tidal theory: see solar system.

theory, in music, discipline involving the construction of cognitive systems to be used as a tool for comprehending musical compositions. The discipline is subdivided into what can be called speculative and analytic theory. Speculative theory engages in reconciling with music certain philosophical observations of man and nature. It can be prescriptive when it imposes these extramusical contentions to establish an aesthetic norm. Music theory tended toward this aspect until the 20th cent. An example is the attempt to assert the superiority of tonal music over other systems by reference to the relationship of the triad to the natural overtone series. Analytic theory, on the other hand, undertakes detailed study of individual pieces. Analyses of compositions of a particular genre are synthesized into a general system, or reference, against which the individuality of these pieces can be perceived. In more general usage the term theory is used to include the study of acoustics, harmony, and ear training. In ancient Greece music theory was mainly concerned with describing different scales (modes) and their emotional character. This theory was transmitted, largely erroneously, to medieval Europe by the Roman philosopher Boethius in his De musica (6th cent. A.D.). Medieval European theory dealt with notation, modal and rhythmic systems, and the relation of music to Christianity. Gioseffo Zarlino (1515-90) was the first to consider the triad as a compositional reference. In the 18th cent. Jean Philippe Rameau sought to show how the major-minor system of tonal harmony derives from the inherent acoustical properties of sound itself, and establish the laws of harmonic progression. The writings of Heinrich Schenker are among the most important in the sphere of tonal theory. Major contemporary theorists are Paul Hindemith, who propounded the idea of non-triadic pitch centrality, and Milton Babbitt, who has published revealing explications of twelve-tone music.

string theory, description of elementary particles based on one-dimensional curves, or "strings," instead of point particles. Superstring theory, which is string theory that contains a kind of symmetry known as supersymmetry, shows promise as a way of unifying the four known fundamental forces of nature. The strings are embedded in a space-time having as many as 10 dimensions—the three ordinary dimensions plus time and seven compactified dimensions. The energy-scale at which the stringlike properties would become evident is so high that it is currently unclear how any of the forms of the theory could be tested.

steady-state theory: see cosmology.

quantum theory, modern physical theory concerned with the emission and absorption of energy by matter and with the motion of material particles; the quantum theory and the theory of relativity together form the theoretical basis of modern physics. Just as the theory of relativity assumes importance in the special situation where very large speeds are involved, so the quantum theory is necessary for the special situation where very small quantities are involved, i.e., on the scale of molecules, atoms, and elementary particles. Aspects of the quantum theory have provoked vigorous philosophical debates concerning, for example, the uncertainty principle and the statistical nature of all the predictions of the theory.

Relationship of Energy and Matter

According to the older theories of classical physics, energy is treated solely as a continuous phenomenon, while matter is assumed to occupy a very specific region of space and to move in a continuous manner. According to the quantum theory, energy is held to be emitted and absorbed in tiny, discrete amounts. An individual bundle or packet of energy, called a quantum (pl. quanta), thus behaves in some situations much like particles of matter; particles are found to exhibit certain wavelike properties when in motion and are no longer viewed as localized in a given region but rather as spread out to some degree.

For example, the light or other radiation given off or absorbed by an atom has only certain frequencies (or wavelengths), as can be seen from the line spectrum associated with the chemical element represented by that atom. The quantum theory shows that those frequencies correspond to definite energies of the light quanta, or photons, and result from the fact that the electrons of the atom can have only certain allowed energy values, or levels; when an electron changes from one allowed level to another, a quantum of energy is emitted or absorbed whose frequency is directly proportional to the energy difference between the two levels.

Dual Nature of Waves and Particles

The restriction of the energy levels of the electrons is explained in terms of the wavelike properties of their motions: electrons occupy only those orbits for which their associated wave is a standing wave (i.e., the circumference of the orbit is exactly equal to a whole number of wavelengths) and thus can have only those energies that correspond to such orbits. Moreover, the electrons are no longer thought of as being at a particular point in the orbit but rather as being spread out over the entire orbit. Just as the results of relativity approximate those of Newtonian physics when ordinary speeds are involved, the results of the quantum theory agree with those of classical physics when very large "quantum numbers" are involved, i.e., on the ordinary large scale of events; this agreement in the classical limit is required by the correspondence principle of Niels Bohr. The quantum theory thus proposes a dual nature for both waves and particles, one aspect predominating in some situations, the other predominating in other situations.

Evolution of Quantum Theory

Early Developments

While the theory of relativity was largely the work of one man, Albert Einstein, the quantum theory was developed principally over a period of thirty years through the efforts of many scientists. The first contribution was the explanation of black body radiation in 1900 by Max Planck, who proposed that the energies of any harmonic oscillator (see harmonic motion), such as the atoms of a black body radiator, are restricted to certain values, each of which is an integral (whole number) multiple of a basic, minimum value. The energy E of this basic quantum is directly proportional to the frequency ν of the oscillator, or E=hν, where h is a constant, now called Planck's constant, having the value 6.63×10^-34 joule-second. In 1905, Einstein proposed that the radiation itself is also quantized according to this same formula, and he used the new theory to explain the photoelectric effect. Following the discovery of the nuclear atom by Rutherford (1911), Bohr used the quantum theory in 1913 to explain both atomic structure and atomic spectra, showing the connection between the electrons' energy levels and the frequencies of light given off and absorbed.

