1000 Sentences With "theorems" | Random Sentence Generator

Why do you think mathematicians keep devising new proofs for certain theorems, when they already know the theorems are true?

She extols the virtues of both complicated theorems and common sense.

Mathematicians use these rules all the time to prove new theorems.

We prove theorems because there is an audience to communicate it to.

These theorems allow researchers to study how systems evolve—even far from equilibrium.

You make theorems about the world that you live in with those rules.

For some theorems, there are different perfect proofs for different types of readers.

They've also proved that theorems proved in one setting work in the others.

An entire program can be tested with the same certainty that mathematicians prove theorems.

Qapital, says Ariely, applies theorems from behavioral science to the world of financial management.

This turned into a whole industry of using topological tools to prove discrete mathematics theorems.

Years later, their so-called lemmas may be pillars of "mathematical canon," while their theorems lie forgotten.

Scholze was bemused by the long theorems with their short proofs, which struck him as valid but insubstantial.

"I'm impressed with the statistical theorems that say that minimal surfaces are all over the place," she said.

International macro people considered trade people boring, obsessed with proving theorems and offering little of real-world use.

American high school students can recite Shakespeare's sonnets, derive advanced calculus theorems, and explain the Chinese spheres of influence.

The Penrose-Hawking "singularity theorems" meant there was no way for space-time to begin smoothly, undramatically at a point.

When he sent a book of theorems to Cambridge, G. H. Hardy wrote back and began a five-year collaboration.

Those theorems don't go through if people all tend to make the same kind of mistakes at a given time.

It's one of those things that are ridiculously difficult to explain unless you're really up on your advanced mathematical theorems.

Professor Shafarevich's work is known throughout the mathematical world, his name enshrined in the Shafarevich-Weil and Golod-Shafarevich theorems.

At five, you incited jealousy in our lab technicians with your facile knowledge of Godel's incompleteness theorems, but you were leery.

Success came relatively late in her career, in her 50s, when she produced her most famous results, known as Ratner's Theorems.

In order to translate infinity categories into objects that could do real mathematical work, Lurie had to prove theorems about them.

Mathematicians have shown that some theorems have proofs so long that it would take the lifetime of the universe to finish them.

There are theorems that have several genuinely different proofs, and each proof tells you something different about the theorem and the structures.

The abc conjecture (in certain forms) would offer new proofs of these two theorems and solve a host of related open problems.

Eskin worked with famous Iranian mathematician Maryam Mirzakhni, before her death in 2017, via Skype to develop theorems of dynamical, moduli spaces.

Despite a head full of relevant theorems and experiments, when he finds himself lakeside surrounded by flat stones, he takes a beginner's attitude.

Fundamental theorems appear in ancient Egyptian work from 1820 BC, and later influences sprout from Babylonian, Ancient Greek, Chinese and Middle Eastern texts.

Perhaps intelligence of many kinds, even the sort that solved theorems and played chess, emerged from the most basic skills—perception, motor control.

Dr. Ratner's theorems are some of the most important in the past half-century, but she never quite received the recognition she deserved.

Reciprocity laws are generalizations of the 200-year-old quadratic reciprocity law, a cornerstone of number theory and one of Scholze's personal favorite theorems.

Cathleen S. Morawetz, a mathematician whose theorems often found use in solving real-world engineering problems, died on Tuesday at her home in Manhattan.

However, even with the fluctuation theorems in hand, the conditions on early Earth or inside a cell are far too complex to predict from first principles.

To demonstrate the technology, the researchers encoded in DNA "The Methods of Mechanical Theorems" written by ancient Greek scientist Archimedes at least two thousand years ago.

To put it very simply, what Noether's theorems show is that anytime there's a continuous symmetry in a physical system, there's a related law of conservation.

This link is a core part of the "Langlands program," a collection of interconnected conjectures and theorems about the relationship between number theory, geometry and analysis.

So we have the proliferation of funds with names intending to suggest ritzy addresses or locales, abstruse theorems and admirable but all-too-rare personal characteristics.

The book, which has been called "a glimpse of mathematical heaven," presents proofs of dozens of theorems from number theory, geometry, analysis, combinatorics and graph theory.

And from Axiom IV and our previous theorems, we can see that the universal energies stirred up by Cupid's difficult interventions belong in the lyrics of choruses.

This first impulse is loosely based on the "fundamental welfare theorems", which say that whatever the initial distribution of wealth, trade will lead to an efficient outcome.

We have these results from logic that say that there are theorems that are true and that have a proof, but they don't have a short proof.

Back on the firmer ground of classical relativity, he proved a number of fundamental theorems about the behavior of black holes and the expansion of the universe.

WHEN he was doing something—simulating on paper how a computer might solve one of Euclid's theorems, say—Marvin Minsky often found himself improvising a nice little tune.

Devlin, a mathematician who has proved a few "largely unremarkable theorems" while distinguishing himself as a popular expositor of mathematics, has more than a little sympathy for Fibonacci.

He, Dr. Penrose and a rotating cast of colleagues published a series of theorems about the behavior of black holes and the dire fate of anything caught in them.

Hardy believed that the only important questions in the field arose internally from this game, that the sole purpose of a mathematician was to create beautiful and "almost wholly useless" theorems.

Let's cherish Hardy's theorems, not his opinions, and recognize mathematics as a field with diverse goals and needs, where people can expect to make useful contributions regardless of gender or age.

Their work, which builds on previous work by Barwick and others, has proved that many of the theorems in Higher Topos Theory hold regardless of which model you apply them in.

" To that point, the palimpsest turned out to contain ideas by Archimedes that had not survived in any other documents, including an entirely new treatise called "The Method of Mechanical Theorems.

In the early days of A.I., intelligence had for the most part been talked about as the ability to do things that A.I. researchers found hard, like proving theorems and playing chess.

"Finding a prime is not going to change any theorems in mathematics, but this is a type of prime that has been interesting to mathematicians since several centuries before Christ," Caldwell said.

The situation changed in the late 1990s, when the physicists Gavin Crooks and Chris Jarzynski derived "fluctuation theorems" that can be used to quantify how much more often certain physical processes happen than reverse processes.

"What you get is these nice theorems that basically show that it's a mathematical fact that the amount by which you do better is gonna be how much more diverse you are," he tells me.

The brilliant codesmetologist thought hard with his brain-box, using Sir Isaac Newton's impeccable physics theorems to unravel the secret nature of Ludwig van Beethoven's most famous sonata in order to save the beautiful woman.

Professor Arrow's theorems set out the precise conditions under which Adam Smith's famous conjecture in "The Wealth of Nations" holds true: that the "invisible hand" of market competition among self-serving individuals serves society well.

The drama follows the journey of the Indian accounting clerk, who was brought to England before World War One by Cambridge University professor Godfrey Harold Hardy (Jeremy Irons), as he seeks to prove himself and his theorems.

When he discussed the papers with her, he told her that he had the feeling that she had written the papers not for other mathematicians to understand but mainly to convince herself that the theorems were correct.

They're especially poorly suited to serving as reference manuals—it's difficult to look up specific theorems, or to check that a specific application of infinity categories that one might encounter in someone else's paper really works out.

There, among walkways and arcades especially designed for thinking, Euclid came to formulate his theorems; Eratosthenes to measure the circumference of the Earth; and Herophilos to prove that the brain, not the heart, was the seat of the intellect.

In Higher Algebra, the latest version of which runs to 1,553 pages, Lurie developed a version of the associative property for infinity categories—along with many other algebraic theorems that collectively established a foundation for the mathematics of equivalence.

Hiding out at home on prize day, she missed a talk by Xin Zhou, of the University of California, Santa Barbara, who began with a prefatory reference to one of Dr. Uhlenbeck's 40-year-old theorems at the top of the chalkboard.

What Ottokar leaves behind, a collection of notes and theorems, calculations toward a major breakthrough in the understanding of time, becomes the obsession of his descendants through the next three generations, haunting them and marking them as standouts or outcasts from the flow of history.

UMi, ostensibly attempting to show off the durability of the London, dropped one off a building, graciously donating one of their phones to help to prove the aforementioned theorems as best as possible and ensure our peace of mind in the existence of physics.

If you throw that into a black hole, the standard story is that the black hole can't keep the magnetic field, because that would violate the black hole "no hair" theorems—the magnetic field would be "hair," so the black hole has to shake it off.

So when someone like Ptolemy did astronomy, they did it a bit like Euclid—effectively taking things like the saros cycle as axioms, and then proving from them often surprisingly elaborate geometrical theorems, such as that there must be at least two solar eclipses in a given year.

The idea of superintelligence is such a poorly defined notion that one could envision it taking almost any form with equal justification: a benevolent genie that solves all the world's problems, or a mathematician that spends all its time proving theorems so abstract that humans can't even understand them.

The goal was either to get credit for the class, which would let him skip ahead to higher-level courses earlier in his high school career, particularly Advanced Placement ones, or to take the course again in the fall and, already familiar with the underlying theorems, be all but guaranteed a top grade.

Lance had always known him to be a wide-eyed scientific optimist, the sort of man who, as far back as the late 1950s, dreamed of building a computer endowed with all the capabilities of a human—a machine that could prove complex mathematical theorems, engage in conversation, and play a decent game of Ping-Pong.

States have upgraded their high school graduation standards to meet the demands of the 21st-century workplace, and colleges have revamped curricula to emphasize applying knowledge to real-world challenges, you need to understand not just the what but the why and when and how to relate those theorems or principles to very different facts or situations.

"Those who knew Hawking would clearly appreciate the dominating presence of a real human being, with an enormous zest for life, great humor, and tremendous determination, yet with normal human weaknesses, as well as his more obvious strengths," writes Roger Penrose, who worked with Hawking on his first major breakthrough; theorems explaining how the Universe might have begun with a singularity.

In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and various other algebraic structures. In universal algebra, the isomorphism theorems can be generalized to the context of algebras and congruences.

The Dubins–Spanier theorems are several theorems in the theory of fair cake- cutting. They were published by Lester Dubins and Edwin Spanier in 1961. Although the original motivation for these theorems is fair division, they are in fact general theorems in measure theory.

One particularly well- known class of closed graph theorems are the closed graph theorems in functional analysis.

In 1967 at University of Wisconsin—Madison, working in the Mathematics Research Center, he produced a technical report New root-location theorems for partitioned matrices.J. L. Brenner (1967) New root-location theorems for partitioned matrices, citation from Defense Technical Information Center In 1968 Brenner, following Alston Householder, published "Gersgorin theorems by Householder’s proof".Brenner (1968) Gersgorin theorems by Householder’s proof, Bulletin of the American Mathematical Society 74:3 , link from Project Euclid In 1970 he published the survey article (21 references) "Gersgorin theorems, regularity theorems, and bounds for determinants of partitioned matrices".Brenner (1970) "Gersgorin theorems, regularity theorems, and bounds for determinants of partitioned matrices", SIAM Journal for Applied Mathematics 19(2) The article was extended with "Some determinantal identities".

In computability theory, there are a number of basis theorems. These theorems show that particular kinds of sets always must have some members that are, in terms of Turing degree, not too complicated. One family of basis theorems concern nonempty effectively closed sets (that is, nonempty \Pi^0_1 sets in the arithmetical hierarchy); these theorems are studied as part of classical computability theory. Another family of basis theorems concern nonempty lightface analytic sets (that is, \Sigma^1_1 in the analytical hierarchy); these theorems are studied as part of hyperarithmetical theory.

The welfare theorems establish a connection between the competitive equilibrium and the social planning problem. The welfare theorems also have practical implications.

400px In geometry, Clifford's theorems, named after the English geometer William Kingdon Clifford, are a sequence of theorems relating to intersections of circles.

Moreover, careful readers have noted a number of nontrivial oversights throughout the text, including missing hypotheses in theorems, inaccurately stated theorems, and proofs that fail to handle all cases.

Gudykunst uses 47 axioms as building blocks for the theorems of AUM. Axioms can be thought of as the lowest common denominators from which all causal theorems are derived.

In mathematics, Ratner's theorems are a group of major theorems in ergodic theory concerning unipotent flows on homogeneous spaces proved by Marina Ratner around 1990. The theorems grew out of Ratner's earlier work on horocycle flows. The study of the dynamics of unipotent flows played a decisive role in the proof of the Oppenheim conjecture by Grigory Margulis. Ratner's theorems have guided key advances in the understanding of the dynamics of unipotent flows.

In economics, the Debreu theorems are several statements about the representation of a preference ordering by a real-valued function. The theorems were proved by Gerard Debreu during the 1950s.

By convention, the value of 0! is defined as 1. This classical factorial function appears prominently in many theorems in number theory. The following are a few of these theorems.

Among other abilities, it can retrieve all MML theorems proved about any particular type or operator.An example of an MML query, yielding all theorems proved on the exponent operator, by the number of times they are cited in subsequent theorems.Another example of an MML query, yielding all theorems proved on sigma fields.

Helmholtz's theorems have application in understanding: :Generation of lift on an airfoil :Starting vortex :Horseshoe vortex :Wingtip vortices. Helmholtz's theorems are now generally proven with reference to Kelvin's circulation theorem. However the Helmholtz's theorems were published in 1858, nine years before the 1867 publication of Kelvin's theorem. There was much communication between the two men on the subject of vortex lines, with many references to the application of their theorems to the study of smoke rings.

In algebra, the first and second fundamental theorems of invariant theory concern the generators and the relations of the ring of invariants in the ring of polynomial functions for classical groups (roughly the first concerns the generators and the second the relations). The theorems are among the most important results of invariant theory. Classically the theorems are proved over the complex numbers. But characteristic-free invariant theory extends the theorems to a field of arbitrary characteristic.

In mathematics, counterexamples are often used to prove the boundaries of possible theorems. By using counterexamples to show that certain conjectures are false, mathematical researchers can then avoid going down blind alleys and learn to modify conjectures to produce provable theorems. It is sometimes said that mathematical development consists primarily in finding (and proving) theorems and counterexamples.

The space hierarchy theorems are separation results that show that both deterministic and nondeterministic machines can solve more problems in (asymptotically) more space, subject to certain conditions. For example, a deterministic Turing machine can solve more decision problems in space n log n than in space n. The somewhat weaker analogous theorems for time are the time hierarchy theorems.

In combinatorics, Hall-type theorems for hypergraphs are several generalization of Hall's marriage theorem from graphs to hypergraphs. Such theorems were proved by Ofra Kessler, Ron Aharoni,, Penny Haxell, Roy Meshulam, and others.

In mathematics, Malmquist's theorem, is the name of any of the three theorems proved by . These theorems restrict the forms of first order algebraic differential equations which have transcendental meromorphic or algebroid solutions.

The following theorems form the foundation of variable structure control.

Theorems in the system are terms of a special "theorem" abstract data type. The general mechanism of abstract data types of ML ensures that theorems are derived using only the inference rules given by the operations of the theorem abstract type. Users can write arbitrarily complex ML programs to compute theorems; the validity of theorems does not depend on the complexity of such programs, but follows from the soundness of the abstract data type implementation and the correctness of the ML compiler.

The incompleteness theorems are among a relatively small number of nontrivial theorems that have been transformed into formalized theorems that can be completely verified by proof assistant software. Gödel's original proofs of the incompleteness theorems, like most mathematical proofs, were written in natural language intended for human readers. Computer-verified proofs of versions of the first incompleteness theorem were announced by Natarajan Shankar in 1986 using Nqthm (Shankar 1994), by Russell O'Connor in 2003 using Coq (O'Connor 2005) and by John Harrison in 2009 using HOL Light (Harrison 2009). A computer-verified proof of both incompleteness theorems was announced by Lawrence Paulson in 2013 using Isabelle (Paulson 2014).

Similar theorems are valid for monoids, vector spaces, modules, and rings.

Deletes the theorem stored at index m in the current proof. This helps to mitigate storage constraints caused by redundant and unnecessary theorems. Deleted theorems can no longer be referenced by the above apply-rule function.

This result known as the Gershgorin circle theorem has been used as a basis for extension. In 1964 Brenner reported on Theorems of Gersgorin Type.Brenner (January 1964) Theorems of Gersgorin type, citation from Defense Technical Information Center.

All theorems of ZFC are also theorems of von Neumann–Bernays–Gödel set theory, but the latter can be finitely axiomatized. The set theory New Foundations can be finitely axiomatized, but only with some loss of elegance.

The log sum inequality is used for proving theorems in information theory.

The theorems in this section simultaneously imply Euclid's theorem and other results.

The base change theorems discussed below are statements of a similar kind.

Classical propositional calculus as described above is equivalent to Boolean algebra, while intuitionistic propositional calculus is equivalent to Heyting algebra. The equivalence is shown by translation in each direction of the theorems of the respective systems. Theorems \phi of classical or intuitionistic propositional calculus are translated as equations \phi = 1 of Boolean or Heyting algebra respectively. Conversely theorems x = y of Boolean or Heyting algebra are translated as theorems (x \to y) \land (y \to x) of classical or intuitionistic calculus respectively, for which x \equiv y is a standard abbreviation.

As Robert Gilmore wrote: :Lie's three theorems provide a mechanism for constructing the Lie algebra associated with any Lie group. They also characterize the properties of a Lie algebra. ¶ The converses of Lie’s three theorems do the opposite: they supply a mechanism for associating a Lie group with any finite dimensional Lie algebra ... Taylor's theorem allows for the construction of a canonical analytic structure function φ(β,α) from the Lie algebra. ¶ These seven theorems – the three theorems of Lie and their converses, and Taylor's theorem – provide an essential equivalence between Lie groups and algebras.

Karamata published 122 scientific papers, 15 monographs and text-books as well as 7 professional-pedagogical papers. Karamata is best known for his work on mathematical analysis. He introduced the notion of regularly varying function, and discovered a new class of theorems of Tauberian type, today known as Karamata's tauberian theorems. He also worked on Mercer's theorems, Frullani integral, and other topics in analysis.

In a model, all the theorems of the system are automatically true statements.

Comparison Theorems in Riemannian Geometry. North-Holland Publishing Company, 1975, pp. 17-18.

There are other theorems that can be deduced simply from the above argument.

The Hurewicz theorems are a key link between homotopy groups and homology groups.

Specifically, he proved equivariant analogs of fundamental theorems such as the localization theorem.

Converses to a theorem like Abel's are called Tauberian theorems: There is no exact converse, but results conditional on some hypothesis. The field of divergent series, and their summation methods, contains many theorems of abelian type and of tauberian type.

There are also "theorems" in science, particularly physics, and in engineering, but they often have statements and proofs in which physical assumptions and intuition play an important role; the physical axioms on which such "theorems" are based are themselves falsifiable.

In mathematics, the Krylov–Bogolyubov theorem (also known as the existence of invariant measures theorem) may refer to either of the two related fundamental theorems within the theory of dynamical systems. The theorems guarantee the existence of invariant measures for certain "nice" maps defined on "nice" spaces and were named after Russian-Ukrainian mathematicians and theoretical physicists Nikolay Krylov and Nikolay Bogolyubov who proved the theorems. Zbl. 16.86.

The connectivity theorems are specific relationships between elasticities and control coefficients. They are useful because they highlight the close relationship between the kinetic properties of individual reactions and the system properties of a pathway. Two basic sets of theorems exists, one for flux and another for concentrations. The concentration connectivity theorems are divided again depending on whether the system species S_n is different from the local species S_m .

The theory of Hilbert algebras can be used to deduce the commutation theorems of Murray and von Neumann; equally well the main results on Hilbert algebras can also be deduced directly from the commutation theorems for traces. The theory of Hilbert algebras was generalised by Takesaki as a tool for proving commutation theorems for semifinite weights in Tomita–Takesaki theory; they can be dispensed with when dealing with states.

In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific law, which is experimental.However, both theorems and scientific law are the result of investigations. See Introduction, The terminology of Archimedes, p. clxxxii:"theorem (θεὼρνμα) from θεωρεἳν to investigate" Many mathematical theorems are conditional statements, whose proof deduces the conclusion from conditions known as hypotheses or premises.

David Hilbert instigated a formalist movement that was eventually tempered by Gödel's incompleteness theorems.

Metamath is a formal language and an associated computer program (a proof checker) for archiving, verifying, and studying mathematical proofs. Several databases of proved theorems have been developed using Metamath covering standard results in logic, set theory, number theory, algebra, topology and analysis, among others. As of July 2020, the set of proved theorems using Metamath is one of the largest bodies of formalized mathematics, containing in particular proofs of 74Metamath 100. of the 100 theorems of the "Formalizing 100 Theorems" challenge, making it third after HOL Light and Isabelle, but before Coq, Mizar, ProofPower, Lean, Nqthm, ACL2, and Nuprl.

It is also similar to some of the theorems outlined in uncertainty reduction theory, from the post-positivist discipline of communication studies. These theorems include constructs of nonverbal expression, perceived similarity, liking, information seeking, and intimacy, and their correlations to one another.

Theorems and Theories , Sam Nelson.Mark C. Chu- Carroll, March 13, 2007:Theorems, Lemmas, and Corollaries. Good Math, Bad Math blog. A physical theory similarly differs from a mathematical theory, in the sense that the word "theory" has a different meaning in mathematical terms.

An elementary introduction to the black hole uniqueness theorems can be found in and in .

One of the basic structure theorems about Milnor fibers is they are parallelizable manifoldspg 75.

In mathematical folklore, the "no free lunch" (NFL) theorem (sometimes pluralized) of David Wolpert and William Macready appears in the 1997 "No Free Lunch Theorems for Optimization".Wolpert, D.H., Macready, W.G. (1997), "No Free Lunch Theorems for Optimization", IEEE Transactions on Evolutionary Computation 1, 67. Wolpert had previously derived no free lunch theorems for machine learning (statistical inference).Wolpert, David (1996), "The Lack of A Priori Distinctions between Learning Algorithms", Neural Computation, pp. 1341–1390.

The incompleteness theorems apply only to formal systems which are able to prove a sufficient collection of facts about the natural numbers. One sufficient collection is the set of theorems of Robinson arithmetic Q. Some systems, such as Peano arithmetic, can directly express statements about natural numbers. Others, such as ZFC set theory, are able to interpret statements about natural numbers into their language. Either of these options is appropriate for the incompleteness theorems.

The time hierarchy theorems are important statements about time-bounded computation on Turing machines. Informally, these theorems say that given more time, a Turing machine can solve more problems. For example, there are problems that can be solved with n2 time but not n time.

Because computer algorithms and programs had been used as early as 1956 to test and validate mathematical theorems, such as the four color theorem, some scholars anticipated that similar computational approaches could "solve" and "prove" analogously formalized problems and theorems of social structures and dynamics.

The role of critical exponents in blowup theorems. SIAM Rev. 32 (1990), no. 2, 262–288.

The Krein–Milman theorem is arguably one of the most well-known theorems about extreme points.

In mathematics, Bôcher's theorem is either of two theorems named after the American mathematician Maxime Bôcher.

This distinguishes Brouwer's result from other fixed-point theorems, such as Stefan Banach's, that guarantee uniqueness.

By considering which theorems of complex analysis are special cases of theorems of potential theory in any dimension, one can obtain a feel for exactly what is special about complex analysis in two dimensions and what is simply the two-dimensional instance of more general results.

Her most important works is on the limit theorems of probability theory, proven integral theorems of local and integral boundary theorems for Markov chains, large deviations of the sum of the sum of random sums, and the sum of random vector amounts. She has determined the random distribution of local time limit distributions and convergence rate estimates. She was the chief researcher at the institute of mathematics and informatics at the Lithuanian Academy of Sciences from 1989 to 2010.

The initial applications were to analogues of the Lefschetz hyperplane theorems. In general such theorems state that homology or cohomology is supported on a hyperplane section of an algebraic variety, except for some 'loss' that can be controlled. These results applied to the algebraic fundamental group and to the Picard group. Another type of application are connectedness theorems such as Grothendieck's connectedness theorem (a local analogue of the Bertini theorem) or the Fulton–Hansen connectedness theorem due to and .

It has been estimated that over a quarter of a million theorems are proved every year.Hoffman 1998, p. 204. The well-known aphorism, "A mathematician is a device for turning coffee into theorems", is probably due to Alfréd Rényi, although it is often attributed to Rényi's colleague Paul Erdős (and Rényi may have been thinking of Erdős), who was famous for the many theorems he produced, the number of his collaborations, and his coffee drinking.Hoffman 1998, p. 7.

The paper is also known for introducing new techniques that Gödel invented to prove the incompleteness theorems.

Her dissertation, supervised by Richard M. Dudley, was Central Limit Theorems for D[0,1]-Valued Random Variables.

As a result, Cohen–Macaulay rings are named after him and Francis Sowerby Macaulay. Cohen and Abraham Seidenberg published their Cohen–Seidenberg theorems, also known as the going-up and going-down theorems. He also coauthored articles with Irving Kaplansky. One of his doctoral students was R. Duncan Luce.

In mathematics, specifically in functional analysis and Hilbert space theory, vector-valued Hahn–Banach theorems are generalizations of the Hahn–Banach theorems from linear functionals (which are always valued in the real numbers ℝ or the complex numbers ℂ) to linear operators valued in topological vector spaces (TVSs).

In mathematical logic, the Friedman translation is a certain transformation of intuitionistic formulas. Among other things it can be used to show that the Π02-theorems of various first-order theories of classical mathematics are also theorems of intuitionistic mathematics. It is named after its discoverer, Harvey Friedman.

He then shows that indirect self-reference is crucial in many of the proofs of Gödel's incompleteness theorems.

The three isomorphism theorems, called homomorphism theorem, and two laws of isomorphism when applied to groups, appear explicitly.

The stars and bars method is often introduced specifically to prove the following two theorems of elementary combinatorics.

Some commentators, such as Rebecca Goldstein, have hypothesized that Gödel developed his logical theorems in opposition to Wittgenstein.

Results that give sufficient conditions for boundedness are known as multiplier theorems. Three such results are given below.

In mathematics, Abelian and Tauberian theorems are theorems giving conditions for two methods of summing divergent series to give the same result, named after Niels Henrik Abel and Alfred Tauber. The original examples are Abel's theorem showing that if a series converges to some limit then its Abel sum is the same limit, and Tauber's theorem showing that if the Abel sum of a series exists and the coefficients are sufficiently small (o(1/n)) then the series converges to the Abel sum. More general Abelian and Tauberian theorems give similar results for more general summation methods. There is not yet a clear distinction between Abelian and Tauberian theorems, and no generally accepted definition of what these terms mean.

This issue is discussed in various prime ideal theorems, which are necessary for many applications that require prime ideals.

Crowley, Aleister. Magick: Liber > ABA, Book 4. Part III (Magick in Theory and Practice). Definition and > Theorems of Magick.

Indeed, according to , the "whole business" of establishing the fundamental theorems of Fourier analysis reduces to the Gaussian integral.

The density of the sum of two or more independent variables is the convolution of their densities (if these densities exist). Thus the central limit theorem can be interpreted as a statement about the properties of density functions under convolution: the convolution of a number of density functions tends to the normal density as the number of density functions increases without bound. These theorems require stronger hypotheses than the forms of the central limit theorem given above. Theorems of this type are often called local limit theorems.

Approximate max-flow min-cut theorems are mathematical propositions in network flow theory. They deal with the relationship between maximum flow rate ("max- flow") and minimum cut ("min-cut") in a multi-commodity flow problem. The theorems have enabled the development of approximation algorithms for use in graph partition and related problems.

Liouville's theorem: several theorems - thus named from number theory, analysis, mechanics and so on. See disambiguation page for full information.

In this context Bost obtains an arithmetic Hodge index theorem and uses this to obtain Lefschetz theorems for arithmetic surfaces.

Recent works include theorems pointing to a new field of nonassociative geometry, noncommutative gravity and (2+1)-dimensional quantum gravity.

The least-upper-bound property of can be used to prove many of the main foundational theorems in real analysis.

Then these six new circles C all pass through a single point. The sequence of theorems can be continued indefinitely.

Many of the preceding hardness results can be explained through meta-theorems about extending preferences over single players to coalitions.

This proof uses only Lebesgue's monotone and dominated convergence theorems. We prove the statement as given above in three steps.

Mathematicians refer to this precision of language and logic as "rigor". Mathematical proof is fundamentally a matter of rigor. Mathematicians want their theorems to follow from axioms by means of systematic reasoning. This is to avoid mistaken "theorems", based on fallible intuitions, of which many instances have occurred in the history of the subject.

Tellegen's theorem is one of the most powerful theorems in network theory. Most of the energy distribution theorems and extremum principles in network theory can be derived from it. It was published in 1952 by Bernard Tellegen. Fundamentally, Tellegen's theorem gives a simple relation between magnitudes that satisfy Kirchhoff's laws of electrical circuit theory.

In a popular book on mathematics, he categorized theorems as beautiful theorems or ugly theorems. He is also known in Japan for speaking out against government reforms in secondary education. He wrote The Dignity of the Nation, which according to Time Asia was the second best selling book in the first six months of 2006 in Japan. In 2006, Fujiwara published Yo ni mo utsukushii sugaku nyumon ("An Introduction to the World's Most Elegant Mathematics") with the writer Yōko Ogawa: it is a dialogue between novelist and mathematician on the extraordinary beauty of numbers.

In fluid mechanics, Helmholtz's theorems, named after Hermann von Helmholtz, describe the three-dimensional motion of fluid in the vicinity of vortex filaments. These theorems apply to inviscid flows and flows where the influence of viscous forces are small and can be ignored. Helmholtz's three theorems are as follows:Kuethe and Schetzer, Foundations of Aerodynamics, Section 2.14 ;Helmholtz's first theorem: :The strength of a vortex filament is constant along its length. ;Helmholtz's second theorem: :A vortex filament cannot end in a fluid; it must extend to the boundaries of the fluid or form a closed path.

Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. Its defining method can briefly be described as "going backwards from the theorems to the axioms", in contrast to the ordinary mathematical practice of deriving theorems from axioms. It can be conceptualized as sculpting out necessary conditions from sufficient ones. The reverse mathematics program was foreshadowed by results in set theory such as the classical theorem that the axiom of choice and Zorn's lemma are equivalent over ZF set theory.

The definition of a functional derivative may be made more mathematically precise and rigorous by defining the space of functions more carefully. For example, when the space of functions is a Banach space, the functional derivative becomes known as the Fréchet derivative, while one uses the Gateaux derivative on more general locally convex spaces. Note that Hilbert spaces are special cases of Banach spaces. The more rigorous treatment allows many theorems from ordinary calculus and analysis to be generalized to corresponding theorems in functional analysis, as well as numerous new theorems to be stated.

In game theory, folk theorems are a class of theorems describing an abundance of Nash equilibrium payoff profiles in repeated games .In mathematics, the term folk theorem refers generally to any theorem that is believed and discussed, but has not been published. Roger Myerson has recommended the more descriptive term "general feasibility theorem" for the game theory theorems discussed here. See Myerson, Roger B. Game Theory, Analysis of conflict, Cambridge, Harvard University Press (1991) The original Folk Theorem concerned the payoffs of all the Nash equilibria of an infinitely repeated game.

In computability theory, Kleene's recursion theorems are a pair of fundamental results about the application of computable functions to their own descriptions. The theorems were first proved by Stephen Kleene in 1938 and appear in his 1952 book Introduction to Metamathematics. A related theorem which constructs fixed points of a computable function is known as Rogers's theorem and is due to Hartley Rogers, Jr. . The recursion theorems can be applied to construct fixed points of certain operations on computable functions, to generate quines, and to construct functions defined via recursive definitions.

A formal system is said to be effectively axiomatized (also called effectively generated) if its set of theorems is a recursively enumerable set (Franzén 2005, p. 112). This means that there is a computer program that, in principle, could enumerate all the theorems of the system without listing any statements that are not theorems. Examples of effectively generated theories include Peano arithmetic and Zermelo–Fraenkel set theory (ZFC). The theory known as true arithmetic consists of all true statements about the standard integers in the language of Peano arithmetic.

The theorems of Ibn al-Haytham, Khayyam and al-Tusi on quadrilaterals, including the Lambert quadrilateral and Saccheri quadrilateral, were "the first few theorems of the hyperbolic and the elliptic geometries". These theorems along with their alternative postulates, such as Playfair's axiom, played an important role in the later development of non- Euclidean geometry. These early attempts at challenging the fifth postulate had a considerable influence on its development among later European geometers, including Witelo, Levi ben Gerson, Alfonso, John Wallis and Saccheri.Boris A. Rosenfeld & Adolf P. Youschkevitch, "Geometry", p.

An empirical statistical law or (in popular terminology) a law of statistics represents a type of behaviour that has been found across a number of datasets and, indeed, across a range of types of data sets.Kitcher & Salmon (2009) p.51 Many of these observances have been formulated and proved as statistical or probabilistic theorems and the term "law" has been carried over to these theorems. There are other statistical and probabilistic theorems that also have "law" as a part of their names that have not obviously derived from empirical observations.

229 It was realised that the theorems that do apply to projective geometry are simpler statements. For example, the different conic sections are all equivalent in (complex) projective geometry, and some theorems about circles can be considered as special cases of these general theorems. During the early 19th century the work of Jean-Victor Poncelet, Lazare Carnot and others established projective geometry as an independent field of mathematics . Its rigorous foundations were addressed by Karl von Staudt and perfected by Italians Giuseppe Peano, Mario Pieri, Alessandro Padoa and Gino Fano during the late 19th century.

Second-order logic, however, fails to retain many desirable properties of first-order logic, such as the completeness and compactness theorems.

Many important theorems in ring theory (especially the theory of commutative rings) rely on the assumptions that the rings are Noetherian.

Within this context, many more techniques from calculus hold. In particular, there are versions of the inverse and implicit function theorems.

By repeating the process a sequence L1, L2, … of logics is obtained, each more complete than the previous one. A logic L can then be constructed in which the provable theorems are the totality of theorems provable with the help of the L1, L2, … etc. Thus Turing showed how one can associate a logic with any constructive ordinal.

Conversely, at some point of the curve, it becomes unrealistically expensive to add additional users. Based on the Paradox the Menz brothers developed the "Menz Theorems of Information and Physical Security". The theorems present two formulas covering access and security of both information systems and physical facilities. They are used to help determine allocation of resources and response levels.

This diagram shows the syntactic entities which may be constructed from formal languages.Dictionary Definition The symbols and strings of symbols may be broadly divided into nonsense and well-formed formulas. A formal language is identical to the set of its well-formed formulas. The set of well-formed formulas may be broadly divided into theorems and non-theorems.

The symbols and strings of symbols may be broadly divided into nonsense and well-formed formulas. A formal language can be thought of as identical to the set of its well-formed formulas. The set of well-formed formulas may be broadly divided into theorems and non-theorems. A theorem may be expressed in a formal language (or "formalized").

Berger, C. 1979. The contact hypothesis in ethnic relations. Oxford: Basil Blackwell. Berger identified 7 axioms (evident truths) and 21 theorems within URT (theoretical statements which are generally accepted, but are used to observe behaviours in need of more proof.) Because of these theorems, we have seen significant growth to the theory, leading to research into AUM.

A large part of the results remain true, or may be generalized to projective geometries for which these theorems do not hold.

Some feel that Gödel's theorems give a negative solution to the problem, while others consider Gentzen's proof as a partial positive solution.

In algebraic group theory, approximation theorems are an extension of the Chinese remainder theorem to algebraic groups G over global fields k.

He is the developer and co- developer of dozens of mathematical theorems and has had a lasting influence in 20th-century mathematics.

Generalisations and extensions are called Jackson-type theorems. A converse to Jackson's inequality is given by Bernstein's theorem. See also constructive function theory.

The statements of the theorems for rings are similar, with the notion of a normal subgroup replaced by the notion of an ideal.

He is also author of the paper Theorems for free! that gave rise to much research on functional language optimization (see also Parametricity).

Acharya Dwivedi's works fall into three major categories: original literature, new literary principles and theorems, and preservation of past literature for future generations.

1 and theorem 2) #T\Vdash Prov(\\#(\rho)) \rightarrow Prov(\\#( eg Prov(\\#(\rho))) (by condition no. 3 and theorem 3) #T\Vdash Prov(\\#(\rho)) \rightarrow eg Prov(\\#(Prov(\\#(\rho))) (by theorems 1 and 4) #T\Vdash Prov(\\#(\rho)) \rightarrow Prov(\\#(Prov(\\#(\rho))) (by condition no. 2) #T\Vdash eg Prov(\\#(\rho)) (by theorems 5 and 6) #T\Vdash eg Prov(\\#(\rho)) \rightarrow \rho (by construction of \rho) #T\Vdash \rho (by theorems 7 and 8) #T\Vdash Prov(\\#(\rho)) (by condition 1 and theorem 9) Thus T proves both Prov(\\#(\rho)) and eg Prov(\\#(\rho)), hence is T inconsistent.

Another reason follows from what are called black-hole uniqueness theorems: over time, black holes retain only a minimal set of distinguishing features (these theorems have become known as "no-hair" theorems), regardless of the starting geometric shape. For instance, in the long term, the collapse of a hypothetical matter cube will not result in a cube-shaped black hole. Instead, the resulting black hole will be indistinguishable from a black hole formed by the collapse of a spherical mass. In its transition to a spherical shape, the black hole formed by the collapse of a more complicated shape will emit gravitational waves.

In mathematics, constructive analysis is mathematical analysis done according to some principles of constructive mathematics. This contrasts with classical analysis, which (in this context) simply means analysis done according to the (more common) principles of classical mathematics. Generally speaking, constructive analysis can reproduce theorems of classical analysis, but only in application to separable spaces; also, some theorems may need to be approached by approximations. Furthermore, many classical theorems can be stated in ways that are logically equivalent according to classical logic, but not all of these forms will be valid in constructive analysis, which uses intuitionistic logic.

