frees space for more mathematical symbols around the numeric pad. Contemporary Macintosh computers, for comparison, lack function keys completely. The mouse has two buttons like Windows, but unlike Windows pressing and holding the right button replaces the system status line at the top of the screen with a Maclike menu bar.
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Much of the research done by the NRC sought to move safety systems from a deterministic viewpoint into a new probabilistic approach. The deterministic approach sought to foresee all problems before they arose. The probabilistic approach uses a more mathematical approach to weigh the risks of potential radiation leaks. Much of the probabilistic safety approach can be drawn from the radiative transfer theory in Physics, which describes how radiation travels in free space and through barriers.
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Williams' early work was on plant physiology, and more specifically on leaf expansion and the function of the Stomata. However, while at Southampton he became interested in more mathematical and statistical research topics such as the spatial distribution of plants. At the same time, he began writing software for his statistical analysis, using the Ferranti Pegasus computer then available at Southampton's computer centre. The centre's director, Godfrey Lance, was later to become a regular research collaborator of Williams.
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The idea of a global coordinate system revolutionized European geographical thought, however, and inspired more mathematical treatment of cartography. Ptolemy's work probably originally came with maps, but none have been discovered. Instead, the present form of the map was reconstructed from Ptolemy's coordinates by Byzantine monks under the direction of Maximus Planudes shortly after 1295. It probably was not that of the original text, as it uses the less favored of the two alternate projections offered by Ptolemy.
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The economic reforms emerged during a period of great ideological debate over economic planning. More mathematical, "cybernetic", viewpoints were at first considered deviant from orthodox Marxist economics, which considered the value of good to derive strictly from labor.Ellman, Soviet Planning Today (1971), p. 4. This doctrine, elaborated in such works as Stalin's 1952 book, Economic Problems of Socialism in the USSR, described the price system as a capitalist relic which would eventually disappear from communist society.Ellman, Soviet Planning Today (1971), p. 30.
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Entrance requirements are similarly more mathematical. On the other hand, coverage of financial and economic theory, and econometrics, while significant, is comparatively secondary.Resolution to Establish a Program Leading to the Master of Science In Financial Economics, Columbia University Senate As mentioned, some Financial Economics programs are substantially quantitative; these differ little from the MQF. The overlap with general finance degrees, such as a Master of Science in Finance (MSF) or an M.B.A. in finance, is further limited, particularly where the Financial Economics program is theory oriented.
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If the result of the calculation is an advice to switch envelopes, whatever amount might be in there, then it would appear that one should switch anyway, without looking. In this case, at Steps 6, 7 and 8 of the reasoning, "A" is any fixed possible value of the amount of money in the first envelope. This interpretation of the two envelopes problem appears in the first publications in which the paradox was introduced in its present-day form, Gardner (1989) and Nalebuff (1989). It is common in the more mathematical literature on the problem.
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He carried out research on granite that showed that it was possible for molten rock to form conformities. He melted basalt in an iron furnace, and demonstrated that it returned to its original form when cooled. He melted limestone in a retort made from a gun barrel, to show that it did not decompose when melted under pressure. His results were published in the Transactions of the Royal Society of Edinburgh and were well received by those like John Playfair who wanted to use Hutton's theory to promote a more mathematical approach to geology.
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Turing equivalence Many machines that might be thought to have more computational capability than a simple universal Turing machine can be shown to have no more power. They might compute faster, perhaps, or use less memory, or their instruction set might be smaller, but they cannot compute more powerfully (i.e. more mathematical functions). (The Church–Turing thesis hypothesizes this to be true: that anything that can be “computed” can be computed by some Turing machine.) The sequential-machine models All of the following are called "sequential machine models" to distinguish them from "parallel machine models".