Quantum Mechanics and Later Developments

Quantum mechanics, the final mathematical formulation of the quantum theory, was developed during the 1920s. In 1924, Louis de Broglie proposed that not only do light waves sometimes exhibit particlelike properties, as in the photoelectric effect and atomic spectra, but particles may also exhibit wavelike properties. This hypothesis was confirmed experimentally in 1927 by C. J. Davisson and L. H. Germer, who observed diffraction of a beam of electrons analogous to the diffraction of a beam of light. Two different formulations of quantum mechanics were presented following de Broglie's suggestion. The wave mechanics of Erwin Schrödinger (1926) involves the use of a mathematical entity, the wave function, which is related to the probability of finding a particle at a given point in space. The matrix mechanics of Werner Heisenberg (1925) makes no mention of wave functions or similar concepts but was shown to be mathematically equivalent to Schrödinger's theory.

Quantum mechanics was combined with the theory of relativity in the formulation of P. A. M. Dirac (1928), which, in addition, predicted the existence of antiparticles. A particularly important discovery of the quantum theory is the uncertainty principle, enunciated by Heisenberg in 1927, which places an absolute theoretical limit on the accuracy of certain measurements; as a result, the assumption by earlier scientists that the physical state of a system could be measured exactly and used to predict future states had to be abandoned. Other developments of the theory include quantum statistics, presented in one form by Einstein and S. N. Bose (the Bose-Einstein statistics) and in another by Dirac and Enrico Fermi (the Fermi-Dirac statistics); quantum electrodynamics, concerned with interactions between charged particles and electromagnetic fields; its generalization, quantum field theory; and quantum electronics.

quantum field theory, study of the quantum mechanical interaction of elementary particles and fields. Quantum field theory applied to the understanding of electromagnetism is called quantum electrodynamics (QED), and it has proved spectacularly successful in describing the interaction of light with matter. The calculations, however, are often complex. They are usually carried out with the aid of Feynman diagrams (named after American physicist Richard P. Feynman), simple graphs that represent possible variations of interactions and provide an elegant shorthand for precise mathematical equations. Quantum field theory applied to the understanding of the strong interactions between quarks and between protons, neutrons, and other baryons and mesons is called quantum chromodynamics (QCD); QCD has a mathematical structure similar to that of QED.

protoplanet theory: see solar system.

planetesimal theory: see solar system.

phlogiston theory, hypothesis regarding combustion. The theory, advanced by J. J. Becher late in the 17th cent. and extended and popularized by G. E. Stahl, postulates that in all flammable materials there is present phlogiston, a substance without color, odor, taste, or weight that is given off in burning. "Phlogisticated" substances are those that contain phlogiston and, on being burned, are "dephlogisticated." The ash of the burned material is held to be the true material. The theory received strong and wide support throughout a large part of the 18th cent. until it was refuted by the work of A. L. Lavoisier, who revealed the true nature of combustion. Joseph Priestley, however, defended the theory throughout his lifetime. Henry Cavendish remained doubtful, but most other chemists of the period, including C. L. Berthollet, rejected it.

number theory, branch of mathematics concerned with the properties of the integers (the numbers 0, 1, -1, 2, -2, 3, -3, …). An important area in number theory is the analysis of prime numbers. A prime number is an integer p<1 divisible only by 1 and p; the first few primes are 2, 3, 5, 7, 11, 13, 17, and 19. Integers that have other divisors are called composite; examples are 4, 6, 8, 9, 10, 12, … . The fundamental theorem of arithmetic, the unique factorization theorem, asserts that any positive integer a is a product (a = p₁ · p₂ · p₃ · · · p_n) of primes that are unique except for the order in which they are listed; e.g., the number 20 is the product 20 = 2 · 2 ·5, and it is unique (disregarding order) since 20 has this and only this product of primes. This theorem was known to the Greek mathematician Euclid, who proved that there are infinitely many primes. Analytic number theory has given a further refinement of Euclid's theorem by determining a function that measures how densely the primes are distributed among all integers. Twin primes are primes having a difference of 2, such as (3,5) and (11,13). The modern theory of numbers made its first great advances through the work of Leonhard Euler, C. F. Gauss, and Pierre de Fermat. It remains a major area of mathematical research, to which the most sophisticated mathematical tools have been applied.

molecular orbital theory, detailed explanation of how electrons are distributed in stable molecules. In the simpler valence theory of the chemical bond, each atom in a molecule is assumed to retain its own electrons. Even when electrons are shared, as in the covalent bond, it is possible to identify which electron came from which atom. The molecular orbital theory, however, treats each electron as associated with the molecule as a whole. Just as a free atom has certain allowed electron orbits and energies, each molecule has its own allowed molecular orbitals. The orbitals give the probability of finding the electron at any point in space. Each orbital can hold a maximum of two valence electrons and the structure of the molecule is built up by filling the lowest energy orbitals first. The calculations involved are extremely complex, and only the simplest molecules can be treated exactly.