The long list of examples in this article indicates that common mathematical constructions are very often adjoint functors. Consequently, general theorems about left/right adjoint functors encode the details of many useful and otherwise non-trivial results. Such general theorems include the equivalence of the various definitions of adjoint functors, the uniqueness of a right adjoint for a given left adjoint, the fact that left/right adjoint functors respectively preserve colimits/limits (which are also found in every area of mathematics), and the general adjoint functor theorems giving conditions under which a given functor is a left/right adjoint.

Gödel gave a series of lectures on his theorems at Princeton in 1933–1934 to an audience that included Church, Kleene, and Rosser. By this time, Gödel had grasped that the key property his theorems required is that the system must be effective (at the time, the term "general recursive" was used). Rosser proved in 1936 that the hypothesis of ω-consistency, which was an integral part of Gödel's original proof, could be replaced by simple consistency, if the Gödel sentence was changed in an appropriate way. These developments left the incompleteness theorems in essentially their modern form.

Authors including the philosopher J. R. Lucas and physicist Roger Penrose have debated what, if anything, Gödel's incompleteness theorems imply about human intelligence. Much of the debate centers on whether the human mind is equivalent to a Turing machine, or by the Church–Turing thesis, any finite machine at all. If it is, and if the machine is consistent, then Gödel's incompleteness theorems would apply to it. Hilary Putnam (1960) suggested that while Gödel's theorems cannot be applied to humans, since they make mistakes and are therefore inconsistent, it may be applied to the human faculty of science or mathematics in general.

Other deductive systems describe term rewriting, such as the reduction rules for λ calculus. The definition of theorems as elements of a formal language allows for results in proof theory that study the structure of formal proofs and the structure of provable formulas. The most famous result is Gödel's incompleteness theorems; by representing theorems about basic number theory as expressions in a formal language, and then representing this language within number theory itself, Gödel constructed examples of statements that are neither provable nor disprovable from axiomatizations of number theory. syntactic entities that can be constructed from formal languages.

What would happen if another axiom schema were added to those listed above? There are two cases: (1) it is a tautology; or (2) it is not a tautology. If it is a tautology, then the set of theorems remains the set of tautologies as before. However, in some cases it may be possible to find significantly shorter proofs for theorems.

According to the Curry-Howard isomorphism, lambda calculus on its own can express theorems in intuitionistic logic only, and several classical logical theorems can't be written at all. However with these new operators one is able to write terms that have the type of, for example, Peirce's law. Semantically these operators correspond to continuations, found in some functional programming languages.

In mathematics, an exotic \R^4 is a differentiable manifold that is homeomorphic but not diffeomorphic to the Euclidean space \R^4. The first examples were found in 1982 by Michael Freedman and others, by using the contrast between Freedman's theorems about topological 4-manifolds, and Simon Donaldson's theorems about smooth 4-manifolds.Kirby (1989), p. 95Freedman and Quinn (1990), p.

Quartic or biquadratic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence x4 ≡ p (mod q) is solvable; the word "reciprocity" comes from the form of some of these theorems, in that they relate the solvability of the congruence x4 ≡ p (mod q) to that of x4 ≡ q (mod p).

A formal system is used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A formal system is essentially an "axiomatic system". Fourth-century BCE philologist Pāṇini is credited with the first use of formal system in Sanskrit grammar.

In classical logic there are theorems that clearly presuppose that there is something in the domain of discourse. Consider the following classically valid theorems. :1. \forall xA \Rightarrow \exists xA :2. \forall x \forall rA(x) \Rightarrow \forall rA(r) :3. \forall rA(r) \Rightarrow \exists xA(x) A valid scheme in the theory of equality which exhibits the same feature is :4.

The 1954 Theorems say, roughly, that every preference relation which is complete, transitive and continuous, can be represented by a continuous ordinal utility function.

There are, a bit weaker, reconstruction theorems from the derived categories of (quasi)coherent sheaves motivating the derived noncommutative algebraic geometry (see just below).

The Chinese remainder theorem has been used to construct a Gödel numbering for sequences, which is involved in the proof of Gödel's incompleteness theorems.

Since then there has been extensive work by other mathematicians on toughness; the recent survey by lists 99 theorems and 162 papers on the subject.

V. Bergelson, A. Leibman, Polynomial extensions of van der Waerden's and Szemerédi's theorems. Journal of the American Mathematical Society, vol. 9 (1996), no. 3, pp.

In mathematics, two Prüfer theorems, named after Heinz Prüfer, describe the structure of certain infinite abelian groups. They have been generalized by L. Ya. Kulikov.

The extension from a circle to a conic having center: The creative method of new theorems, International Journal of Computer Discovered Mathematics, pp.21-32.

In an effort to avoid naming everything after Euler, some discoveries and theorems are attributed to the first person to have proved them after Euler.

Flat modules have increased importance in constructive mathematics, where projective modules are less useful. For example, that all free modules are projective is equivalent to the full axiom of choice, so theorems about projective modules, even if proved constructively, do not necessarily apply to free modules. In contrast, no choice is needed to prove that free modules are flat, so theorems about flat modules can still apply.

470–410 BCE) gave some of the first known proofs of theorems in geometry. Eudoxus (408–355 BCE) and Theaetetus (417–369 BCE) formulated theorems but did not prove them. Aristotle (384–322 BCE) said definitions should describe the concept being defined in terms of other concepts already known. Mathematical proof was revolutionized by Euclid (300 BCE), who introduced the axiomatic method still in use today.

In 1930 Gödel attended the Second Conference on the Epistemology of the Exact Sciences, held in Königsberg, 5–7 September. Here he delivered his incompleteness theorems. Gödel published his incompleteness theorems in (called in English "On Formally Undecidable Propositions of and Related Systems"). In that article, he proved for any computable axiomatic system that is powerful enough to describe the arithmetic of the natural numbers (e.g.

NBG is not logically equivalent to ZFC because its language is more expressive: it can make statements about classes, which cannot be made in ZFC. However, NBG and ZFC imply the same statements about sets. Therefore, NBG is a conservative extension of ZFC. NBG implies theorems that ZFC does not imply, but since NBG is a conservative extension, these theorems must involve proper classes.

Partial converses to Abelian theorems are called Tauberian theorems. The original result of stated that if we assume also :an = o(1/n) (see Little o notation) and the radial limit exists, then the series obtained by setting z = 1 is actually convergent. This was strengthened by John Edensor Littlewood: we need only assume O(1/n). A sweeping generalization is the Hardy–Littlewood Tauberian theorem.

The main results established are Gödel's first and second incompleteness theorems, which have had an enormous impact on the field of mathematical logic. These appear as theorems VI and XI, respectively, in the paper. In order to prove these results, Gödel introduced a method now known as Gödel numbering. In this method, each sentence and formal proof in first-order arithmetic is assigned a particular natural number.

Regardless, the role of axioms in mathematics and in the above- mentioned sciences is different. In mathematics one neither "proves" nor "disproves" an axiom for a set of theorems; the point is simply that in the conceptual realm identified by the axioms, the theorems logically follow. In contrast, in physics a comparison with experiments always makes sense, since a falsified physical theory needs modification.

Originally algorithmic induction methods extrapolated ordered sequences of strings. Methods were needed for dealing with other kinds of data. A 1999 report,"Two Kinds of Probabilistic Induction," The Computer Journal, Vol 42, No. 4, 1999. (pdf version) generalizes the Universal Distribution and associated convergence theorems to unordered sets of strings and a 2008 report,"Three Kinds of Probabilistic Induction, Universal Distributions and Convergence Theorems" 2008.

Among hundreds of fixed-point theorems,E.g. F & V Bayart Théorèmes du point fixe on [email protected] Brouwer's is particularly well known, due in part to its use across numerous fields of mathematics. In its original field, this result is one of the key theorems characterizing the topology of Euclidean spaces, along with the Jordan curve theorem, the hairy ball theorem and the Borsuk–Ulam theorem.

Many treatments of predicate logic don't allow functional predicates, only relational predicates. This is useful, for example, in the context of proving metalogical theorems (such as Gödel's incompleteness theorems), where one doesn't want to allow the introduction of new functional symbols (nor any other new symbols, for that matter). But there is a method of replacing functional symbols with relational symbols wherever the former may occur; furthermore, this is algorithmic and thus suitable for applying most metalogical theorems to the result. Specifically, if F has domain type T and codomain type U, then it can be replaced with a predicate P of type (T,U).

He investigated the completeness of systems of own elements of one class of not self-interfaced operators, polynomially – and rationally – depending on parameter. His theorems of completeness and basis were proven, is given definitions of the best approach of linear operators finite-dimensional operators, its exact expression is found, given definitions of repeated completeness of systems of elements in linear spaces and many thorough theorems in this direction are proved. Theorems of new type of completeness and basis systems of own and attached elements in Banach and Hilbert spaces are proved. Necessary and sufficient conditions closure, limitations and quite continuity of operators in multiplicative terms are found.

"Majority Rule and Impossibility Theorems." Social Science Quarterly 73: 511-522. 1992."The General Will and Social Choice Theory." Review of Politics 54: 34-49. 1992.

When the definition of the classifying space takes place within the homotopy category of CW complexes, existence theorems for universal bundles arise from Brown's representability theorem.

In complex analysis, Picard's great theorem and Picard's little theorem are related theorems about the range of an analytic function. They are named after Émile Picard.

Kim has made contributions to the application of arithmetic homotopy theory to the study of Diophantine problems, especially to finiteness theorems of the Faltings–Siegel type.

Installation was completed in September, 2012.Helaman Ferguson, "Two Theorems, Two Sculptures, Two Posters", American Mathematical Monthly, Volume 97, Number 7,August-September 1990, pages 589-610.

A first-order theory is a set of first-order sentences (theorems) recursively obtained by the inference rules of the system applied to the set of axioms.

In mathematics, an autonomous convergence theorem is one of a family of related theorems which specify conditions guaranteeing global asymptotic stability of a continuous autonomous dynamical system.

On quadratic forms :Chapter 5. Determination of the class number of binary quadratic forms :Supplement I. Some theorems from Gauss's theory of circle division :Supplement II. On the limiting value of an infinite series :Supplement III. A geometric theorem :Supplement IV. Genera of quadratic forms :Supplement V. Power residues for composite moduli :Supplement VI. Primes in arithmetic progressions :Supplement VII. Some theorems from the theory of circle division :Supplement VIII.

It is a natural question to ask: under which conditions can two categories be considered essentially the same, in the sense that theorems about one category can readily be transformed into theorems about the other category? The major tool one employs to describe such a situation is called equivalence of categories, which is given by appropriate functors between two categories. Categorical equivalence has found numerous applications in mathematics.

Singularity theorems which are premised on and formulated within the setting of Riemannian geometry (e.g. Penrose–Hawking singularity theorems) need not hold in Riemann–Cartan geometry. Consequently, Einstein–Cartan theory is able to avoid the general-relativistic problem of the singularity at the Big Bang. The minimal coupling between torsion and Dirac spinors generates an effective nonlinear spin–spin self-interaction, which becomes significant inside fermionic matter at extremely high densities.

Note that P need not be itself convex. A consequence of this is that P′ can always be extremal in P, as non-extremal points can be removed from P without changing the membership of x in the convex hull. The similar theorems of Helly and Radon are closely related to Carathéodory's theorem: the latter theorem can be used to prove the former theorems and vice versa. See in particular p.

The theorem was proven by Bing in 1951 and was an independent discovery with the Nagata–Smirnov metrization theorem that was proved independently by both Nagata (1950) and Smirnov (1951). Both theorems are often merged in the Bing-Nagata-Smirnov metrization theorem. It is a common tool to prove other metrization theorems, e.g. the Moore metrization theorem – a collectionwise normal, Moore space is metrizable – is a direct consequence.

Priority and attribution of mathematical discovery are important to professional practice, even as some theorems bear the name of the person making the conjecture rather than finding the proof. Folk theorems, or mathematical folklore cannot be attributed to an individual, and may not have an agreed proof, despite being an accepted result, potentially leading to injusticevan Bendegem, J., Rittberg, C. & Tanswell, F. (2018) Epistemic Injustice in Mathematics, Synthese.

In connection with Cauchy problems, usually a linear operator A is given and the question is whether this is the generator of a strongly continuous semigroup. Theorems which answer this question are called generation theorems. A complete characterization of operators that generate strongly continuous semigroups is given by the Hille–Yosida theorem. Of more practical importance are however the much easier to verify conditions given by the Lumer–Phillips theorem.

Before this result was established it was not known whether there could be other examples of Hilbert spaces with operators invoking the same loop algebra – other realizations not equivalent to the one that had been used so far. These uniqueness theorems imply no others exist, so if LQG does not have the correct semiclassical limit then the theorems would mean the end of the loop representation of quantum gravity altogether.

In mathematics, particularly in functional analysis and topology, closed graph is a property of functions. A function between topological spaces has a closed graph if its graph is a closed subset of the product space . A related property is open graph. This property is studied because there are many theorems, known as closed graph theorems, giving conditions under which a function with a closed graph is necessarily continuous.

These axiom systems describe the space via primitive notions (such as "point", "between", "congruent") constrained by a number of axioms. Analytic geometry made great progress and succeeded in replacing theorems of classical geometry with computations via invariants of transformation groups. Since that time, new theorems of classical geometry have been of more interest to amateurs than to professional mathematicians. However, the heritage of classical geometry was not lost.

This set consists of all wffs for which there is a proof. Thus all axioms are considered theorems. Unlike the grammar for wffs, there is no guarantee that there will be a decision procedure for deciding whether a given wff is a theorem or not. The notion of theorem just defined should not be confused with theorems about the formal system, which, in order to avoid confusion, are usually called metatheorems.

Schematic describing Anxiety/Uncertainty Management theory Gudykunst uses two types of theoretical statements to construct his theory; axioms and theorems. Axioms are "propositions that involve variables that are taken to be directly linked causally; axioms should therefore be statements that imply direct causal links among variables" Some axioms do not apply in all situations. Boundary conditions specify when the axioms hold. The axioms can be combined to derive theorems.

He published several mathematical theorems. In addition to his academic career, he also worked as an employee for newspaper Svenska Dagbladet and Sveriges Radio, the national Swedish radio.

The minimax values are very important in the theory of repeated games. One of the central theorems in this theory, the folk theorem, relies on the minimax values.

The property converse to completeness is called soundness: a system is sound with respect to a property (mostly semantical validity) if each of its theorems has that property.

La Ceppède died, loaded down with honors, at Avignon in July 1623. Keith Bosley (1983), From the Theorems of Master Jean de La Ceppède: LXX Sonnets, page 5.

In 1951, Arrow presented the first and second fundamental theorems of welfare economics and their proofs without requiring differentiability of utility, consumption, or technology, and including corner solutions.

Smullyan, R M (2001) "Gödel's Incompleteness Theorems" in Goble, Lou, ed., The Blackwell Guide to Philosophical Logic. Blackwell (). Smullyan wrote many books about recreational mathematics and recreational logic.

Gödel's incompleteness theorems show that even elementary axiomatic systems such as Peano arithmetic are either self- contradicting or contain logical propositions that are impossible to prove or disprove.

Later results showed that stronger determinacy theorems cannot be proven in Zermelo–Fraenkel set theory, although they are relatively consistent with it, if certain large cardinals are consistent.

He received his Ph.D. in physics from Princeton University in 1985 after completing a doctoral dissertation, titled "Symmetries, inequalities and index theorems", under the supervision of Edward Witten.

This assumption holds well in most cases.Linearity of fisheries acoustics, with additional theorems. Kenneth G. Foote, 1983. Journal of the Acoustical Society of America 73, pp. 1932-1940.

When combined the axioms and theorems form a "casual process" theoryReynolds, P. D. 1971. A primer in theory construction. Indianapolis: Bobbs-Merrill Co., Inc. that explains effective communication.

The IPS was introduced as part of the Dubins–Spanier theorems and used in the proof of Weller's theorem. The term "Individual Pieces set" was coined by Julius Barbanel.

Many of the Sobolev embedding theorems require that the domain of study be a Lipschitz domain. Consequently, many partial differential equations and variational problems are defined on Lipschitz domains.

In , Fredholm introduced and analysed a class of integral equations now called Fredholm equations. His analysis included the construction of Fredholm determinants, and the proof of the Fredholm theorems.

86 no. 1 (2010), pp. 163–188, ISSN 0022-040X [arXiv:0808.0667] [abs] 8\. M Stern, Fixed point theorems from a de Rham perspective, Asian Journal of Mathematics, vol.

Gödel shows that in some possible world a Godlike object exists (theorem 2), called "God" in the following.By removing all modal operators from axioms, definitions, proofs, and theorems, a modified version of theorem 2 is obtained saying "∃x G(x)", i.e. "There exists an object which has all positive, but no negative properties". Nothing more than axioms 1-3, definition 1, and theorems 1-2 needs to be considered for this result.

In Euclid's Elements, the first 28 propositions and Proposition I.31 avoid using the parallel postulate, and therefore are valid theorems in absolute geometry. Proposition I.31 proves the existence of parallel lines (by construction). Also, the Saccheri–Legendre theorem, which states that the sum of the angles in a triangle is at most 180°, can be proved. The theorems of absolute geometry hold in hyperbolic geometry as well as in Euclidean geometry.

Every order theoretic definition has its dual: it is the notion one obtains by applying the definition to the inverse order. Since all concepts are symmetric, this operation preserves the theorems of partial orders. For a given mathematical result, one can just invert the order and replace all definitions by their duals and one obtains another valid theorem. This is important and useful, since one obtains two theorems for the price of one.

The theorems of Alhacen, Khayyam and al-Tūsī on quadrilaterals, including the Ibn al-Haytham–Lambert quadrilateral and Khayyam–Saccheri quadrilateral, were the first theorems on hyperbolic geometry. Their works on hyperbolic geometry had a considerable influence on its development among later European geometers, including Witelo, Gersonides, Alfonso, John Wallis and Saccheri.Boris A. Rosenfeld and Adolf P. Youschkevitch (1996), "Geometry", in Roshdi Rashed, ed., Encyclopedia of the History of Arabic Science, Vol.

In topology, a branch of mathematics, Quillen's Theorem A gives a sufficient condition for the classifying spaces of two categories to be homotopy equivalent. Quillen's Theorem B gives a sufficient condition for a square consisting of classifying spaces of categories to be homotopy Cartesian. The two theorems play central roles in Quillen's Q-construction in algebraic K-theory and are named after Daniel Quillen. The precise statements of the theorems are as follows.

The set of well-formed formulas may be broadly divided into theorems and non-theorems. However, according to Hofstadter, a formal system often simply defines all its well-formed formula as theorems.Hofstadter 1980 Different sets of derivation rules give rise to different interpretations of what it means for an expression to be a theorem. Some derivation rules and formal languages are intended to capture mathematical reasoning; the most common examples use first-order logic.

Two works by Motot have been preserved. One is a treatise on algebra, entitled "Sefer ha- Alzibra," or "Kelale me-Ḥeshbon ha-Aljibra." In it, Motot claims to have studied several mathematical works written by Christians and to have found among them one containing theorems without demonstrations. This book he chose as his basic work and translated it, supplying the demonstrations from other mathematical sources, and adding some theorems of his own.

Cantor's intersection theorem refers to two closely related theorems in general topology and real analysis, named after Georg Cantor, about intersections of decreasing nested sequences of non-empty compact sets.

He also preserved the writings of Menelaus of Alexandria and reworked many of the Greeks theorems. He died in the Ghaznavid Empire (modern-day Afghanistan) near the city of Ghazna.

300px The action of the pentagram map on pentagons and hexagons is similar in spirit to classical configuration theorems in projective geometry such as Pascal's theorem, Desargues's theorem and others.

Important work was done by Errett Bishop, who managed to prove versions of the most important theorems in real analysis as constructive analysis in his 1967 Foundations of Constructive Analysis.

Discrete fixed-point theorems have been used to prove the existence of a Nash equilibrium in a discrete game, and the existence of a Walrasian equilibrium in a discrete market.

Milgrom and Segal (2002) demonstrate that the generalized version of the envelope theorems can also be applied to convex programming, continuous optimization problems, saddle-point problems, and optimal stopping problems.

In particular, no theory extending ZF can prove either the completeness or compactness theorems over arbitrary (possibly uncountable) languages without also proving the ultrafilter lemma on a set of same cardinality.

It is among the most notable theorems in the history of mathematics, and prior to its proof it was in the Guinness Book of World Records for "most difficult mathematical problems".

In the history of mathematics, Alfred Tarski (1901–1983) is one of the most important logicians. His name is now associated with a number of theorems and concepts in that field.

Bosley added, however, that after d'Aubigné's death, he, "was forgotten until the Romantics rediscovered him."Keith Bosley (1983), From the Theorems of Master Jean de La Ceppède: LXX Sonnets, page 4.

See the page on direction- preserving function for definitions. Continuous fixed-point theorems often require a convex set. The analogue of this property for discrete sets is an integrally-convex set.

In computational complexity theory and cryptography, averaging argument is a standard argument for proving theorems. It usually allows us to convert probabilistic polynomial-time algorithms into non-uniform polynomial-size circuits.

Huge sets of this nature are possible if ZF is augmented with Tarski's axiom. Assuming that axiom turns the axioms of infinity, power set, and choice (7 – 9 above) into theorems.

Gödel's incompleteness theorems are two fundamental theorems of mathematical logic which state inherent limitations of sufficiently powerful axiomatic systems for mathematics. The theorems were proven by Kurt Gödel in 1931, and are important in the philosophy of mathematics. Roughly speaking, in proving the first incompleteness theorem, Gödel used a modified version of the liar paradox, replacing "this sentence is false" with "this sentence is not provable", called the "Gödel sentence G". His proof showed that for any sufficiently powerful theory T, G is true, but not provable in T. The analysis of the truth and provability of G is a formalized version of the analysis of the truth of the liar sentence. To prove the first incompleteness theorem, Gödel represented statements by numbers.

In mathematics, an elementary proof is a mathematical proof that only uses basic techniques. More specifically, the term is used in number theory to refer to proofs that make no use of complex analysis. Historically, it was once thought that certain theorems, like the prime number theorem, could only be proved by invoking "higher" mathematical theorems or techniques. However, as the time progresses, many of these results have also been subsequently reproven using only elementary techniques.

The theorem applied to an open cylinder, cone and a sphere to obtain their surface areas. The centroids are at a distance a (in red) from the axis of rotation. In mathematics, Pappus's centroid theorem (also known as the Guldinus theorem, Pappus–Guldinus theorem or Pappus's theorem) is either of two related theorems dealing with the surface areas and volumes of surfaces and solids of revolution. The theorems are attributed to Pappus of Alexandria and Paul Guldin.

Gödel's incompleteness theorems also imply the existence of non-standard models of arithmetic. The incompleteness theorems show that a particular sentence G, the Gödel sentence of Peano arithmetic, is not provable nor disprovable in Peano arithmetic. By the completeness theorem, this means that G is false in some model of Peano arithmetic. However, G is true in the standard model of arithmetic, and therefore any model in which G is false must be a non-standard model.

During the French Wars of Religion, La Ceppède belonged to a Royalist literary circle which included the son of Nostradamus. During the same period, he began writing the sonnets that appear in the Theorems. Keith Bosley (1983), From the Theorems of Master Jean de La Ceppède: LXX Sonnets, page 5. As France was increasingly reunified by the armies of King Henri IV, La Ceppède published his first collection of poems, which was an imitation of the Seven Penitential Psalms.

In the theory of finite groups the Sylow theorems imply that, if a power of a prime number p^n divides the order of a group, then the group has a subgroup of order p^n. By Lagrange's theorem, any group of prime order is a cyclic group, and by Burnside's theorem any group whose order is divisible by only two primes is solvable. For the Sylow theorems see p. 43; for Lagrange's theorem, see p.

In other systems, such as set theory, only some sentences of the formal system express statements about the natural numbers. The incompleteness theorems are about formal provability within these systems, rather than about "provability" in an informal sense. There are several properties that a formal system may have, including completeness, consistency, and the existence of an effective axiomatization. The incompleteness theorems show that systems which contain a sufficient amount of arithmetic cannot possess all three of these properties.

In Euclidean plane geometry, a tangent line to a circle is a line that touches the circle at exactly one point, never entering the circle's interior. Tangent lines to circles form the subject of several theorems, and play an important role in many geometrical constructions and proofs. Since the tangent line to a circle at a point P is perpendicular to the radius to that point, theorems involving tangent lines often involve radial lines and orthogonal circles.

Reactions in chemical processes are either unimolecular or bimolecular. The rate of a unimolecular reaction is an average over a vast ensemble of the rate coefficients for the microscopic events of collisional energy transfer and of reaction of a completely isolated molecule. Gilbert's work in the field of unimolecular processes started with the development of theorems for this relationship. These theorems are elegant developments in matrix algebra, proving relations that had been previously known only for particular cases.

To what extent do they correspond to an experimental reality? This important physical problem no longer has anything to do with mathematics. Even if a "geometry" does not correspond to an experimental reality, its theorems remain no less "mathematical truths". A Euclidean model of a non-Euclidean geometry is a choice of some objects existing in Euclidean space and some relations between these objects that satisfy all axioms (and therefore, all theorems) of the non-Euclidean geometry.

There are many ways AUM theory can be applied. It can be effective in studying the behavior of a stranger adjusting to a new culture, as well as in examining how individuals communicate with strangers and often accurately predict their behavior; this is done when we are mindful. Gudykunst explains that some axioms can be combined to form theorems. These theorems that are generated might be consistent with previous research, while others might be useful for future study.

In general relativity, energy conditions are often used (and required) in proofs of various important theorems about black holes, such as the no hair theorem or the laws of black hole thermodynamics.

Friedhelm Waldhausen's theorems on topological rigidity say that certain 3-manifolds (such as those with an incompressible surface) are homeomorphic if there is an isomorphism of fundamental groups which respects the boundary.

A series of twelve articles, published between 1946–1952, established the full results, and also simplified and extended related theorems in the analytic theory of differential equations by Poincaré, Cayley and others.

The supersymmetry of this theory means that, unlike QCD, one may use nonrenormalization theorems to analytically demonstrate the existence of these phenomena and even calculate the condensate which breaks the chiral symmetry.

Additionally, there should be at least a possibility (if not a guarantee) that the partner receives more than 1/n; this explains the importance of the existence theorems of super-proportional division.

The welfare theorems can be extended to the above generalized abstract economies. A further generalization of these equilibrium concepts for a general model without ordered preferences can be found in Barabolla (1985).

With James Glimm, he founded the subject called constructive quantum field theory. Their major achievement was to establish existence theorems for two- and three-dimensional examples of non-linear, relativistic quantum fields.

Budan's and Fourier's theorems were soon considered of a great importance, although they do not solve completely the problem of counting the number of real roots of a polynomial in an interval. This problem was completely solved in 1827 by Sturm. Although Sturm's theorem is not based on Descartes' rule of signs, Sturm's and Fourier's theorems are related not only by the use of the number of sign variations of a sequence of numbers, but also by a similar approach of the problem. Sturm himself acknowledged having been inspired by Fourier's methods: « C'est en m'appuyant sur les principes qu'il a posés, et en imitant ses démonstrations, que j'ai trouvé les nouveaux théorèmes que je vais énoncer. » which translates into « It is by relying upon the principles he has laid out and by imitating his proofs that I have found the new theorems which I am about to present. » Because of this, during the 19th century, Fourier's and Sturm's theorems appeared together in almost all books on the theory of equations.

Main scientific papers are dedicated to differential equations, elliptic and hypoelliptic equations, the study of the properties of functions in different multianisotropic spaces, integral representations and embedding theorems for functions in multianisotropic spaces.

One of the useful structure theorems for vector spaces over locally compact fields is that the finite dimensional vector spaces have only an equivalence class of norm: the sup norm pg. 58-59.

The first result of this kind may have been the theorem of Hilbert and Hurwitz dealing with the case g = 0. The theory consists both of theorems and many conjectures and open questions.

He became well known for his contributions on time series and Markov processes. He conducted seminal work on density estimation, central limit theorems under strong mixing, spectral domain methods and long memory processes.

He also dealt with probability theory and Tauberian theorems in analysis. In 1966, with Vinogradov, he was a Plenary Speaker of the ICM in Moscow with talk Recent developments in analytic number theory.

In computational complexity theory and cryptography, the existence of pseudorandom generators is related to the existence of one-way functions through a number of theorems, collectively referred to as the pseudorandom generator theorem.

One of the fundamental theorems for the birational geometry of surfaces is Castelnuovo's theorem. This states that any birational map between algebraic surfaces is given by a finite sequence of blowups and blowdowns.

Before the Disquisitiones was published, number theory consisted of a collection of isolated theorems and conjectures. Gauss brought the work of his predecessors together with his own original work into a systematic framework, filled in gaps, corrected unsound proofs, and extended the subject in numerous ways. The logical structure of the Disquisitiones (theorem statement followed by proof, followed by corollaries) set a standard for later texts. While recognising the primary importance of logical proof, Gauss also illustrates many theorems with numerical examples.

In mathematics, a decomposable measure is a measure that is a disjoint union of finite measures. This is a generalization of σ-finite measures, which are the same as those that are a disjoint union of countably many finite measures. There are several theorems in measure theory such as the Radon–Nikodym theorem that are not true for arbitrary measures but are true for σ-finite measures. Several such theorems remain true for the more general class of decomposable measures.

In the mathematical field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the convergence of monotonic sequences (sequences that are non-decreasing or non-increasing) that are also bounded. Informally, the theorems state that if a sequence is increasing and bounded above by a supremum, then the sequence will converge to the supremum; in the same way, if a sequence is decreasing and is bounded below by an infimum, it will converge to the infimum.

Arunava Sen has provided simple proofs of three important theorems in mechanism design. In his work, he uses induction on the number of agents to provide a simple proof of the Gibbard-Satterthwaite (GS) theorem. The induction technique in proving the GS theorem is quite easily extendible to other settings where such theorems hold. For instance, in his work with Dipjyoti Majumdar, he uses similar induction techniques to prove an analogue of the GS theorem using a weaker notion of incentive compatibility.

Sylow was a high school teacher at Hartvig Nissen School, later before becoming a headmaster in Halden from 1858 to 1898. He was a substitute lecturer at University of Christiania in 1862, covering Galois theory. It was then that he posed the question that led to his theorems regarding Sylow subgroups. Sylow published the Sylow theorems in 1872, and subsequently devoted eight years of his life, with Sophus Lie, to the project of editing the mathematical works of his countryman, Niels Henrik Abel.

The Second Conference on the Epistemology of the Exact Sciences () was held on 5–7 September 1930 in Königsberg, then located in East Prussia. It was at this conference that Kurt Gödel first presented his incompleteness theorems, though just "in an off-hand remark during a general discussion on the last day".Mancosu, Paolo "Between Vienna and Berlin: The immidiate reception of Gödel's incompleteness theorems", History and Philosophy of Logic, 20, 1999, 33-45. The real first presentation took place in Vienna.

Sometimes an economic hypothesis is only qualitative, not quantitative. Expositions of economic reasoning often use two-dimensional graphs to illustrate theoretical relationships. At a higher level of generality, Paul Samuelson's treatise Foundations of Economic Analysis (1947) used mathematical methods beyond graphs to represent the theory, particularly as to maximizing behavioural relations of agents reaching equilibrium. The book focused on examining the class of statements called operationally meaningful theorems in economics, which are theorems that can conceivably be refuted by empirical data.

This is because the payoff in a repeated game is just a weighted average of payoffs in the basic games. Folk theorems are partially converse claims: they say that, under certain conditions (which are different in each folk theorem), every payoff profile that is both individually rational and feasible can be realized as a Nash equilibrium payoff profile of the repeated game. There are various folk theorems; some relate to finitely-repeated games while others relate to infinitely-repeated games.

From 1943–1959 Bernstein taught at the University of Rochester, where she worked on existence theorems for partial differential equations. Her work was motivated by non-linear problems that were just being tackled by high-speed digital computers. In 1950, Princeton University Press published her book, Existence Theorems in Partial Differential Equations. She spent 1959–1979 as a professor of mathematics at Goucher College, where she was chairman of the mathematics department for most of that time (1960–70, 1974–79).

The best vertex degree characterization of Hamiltonian graphs was provided in 1972 by the Bondy–Chvátal theorem, which generalizes earlier results by G. A. Dirac (1952) and Øystein Ore. Both Dirac's and Ore's theorems can also be derived from Pósa's theorem (1962). Hamiltonicity has been widely studied with relation to various parameters such as graph density, toughness, forbidden subgraphs and distance among other parameters. Dirac and Ore's theorems basically state that a graph is Hamiltonian if it has enough edges.

The treatise which has given rise to this subject is the Porisms of Euclid, the author of the Elements. As much as is known of this lost treatise is due to the Collection of Pappus of Alexandria, who mentions it along with other geometrical treatises, and gives a number of lemmas necessary for understanding it. Pappus states: :The porisms of all classes are neither theorems nor problems, but occupy a position intermediate between the two, so that their enunciations can be stated either as theorems or problems, and consequently some geometers think that they are really theorems, and others that they are problems, being guided solely by the form of the enunciation. But it is clear from the definitions that the old geometers understood better the difference between the three classes.

In mathematical logic, a conservative extension is a supertheory of a theory which is often convenient for proving theorems, but proves no new theorems about the language of the original theory. Similarly, a non-conservative extension is a supertheory which is not conservative, and can prove more theorems than the original. More formally stated, a theory T_2 is a (proof theoretic) conservative extension of a theory T_1 if every theorem of T_1 is a theorem of T_2, and any theorem of T_2 in the language of T_1 is already a theorem of T_1. More generally, if \Gamma is a set of formulas in the common language of T_1 and T_2, then T_2 is \Gamma-conservative over T_1 if every formula from \Gamma provable in T_2 is also provable in T_1.

This implies that a trigonometric number is an algebraic number, and twice a trigonometric number is an algebraic integer. Ivan Niven gave proofs of theorems regarding these numbers.Niven, Ivan. Numbers: Rational and Irrational, 1961.

16, no 2, pp. 97–159. on gene frequency in 1931 and YaglomYAGLOM, Akiva M. Certain limit theorems of the theory of branching random processes. In : Doklady Akad. Nauk SSSR (NS). 1947. p. 3.

Thus the statements of duality theorems such as Serre duality or Grothendieck local duality for Gorenstein or Cohen–Macaulay schemes retain some of the simplicity of what happens for regular schemes or smooth varieties.

Any axiomatizable theory, such as ST and GST, whose theorems include the Q axioms is likewise incomplete. Moreover, the consistency of GST cannot be proved within GST itself, unless GST is in fact inconsistent.

Georges Jean Marie Valiron (7 September 1884 – 17 March 1955) was a French mathematician, notable for his contributions to analysis, in particular, the asymptotic behaviour of entire functions of finite order and Tauberian theorems.

Similar theorems describe the degree sequences of simple graphs and simple directed graphs. The first problem is characterized by the Erdős–Gallai theorem. The latter case is characterized by the Fulkerson–Chen–Anstee theorem.

M Stern, Lefschetz formulae for arithmetic varieties, Inventiones Mathematicae, vol. 115 no. 1 (1994), pp. 241–296, ISSN 0020-9910 [doi] 25\. M Stern, L2-index theorems on locally symmetric spaces, Inventiones Mathematicae, vol.

While this seems at first to be a weakness in the concept of the word metric, it can be exploited to prove theorems about geometric properties of groups, as is done in geometric group theory.

10, p. 285 He also created the Hα, β, γ function space and proved some theorems for nonlinear singular integral equations with Cauchy kernel within that space. Huseynov held positions in some Azerbaijani scientific institutions.

Although it was based on the proof methods of logic, Planner, developed at MIT, was the first language to emerge within this proceduralist paradigm.Carl Hewitt. "Planner: A Language for Proving Theorems in Robots". IJCAI 1969.

Z. Ruzsa, G.J. Székely, "Algebraic probability theory" , Wiley (1988)) of topological semi-groups is known, including the convolution semi- group of distributions on the line, in which factorization theorems analogous to Khinchin's theorem are valid.

Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus (1965) by Michael Spivak is a brief, rigorous, and modern textbook of multivariable calculus, differential forms, and integration on manifolds for advanced undergraduates.

Littlewood's theorem follows from the later Hardy–Littlewood tauberian theorem, which is in turn a special case of Wiener's tauberian theorem, which itself is a special case of various abstract Tauberian theorems about Banach algebras.

LeVeque, Randall (2002), Finite Volume Methods for Hyperbolic Problems, Cambridge University Press. The theoretical justification of these methods often involves theorems from functional analysis. This reduces the problem to the solution of an algebraic equation.

This theory is consistent, and complete, and contains a sufficient amount of arithmetic. However it does not have a recursively enumerable set of axioms, and thus does not satisfy the hypotheses of the incompleteness theorems.

It also includes in a chapter of supplementary material the translations of three related articles by Volkov and Shor, including a simplified proof of Pogorelov's theorems generalizing Alexandrov's uniqueness theorem to non-polyhedral convex surfaces.

Lance Fortnow. My Favorite Ten Complexity Theorems of the Past Decade. Foundations of Software Technology and Theoretical Computer Science: Proceedings of the 14th Conference, Madras, India, December 15–17, 1994. P. S. Thiagarajan (editor), pp.

In 1918, David Hilbert wrote about the difficulty in assigning an energy to a "field" and "the failure of the energy theorem" in a correspondence with Klein. In this letter, Hilbert conjectured that this failure is a characteristic feature of the general theory, and that instead of "proper energy theorems" one had 'improper energy theorems'. This conjecture was soon proved to be correct by one of Hilbert's close associates, Emmy Noether. Noether's theorem applies to any system which can be described by an action principle.

She received her BA from Goucher College in 1915 and went on to attend University of Pennsylvania for doctoral work. She was Robert Lee Moore's third student, graduating in 1922 with a dissertation entitled Certain Theorems Relating to Plane Connected Point Sets. Her dissertation was published that year in Transactions of the American Mathematical SocietyMullikin, A. (1922) Certain Theorems Relating to Plane Connected Point Sets, Transactions of the American Mathematical Society 24, 144-162. and subsequently became the catalyst for significant advances in the field.