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But when its concentration exceeds several atomic % , an alloy is formed and the interaction among the added atoms can no longer be neglected. Before giving a more mathematical outline of the RBM it is convenient to give somewhat of a visualization of what happens to a metal upon alloying it. In a pure metal, we'll take silver as an example, all lattice sites are occupied by silver atoms. When different kind of atoms are dissolved into it, for example 10% of copper, some random lattice sites become occupied by copper atoms.
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In physics, the Clebsch–Gordan (CG) coefficients are numbers that arise in angular momentum coupling in quantum mechanics. They appear as the expansion coefficients of total angular momentum eigenstates in an uncoupled tensor product basis. In more mathematical terms, the CG coefficients are used in representation theory, particularly of compact Lie groups, to perform the explicit direct sum decomposition of the tensor product of two irreducible representations (i.e., a reducible representation into irreducible representations, in cases where the numbers and types of irreducible components are already known abstractly).
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In this model (from 1944) the companion went nova with ejected material captured by the Sun and planets forming from this material. In a version a year later it was a supernova. In 1955 he proposed a similar system to Laplace, and with more mathematical detail in 1960. It differs from Laplace in that a magnetic torque occurs between the disk and the Sun, which comes into effect immediately or else more and more matter would be ejected resulting in a much too massive planetary system, one comparable to the Sun.
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The 20th century saw mathematics become a major profession. As in most areas of study, the explosion of knowledge in the scientific age has led to specialization: by the end of the century there were hundreds of specialized areas in mathematics and the Mathematics Subject Classification was dozens of pages long.Mathematics Subject Classification 2000 Every year, thousands of new Ph.D.s in mathematics were awarded, and jobs were available in both teaching and industry. More and more mathematical journals were published and, by the end of the century, the development of the World Wide Web led to online publishing.
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The Hartmann Neuron was an electronic musical instrument designed and built by industry designer Axel Hartmann of the German company Hartmann Music in the years 2001–2005. The Neuron synthesizer used an Artificial Neural Network to create a digital computer model from sampled sounds, implementing a new synthesis technology which was developed by Stephan Bernsee at Prosoniq. This computer model could then be used to resynthesize a new sound, giving the player control over "musical" aspects such as instrument body size, material and other acoustic properties. This parametric approach is very different from the more mathematical approaches of other synthesizers.
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In a later issue, Carroll gives the solution to a Knot and discusses readers' answers. The mathematical interpretations of the Knots are not always straightforward. The ribbing of readers answering wrongly – giving their names – was not always well received (see Knot VI below). In the December 1885 book preface Carroll writes: :The writer's intention was to embody in each Knot (like medicine so dexterously, but ineffectually, concealed in the jam of our early childhood) one or more mathematical questions – in Arithmetic, Algebra, or Geometry, as the case might be – for the amusement, and possible edification, of the fair readers of that magazine.
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The structure of the eye is presented as a hierarchy, starting first with the part that is most directly responsible for sight, and working down towards the parts which exists only to facilitate this function. The lens, described as white, transparent, and luminous have a composition which lends itself to quickly receive colors. As opposed to Galen's more mathematical conception of flat-like lens, Hunayn opts for a more spherical shape which allows for a larger field of vision. Hunayn repeatedly emphasized that he believed the crystalline lens to be in the exact center of the eye.
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Charles Grafton Page (in Salem, Massachusetts January 25, 1812 – May 5, 1868 in Washington, D.C.) was an American electrical experimenter and inventor, physician, patent examiner, patent advocate, and professor of chemistry. Like his more famous contemporaries Michael Faraday and Joseph Henry, Page began his career as an astute natural philosopher who developed innovative work with natural phenomena through direct observation and experimenting. Toward the later part of their careers, the science of the day had moved on to a more mathematical emphasis in which these scientists did not participate. Through his exploratory experiments and distinctive inventions, Page developed a deep understanding of electromagnetism.