kinetic-molecular theory of gases, physical theory that explains the behavior of gases on the basis of the following assumptions: (1) Any gas is composed of a very large number of very tiny particles called molecules; (2) The molecules are very far apart compared to their sizes, so that they can be considered as points; (3) The molecules exert no forces on one another except during rare collisions, and these collisions are perfectly elastic, i.e., they take place within a negligible span of time and in accordance with the laws of mechanics. A gas corresponding to these assumptions is called an ideal gas; as the temperature of a real gas is lowered, or its pressure is raised, its behavior no longer resembles that of an ideal gas because one or more of the assumptions of the theory is no longer valid. The analysis of the behavior of an ideal gas according to the laws of mechanics leads to the general gas law, or ideal gas law: The product of the pressure and volume of an ideal gas is directly proportional to its absolute temperature, or PV = kT (see gas laws). Boyle's law, Charles's law, and Gay-Lussac's law, which are special cases of the general gas law, may also be easily derived. The theory further shows that the absolute temperature is directly proportional to the average kinetic energy of the molecules, thus providing an interpretation of the nature of temperature in general in terms of the detailed structure of matter (see temperature; Kelvin temperature scale). Pressure is seen to be the result of large numbers of collisions between the molecules and the walls of the container in which the gas is held. See thermodynamics.

information theory or communication theory, mathematical theory formulated principally by the American scientist Claude E. Shannon to explain aspects and problems of information and communication. While the theory is not specific in all respects, it proves the existence of optimum coding schemes without showing how to find them. For example, it succeeds remarkably in outlining the engineering requirements of communication systems and the limitations of such systems.

In information theory, the term information is used in a special sense; it is a measure of the freedom of choice with which a message is selected from the set of all possible messages. Information is thus distinct from meaning, since it is entirely possible for a string of nonsense words and a meaningful sentence to be equivalent with respect to information content.

Measurement of Information Content

Numerically, information is measured in bits (short for binary digit; see binary system). One bit is equivalent to the choice between two equally likely choices. For example, if we know that a coin is to be tossed but are unable to see it as it falls, a message telling whether the coin came up heads or tails gives us one bit of information. When there are several equally likely choices, the number of bits is equal to the logarithm of the number of choices taken to the base two. For example, if a message specifies one of sixteen equally likely choices, it is said to contain four bits of information. When the various choices are not equally probable, the situation is more complex.

Interestingly, the mathematical expression for information content closely resembles the expression for entropy in thermodynamics. The greater the information in a message, the lower its randomness, or "noisiness," and hence the smaller its entropy. Since the information content is, in general, associated with a source that generates messages, it is often called the entropy of the source. Often, because of constraints such as grammar, a source does not use its full range of choice. A source that uses just 70% of its freedom of choice would be said to have a relative entropy of 0.7. The redundancy of such a source is defined as 100% minus the relative entropy, or, in this case, 30%. The redundancy of English is estimated to be about 50%; i.e., about half of the elements used in writing or speaking are freely chosen, and the rest are required by the structure of the language.

Analysis of the Transfer of Messages through Channels

A message proceeds along a channel from the source to the receiver; information theory defines for any given channel a limiting capacity or rate at which it can carry information, expressed in bits per second. In general, it is necessary to process, or encode, information from a source before transmitting it through a given channel. For example, a human voice must be encoded before it can be transmitted by telephone. An important theorem of information theory states that if a source with a given entropy feeds information to a channel with a given capacity, and if the source entropy is less than the channel capacity, a code exists for which the frequency of errors may be reduced as low as desired. If the channel capacity is less than the source entropy, no such code exists.

The theory further shows that noise, or random disturbance of the channel, creates uncertainty as to the correspondence between the received signal and the transmitted signal. The average uncertainty in the message when the signal is known is called the equivocation. It is shown that the net effect of noise is to reduce the information capacity of the channel. However, redundancy in a message, as distinguished from redundancy in a source, makes it more likely that the message can be reconstructed at the receiver without error. For example, if something is already known as a certainty, then all messages about it give no information and are 100% redundant, and the information is thus immune to any disturbances of the channel. Using various mathematical means, Shannon was able to define channel capacity for continuous signals, such as music and speech.

games, theory of, group of mathematical theories first developed by John Von Neumann and Oskar Morgenstern. A game consists of a set of rules governing a competitive situation in which from two to n individuals or groups of individuals choose strategies designed to maximize their own winnings or to minimize their opponent's winnings; the rules specify the possible actions for each player, the amount of information received by each as play progresses, and the amounts won or lost in various situations. Von Neumann and Morgenstern restricted their attention to zero-sum games, that is, to games in which no player can gain except at another's expense.