Characterization theorems in probability theory and mathematical statistics are such theorems that establish a connection between the type of the distribution of random variables or random vectors and certain general properties of functions in them. For example, the assumption that two linear (or non-linear) statistics are identically distributed (or independent, or have a constancy regression and so on) can be used to characterize various populations. Verification of conditions of this or that characterization theorem in practice is possible only with some error, i.e.

In the paper of :Edward F. Moore, (n; m; p), an automat (or a machine) S, is defined as a device with n states, m input symbols and p output symbols. Nine theorems on the structure of S and experiments with S are proved. Later such S machines got the name of Moore machines. At the end of the paper, in the chapter «New problems», Moore formulates the problem of improving the estimates which he obtained in Theorems 8 and 9: : Theorem 8 (Moore).

Some scholars have debated over what, if anything, Gödel's incompleteness theorems imply about anthropic mechanism. Much of the debate centers on whether the human mind is equivalent to a Turing machine, or by the Church-Turing thesis, any finite machine at all. If it is, and if the machine is consistent, then Gödel's incompleteness theorems would apply to it. Gödelian arguments claim that a system of human mathematicians (or some idealization of human mathematicians) is both consistent and powerful enough to recognize its own consistency.

The problem of counting and locating the real roots of a polynomial started to be systematically studied only in the beginning of the 19th. In 1807, François Budan de Boislaurent discovered a method for extending Descartes' rule of signs—valid for the interval —to any interval. Joseph Fourier published a similar theorem in 1820, on which he worked for more than twenty years. Because of the similarity between the two theorems, there was a priority controversy, despite the fact that the two theorems were discovered independently.

He almost singlehandedly founded complex analysis and the study of permutation groups in abstract algebra. A profound mathematician, Cauchy had a great influence over his contemporaries and successors; Hans Freudenthal stated: "More concepts and theorems have been named for Cauchy than for any other mathematician (in elasticity alone there are sixteen concepts and theorems named for Cauchy)." Cauchy was a prolific writer; he wrote approximately eight hundred research articles and five complete textbooks on a variety of topics in the fields of mathematics and mathematical physics.

While the theorems of Gödel and Gentzen are now well understood by the mathematical logic community, no consensus has formed on whether (or in what way) these theorems answer Hilbert's second problem. Simpson (1988:sec. 3) argues that Gödel's incompleteness theorem shows that it is not possible to produce finitistic consistency proofs of strong theories. Kreisel (1976) states that although Gödel's results imply that no finitistic syntactic consistency proof can be obtained, semantic (in particular, second-order) arguments can be used to give convincing consistency proofs.

Gentzen published his consistency proof for first-order arithmetic in 1936. Hilbert accepted this proof as "finitary" although (as Gödel's theorem had already shown) it cannot be formalized within the system of arithmetic that is being proved consistent. The impact of the incompleteness theorems on Hilbert's program was quickly realized. Bernays included a full proof of the incompleteness theorems in the second volume of Grundlagen der Mathematik (1939), along with additional results of Ackermann on the ε-substitution method and Gentzen's consistency proof of arithmetic.

In some cases, one might even be able to substantiate a theorem by using a picture as its proof. Because theorems lie at the core of mathematics, they are also central to its aesthetics. Theorems are often described as being "trivial", or "difficult", or "deep", or even "beautiful". These subjective judgments vary not only from person to person, but also with time and culture: for example, as a proof is obtained, simplified or better understood, a theorem that was once difficult may become trivial.

When used in conjunction with one of Fermat's theorems, the Brahmagupta–Fibonacci identity proves that the product of a square and any number of primes of the form 4n + 1 is a sum of two squares.

Further insights result when categories of orders are found categorically equivalent to other categories, for example of topological spaces. This line of research leads to various representation theorems, often collected under the label of Stone duality.

M Stern and P Yi, Counting Yang-Mills dyons with index theorems, Physical Review D - Particles, Fields, Gravitation and Cosmology, vol. 62 no. 12 (2000), pp. 1–15, ISSN 0556-2821 [hep-th/0005275] [abs] 15\.

A set of formal theorems may be referred to as a formal theory. A theorem whose interpretation is a true statement about a formal system (as opposed to of a formal system) is called a metatheorem.

A validity is a formula that is true under any possible interpretation (for example, in classical propositional logic, validities are tautologies). A formal system is considered semantically complete when all of its theorems are also tautologies.

Agda has an extensive de facto standard library, which includes many useful definitions and theorems about basic data structures, such as natural numbers, lists, and vectors. The library is in beta, and is under active development.

Above we showed how to prove the Borsuk–Ulam theorem from Tucker's lemma. The converse is also true: it is possible to prove Tucker's lemma from the Borsuk–Ulam theorem. Therefore, these two theorems are equivalent.

Deficiency is a concept in graph theory that is used to refine various theorems related to perfect matching in graphs, such as Hall's marriage theorem. Is was first studied by Øystein Ore. A related property is surplus.

C. Lim, Remarks on some fixed point theorems, Proc. Amer. Math. Soc. 60 (1976), 179–182. and, soon after, under the name of almost convergence, by Tadeusz Kuczumow.T. Kuczumow, An almost convergence and its applications, Ann. Univ.

For the generalized case, the Ornstein isomorphism theorem still holds if the group G is a countably infinite amenable group. D. Ornstein and B. Weiss. "Entropy and isomorphism theorems for actions of amenable groups." J. Analyse Math.

One important use of these inequalities is to prove convergence of families of harmonic functions or sub-harmonic functions, see Harnack's theorem. These convergence theorems are used to prove the existence of harmonic functions with particular properties.

5, 21-29. There Samuelson identifies qualitative restrictions and the hypotheses of maximization and stability of equilibrium as the three fundamental sources of meaningful theorems — hypotheses about empirical data that could conceivably be refuted by empirical data.

By comparing the rate of orbital precession of two stars on different orbits, it is possible in principle to test the no-hair theorems of general relativity, in addition to measuring the spin of the black hole.

This even allows one to prove the truth of otherwise unprovable theorems such as the well-ordering theorem and the falsity of others such as the continuum hypothesis. There are also dialetheic solutions to the sorites paradox.

Next is the Marx’s propositions of dynamic movement of capitalistic economy. In the paper “A Formal Proof of Marx’s Two Theorems” he tried to prove Marx’s two theorems; first, the tendencial falling rate of profit and, second, the tendencial increase in unemployment. By “formal” Okishio meant whether we can deduct two propositions from Marx’s presumptions of increasing organic composition of production. He showed that if new technologies with increasing organic composition of production are continuously introduced, then the rate of profit must fall and the unemployment must increase.

Many subsequent authors, such as Theon of Alexandria, made their own editions, with alterations, comments, and new theorems or lemmas. Many mathematicians were influenced and inspired by Euclid's masterpiece. For example, Archimedes of Syracuse and Apollonius of Perga, the greatest mathematicians of their time, received their training from Euclid's students and his Elements and were able to solve many open problems at the time of Euclid. It is a prime example of how to write a text in pure mathematics, featuring simple and logical axioms, precise definitions, clearly stated theorems, and logical deductive proofs.

However, as mentioned above, the continuity of the Wiener-Rosenblueth medium has not so far allowed as precise a theorem about persistence of patterns as the one for GH which is described below. On the other hand, several theorems are stated in which are similar to those proved in, though the proofs given for the theorems in are less clear than those in because of the natures of the respective models. See also for an often cited computer study based on a model which is similar to that of Wiener and Rosenblueth.

In 2005, Wolpert and Macready themselves indicated that the first theorem in their paper "state[s] that any two optimization algorithms are equivalent when their performance is averaged across all possible problems".Wolpert, D.H., and Macready, W.G. (2005) "Coevolutionary free lunches", IEEE Transactions on Evolutionary Computation, 9(6): 721–735 The "no free lunch" (NFL) theorem is an easily stated and easily understood consequence of theorems Wolpert and Macready actually prove. It is weaker than the proven theorems, and thus does not encapsulate them. Various investigators have extended the work of Wolpert and Macready substantively.

It starts with undefined terms and axioms, propositions concerning the undefined terms which are assumed to be self-evidently true (from Greek "axios", something worthy). From this basis, the method proves theorems using deductive logic. Euclid's book, the Elements, was read by anyone who was considered educated in the West until the middle of the 20th century. In addition to theorems of geometry, such as the Pythagorean theorem, the Elements also covers number theory, including a proof that the square root of two is irrational and a proof that there are infinitely many prime numbers.

In mathematical logic, structural proof theory is the subdiscipline of proof theory that studies proof calculi that support a notion of analytic proof, a kind of proof whose semantic properties are exposed. When all the theorems of a logic formalised in a structural proof theory have analytic proofs, then the proof theory can be used to demonstrate such things as consistency, provide decision procedures, and allow mathematical or computational witnesses to be extracted as counterparts to theorems, the kind of task that is more often given to model theory.

Often, a theorem is called "Abelian" if it shows that some summation method gives the usual sum for convergent series, and is called "Tauberian" if it gives conditions for a series summable by some method that allows it to be summable in the usual sense. In the theory of integral transforms Abelian theorems give the asymptotic behaviour of the transform based on properties of the original function. Conversely Tauberian theorems give the asymptotic behaviour of the original function based on properties of the transform but usually require some restrictions on the original function.

There are various theorems, often provided in the form of memory models, that provide SC for DRF guarantees given various contexts. The premises of these theorems typically place constraints upon both the memory model (and therefore upon the implementation), and also upon the programmer; that is to say, typically it is the case that there are programs which do not meet the premises of the theorem and which could not be guaranteed to execute in a sequentially consistent manner. The DRF1 memory modelAdve, Sarita. (1994). Designing Memory Consistency Models For Shared-Memory Multiprocessors.

This also implies that there may always be a better compiler since the proof that one has the best compiler cannot exist. Therefore, compiler writers will always be able to speculate that they have something to improve. A similar example in practical computer science is the idea of no free lunch in search and optimization, which states that no efficient general-purpose solver can exist, and hence there will always be some particular problem whose best known solution might be improved. Similarly, Gödel's incompleteness theorems have been called full employment theorems for mathematicians.

The Nash embedding theorems (or imbedding theorems), named after John Forbes Nash, state that every Riemannian manifold can be isometrically embedded into some Euclidean space. Isometric means preserving the length of every path. For instance, bending without stretching or tearing a page of paper gives an isometric embedding of the page into Euclidean space because curves drawn on the page retain the same arclength however the page is bent. The first theorem is for continuously differentiable (C1) embeddings and the second for analytic embeddings or embeddings that are smooth of class Ck, 3 ≤ k ≤ ∞.

The axiomatic method of Euclid's Elements was influential in the development of Western science. Mathematical practice comprises the working practices of professional mathematicians: selecting theorems to prove, using informal notations to persuade themselves and others that various steps in the final proof are convincing, and seeking peer review and publication, as opposed to the end result of proven and published theorems. Philip Kitcher has proposed a more formal definition of a mathematical practice, as a quintuple. His intention was primarily to document mathematical practice through its historical changes.

He appealed through the courts, won his case and gained tenure in 1995, and was promoted to full professor one year later. In doing so he became the first African-American tenured associate professor and the first African-American full professor at UNC, as well as the only mathematician there to be promoted from associate to full so quickly. Assani's research concerns ergodic theory. He is the author of the research monograph Wiener Wintner Ergodic Theorems (World Scientific, 2003),Review of Wiener Wintner Ergodic Theorems by U. Krengel (2004), .

Compared to the theorems stated in Gödel's 1931 paper, many contemporary statements of the incompleteness theorems are more general in two ways. These generalized statements are phrased to apply to a broader class of systems, and they are phrased to incorporate weaker consistency assumptions. Gödel demonstrated the incompleteness of the system of Principia Mathematica, a particular system of arithmetic, but a parallel demonstration could be given for any effective system of a certain expressiveness. Gödel commented on this fact in the introduction to his paper, but restricted the proof to one system for concreteness.

HOL (Higher Order Logic) denotes a family of interactive theorem proving systems using similar (higher-order) logics and implementation strategies. Systems in this family follow the LCF approach as they are implemented as a library in some programming language. This library implements an abstract data type of proven theorems so that new objects of this type can only be created using the functions in the library which correspond to inference rules in higher-order logic. As long as these functions are correctly implemented, all theorems proven in the system must be valid.

In this class W, V is just a set, closed under all the set-forming operations of A. In other words the universe W contains a set V which resembles W in that it is closed under all the methods A. We can use this informal argument in two ways. We can try to formalize it in (say) ZF set theory; by doing this we obtain some theorems of ZF set theory, called reflection theorems. Alternatively we can use this argument to motivate introducing new axioms for set theory.

Theories used in applications are abstractions of observed phenomena and the resulting theorems provide solutions to real-world problems. Obvious examples include arithmetic (abstracting concepts of number), geometry (concepts of space), and probability (concepts of randomness and likelihood). Gödel's incompleteness theorem shows that no consistent, recursively enumerable theory (that is, one whose theorems form a recursively enumerable set) in which the concept of natural numbers can be expressed, can include all true statements about them. As a result, some domains of knowledge cannot be formalized, accurately and completely, as mathematical theories.

Some, on the other hand, may be called "deep", because their proofs may be long and difficult, involve areas of mathematics superficially distinct from the statement of the theorem itself, or show surprising connections between disparate areas of mathematics. A theorem might be simple to state and yet be deep. An excellent example is Fermat's Last Theorem, and there are many other examples of simple yet deep theorems in number theory and combinatorics, among other areas. Other theorems have a known proof that cannot easily be written down.

Specifically, a formal theorem is always the last formula of a derivation in some formal system, each formula of which is a logical consequence of the formulas that came before it in the derivation. The initially-accepted formulas in the derivation are called its axioms, and are the basis on which the theorem is derived. A set of theorems is called a theory. What makes formal theorems useful and interesting is that they can be interpreted as true propositions and their derivations may be interpreted as a proof of the truth of the resulting expression.

However, that argument appears not to acknowledge the distinction between theorems of first-order logic and theorems of higher- order logic. The former can be proven using finistic methods, while the latter — in general — cannot. Tarski's undefinability theorem shows that Gödel numbering can be used to prove syntactical constructs, but not semantic assertions. Therefore, the claim that logicism remains a valid programme may commit one to holding that a system of proof based on the existence and properties of the natural numbers is less convincing than one based on some particular formal system.

In mathematics, affiliated operators were introduced by Murray and von Neumann in the theory of von Neumann algebras as a technique for using unbounded operators to study modules generated by a single vector. Later Atiyah and Singer showed that index theorems for elliptic operators on closed manifolds with infinite fundamental group could naturally be phrased in terms of unbounded operators affiliated with the von Neumann algebra of the group. Algebraic properties of affiliated operators have proved important in L2 cohomology, an area between analysis and geometry that evolved from the study of such index theorems.

In mathematics, an existence theorem is purely theoretical if the proof given for it does not indicate a construction of the object whose existence is asserted. Such a proof is non-constructive, since the whole approach may not lend itself to construction. In terms of algorithms, purely theoretical existence theorems bypass all algorithms for finding what is asserted to exist. These are to be contrasted with the so-called "constructive" existence theorems, which many constructivist mathematicians working in extended logics (such as intuitionistic logic) believe to be intrinsically stronger than their non- constructive counterparts.

Marina Evseevna Ratner (; October 30, 1938 – July 7, 2017) was a professor of mathematics at the University of California, Berkeley who worked in ergodic theory. Around 1990, she proved a group of major theorems concerning unipotent flows on homogeneous spaces, known as Ratner's theorems. Ratner was elected to the American Academy of Arts and Sciences in 1992, awarded the Ostrowski Prize in 1993 and elected to the National Academy of Sciences the same year. In 1994, she was awarded the John J. Carty Award from the National Academy of Sciences.

Network graphs: matrices associated with graphs; incidence, fundamental cut set and fundamental circuit matrices. Solution methods: nodal and mesh analysis. Network theorems: superposition, Thevenin and Norton's maximum power transfer, Wye-Delta transformation. Steady state sinusoidal analysis using phasors.

They are the building blocks of geometric concepts, since they specify the properties that the primitives have. # The laws of logic. # The theoremsIn this context no distinction is made between different categories of theorems. Propositions, lemmas, corollaries, etc.

The theory of non-standard analysis is rich enough to be applied in many branches of mathematics. As such, books and articles dedicated solely to the traditional theorems of calculus often go by the title non-standard calculus.

The first examples of non-projective complete varieties were given by Masayoshi NagataExistence theorems for nonprojective complete algebraic varieties, Illinois J. Math. 2 (1958) 490–498. and Heisuke Hironaka. An affine space of positive dimension is not complete.

Takes in the index k of an inference rule (such as Modus tollens, Modus ponens), and attempts to apply it to the two previously proved theorems m and n. The resulting theorem is then added to the proof.

In order to solve this, an extension of the notions of fuzzy grammar and fuzzy Turing machine are necessary. Another open question is to start from this notion to find an extension of Gödel's theorems to fuzzy logic.

Ideally, the sequence in which the theorems are presented is as easy to understand as possible, although illuminating a branch of mathematics is the purpose of textbooks, rather than the mathematical theory they might be written to cover.

While math dominates much of his art, its role, and how it is utilized vary (metaphor, analogy, play on words, etc). The topics incorporated are also diverse ranging from famous classic results to fundamental 20th-century concepts and theorems.

Among them, his text Calculus (W. A. Benjamin Inc., 1967; Publish or Perish, 4th ed., 2008) takes a rigorous and theoretical approach to introductory calculus and includes proofs of many theorems taken on faith in most other introductory textbooks.

This was in 1935. According to Bohr, this new theory should be probabilistic, whereas according to Einstein it should be deterministic. Notably, the underlying quantum mechanical theory, i.e. the set of "theorems" derived by it, seemed to be identical.

Many of the theorems in Donaldson theory can now be proved more easily using Seiberg-Witten theory, though there are a number of open problems remaining in Donaldson theory, such as the Witten conjecture and the Atiyah–Floer conjecture.

Many mathematical theorems can be reduced to more straightforward computation, including polynomial identities, trigonometric identitiesSuch as the derivation of the formula for \tan (\alpha + \beta) from the addition formulas of sine and cosine. and hypergeometric identities.Petkovsek et al. 1996.

In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow (1872) that give detailed information about the number of subgroups of fixed order that a given finite group contains. The Sylow theorems form a fundamental part of finite group theory and have very important applications in the classification of finite simple groups. For a prime number p, a Sylow p-subgroup (sometimes p-Sylow subgroup) of a group G is a maximal p-subgroup of G, i.e., a subgroup of G that is a p-group (so that the order of every group element is a power of p) that is not a proper subgroup of any other p-subgroup of G. The set of all Sylow p-subgroups for a given prime p is sometimes written Sylp(G).

The field of welfare economics is associated with two fundamental theorems. The first states that given certain assumptions, competitive markets (price equilibria with transfers, e.g. Walrasian equilibria) produce Pareto efficient outcomes. The assumptions required are generally characterised as "very weak".

0- and 2-dimensional Bessel processes are related to local times of Brownian motion via the Ray-Knight theorems. The law of a Brownian motion near x-extrema is the law of a 3-dimensional Bessel process (theorem of Tanaka).

General set theory (GST) is George Boolos's (1998) name for a fragment of the axiomatic set theory Z. GST is sufficient for all mathematics not requiring infinite sets, and is the weakest known set theory whose theorems include the Peano axioms.

In mathematics, particularly in functional analysis, the Krein-Smuilian theorem can refer to two theorems relating the closed convex hull and compactness in the weak topology. They are named after Mark Krein and Vitold Shmulyan, who published them in 1940.

Vasily Sergeyevich Vladimirov (; 9 January 1923 – 3 November 2012) was a Soviet mathematician and mathematical physicist working in the fields of number theory, mathematical physics, quantum field theory, numerical analysis, generalized functions, several complex variables, p-adic analysis, multidimensional tauberian theorems.

We can estimate P, or a related distribution function F by means of the empirical measure or empirical distribution function, respectively. These are uniformly good estimates under certain conditions. Theorems in the area of empirical processes provide rates of this convergence.

Since 1986, λProlog has received numerous implementations. As of 2013, the language and its implementations are still actively being developed. The Abella theorem prover has been designed to provide an interactive environment for proving theorems about the declarative core of λProlog.

If the intended model is infinite and the language is first-order, then the Löwenheim–Skolem theorems guarantee the existence of non-standard models. The non-standard models can be chosen as elementary extensions or elementary substructures of the intended model.

In discrete mathematics, Schur's theorem is any of several theorems of the mathematician Issai Schur. In differential geometry, Schur's theorem is a theorem of Axel Schur. In functional analysis, Schur's theorem is often called Schur's property, also due to Issai Schur.

Network graphs: matrices associated with graphs; incidence, fundamental cut set, and fundamental circuit matrices. Solution methods: nodal and mesh analysis. Network theorems: superposition, Thevenin and Norton's maximum power transfer, Wye-Delta transformation.J. O. Bird Electrical Circuit Theory and Technology, pp.

Victor Lenard Shapiro (16 October 1924, Chicago – 1 March 2013, Riverside, California) was an American mathematician, specializing in trigonometric series and differential equations. He is known for his two theorems (published in 1957) on the uniqueness of multiple Fourier series.

In complex analysis, an area of mathematics, Montel's theorem refers to one of two theorems about families of holomorphic functions. These are named after French mathematician Paul Montel, and give conditions under which a family of holomorphic functions is normal.

Local rigidity theorems in the theory of discrete subgroups of Lie groups are results which show that small deformations of certain such subgroups are always trivial. It is different from Mostow rigidity and weaker (but holds more frequently) than superrigidity.

William Bown. New-wave mathematics: A new generation of mathematicians is rebelling against the ancient tradition of theorem and proof.New Scientist. August 3, 1991 Some critics of the new journal suggested that it be renamed as the "Journal of Unproved Theorems".

And so on indefinitely. The theorem has been compared to Clifford's circle theorems since they both are an infinite chain of theorems. In 1941 Richmond argued that Cox's chain was superior: :Cox's interest lay in the discovery of applications of Grassmann's Ausdehnungslehre and he uses the chain to that end. Any present-day geometer (to whom many of Cox's properties of circles in a plane must appear not a little artificial) would agree that his figure of points and planes in space is simpler and more fundamental than that of circles in a plane which he derives from it.

The rules proposed by Thomas have inspired various mathematicians, who translated them into rigorous theorems, first referring to ordinary differential equations, but also referring to Boolean and multilevel logical formalisms. This is one of the few cases where biological studies led to the formulation and demonstration of general mathematical theorems. The theoretical studies by Thomas on the properties of genetic regulatory circuits were also accompanied by practical considerations regarding the synthesis of novel circuits, with specific properties, in the bacterium E. coli. However, due to various technical problems, the attempts of Thomas' group to build synthetic gene circuits were unsuccessful.

Wolpert had previously derived no free lunch theorems for machine learning (statistical inference). Before Wolpert's article was published, Cullen Schaffer independently proved a restricted version of one of Wolpert's theorems and used it to critique the current state of machine learning research on the problem of induction. In the "no free lunch" metaphor, each "restaurant" (problem-solving procedure) has a "menu" associating each "lunch plate" (problem) with a "price" (the performance of the procedure in solving the problem). The menus of restaurants are identical except in one regard – the prices are shuffled from one restaurant to the next.

These semantics permit a translation between tautologies of propositional logic and equational theorems of Boolean algebra. Every tautology Φ of propositional logic can be expressed as the Boolean equation Φ = 1, which will be a theorem of Boolean algebra. Conversely every theorem Φ = Ψ of Boolean algebra corresponds to the tautologies (Φ∨¬Ψ) ∧ (¬Φ∨Ψ) and (Φ∧Ψ) ∨ (¬Φ∧¬Ψ). If → is in the language these last tautologies can also be written as (Φ→Ψ) ∧ (Ψ→Φ), or as two separate theorems Φ→Ψ and Ψ→Φ; if ≡ is available then the single tautology Φ ≡ Ψ can be used.

The major accomplishment of Hippocrates is that he was the first to write a systematically organized geometry textbook, called Elements (Στοιχεῖα, Stoicheia), that is, basic theorems, or building blocks of mathematical theory. From then on, mathematicians from all over the ancient world could, at least in principle, build on a common framework of basic concepts, methods, and theorems, which stimulated the scientific progress of mathematics. Only a single, famous fragment of Hippocrates' Elements is existent, embedded in the work of Simplicius. In this fragment the area is calculated of some so-called Hippocratic lunes -- see Lune of Hippocrates.

He has worked on mechanism design in various settings including optimal and dynamic contract design, the existence of Nash equilibrium in discontinuous games, auction with participation costs, matching issues, full characterization on fixed-point theorems and related theorems, and China's transition to a modern market economy. He has published over 90 articles in international journals and over 90 articles in Chinese journals. In 2015, Tian received the Sun Yefang Award, the highest honor in economic science in China for his book China's Reform: History, Logic and Future. He received the Annual Great Thinker award by the China Business Network in 2016.

Nevertheless, the minimum length of proofs of theorems will remain unbounded, that is, for any natural number n there will still be theorems which cannot be proved in n or fewer steps. If the new axiom schema is not a tautology, then every formula becomes a theorem (which makes the concept of a theorem useless in this case). What is more, there is then an upper bound on the minimum length of a proof of every formula, because there is a common method for proving every formula. For example, suppose the new axiom schema were ((B→C)→C)→B.

Plastic limit theorems in continuum mechanics provide two bounds that can be used to determine whether material failure is possible by means of plastic deformation for a given external loading scenario. According to the theorems, to find the range within which the true solution must lie, it is necessary to find both a stress field that balances the external forces and a velocity field or flow pattern that corresponds to those stresses. If the upper and lower bounds provided by the velocity field and stress field coincide, the exact value of the collapse load is determined.

Attempts have also been made in the area of artificial intelligence research to create smaller, explicit, new proofs of mathematical theorems from the bottom up using machine reasoning techniques such as heuristic search. Such automated theorem provers have proved a number of new results and found new proofs for known theorems. Additionally, interactive proof assistants allow mathematicians to develop human-readable proofs which are nonetheless formally verified for correctness. Since these proofs are generally human-surveyable (albeit with difficulty, as with the proof of the Robbins conjecture) they do not share the controversial implications of computer-aided proofs-by-exhaustion.

All major theories in population ethics tend to produce counterintuitive results Hilary Greaves, Oxford Professor of Philosophy and Director of the Global Priorities Institute, explains that this is no coincidence, as academics have proved a series of impossibility theorems for the field in recent decades. These impossibility theorems are formal results showing that "for various lists of prima facie intuitively compelling desiderata, ... no axiology can simultaneously satisfy all the desiderata on the list." She concludes that choosing a theory in population ethics comes down to choosing which moral intuition one is least unwilling to give up.

The most important are those relating to algebraic curves and surfaces, especially the short paper Allgemeine Eigenschaften algebraischer Curven. This contains only results, and there is no indication of the method by which they were obtained, so that, according to L. O. Hosse, they are, like Fermat's theorems, riddles to the present and future generations. Eminent analysts succeeded in proving some of the theorems, but it was reserved to Luigi Cremona to prove them all, and that by a uniform synthetic method, in his book on algebraic curves. Other important investigations relate to maxima and minima.

The first result had already been determined by G. Bauer in 1859. The second was new to Hardy, and was derived from a class of functions called hypergeometric series, which had first been researched by Euler and Gauss. Hardy found these results "much more intriguing" than Gauss's work on integrals. After seeing Ramanujan's theorems on continued fractions on the last page of the manuscripts, Hardy said the theorems "defeated me completely; I had never seen anything in the least like them before", and that they "must be true, because, if they were not true, no one would have the imagination to invent them".

The theorem of the gnomon was described as early as in Euclid's Elements (around 300 BC) and there it plays an important role in the derivation of other theorems. It is given as proposition 43 in Book I of the Elements, where it is phrased as a statement about parallelograms without using the term gnomon. The latter is introduced by Euclid as the second definition of the second book of Elements. Further theorems for which the gnomon and its properties play an important role are proposition 6 in Book II, proposition 29 in Book VI and propositions 1 to 4 in Book XIII.

Most universal approximation theorems can be parsed into two classes. The first quantifies the approximation capabilities of neural networks with an arbitrary number of artificial neurons ("arbitrary width" case) and the second focuses on the case with an arbitrary number of hidden layers, each containing a limited number of artificial neurons ("arbitrary depth" case). Universal approximation theorems imply that neural networks can represent a wide variety of interesting functions when given appropriate weights. On the other hand, they typically do not provide a construction for the weights, but merely state that such a construction is possible.

All other assertions (theorems, in the case of mathematics) must be proven with the aid of these basic assumptions. However, the interpretation of mathematical knowledge has changed from ancient times to the modern, and consequently the terms axiom and postulate hold a slightly different meaning for the present day mathematician, than they did for Aristotle and Euclid. The ancient Greeks considered geometry as just one of several sciences, and held the theorems of geometry on par with scientific facts. As such, they developed and used the logico-deductive method as a means of avoiding error, and for structuring and communicating knowledge.

The decision must be made on other grounds. One argument given in favor of using the axiom of choice is that it is convenient to use it because it allows one to prove some simplifying propositions that otherwise could not be proved. Many theorems which are provable using choice are of an elegant general character: every ideal in a ring is contained in a maximal ideal, every vector space has a basis, and every product of compact spaces is compact. Without the axiom of choice, these theorems may not hold for mathematical objects of large cardinality.

In practical terms, this means that theorems of algebra and combinatorics are restricted to countable structures, while theorems of analysis and topology are restricted to separable spaces. Many principles that imply the axiom of choice in their general form (such as "Every vector space has a basis") become provable in weak subsystems of second-order arithmetic when they are restricted. For example, "every field has an algebraic closure" is not provable in ZF set theory, but the restricted form "every countable field has an algebraic closure" is provable in RCA0, the weakest system typically employed in reverse mathematics.

In his second monograph on biquadratic reciprocity Gauss displays some examples and makes conjectures that imply the theorems listed above for the biquadratic character of small primes. He makes some general remarks, and admits there is no obvious general rule at work. He goes on to say > The theorems on biquadratic residues gleam with the greatest simplicity and > genuine beauty only when the field of arithmetic is extended to imaginary > numbers, so that without restriction, the numbers of the form a + bi > constitute the object of study ... we call such numbers integral complex > numbers.Gauss, BQ, § 30, translation in Cox, p.

There will also be primitive terms which are not defined, as they cannot be defined without circularity. For example, one can define a line as a set of points, but to then define a point as the intersection of two lines would be circular. Because of these interesting characteristics of formal systems, Bertrand Russell humorously referred to mathematics as "the field where we don't know what we are talking about, nor whether or not what we say is true". All theorems and corollaries are proven by exploring the implications of the axiomata and other theorems that have previously been developed.

Miller theorems are not only pure mathematical expressions. These arrangements explain important circuit phenomena about modifying impedance (Miller effect, virtual ground, bootstrapping, negative impedance, etc.) and help in designing and understanding various commonplace circuits (feedback amplifiers, resistive and time-dependent converters, negative impedance converters, etc.). The theorems are useful in 'circuit analysis' especially for analyzing circuits with feedback and certain transistor amplifiers at high frequencies. There is a close relationship between Miller theorem and Miller effect: the theorem may be considered as a generalization of the effect and the effect may be thought as of a special case of the theorem.

In category theory, an abstract branch of mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics. Establishing an equivalence involves demonstrating strong similarities between the mathematical structures concerned. In some cases, these structures may appear to be unrelated at a superficial or intuitive level, making the notion fairly powerful: it creates the opportunity to "translate" theorems between different kinds of mathematical structures, knowing that the essential meaning of those theorems is preserved under the translation.

One of the most interesting theorems in this form of integral geometry is Hadwiger's theorem in the Euclidean setting. Subsequently Hadwiger-type theorems were established in various settings, notably in hermitian geometry, using advanced tools from valuation theory. The more recent meaning of integral geometry is that of Sigurdur HelgasonSigurdur Helgason (2000) Groups and Geometric Analysis: integral geometry, invariant differential operators, and spherical functions, American Mathematical Society Sigurdur Helgason (2011) Integral Geometry and Radon Transforms, Springer, and Israel Gelfand.I.M. Gel’fand (2003) Selected Topics in Integral Geometry, American Mathematical Society It deals more specifically with integral transforms, modeled on the Radon transform.

Theorems in mathematics and theories in science are fundamentally different in their epistemology. A scientific theory cannot be proved; its key attribute is that it is falsifiable, that is, it makes predictions about the natural world that are testable by experiments. Any disagreement between prediction and experiment demonstrates the incorrectness of the scientific theory, or at least limits its accuracy or domain of validity. Mathematical theorems, on the other hand, are purely abstract formal statements: the proof of a theorem cannot involve experiments or other empirical evidence in the same way such evidence is used to support scientific theories.

Gödel's result suggests that in order to maintain a logicist position, while still retaining as much as possible of classical mathematics, one must accept some axiom of infinity as part of logic. On the face of it, this damages the logicist programme also, albeit only for those already doubtful concerning 'infinitary methods'. Nonetheless, positions deriving from both logicism and from Hilbertian finitism have continued to be propounded since the publication of Gödel's result. One argument that programmes derived from logicism remain valid might be that the incompleteness theorems are 'proved with logic just like any other theorems'.

Pures Appl., 68 (1989), p. 261–295Ciarlet, P.G., Plates and Junctions in Elastic Multi-Structures : An Asymptotic Analysis, Paris et Heidelberg, Masson & Springer-Verlag, 1990 Modeling and mathematical analysis of "general" shells: Philippe Ciarlet established the first existence theorems for two-dimensional linear shell models, such as those of W.T. Koiter and P.M. Naghdi,Bernadou, M. ; Ciarlet, P.G. ; Miara, B., « Existence theorems for two- dimensional linear shell theories », J. Elasticity, 34 (1994), p. 111–138 and justified the equations of the "bending" and "membrane" shell;Ciarlet, P.G. ; Lods, V., « Asymptotic analysis of linearly elastic shells. I. Justification of membrane shells equations », Arch.

By the fundamental theorems of welfare economics, any CE allocation is Pareto efficient, and any efficient allocation can be sustainable by a competitive equilibrium. Furthermore, by Varian's theorems, a CE allocation in which all agents have the same income is also envy-free. At the competitive equilibrium, the value society places on a good is equivalent to the value of the resources given up to produce it (marginal benefit equals marginal cost). This ensures allocative efficiency: the additional value society places on another unit of the good is equal to what society must give up in resources to produce it.

Conversely, for many deductive systems, it is possible to prove the completeness theorem as an effective consequence of the compactness theorem. The ineffectiveness of the completeness theorem can be measured along the lines of reverse mathematics. When considered over a countable language, the completeness and compactness theorems are equivalent to each other and equivalent to a weak form of choice known as weak König's lemma, with the equivalence provable in RCA0 (a second- order variant of Peano arithmetic restricted to induction over Σ01 formulas). Weak König's lemma is provable in ZF, the system of Zermelo–Fraenkel set theory without axiom of choice, and thus the completeness and compactness theorems for countable languages are provable in ZF. However the situation is different when the language is of arbitrary large cardinality since then, though the completeness and compactness theorems remain provably equivalent to each other in ZF, they are also provably equivalent to a weak form of the axiom of choice known as the ultrafilter lemma.

With the definitions of integration and derivatives, key theorems can be formulated, including the fundamental theorem of calculus integration by parts, and Taylor's theorem. Evaluating a mixture of integrals and derivatives can be done by using theorem differentiation under the integral sign.

The advisor of many PhD. His main research areas are measure theory, functional analysis, foundations of mathematics and probability theory. Several theorems bear his name: the Ryll-Nardzewski fixed point theorem, “9. Theorem of Ryll-Nardzewski” (p. 171), “(9.6) Theorem (Ryll- Nardzewski)” (p.

Review by Bojan Mohar (1995), .Review by Wessel (1996), Journal of Applied Mathematics and Mechanics 76 (10): 144, .Review by P. Rowlinson (1996), Proceedings of the Edinburgh Mathematical Society (Series 2) 39: 188–189, . Two theorems in graph theory bear his name.

Theorems and results within analytic number theory tend not to be exact structural results about the integers, for which algebraic and geometrical tools are more appropriate. Instead, they give approximate bounds and estimates for various number theoretical functions, as the following examples illustrate.

Holden defends his actions with an interpretation of Gödel's incompleteness theorems: to understand our own humanity we must study what is as radically different as possible. The Ihrdizu are much too similar to humans in their biology and mentality to serve this purpose.

It is among the most notable theorems in the history of mathematics and prior to its proof was in the Guinness Book of World Records as the "most difficult mathematical problem" in part because the theorem has the largest number of unsuccessful proofs.

DCC may not be very effective on theorems with only propositional clauses. How does DCC work? After every application of an inference rule, certain variables may have to be substituted by terms (e.g. x-> f(a)) and thus a substitution set is formed.

In quantum information theory, the idea of a typical subspace plays an important role in the proofs of many coding theorems (the most prominent example being Schumacher compression). Its role is analogous to that of the typical set in classical information theory.

Historically the theorems above were pointers to the following result, at one time known as the \alpha + \beta hypothesis. It was used by Edmund Landau and was finally proved by Henry Mann in 1942. > Theorem. Let A and B be subsets of \N.

Enderton, Herbert. Elements of Set Theory. Academic Press. 1977. Russell wrote (in Portraits from Memory, 1956) of his reaction to Gödel's 'Theorems of Undecidability': Evidence of Russell's influence on Wittgenstein can be seen throughout the Tractatus, which Russell was instrumental in having published.