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Once Curto gained her Ph.D, she moved to Rutgers University in New Jersey, working as a Postdoctoral Associate in a neuroscience lab at the Center for Molecular and Behavioral Neuroscience. After three years at Rutgers, she returned to working in mathematics, focusing on neuroscience from a more mathematical perspective, and moved to the Courant Institute of Mathematics at New York University, then to the University of Nebraska- Lincoln, followed by the Pennsylvania State University. Curto is a Professor of Mathematics, the co-head of their mathematical neuroscience lab, and a member of the Center for Neural Engineering.
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SEM image of a pentamode metamaterial (with a size of roughly 300μm) A pentamode metamaterial is an artificial three- dimensional structure which, despite being a solid, ideally behaves like a fluid. Thus, it has a finite bulk but vanishing shear modulus, or in other words it is hard to compress yet easy to deform. Speaking in a more mathematical way, pentamode metamaterials have an elasticity tensor with only one non-zero eigenvalue and five (penta) vanishing eigenvalues. Pentamode structures have been proposed theoretically by Graeme Milton and Andrej Cherkaev in 1995 but have not been fabricated until early 2012.
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These branches of natural science may be further divided into more specialized branches (also known as fields). As empirical sciences, natural sciences use tools from the formal sciences, such as mathematics and logic, converting information about nature into measurements which can be explained as clear statements of the "laws of nature". Modern natural science succeeded more classical approaches to natural philosophy, usually traced to Taoists traditions in Asia and in the Occident to ancient Greece. Galileo, Descartes, Bacon, and Newton debated the benefits of using approaches which were more mathematical and more experimental in a methodical way.
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"Figure 8" or bi-directional microphones receive sound equally from both the front and back of the element. Most ribbon microphones are of this pattern. In principle they do not respond to sound pressure at all, only to the change in pressure between front and back; since sound arriving from the side reaches front and back equally there is no difference in pressure and therefore no sensitivity to sound from that direction. In more mathematical terms, while omnidirectional microphones are scalar transducers responding to pressure from any direction, bi-directional microphones are vector transducers responding to the gradient along an axis normal to the plane of the diaphragm.
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The incredible precision of mathematical expression permits a vast applicability of mental abstractions to real life situations. Many more mathematical formulas describe shapes that are unfamiliar, or do not necessarily correspond to objects in the real world. For example, the Klein bottleBurger, E. B., & Starbird, M., "A One-Sided, Sealed Surface—The Klein Bottle", in E. B. Burger, M. Starbird, T. Stonebarger, & T. Dunton, Joy of Thinking: The Beauty and Power of Classical Mathematical Ideas (Chantilly, VA: The Teaching Company, 2003). is a one-sided, sealed surface with no inside or outside (in other words, it is the three-dimensional equivalent of the Möbius strip).
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Some of the most popular and influential djent bands were Periphery, After the Burial, Animals as Leaders, Tesseract and Born of Osiris. Djent's popularity led to more bands including more mathematical elements in their music with it eventually leading to elements in mainstream pop music. Another form of progressive rock in this decade was post-progressive, which drew influences from bands such as Radiohead and Porcupine Tree from the 2000s. This genre relies far more on simple melodic patterns combined with complex structure, texture and harmonies to create deep, emotionally driven, musical journeys in an effort to return the genre to its roots and make it relevant in the decade.
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Gottlob Frege began the program of reducing mathematics to logic, continued it with Bertrand Russell, but lost interest in this logicism, and Russell continued it with Alfred North Whitehead in their Principia Mathematica, inspiring some of the more mathematical logical posivists, such as Hans Hahn and Rudolf Carnap. Carnap's early anti- metaphysical works employed Russell's theory of types.See Rudolf Carnap, "The elimination of metaphysics through logical analysis of language", Erkenntnis, 1932;2, reprinted in Logical Positivism, Alfred Jules Ayer, ed, (New York: Free Press, 1959), pp. 60–81. Carnap envisioned a universal language that could reconstruct mathematics and thereby encode physics.Jaako Hintikka, "Logicism", in Andrew D Irvine, ed, Philosophy of Mathematics (Burlington MA: North Holland, 2009), pp. 283–84.