This restriction was overcome by the work of John F. Nash during the early 1950s. Nash mathematically clarified the distinction between cooperative and noncooperative games. In noncooperative games, unlike cooperative ones, no outside authority assures that players stick to the same predetermined rules, and binding agreements are not feasible. Further, he recognized that in noncooperative games there exist sets of optimal strategies (so-called Nash equilibria) used by the players in a game such that no player can benefit by unilaterally changing his or her strategy if the strategies of the other players remain unchanged. Because noncooperative games are common in the real world, the discovery revolutionized game theory. Nash also recognized that such an equilibrium solution would also be optimal in cooperative games. He suggested approaching the study of cooperative games via their reduction to noncooperative form and proposed a methodology, called the Nash program, for doing so. Nash also introduced the concept of bargaining, in which two or more players collude to produce a situation where failure to collude would make each of them worse off.

The theory of games applies statistical logic to the choice of strategies. It is applicable to many fields, including military problems and economics. The Nobel Memorial Prize in Economic Sciences was awarded to Nash, John Harsanyi, and Reinhard Selten (1994) and to Robert J. Aumann and Thomas C. Schelling (2005) for work in applying game theory to aspects of economics.

encounter theory, in astronomy, see solar system.

electroweak theory, a unified field theory that describes two of the fundamental forces in nature, electromagnetism (see electromagnetic radiation) and the weak interaction. The electroweak theory derived from efforts to produce a theory for the weak force analogous to quantum electrodynamics (QED), the quantum theory of the electromagnetic force. Although the weak force fails to meet a requirement for that theory—that it behave the same way at different points in space and time—because it acts only across distances smaller than an atomic nucleus, it was shown that the electromagnetic force, which can extend across interstellar distances, and the weak force are but different manifestations of a more fundamental force, the electroweak force. This made it possible to formulate a unified model that predicted the existence of mediating, or messenger, particles. The electroweak theory, for which Sheldon Glashow, Abdus Salam, and Steven Weinberg shared the 1979 Nobel Prize in Physics, was confirmed in 1983 by the discovery of the W and Z particles, two of a number of elementary particles it predicted.

domino theory, the notion that if one country becomes Communist, other nations in the region will probably follow, like dominoes falling in a line. The analogy, first applied (1954) to Southeast Asia by President Dwight Eisenhower, was adopted in the 1960s by supporters of the U.S. role in the Vietnam War. The theory was revived in the 1980s to characterize the threat perceived from leftist unrest in Central America.

communication theory: see information theory.

chaos theory, in mathematics, physics, and other fields, a set of ideas that attempts to reveal structure in aperiodic, unpredictable dynamic systems such as cloud formation or the fluctuation of biological populations. Although chaotic systems obey certain rules that can be described by mathematical equations, chaos theory shows the difficulty of predicting their long-range behavior. In the last half of the 20th cent., theorists in various scientific disciplines began to believe that the type of linear analysis used in classical applied mathematics presumes an orderly periodicity that rarely occurs in nature; in the quest to discover regularities, disorder had been ignored. Thus, chaos theorists have set about constructing deterministic, nonlinear dynamic models that elucidate irregular, unpredictable behavior (see nonlinear dynamics). Some of the early investigators of chaos were the American physicist Mitchell Feigenbaum; the Polish-born mathematician and inventor of fractals (see fractal geometry) Benoit Mandelbrot; the American mathematician James Yorke, who popularized the term "chaos"; and the American meteorologist Edward Lorenz.

big-bang theory: see cosmology.

Young-Helmholtz theory: see vision.

BCS theory: see superconductivity.

Attempt to describe all fundamental interactions between elementary particles in terms of a single theoretical framework (a “theory of everything”) based on quantum field theory. So far, the weak force and the electromagnetic force have been successfully united in electroweak theory, and the strong force is described by a similar quantum field theory called quantum chromodynamics. However, attempts to unite the strong and electroweak theories in a grand unified theory have failed, as have attempts at a self-consistent quantum field theory of gravitation.

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In logic, a theory introduced by Bertrand Russell and Alfred North Whitehead in their Principia Mathematica (1910–13) to deal with logical paradoxes arising from the unrestricted use of propositional functions as variables. The type of a propositional function is determined by the number and type of its arguments (the distinct variables it contains). By not allowing propositional functions to be applied to arguments of equal or higher type, contradictions within the system are avoided.

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or value theory

Philosophical theory of value. Axiology is the study of value, or goodness, in its widest sense. The distinction is commonly made between intrinsic and extrinsic value—i.e., between that which is valuable for its own sake and that which is valuable only as a means to something else, which itself may be extrinsically or intrinsically valuable. Many different answers have been given to the question “What is intrinsically valuable?” For hedonists, it is pleasure; for pragmatists, it is satisfaction, growth, or adjustment; for Kantians, it is a good will. Pluralists such as G.E. Moore and William David Ross assert that there are any number of intrinsically valuable things. According to subjective theories of value, things are valuable only insofar as they are desired; objective theories hold that there are at least some things that are valuable independently of people's interest in or desire for them. Cognitive theories of value assert that ascriptions of value function logically as statements of fact, whereas noncognitive theories assert that they are merely expressions of feeling (see emotivism) or prescriptions or commendations (see prescriptivism). According to naturalists, expressions such as “intrinsically good” can be analyzed as referring to natural, or non-ethical, properties, such as being pleasant. Moore famously denied this, holding that “good” refers to a simple (unanalyzable) non-natural property. Seealso fact-value distinction; naturalistic fallacy.