See fixed-point theorems in infinite-dimensional spaces. The collage theorem in fractal compression proves that, for many images, there exists a relatively small description of a function that, when iteratively applied to any starting image, rapidly converges on the desired image.

Automated theorem proving (also known as ATP or automated deduction) is a subfield of automated reasoning and mathematical logic dealing with proving mathematical theorems by computer programs. Automated reasoning over mathematical proof was a major impetus for the development of computer science.

Manevitz, Larry M.; Weinberger, Shmuel: Discrete circle actions: a note using nonstandard analysis. Israel J. Math. 94 (1996), 147--155. The real contributions of nonstandard analysis lie however in the concepts and theorems that utilize the new extended language of nonstandard set theory.

Magda Peligrad is a Romanian mathematician and mathematical statistician known for her research in probability theory, and particularly on central limit theorems and stochastic processes. She works at the University of Cincinnati, where she is Distinguished Charles Phelps Taft Professor of Mathematical Sciences.

Description and contents preview. Economic equilibrium is studied in optimization theory as a key ingredient of economic theorems that in principle could be tested against empirical data.• Samuelson, Paul A., 1998. "How Foundations Came to Be", Journal of Economic Literature, 36(3), pp. 1375–1386.

56 (2000), 167-188. Theorems 5.2 and 5.4. In this sense, X is decomposed into a family of varieties of Kodaira dimension zero over a base (B, Δ) of general type. (Note that the variety B by itself need not be of general type.

In mathematics, a pre-measure is a function that is, in some sense, a precursor to a bona fide measure on a given space. Indeed, one of the fundamental theorems in measure theory states that a pre-measure can be extended to a measure.

Following Uzawa's theorem, many mathematical economists consider proving existence a deeper result than proving the two Fundamental Theorems. Another method of proof of existence, global analysis, uses Sard's lemma and the Baire category theorem; this method was pioneered by Gérard Debreu and Stephen Smale.

Similar theorems describe the degree sequences of simple directed graphs, simple directed graphs with loops, and simple bipartite graphs . The first problem is characterized by the Fulkerson–Chen–Anstee theorem. The latter two cases, which are equivalent, are characterized by the Gale–Ryser theorem.

The blocks method helps proving limit theorems in the case of dependent random variables. The blocks method was introduced by S. Bernstein: Bernstein S.N. (1926) Sur l'extension du théorème limite du calcul des probabilités aux sommes de quantités dépendantes. Math. Annalen, v. 97, 1-59.

In 1917, Bernstein suggested the first axiomatic foundation of probability theory, based on the underlying algebraic structure. It was later superseded by the measure-theoretic approach of Kolmogorov. In the 1920s, he introduced a method for proving limit theorems for sums of dependent random variables.

Some have been answered definitively; some have not yet been solved; a few have been shown to be impossible to answer with mathematical rigor. In 1931, Gödel's incompleteness theorems showed that some mathematical questions cannot be answered in the manner we would usually prefer.

Knowledge about the structure of the group can be obtained by studying the adjacency matrix of the graph and in particular applying the theorems of spectral graph theory. The genus of a group is the minimum genus for any Cayley graph of that group.

In an affine space such as the Euclidean plane a similar statement is true, but only if one lists various exceptions involving parallel lines. Desargues's theorem is therefore one of the simplest geometric theorems whose natural home is in projective rather than affine space.

Generalizations of the Farkas' lemma are about the solvability theorem for convex inequalities, i.e., infinite system of linear inequalities. Farkas' lemma belongs to a class of statements called "theorems of the alternative": a theorem stating that exactly one of two systems has a solution.

The above theorems are very useful to design approximation algorithms for NP-hard problems, such as the graph partition problem and its variations. Here below we briefly introduce a few examples, and the in-depth elaborations can be found in Leighton and Rao (1999).

More expressive logics, such as Higher-order logics, allow the convenient expression of a wider range of problems than first order logic, but theorem proving for these logics is less well developed.Kerber, Manfred. "How to prove higher order theorems in first order logic." (1999).

Hastad and Näslund proved KFB to be secure in terms of the theorems of complexity theory introduced by Blum, Micali, Levin and Goldreich, giving a quantitative relation between the effort of distinguishing the keystream from true randomness to the effort of retrieving the secret key.

This type of decomposition of a distribution is used in probability and statistics to find families of probability distributions that might be natural choices for certain models or applications. Infinitely divisible distributions play an important role in probability theory in the context of limit theorems.

Similar theorems describe the degree sequences of simple graphs, simple directed graphs with loops, and simple bipartite graphs. The first problem is characterized by the Erdős–Gallai theorem. The latter two cases, which are equivalent see Berger, are characterized by the Gale–Ryser theorem.

These include (but are not limited to) entanglement, noncommutativity of measurements, teleportation, interference, the no-cloning and no-broadcasting theorems, and unsharp measurements. The toy model cannot, however, reproduce quantum nonlocality and quantum contextuality, as it is a local and non-contextual hidden-variable theory.

The situation with representations over a field of positive characteristic, so- called "modular representations", is more delicate, but Richard Brauer developed a powerful theory of characters in this case as well. Many deep theorems on the structure of finite groups use characters of modular representations.

Theorem provers use automated reasoning techniques to determine proofs of mathematical theorems. They may also be used to verify existing proofs. In addition to academic use, typical applications of theorem provers include verification of the correctness of integrated circuits, software programs, engineering designs, etc.

In the 1960s topological interpretations of class field theory were given by John TateJ. Tate, Duality theorems in Galois cohomology over number fields, (Proc. Intern. Cong. Stockholm, 1962, p. 288-295). based on Galois cohomology, and also by Michael Artin and Jean-Louis VerdierM.

These assumptions are the elementary theorems of the particular theory, and can be thought of as the axioms of that field. Some commonly known examples include set theory and number theory; however literary theory, critical theory, and music theory are also of the same form.

His pedagogy was influenced by the ideas of N. F. S. Grundtvig, where theology and learning was seen as a voluntary act, and obligatory exams were replaced by voluntary self- evaluation. He was also a spokesman of the theorems of American economist Henry George.

Kőnig's theorem is equivalent to numerous other min-max theorems in graph theory and combinatorics, such as Hall's marriage theorem and Dilworth's theorem. Since bipartite matching is a special case of maximum flow, the theorem also results from the max-flow min-cut theorem.

In number theory, Mertens' theorems are three 1874 results related to the density of prime numbers proved by Franz Mertens.F. Mertens. J. reine angew. Math. 78 (1874), 46–62 Ein Beitrag zur analytischen Zahlentheorie "Mertens' theorem" may also refer to his theorem in analysis.

Systems for which the Poincaré recurrence theorem holds are conservative systems; thus all ergodic systems are conservative. More precise information is provided by various ergodic theorems which assert that, under certain conditions, the time average of a function along the trajectories exists almost everywhere and is related to the space average. Two of the most important theorems are those of Birkhoff (1931) and von Neumann which assert the existence of a time average along each trajectory. For the special class of ergodic systems, this time average is the same for almost all initial points: statistically speaking, the system that evolves for a long time "forgets" its initial state.

Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries. Perhaps the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, analysis, differential geometry, matrix theory, number theory, and statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show in part that any consistent axiomatic system—if powerful enough to describe arithmetic—will contain true propositions that cannot be proved. Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both.

Pseudo-reductive groups arise naturally in the study of algebraic groups over function fields of positive- dimensional varieties in positive characteristic (even over a perfect field of constants). gives an exposition of Tits' results on pseudo-reductive groups, while builds on Tits' work to develop a general structure theory, including more advanced topics such as construction techniques, root systems and root groups and open cells, classification theorems, and applications to rational conjugacy theorems for smooth connected affine groups over arbitrary fields. The general theory (with applications) as of 2010 is summarized in , and later work in the second edition and in provides further refinements.

In one instance Iyer submitted some of Ramanujan's theorems on summation of series to the journal, adding, "The following theorem is due to S. Ramanujan, the mathematics student of Madras University." Later in November, British Professor Edward B. Ross of Madras Christian College, whom Ramanujan had met a few years before, stormed into his class one day with his eyes glowing, asking his students, "Does Ramanujan know Polish?" The reason was that in one paper, Ramanujan had anticipated the work of a Polish mathematician whose paper had just arrived in the day's mail. In his quarterly papers Ramanujan drew up theorems to make definite integrals more easily solvable.

One of Wolpert’s most discussed achievements is known as No free lunch in search and optimization.Wolpert, D.H., Macready, W.G. (1995), No Free Lunch Theorems for Search, Technical Report SFI-TR-95-02-010 (Santa Fe Institute).Wolpert, David (1996), The Lack of A Priori Distinctions between Learning Algorithms, Neural Computation, pp. 1341–1390.David H. Wolpert, What the No Free Lunch Theorems Really Mean; How to Improve Search Algorithms, SFI Working Paper 2012-10-017, Santa Fe Institute 2012 By this theorem, all algorithms for search and optimization perform equally well averaged over all problems in the class with which they are designed to deal.

Because of the connection between separators and expansion, every minor-closed graph family, including the family of planar graphs, has polynomial expansion. The same is true for 1-planar graphs, and more generally the graphs that can be embedded onto surfaces of bounded genus with a bounded number of crossings per edge, as well as the biclique-free string graphs, since these all obey similar separator theorems to the planar graphs., 14.2 Crossing Number, pp. 319–321.... In higher dimensional Euclidean spaces, intersection graphs of systems of balls with the property that any point of space is covered by a bounded number of balls also obey separator theorems.

The parallel postulate (Postulate 5): If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. Euclidean geometry is an axiomatic system, in which all theorems ("true statements") are derived from a small number of simple axioms. Until the advent of non-Euclidean geometry, these axioms were considered to be obviously true in the physical world, so that all the theorems would be equally true. However, Euclid's reasoning from assumptions to conclusions remains valid independent of their physical reality.

Informally, model theory can be divided into classical model theory, model theory applied to groups and fields, and geometric model theory. A missing subdivision is computable model theory, but this can arguably be viewed as an independent subfield of logic. Examples of early theorems from classical model theory include Gödel's completeness theorem, the upward and downward Löwenheim–Skolem theorems, Vaught's two-cardinal theorem, Scott's isomorphism theorem, the omitting types theorem, and the Ryll-Nardzewski theorem. Examples of early results from model theory applied to fields are Tarski's elimination of quantifiers for real closed fields, Ax's theorem on pseudo-finite fields, and Robinson's development of non-standard analysis.

It has been claimed that formalists, such as David Hilbert (1862-1943), hold that mathematics is only a language and a series of games. Indeed, he used the words "formula game" in his 1927 response to L. E. J. Brouwer's criticisms: Thus Hilbert is insisting that mathematics is not an arbitrary game with arbitrary rules; rather it must agree with how our thinking, and then our speaking and writing, proceeds. The foundational philosophy of formalism, as exemplified by David Hilbert, is a response to the paradoxes of set theory, and is based on formal logic. Virtually all mathematical theorems today can be formulated as theorems of set theory.

The incompleteness theorems apply to formal systems that are of sufficient complexity to express the basic arithmetic of the natural numbers and which are consistent, and effectively axiomatized, these concepts being detailed below. Particularly in the context of first-order logic, formal systems are also called formal theories. In general, a formal system is a deductive apparatus that consists of a particular set of axioms along with rules of symbolic manipulation (or rules of inference) that allow for the derivation of new theorems from the axioms. One example of such a system is first-order Peano arithmetic, a system in which all variables are intended to denote natural numbers.

Although Gödel's theorems are usually studied in the context of classical logic, they also have a role in the study of paraconsistent logic and of inherently contradictory statements (dialetheia). Graham Priest (1984, 2006) argues that replacing the notion of formal proof in Gödel's theorem with the usual notion of informal proof can be used to show that naive mathematics is inconsistent, and uses this as evidence for dialetheism. The cause of this inconsistency is the inclusion of a truth predicate for a system within the language of the system (Priest 2006:47). Stewart Shapiro (2002) gives a more mixed appraisal of the applications of Gödel's theorems to dialetheism.

Theories are distinct from theorems. A theorem is derived deductively from axioms (basic assumptions) according to a formal system of rules, sometimes as an end in itself and sometimes as a first step toward being tested or applied in a concrete situation; theorems are said to be true in the sense that the conclusions of a theorem are logical consequences of the axioms. Theories are abstract and conceptual, and are supported or challenged by observations in the world. They are 'rigorously tentative', meaning that they are proposed as true and expected to satisfy careful examination to account for the possibility of faulty inference or incorrect observation.

Various studies suggest that the drawing experts knew specific rules of "chaining" and "elimination" relating to the systematic construction of monolinear figures. Studies suggest that the "drawing experts" who invented these rules knew why they were valid, and could prove in one way or another the validity of the theorems that these rules express. It is difficult to find accounts of theorems developed by the drawing experts to generalize specific patterns relating to dimension and monolinearity/polylinearity, as this tradition was secret and in extinction when it started to be recorded. However, the drawing experts possibly knew that rectangles with relatively prime dimensions give one-line drawings.

Georg Cantor, 1870 Cantor's first set theory article contains Georg Cantor's first theorems of transfinite set theory, which studies infinite sets and their properties. One of these theorems is his "revolutionary discovery" that the set of all real numbers is uncountably, rather than countably, infinite.. This theorem is proved using Cantor's first uncountability proof, which differs from the more familiar proof using his diagonal argument. The title of the article, "On a Property of the Collection of All Real Algebraic Numbers" ("Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen"), refers to its first theorem: the set of real algebraic numbers is countable. Cantor's article was published in 1874.

In his 1941 book The calculus of extensions, Henry Forder published numerous examples in vector analysis taken from Genese's posthumous notes. (Genese's notes were left to the Mathematical Association and then given in 1929 to Forder by E. H. Neville.) Genese was an Invited Speaker of the ICM in 1904 in Heidelberg with talk On some useful theorems in the continued multiplication of a regressive product in real four-point spaceOn some useful theorems in the continued multiplication of a regressive product in real four- point space, Proceedings of the ICM in 1904, Heidelberg and in 1908 in Rome with talk The method of reciprocal polars applied to forces in space.

The magnetic fields of planets with slow rotation periods and/or solid cores, such as Mercury, Venus and Mars, have dissipated to almost nothing by comparison. The impact of the known anti-dynamo theorems is that successful dynamos do not possess a high degree of symmetry.

Geometry from the Land of the Incas is divided into nine major categories: Geometry theorems and problems, Inca Geometry, Quizzes, Puzzles, Quotations, Inspiration, Landscapes, Mindmaps, and Geometric art. This site is loaded with advertisements and very little, if any, interesting exploration of the Inca's understanding of geometry.

379–423, 623–656. Hugh Everett Theory of the Universal Wavefunction, Thesis, Princeton University, (1956, 1973), Appendix I, pp 121 ff. In his thesis, Everett used the term "detailed balance" unconventionally, instead of balance equation These theorems may be considered as simplifications of the Boltzmann result.

In abstract algebra, the fundamental theorem on homomorphisms, also known as the fundamental homomorphism theorem, relates the structure of two objects between which a homomorphism is given, and of the kernel and image of the homomorphism. The homomorphism theorem is used to prove the isomorphism theorems.

It follows that the original "no free lunch" theorem does not apply to what can be stored in a physical computer; instead the so-called "tightened" no free lunch theorems need to be applied. It has also been shown that NFL results apply to incomputable functions .

Shalev (partly with Larsen) has been researching the behavior of word maps on groups, proving Waring type theorems; he also proved, together with Liebeck, O’Brien and Tiep, the Ore conjecture from 1951, according to which every element in a non-cyclic finite simple group is a commutator.

Condition of asymptotic completeness in the Delta-compactness theorem is satisfied by uniformly convex Banach spaces, and more generally, by uniformly rotund metric spaces as defined by J. Staples.J. Staples, Fixed point theorems in uniformly rotund metric spaces, Bull. Austral. Math. Soc. 14 (1976), 181–192.

All the above conjectures and theorems are consequences of the unproven extension of Baker's theorem, that logarithms of algebraic numbers that are linearly independent over the rational numbers are automatically algebraically independent too. The diagram on the right shows the logical implications between all these results.

Alternatively, one may restrict or forbid the use of some of the structural rules. This yields a variety of substructural logic systems. They are generally weaker than LK (i.e., they have fewer theorems), and thus not complete with respect to the standard semantics of first-order logic.

He worked on functional analysis, harmonic analysis, ergodic theory, mean value theorems, and numerical integration. Eberlein also worked on spacetime models, internal symmetries in gauge theory, and spinors. His name is attached to the Eberlein–Šmulian theorem in functional analysis. and the Eberlein compacta in topology..

Although the definition of a manifold does not require that its model space should be , this choice is the most common, and almost exclusive one in differential geometry. On the other hand, Whitney embedding theorems state that any real differentiable -dimensional manifold can be embedded into .

Nearly all of the important theorems in the traditional theory of the Lebesgue integral, such as Lebesgue's dominated convergence theorem, the Riesz–Fischer theorem, Fatou's lemma, and Fubini's theorem may also readily be proved using this construction. Its properties are identical to the traditional Lebesgue integral.

In his honor, completely regular topological spaces are also named Tychonoff spaces. In mathematical physics, he proved the fundamental uniqueness theorems for the heat equation and studied Volterra integral equations. He founded the theory of asymptotic analysis for differential equations with small parameter in the leading derivative.

Errett Albert Bishop (July 14, 1928 – April 14, 1983)UCSD Obituary was an American mathematician known for his work on analysis. He expanded constructive analysis in his 1967 Foundations of Constructive Analysis, where he proved most of the important theorems in real analysis by constructive methods.

In modern times, due to wider international cooperation in mathematics, the wider world has taken notice of the work. For example, both Oxford University and the Royal Society of Great Britain have given attribution to pioneering mathematical theorems of Indian origin that predate their Western counterparts.

In mathematics, the Schneider–Lang theorem is a refinement by of a theorem of about the transcendence of values of meromorphic functions. The theorem implies both the Hermite–Lindemann and Gelfond–Schneider theorems, and implies the transcendence of some values of elliptic functions and elliptic modular functions.

It is a time and resource-consuming strategy, affecting performance. The scope is known. It cannot be successful if not supported by other strategies. Claude Shannon's theorems show that if the encryption key is smaller than the secured information, the information-theoretic security can not be achieved.

1.1; Conrad (2014), Theorems 6.1.16 and 6.1.17. This statement includes the existence of Chevalley groups as group schemes over Z, and it says that every split reductive group over a scheme S is isomorphic to the base change of a Chevalley group from Z to S.

In mathematics, more specifically in functional analysis, a subset T of a topological vector space X is said to be a total subset of X if the linear span of T is a dense subset of X. This condition arises frequently in many theorems of functional analysis.

However, in the case of the two graphs K5 and K3,3, it is straightforward to prove that a graph that has at least one of these two graphs as a minor also has at least one of them as a subdivision, so the two theorems are equivalent..

The single cell experiments used intracranial electrodes in the medial temporal lobe (the hippocampus and surrounding cortex). Modern development of concentration of measure theory (stochastic separation theorems) with applications to artificial neural networks give mathematical background to unexpected effectiveness of small neural ensembles in high-dimensional brain.

Later results, particularly of Alan Baker, changed the position. Qualitatively speaking, Baker's theorems look weaker, but they have explicit constants and can actually be applied, in conjunction with machine computation, to prove that lists of solutions (suspected to be complete) are actually the entire solution set.

To quote Nagel and Newman (p. 68), "Gödel's paper is difficult. Forty-six preliminary definitions, together with several important preliminary theorems, must be mastered before the main results are reached" (p. 68). In fact, Nagel and Newman required a 67-page introduction to their exposition of the proof.

Some concepts in math with specific aesthetic application include sacred ratios in geometry, the intuitiveness of axioms, the complexity and intrigue of fractals, the solidness and regularity of polyhedra, and the serendipity of relating theorems across disciplines. There is a developed aesthetic and theory of humor in mathematical humor.

The lemma states that, under certain conditions, an event will have probability of either zero or one. Accordingly, it is the best-known of a class of similar theorems, known as zero-one laws. Other examples include Kolmogorov's zero–one law and the Hewitt–Savage zero–one law.

By changing the integration variable from t to it, Abel found that . This elliptic function could thus be found for purely imaginary values of the argument. In particular one has . Using the addition theorems one can then calculate the functions for a general complex argument of the form .

Elements of vector calculus: divergence and curl; Gauss' and Stokes' theorems, Maxwell's equations: differential and integral forms. Wave equation, Poynting vector. Plane waves: propagation through various media; reflection and refraction; phase and group velocity; skin depth. Transmission lines: characteristic impedance; impedance transformation; Smith chart; impedance matching; pulse excitation.

The geometry of the fourth space has no > more secret for me. Previously I had only intuitions, now I have certainty. > I have made a whole series of theorems on the laws of displacement > [déplacement], of reversal [retournement] etc. I have read Schoute, Rieman > (sic), Argand, Schlegel etc.

An important variation is the truncated moment problem, which studies the properties of measures with fixed first k moments (for a finite k). Results on the truncated moment problem have numerous applications to extremal problems, optimisation and limit theorems in probability theory. See also: Chebyshev–Markov–Stieltjes inequalities and .

The development of the field of Tauberian theorems received a fresh turn with Norbert Wiener's very general results, namely Wiener's Tauberian theorem and its large collection of corollaries. The central theorem can now be proved by Banach algebra methods, and contains much, though not all, of the previous theory.

One of the applications of cyclic homology is to find new proofs and generalizations of the Atiyah-Singer index theorem. Among these generalizations are index theorems based on spectral triplesAlain Connes and Henri Moscovici. The local index formula in noncommutative geometry. Geom. Funct. Anal., 5(2):174–243, 1995.

The geometry of the fourth space has no > more secret for me. Previously I had only intuitions, now I have certainty. > I have made a whole series of theorems on the laws of displacement > [déplacement], of reversal [retournement] etc. I have read Schoute, Rieman > (sic), Argand, Schlegel etc.

In functional analysis, a branch of mathematics, a selection theorem is a theorem that guarantees the existence of single-valued selection function from a given multi-valued map. There are various selection theorems, and they are important in the theories of differential inclusions, optimal control, and mathematical economics.

Algebraically, hyperbolic and spherical geometry have the same structure. This allows us to apply concepts and theorems to one geometry to the other. Applying hyperbolic geometry to spherical geometry can make it easier to understand because spheres are much more concrete, which then makes spherical geometry easier to conceptualize.

In mathematics and logic, a direct proof is a way of showing the truth or falsehood of a given statement by a straightforward combination of established facts, usually axioms, existing lemmas and theorems, without making any further assumptions.Cupillari, Antonella. The Nuts and Bolts of Proofs. Academic Press, 2001.

The Darmois–Skitovich theorem is one of the most famous characterization theorems of mathematical statistics. It characterizes the normal distribution (the Gaussian distribution) by the independence of two linear forms from independent random variables. This theorem was proved independently by G. Darmois and V. P. Skitovich in 1953.

As another example, the inscribed angle theorem is the basis for several theorems related to the power of a point with respect to a circle. Further, it allows one to prove that when two chords intersect in a circle, the products of the lengths of their pieces are equal.

Elements of vector calculus: divergence and curl; Gauss' and Stokes' theorems, Maxwell's equations: differential and integral forms. Wave equation, Poynting vector. Plane waves: propagation through various media; reflection and refraction; phase and group velocity; skin depth. Transmission lines: characteristic impedance; impedance transformation; Smith chart; impedance matching; pulse excitation.

Theorem 5 of the Brillhart, Lehmer, and Selfridge paper allows a primality proof when the factored part has reached only (N/2)^{1/3}. Many additional such theorems are presented that allow one to prove the primality of based on the partial factorization of N - 1 and N + 1.

A palimpsest stolen from the Greek Orthodox Church in the early 20th century, which reappeared at auction in 1998, contained many of Archimedes works, including The Method of Mechanical Theorems, in which he describes a method to determine volumes which involves balances, centers of mass and infinitesimal slices.

The compactness theorem first appeared as a lemma in Gödel's proof of the completeness theorem, and it took many years before logicians grasped its significance and began to apply it routinely. It says that a set of sentences has a model if and only if every finite subset has a model, or in other words that an inconsistent set of formulas must have a finite inconsistent subset. The completeness and compactness theorems allow for sophisticated analysis of logical consequence in first-order logic and the development of model theory, and they are a key reason for the prominence of first-order logic in mathematics. Gödel's incompleteness theorems (Gödel 1931) establish additional limits on first- order axiomatizations.

Nikolay Krylov developed new methods for analysis of equations of mathematical physics, which can be used not only for proving the existence of solutions but also for their construction. Since 1932, he worked together with his student Nikolay Bogoliubov on mathematical problems of non-linear mechanics. In this period, they invented certain asymptotic methods for integration of non-linear differential equations, studied dynamical systems, and made significant contributions to the foundations of non-linear mechanics. They proved the first theorems on existence of invariant measures known as Krylov–Bogolyubov theorems, introduced the Krylov-Bogoliubov averaging method and, together with Yurii Mitropolskiy, developed the Krylov–Bogoliubov–Mitropolskiy asymptotic method for approximate solving equations of non-linear mechanics.

The isomorphism theorems were formulated in some generality for homomorphisms of modules by Emmy Noether in her paper Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern, which was published in 1927 in Mathematische Annalen. Less general versions of these theorems can be found in work of Richard Dedekind and previous papers by Noether. Three years later, B.L. van der Waerden published his influential Algebra, the first abstract algebra textbook that took the groups-rings-fields approach to the subject. Van der Waerden credited lectures by Noether on group theory and Emil Artin on algebra, as well as a seminar conducted by Artin, Wilhelm Blaschke, Otto Schreier, and van der Waerden himself on ideals as the main references.

In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of A is B, then the dual of B is A. Such involutions sometimes have fixed points, so that the dual of A is A itself. For example, Desargues' theorem is self-dual in this sense under the standard duality in projective geometry. In mathematical contexts, duality has numerous meanings. It has been described as "a very pervasive and important concept in (modern) mathematics" and "an important general theme that has manifestations in almost every area of mathematics".

Then he used these theorems to give rigorous proofs of theorems proven by Fisher and Hotelling related to Fisher's maximum likelihood estimator for estimating a parameter of a distribution. After writing a series of papers on the foundations of probability and stochastic processes including martingales, Markov processes, and stationary processes, Doob realized that there was a real need for a book showing what is known about the various types of stochastic processes, so he wrote the book Stochastic Processes.Doob J.L., Stochastic Processes It was published in 1953 and soon became one of the most influential books in the development of modern probability theory. Beyond this book, Doob is best known for his work on martingales and probabilistic potential theory.

The incompleteness theorem is sometimes thought to have severe consequences for the program of logicism proposed by Gottlob Frege and Bertrand Russell, which aimed to define the natural numbers in terms of logic (Hellman 1981, p. 451–468). Bob Hale and Crispin Wright argue that it is not a problem for logicism because the incompleteness theorems apply equally to first order logic as they do to arithmetic. They argue that only those who believe that the natural numbers are to be defined in terms of first order logic have this problem. Many logicians believe that Gödel's incompleteness theorems struck a fatal blow to David Hilbert's second problem, which asked for a finitary consistency proof for mathematics.

In common mathematical parlance, a mathematical result is called folklore if it is an unpublished result with no clear originator, but which is well- circulated and believed to be true among the specialists. More specifically, folk mathematics, or mathematical folklore, is the body of theorems, definitions, proofs, facts or techniques that circulate among mathematicians by word of mouth, but have not yet appeared in print, either in books or in scholarly journals. Knowledge of folklore is the coin of the realm of academic mathematics. Quite important at times for researchers are folk theorems, which are results known, at least to experts in a field, and are considered to have established status, though not published in complete form.

Its significance was not recognised until 1906, when it was examined by Danish professor Johan Ludvig Heiberg. The palimpsest contained an extended version of Stomachion, and a treatise entitled The Method of Mechanical Theorems that had previously been thought lost. These works have been a focus of research by later scholars.

The property of completeness is crucial in advanced treatments and applications of quantum mechanics. For instance, the existence of projection operators or orthogonal projections relies on the completeness of the space. These projection operators, in turn, are essential for the statement and proof of many useful theorems, e.g. the spectral theorem.

Definitions and properties of Laplace transform, continuous-time and discrete-time Fourier series, continuous-time and discrete-time Fourier Transform, z-transform. Sampling theorems. Linear Time-Invariant Systems: definitions and properties; casualty, stability, impulse response, convolution, poles and zeros frequency response, group delay, phase delay. Signal transmission through LTI systems.

468 The year 1908 was important for Peano. The fifth and final edition of the Formulario project, titled Formulario mathematico, was published. It contained 4200 formulae and theorems, all completely stated and most of them proved. The book received little attention since much of the content was dated by this time.

A bipartite graph B = (X,Y,E) is chordal bipartite if and only if every induced subgraph of B has a maximum X-neighborhood ordering and a maximum Y-neighborhood ordering. Various results describe the relationship between chordal bipartite graphs and totally balanced neighborhood hypergraphs of bipartite graphs. , Theorems 8.2.5, 8.2.

Gödel's theorems do not hold when any one of the seven axioms above is dropped. These fragments of Q remain undecidable, but they are no longer essentially undecidable: they have consistent decidable extensions, as well as uninteresting models (i.e., models which are not end-extensions of the standard natural numbers).

The use of this fact forms the basis of a proof technique called proof by contradiction, which mathematicians use extensively to establish the validity of a wide range of theorems. This applies only in a logic where the law of excluded middle A\vee eg A is accepted as an axiom.

In all of the following theorems we assume some local behavior of the space (usually formulated using curvature assumption) to derive some information about the global structure of the space, including either some information on the topological type of the manifold or on the behavior of points at "sufficiently large" distances.

Fermat was not the first mathematician so moved to write in his own marginal notes to Diophantus; the Byzantine scholar John Chortasmenos (1370–1437) had written "Thy soul, Diophantus, be with Satan because of the difficulty of your other theorems and particularly of the present theorem" next to the same problem.

René-Louis Baire (; 21 January 1874 – 5 July 1932) was a French mathematician most famous for his Baire category theorem, which helped to generalize and prove future theorems. His theory was published originally in his dissertation Sur les fonctions de variable réelles ("On the Functions of Real Variables") in 1899.

A Sangaku dedicated to Konnoh Hachimangu (Shibuya, Tokyo) in 1859. Sangaku or San Gaku (算額; lit. translation: calculation tablet) are Japanese geometrical problems or theorems on wooden tablets which were placed as offerings at Shinto shrines or Buddhist temples during the Edo period by members of all social classes.

Ion Pătraşcu (2010), A generalization of Kosnita's theorem (in Romanian) John Rigby (1997), Brief notes on some forgotten geometrical theorems. Mathematics and Informatics Quarterly, volume 7, pages 156-158 (as cited by Kimberling). Darij Grinberg (2003), On the Kosnita Point and the Reflection Triangle. Forum Geometricorum, volume 3, pages 105–111.

Mathematical statements have their own moderately complex taxonomy, being divided into axioms, conjectures, propositions, theorems, lemmas and corollaries. And there are stock phrases in mathematics, used with specific meanings, such as "", "" and "without loss of generality". Such phrases are known as mathematical jargon. The vocabulary of mathematics also has visual elements.

Definitions and properties of Laplace transform, continuous-time and discrete-time Fourier series, continuous-time and discrete-time Fourier Transform, z-transform. Sampling theorems. Linear Time- Invariant (LTI) Systems: definitions and properties; causality, stability, impulse response, convolution, poles and zeros frequency response, group delay, phase delay. Signal transmission through LTI systems.

Scientists: Extraordinary People Who Altered the Course of History. New York: Metro Books. g. 12. In the Elements, Euclid deduced the theorems of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory, and mathematical rigour.

Given any desired positive integer c, this theorem shows that one can find an algebraic solution approximating a formal power series solution up to the degree specified by c. This leads to theorems that deduce the existence of certain formal moduli spaces of deformations as schemes. See also: Artin's criterion.

Marjorie "Molly" Greene Hahn (born December 30, 1948) is an American mathematician and tennis player. In mathematics and mathematical statistics she is known for her research in probability theory, including work on central limit theorems, stochastic processes, and stochastic differential equations. She is a professor emeritus of mathematics at Tufts University.

In mathematics and probability theory, Skorokhod's embedding theorem is either or both of two theorems that allow one to regard any suitable collection of random variables as a Wiener process (Brownian motion) evaluated at a collection of stopping times. Both results are named for the Ukrainian mathematician A. V. Skorokhod.

Automatons were quite a novelty. In the days before computers and electronics, some were very sophisticated, using pneumatics, mechanics, and hydraulics. The first automata were conceived during the third and second centuries BC and these were demonstrated by the theorems of Hero of Alexandria, which included sophisticated mechanical and hydraulic solutions.Droz, Edmond.

He is known for his work on the embedding problem in algebraic number theory, the Báyer–Neukirch theorem on special values of L-functions, arithmetic Riemann existence theorems and the Neukirch–Uchida theorem in birational anabelian geometry. He gave a simple description of the reciprocity maps in local and global class field theory.

Turing's thesis that every function which would naturally be regarded as computable under his definition, i.e. by one of his machines, is equivalent to Church's thesis by Theorem XXX." Indeed immediately before this statement, Kleene states the Theorem XXX: ::"Theorem XXX (= Theorems XXVIII + XXIX). The following classes of partial functions are coextensive, i.e.

In mathematics, categorification is the process of replacing set-theoretic theorems by category-theoretic analogues. Categorification, when done successfully, replaces sets by categories, functions with functors, and equations by natural isomorphisms of functors satisfying additional properties. The term was coined by Louis Crane. The reverse of categorification is the process of decategorification.

Posidonius was one of the first to attempt to prove Euclid's fifth postulate of geometry. He suggested changing the definition of parallel straight lines to an equivalent statement that would allow him to prove the fifth postulate. From there, Euclidean geometry could be restructured, placing the fifth postulate among the theorems instead.Trudeau, Richard.

W. Lawvere, "The Category of Categories as a Foundation for Mathematics". Proceedings of the Conference on Categorical Algebra (La Jolla, Calif., 1965), pp. 1–20. Springer-Verlag, New York (1966) The incompleteness theorems of Kurt Gödel, published in 1931, caused doubt about the attainability of an axiomatic foundation for all of mathematics.

In mathematics, profinite groups are topological groups that are in a certain sense assembled from finite groups. They share many properties with their finite quotients: for example, both Lagrange's theorem and the Sylow theorems generalise well to profinite groups. A non-compact generalization of a profinite group is a locally profinite group.

Apex-minor- free graph families obey a strengthened version of the graph structure theorem, leading to additional approximation algorithms for graph coloring and the travelling salesman problem.. However, some of these results can also be extended to arbitrary minor-closed graph families via structure theorems relating them to apex-minor-free graphs..

The company was called Mathematics, Inc., a company that he imagined having commercialized the production of mathematical theorems in the same way that software companies had commercialized the production of computer programs. He invented a number of activities and challenges of Mathematics Inc. and documented them in several papers in the EWD series.

Eames was the youngest daughter of Lorenzo and Katie Bridenstine, who had four more children. She grew up in Hoisington, Kansas. As a child she would be in every play she could. Her mother, who gave poetry readings, painted china and theorems, insisted that each of her children learn a musical instrument.

He has likened the band's sense of silliness to that of "Weird Al" Yankovic. Townsend's lyrical influences covered a wide range of themes, including warfare, mathematical theorems, and movies. He also used the technique of cross-referencing, repeating lines from his own works, such as older Strapping Young Lad, or solo material.

According to the complete class theorems, under mild conditions every admissible rule is a (generalized) Bayes rule (with respect to some prior \pi(\theta)\,\\!—possibly an improper one—that favors distributions \theta\,\\! where that rule achieves low risk). Thus, in frequentist decision theory it is sufficient to consider only (generalized) Bayes rules.

He further contributed significantly to the understanding of perfect numbers, which had fascinated mathematicians since Euclid. Euler made progress toward the prime number theorem and conjectured the law of quadratic reciprocity. The two concepts are regarded as the fundamental theorems of number theory, and his ideas paved the way for Carl Friedrich Gauss.

OMDoc allows for mathematical expressions on three levels: ;Object level:Formulae, written in Content MathML (the non-presentational subset of MathML), OpenMath or languages for mathematical logic. ;Statement level:Definitions, theorems, proofs, examples and the relations between them (e.g. “this proof proves that theorem”). ;Theory level:A theory is a set of contextually related statements.

In mathematics, especially in algebraic geometry and the theory of complex manifolds, the adjunction formula relates the canonical bundle of a variety and a hypersurface inside that variety. It is often used to deduce facts about varieties embedded in well-behaved spaces such as projective space or to prove theorems by induction.

The adjunction formula is false when the conormal exact sequence is not a short exact sequence. However, it is possible to use this failure to relate the singularities of X with the singularities of D. Theorems of this type are called inversion of adjunction. They are an important tool in modern birational geometry.

Formalism also more precisely refers to a certain school in the philosophy of mathematics, stressing axiomatic proofs through theorems, specifically associated with David Hilbert. In the philosophy of mathematics, therefore, a formalist is a person who belongs to the school of formalism, which is a certain mathematical-philosophical doctrine descending from Hilbert.

A Frobenioid consists of a category C together with a functor to an elementary Frobenioid, satisfying some complicated conditions related to the behavior of line bundles and divisors on models of global fields. One of Mochizuki's fundamental theorems states that under various conditions a Frobenioid can be reconstructed from the category C.

Thus, one can regard a finite vector space as a q-generalization of a set, and the subspaces as the q-generalization of the subsets of the set. This has been a fruitful point of view in finding interesting new theorems. For example, there are q-analogs of Sperner's theorem and Ramsey theory.

In particular, in the N=4 case with SU(2) gauge symmetry, the metric on the moduli space was found by Nathan Seiberg and Edward Witten using holomorphy and supersymmetric nonrenormalization theorems several days before Intriligator and Seiberg's 3-dimensional mirror symmetry paper appeared. Their results were reproduced using standard instanton techniques.