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In particular they are used in combinations, such as LC, to mean, for instance, a network consisting only of inductors and capacitors. Z is used for electrical impedance, any 2-terminalA terminal of a network is a connection point where current can enter or leave the network from the world outside. This is often called a pole in the literature, especially the more mathematical, but is not to be confused with a pole of the transfer function which is a meaning also used in this article. A 2-terminal network amounts to a single impedance (although it may consist of many elements connected in a complicated set of meshes) and can also be described as a one-port network.
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According to Plato, the Earth was a sphere, stationary at the center of the universe. The stars and planets were carried around the Earth on spheres or circles, arranged in the order (outwards from the center): Moon, Sun, Venus, Mercury, Mars, Jupiter, Saturn, fixed stars, with the fixed stars located on the celestial sphere. In his "Myth of Er", a section of the Republic, Plato describes the cosmos as the Spindle of Necessity, attended by the Sirens and turned by the three Fates. Eudoxus of Cnidus, who worked with Plato, developed a less mythical, more mathematical explanation of the planets' motion based on Plato's dictum stating that all phenomena in the heavens can be explained with uniform circular motion.
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Until the release of version 16, Stata could only open a single dataset at any one time. Stata holds datasets in (random- access or virtual) memory, which limits its use with extremely large datasets. This is mitigated to some extent by efficient internal storage, as there are integer storage types which occupy only one or two bytes rather than four, and single-precision (4 bytes) rather than double-precision (8 bytes) is the default for floating-point numbers. The dataset is always rectangular in format, that is, all variables hold the same number of observations (in more mathematical terms, all vectors have the same length, although some entries may be missing values).
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Hanski's model predicted that the power law exponent would be constrained to range closely about the value of 2, which seemed inconsistent with many reported values. Anderson et al formulated a simple stochastic birth, death, immigration and emigration model that yielded a quadratic variance function. As a response to this model Taylor argued that such a Markov process would predict that the power law exponent would vary considerably between replicate observations, and that such variability had not been observed. About this time concerns were, however, raised regarding the statistical variability with measurements of the power law exponent, and the possibility that observations of a power law might reflect more mathematical artifact than a mechanistic process.
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Tale of Genji chapter symbols, including diagrams of the first 52 set partitions The number of different possible rhyme schemes for an n-line poem is given by the Bell numbers,. Reprinted with an addendum as "The Tinkly Temple Bells", Chapter 2 of Fractal Music, Hypercards, and more ... Mathematical Recreations from Scientific American, W. H. Freeman, 1992, pp. 24–38. which for n = 1, 2, 3, ... are :1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, .. . Examples: We find one rhyme scheme for a one-line poem (A), two different rhyme schemes for a two-line poem (AA, AB), and five for a three-line poem: AAA, AAB, ABA, ABB, and ABC.
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Hamilton enrolled in an MSc course in demography at the London School of Economics (LSE), under Norman Carrier, who helped secure various grants for his studies. Later, when his work became more mathematical and genetical, he had his supervision transferred to John Hajnal of the LSE and Cedric Smith of University College London (UCL). Both Fisher and J. B. S. Haldane had seen a problem in how organisms could increase the fitness of their own genes by aiding their close relatives, but not recognised its significance or properly formulated it. Hamilton worked through several examples, and eventually realised that the number that kept falling out of his calculations was Sewall Wright's coefficient of relationship.
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The book aims to provide an explanation of quantum mechanics and its impact on the modern world that is accessible to a general reader. The authors say that "our goal in writing this book is to demystify quantum theory". Starting with the concepts of wave–particle duality and a non-technical description of the path integral formulation of quantum mechanics, the book explains the uncertainty principle, energy levels in atoms, the physics of semi-conductors and transistors, and the Standard Model of particle physics. A more mathematical Epilogue discusses the role of quantum mechanics in models of stellar evolution, and derives the Chandrasekhar limit for the maximum mass of a stable white dwarf.