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Study of the origin, nature, and limits of human knowledge. Nearly every great philosopher has contributed to the epistemological literature. Some historically important issues in epistemology are: (1) whether knowledge of any kind is possible, and if so what kind; (2) whether some human knowledge is innate (i.e., present, in some sense, at birth) or whether instead all significant knowledge is acquired through experience (see empiricism; rationalism); (3) whether knowledge is inherently a mental state (see behaviourism); (4) whether certainty is a form of knowledge; and (5) whether the primary task of epistemology is to provide justifications for broad categories of knowledge claim or merely to describe what kinds of things are known and how that knowledge is acquired. Issues related to (1) arise in the consideration of skepticism, radical versions of which challenge the possibility of knowledge of matters of fact, knowledge of an external world, and knowledge of the existence and natures of other minds.

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Theory of circuits made up of ideal digital devices, including their structure, behaviour, and design. It incorporates Boolean logic (see Boolean algebra), a basic component of modern digital switching systems. Switching is essential to telephone, telegraph, data processing, and other technologies in which it is necessary to make rapid decisions about routing information. Seealso queuing theory.

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Any of a number of theories in particle physics that treat elementary particles (see subatomic particle) as infinitesimal one-dimensional “stringlike” objects rather than dimensionless points in space-time. Different vibrations of the strings correspond to different particles. Introduced in the early 1970s in attempts to describe the strong force, string theories became popular in the 1980s when it was shown that they might provide a fully self-consistent quantum field theory that could describe gravitation as well as the weak, strong, and electromagnetic forces. The development of a unified quantum field theory is a major goal in theoretical particle physics, but inclusion of gravity usually leads to difficult problems with infinite quantities in the calculations. The most self-consistent string theories propose 11 dimensions; 4 correspond to the 3 ordinary spatial dimensions and time, while the rest are curled up and not perceptible.

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Concept of an expanding universe whose average density remains constant, matter being continuously created throughout it to form new stars and galaxies at the same rate that old ones recede from sight. A steady-state universe has no beginning or end, and its average density and arrangement of galaxies are the same as seen from every point. Galaxies of all ages are intermingled. The theory was first put forward by William Macmillan (1861–1948) in the 1920s and modified by Fred Hoyle to deal with problems that had arisen in connection with the big-bang model. Much evidence obtained since the 1950s contradicts the steady-state theory and supports the big-bang model.

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In statistics and related subfields of philosophy, the theory and method of formulating and solving general decision problems. Such a problem is specified by a set of possible states of the environment or possible initial conditions; a set of available experiments and a set of possible outcomes for each experiment, giving information about the state of affairs preparatory to making a decision; a set of available acts depending on the experiments made and their consequences; and a set of possible consequences of the acts, in which each possible act assigns to each possible initial state some particular consequence. The problem is dealt with by assessing probabilities of consequences conditional on different choices of experiments and acts and by assigning a utility function to the set of consequences according to some scheme of value or preference of the decision maker. An optimal solution consists of an optimal decision function, which assigns to each possible experiment an optimal act that maximizes the utility, or value, and a choice of an optimal experiment. See also cost-benefit analysis, game theory.

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Branch of mathematics that deals with the properties of sets. It is most valuable as applied to other areas of mathematics, which borrow from and adapt its terminology and concepts. These include the operations of union (∪), and intersection (∩). The union of two sets is a set containing all the elements of both sets, each listed once. The intersection is the set of all elements common to both original sets. Set theory is useful in analyzing difficult concepts in mathematics and logic. It was placed on a firm theoretical footing by Georg Cantor, who discovered the value of clearly formulated sets in the analysis of problems in symbolic logic and number theory.

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Study of the behaviour of queues (waiting lines) and their elements. Queuing theory is a tool for studying several performance parameters of computer systems and is particularly useful in locating the reasons for “bottlenecks,” compromised computer performance caused by too much data waiting to be acted on at a particular phase. Queue size and waiting time can be looked at, or items within queues can be studied and manipulated according to factors such as priority, size, or time of arrival.

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Branch of mathematical physics that deals with atomic and subatomic systems. It is concerned with phenomena that are so small-scale that they cannot be described in classical terms, and it is formulated entirely in terms of statistical probabilities. Considered one of the great ideas of the 20th century, quantum mechanics was developed mainly by Niels Bohr, Erwin Schrödinger, Werner Heisenberg, and Max Born and led to a drastic reappraisal of the concept of objective reality. It explained the structure of atoms, atomic nuclei (see nucleus), and molecules; the behaviour of subatomic particles; the nature of chemical bonds (see bonding); the properties of crystalline solids (see crystal); nuclear energy; and the forces that stabilize collapsed stars. It also led directly to the development of the laser, the electron microscope, and the transistor.