John Nash used the theorem in game theory to prove the existence of an equilibrium strategy profile. The theorem proved its worth in more than one way. During the 20th century numerous fixed-point theorems were developed, and even a branch of mathematics called fixed-point theory.V. I. Istratescu Fixed Point Theory.

Wolchover writes on topics within the physical sciences, such as high-energy physics, particle physics, AdS/CFT, quantum computing, gravitational waves, astrophysics, climate change, and Gödel's incompleteness theorems. Notable interviews include the highly cited theorists in high energy physics Ed Witten, Lisa Randall, Eva Silverstein, Juan Maldecena, Joe Polchinski, and Nima Arkani-Hamed.

The study of computability came to be known as recursion theory or computability theory, because early formalizations by Gödel and Kleene relied on recursive definitions of functions.A detailed study of this terminology is given by Soare (1996). When these definitions were shown equivalent to Turing's formalization involving Turing machines, it became clear that a new concept - the computable function - had been discovered, and that this definition was robust enough to admit numerous independent characterizations. In his work on the incompleteness theorems in 1931, Gödel lacked a rigorous concept of an effective formal system; he immediately realized that the new definitions of computability could be used for this purpose, allowing him to state the incompleteness theorems in generality that could only be implied in the original paper.

Kennedy and Roman Kossak are the editors of Set Theory, Arithmetic, and Foundations of Mathematics: Theorems, Philosophies, published as Book 36 in the series Lecture Notes in Logic in 2012 by Cambridge University Press. Kennedy is the editor of Interpreting Gödel: Critical Essays, published in 2014 by Cambridge University Press and reprinted in 2017. In the book Kennedy brought together leading contemporary philosophers and mathematicians to explore the impact of Gödel's work on the foundations and philosophy of mathematics. The logician Kurt Gödel has in 1931 formulated the incompleteness theorems, which among other things prove that within any formal system with resources sufficient to code arithmetic, questions exist which are neither provable nor disprovable on the basis of the axioms which define the system.

The object ■n□ demonstrates the use of "abbreviation", a way to simplify the denoting of objects, and consequently discussions about them, once they have been created "officially". Done correctly the definition would proceed as follows: ::: ■□ ≡ ■1□, ■■□ ≡ ■2□, ■■■□ ≡ ■3□, etc, where the notions of ≡ ("defined as") and "number" are presupposed to be understood intuitively in the metatheory. Kurt Gödel 1931 virtually constructed the entire proof of his incompleteness theorems (actually he proved Theorem IV and sketched a proof of Theorem XI) by use of this tactic, proceeding from his axioms using substitution, concatenation and deduction of modus ponens to produce a collection of 45 "definitions" (derivations or theorems more accurately) from the axioms. A more familiar tactic is perhaps the design of subroutines that are given names, e.g.

Mapping cylinders are quite common homotopical tools. One use of mapping cylinders is to apply theorems concerning inclusions of spaces to general maps, which might not be injective. Consequently, theorems or techniques (such as homology, cohomology or homotopy theory) which are only dependent on the homotopy class of spaces and maps involved may be applied to f\colon X\rightarrow Y with the assumption that X \subset Y and that f is actually the inclusion of a subspace. Another, more intuitive appeal of the construction is that it accords with the usual mental image of a function as "sending" points of X to points of Y, and hence of embedding X within Y, despite the fact that the function need not be one-to-one.

This observation took on a life of its own as what Shell called the Philadelphia Pholk theorem: if the first welfare theorem doesn't hold, then you can find an economy where sunspots matter. In addition to raising troubling questions about what the right state space was for dynamic stochastic economies, the notion of sunspot equilibrium raised a number of deep questions about the overall determinacy of economic equilibria and the role of the welfare theorems in the occurrence or non-occurrence of sunspot equilibria. These questions spawned a large literature on determinacy in dynamic economies in which the welfare theorems broke down. These include overlapping generations models, growth models with externalities or taxes, and models in which asset markets were incomplete.

The project was to fix a finite number of symbols (essentially the numerals 1, 2, 3, ... the letters of alphabet and some special symbols like "+", "⇒", "(", ")", etc.), give a finite number of propositions expressed in those symbols, which were to be taken as "foundations" (the axioms), and some rules of inference which would model the way humans make conclusions. From these, regardless of the semantic interpretation of the symbols the remaining theorems should follow formally using only the stated rules (which make mathematics look like a game with symbols more than a science) without the need to rely on ingenuity. The hope was to prove that from these axioms and rules all the theorems of mathematics could be deduced. That aim is known as logicism.

In mathematics, Kronecker's lemma (see, e.g., ) is a result about the relationship between convergence of infinite sums and convergence of sequences. The lemma is often used in the proofs of theorems concerning sums of independent random variables such as the strong Law of large numbers. The lemma is named after the German mathematician Leopold Kronecker.

Theoretical physicist Roger Penrose and anaesthesiologist Stuart Hameroff collaborated to produce the theory known as Orchestrated Objective Reduction (Orch-OR). Penrose and Hameroff initially developed their ideas separately and later collaborated to produce Orch-OR in the early 1990s. They reviewed and updated their theory in 2013. Penrose's argument stemmed from Gödel's incompleteness theorems.

Numerous topics relating to probability are named after him, including Feller processes, Feller's explosion test, Feller–Brown movement, and the Lindeberg–Feller theorem. Feller made fundamental contributions to renewal theory, Tauberian theorems, random walks, diffusion processes, and the law of the iterated logarithm. Feller was among those early editors who launched the journal Mathematical Reviews.

This allowed Liouville, in 1844, to produce the first explicit transcendental number. Later, the proofs that and e are transcendental were obtained with a similar method. Thus Diophantine approximations and transcendental number theory are very close areas that share many theorems and methods. Diophantine approximations also have important applications in the study of Diophantine equations.

For example, if R is Noetherian, then so is the polynomial ring (by Hilbert's basis theorem), any localization S−1R, and also any factor ring R / I. Any non-noetherian ring R is the union of its Noetherian subrings. This fact, known as Noetherian approximation, allows the extension of certain theorems to non-Noetherian rings.

Among the core results of coherent sheaf cohomology are results on finite-dimensionality of cohomology, results on the vanishing of cohomology in various cases, duality theorems such as Serre duality, relations between topology and algebraic geometry such as Hodge theory, and formulas for Euler characteristics of coherent sheaves such as the Riemann–Roch theorem.

In mathematics, Riemann–Hilbert problems, named after Bernhard Riemann and David Hilbert, are a class of problems that arise in the study of differential equations in the complex plane. Several existence theorems for Riemann–Hilbert problems have been produced by Mark Krein, Israel Gohberg and others (see the book by Clancey and Gohberg (1981)).

The Metamath website also hosts a few older databases which are not maintained anymore, such as the "Hilbert Space Explorer", which presents theorems pertaining to Hilbert space theory which have now been merged into the Metamath Proof Explorer, and the "Quantum Logic Explorer", which develops quantum logic starting with the theory of orthomodular lattices.

In 2012, Short co-founded Setem Technologies, where he continues to serve as Chief Technical Officer. Another UNH spinoff company, Setem seeks to use Short’s mathematical theorems and signal separation technology to enhance the voice clarity and audio signals in today's voice and speech recognition products (i.e.-cell phones, headsets, hearing aids, voice-activated electronics).

In measure theory and probability, the monotone class theorem connects monotone classes and sigma-algebras. The theorem says that the smallest monotone class containing an algebra of sets G is precisely the smallest σ-algebra containing G. It is used as a type of transfinite induction to prove many other theorems, such as Fubini's theorem.

Mark A. Stern, Eta invariants and hermitian locally symmetric spaces, J. Diff. Geom. 31 (1990), 771-789 34\. L. Saper and Mark A. Stern, L^2 cohomology of arithmetic varieties, Annals of Mathematics 132 (1990), 1-69 35\. Mark A. Stern, L^2 index theorems on locally symmetric spaces, Inventiones 96 (1989), 231-282 36\.

In mathematics -- specifically, in measure theory -- Malliavin's absolute continuity lemma is a result due to the French mathematician Paul Malliavin that plays a foundational rôle in the regularity (smoothness) theorems of the Malliavin calculus. Malliavin's lemma gives a sufficient condition for a finite Borel measure to be absolutely continuous with respect to Lebesgue measure.

Rippling continues to be developed at Edinburgh, and elsewhere, as of 2007. Rippling has been applied to many problems traditionally viewed as being hard in the inductive theorem proving community, including Bledsoe's limit theorems and a proof of the Gordon microprocessor, a miniature computer developed by Michael J. C. Gordon and his team at Cambridge.

In 1974 he retired as professor emeritus. At the University of Uppsala, Harald Bergström did research mainly on algebraic number fields and related topics. After his move to Chalmers University of Technology, he focused on probability theory and statistics. He made important contributions to central limit theorems and the theory of alpha stable distributions.

In the terms of Felix Klein's Erlangen programme, we read off from this that Euclidean geometry, the geometry of the Euclidean group of symmetries, is, therefore, a specialisation of affine geometry. All affine theorems apply. The origin of Euclidean geometry allows definition of the notion of distance, from which angle can then be deduced.

Hence, in the two- dimensional case only, it can also be referred to as the Markus–Yamabe theorem. Related mathematical results concerning global asymptotic stability, which are applicable in dimensions higher than two, include various autonomous convergence theorems. Analog of the conjecture for nonlinear control system with scalar nonlinearity is known as Kalman's conjecture.

In mathematics, the analytic Fredholm theorem is a result concerning the existence of bounded inverses for a family of bounded linear operators on a Hilbert space. It is the basis of two classical and important theorems, the Fredholm alternative and the Hilbert–Schmidt theorem. The result is named after the Swedish mathematician Erik Ivar Fredholm.

Jean Jacod (born 1944) is a French mathematician specializing in Stochastic processes and probability theory. He has been a professor at the Université Pierre et Marie Curie. He has made fundamental contributions to a wide range of topics in probability theory including stochastic calculus, limit theorems, martingale problems, Malliavin calculus and statistics of stochastic processes.

This may come as a surprise, but is only due to a logically stringent interpretation of the utility theorem and other theorems of decision and game theory. The usual economic interpretation is only one among many others and in fact, this interpretation is incompatible with most of the practical reasons that we take to be indispensable.

She especially notes the difference between scientific and social knowledge. In regards to sociology for example it is difficult to determine the “truth” when constructing an analysis. Scholars are heavily influenced by their social and cultural backgrounds which add to the subjectivity found in sociology. Scientific knowledge relies on such pragmatic instruments for analysis like theorems and formulas.

Weak dependence is a sufficient weak condition that many natural instances of stochastic processes exhibit it. In particular, weak dependence is a natural condition for the ergodic theory of random functions. A sufficient substitute for independence in the Lindeberg–Lévy central limit theorem is weak dependence. For this reason, specializations often appear in the probability literature on limit theorems.

In his thesis (Carnegie Mellon University, 1969), which concerned the extraction of computer programs from proofs of theorems, he found that the application of the resolution rule accounted for the appearance of a conditional branch in the extracted program, while the use of the mathematical induction principle caused the introduction of recursion and other repetitive constructs.

The majority of the theorems mentioned in the sections Galois theory, Constructing fields and Elementary notions can be found in Steinitz's work. linked the notion of orderings in a field, and thus the area of analysis, to purely algebraic properties. Emil Artin redeveloped Galois theory from 1928 through 1942, eliminating the dependency on the primitive element theorem.

In particular, if is a subset of then is a barrel in if and only if it is a barrel in . The following theorem shows that barrels (i.e. closed absorbing disks) are exactly the polars of weakly bounded subsets. All of this leads to Mackey's theorem, which is one of the central theorems in the theory of dual systems.

The essence of network synthesis is to start with a required filter response and produce a network that delivers that response, or approximates to it within a specified boundary. This is the inverse of network analysis which starts with a given network and by applying the various electric circuit theorems predicts the response of the network.Cauer et al., p.

In mathematics, coarse functions are functions that may appear to be continuous at a distance, but in reality are not necessarily continuous.Chul- Woo Lee and Jared Duke (2007), Coarse Function Value Theorems. Rose-Hulman Undergraduate Mathematics Journal 8 (2) Although continuous functions are usually observed on a small scale, coarse functions are usually observed on a large scale.

Typographical Number Theory (TNT) is a formal axiomatic system describing the natural numbers that appears in Douglas Hofstadter's book Gödel, Escher, Bach. It is an implementation of Peano arithmetic that Hofstadter uses to help explain Gödel's incompleteness theorems. Like any system implementing the Peano axioms, TNT is capable of referring to itself (it is self-referential).

The Mizar Mathematical Library (MML) includes all theorems to which authors can refer in newly written articles. Once approved by the proof checker they are further evaluated in a process of peer-review for appropriate contribution and style. If accepted they are published in the associated Journal of Formalized MathematicsJournal of Formalized Mathematics and added to the MML.

It is possible to find the hyperplanes guaranteed by the above theorems in O(Nd) steps. Also, if the 2d lists of the lower and upper endpoints of the intervals defining the boxes's ith coordinates are pre- sorted, then the best such hyperplane (according to a wide variety of optimality measures) may be found in O(Nd) steps.

There are collineations besides the homographies. In particular, any field automorphism σ of a field F induces a collineation of every projective space over F by applying σ to all homogeneous coordinates (over a projective frame) of a point. These collineations are called automorphic collineations. The fundamental theorem of projective geometry consists of the three following theorems.

The Leroy P Steele Prize of the AMS, MacTutor history of mathematics archive, retrieved 2015-04-24. He proved theorems in Schubert calculus about singularities of Schubert varieties. The Carrell–Liebermann theorem on the zero set of a holomorphic vector field is used in complex algebraic geometry. He is a fellow of the American Mathematical Society.

Recently, the 15 and 290 theorems have completely characterized universal integral quadratic forms: if all coefficients are integers, then it represents all positive integers if and only if it represents all integers up through 290; if it has an integral matrix, it represents all positive integers if and only if it represents all integers up through 15.

This concept stems from two fundamental theorems of quantum mechanics: the no-cloning theorem and the no-deleting theorem. But the no-hiding theorem is the ultimate proof of the conservation of quantum information. The importance of the no-hiding theorem is that it proves the conservation of wave function in quantum theory. This has never been proved earlier.

Theorems showing that certain objects of interest are the dual spaces (in the sense of linear algebra) of other objects of interest are often called dualities. Many of these dualities are given by a bilinear pairing of two K-vector spaces :A ⊗ B → K. For perfect pairings, there is, therefore, an isomorphism of A to the dual of B.

In mathematics, the HNN extension is an important construction of combinatorial group theory. Introduced in a 1949 paper Embedding Theorems for Groups by Graham Higman, Bernhard Neumann, and Hanna Neumann, it embeds a given group G into another group G' , in such a way that two given isomorphic subgroups of G are conjugate (through a given isomorphism) in G' .

From 1932 to 1933 she traveled on a fellowship to the University of Göttingen in Germany; she returned to Brown, and completed her Ph.D. there in 1934. Her dissertation, on the mathematical analysis of hyperbolic partial differential equations, was Some General Existence Theorems for Partial Differential Equations of Hyperbolic Type; her doctoral advisor was Jacob Tamarkin.

Alexander describes his skein relation toward the end of his paper under the heading "miscellaneous theorems", which is possibly why it got lost. Joan Birman mentions in her paper New points of view in knot theory (Bull. Amer. Math. Soc. (N.S.) 28 (1993), no. 2, 253–287) that Mark Kidwell brought her attention to Alexander's relation in 1970.

He was the first ever winner of the Canadian Mathematical Olympiad in 1969. He continued his education at Harvard University and the Massachusetts Institute of Technology in Cambridge, receiving a PhD from the latter institution in 1977. His dissertation was entitled "Witt Theorems for Lattices over Discrete Valuation Rings". He worked as a corporate planner and financial analyst.

In mathematics, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex analysis about holomorphic functions from the open unit disk to itself. The lemma is less celebrated than stronger theorems, such as the Riemann mapping theorem, which it helps to prove. It is, however, one of the simplest results capturing the rigidity of holomorphic functions.

Letter, Ramanujan to Hardy, 22 January 1914. In accordance with his Brahmin upbringing, Ramanujan refused to leave his country to "go to a foreign land". Meanwhile, he sent Hardy a letter packed with theorems, writing, "I have found a friend in you who views my labour sympathetically."Letter, Ramanujan to Hardy, 27 February 1913, Cambridge University Library.

Practical applications of scientific principles and abstract theorems, however, are not excluded from patentability. Therefore, while Newton's law of universal gravitation may not be patentable, a patent may be granted for the practical application of the theory, such as an improved gravity pump.David Vaver, Intellectual Property Law: Copyright, Patents, Trade-Marks, 2nd ed (Toronto: Irwin Law, 2011) at 308.

David Hilton Wolpert is an American mathematician, physicist and computer scientist. He is a professor at Santa Fe Institute. He is the author of three books, three patents, over one hundred refereed papers, and has received numerous awards. His name is particularly associated with a group of theorems in computer science known as "no free lunch".

In the context of combinatorial mathematics, stars and bars is a graphical aid for deriving certain combinatorial theorems. It was popularized by William Feller in his classic book on probability. It can be used to solve many simple counting problems, such as how many ways there are to put n indistinguishable balls into k distinguishable bins.

Various approaches to geometry have based exercises on relations of angles, segments, and triangles. The topic of trigonometry gains many of its exercises from the trigonometric identities. In college mathematics exercises often depend on functions of a real variable or application of theorems. The standard exercises of calculus involve finding derivatives and integrals of specified functions.

The Metamath language design is focused on simplicity; the language, employed to state the definitions, axioms, inference rules and theorems is only composed of a handful of keywords, and all the proofs are checked using one simple algorithm based on the substitution of variables (with optional provisos for what variables must remain distinct after a substitution is made).

The Metamath website hosts several databases that store theorems derived from various axiomatic systems. Most databases (.mm files) have an associated interface, called an "Explorer", which allows one to navigate the statements and proofs interactively on the website, in a user-friendly way. Most databases use a Hilbert system of formal deduction though this is not a requirement.

What follows is an incomplete list of the most classical theorems in Riemannian geometry. The choice is made depending on its importance and elegance of formulation. Most of the results can be found in the classic monograph by Jeff Cheeger and D. Ebin (see below). The formulations given are far from being very exact or the most general.

However, there are also other ω-models; for example, RCA0 has a minimal ω-model where S consists of the recursive subsets of ω. A β model is an ω model that is equivalent to the standard ω-model for Π and Σ sentences (with parameters). Non-ω models are also useful, especially in the proofs of conservation theorems.

The isoperimetric problem is to determine a figure with the largest area, amongst those having a given perimeter. The solution is intuitive; it is the circle. In particular, this can be used to explain why drops of fat on a broth surface are circular. This problem may seem simple, but its mathematical proof requires some sophisticated theorems.

In mathematics, Budan's theorem is a theorem for bounding the number of real roots of a polynomial in an interval, and computing the parity of this number. It was published in 1807 by François Budan de Boislaurent. A similar theorem was published independently by Joseph Fourier in 1820. Each of these theorems are a corollary of the other.

In 1990 Topping was awarded a Telford Premium by the Institution of Civil Engineers for his paper Theorems Of Geometric Variation For Engineering Structures. In 2001 Topping received the Best Paper Award by the Journal of Computing in Civil Engineering (American Society of Civil Engineers) in respect of his paper Transient Dynamic Nonlinear Analysis Using MIMD Computer Architectures.

In 1829 he defined for the first time a complex function of a complex variable in another textbook. In spite of these, Cauchy's own research papers often used intuitive, not rigorous, methods; thus one of his theorems was exposed to a "counter-example" by Abel, later fixed by the introduction of the notion of uniform continuity.

For some classes of matrices with non-commutative elements, one can define the determinant and prove linear algebra theorems that are very similar to their commutative analogs. Examples include the q-determinant on quantum groups, the Capelli determinant on Capelli matrices, and the Berezinian on supermatrices. Manin matrices form the class closest to matrices with commutative elements.

In probability theory, Lévy’s continuity theorem, or Lévy's convergence theorem, named after the French mathematician Paul Lévy, connects convergence in distribution of the sequence of random variables with pointwise convergence of their characteristic functions. This theorem is the basis for one approach to prove the central limit theorem and it is one of the major theorems concerning characteristic functions.

The branch of trade theory which is conventionally categorized as "classical" consists mainly of the application of deductive logic, originating with Ricardo's Theory of Comparative Advantage and developing into a range of theorems that depend for their practical value upon the realism of their postulates. "Modern" trade analysis, on the other hand, depends mainly upon empirical analysis.

The Penrose–Hawking singularity theorems define a singularity to have geodesics that cannot be extended in a smooth manner. The termination of such a geodesic is considered to be the singularity. The initial state of the universe, at the beginning of the Big Bang, is also predicted by modern theories to have been a singularity.Wald, p.

This theory is a key technical tool in algebraic geometry. Among the main theorems are results on the vanishing of cohomology in various situations, results on finite-dimensionality of cohomology, comparisons between coherent sheaf cohomology and singular cohomology such as Hodge theory, and formulas on Euler characteristics in coherent sheaf cohomology such as the Riemann–Roch theorem.

Further developments of these theorems on planetary motion were given in his two memoirs of 1788 and 1789, but with the aid of Laplace's discoveries, the tables of the motions of Jupiter and Saturn could at last be made much more accurate. It was on the basis of Laplace's theory that Delambre computed his astronomical tables.

179, Belmont: Wadsworth More generally, the term motion is a synonym for surjective isometry in metric geometry,M.A. Khamsi & W.A. Kirk (2001) An Introduction to Metric Spaces and Fixed Point Theorems, p. 15, John Wiley & Sons including elliptic geometry and hyperbolic geometry. In the latter case, hyperbolic motions provide an approach to the subject for beginners.

It is also a semi-simple group, in fact a simple group with the exception SO(4).; The relevance of this is that all theorems and all machinery from the theory of analytic manifolds (analytic manifolds are in particular smooth manifolds) apply and the well-developed representation theory of compact semi-simple groups is ready for use.

Some consequences of the RH are also consequences of its negation, and are thus theorems. In their discussion of the Hecke, Deuring, Mordell, Heilbronn theorem, say > The method of proof here is truly amazing. If the generalized Riemann > hypothesis is true, then the theorem is true. If the generalized Riemann > hypothesis is false, then the theorem is true.

The mountain pass theorem is an existence theorem from the calculus of variations, originally due to Antonio Ambrosetti and Paul Rabinowitz. Given certain conditions on a function, the theorem demonstrates the existence of a saddle point. The theorem is unusual in that there are many other theorems regarding the existence of extrema, but few regarding saddle points.

The Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex analysis about holomorphic functions from the open unit disk to itself. The lemma is less celebrated than stronger theorems, such as the Riemann mapping theorem, which it helps to prove. It is however one of the simplest results capturing the rigidity of holomorphic functions.

The procedures developed by Sprague and Grundy for using their function to analyse impartial games are collectively called Sprague–Grundy theory, and at least two different theorems concerning these procedures have been called Sprague–Grundy theorems.The theorem given that name by Smith (2015, p.340) is one which was in fact proved by Sprague and Grundy.

A set of sentences is called a theory; thus, individual sentences may be called theorems. To properly evaluate the truth (or falsehood) of a sentence, one must make reference to an interpretation of the theory. For first-order theories, interpretations are commonly called structures. Given a structure or interpretation, a sentence will have a fixed truth value.

In the study of graph algorithms, Courcelle's theorem is the statement that every graph property definable in the monadic second-order logic of graphs can be decided in linear time on graphs of bounded treewidth.... The result was first proved by Bruno Courcelle in 1990 and independently rediscovered by .. It is considered the archetype of algorithmic meta-theorems...

Thales is supposed to have used geometry to solve problems such as calculating the height of pyramids based on the length of shadows, and the distance of ships from the shore. He is also credited by tradition with having made the first proof of two geometric theorems—the "Theorem of Thales" and the "Intercept theorem" described above.

They had three children, Katherine, Peter, and Barbara. His wife died in 2002. From 1950 to 1955, Simon studied mathematical economics and during this time, together with David Hawkins, discovered and proved the Hawkins–Simon theorem on the "conditions for the existence of positive solution vectors for input-output matrices". He also developed theorems on near-decomposability and aggregation.

In the 1930s G. A. Hedlund proved that the horocycle flow on a compact hyperbolic surface is minimal and ergodic. Unique ergodicity of the flow was established by Hillel Furstenberg in 1972. Ratner's theorems provide a major generalization of ergodicity for unipotent flows on the homogeneous spaces of the form Γ \ G, where G is a Lie group and Γ is a lattice in G. In the last 20 years, there have been many works trying to find a measure-classification theorem similar to Ratner's theorems but for diagonalizable actions, motivated by conjectures of Furstenberg and Margulis. An important partial result (solving those conjectures with an extra assumption of positive entropy) was proved by Elon Lindenstrauss, and he was awarded the Fields medal in 2010 for this result.

Although density functional theory has its roots in the Thomas–Fermi model for the electronic structure of materials, DFT was first put on a firm theoretical footing by Walter Kohn and Pierre Hohenberg in the framework of the two Hohenberg–Kohn theorems (H–K). The original H–K theorems held only for non- degenerate ground states in the absence of a magnetic field, although they have since been generalized to encompass these. The first H–K theorem demonstrates that the ground-state properties of a many-electron system are uniquely determined by an electron density that depends on only three spatial coordinates. It set down the groundwork for reducing the many-body problem of electrons with spatial coordinates to three spatial coordinates, through the use of functionals of the electron density.

Khayyam then considered the three cases right, obtuse, and acute that the summit angles of a Saccheri quadrilateral can take and after proving a number of theorems about them, he (correctly) refuted the obtuse and acute cases based on his postulate and hence derived the classic postulate of Euclid. It was not until 600 years later that Giordano Vitale made an advance on Khayyam in his book Euclide restituo (1680, 1686), when he used the quadrilateral to prove that if three points are equidistant on the base AB and the summit CD, then AB and CD are everywhere equidistant. Saccheri himself based the whole of his long and ultimately flawed proof of the parallel postulate around the quadrilateral and its three cases, proving many theorems about its properties along the way.

The book begins with five chapters that discuss the field of reverse mathematics, which has the goal of classifying mathematical theorems by the axiom schemes needed to prove them, the big five subsystems of second-order arithmetic into which many theorems of mathematics have been classified. These chapters also review some of the tools needed in this study, including computability theory, forcing, and the low basis theorem. Chapter six, "the real heart of the book", applies this method to an infinitary form of Ramsey's theorem: every edge coloring of a countably infinite complete graph or complete uniform hypergraph, using finitely many colors, contains a monochromatic infinite induced subgraph. The standard proof of this theorem uses the arithmetical comprehension axiom, falling into one of the big five subsystems, ACA0.

Mirsky was inspired by Dilworth's theorem, stating that, for every partially ordered set, the maximum size of an antichain equals the minimum number of chains in a partition of the set into chains. For sets of order dimension two, the two theorems coincide (a chain in the majorization ordering of points in general position in the plane is an antichain in the set of points formed by a 90° rotation from the original set, and vice versa) but for more general partial orders the two theorems differ, and (as Mirsky observes) Dilworth's theorem is more difficult to prove. Mirsky's theorem and Dilworth's theorem are also related to each other through the theory of perfect graphs. An undirected graph is perfect if, in every induced subgraph, the chromatic number equals the size of the largest clique.

Finiteness theorems for discrete subgroups of bounded covolume in semi-simple groups, Publ.Math.IHES 69(1989), 119-171; Addendum: ibid, 71(1990); with A.Borel. [7]. Values of isotropic quadratic forms at S-integral points, Compositio Mathematica, 83 (1992), 347-372; with A.Borel. [8]. Unrefined minimal K-types for p-adic groups, Inventiones Math. 116(1994), 393-408; with Allen Moy. [9].

The complexity class F(NP \cap coNP) can be defined in two different ways, and those ways are not known to be equivalent. One way applies F to the machine model for NP \cap coNP. It is known that with this definition, F(NP \cap coNP) coincides with TFNPMegiddo and Papadimitriou. A Note on Total Functions, Existence Theorems and Computational Complexity.

In geometry Mukhopadhyaya's theorem may refer to one of several closely related theorems about the number of vertices of a curve due to . One version, called the Four-vertex theorem, states that a simple convex curve in the plane has at least 4 vertices, and another version states that a simple convex curve in the affine plane has at least 6 affine vertices.

In the study of Lorentzian manifold spacetimes there exists a hierarchy of causality conditions which are important in proving mathematical theorems about the global structure of such manifolds. These conditions were collected during the late 1970s.E. Minguzzi and M. Sanchez, The causal hierarchy of spacetimes in H. Baum and D. Alekseevsky (eds.), vol. Recent developments in pseudo-Riemannian geometry, ESI Lect. Math. Phys.

In 1926, Wiener returned to Europe as a Guggenheim scholar. He spent most of his time at Göttingen and with Hardy at Cambridge, working on Brownian motion, the Fourier integral, Dirichlet's problem, harmonic analysis, and the Tauberian theorems. In 1926, Wiener's parents arranged his marriage to a German immigrant, Margaret Engemann; they had two daughters. His sister, Constance, married Philip Franklin.

At the Institute for Advanced Study, Poe played guitar and sang in a loud rock and roll band called "Do Not Erase," consisting entirely of fellows at the Institute. The name of the band is taken from what mathematicians write under their long theorems and proofs on chalk boards, so that janitors won't erase them, especially if their equations have discovered something new.

Finite-dimensional vector spaces over local fields and division algebras under the topology uniquely determined by the field's topology are studied, and lattices are defined topologically, an analogue of Minkowski's theorem is proved in this context, and the main theorems about character groups of these vector spaces, which in the commutative one-dimensional case reduces to `self duality’ for local fields, are shown.

It was to be an "Encyclopedia of Mathematics", containing all known formulae and theorems of mathematical science using a standard notation invented by Peano. In 1897, the first International Congress of Mathematicians was held in Zürich. Peano was a key participant, presenting a paper on mathematical logic. He also started to become increasingly occupied with Formulario to the detriment of his other work.

Absolute geometry is an extension of ordered geometry, and thus, all theorems in ordered geometry hold in absolute geometry. The converse is not true. Absolute geometry assumes the first four of Euclid's Axioms (or their equivalents), to be contrasted with affine geometry, which does not assume Euclid's third and fourth axioms. Ordered geometry is a common foundation of both absolute and affine geometry.

When a theorem is proven, the system produces a verifiable proof, which validates both the phase and the refutation of the conjunctive normal form. Along with proving theorems, Vampire has other related functionalities such as generating interpolants. Executables can be obtained from the system website. A somewhat outdated version is available under the GNU Lesser General Public License as part of Sigma KEE.

Two theorems in the mathematical field of Riemannian geometry bear the name Myers–Steenrod theorem, both from a 1939 paper by Myers and Steenrod. The first states that every distance-preserving map (i.e., an isometry of metric spaces) between two connected Riemannian manifolds is actually a smooth isometry of Riemannian manifolds. A simpler proof was subsequently given by Richard Palais in 1957.

See also pages 188, 250., gives a very brief proof of the cut-elimination theorem. a result with far-reaching meta-theoretic consequences, including consistency. Gentzen further demonstrated the power and flexibility of this technique a few years later, applying a cut- elimination argument to give a (transfinite) proof of the consistency of Peano arithmetic, in surprising response to Gödel's incompleteness theorems.

Prof Sergei Lvovich Sobolev () HFRSE (6 October 1908 – 3 January 1989) was a Soviet mathematician working in mathematical analysis and partial differential equations. Sobolev introduced notions that are now fundamental for several areas of mathematics. Sobolev spaces can be defined by some growth conditions on the Fourier transform. They and their embedding theorems are an important subject in functional analysis.

Although others before him proved theorems via the probabilistic method (for example, Szele's 1943 result that there exist tournaments containing a large number of Hamiltonian cycles), many of the most well known proofs using this method are due to Erdős. The first example below describes one such result from 1947 that gives a proof of a lower bound for the Ramsey number .

Soddy's hexlet problem in japanese mathematics book Kokonsankan (1832). Replica of Sangaku at Hōtoku museum in Samukawa Shrine. Japanese mathematicians discovered the same hexlet over one hundred years before Soddy did. They analysed the packing problems in which circles and polygons, balls and polyhedrons come into contact and often found the relevant theorems independently before their discovery by Western mathematicians.

With the introduction of matrices, the Euler theorems were rewritten. The rotations were described by orthogonal matrices referred to as rotation matrices or direction cosine matrices. When used to represent an orientation, a rotation matrix is commonly called orientation matrix, or attitude matrix. The above- mentioned Euler vector is the eigenvector of a rotation matrix (a rotation matrix has a unique real eigenvalue).

With the introduction of matrices the Euler theorems were rewritten. The rotations were described by orthogonal matrices referred to as rotation matrices or direction cosine matrices. When used to represent an orientation, a rotation matrix is commonly called orientation matrix, or attitude matrix. The above-mentioned Euler vector is the eigenvector of a rotation matrix (a rotation matrix has a unique real eigenvalue).

These theorems are for Banach spaces with the Radon–Nikodym property. A theorem of Joram Lindenstrauss states that, in a Banach space with the Radon–Nikodym property, a nonempty closed and bounded set has an extreme point. (In infinite-dimensional spaces, the property of compactness is stronger than the joint properties of being closed and being bounded). Edgar's theorem implies Lindenstrauss's theorem.

Usually that refers to which is a Hilbert space, or to and . Therefore one may prove theorems about the more complicated cases by proving them in two simple cases and then using the Riesz–Thorin theorem to pass from the simple cases to the complicated cases. The Marcinkiewicz theorem is similar but applies also to a class of non-linear maps.

The Journal of Automated Reasoning was established in 1983 by Larry Wos, who was its editor in chief until 1992. It covers research and advances in automated reasoning - mechanical verification of theorems and other deductions in classical and non-classical logic. The journal is published by Springer Science+Business Media. As of 2010, the editor-in-chief is Tobias Nipkow.

She was invited to present this work at the Geometry Festival. She was a Bunting Scholar at the Radcliffe Institute for Advanced Study in 1991. For the subsequent decades, Lesley Sibner focussed on gauge theory and gravitational instantons. Although the research sounds very physical, in fact throughout her career, Lesley Sibner applied physical intuition to prove important geometric and topological theorems.

Robustness curves for alternative decisions may cross as a function of aspiration, implying reversal of preference. Various theorems identify conditions where larger info-gap robustness implies larger probability of success, regardless of the underlying probability distribution. However, these conditions are technical, and do not translate into any common-sense, verbal recommendations, limiting such applications of info-gap theory by non-experts.

The Mathematics Department split from the Institute in 1934. It is now known as Steklov Institute of Mathematics. Steklov's primary scientific contribution was in the area of orthogonal functional sets. He introduced a class of closed orthogonal sets, developed the asymptotic Liouville–Steklov method for orthogonal polynomials, proved theorems on generalized Fourier series, and developed an approximation technique later named Steklov function.

Schouten's name appears in various mathematical entities and theorems, such as the Schouten tensor, the Schouten bracket and the Weyl–Schouten theorem. He wrote Der Ricci-Kalkül in 1922 surveying the field of tensor analysis. In 1931 he wrote a treatise on tensors and differential geometry. The second volume, on applications to differential geometry, was authored by his student Dirk Jan Struik.

There are general existence theorems that apply; the most basic of them guarantees that :Whenever C is a variety, then for every set X there is a free object F(X) in C. Here, a variety is a synonym for a finitary algebraic category, thus implying that the set of relations are finitary, and algebraic because it is monadic over Set.

The key observation that leads to Lie sphere geometry is that theorems of Euclidean geometry in the plane (resp. in space) which only depend on the concepts of circles (resp. spheres) and their tangential contact have a more natural formulation in a more general context in which circles, lines and points (resp. spheres, planes and points) are treated on an equal footing.

In mathematics, Toda–Smith complexes are spectra characterized by having a particularly simple BP-homology, and are useful objects in stable homotopy theory. Toda–Smith complexes provide examples of periodic self maps. These self maps were originally exploited in order to construct infinite families of elements in the homotopy groups of spheres. Their existence pointed the way towards the nilpotence and periodicity theorems.

Learners can construct geometric proofs at a secondary school level and understand their meaning. They understand the role of undefined terms, definitions, axioms and theorems in Euclidean geometry. However, students at this level believe that axioms and definitions are fixed, rather than arbitrary, so they cannot yet conceive of non-Euclidean geometry. Geometric ideas are still understood as objects in the Euclidean plane.

In mathematics, the Barban–Davenport–Halberstam theorem is a statement about the distribution of prime numbers in an arithmetic progression. It is known that in the long run primes are distributed equally across possible progressions with the same difference. Theorems of the Barban–Davenport–Halberstam type give estimates for the error term, determining how close to uniform the distributions are.

This method was introduced by J. Barkley Rosser in 1936, as an improvement of Gödel's original proof of the incompleteness theorems that was published in 1931. While Gödel's original proof uses a sentence that says (informally) "This sentence is not provable", Rosser's trick uses a formula that says "If this sentence is provable, there is a shorter proof of its negation".

It has native support for Unicode symbols. (These can be typed using LaTeX-like sequences, such as "\times" for "×".) Lean uses its own language for meta-programming. So, if the user wants to write a function that automatically proves some theorems, they write that function in Lean's own language. Lean has gotten attention from mathematicians Thomas Hales and Kevin Buzzard.

As previously mentioned, complements need not exist. A p-complement is a complement to a Sylow p-subgroup. Theorems of Frobenius and Thompson describe when a group has a normal p-complement. Philip Hall characterized finite soluble groups amongst finite groups as those with p-complements for every prime p; these p-complements are used to form what is called a Sylow system.