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The authors of this study conclude that individuals high in psychopathic traits are less influenced by their emotions and therefore make more "mathematical" decisions and choose the option that leads to the lowest number of deaths. The theories discussed above are not intended to be a comprehensive list but instead to provide a sense of how psychopaths differ in their approach to social interactions. As with most psychological/social phenomena, it is likely a combination of these factors that leads to psychopaths' lack of prosocial behavior. Further research is needed to determine the causal nature of any one of these individual deficits as well as if there is any way to help these individuals develop more prosocial patterns of behavior.
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Clausius's PhD thesis concerning the refraction of light proposed that we see a blue sky during the day, and various shades of red at sunrise and sunset (among other phenomena) due to reflection and refraction of light. Later, Lord Rayleigh would show that it was in fact due to the scattering of light, but regardless, Clausius used a far more mathematical approach than some have used. His most famous paper, Ueber die bewegende Kraft der Wärme ("On the Moving Force of Heat and the Laws of Heat which may be Deduced Therefrom"). See English Translation: On the Moving Force of Heat, and the Laws regarding the Nature of Heat itself which are deducible therefrom. Phil. Mag. (1851), series 4, 2, 1–21, 102–119.
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Haertling's designs reflect an eclectic mix of different architectural styles and philosophies, incorporating elements of modernism and organic architecture, and drawing inspiration from the Usonian principles of Frank Lloyd Wright and the works of Bruce Goff, among others. He often experimented with forms found in nature, including leaves (Leaneagh House, 1980), mushrooms and barnacles (Brenton House, 1969), yucca plants (Warburton House, 1963), and other natural shapes. At the same time, he also designed many structures according to more mathematical and geometric themes, as in the examples of the Willard House (1962) and the multifaceted Jourgensen House (1971). For his St. Stephens Church in Northglenn, Colorado (1964), Haertling employed a hyperbolic paraboloid roof, evocative of similar curved concrete designs by Oscar Niemeyer.
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The MSF deals with these topics conceptually, as opposed to technically, and the overlap is therefore slight: although practical, these topics are too technical for a generalist finance degree, and the exposure will be limited to the generalist level. The M.Fin / MSc, on the other hand, cover these topics in a substantially mathematical fashion, and the treatment is often identical. The distinction here though, is that these place relatively more emphasis on financial theory than the MQF, and also allow for electives outside of quantitative finance; at the same time, the range of quantitative electives is often smaller. Entrance requirements to the MQF are significantly more mathematical than for the MSF, while for the M.Fin / MSc the requirements may be identical.
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The Precious Mirror also contains a diagram of Pascal's triangle with coefficients of binomial expansions through the eighth power, though both appear in Chinese works as early as 1100. The Chinese also made use of the complex combinatorial diagram known as the magic square and magic circles, described in ancient times and perfected by Yang Hui (AD 1238–1298). Even after European mathematics began to flourish during the Renaissance, European and Chinese mathematics were separate traditions, with significant Chinese mathematical output in decline from the 13th century onwards. Jesuit missionaries such as Matteo Ricci carried mathematical ideas back and forth between the two cultures from the 16th to 18th centuries, though at this point far more mathematical ideas were entering China than leaving.
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The work of Chomsky in generative linguistics apparently inspired much more confidence in philosophers and logicians to assert that perhaps natural languages weren't as unsystematic and misleading as their philosophical predecessors had made them out to be ... at the end of 1960s formal semantics began to flourish." writes: "Recent work by Chomsky and others is doing much to bring the complexities of natural languages within the scope of serious semantic theory". ;Computer science With its formal and logical treatment of language, Syntactic Structures also brought linguistics and the new field of computer science closer together. Computer scientist Donald Knuth (winner of the Turing Award) recounted that he read Syntactic Structures in 1961 and was influenced by it.From the preface of : "... researchers in linguistics were beginning to formulate rules of grammar that were considerably more mathematical than before.