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Theory that brings quantum mechanics and special relativity together to account for subatomic phenomena. In particular, the interactions of subatomic particles are described in terms of their interactions with fields, such as the electromagnetic field. However, the fields are quantized and represented by particles, such as photons for the electromagnetic field. Quantum electrodynamics is the quantum field theory that describes the interaction of electrically charged particles via electromagnetic fields. Quantum chromodynamics describes the action of the strong force. The electroweak theory, a unified theory of electromagnetic and weak forces, has considerable experimental support, and can likely be extended to include the strong force. Theories that include the gravitational force (see gravitation) are more speculative. Seealso grand unified theory, unified field theory.

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Quantum theory of the interactions of charged particles with the electromagnetic field. It describes the interactions of light with matter as well as those of charged particles with each other. Its foundations were laid by P. A. M. Dirac when he discovered an equation describing the motion and spin of electrons that incorporated both quantum mechanics and the theory of special relativity. The theory, as refined and developed in the late 1940s, rests on the idea that charged particles interact by emitting and absorbing photons. It has become a model for other quantum field theories.

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Experimental method of computing that makes use of quantum-mechanical phenomena. It incorporates quantum theory and the uncertainty principle. Quantum computers would allow a bit to store a value of 0 and 1 simultaneously. They could pursue multiple lines of inquiry simultaneously, with the final output dependent on the interference pattern generated by the various calculations. Seealso DNA computing, quantum mechanics.

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Theory that describes the action of the strong force. The strong force acts only on certain particles, principally quarks that are bound together in the protons and neutrons of the atomic nucleus, as well as in less stable, more exotic forms of matter. Quantum chromodynamics has been built on the concept that quarks interact via the strong force because they carry a form of “strong charge,” which has been given the name “colour.” The three types of charge are called red, green, and blue, in analogy to the primary colours of light, though there is no connection with the usual sense of colour.

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In physics, a discrete natural unit, or packet, of energy, charge, angular momentum, or other physical property. Light, for example, which appears in some respects as a continuous electromagnetic wave, on the submicroscopic level is emitted and absorbed in discrete amounts, or quanta; for light of a given wavelength, the magnitude of all the quanta emitted or absorbed is the same in both energy and momentum. These particlelike packets of light are called photons, a term also applicable to quanta of other forms of electromagnetic energy such as X rays and gamma rays. Submicroscopic mechanical vibrations in the layers of atoms comprising crystals also give up or take on energy and momentum in quanta called phonons. Seealso quantum mechanics.

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Economic theory relating changes in the price level to changes in the quantity of money. It has often been used to analyze the factors underlying inflation and deflation. The quantity theory was developed in the 17th and 18th centuries by philosophers such as John Locke and David Hume and was intended as a weapon against mercantilism. Drawing a distinction between money and wealth, advocates of the quantity theory argued that if the accumulation of money by a nation merely raised prices, the mercantilist emphasis on a favourable balance of trade would only increase the supply of money without increasing wealth. The theory contributed to the ascendancy of free trade over protectionism. In the 19th–20th centuries it played a part in the analysis of business cycles and in the theory of rates of foreign exchange.

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Branch of mathematics that deals with analysis of random events. Probability is the numerical assessment of likelihood on a scale from 0 (impossibility) to 1 (absolute certainty). Probability is usually expressed as the ratio between the number of ways an event can happen and the total number of things that can happen (e.g., there are 13 ways of picking a diamond from a deck of 52 cards, so the probability of picking a diamond is 1352, or 14). Probability theory grew out of attempts to understand card games and gambling. As science became more rigorous, analogies between certain biological, physical, and social phenomena and games of chance became more evident (e.g., the sexes of newborn infants follow sequences similar to those of coin tosses). As a result, probability became a fundamental tool of modern genetics and many other disciplines. Probability theory is also the basis of the insurance industry, in the form of actuarial statistics.

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Branch of mathematics concerned with properties of and relations among integers. It is a popular subject among amateur mathematicians and students because of the wealth of seemingly simple problems that can be posed. Answers are much harder to come up with. It has been said that any unsolved mathematical problem of any interest more than a century old belongs to number theory. One of the best examples, recently solved, is Fermat's last theorem.

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In mathematics, a generalization of the concepts of length and area (see length, area, and volume) to arbitrary sets of points not composed of line segments or rectangles. A measure is any rule for associating a number with a set. The result must be nonnegative and also additive, meaning that the measure of two nonoverlapping sets equals the sum of their individual measures. This is simple enough for sets consisting of line segments or rectangles, but the measure of sets such as curved regions or intervals with missing points requires more abstract methods, including limits and upper and lower bounds.

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Set of numbers arranged in rows and columns to form a rectangular array. Matrix elements may also be differential operators, vectors, or functions. Matrices have wide applications in engineering, physics, economics, and statistics, as well as in various branches of mathematics. They are usually first encountered in the study of systems of equations represented by matrix equations of the form math.Amath.x = math.B, which may be solved by finding the inverse of matrix math.A or by using an algebraic method based on its determinant.