Johannes Lambertus Adriana van de Snepscheut (; 12 September 195323 February 1994) was a computer scientist and educator. He was a student of Martin Rem and Edsger Dijkstra. At the time of his death he was the executive officer of the computer science department at the California Institute of Technology. He was also developing an editor for proving theorems called "Proxac".

HoTT allows mathematical proofs to be translated into a computer programming language for computer proof assistants much more easily than before. This approach offers the potential for computers to check difficult proofs. One goal of mathematics is to formulate axioms from which virtually all mathematical theorems can be derived and proven unambiguously. Correct proofs in mathematics must follow the rules of logic.

Farlow earned bachelor's and master's degrees in mathematics at the University of Iowa. He completed his Ph.D. in mathematics at Oregon State University in 1967. His doctoral supervisor was Ronald Bernard Guenther, and his doctoral dissertation was on Existence Theorems for Periodic Solutions of Parabolic Partial Differential Equations. He is currently a professor of mathematics at the University of Maine.

Baron Augustin-Louis Cauchy (;"Cauchy". Random House Webster's Unabridged Dictionary. ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He was one of the first to state and rigorously prove theorems of calculus, rejecting the heuristic principle of the generality of algebra of earlier authors.

Peripheral cycles appear in the theory of polyhedral graphs, that is, 3-vertex-connected planar graphs. For every planar graph G, and every planar embedding of G, the faces of the embedding that are induced cycles must be peripheral cycles. In a polyhedral graph, all faces are peripheral cycles, and every peripheral cycle is a face., Theorems 2.7 and 2.8.

The MU puzzle is a puzzle stated by Douglas Hofstadter and found in Gödel, Escher, Bach involving a simple formal system called "MIU". Hofstadter's motivation is to contrast reasoning within a formal system (ie., deriving theorems) against reasoning about the formal system itself. MIU is an example of a Post canonical system and can be reformulated as a string rewriting system.

By a “scientific theology” McGrath does not mean an attempt to reconcile particular Christian beliefs with particular scientific theorems. Such efforts are regarded by him as pointless because they become outdated with scientific progress.McGrath (2004), pp.27-31 Rather, McGrath seeks to draw upon the proven assumptions and methods of the natural sciences in order to inform the practice of Christian theology.

The theorems were stated without proof, but proofs for the series for sine, cosine, and inverse tangent were provided a century later in the work Yuktibhasa (), written in Malayalam, by Jyesthadeva, and also in a commentary on Tantrasangraha.Roy, Ranjan. 1990. "Discovery of the Series Formula for \pi by Leibniz, Gregory, and Nilakantha." Mathematics Magazine (Mathematical Association of America) 63(5):291–306.

Marmion's great gift for teaching came into full bloom during this period. His lectures were distinguished by, "on the one hand, his extreme clearness, and on the other his happy and fluent application of doctrine to the inner life."Thibaut, p. 125. Rather than presenting "revealed truths like mere theorems of geometry having no bearing on the interior life,"Thibaut, p. 127.

Mathematical logic generally does not allow explicit reference to its own sentences. However the heart of Gödel's incompleteness theorems is the observation that a different form of self-reference can be added; see Gödel number. The axiom of unrestricted comprehension adds the ability to construct a recursive definition in set theory. This axiom is not supported by modern set theory.

Much of group theory can be formulated in the context of the more general group objects. The notions of group homomorphism, subgroup, normal subgroup and the isomorphism theorems are typical examples. However, results of group theory that talk about individual elements, or the order of specific elements or subgroups, normally cannot be generalized to group objects in a straightforward manner.

In some form these considerations have a history of centuries in mathematics, but also in physics and engineering. For example, in the geometry of numbers a class of results called isolation theorems was recognised, with the topological interpretation of an open orbit (of a group action) around a given solution. Perturbation theory also looks at deformations, in general of operators.

In mathematics, the Riemann–Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes. The definition of this integral was first published in 1894 by Stieltjes. It serves as an instructive and useful precursor of the Lebesgue integral, and an invaluable tool in unifying equivalent forms of statistical theorems that apply to discrete and continuous probability.

For his results in classical and quantum transport in low dimensional systems and contributions to non-equilibrium fluctuation theorems, he was awarded the ICTP prize in 2008 and Shanti Swarup Bhatnagar award for Physical Sciences in 2009. He became a fellow of the Indian Academy of Sciences in 2012. He was elected to the National Academy of Sciences in 2018.

This arc length must be greater than the straight-line distance between the same two centers, so the two circles have centers closer together than the difference of their radii, from which the theorem follows. Analogous disjointness theorems can be proved for the family of Taylor polynomials of a given smooth function, and for the osculating conics to a given smooth curve.

The classical results of the theory are Fredholm's theorems, one of which is the Fredholm alternative. One of the important results from the general theory is that the kernel is a compact operator when the space of functions are equicontinuous. A related celebrated result is the Atiyah–Singer index theorem, pertaining to index (dim ker – dim coker) of elliptic operators on compact manifolds.

Logic Theorist is a computer program written in 1956 by Allen Newell, Herbert A. Simon and Cliff Shaw. , and It was the first program deliberately engineered to perform automated reasoning and is called "the first artificial intelligence program". It would eventually prove 38 of the first 52 theorems in Whitehead and Russell's Principia Mathematica, and find new and more elegant proofs for some.

He also calculated a 7-digit logarithm table and extended a table of integer factorizations from 6,000,000 to 9,000,000. Dase had very little knowledge of mathematical theory. The mathematician Julius Petersen tried to teach him some of Euclid's theorems, but gave up the task once he realized that their comprehension was beyond Dase's capabilities. Preston, Richard, 2008, Panic in Level 4, p. 32.

First Rejewski tackled the problem of discovering the wiring of the rotors. To do this, according to historian David Kahn, he pioneered the use of pure mathematics in cryptanalysis. Previous methods had largely exploited linguistic patterns and the statistics of natural-language texts—letter-frequency analysis. Rejewski applied techniques from group theory—theorems about permutations—in his attack on Enigma.

A formal theorem is the purely formal analogue of a theorem. In general, a formal theorem is a type of well-formed formula that satisfies certain logical and syntactic conditions. The notation S is often used to indicate that S is a theorem. Formal theorems consist of formulas of a formal language and the transformation rules of a formal system.

The concept of a formal theorem is fundamentally syntactic, in contrast to the notion of a true proposition, which introduces semantics. Different deductive systems can yield other interpretations, depending on the presumptions of the derivation rules (i.e. belief, justification or other modalities). The soundness of a formal system depends on whether or not all of its theorems are also validities.

It is called a comparison theorem as it is an analogue for Arakelov theory of comparison theorems in cohomology relating de Rham cohomology to singular cohomology of complex varieties or étale cohomology of p-adic varieties. In and he pointed out that arithmetic Kodaira–Spencer map and Gauss–Manin connection may give some important hints for Vojta's conjecture, ABC conjecture and so on.

In the mathematical field of group theory, the transfer defines, given a group G and a subgroup of finite index H, a group homomorphism from G to the abelianization of H. It can be used in conjunction with the Sylow theorems to obtain certain numerical results on the existence of finite simple groups. The transfer was defined by and rediscovered by .

Multivariate analysis, design of experiments, and survey sampling By Subir Ghosh, Jagdish Narain Srivastava, CRC Press, 1999 Srivastava code was invented by him. Gödel's incompleteness theorems inspired him to recognize the limitations of science. He slowly turned toward spirituality and studied all the major religions of the world. This led him to obtain his 1991 joint appointment in the philosophy department of CSU.

The oldest existent work on geometry in China comes from the philosophical Mohist canon c. 330 BC, compiled by the followers of Mozi (470–390 BC). The Mo Jing described various aspects of many fields associated with physical science, and provided a small number of geometrical theorems as well. It also defined the concepts of circumference, diameter, radius, and volume.

Some reject the validity of the argument pointing out various flaws, such as a category error involved in the first premise of the argument, namely that just because there's a statement that's universally true it won't make that statement a part of reality in itself. Another issue pointed out is that it's not needed to have a god to have logic or morality. In particular the existence of multiple logic systems with differing axioms such as non- classical logic as well as multiple radically different moral systems constitutes evidence against the idea that logic and morality are actually universals. Furthermore, the existence of theorems like Goedel's completeness theorem and the soundness theorems for classical logic provide justification for some logic systems like classical propositional logic without using any god hypotheses thus contradicting the first premise of the argument.

Doron Zeilberger considers a time when computers become so powerful that the predominant questions in mathematics change from proving things to determining how much it would cost: "As wider classes of identities, and perhaps even other kinds of classes of theorems, become routinely provable, we might witness many results for which we would know how to find a proof (or refutation), but we would be unable, or unwilling, to pay for finding such proofs, since “almost certainty” can be bought so much cheaper. I can envision an abstract of a paper, c. 2100, that reads : “We show, in a certain precise sense, that the Goldbach conjecture is true with probability larger than 0.99999, and that its complete truth could be determined with a budget of $10B.”"Doron Zeilberger (1994). "Theorems for a Price: Tomorrow’s Semi-Rigorous Mathematical Culture".

In reverse mathematics, one starts with a framework language and a base theory—a core axiom system—that is too weak to prove most of the theorems one might be interested in, but still powerful enough to develop the definitions necessary to state these theorems. For example, to study the theorem “Every bounded sequence of real numbers has a supremum” it is necessary to use a base system which can speak of real numbers and sequences of real numbers. For each theorem that can be stated in the base system but is not provable in the base system, the goal is to determine the particular axiom system (stronger than the base system) that is necessary to prove that theorem. To show that a system S is required to prove a theorem T, two proofs are required.

In mathematics the use of the term theory is different, necessarily so, since mathematics contains no explanations of natural phenomena, per se, even though it may help provide insight into natural systems or be inspired by them. In the general sense, a mathematical theory is a branch of or topic in mathematics, such as Set theory, Number theory, Group theory, Probability theory, Game theory, Control theory, Perturbation theory, etc., such as might be appropriate for a single textbook. In the same sense, but more specifically, the word theory is an extensive, structured collection of theorems, organized so that the proof of each theorem only requires the theorems and axioms that preceded it (no circular proofs), occurs as early as feasible in sequence (no postponed proofs), and the whole is as succinct as possible (no redundant proofs).

The laws are named after Augustus De Morgan (1806–1871),DeMorgan’s Theorems at mtsu.edu who introduced a formal version of the laws to classical propositional logic. De Morgan's formulation was influenced by algebraization of logic undertaken by George Boole, which later cemented De Morgan's claim to the find. Nevertheless, a similar observation was made by Aristotle, and was known to Greek and Medieval logicians.

The first wave of proofs of the central theorems of class field theory was structured as consisting of two 'inequalities' (the same structure as in the proofs now given of the fundamental theorem of Galois theory, though much more complex). One of the two inequalities involved an argument with L-functions.In today's terminology, that is the second inequality. See class formation for a contemporary presentation.

In probability, weak dependence of random variables is a generalization of independence that is weaker than the concept of a martingale. A (time) sequence of random variables is weakly dependent if distinct portions of the sequence have a covariance that asymptotically decreases to 0 as the blocks are further separated in time. Weak dependence primarily appears as a technical condition in various probabilistic limit theorems.

The class of locally finite groups is closed under subgroups, quotients, and extensions . Locally finite groups satisfy a weaker form of Sylow's theorems. If a locally finite group has a finite p-subgroup contained in no other p-subgroups, then all maximal p-subgroups are finite and conjugate. If there are finitely many conjugates, then the number of conjugates is congruent to 1 modulo p.

Abelian categories are the most general setting for homological algebra. All of the constructions used in that field are relevant, such as exact sequences, and especially short exact sequences, and derived functors. Important theorems that apply in all abelian categories include the five lemma (and the short five lemma as a special case), as well as the snake lemma (and the nine lemma as a special case).

In: Journal of Applied Probability. 36(4)/1999. Applied Probability Trust, pp. 1256, In the context of limit theorems for superpositions of point processes he came to the problem of infinite divisibility of point processes (following a suggestion by Boris Vladimirovich Gnedenko). Together with his coworkers he investigated systematically the structure of infinitely divisible distributions, which culminated in the monograph "Infinitely Divisible Point Processes".

Some philosophers and logicians disagree with the philosophical conclusions that Chaitin has drawn from his theorems related to what Chaitin thinks is a kind of fundamental arithmetic randomness.Panu Raatikainen, "Exploring Randomness and The Unknowable" Notices of the American Mathematical Society Book Review October 2001. The logician Torkel Franzén criticized Chaitin's interpretation of Gödel's incompleteness theorem and the alleged explanation for it that Chaitin's work represents.

In mathematics, the Fredholm alternative, named after Ivar Fredholm, is one of Fredholm's theorems and is a result in Fredholm theory. It may be expressed in several ways, as a theorem of linear algebra, a theorem of integral equations, or as a theorem on Fredholm operators. Part of the result states that a non- zero complex number in the spectrum of a compact operator is an eigenvalue.

For general point processes, Campbell's theorem is only for sums of functions of a single point of the point process. To calculate the sum of a function of a single point as well as the entire point process, then generalized Campbell's theorems are required using the Palm distribution of the point process, which is based on the branch of probability known as Palm theory or Palm calculus.

Equivalence here means that in the presence of the other axioms of the geometry each of these theorems can be assumed to be true and the parallel postulate can be proved from this altered set of axioms. This is not the same as logical equivalence.An appropriate example of logical equivalence is given by Playfair's axiom and Euclid I.30 (see Playfair's axiom#Transitivity of parallelism).

A patent gives inventors the right to exclude others from making, using, or selling an invention. A patented invention must be something new, useful, and ingenious. Patents can be obtained for products, apparatuses, manufacturing processes, chemical compositions, and significant improvements to existing inventions. Patents may not generally be obtained for scientific principles, abstract theorems, ideas, methods of conducting business, computer programs, and medical treatments.

Letter from Goldbach to Euler, 1742 Goldbach is most noted for his correspondence with Leibniz, Euler, and Bernoulli, especially in his 1742 letter to Euler stating his Goldbach's conjecture. He also studied and proved some theorems on perfect powers, such as the Goldbach–Euler theorem, and made several notable contributions to analysis. He also proved a result concerning Fermat numbers that is called Goldbach's theorem.

In 1978 Gabber received a Ph.D. from Harvard University for the thesis Some theorems on Azumaya algebras, written under the supervision of Barry Mazur. He has been at the Institut des Hautes Études Scientifiques in Bures-sur-Yvette in Paris since 1984 as a CNRS senior researcher. He won the Erdős Prize in 1981 and the Prix Thérèse Gautier from the French Academy of Sciences in 2011.

An elementary proof is a proof which only uses basic techniques. More specifically, the term is used in number theory to refer to proofs that make no use of complex analysis. For some time it was thought that certain theorems, like the prime number theorem, could only be proved using "higher" mathematics. However, over time, many of these results have been reproved using only elementary techniques.

While the statement of this theorem seems to be intuitively obvious, it takes some ingenuity to prove it by elementary means. "Although the JCT is one of the best known topological theorems, there are many, even among professional mathematicians, who have never read a proof of it." (). More transparent proofs rely on the mathematical machinery of algebraic topology, and these lead to generalizations to higher-dimensional spaces.

Buss (1986) proved that \Sigma^b_1 theorems of S^1_2 are witnessed by polynomial-time functions. > Theorem (Buss 1986) Assume that S^1_2\vdash\forall x\exists y \phi(x,y), > with \phi\in\Sigma^b_1. Then, there exists a PV-function symbol f such that > PV\vdash \forall x \phi(x,f(x)). Moreover, S^1_2 can \Sigma^b_1-define all polynomial-time functions.

A number of famous conjectures and theorems in number theory would follow immediately from the abc conjecture or its versions. described the abc conjecture as "the most important unsolved problem in Diophantine analysis". The abc conjecture originated as the outcome of attempts by Oesterlé and Masser to understand the Szpiro conjecture about elliptic curves,. which involves more geometric structures in its statement than the abc conjecture.

Real Magnets usually do not have a continuous symmetry, since the spin-orbit coupling of the electrons imposes an anisotropy. For atomic systems like graphene, one can show that monolayers of cosmological (or at least continental) size are necessary to measure a significant size of the amplitudes of fluctuations. A recent discussion about the Mermin-Wagner-Hohenberg-Theorems and its limitations is given by Bertrand Halperin.

The mathematical disciplines of combinatorics and dynamical systems interact in a number of ways. The ergodic theory of dynamical systems has recently been used to prove combinatorial theorems about number theory which has given rise to the field of arithmetic combinatorics. Also dynamical systems theory is heavily involved in the relatively recent field of combinatorics on words. Also combinatorial aspects of dynamical systems are studied.

No similar result is known for the time complexity classes, and indeed it is conjectured that NP is not equal to co-NP. The principle used to prove the theorem has become known as inductive counting. It has also been used to prove other theorems in computational complexity, including the closure of LOGCFL under complementation and the existence of error-free randomized logspace algorithms for USTCON..

With the definitions of multiple integration and partial derivatives, key theorems can be formulated, including the fundamental theorem of calculus in several real variables (namely Stokes' theorem), integration by parts in several real variables, the symmetry of higher partial derivatives and Taylor's theorem for multivariable functions. Evaluating a mixture of integrals and partial derivatives can be done by using theorem differentiation under the integral sign.

During this time, he developed a theorem commonly known as the John Hope Rule. It consists of two sub-theorems. One, that if a system is not a bona fide tropical storm before crossing the Windward Islands, or the Lesser Antilles, it will not survive the trek across the Eastern Caribbean Sea. If the wave is still present, formation in the Western Caribbean is possible.

Wos studied at the University of Chicago, receiving a bachelor's degree in 1950 and a master's in mathematics in 1954, and went on for doctoral studies at the University of Illinois at Urbana-Champaign. He joined Argonne in 1957, and began using computers to prove mathematical theorems in 1963.. Wos was congenitally blind. He was an avid bowler, the best male blind bowler in the US...

In his work on transformation groups, Sophus Lie proved three theorems relating the groups and algebras that bear his name. The first theorem exhibited the basis of an algebra through infinitesimal transformations. The second theorem exhibited structure constants of the algebra as the result of commutator products in the algebra. The third theorem showed these constants are anti-symmetric and satisfy the Jacobi identity.

In 1973, Kobayashi and Takushiro Ochiai proved some rigidity theorems for Kähler manifolds. In particular, if is a closed Kähler manifold and there exists in such that :c_1(M)\geq(n+1)\alpha, then must be biholomorphic to complex projective space. This forms the final part of Yum-Tong Siu and Shing-Tung Yau's proof of the Frankel conjecture.Yum Tong Siu and Shing Tung Yau.

Along with Samuel L. Braunstein, he proved the quantum no-deleting theorem. Similar to the no-cloning theorem, the no-deleting theorem is a fundamental consequence of the linearity of quantum mechanics. This proves that given two copies of an unknown quantum state we cannot delete one copy. The no-cloning and the no-deleting theorems suggest that we can neither create nor destroy quantum information.

Giuseppe Moletti (1531–1588) was an Italian mathematician best known for his Dialogo intorno alla Meccanica (Dialogue on Mechanics). Though an obscure figure today, he was a renowned mathematician during his lifetime, and was even consulted by Pope Gregory XIII on his new calendar. He held the mathematics chair at the University of Padua, preceding Galileo, who had sent him his theorems on the centre of gravity.

In information theory, the Cheung-Marks theorem,J.L. Brown and S.D.Cabrera, "On well-posedness of the Papoulis generalized sampling expansion," IEEE Transactions on Circuits and Systems, May 1991 Volume: 38 , Issue 5, pp. 554-556 named after K. F. Cheung and Robert J. Marks II, specifies conditionsK.F. Cheung and R. J. Marks II, "Ill-posed sampling theorems", IEEE Transactions on Circuits and Systems, vol.

Mel O'Cat designed a system called Mmj2, which provides a graphic user interface for proof entry. The initial aim of Mel O'Cat was to allow the user to enter the proofs by simply typing the formulas and letting Mmj2 find the appropriate inference rules to connect them. In Metamath on the contrary you may only enter the theorems names. You may not enter the formulas directly.

Dewar graduated summa cum laude in 1968 from Saint Louis University, and earned her Ph.D. from the University of Southern California in 1973. Her dissertation, Coincidence Theorems for Set Valued Mappings, was supervised by James Dugundji. She was on the faculty of Loyola Marymount from 1973 until her retirement in 2013, and chaired the mathematics department there from 1983 to 1986 and again from 2005 to 2006.

The hierarchy makes proving certain kinds of theorems about Haken manifolds a matter of induction. One proves the theorem for 3-balls. Then one proves that if the theorem is true for pieces resulting from a cutting of a Haken manifold, then it is true for that Haken manifold. The key here is that the cutting takes place along a surface that was very "nice", i.e.

Wigner and Hermann Weyl were responsible for introducing group theory into physics, particularly the theory of symmetry in physics. Along the way he performed ground-breaking work in pure mathematics, in which he authored a number of mathematical theorems. In particular, Wigner's theorem is a cornerstone in the mathematical formulation of quantum mechanics. He is also known for his research into the structure of the atomic nucleus.

Autònoma de Barcelona, 2006. p.48 and on In mathematics, a definition is used to give a precise meaning to a new term, by describing a condition which unambiguously qualifies what a mathematical term is and is not. Definitions and axioms form the basis on which all of modern mathematics is to be constructed.Richard J. Rossi (2011) Theorems, Corollaries, Lemmas, and Methods of Proof.

In abstract algebra, a branch of mathematics, the algebraic structure group with operators or Ω-group can be viewed as a group with a set Ω that operates on the elements of the group in a special way. Groups with operators were extensively studied by Emmy Noether and her school in the 1920s. She employed the concept in her original formulation of the three Noether isomorphism theorems.

Every algebraic extension of a field of characteristic zero is separable, and every algebraic extension of a finite field is separable.Isaacs, Theorem 18.11, p. 281 It follows that most extensions that are considered in mathematics are separable. Nevertheless, the concept of separability is important, as the existence of inseparable extensions is the main obstacle for extending many theorems proved in characteristic zero to non-zero characteristic.

Model theory is the branch of mathematical logic that deals with the relation between a formal theory and its interpretations, called models.Chang and Keisler, p. 1 A theory consists of a set of sentences in a formal language, which consists generally of the axioms of the theory, and all theorems that can be deduced from them. A model is a realization of the theory inside another theory.

Herbrand's theorem refers to either of two completely different theorems. One is a result from his doctoral thesis in proof theory, and the other one half of the Herbrand–Ribet theorem. The Herbrand quotient is a type of Euler characteristic, used in homological algebra. He contributed to Hilbert's program in the foundations of mathematics by providing a constructive consistency proof for a weak system of arithmetic.

He is the author of four monographs and more than 90 scientific articles. Feldman specializes in the field of abstract harmonic analysis and algebraic probability theory. He constructed a theory of decompositions of random variables and proved analogs of the classical characterization theorems of mathematical statistics in the case when random variables take values in various classes of locally compact Abelian groups (discrete, compact, and others).

In 1931, Kurt Gödel published the incompleteness theorems, which he proved in part by showing how to represent the syntax of formal logic within first- order arithmetic. Each expression of the formal language of arithmetic is assigned a distinct number. This procedure is known variously as Gödel numbering, coding and, more generally, as arithmetization. In particular, various sets of expressions are coded as sets of numbers.

In the years following Gödel's theorems, as it became clear that there is no hope of proving consistency of mathematics, and with development of axiomatic set theories such as Zermelo–Fraenkel set theory and the lack of any evidence against its consistency, most mathematicians lost interest in the topic. Today most classical mathematicians are considered Platonist and readily use infinite mathematical objects and a set-theoretical universe.

She briefly worked as a school teacher before she decided to continue her studies at the University of Łódź where she became a lecturer at the Faculty of Mathematics. She obtained her PhD degree in mathematics in 2001 after she wrote her doctoral dissertation entitled Limit Theorems in Quantum Probability. She later assumed the position of assistant professor at the Department of Probability Theory and Statistics.

Starting in 1998, the application of the Rayleigh theorem for eigenvalues has led to mostly accurate, calculated band gaps of materials, using LDA potentials. A misunderstanding of the second theorem of DFT appears to explain most of the underestimation of band gap by LDA and GGA calculations, as explained in the description of density functional theory, in connection with the statements of the two theorems of DFT.

He also proved several theorems concerning convergence of sequences of measurable and holomorphic functions. The Vitali convergence theorem generalizes Lebesgue's dominated convergence theorem. Another theorem bearing his name gives a sufficient condition for the uniform convergence of a sequence of holomorphic functions on an open domain. This result has been generalized to normal families of meromorphic functions, holomorphic functions of several complex variables, and so on.

Except for metals, most of these other materials have an energy or a band gap, i.e., the difference between the lowest, unoccupied energy and the highest, occupied energy. For crystals, the energy spectrum is in bands and there is a band gap, if any, as opposed to energy gap. Given the diverse contributions of Lord Rayleigh, his name is associated with other theorems, including Parseval's theorem.

S Sethi and M Stern, Invariance theorems for supersymmetric Yang-Mills theories, Advances in Theoretical and Mathematical Physics, vol. 4 no. 2 (2000), pp. 1–12, ISSN 1095-0761 [hep-th/0001189] [abs] 16\. S Sethi and M Stern, The structure of the D0-D4 bound state, Nuclear Physics B, vol. 578 no. 1-2 (2000), pp. 163–198 [hep-th/0002131] [abs] 17\.

Paul Finsler (1926) used a version of Richard's paradox to construct an expression that was false but unprovable in a particular, informal framework he had developed. Gödel was unaware of this paper when he proved the incompleteness theorems (Collected Works Vol. IV., p. 9). Finsler wrote to Gödel in 1931 to inform him about this paper, which Finsler felt had priority for an incompleteness theorem.

PLoS ONE (in press): e24274. uses a linguistic version of category theory to model a given situation. Akin to entity-relationship models, custom categories or sketches can be directly translated into database schemas. The difference is that logic is replaced by category theory, which brings powerful theorems to bear on the subject of modeling, especially useful for translating between disparate models (as functors between categories).

Poisson structures are one instance of Jacobi structures introduced by André Lichnerowicz in 1977. They were further studied in the classical paper of Alan Weinstein, where many basic structure theorems were first proved, and which exerted a huge influence on the development of Poisson geometry — which today is deeply entangled with non-commutative geometry, integrable systems, topological field theories and representation theory, to name a few.

His theorems also became the basis for numerical methods that he developed to perform the requisite calculations. For this purpose, he created a computer code, UNIMOL, which is widely used by researchers. He developed, with Prof J Troe, easily used approximate solutions for the pressure dependence of the rate coefficient. He provided the first solutions for cases where angular momentum conservation needs to be incorporated.

William Forrest "Woody" Stinespring (16 March 1929, Charlottesville, Virginia – 15 May 2012) was an American mathematician, specializing in operator theory. He is known for the Stinespring factorization theorem. After graduating with a bachelor's degree from Harvard University, Stinespring received his Ph.D. from the University of Chicago in 1957. His thesis Integration for gages and duality theorems was written under the supervision of Irving Segal.

Pugh's closing lemma means, for example, that any chaotic set in a bounded continuous dynamical system corresponds to a periodic orbit in a different but closely related dynamical system. As such, an open set of conditions on a bounded continuous dynamical system that rules out periodic behaviour also implies that the system cannot behave chaotically; this is the basis of some autonomous convergence theorems.

In 1942, he provided one of the first proofs of the First and Second Welfare Theorems. He initiated the analysis of stability of general equilibrium (1942, 1944). His critique of the quantity theory of money (1942) prompted his student Don Patinkin to develop his remarkable "integration" of money into general equilibrium theory. Lange made several seminal contributions to the development of neoclassical synthesis (1938, 1943, 1944).

Another mechanism for proof automation is proof search action in emacs mode. It enumerates possible proof terms (limited to 5 seconds), and if one of the terms fits the specification, it will be put in the meta variable where the action is invoked. This action accepts hints, e.g., which theorems and from which modules can be used, whether the action can use pattern matching, etc.

The Elements ( Stoicheia) is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt c. 300 BC. It is a collection of definitions, postulates, propositions (theorems and constructions), and mathematical proofs of the propositions. The books cover plane and solid Euclidean geometry, elementary number theory, and incommensurable lines. Elements is the oldest extant large-scale deductive treatment of mathematics.

Reciprocity is also a basic lemma that is used to prove other theorems about electromagnetic systems, such as the symmetry of the impedance matrix and scattering matrix, symmetries of Green's functions for use in boundary-element and transfer-matrix computational methods, as well as orthogonality properties of harmonic modes in waveguide systems (as an alternative to proving those properties directly from the symmetries of the eigen-operators).

Some consequences of AD followed from theorems proved earlier by Stefan Banach and Stanisław Mazur, and Morton Davis. Mycielski and Stanisław Świerczkowski contributed another one: AD implies that all sets of real numbers are Lebesgue measurable. Later Donald A. Martin and others proved more important consequences, especially in descriptive set theory. In 1988, John R. Steel and W. Hugh Woodin concluded a long line of research.

Ivan Vidav 1963 Ivan Vidav (January 17, 1918 – October 6, 2015) was a Slovenian mathematician. Ivan Vidav was born in Opčine near Trieste (Slovenian Trst), Italy. He was Josip Plemelj's student. Vidav got his Ph.D. under Plemelj's advisory in 1941 at the University of Ljubljana with a dissertation Kleinovi teoremi v teoriji linearnih diferencialnih enačb (Klein's theorems in the theory of linear differential equations).

He was born in Paris on 24 July 1856 and educated there at the Lycée Henri-IV. He then studied Mathematics at the École Normale Supérieure. Picard's mathematical papers, textbooks, and many popular writings exhibit an extraordinary range of interests, as well as an impressive mastery of the mathematics of his time. Modern students of complex variables are probably familiar with two of his named theorems.

The front cover has a picture of the handwritten Poisson's equations for electricity and magnetism on a chalkboard. The first inner cover contains vector identities, vector derivatives in Cartesian, spherical, and cylindrical coordinates, and the fundamental theorems of vector calculus. The second inner cover contains the basic equations of electrodynamics, the accepted values of some fundamental constants, and the transformation equations for spherical and cylindrical coordinates.

Wiener's Tauberian theorem, a 1932 result of Wiener, developed Tauberian theorems in summability theory, on the face of it a chapter of real analysis, by showing that most of the known results could be encapsulated in a principle taken from harmonic analysis. In its present formulation, the theorem of Wiener does not have any obvious association with Tauberian theorems, which deal with infinite series; the translation from results formulated for integrals, or using the language of functional analysis and Banach algebras, is however a relatively routine process. The Paley–Wiener theorem relates growth properties of entire functions on Cn and Fourier transformation of Schwartz distributions of compact support. The Wiener–Khinchin theorem, (also known as the Wiener – Khintchine theorem and the Khinchin – Kolmogorov theorem), states that the power spectral density of a wide-sense-stationary random process is the Fourier transform of the corresponding autocorrelation function.

Propositional logic is a logical system that is intimately connected to Boolean algebra. Many syntactic concepts of Boolean algebra carry over to propositional logic with only minor changes in notation and terminology, while the semantics of propositional logic are defined via Boolean algebras in a way that the tautologies (theorems) of propositional logic correspond to equational theorems of Boolean algebra. Syntactically, every Boolean term corresponds to a propositional formula of propositional logic. In this translation between Boolean algebra and propositional logic, Boolean variables x,y... become propositional variables (or atoms) P,Q,..., Boolean terms such as x∨y become propositional formulas P∨Q, 0 becomes false or ⊥, and 1 becomes true or T. It is convenient when referring to generic propositions to use Greek letters Φ, Ψ,... as metavariables (variables outside the language of propositional calculus, used when talking about propositional calculus) to denote propositions.

An exotic R4 is a differentiable manifold that is homeomorphic but not diffeomorphic to the Euclidean space R4. The first examples were found in the early 1980s by Michael Freedman, by using the contrast between Freedman's theorems about topological 4-manifolds, and Simon Donaldson's theorems about smooth 4-manifolds.. There is a continuum of non-diffeomorphic differentiable structures of R4, as was shown first by Clifford Taubes.Theorem 1.1 of Prior to this construction, non-diffeomorphic smooth structures on spheres—exotic spheres—were already known to exist, although the question of the existence of such structures for the particular case of the 4-sphere remained open (and still remains open as of 2018). For any positive integer n other than 4, there are no exotic smooth structures on Rn; in other words, if n ≠ 4 then any smooth manifold homeomorphic to Rn is diffeomorphic to Rn.Corollary 5.2 of .

The main directions of I. A. Panin's work are the theory of oriented cohomology on algebraic varieties, algebraic K-theory of homogeneous varieties, Gersten's conjecture, the Grothendieck-Serre conjecture on principal G-bundles, and purity in algebraic geometry.. I. A. Panin proved (together with A. L. Smirnov) theorems of the Riemann-Roch type for oriented cohomology theories and Riemann-Roch type theorems for the Adams operation. Panin found a proof of Gersten's conjecture in the case of equal characteristic and an affirmative solution (jointly with Manuel Ojanguren) of the "purity" problem for quadratic forms. Panin computed the algebraic K-groups of all twisted forms of flag varieties and all principal homogeneous spaces over the inner forms of semisimple algebraic groups. He, jointly with A. S. Merkurjev and A. R. Wadsworth, generalized, to arbitrary Borel varieties, results proved by David Tao concerning index reduction formulas for the function fields of involution varieties.

' To Newton and to Newton's dog Diamond, what a different pair of Universes; while the painting on the optical retina of both was, most likely, the same!" Nevertheless, Diamond is the subject of several anecdotes concerning Newton. In another tale, Newton is said to have claimed that the dog discovered two theorems in a single morning. He added, however, that "one had a mistake and the other had a pathological exception.

PACELC builds on the CAP theorem. Both theorems describe how distributed databases have limitations and tradeoffs regarding consistency, availability, and partition tolerance. PACELC however goes further and states that another trade-off also exists: this time between latency and consistency, even in absence of partitions, thus providing a more complete portrayal of the potential consistency tradeoffs for distributed systems. A high availability requirement implies that the system must replicate data.

The Hilbert–Bernays provability conditions, combined with the diagonal lemma, allow proving both of Gödel's incompleteness theorems shortly. Indeed the main effort of Godel's proofs lied in showing that these conditions (or equivalent ones) and the diagonal lemma hold for Peano arithmetics; once these are established the proof can be easily formalized. Using the diagonal lemma, there is a formula \rho such that T \Vdash \rho \leftrightarrow eg Prov(\\#(\rho)).

2 #8 Dr. Strange is in the middle of investigating the Empirikul. He gives Beast the Third Eye of Horus to show him the realm between science and mysticism. Beast then shows him that even magic has a scientific base and points out his spells are based on mathematical theorems and there are flaws in the computations. Dr. Strange thanks him for enlightening him and leaves him the mask.

A spiral similarity taking triangle ABC to triangle A'B'C'. Spiral similarity is a plane transformation in mathematics composed of a rotation and a dilation. It is used widely in Euclidean geometry to facilitate the proofs of many theorems and other results in geometry, especially in mathematical competitions and Olympiads. Though the origin of this idea is not known, it was documented in 1967 by Coxeter in his book Geometry Revisited.

Network synthesis on the other hand, takes care of the termination impedances simply by incorporating them into the network being designed.Matthaei, pp.83–84 The development of network analysis needed to take place before network synthesis was possible. The theorems of Gustav Kirchhoff and others and the ideas of Charles Steinmetz (phasors) and Arthur Kennelly (complex impedance)Arthur E. Kennelly, 1861 – 1939 IEEE biography, retrieved 13 June 2009 laid the groundwork.

As of July 2012, the MML included 1150 articles written by 241 authors.The MML Query search engine In aggregate, these contain more than 10,000 formal definitions of mathematical objects and about 52,000 theorems proved on these objects. More than 180 named mathematical facts have so benefited from formal codification. Some examples are the Hahn–Banach theorem, Kőnig's lemma, Brouwer fixed point theorem, Gödel's completeness theorem and Jordan curve theorem.

It is NP-complete to test whether a graph is pancyclic, even for the special case of 3-connected cubic graphs, and it is also NP-complete to test whether a graph is node-pancyclic, even for the special case of polyhedral graphs., Theorems 2.3 and 2.4. It is also NP- complete to test whether the square of a graph is Hamiltonian, and therefore whether it is pancyclic.

George Sterman's research focuses on quantum field theory and its applications in quantum chromodynamics. With Steven Weinberg he proved the infrared finiteness of jet cross sections, thus proving that perturbation theory is a safe method in that regime. He also worked on reformulation and proof of factorization theorems with Stephen Libby, John C. Collins and Davison E. Soper. He authored a textbook entitled An Introduction to Quantum Field Theory in 1993.

In mathematics, specifically the field of transcendental number theory, the four exponentials conjecture is a conjecture which, given the right conditions on the exponents, would guarantee the transcendence of at least one of four exponentials. The conjecture, along with two related, stronger conjectures, is at the top of a hierarchy of conjectures and theorems concerning the arithmetic nature of a certain number of values of the exponential function.

Baker's theorem, while the lower two rows are detailed at the six exponentials theorem article. The strongest result that has been conjectured in this circle of problems is the strong four exponentials conjecture.Waldschmidt, (2000), conjecture 11.17. This result would imply both aforementioned conjectures concerning four exponentials as well as all the five and six exponentials conjectures and theorems, as illustrated to the right, and all the three exponentials conjectures detailed below.

Afterward, they moved on to New York (Brooklyn) and New Jersey, where Baldor continued teaching at Saint Peter's College in Jersey City. He also taught daily classes in mathematics at the now defunct Stevens Academy, in Hoboken, New Jersey. He spent much time writing mathematical theorems and exercises. Once a tall and imposing man weighing 100 kg (220 lbs), Baldor slowly began losing weight as his health declined.