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The more mathematical approach uses an index-free notation, emphasizing the geometric and algebraic structure of the gauge theory and its relationship to Lie algebras and Riemannian manifolds; for example, treating gauge covariance as equivariance on fibers of a fiber bundle. The index notation used in physics makes it far more convenient for practical calculations, although it makes the overall geometric structure of the theory more opaque. The physics approach also has a pedagogical advantage: the general structure of a gauge theory can be exposed after a minimal background in multivariate calculus, whereas the geometric approach requires a large investment of time in the general theory of differential geometry, Riemannian manifolds, Lie algebras, representations of Lie algebras and principle bundles before a general understanding can be developed. In more advanced discussions, both notations are commonly intermixed.
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Despite his fame, Anders cared little for the glamorous aspects of his life as a Pyramid star, instead revealing in an interview that he took pleasure in the subtler aspects of the game, the joys of creating an impressive play, relating the sport to more mathematical concepts of physics and precision. Although highly successful at the game, he revealed that winning championships was of comparatively little interest to him compared to these more nebulous aspects. As a resistance leader, Anders showed a talent for organization and a good tactical ability, mounting a number of highly successful strikes, even some deep inside Cylon occupied cities. He understandably carried a deep hatred for the Cylons, but also had a pragmatic attitude, such as his willingness to accept that the Number Eight copy who had been traveling with Karl Agathon was on their side.
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In reviewing the book, mathematicians Thomas Sonar and James J. Tattersall recommend the book to a general audience interested in mathematics and its history, and reviewer James J. Tattersall writes that it contains "a wealth of mathematical and historical information on a wide selection of topics". Reviewer William J. Satzer, also a mathematician, disagrees on the target audience, writing that although the book would be of interest to historians of mathematics, it would make difficult reading for others because its topics are too loosely connected. Satzer also faults the book for its omission of Japanese and Chinese threads in its tapestry. On the other hand, Argentine origami book author Laura Rozenberg, despite admitting to skipping over the more mathematical parts of the story, says it "can be read by the non-mathematician without pausing", writing that it felt that "Friedman had read our minds and had decided to indulge us with answers to problems that have beset paperfolding aficionados for years".
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This example is an extension of the 'rolling wheel' problem considered above, with a more mathematical treatment. Consider a three-dimensional orthogonal Cartesian coordinate frame, for example, a level table top with a point marked on it for the origin, and the x and y axes laid out with pencil lines. Take a sphere of unit radius, for example, a ping-pong ball, and mark one point B in blue. Corresponding to this point is a diameter of the sphere, and the plane orthogonal to this diameter positioned at the center C of the sphere defines a great circle called the equator associated with point B. On this equator, select another point R and mark it in red. Position the sphere on the z = 0 plane such that the point B is coincident with the origin, C is located at x = 0, y = 0, z = 1, and R is located at x = 1, y = 0, and z = 1, i.e.
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RCA Mark II Sound Synthesizer In 1958, Columbia-Princeton developed the RCA Mark II Sound Synthesizer, the first programmable synthesizer.. Prominent composers such as Vladimir Ussachevsky, Otto Luening, Milton Babbitt, Charles Wuorinen, Halim El-Dabh, Bülent Arel and Mario Davidovsky used the RCA Synthesizer extensively in various compositions.. One of the most influential composers associated with the early years of the studio was Egypt's Halim El-Dabh who,. after having developed the earliest known electronic tape music in 1944, became more famous for Leiyla and the Poet, a 1959 series of electronic compositions that stood out for its immersion and seamless fusion of electronic and folk music, in contrast to the more mathematical approach used by serial composers of the time such as Babbitt. El-Dabh's Leiyla and the Poet, released as part of the album Columbia-Princeton Electronic Music Center in 1961, would be cited as a strong influence by a number of musicians, ranging from Neil Rolnick, Charles Amirkhanian and Alice Shields to rock musicians Frank Zappa and The West Coast Pop Art Experimental Band.
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