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In economics, the theory that firms will pay a productive agent only what he or she adds to the financial earnings of the firm. Developed by writers such as John Bates Clark and Philip Henry Wicksteed at the end of the 19th century, marginal productivity theory holds that it is unprofitable to buy, for example, a man-hour of labour if it costs more than it contributes to its buyer's income. The amount in excess of costs that a productive input yields is the value of its marginal product; the theory posits that every type of input should be paid the value of its marginal product.

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Mathematical theory of closed curves in three-dimensional space. The number of times and the manner in which a curve crosses itself distinguish different knots. The fewest possible crossings is three, for the overhand (trefoil) knot, which occurs in two mirror versions according to the directions in which the curve crosses itself. Knot theory has been used to understand both atomic and molecular structures (protein folding).

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Theory based on a simple description of a gas, from which many properties of gases can be derived. Established primarily by James Clerk Maxwell and Ludwig Boltzmann, the theory is one of the most important concepts in modern science. The simplest kinetic model is based on the assumptions that (1) a gas is composed of a large number of identical molecules moving in random directions, separated by distances that are large compared to their size; (2) the molecules undergo perfectly elastic (no energy loss) collisions with each other and with the walls of the container; and (3) the transfer of kinetic energy between molecules is heat. This model describes a perfect gas but is a reasonable approximation to a real gas. Using the kinetic theory, scientists can relate the independent motion of molecules of gases to their pressure, volume, temperature, viscosity, and heat conductivity.

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Set of conditions under which a resort to war is morally legitimate (jus ad bellum); also, rules for the moral conduct of war (jus in bello). Among the proposed conditions for the just resort to war are that the cause be just (e.g., self-defense against an attack or the threat of imminent attack), that the authority undertaking the war be competent, that all peaceful alternatives be exhausted, and that there be a reasonable hope of success. Two of the most important conditions for the just conduct of war are that the force used be “proportional” to the just cause the war is supposed to serve (in the sense that the evil created by the war must not outweigh the good represented by the just cause) and that military personnel be discriminated from innocents (noncombatant civilians), who may not be killed. The concept of just war was developed in the early Christian church; it was discussed by St. Augustine in the 4th century and was still accepted by Hugo Grotius in the 17th century. Interest in the concept thereafter declined, though it was revived in the 20th century in connection with the development of nuclear weapons (the use of which, according to some, would violate the conditions of proportionality and discrimination) and the advent of “humanitarian intervention” to put an end to acts of genocide and other crimes committed within the borders of a single state.

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Field of mathematics that studies the problems of signal transmission, reception, and processing. It stems from Claude E. Shannon's mathematical methods for measuring the degree of order (nonrandomness) in a signal, which drew largely on probability theory and stochastic processes and led to techniques for determining a source's rate of information production, a channel's capacity to handle information, and the average amount of information in a given type of message. Crucial to the design of communications systems, these techniques have important applications in linguistics, psychology, and even literary theory.

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In the philosophy of mind, the doctrine that mental events are identical to physico-chemical events in the brain. So-called “type” identity theory asserts that each type of mental event, such as pain, is identical to some type of event in the brain, such as the firing of c-fibres. In response to objections based on the assumed “multiple realizability” of mental states, “token” identity theory makes the weaker claim that each token of a mental event, such as a particular pain, is identical to some token of a brain event of some type. Seealso mind-body problem.

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Mathematical theory of networks. A graph consists of vertices (also called points or nodes) and edges (lines) connecting certain pairs of vertices. An edge that connects a node to itself is called a loop. In 1735 Leonhard Euler published an analysis of an old puzzle concerning the possibility of crossing every one of seven bridges (no bridge twice) that span a forked river flowing past an island. Euler's proof that no such path exists and his generalization of the problem to all possible networks are now recognized as the origin of both graph theory and topology. Since the mid-20th century, graph theory has become a standard tool for analyzing and designing communications networks, power transmission systems, transportation networks, and computer architectures.

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Concept of the physical basis of heredity expressed by the biologist August Weismann (1834–1914). It claimed that germ plasm, which Weismann believed to be independent from all other cells of the body, was the essential element of germ cells (eggs and sperm) and was the hereditary material passed from generation to generation. First proposed in 1883, his view contradicted Lamarck's then-prevalent theory of acquired characteristics. Though its details have been altered, its idea of the stability of hereditary material is the basis of the modern understanding of physical inheritance.

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Theory that certain diseases are caused by invasion of the body by microorganisms. Louis Pasteur, Joseph Lister, and Robert Koch are given much of the credit for its acceptance in the later 19th century. Pasteur showed that organisms in the air cause fermentation and spoil food; Lister was first to use an antiseptic to exclude germs in the air to prevent infection; and Koch first linked a specific organism with a disease (anthrax). The full implications of germ theory for medical practice were not immediately apparent after it was proven; surgeons operated without masks or head coverings as late as the 1890s.