Manin works were influenced by the quantum group theory. He discovered that quantized algebra of functions Funq(GL) can be defined by the requirement that T and Tt are simultaneously q-Manin matrices. In that sense it should be stressed that (q)-Manin matrices are defined only by half of the relations of related quantum group Funq(GL), and these relations are enough for many linear algebra theorems.

E Witten, Supersymmetry and Morse theory. J.Diff.Geom.17:661–692,1982. The electromagnetic field of a plane wave without sources is nilpotent when it is expressed in terms of the algebra of physical space.Rowlands, P. Zero to Infinity: The Foundations of Physics, London, World Scientific 2007, More generally, the technique of microadditivity used to derive theorems makes use of nilpotent or nilsquare infinitesimals, and is part smooth infinitesimal analysis.

Over 30 works were published under the name, including whimsical poetry and mathematical humour, but some serious mathematical results as well. Many of these publications appeared in Eureka, a mathematical student magazine in Cambridge. Notably, the foursome proved several theorems in mathematical tessellation. In particular, they solved the problem of squaring the square, showing that a square can be divided into smaller squares, no two of which are the same.

Carl Eddie Hewitt () is an American computer scientist who designed the Planner programming language for automated planningCarl Hewitt. PLANNER: A Language for Proving Theorems in Robots IJCAI. 1969. and the actor model of concurrent computation, which have been influential in the development of logic, functional and object-oriented programming. Planner was the first programming language based on procedural plans invoked using pattern-directed invocation from assertions and goals.

Therefore, a theory is needed that integrates relativity theory and quantum theory.Stephen Hawking wrote 1999: So what the singularity theorems are really telling us, is that the universe had a quantum origin, and that we need a theory of quantum cosmology, if we are to predict the present state of the universe. Such an approach is attempted for instance with loop quantum gravity, string theory and causal set theory.

Many fixed-point theorems yield algorithms for locating the least fixed point. Least fixed points often have desirable properties that arbitrary fixed points do not. In mathematical logic and computer science, the least fixed point is related to making recursive definitions (see domain theory and/or denotational semantics for details). Immerman N. Immerman, Relational queries computable in polynomial time, Information and Control 68 (1–3) (1986) 86–104.

In 1968 he was awarded the D.Phil. by the University of Sussex for a thesis on "Structure Theorems for Linear Groups" done under advisor Walter Ledermann. He immediately joined the staff at University College Dublin, from which he officially retired in 2009, but he has continued to publish regularly. His research has focussed on group theory, and later linear algebra too, and he has supervised five Ph.D. students.

In mathematics, the theorem of Bertini is an existence and genericity theorem for smooth connected hyperplane sections for smooth projective varieties over algebraically closed fields, introduced by Eugenio Bertini. This is the simplest and broadest of the "Bertini theorems" applying to a linear system of divisors; simplest because there is no restriction on the characteristic of the underlying field, while the extensions require characteristic 0.Hartshorne, Ch. III.10.

Ideals and quotient rings can be defined for rngs in the same manner as for rings. The ideal theory of rngs is complicated by the fact that a nonzero rng, unlike a nonzero ring, need not contain any maximal ideals. Some theorems of ring theory are false for rngs. A rng homomorphism maps any idempotent element to an idempotent element; this applies in particular to 1R if it exists.

Arrow's and Gibbard's theorems prove that no system using ranked voting or cardinal voting, can meet all such criteria simultaneously. Instead of debating the importance of different criteria, another method is to simulate many elections with different electoral systems, and estimate the typical overall happiness of the population with the results, their vulnerability to strategic voting, their likelihood of electing the candidate closest to the average voter, etc.

His doctoral advisor was Witold Hurewicz. His dissertation thesis was titled Some Fixed Point Theorems He has worked as a professor of mathematics at UCLA, where supervised the PhDs of eight doctoral students. He made a foundational contribution to the theory of Riemannian submersions, showing how geometric quantities on the total space and on the base are related to one another. "O'Neill's formula" refers to the relation between the sectional curvatures.

Tang's research was mainly focused on quantum chemistry, polymer chemistry, and polymer physics. In the 1950s, he pioneered a method to calculate the "potential function of molecular internal rotation". He later made contributions to the ligand field theory and developed three graph theorems of molecular orbital. He co-authored eight monographs and was conferred four consecutive State Natural Science Awards (including two first-class awards), an unprecedented achievement.

Even though every equilibrium is efficient, neither of the above two theorems say anything about the equilibrium existing in the first place. To guarantee that an equilibrium exists, it suffices that consumer preferences be strictly convex. With enough consumers, the convexity assumption can be relaxed both for existence and the second welfare theorem. Similarly, but less plausibly, convex feasible production sets suffice for existence; convexity excludes economies of scale.

Unusually for the time, instead of students merely reporting on the content of courses, Kronrod made his students undertake training exercises, even proving basic theorems themselves. The preparation required for this reduced the numbers of participants, but those who remained, including R. A. Minlos and A. G. Vitushkin, derived great benefit. Vitushkin described him as "witty and friendly". At his own request, Kronrod was called simply "Sasha" by his students.

They also form the background for parameter estimation. In the case of extremum estimators for parametric models, a certain objective function is maximized or minimized over the parameter space. Theorems of existence and consistency of such estimators require some assumptions about the topology of the parameter space. For instance, compactness of the parameter space, together with continuity of the objective function, suffices for the existence of an extremum estimator.

Varian's theorems study it in the context of dividing homogeneous goods. Under mild restrictions on the agents' utility functions, there exist allocations which are both PE and EF. The proof uses a previous result on the existence of a competitive equilibrium from equal incomes (CEEI). David Gale proved a similar existence result for agents with linear utilities. Cake-cutting is more challenging than homogeneous good allocation, since a cake is heterogeneous.

These theorems would indicate that certain types of Tweedie models should have a role as equilibrium distributions in natural systems. They can be used to explain the origin of Taylor's law as well as 1/f noise and multifractality. Consequent to Jørgensen's work the Tweedie distributions and their convergence theorem have provided mechanistic insight into complicated natural systems that manifest features of self-organized criticality and random fractals.

He also served as the college's bursar for a time. Griffiths was particularly interested in analytical geometry, publishing numerous papers in mathematical journals and two tracts on theorems connected with the geometry of the triangle. He was described in his obituary in The Times as "a man of a very sociable and affectionate nature [but] excessively shy". He died in May 1916 in his native village in Carmarthenshire.

He has authored several mathematics textbooks, including the Lectures on Geometric Measure TheoryThis is basically a textbook describing many results in geometric measure theory and the mathematical tools used in this field: see . and An Introduction to Multivariable Mathematics. He published the monograph Theorems on regularity and singularity of energy minimising maps in 1996, based in part on lectures he gave at Eidgenössische Technische Hochschule (ETH) in Zürich.

The max-flow min-cut theorem is a special case of the strong duality theorem: flow-maximization is the primal LP, and cut-minimization is the dual LP. See Max-flow min-cut theorem#Linear program formulation. Other graph-related theorems can be proved using the strong duality theorem, in particular, Konig's theorem. The Minimax theorem for zero-sum games can be proved using the strong-duality theorem.

The Metamath language is a metalanguage, suitable for developing a wide variety of formal systems. The Metamath language has no specific logic embedded in it. Instead, it can simply be regarded as a way to prove that inference rules (asserted as axioms or proven later) can be applied. The largest database of proved theorems follows conventional ZFC set theory and classic logic, but other databases exist and others can be created.

Hadamard's and de la Vallée Poussin's original proofs are long and elaborate; later proofs introduced various simplifications through the use of Tauberian theorems but remained difficult to digest. A short proof was discovered in 1980 by American mathematician Donald J. Newman. Newman's proof is arguably the simplest known proof of the theorem, although it is non-elementary in the sense that it uses Cauchy's integral theorem from complex analysis.

Depending on the prime factorization of n, some restrictions may be placed on the structure of groups of order n, as a consequence, for example, of results such as the Sylow theorems. For example, every group of order pq is cyclic when are primes with not divisible by q. For a necessary and sufficient condition, see cyclic number. If n is squarefree, then any group of order n is solvable.

These two theorems are very different from each other. The first theorem has a very simple proof but leads to some counterintuitive conclusions, while the second theorem has a technical and counterintuitive proof but leads to a less surprising result. The C1 theorem was published in 1954, the Ck-theorem in 1956. The real analytic theorem was first treated by Nash in 1966; his argument was simplified considerably by .

This implies that there are more quadratic residues than nonresidues among the numbers 1, 2, ..., (q − 1)/2. > For example, modulo 11 there are four residues less than 6 (namely 1, 3, 4, > and 5), but only one nonresidue (2). An intriguing fact about these two theorems is that all known proofs rely on analysis; no-one has ever published a simple or direct proof of either statement.

The perfect graphs include many important families of graphs and serve to unify results relating colorings and cliques in those families. For instance, in all perfect graphs, the graph coloring problem, maximum clique problem, and maximum independent set problem can all be solved in polynomial time. In addition, several important min-max theorems in combinatorics, such as Dilworth's theorem, can be expressed in terms of the perfection of certain associated graphs.

" Maudlin's probability criticism confused the transactional interpretation with Heisenberg's knowledge interpretation. However, he raised a valid point concerning causally connected possible outcomes, which led Cramer to add hierarchy to the pseudo-time description of transaction formation.Berkovitz, J. (2002). ``On Causal Loops in the Quantum Realm," in T. Placek and J. Butterfield (Ed.), Proceedings of the NATO Advanced Research Workshop on Modality, Probability and Bell's Theorems, Kluwer, 233–255.

Fondements de la Géometrie Algébrique (FGA) is a book that collected together seminar notes of Alexander Grothendieck. It is an important source for his pioneering work on scheme theory, which laid foundations for algebraic geometry in its modern technical developments. The title is a translation of the title of André Weil's book Foundations of Algebraic Geometry. It contained material on descent theory, and existence theorems including that for the Hilbert scheme.

Intuition why the above theorem should be true, is only partially true and sometimes completely wrong (explicit counterexamples). This is why this result has attracted much attention. The mathematical proof does not use new mathematical methods but is subtle. Apart from a classical result on so-called complete convergence, it is mainly based on theorems for stopping times on sums of independent and identically distributed order statistics (ref.

His collected works (Gesammelte Abhandlungen) have been published several times. The original versions of his papers contained "many technical errors of varying degree";Reid, chap.13 when the collection was first published, the errors were corrected and it was found that this could be done without major changes in the statements of the theorems, with one exception—a claimed proof of the continuum hypothesis.Page 284f in: Rota G.-C.

These concerns, however, are by no means mutually exclusive. A "methodological" approach is concerned with practicing theology in a "scientific" manner and focuses on clearly articulating the assumptions, methods, and related thought-forms to be taken into account in the construction of dogmatic formulations. A "doctrinal" approach is concerned with the inter-relationship of scientific and doctrinal content and focuses on formulating Christian theology against a framework of specific scientific theorems.

In mathematics, a Paley–Wiener theorem is any theorem that relates decay properties of a function or distribution at infinity with analyticity of its Fourier transform. The theorem is named for Raymond Paley (1907–1933) and Norbert Wiener (1894–1964). The original theorems did not use the language of distributions, and instead applied to square-integrable functions. The first such theorem using distributions was due to Laurent Schwartz.

In 1932, Hohenemser and Prager proposed the first model for slow viscoplastic flow. This model provided a relation between the deviatoric stress and the strain rate for an incompressible Bingham solidBingham, E. C. (1922) Fluidity and plasticity. McGraw-Hill, New York. However, the application of these theories did not begin before 1950, where limit theorems were discovered. In 1960, the first IUTAM Symposium “Creep in Structures” organized by HoffHoff, ed.

Thus, regular spaces are usually studied to find properties and theorems, such as the ones below, that are actually applied to completely regular spaces, typically in analysis. There exist Hausdorff spaces that are not regular. An example is the set R with the topology generated by sets of the form U — C, where U is an open set in the usual sense, and C is any countable subset of U.

Hyperbolic geometry is more closely related to Euclidean geometry than it seems: the only axiomatic difference is the parallel postulate. When the parallel postulate is removed from Euclidean geometry the resulting geometry is absolute geometry. There are two kinds of absolute geometry, Euclidean and hyperbolic. All theorems of absolute geometry, including the first 28 propositions of book one of Euclid's Elements, are valid in Euclidean and hyperbolic geometry.

Bogart was originally from Cincinnati, and was a 1965 graduate of Marietta College. He earned his Ph.D. in 1968 at the California Institute of Technology. His dissertation, Structure Theorems for Local Noether Lattices, was supervised by Robert P. Dilworth. He joined the faculty of the Dartmouth College mathematics department in 1968, was promoted to full professor in 1980, and was chair of the department from 1989 to 1995.

For instance, she sought to disprove the validity of some of Husserlian theories through her investigations on dependency. In the paper, Zur Husserlschen Lehre von den Ganzen und den Teilen (On Husserl's Theory of Wholes and Parts), Ginsberg discussed six of Husserl's theories. She offered proofs to theorems 1 and 3, validated theorem 5, but countered the three others. She also developed theories on descriptive psychology based on Husserlian thought.

A logical machine is a tool containing a set of parts that uses energy to perform formal logic operations. Early logical machines were mechanical devices that performed basic operations in Boolean logic. Contemporary logical machines are computer-based electronic programs that perform proof assistance with theorems in mathematical logic. In the 21st century, these proof assistant programs have given birth to a new field of study called mathematical knowledge management.

It is difficult to determine what methodology the Oxford 'Calculators' used when they were conjecturing and postulating theorems by way of abstraction (i.e., without empirical investigation). This criticism is not expressly made toward Dumbleton's conjectures but more broadly aimed at the methodology of the whole group of Mertonian physicists. One suggestion is that they may have been trying to create a mathematical picture of the Aristotelian world-view.

Although the above correspondence with holomorphic functions only holds for functions of two real variables, harmonic functions in n variables still enjoy a number of properties typical of holomorphic functions. They are (real) analytic; they have a maximum principle and a mean- value principle; a theorem of removal of singularities as well as a Liouville theorem holds for them in analogy to the corresponding theorems in complex functions theory.

Similarly it is a "coincidence" if ab = ba, of any other law of algebra holds. Fortunately, we can show that the required coincidences actually occur, because they are implied by certain geometric coincidences, namely the Pappus and Desargues theorems. If one interprets von Staudt’s work as a construction of the real numbers, then it is incomplete. One of the required properties is that a bounded sequence has a cluster point.

He warned against interpreting "positive" as being morally or aesthetically "good" (the greatest advantage and least disadvantage), as this includes negative characteristics. Instead, he suggested that "positive" should be interpreted as being perfect, or "purely good", without negative characteristics. Gödel's listed theorems follow from the axioms, so most criticisms of the theory focus on those axioms or the assumptions made. Oppy argued that Gödel gives no definition of "positive properties".

In mathematical logic, a formula is said to be absolute if it has the same truth value in of structures (also called models). Theorems about absoluteness typically establish relationships between the absoluteness of formulas and their syntactic form. There are two weaker forms of partial absoluteness. If the truth of a formula in each substructure N of a structure M follows from its truth in M, the formula is downward absolute.

Theorems have generally a title or label in bold type, and might even identify its originator (e.g., ""). This is immediately followed by the statement of the theorem, which in turn is usually set in italics. The proof of a theorem is usually clearly delimited, starting with the word Proof, while the end of the proof is indicated by a tombstone ("∎ or □") or another symbol, or by the letters Q.E.D..

In differential geometry, the Atiyah–Singer index theorem, proved by , states that for an elliptic differential operator on a compact manifold, the analytical index (related to the dimension of the space of solutions) is equal to the topological index (defined in terms of some topological data). It includes many other theorems, such as the Chern–Gauss–Bonnet theorem and Riemann–Roch theorem, as special cases, and has applications to theoretical physics.

In mathematics, the Artin approximation theorem is a fundamental result of in deformation theory which implies that formal power series with coefficients in a field k are well-approximated by the algebraic functions on k. More precisely, Artin proved two such theorems: one, in 1968, on approximation of complex analytic solutions by formal solutions (in the case k = \Complex); and an algebraic version of this theorem in 1969.

No longer satisfied with establishing properties of concrete objects, mathematicians started to turn their attention to general theory. Formal definitions of certain algebraic structures began to emerge in the 19th century. For example, results about various groups of permutations came to be seen as instances of general theorems that concern a general notion of an abstract group. Questions of structure and classification of various mathematical objects came to forefront.

1832); (2)' On the Determination of the Attractions of Ellipsoids of Variable Densities ' (6 May 1833). Both papers display great analytical power, but are rather curious than practically interesting. Green's 1828 essay was neglected by mathematicians till 1846, and before that time most of its important theorems had been rediscovered by Gauss, Chasles, Sturm, and Thomson J.Maxwell, J. C. (1881). A treatise on electricity and magnetism. p. 14.

Chapter one is titled "Preliminary Notions". The ten sections explicate notions of set theory, vector spaces, homomorphisms, duality, linear equations, group theory, field theory, ordered fields and valuations. On page vii Artin says "Chapter I should be used mainly as a reference chapter for the proofs of certain isolated theorems." Pappus's hexagon theorem holds if and only if k is commutative Chapter two is titled "Affine and Projective Geometry".

He concluded "that if we humans say anything authentic about God, we can do so only on the basis of divine self-revelation; all other God-talk is conjectural." In his magnum opus he presented a version of Christian apologetics called presuppositional apologetics. Henry regarded all truth as propositional, and Christian doctrine as "the theorems derived from the axioms of revelation." His autobiography, Confessions of a Theologian, was published in 1986.

The method of coordinates (analytic geometry) was adopted by René Descartes in 1637. At that time, geometric theorems were treated as absolute objective truths knowable through intuition and reason, similar to objects of natural science; and axioms were treated as obvious implications of definitions. Two equivalence relations between geometric figures were used: congruence and similarity. Translations, rotations and reflections transform a figure into congruent figures; homotheties — into similar figures.

In mathematics, especially homological algebra and other applications of abelian category theory, the five lemma is an important and widely used lemma about commutative diagrams. The five lemma is not only valid for abelian categories but also works in the category of groups, for example. The five lemma can be thought of as a combination of two other theorems, the four lemmas, which are dual to each other.

For the case of perfect fractional matchings, both the above theorems can derived from the colorful Caratheodory theorem in the previous section. For a general r-uniform hypergraph (admitting a perfect matching of size n), the vectors 1e live in a (rn)-dimensional space. For an r-uniform r-partite hypergraph, the r-partiteness constraints imply that the vectors 1e live in a (rn-r+1)-dimensional space.

189-190 claims that he, Al Newell, and Cliff Shaw are "commonly adjudged to be the parents of [the] artificial intelligence [field]", for writing Logic Theorist, a program which proved theorems from Principia Mathematica automatically. In order to accomplish this, they had to invent a language and a paradigm which, which viewed retrospectively, embeds functional programming. were both "impure" functional languages by the current definition. Purely functional data structures are persistent.

In 1920, after receiving a scholarship from Emmanouíl Benákis, he was sent to Paris to continue his studies. There, by perfecting the theorems of Georgios Remoundos, he began to send scientific papers to the French Academy of Sciences. These works contributed to the decision of University of Paris to give him the possibility of obtaining a doctoral degree on the basis of his dissertation alone, without taking examinations.

The following theorems can be regarded as directed versions: :Ghouila-Houiri (1960). A strongly connected simple directed graph with n vertices is Hamiltonian if every vertex has a full degree greater than or equal to n. :Meyniel (1973). A strongly connected simple directed graph with n vertices is Hamiltonian if the sum of full degrees of every pair of distinct non-adjacent vertices is greater than or equal to .

His art is based on mathematical principles like tessellations, spherical geometry, the Möbius strip, unusual perspectives, visual paradoxes and illusions, different kinds of symmetries and impossible objects. Gödel, Escher, Bach by Douglas Hofstadter discusses the ideas of self-reference and strange loops, drawing on a wide range of artistic and scientific work, including Escher's art and the music of J. S. Bach, to illustrate ideas behind Gödel's incompleteness theorems.

Coxeter 2003, p. 14 Projective geometry, like affine and Euclidean geometry, can also be developed from the Erlangen program of Felix Klein; projective geometry is characterized by invariants under transformations of the projective group. After much work on the very large number of theorems in the subject, therefore, the basics of projective geometry became understood. The incidence structure and the cross-ratio are fundamental invariants under projective transformations.

Wittgenstein in the Remarks adopts an attitude of doubt in opposition to much orthodoxy in the philosophy of mathematics. Particularly controversial in the Remarks was Wittgenstein's "notorious paragraph", which contained an unusual commentary on Gödel's incompleteness theorems. Multiple commentators read Wittgenstein as misunderstanding Gödel. In 2000 Juliet Floyd and Hilary Putnam suggested that the majority of commentary misunderstands Wittgenstein but their interpretation has not been met with approval.

Such theorems provide no indication as to how to construct (or exhibit) the object whose existence is being claimed. From a constructivist viewpoint, such approaches are not viable as it lends to mathematics losing its concrete applicability,See the section on nonconstructive proofs of the entry "Constructive proof". while the opposing viewpoint is that abstract methods are far-reaching (meaning what?), in a way that numerical analysis cannot be.

He applies Penrose's theorems for collapsing stars to the universe itself. Justifying Sciama's faith in him, he produces a PhD of real brilliance and profound implications. In theory, at least, the big bang could have happened. Two years after his initial diagnosis, Stephen is not only still very much alive, but has played a part in a great scientific breakthrough which revolutionises the way we think about the universe.

Since filters are composed of subsets of the very topological space that is under consideration, topological set operations (such as closure or interior) may be applied to the sets that constitute the filter. Taking the closure of the all sets in a filter is sometimes useful in Functional Analysis for instance. Theorems about images or preimages of sets (e.g. continuity) under functions may also be applied to filters.

Such an approach is called a spectral method. DCTs are also widely employed in solving partial differential equations by spectral methods, where the different variants of the DCT correspond to slightly different even/odd boundary conditions at the two ends of the array. Laplace transforms are used to solve partial differential equations. The general theory for obtaining solutions in this technique is developed by theorems on Laplace transform in n dimensions.

In mathematics, Hilbert's syzygy theorem is one of the three fundamental theorems about polynomial rings over fields, first proved by David Hilbert in 1890, which were introduced for solving important open questions in invariant theory, and are at the basis of modern algebraic geometry. The two other theorems are Hilbert's basis theorem that asserts that all ideals of polynomial rings over a field are finitely generated, and Hilbert's Nullstellensatz, which establishes a bijective correspondence between affine algebraic varieties and prime ideals of polynomial rings. Hilbert's syzygy theorem concern the relations, or syzygies in Hilbert's terminology, between the generators of an ideal, or, more generally, a module. As the relations form a module, one may consider the relations between the relations; Hilbert's syzygy theorem asserts that, if one continues in this way, starting with a module over a polynomial ring in indeterminates over a field, one eventually finds a zero module of relations, after at most steps.

Mellon had read a 1940 article about St. John's in Life Magazine, and wrote in his autobiography that after reading the article he drove to Annapolis :to offer financial assistance for the project, but I got so interested in it—this curriculum rooted in the medieval system of the trivium and the quadrivium—that I decided to sign on as a student.... I started in the autumn of 1940 as a mature student, being about fourteen years older than my fellow freshmen.... Mathematics proved a big problem. Purely by memorizing theorems at Choate, I had done well in plane geometry and had got a perfect score on my College Board examination, but at St. John’s the students were assigned some ten theorems a day. We were supposed to work them out to their QED solely by logic. When asked to prove one at the blackboard early in my first term, I was flabbergasted and unable to go beyond the first segment.

Gödel's second incompleteness theorem (see Gödel's incompleteness theorems), another celebrated result, shows that there are inherent limitations in what can be achieved with formal proofs in mathematics. The name for the incompleteness theorem refers to another meaning of complete (see model theory – Using the compactness and completeness theorems): A theory T is complete (or decidable) if for every formula f in the language of T either T\vdash f or T\vdash eg f. Gödel's second incompleteness theorem states that in any consistent effective theory T containing Peano arithmetic (PA), a formula CT like CT = eg (0 = 1) expressing the consistency of T cannot be proven within T. The completeness theorem implies the existence of a model of T in which the formula CT is false. Such a model (precisely, the set of "natural numbers" it contains) is necessarily a non-standard model, as it contains the code number of a proof of a contradiction of T. But T is consistent when viewed from the outside.

Calculus on Manifolds is a brief monograph on the theory of vector-valued functions of several real variables (f : Rn→Rm) and differentiable manifolds in Euclidean space. In addition to extending the concepts of differentiation (including the inverse and implicit function theorems) and Riemann integration (including Fubini's theorem) to functions of several variables, the book treats the classical theorems of vector calculus, including those of Cauchy–Green, Ostrogradsky–Gauss (divergence theorem), and Kelvin–Stokes, in the language of differential forms on differentiable manifolds embedded in Euclidean space, and as corollaries of the generalized Stokes' theorem on manifolds-with-boundary. The book culminates with the statement and proof of this vast and abstract modern generalization of several classical results: The cover of Calculus on Manifolds features snippets of a July 2, 1850 letter from Lord Kelvin to Sir George Stokes containing the first disclosure of the classical Stokes' theorem (i.e., the Kelvin–Stokes theorem).

But it is undoubtedly also true that a Greek geometer versed in the fourteen theorems of Euclid's "algebra" was far more adept in applying these theorems to practical mensuration than is an experienced geometer of today. Ancient geometric "algebra" was not an ideal tool, but it was far from ineffective. Euclid's statement (Proposition 4), "If a straight line be cut at random, the square on the whole is equal to the squares on the segments and twice the rectangle contained by the segments, is a verbose way of saying that (a + b)^2 = a^2 + 2ab + b^2," Many basic laws of addition and multiplication are included or proved geometrically in the Elements. For instance, proposition 1 of Book II states: :If there be two straight lines, and one of them be cut into any number of segments whatever, the rectangle contained by the two straight lines is equal to the rectangles contained by the uncut straight line and each of the segments.

Before he left Padua, Gregory published Vera Circuli et Hyperbolae Quadratura (1667) in which he approximated the areas of the circle and hyperbola with convergent series: :[James Gregory] cannot be denied the authorship of many curious theorems on the relation of the circle to inscribed and circumscribed polygons, and their relation to each other. By means of these theorems he gives with infinitely less trouble than by the usual calculations, … the measure of the circle and hyperbola (and consequently the construction of logarithms) to more than twenty decimal places. Following the example of Huygens, he also gave constructions of straight lines equal to the arcs of the circle, and whose error is still less.Jean Montucla (1873) History of the Quadrature of the Circle, J. Babin translator, William Alexander Myers editor, page 23, link from HathiTrust "The first proof of the fundamental theorem of calculus and the discovery of the Taylor series can both be attributed to him."W.

Work in set theory showed that almost all ordinary mathematics can be formalized in terms of sets, although there are some theorems that cannot be proven in common axiom systems for set theory. Contemporary work in the foundations of mathematics often focuses on establishing which parts of mathematics can be formalized in particular formal systems (as in reverse mathematics) rather than trying to find theories in which all of mathematics can be developed.

Many definitions and theorems about monoids can be generalised to small categories with more than one object. For example, a quotient of a category with one object is just a quotient monoid. Monoids, just like other algebraic structures, also form their own category, Mon, whose objects are monoids and whose morphisms are monoid homomorphisms. There is also a notion of monoid object which is an abstract definition of what is a monoid in a category.

Seidenberg was known for his research in commutative algebra, algebraic geometry, differential algebra, and the history of mathematics. He published Prime ideals and integral dependence written jointly with Irvin Cohen, which greatly simplified the existing proofs of the going-up and going-down theorems of ideal theory. He also made important contributions to algebraic geometry. In 1950, he published a paper called The hyperplane sections of normal varieties, which has proved fundamental in later advances.

This includes different branches of mathematical analysis, which are based on fields with additional structure. Basic theorems in analysis hinge on the structural properties of the field of real numbers. Most importantly for algebraic purposes, any field may be used as the scalars for a vector space, which is the standard general context for linear algebra. Number fields, the siblings of the field of rational numbers, are studied in depth in number theory.

Pitowsky uses Gleason's theorem to argue that quantum mechanics represents a new theory of probability, one in which the structure of the space of possible events is modified from the classical, Boolean algebra thereof. He regards this as analogous to the way that special relativity modifies the kinematics of Newtonian mechanics. The Gleason and Kochen–Specker theorems have been cited in support of various philosophies, including perspectivism, constructive empiricism and agential realism.

The Institute is now associated with the University of Madras, after it was merged with the Department of Mathematics at the University in 1967. Rajagopal conducted research on sequences, series, summability, and published more than 80 papers but is most noted for his work in the area of generalising and unifying Tauberian theorems. He also did research in many other mathematical topics. Rajagopal also conducted research in the history of medieval Indian mathematics.

His method improves Remak's use of idempotents to create the appropriate central automorphisms. Both Remak and Schmidt published subsequent proofs and corollaries to their theorems. Wolfgang Krull (Über verallgemeinerte endliche Abelsche Gruppen, M. Z. 23 (1925) 161–196), returned to G.A. Miller's original problem of direct products of abelian groups by extending to abelian operator groups with ascending and descending chain conditions. This is most often stated in the language of modules.

Elliptical distributions are important in portfolio theory because, if the returns on all assets available for portfolio formation are jointly elliptically distributed, then all portfolios can be characterized completely by their location and scale - that is, any two portfolios with identical location and scale of portfolio return have identical distributions of portfolio return. Various features of portfolio analysis, including mutual fund separation theorems and the Capital Asset Pricing Model, hold for all elliptical distributions.

In 1974, Biggs published Algebraic Graph Theory which articulates properties of graphs in algebraic terms, then works out theorems regarding them. In the first section, he tackles the applications of linear algebra and matrix theory; algebraic constructions such as adjacency matrix and the incidence matrix and their applications are discussed in depth. Next, there is and wide-ranging description of the theory of chromatic polynomials. The last section discusses symmetry and regularity properties.

Electrical engineering uses drawn symbols and connect them with lines that stand for the mathematicals act of substitution and replacement. They then verify their drawings with truth tables and simplify the expressions as shown below by use of Karnaugh maps or the theorems. In this way engineers have created a host of "combinatorial logic" (i.e. connectives without feedback) such as "decoders", "encoders", "mutifunction gates", "majority logic", "binary adders", "arithmetic logic units", etc.

In 1937--1941, he was an editor of the journal Deutsche Mathematik. Feigl's main areas of work were the foundations of geometry and topology, where he studied fixed point theorems for n-dimensional manifolds. Feigl was one of the initial authors of the Mathematisches Wörterbuch ("Mathematical dictionary"). Because of the impending siege by the Red Army he was forced to leave Breslau in January 1945 with his family and other members of the Mathematical Institute.

There are various proofs of this theorem, by either analytic methods such as Liouville's theorem, or topological ones such as the winding number, or a proof combining Galois theory and the fact that any real polynomial of odd degree has at least one real root. Because of this fact, theorems that hold for any algebraically closed field apply to C. For example, any non-empty complex square matrix has at least one (complex) eigenvalue.

Synthetic proofs of geometric theorems make use of auxiliary constructs (such as helping lines) and concepts such as equality of sides or angles and similarity and congruence of triangles. Examples of such proofs can be found in the articles Butterfly theorem, Angle bisector theorem, Apollonius' theorem, British flag theorem, Ceva's theorem, Equal incircles theorem, Geometric mean theorem, Heron's formula, Isosceles triangle theorem, Law of cosines, and others that are linked to here.

Since this early work, sequent calculi, also called Gentzen systems,, calls Gentzen systems LC systems. Curry's emphasis is more on theory than on practical logic proofs.. This book is much more concerned with the theoretical, metamathematical implications of Gentzen-style sequent calculus than applications to practical logic proofs., defines Gentzen systems and proves various theorems within these systems, including Gödel's completeness theorem and Gentzen's theorem., gives a brief theoretical presentation of Gentzen systems.

The theorems are those formulae that appear as the concluding judgment in a valid proof. A Hilbert-style system needs no distinction between formulae and judgments; we make one here solely for comparison with the cases that follow. The price paid for the simple syntax of a Hilbert-style system is that complete formal proofs tend to get extremely long. Concrete arguments about proofs in such a system almost always appeal to the deduction theorem.

In mathematics, Manin matrices, named after Yuri Manin who introduced them around 1987–88, are a class of matrices with elements in a not-necessarily commutative ring, which in a certain sense behave like matrices whose elements commute. In particular there is natural definition of the determinant for them and most linear algebra theorems like Cramer's rule, Cayley–Hamilton theorem, etc. hold true for them. Any matrix with commuting elements is a Manin matrix.

Furthermore, techniques such as partial summation and Tauberian theorems can be used to get information about the coefficients from analytic information about the Dirichlet series. Thus a common method for estimating a multiplicative function is to express it as a Dirichlet series (or a product of simpler Dirichlet series using convolution identities), examine this series as a complex function and then convert this analytic information back into information about the original function.

Not only that, but they will also correspond with any other inference of this form, which will be valid on the same basis this inference is. Propositional logic may be studied through a formal system in which formulas of a formal language may be interpreted to represent propositions. A system of axioms and inference rules allows certain formulas to be derived. These derived formulas are called theorems and may be interpreted to be true propositions.

Some authors add the requirement that a Dedekind domain not be a field. Many more authors state theorems for Dedekind domains with the implicit proviso that they may require trivial modifications for the case of fields. An immediate consequence of the definition is that every principal ideal domain (PID) is a Dedekind domain. In fact a Dedekind domain is a unique factorization domain (UFD) if and only if it is a PID.

Thomas Pynchon introduced the fictional character, Sammy Hilbert-Spaess (a pun on "Hilbert Space"), in his 1973 novel, Gravity's Rainbow. Hilbert-Spaess is first described as a "a ubiquitous double agent" and later as "at least a double agent". The novel had earlier referenced the work of fellow German mathematician Kurt Gödel's Incompleteness Theorems, which showed that Hilbert's Program, Hilbert's formalized plan to unify mathematics into a single set of axioms, was not possible.

In 1918, he became a Docent in Mathematics and was elected to the Norwegian Academy of Science and Letters. Skolem did not at first formally enroll as a Ph.D. candidate, believing that the Ph.D. was unnecessary in Norway. He later changed his mind and submitted a thesis in 1926, titled Some theorems about integral solutions to certain algebraic equations and inequalities. His notional thesis advisor was Axel Thue, even though Thue had died in 1922.

The term's common usage today is another corollary to Huygens's result. The concept may be stated as an ironic paradox: Persistently taking beneficial chances is never beneficial at the end. This paradoxical form of gambler's ruin should not be confused with the gambler's fallacy, a different concept. The concept has specific relevance for gamblers; however it also leads to mathematical theorems with wide application and many related results in probability and statistics.

Conversely any Lie algebra is obviously a Leibniz algebra. In this sense, Leibniz algebras can be seen as a non-commutative generalization of Lie algebras. The investigation of which theorems and properties of Lie algebras are still valid for Leibniz algebras is a recurrent theme in the literature. For instance, it has been shown that Engel's theorem still holds for Leibniz algebras and that a weaker version of Levi-Malcev theorem also holds.

Church's paper An Unsolvable Problem of Elementary Number Theory (1936) proved that the Entscheidungsproblem was undecidable within the λ-calculus and Gödel-Herbrand's general recursion; moreover Church cites two theorems of Kleene's that proved that the functions defined in the λ-calculus are identical to the functions defined by general recursion: :"Theorem XVI. Every recursive function of positive integers is λ-definable.16 :"Theorem XVII. Every λ-definable function of positive integers is recursive.

Since 1978 Roquette is member of the Heidelberg Academy of Sciences and since 1985, the German Academy of Sciences Leopoldina. He has an honorary doctorate from the University of Duisburg-Essen and is honorary member of the Mathematical Society of Hamburg. In 1958 he was an invited speaker at the International Congress of Mathematicians in Edinburgh (on the topic of Some fundamental theorems on abelian function fields). His doctoral students include Gerhard Frey and .

The Hertwig brothers were the most eminent scholars of Ernst Haeckel (and Carl Gegenbaur) from the University of Jena. They were independent of Haeckel's philosophical speculations but took his ideas in a positive way to widen their concepts in zoology. Initially, between 1879–1883, they performed embryological studies, especially on the theory of the coelom (1881), the fluid-filled body cavity. These problems were based on the phylogenetic theorems of Haeckel, i.e.

Continuous fixed-point theorems often require a continuous function. Since continuity is not meaningful for functions on discrete sets, it is replaced by conditions such as a direction-preserving function. Such conditions imply that the function does not change too drastically when moving between neighboring points of the integer grid. There are various direction-preservation conditions, depending on whether neighboring points are considered points of a hypercube (HGDP), of a simplex (SGDP) etc.

Proofs of the existence of equilibrium traditionally rely on fixed-point theorems such as Brouwer fixed- point theorem for functions (or, more generally, the Kakutani fixed-point theorem for set-valued functions). See Competitive equilibrium#Existence of a competitive equilibrium. The proof was first due to Lionel McKenzie, and Kenneth Arrow and Gérard Debreu. In fact, the converse also holds, according to Uzawa's derivation of Brouwer's fixed point theorem from Walras's law.

As a post- doctoral student, he was at the Mathematical Sciences Research Institute (MSRI) in Berkeley, and then a tutor at Jesus College, Oxford. From 1998 until shortly before his death he was a professor at the Pennsylvania State University. His research interests center around index theorems, coarse geometry, operator algebras, noncommutative geometry, and the Novikov conjecture in differential topology. He was an editor of the Journal of Noncommutative Geometry and the Journal of Topology.