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Branch of applied mathematics devised to analyze certain situations in which there is an interplay between parties that may have similar, opposed, or mixed interests. Game theory was originally developed by John von Neumann and Oscar Morgenstern in their book The Theory of Games and Economic Behavior (1944). In a typical game, or competition with fixed rules, “players” try to outsmart one another by anticipating the others' decisions, or moves. A solution to a game prescribes the optimal strategy or strategies for each player and predicts the average, or expected, outcome. Until a highly contrived counterexample was devised in 1967, it was thought that every contest had at least one solution. Seealso decision theory; prisoner's dilemma.

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In mathematics, the study of the structure of a set of objects (e.g., numbers) with two combining operations (e.g., addition and multiplication). Such a system, known as a field, must satisfy certain properties: associative law, commutative law, distributive law, an additive identity (“zero”), a muliplicative identity (“one”), additive inverses (see inverse function), and multiplicative inverses for nonzero elements. The sets of rational numbers, real numbers, and complex numbers are fields under ordinary addition and multiplication. The investigation of polynomial equations and their solutions led to the development of field theory.

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Theory that describes both the electromagnetic force and the weak force. Though the forces appear to be different, they are actually different facets of a more fundamental force. This theory, formulated in the 1960s by Sheldon Glashow (born 1932), Steven Weinberg (born 1933), and Abdus Salam (born 1926), represents a 20th-century scientific landmark and won its authors a 1979 Nobel Prize. It was validated in the 1980s with the discovery of the W particle and Z particle, which it had predicted. Seealso fundamental interaction, unified field theory.

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or domino effect

Doctrine of U.S. foreign policy during the Cold War, according to which the fall of a noncommunist state to communism would precipitate the fall of other neighbouring noncommunist states. The theory was first enunciated by Pres. Harry Truman, who used it to justify sending U.S. military aid to Greece and Turkey in the late 1940s. Dwight D. Eisenhower, John F. Kennedy, and Lyndon B. Johnson invoked it to justify U.S. military involvement in Southeast Asia, especially the prosecution of the Vietnam War.

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Field of applied mathematics relevant to the control of certain physical processes and systems. It became a field in its own right in the late 1950s and early '60s. After World War II, problems arising in engineering and economics were recognized as variants of problems in differential equations and in the calculus of variations, though they were not covered by existing theories. Special modifications of classical techniques and theories were devised to solve individual problems, until it was recognized that these seemingly diverse problems all had the same mathematical structure, and control theory emerged. Seealso control system.

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Behaviour in a complex system that appears irregular or unpredictable but is actually determinate. The apparently random or unpredictable behaviour in systems governed by complicated (nonlinear) deterministic laws is the result of high sensitivity to initial conditions. For example, Edward Lorenz discovered that a simple model of heat convection exhibits chaotic behaviour. In a now-classic example of such sensitivity to initial conditions, he suggested that the mere flapping of a butterfly's wings could eventually result in large-scale changes in the weather (the “butterfly effect”).

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Branch of mathematics (considered a branch of geometry) that explores how gradual changes to a system produce sudden, drastic results (though usually not as dire as the name suggests). A simple example is how a plastic coffee stirrer subjected to gradually increasing pressure from both ends will suddenly buckle in one direction or another. Other “catastrophes” include optical phenomena such as reflection or refraction of light through moving water. More speculatively, ideas from catastrophe theory have been applied by social scientists to such situations as the sudden eruption of mob violence.

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In chemistry and physics, a theoretical model describing the states of electrons in solid materials, which can have energy values only within certain specific ranges, called bands. Ranges of energy between two allowed bands are called forbidden bands. As electrons in an atom move from one energy level to another, so can electrons in a solid move from an energy level in one band to another in the same band or in another band. The band theory accounts for many of the electrical and thermal properties of solids and forms the basis of the technology of devices such as semiconductors, heating elements, and capacitors (see capacitance).

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Body of physical and logical principles underlying the operation of any electromechanical device (an automaton) that converts information input in one form into another, or into some action, according to an algorithm. Norbert Wiener and Alan M. Turing are regarded as pioneers in the field. In computer science, automata theory is concerned with the construction of robots (see robotics) from basic building blocks of automatons. The best example of a general automaton is an electronic digital computer. Networks of automata may be designed to mimic human behaviour. Seealso artificial intelligence; Turing machine.

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Theory that holds that a film's director is its “author” (French, auteur). It originated in France in the 1950s and was promoted by Francsubcommaois Truffaut and Jean-Luc Godard and the journal Cahiers du Cinéma. The director oversees and “writes” the film's audio and visual scenario and therefore is considered more responsible for its content than the screenwriter. Supporters maintain that the most successful films bear the distinctive imprint of their director.

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Comprehensive theory that explains the behaviour of superconducting materials. It was developed in 1957 by John Bardeen, Leon Cooper, and J. Robert Schrieffer (b. 1931), whose surname initials provide its name. Cooper discovered that electrons in a superconductor are grouped in pairs (Cooper pairs) and that the motions of all the pairs within a single superconductor constitute a system that functions as a single entity. An electric voltage applied to the superconductor causes all Cooper pairs to move, forming an electric current. When the voltage is removed, the current continues to flow because the pairs encounter no opposition. Seealso superconductivity.

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