But by examining vector fields in a sufficiently small neighborhood of a source or sink, we see that sources and sinks contribute integer amounts (known as the index) to the total, and they must all sum to 0. This result may be considered one of the earliest of a whole series of theorems establishing deep relationships between geometric and analytical or physical concepts. They play an important role in the modern study of both fields.

This is the study into fundamental computer algorithms, which are the basis to computer programs. Without algorithms, no computer programs would exist. This also involves the process of looking into various mathematical functions behind computational algorithms, basic theory and functional (low level) programming. In an academic setting, this area would introduce the fundamental mathematical theorems and functions behind theoretical computer science which are the building blocks for other areas in the field.

The stable stationary state has a local maximum of entropy and is locally the most reproducible state of the system. There are theorems about the irreversible dissipation of fluctuations. Here 'local' means local with respect to the abstract space of thermodynamic coordinates of state of the system. If the stationary state is unstable, then any fluctuation will almost surely trigger the virtually explosive departure of the system from the unstable stationary state.

All abelian categories are exact categories, but not all exact categories are abelian. Because Quillen was able to work in this more general situation, he was able to use exact categories as tools in his proofs. This technique allowed him to prove many of the basic theorems of algebraic K-theory. Additionally, it was possible to prove that the earlier definitions of Swan and Gersten were equivalent to Quillen's under certain conditions.

He is the author of more than 250 papers and 18 books (monographs and course notes): his work concerns mainly the fields of pure and applied mathematics listed below. A common characteristic to all of his research is the use of the methods of functional analysis to prove existence, uniqueness and approximation theorems for the various problems he studied, and also a high consideration of the analytic problems related to problems in applied mathematics.

Consequently, we rely on the stronger definition above which implies that the irrationals are typical and the rationals are not. For applications, if a property holds on a residual set, it may not hold for every point, but perturbing it slightly will generally land one inside the residual set (by nowhere density of the components of the meagre set), and these are thus the most important case to address in theorems and algorithms.

PEEF allocations exist even when agents' preferences are not convex. There are several sufficient conditions that are related to the shape of the set of allocations corresponding to a specific efficient utility profile. GIven a utility-vector u, define A(u) = the set of all allocations for which the utility-profile is u. The following successively more general theorems were proved by different authors: Theorem 2 (Varian): Suppose all agents' preferences are strongly monotone.

With equilibrium defined as ‘competitive equilibrium’, the first fundamental theorem can be proved even if indifference curves need not be convex: any competitive equilibrium is (globally) Pareto optimal. However the proof is no longer obvious, and the reader is referred to the article on Fundamental theorems of welfare economics. The same result would not have been considered to hold (with non- convex indifference curves) under the tangency definition of equilibrium. The point x of Fig.

In graph theory, a family of graphs is said to have bounded expansion if all of its shallow minors are sparse graphs. Many natural families of sparse graphs have bounded expansion. A closely related but stronger property, polynomial expansion, is equivalent to the existence of separator theorems for these families. Families with these properties have efficient algorithms for problems including the subgraph isomorphism problem and model checking for the first order theory of graphs.

The Kaplansky conjecture predicts that for an integral domain R and a torsionfree group G the only idempotents in R[G] are 0,1. Each such idempotent p gives a projective R[G] module by taking the image of the right multiplication with p. Hence there seems to be a connection between the Kaplansky conjecture and the vanishing of K_0(R[G]). There are theorems relating the Kaplansky conjecture to the Farrell–Jones conjecture (compare ).

Emanuel Sperner (9 December 1905 – 31 January 1980) was a German mathematician, best known for two theorems. He was born in Waltdorf (near Neiße, Upper Silesia, now Nysa, Poland), and died in Sulzburg-Laufen, West Germany. He was a student at Carolinum in Nysa and then Hamburg University where his advisor was Wilhelm Blaschke. He was appointed Professor in Königsberg in 1934, and subsequently held posts in a number of universities until 1974.

The constant can be lowered to , but at the expense of replacing with the worse constant of . The motivation of Leighton in studying crossing numbers was for applications to VLSI design in theoretical computer science. Later, Székely also realized that this inequality yielded very simple proofs of some important theorems in incidence geometry, such as Beck's theorem and the Szemerédi-Trotter theorem, and Tamal Dey used it to prove upper bounds on geometric k-sets.

While the first half treats established subjects, the second half deals with modern research areas like commutative algebra and spectral theory. This divide in the work is related to a historical change in the intent of the treatise. The Éléments' content consists of theorems, proofs, exercises and related commentary, common material in math textbooks. Despite this presentation, the first half was not written as original research but rather as a reorganized presentation of established knowledge.

This included the theory of positive-definite continued fractions, convergence results for continued fractions, parabola theorems, Hausdorff moments, and Hausdorff summability. He studied the polynomials now named Wall polynomials after him. While at Northwestern he started a collaboration with Ernst Hellinger, and he was very interested in Hellinger integrals throughout his career, but did publish anything on them. While at Texas Wall was a prominent practitioner of the Moore method of teaching.

Thus the sequences considered in Fourier's theorem and in Budan's theorem have the same number of sign variations. This strong relationship between the two theorems may explain the priority controversy that occurred in 19th century, and the use of several names for the same theorem. In modern usage, for computer computation, Budan's theorem is generally preferred since the sequences have much larger coefficients in Fourier's theorem than in Budan's, because of the factorial factor.

Roughly speaking, Vincent's theorem consists of using continued fractions for replacing Budan's linear transformations of the variable by Möbius transformations. Budan's, Fourier's and Vincent theorem sank into oblivion at the end of 19th century. The last author mentioning these theorems before the second half of 20th century Joseph Alfred Serret. They were introduced again in 1976 by Collins and Akritas, for providing, in computer algebra, an efficient algorithm for real roots isolation on computers.

Noam Chomsky once wrote, "Martin Gardner's contribution to contemporary intellectual culture is unique--in its range, its insight, and understanding of hard questions that matter."Brown (2010)MacTutor: Gardner has produced a number of mathematical papers, written with leading mathematicians. Gardner repeatedly alerted the public (and other mathematicians) to recent discoveries in mathematics–recreational and otherwise.Malkevitch (2014): The range of wonderful problems, examples, and theorems that Gardner treated over the years is enormous.

Among his notable results is his work with Marc Culler, John Luecke, and Peter Shalen on the cyclic surgery theorem. This was an important ingredient in his work with Luecke showing that knots were determined by their complement. Gordon was also involved in the resolution of the Smith conjecture. Andrew Casson and Gordon defined and proved basic theorems regarding strongly irreducible Heegaard splittings, an important concept in the modernization of Heegaard splitting theory.

To give it a meaning, infinitesimals are often compared to other infinitesimals of similar size (as in a derivative). Infinitely many infinitesimals are summed to produce an integral. The concept of infinitesimals was originally introduced around 1670 by either Nicolaus Mercator or Gottfried Wilhelm Leibniz. Archimedes used what eventually came to be known as the method of indivisibles in his work The Method of Mechanical Theorems to find areas of regions and volumes of solids.

For example, integers can be represented in binary notation, and graphs can be encoded directly via their adjacency matrices, or by encoding their adjacency lists in binary. Even though some proofs of complexity-theoretic theorems regularly assume some concrete choice of input encoding, one tries to keep the discussion abstract enough to be independent of the choice of encoding. This can be achieved by ensuring that different representations can be transformed into each other efficiently.

During his youth, Archimedes may have studied in Alexandria, Egypt, where Conon of Samos and Eratosthenes of Cyrene were contemporaries. He referred to Conon of Samos as his friend, while two of his works (The Method of Mechanical Theorems and the Cattle Problem) have introductions addressed to Eratosthenes.In the preface to On Spirals addressed to Dositheus of Pelusium, Archimedes says that "many years have elapsed since Conon's death." Conon of Samos lived c.

It shows that no sufficiently rich interpreted language can represent its own semantics. A corollary is that any metalanguage capable of expressing the semantics of some object language must have expressive power exceeding that of the object language. The metalanguage includes primitive notions, axioms, and rules absent from the object language, so that there are theorems provable in the metalanguage not provable in the object language. The undefinability theorem is conventionally attributed to Alfred Tarski.

He held positions at the University of Minnesota and Purdue University before joining the faculty at Michigan in 1977. Hochster's work is primarily in commutative algebra, especially the study of modules over local rings. He has established classic theorems concerning Cohen–Macaulay rings, invariant theory and homological algebra. For example, the Hochster–Roberts theorem states that the invariant ring of a linearly reductive group acting on a regular ring is Cohen–Macaulay.

His major publications deal with fundamental properties of string theories, and include the conformal invariance of supersymmetric two- dimensional field theories which describe the world-sheet dynamics of strings, the study of supersymmetric solitons using index theorems, the discovery of a new duality between string theory and M-theory, the identification of string networks as supersymmetric states and the discovery of a novel Higgs mechanism in the worldvolume theory of M-theory membranes.

In axiomatic set theory, a mathematical discipline, a morass is an infinite combinatorial structure, used to create "large" structures from a "small" number of "small" approximations. They were invented by Ronald Jensen for his proof that cardinal transfer theorems hold under the axiom of constructibility. A far less complex but equivalent variant known as a simplified morass was introduced by Velleman, and the term morass is now often used to mean these simpler structures.

With PCF theory, he showed that in spite of the undecidability of the most basic questions of cardinal arithmetic (such as the continuum hypothesis), there are still highly nontrivial ZFC theorems about cardinal exponentiation. Shelah constructed a Jónsson group, an uncountable group for which every proper subgroup is countable. He showed that Whitehead's problem is independent of ZFC. He gave the first primitive recursive upper bound to van der Waerden's numbers V(C,N).

In mathematics, a (left) coherent ring is a ring in which every finitely generated left ideal is finitely presented. Many theorems about finitely generated modules over Noetherian rings can be extended to finitely presented modules over coherent rings. Every left Noetherian ring is left-coherent. The ring of polynomials in an infinite number of variables over a left Noetherian ring is an example of a left-coherent ring that is not left Noetherian.

E.F.) to close propositions that were not proofs of theorems, but constructions of geometric objects. For example, Euclid's first proposition showing how to construct an equilateral triangle, given one side, is concluded this way. Many times, mathematicians will only utilize () faciendia as a result of the results of previous definitions or demonstradums. An idea of this is expressed within Topics (Aristotle), where he goes over the difference between a proposition and a problem.

Around this same time, Debicki also starred in a 13-minute short film called "Gödel Incomplete" (see also: Gödel's incompleteness theorems) and made an appearance as a guest star in the third season of the Australian television series Rake. In 2015, Debicki played supporting roles in three major motion pictures. She played the villain in the Guy Ritchie-directed film adaptation of The Man from U.N.C.L.E. (2015), learning to drive on set.

The theorem is actually a collection of related theorems. The first theorem states that if two different Bernoulli shifts have the same Kolmogorov entropy, then they are isomorphic as dynamical systems. The third theorem extends this result to flows: namely, that there exists a flow T_t such that T_1 is a Bernoulli shift. The fourth theorem states that, for a given fixed entropy, this flow is unique, up to a constant rescaling of time.

Dobb also criticized the motives behind marginal utility theory. Jevons wrote, for example, "so far as is consistent with the inequality of wealth in every community, all commodities are distributed by exchange so as to produce the maximum social benefit." (See Fundamental theorems of welfare economics.) Dobb contended that this statement indicated that marginalism is intended to insulate market economics from criticism by making prices the natural result of the given income distribution.

Logic, especially in the field of proof theory, considers theorems as statements (called formulas or well formed formulas) of a formal language. The statements of the language are strings of symbols and may be broadly divided into nonsense and well-formed formulas. A set of deduction rules, also called transformation rules or rules of inference, must be provided. These deduction rules tell exactly when a formula can be derived from a set of premises.

160–188, Theorems 7 and 8. In Theorem 7 Euler proves the formula in the special case s=1, and in Theorem 8 he proves it more generally. In the first corollary to his Theorem 7 he notes that \zeta(1)=\log\infty, and makes use of this latter result in his Theorem 19, in order to show that the sum of the inverses of the prime numbers is \log\log\infty.

A metatheory is a theory whose subject matter is some other theory (a theory about a theory). Statements made in the metatheory about the theory are called metatheorems. A metatheorem is a true statement about a formal system expressed in a metalanguage. Unlike theorems proved within a given formal system, a metatheorem is proved within a metatheory, and may reference concepts that are present in the metatheory but not the object theory.

In theoretical physics a nonrenormalization theorem is a limitation on how a certain quantity in the classical description of a quantum field theory may be modified by renormalization in the full quantum theory. Renormalization theorems are common in theories with a sufficient amount of supersymmetry, usually at least 4 supercharges. Perhaps the first nonrenormalization theorem was introduced by Marcus T. Grisaru, Martin Rocek and Warren Siegel in their 1979 paper Improved methods for supergraphs.

From a module-theoretic point of view this was integrated into the Cartan–Eilenberg theory of homological algebra in the early 1950s. The application in algebraic number theory to class field theory provided theorems valid for general Galois extensions (not just abelian extensions). The cohomological part of class field theory was axiomatized as the theory of class formations. In turn, this led to the notion of Galois cohomology and étale cohomology (which builds on it) .

This follows from putting D=K in the theorem. In particular, as long as D has degree at least 2g-1, the correction term is 0, so that :\ell(D) = \deg(D) - g + 1. The theorem will now be illustrated for surfaces of low genus. There are also a number other closely related theorems: an equivalent formulation of this theorem using line bundles and a generalization of the theorem to algebraic curves.

Discontinuities of Green Functions in Field Theory at Finite Temperature and Density. R.L. Kobes, G.W. Semenoff, Nuclear Physics B260:714-746, 1985.Discontinuities of Green Functions in Field Theory at Finite Temperature and Density. 2, R.L. Kobes, G.W. Semenoff, Nuclear Physics B272:329-364, 1986. the application of index theorems and their generalizations in quantum field theoryFermion Number Fractionization in Quantum Field Theory. A.J. Niemi, G.W. Semenoff, Physics Reports 135:99, 1986.

Besides the fixed-point theorems for more or less contracting functions, there are many that have emerged directly or indirectly from the result under discussion. A continuous map from a closed ball of Euclidean space to its boundary cannot be the identity on the boundary. Similarly, the Borsuk–Ulam theorem says that a continuous map from the n-dimensional sphere to Rn has a pair of antipodal points that are mapped to the same point.

The central problem regarding constant-weight codes is the following: what is the maximum number of codewords in a binary constant-weight code with length n, Hamming distance d, and weight w? This number is called A(n,d,w). Apart from some trivial observations, it is generally impossible to compute these numbers in a straightforward way. Upper bounds are given by several important theorems such as the first and second Johnson bounds,See pp.

670–674 > But the main difference reflects the philosophy above: we are interested not > only in theorems and proofs but also in the way in which they have been or > can be reached. Note that we do value proofs: experimentally inspired > results that can be proved are more desirable than conjectural ones. > However, we do publish significant conjectures or explorations in the hope > of inspiring other, perhaps better-equipped researchers to carry on the > investigation.

He made numerous contributions to the study of topology, graph theory, calculus, combinatorics, and complex analysis, as evidenced by the multitude of theorems and notations named for him. Other important European mathematicians of the 18th century included Joseph Louis Lagrange, who did pioneering work in number theory, algebra, differential calculus, and the calculus of variations, and Laplace who, in the age of Napoleon, did important work on the foundations of celestial mechanics and on statistics.

The method can be viewed as the inverse problem of network analysis. Network analysis starts with a network and by applying the various electric circuit theorems predicts the response of the network. Network synthesis on the other hand, starts with a desired response and its methods produce a network that outputs, or approximates to, that response. Network synthesis was originally intended to produce filters of the kind formerly described as wave filters but now usually just called filters.

A presentation of class field theory in terms of group cohomology was carried out by Claude Chevalley, Emil Artin and others, mainly in the 1940s. This resulted in a formulation of the central results by means of the group cohomology of the idele class group. The theorems of the cohomological approach are independent of whether or not the Galois group G of L/K is abelian. This theory has never been regarded as the sought-after non-abelian theory.

In 1958 he graduated from Moscow State University. There he received in 1961 his Ph.D. under Yuri Prokhorov with thesis "Распределения вероятностей и характеристические функционалы" (Probability distributions and characteristic functionals). Sazonov worked in the Steklov Institute of Mathematics from 1958 to 2002. In 1968 he received his Russian doctorate of sciences (Doctor Nauk) with thesis "Исследования по многомерным и бесконечномерным предельным теоремам теории вероятностей" (Investigations of multidimensional, infinite-dimensional and limit theorems of the theory of probabilities).

Other similarly-related findings are those of the Gödel's incompleteness theorems, which uncovers some fundamental limitations in the provability of formal systems. In computational complexity theory, techniques like relativization (see oracle machine) provide "weak" proofs of impossibility excluding certain proof techniques. Other techniques, such as proofs of completeness for a complexity class, provide evidence for the difficulty of problems, by showing them to be just as hard to solve as other known problems that have proved intractable.

The following is one of the most important theorems in duality theory. It follows that the Mackey topology , which recall is the polar topology generated by all -compact disks in , is the strongest locally convex topology on that is compatible with the pairing . A locally convex space whose given topology is identical to the Mackey topology is called a Mackey space. The following consequence of the above Mackey-Arens theorem is also called the Mackey-Arens theorem.

The simplest generalized Mersenne primes are prime numbers of the form , where is a low-degree polynomial with small integer coefficients. An example is , in this case, , and ; another example is , in this case, , and . It is also natural to try to generalize primes of the form to primes of the form (for and ). However (see also theorems above), is always divisible by , so unless the latter is a unit, the former is not a prime.

The notions of unibranch and geometrically unibranch points are used in some theorems in algebraic geometry. For example, there is the following result: Theorem Let X and Y be two integral locally noetherian schemes and f \colon X \to Y a proper dominant morphism. Denote their function fields by K(X) and K(Y), respectively. Suppose that the algebraic closure of K(Y) in K(X) has separable degree n and that y \in Y is unibranch.

Early in his career, Brouwer proved a number of theorems in the emerging field of topology. The most important were his fixed point theorem, the topological invariance of degree, and the topological invariance of dimension. Among mathematicians generally, the best known is the first one, usually referred to now as the Brouwer Fixed Point Theorem. It is a simple corollary to the second, concerning the topological invariance of degree, which is the best known among algebraic topologists.

Much of his work concerns the Geometry of Numbers, Hausdorff Measures, Analytic Sets, Geometry and Topology of Banach Spaces, Selection Theorems and Finite Dimensionsl Convex Geometry. In the theory of Banach spaces and summability, he proved the Dvoretzky-Rogers lemma and the Dvoretzky-Rogers theorem, both with Aryeh Dvoretzky. He constructed a counterexample to a conjecture related to the Busemann–Petty problem. In the geometry of numbers, the Rogers bound is a bound for dense packings of spheres.

Her collection included: paintings, weathervanes, shop signs, pottery, quilts, and other decorative household items. Some of her favorite types of items in her collection were: children's portraits and student art, in the form of calligraphy, memorials, and theorems. The original pieces of the collection were largely sourced from New England and Pennsylvania, though later it would expand to include Virginia, North Carolina, South Carolina, and Georgia. Rockefeller's collection grew with the help of Holger Cahill and Edith Halpert.

He also upheld the view that social sciences are scientific, and should adopt the same standards as natural sciences. Nagel wrote An Introduction to Logic and the Scientific Method with Morris Raphael Cohen, his CCNY teacher in 1934. In 1958, he published with James R. Newman Gödel's proof, a short book explicating Gödel's incompleteness theorems to those not well trained in mathematical logic. He edited the Journal of Philosophy (1939–1956) and the Journal of Symbolic Logic (1940-1946).

One can proceed to prove theorems about groups by making logical deductions from the set of axioms defining groups. For example, it is immediately proven from the axioms that the identity element of a group is unique. Instead of focusing merely on the individual objects (e.g., groups) possessing a given structure, category theory emphasizes the morphisms – the structure-preserving mappings – between these objects; by studying these morphisms, one is able to learn more about the structure of the objects.

Here Ravenel uses localization in the sense of Aldridge K. Bousfield in a crucial way. All but one of the Ravenel conjectures were proved by Ethan Devinatz, Michael J. Hopkins and Jeff Smith not long after the article got published. Frank Adams said on that occasion: In further work, Ravenel calculates the Morava K-theories of several spaces and proves important theorems in chromatic homotopy theory together with Hopkins. He was also one of the founders of elliptic cohomology.

The preceding alternative calculus is an example of a Hilbert-style deduction system. In the case of propositional systems the axioms are terms built with logical connectives and the only inference rule is modus ponens. Equational logic as standardly used informally in high school algebra is a different kind of calculus from Hilbert systems. Its theorems are equations and its inference rules express the properties of equality, namely that it is a congruence on terms that admits substitution.

TLA+ is also used to write machine-checked proofs of correctness both for algorithms and mathematical theorems. The proofs are written in a declarative, hierarchical style independent of any single theorem prover backend. Both formal and informal structured mathematical proofs can be written in TLA+; the language is similar to LaTeX, and tools exist to translate TLA+ specifications to LaTeX documents. TLA+ was introduced in 1999, following several decades of research into a verification method for concurrent systems.

He proved that planar polynomial vector fields have only finitely many limit cycles. Jean Écalle independently proved the same result, and an earlier attempted proof by Henri Dulac (in 1923) was shown to be defective by Ilyashenko in the 1970s. He was an Invited Speaker of the ICM in 1978 at Helsinki and in 1990 with talk Finiteness theorems for limit cycles at Kyoto. In 2017 he was elected a Fellow of the American Mathematical Society.

The full Taniyama–Shimura–Weil conjecture was finally proved by , , and who, building on Wiles's work, incrementally chipped away at the remaining cases until the full result was proved. The now fully proved conjecture became known as the modularity theorem. Several other theorems in number theory similar to Fermat's Last Theorem also follow from the same reasoning, using the modularity theorem. For example: no cube can be written as a sum of two coprime n-th powers, n ≥ 3\.

M. Hasegawa, M. Hofmann and G. Plotkin, "Finite dimensional vector spaces are complete for traced symmetric monoidal categories", LNCS 4800, (2008), pp. 367–385. i.e. an equational statement in the language of dagger compact categories holds if and only if it can be derived in the concrete category of finite dimensional Hilbert spaces and linear maps. There is no analogous completeness for Rel or nCob. This completeness result implies that various theorems from Hilbert spaces extend to this category.

See theorems 5.1 and 5.2 of . However, every graph of maximum degree three has slope number at most four;, improving an earlier result of ; theorem 5.3 of . the result of for the complete graph shows that this is tight. Not every set of four slopes is suitable for drawing all degree-3 graphs: a set of slopes is suitable for this purpose if and only it forms the slopes of the sides and diagonals of a parallelogram.

Imprecise calculations with infinitesimals were widely replaced with the rigorous (ε, δ)-definition of limit starting in the 1870s. Meanwhile, calculations with infinitesimals persisted and often led to correct results. This led Abraham Robinson to investigate if it were possible to develop a number system with infinitesimal quantities over which the theorems of calculus were still valid. In 1960, building upon the work of Edwin Hewitt and Jerzy Łoś, he succeeded in developing non-standard analysis.

Arunava Sen has done fundamental contributions to the theory of strategic voting. The starting point of this theory is an impossibility result due to Gibbard and Satterthwaite: the Gibbard-Satterthwaite (GS) impossibility theorem and Gibbard's theorem. Roughly, it states that there is no voting rule which is unanimous, non-dictatorial, and non-manipulable (strategyproof) if the preferences of voters are unrestricted. Arunava Sen's work in this area identifies environments where such theorems hold or well-behaved voting rules exist.

The sociology of emotion applies sociological theorems and techniques to the study of human emotions. As sociology emerged primarily as a reaction to the negative effects of modernity, many normative theories deal in some sense with emotion without forming a part of any specific subdiscipline: Karl Marx described capitalism as detrimental to personal 'species-being', Georg Simmel wrote of the deindividualizing tendencies of 'the metropolis', and Max Weber's work dealt with the rationalizing effect of modernity in general.

Most of the no-go theorems constrain interactions in the flat space. One of the most well-known is the Weinberg low energy theorem that explains why there are no macroscopic fields corresponding to particles of spin 3 or higher. The Weinberg theorem can be interpreted in the following way: Lorentz invariance of the S-matrix is equivalent, for massless particles, to decoupling of longitudinal states. The latter is equivalent to gauge invariance under the linearised gauge symmetries above.

In a 1938 paper, Kolmogorov "established the basic theorems for smoothing and predicting stationary stochastic processes"—a paper that had major military applications during the Cold War.Salsburg, p. 139. In 1939, he was elected a full member (academician) of the USSR Academy of Sciences. During World War II Kolmogorov contributed to the Russian war effort by applying statistical theory to artillery fire, developing a scheme of stochastic distribution of barrage balloons intended to help protect Moscow from German bombers.

The connected, projective variety examples are indeed exhausted by abelian functions, as is shown by a number of results characterising an abelian variety by rather weak conditions on its group law. The so-called quasi- abelian functions are all known to come from extensions of abelian varieties by commutative affine group varieties. Therefore, the old conclusions about the scope of global algebraic addition theorems can be said to hold. A more modern aspect is the theory of formal groups.

Laruelle believes that both philosophy and non-philosophy are performative. However, philosophy merely performatively legitimates the decisional structure which, as already noted, it is unable to fully grasp, in contrast to non-philosophy which collapses the distinction (present in philosophy) between theory and action. In this sense, non- philosophy is radically performative because the theorems deployed in accordance with its method constitute fully-fledged scientific actions. Non- philosophy, then, is conceived as a rigorous and scholarly discipline.

He combined ideas of modern functional analysis with classical analysis. Together with his student Simson Baron he started to describe the summability factors for double series. Considering applications to orthogonal series and Tauberian theorems, Kangro created a theory of summability with speed based on functional analysis, which helped him to solve several problems in function and summability theory. In addition to laying the basis for the new theory he also pointed out main directions for applications.

Grothendieck nevertheless wrote a revised version of EGA I which was published by Springer-Verlag. It updates the terminology, replacing "prescheme" by "scheme" and "scheme" by "separated scheme", and heavily emphasizes the use of representable functors. The new preface of the second edition also includes a slightly revised plan of the complete treatise, now divided into twelve chapters. Grothendieck's EGA 5 which deals with Bertini type theorems is to some extent available from the Grothendieck Circle website.

The book contains accessible popular expositions on the mathematical theory of infinity, and a number of related topics. These include Gödel's incompleteness theorems and their relationship to concepts of artificial intelligence and the human mind, as well as the conceivability of some unconventional cosmological models. The material is approached from a variety of viewpoints, some more conventionally mathematical and others being nearly mystical. There is a brief account of the author's personal contact with Kurt Gödel.

Harry Ernest Rauch (November 9, 1925 - June 18, 1979) was an American mathematician, who worked on complex analysis and differential geometry. He was born in Trenton, New Jersey, and died in White Plains, New York. Rauch earned his PhD in 1948 from Princeton University under Salomon Bochner with thesis Generalizations of Some Classic Theorems to the Case of Functions of Several Variables. From 1949 to 1951 he was a visiting member of the Institute for Advanced Study.

Weierstrass and Runge's theorems were put forward in 1885, while Mergelyan's theorem dates from 1951. This rather large time difference is not surprising, as the proof of Mergelyan's theorem is based on a new powerful method created by Mergelyan. After Weierstrass and Runge, many mathematicians (in particular Walsh, Keldysh, and Lavrentyev) had been working on the same problem. The method of the proof suggested by Mergelyan is constructive, and remains the only known constructive proof of the result.

The Mathematical Intelligencer 16:4, pages 11–18, December 1994. Some people strongly disagree with Zeilberger's prediction, for example it has been described as provocative and quite wrongheaded,Proof and other dilemmas: mathematics and philosophy, Bonnie Gold, Roger A. Simons, MAA, 2008, whereas it has also been stated that choosing which theorems are interesting enough to pay for, already happens as a result of funding bodies making decisions as to which areas of research to invest in.

For , if is analytic on the unit disk, fixes , and , then Gabriel Koenigs showed in 1884 that there is an analytic (non-trivial) satisfying Schröder's equation. This is one of the first steps in a long line of theorems fruitful for understanding composition operators on analytic function spaces, cf. Koenigs function. Equations such as Schröder's are suitable to encoding self- similarity, and have thus been extensively utilized in studies of nonlinear dynamics (often referred to colloquially as chaos theory).

Every WUM division with positive weights is obviously Pareto-efficient. This is because, if a division Y Pareto-dominates a division X, then the weighted sum-of-utilities in Y is strictly larger than in X, so X cannot be a WUM division. What's more surprising is that every Pareto-efficient division is WUM for some selection of weights.. See also Weller's theorem. For a similar result related to the problem of homogeneous resource allocation, see Varian's theorems.

After introducing, via the Eilenberg–Steenrod axioms, the abstract approach to homology theory, he and Eilenberg originated category theory in 1945. He is especially known for his work on coherence theorems. A recurring feature of category theory, abstract algebra, and of some other mathematics as well, is the use of diagrams, consisting of arrows (morphisms) linking objects, such as products and coproducts. According to McLarty (2005), this diagrammatic approach to contemporary mathematics largely stems from Mac Lane (1948).

In mathematics, incidence geometry is the study of incidence structures. A geometric structure such as the Euclidean plane is a complicated object that involves concepts such as length, angles, continuity, betweenness, and incidence. An incidence structure is what is obtained when all other concepts are removed and all that remains is the data about which points lie on which lines. Even with this severe limitation, theorems can be proved and interesting facts emerge concerning this structure.

A lesson learned by mathematics in the last 150 years is that it is useful to strip the meaning away from the mathematical assertions (axioms, postulates, propositions, theorems) and definitions. One must concede the need for primitive notions, or undefined terms or concepts, in any study. Such abstraction or formalization makes mathematical knowledge more general, capable of multiple different meanings, and therefore useful in multiple contexts. Alessandro Padoa, Mario Pieri, and Giuseppe Peano were pioneers in this movement.

He received a knighthood in the 1991 Birthday Honours for "mathematical excellence and service to British mathematics and mathematics education". He was invited to become President of The Mathematical Association in 2003 and based his book Three-dimensional Theorems for Schools on his 2004 Presidential Address. On Friday 6 May 2005, the University of Warwick's new Mathematics and Statistics building was named the Zeeman building in his honour. He became an Honorary Member of The Mathematical Association in 2006.

Snell studied mathematics at the University of Illinois with Joseph L. Doob from 1948 through 1951; Doob introduced him to martingales, an aspect of probability theory. Doob assigned such topics by having students attempt to solve a series of problems that he kept on file cards.J.L. Snell (2005) "Obituary: Joseph L. Doob", Journal of Applied Probability 42(1): 247–56 Snell earned his Ph.D. in 1951 ("Applications of Martingale System Theorems"), with Doob as his supervisor.

In this, he proved that the constructible universe is an inner model of ZF set theory, and also that the axiom of choice and the generalized continuum hypothesis are true in the constructible universe. This shows that both propositions are consistent with the basic axioms of set theory, if ZF itself is consistent. Since many other theorems only hold in systems in which one or both of the propositions is true, their consistency is an important result.

Uniform dimension generalizes some, but not all, aspects of the notion of the dimension of a vector space. Finite uniform dimension was a key assumption for several theorems by Goldie, including Goldie's theorem, which characterizes which rings are right orders in a semisimple ring. Modules of finite uniform dimension generalize both Artinian modules and Noetherian modules. In the literature, uniform dimension is also referred to as simply the dimension of a module or the rank of a module.

If S is finite, then a group is called finitely generated. The structure of finitely generated abelian groups in particular is easily described. Many theorems that are true for finitely generated groups fail for groups in general. It has been proven that if a finite group is generated by a subset S, then each group element may be expressed as a word from the alphabet S of length less than or equal to the order of the group.

See If the answer is yes, many important problems can be shown to have more efficient solutions. These include various types of integer programming problems in operations research, many problems in logistics, protein structure prediction in biology, and the ability to find formal proofs of pure mathematics theorems. The P versus NP problem is one of the Millennium Prize Problems proposed by the Clay Mathematics Institute. There is a US$1,000,000 prize for resolving the problem.

In general, repeated games are easily solved using strategies provided by folk theorems. Complex repeated games can be solved using various techniques most of which rely heavily on linear algebra and the concepts expressed in fictitious play. It may be deducted that you can determine the characterization of equilibrium payoffs in infinitely repeated games. Through alternation between two payoffs, say a and f, the average payoff profile may be a weighted average between a and f.

As a philosopher, Jeffrey specialized in epistemology and decision theory. He is perhaps best known for defending and developing the Bayesian approach to probability. Jeffrey also wrote, or co-wrote, two widely used and influential logic textbooks: Formal Logic: Its Scope and Limits, a basic introduction to logic, and Computability and Logic, a more advanced text dealing with, among other things, the famous negative results of twentieth century logic such as Gödel's incompleteness theorems and Tarski's indefinability theorem.

Slicing the Truth: On the Computability Theoretic and Reverse Mathematical Analysis of Combinatorial Principles is a book on reverse mathematics in combinatorics, the study of the axioms needed to prove combinatorial theorems. It was written by Denis R. Hirschfeldt, based on a course given by Hirschfeldt at the National University of Singapore in 2010, and published in 2014 by World Scientific, as volume 28 of the Lecture Notes Series of the Institute for Mathematical Sciences, National University of Singapore.

Chapter nine discusses ways to weaken Ramsey's theorem, and the final chapter discusses stronger theorems in combinatorics including the Dushnik–Miller theorem on self-embedding of infinite linear orderings, Kruskal's tree theorem, Laver's theorem on order embedding of countable linear orders, and Hindman's theorem on IP sets. An appendix provides a proof of a theorem of Jiayi Liu, part of the collection of results showing that the graph Ramsey theorem does not fall into the big five subsystems.

In mathematics, a solvmanifold is a homogeneous space of a connected solvable Lie group. It may also be characterized as a quotient of a connected solvable Lie group by a closed subgroup. (Some authors also require that the Lie group be simply-connected, or that the quotient be compact.) A special class of solvmanifolds, nilmanifolds, was introduced by Anatoly Maltsev, who proved the first structural theorems. Properties of general solvmanifolds are similar, but somewhat more complicated.

In mathematics, geometry and topology is an umbrella term for the historically distinct disciplines of geometry and topology, as general frameworks allow both disciplines to be manipulated uniformly, most visibly in local to global theorems in Riemannian geometry, and results like the Gauss–Bonnet theorem and Chern–Weil theory. Sharp distinctions between geometry and topology can be drawn, however, as discussed below. It is also the title of a journal Geometry & Topology that covers these topics.

Because of this, the term "coffee table book" can be used pejoratively to indicate a superficial approach to the subject. In the field of mathematics, a coffee table book is usually a notebook containing a number of mathematical problems and theorems contributed by a community meeting in a particular place, or connected by a common scientific interest. An example of this was the Scottish Book created by mathematicians at Lviv University in the 1930s and 1940s.

Grünbaum authored over 200 papers, mostly in discrete geometry, an area in which he is known for various classification theorems. He wrote on the theory of abstract polyhedra. His paper on line arrangements may have inspired a paper by N. G. de Bruijn on quasiperiodic tilings (the most famous example of which is the Penrose tiling of the plane). This paper is also cited by the authors of a monograph on hyperplane arrangements as having inspired their research.

Probability theory became measure theory with its own problems and terminology. Doob recognized that this would make it possible to give rigorous proofs for existing probability results, and he felt that the tools of measure theory would lead to new probability results. Doob's approach to probability was evident in his first probability paper,J.L. Doob Probability and statistics in which he proved theorems related to the law of large numbers, using a probabilistic interpretation of Birkhoff's ergodic theorem.

This theorem showed that axiom systems were limited when reasoning about the computation that deduces their theorems. Church and Turing independently demonstrated that Hilbert's (decision problem) was unsolvable, thus identifying the computational core of the incompleteness theorem. This work, along with Gödel's work on general recursive functions, established that there are sets of simple instructions, which, when put together, are able to produce any computation. The work of Gödel showed that the notion of computation is essentially unique.

Mishura earned a Ph.D. in 1978 from the Taras Shevchenko National University of Kyiv with a dissertation on Limit Theorems for Functionals from Stochastic Fields supervised by Dmitrii Sergeevich Silvestrov. She earned a Dr. Sci. from the National Academy of Sciences of Ukraine in 1990 with a dissertation Martingale Methods in the Theory of Stochastic Fields. She became an assistant professor in the Faculty of Mechanics and Mathematics at National Taras Shevchenko University of Kyiv in 1976.

The Almagest is one of the most influential books in the history of Western astronomy. In this book, Ptolemy explained how to predict the behavior of the planets, as Hipparchus could not, with the introduction of a new mathematical tool, the equant. The Almagest gave a comprehensive treatment of astronomy, incorporating theorems, models, and observations from many previous mathematicians. This fact may explain its survival, in contrast to more specialized works that were neglected and lost.

The first branch of algebraic graph theory involves the study of graphs in connection with linear algebra. Especially, it studies the spectrum of the adjacency matrix, or the Laplacian matrix of a graph (this part of algebraic graph theory is also called spectral graph theory). For the Petersen graph, for example, the spectrum of the adjacency matrix is (−2, −2, −2, −2, 1, 1, 1, 1, 1, 3). Several theorems relate properties of the spectrum to other graph properties.

A theory can be either descriptive as in science, or prescriptive (normative) as in philosophy. The latter are those whose subject matter consists not of empirical data, but rather of ideas. At least some of the elementary theorems of a philosophical theory are statements whose truth cannot necessarily be scientifically tested through empirical observation. A field of study is sometimes named a "theory" because its basis is some initial set of assumptions describing the field's approach to the subject.

Hence S+ + Extensionality has the power of ZF. Boolos also argued that the axiom of choice does not follow from the iterative conception, but did not address whether Choice could be added to S in some way.Boolos (1998: 97). Hence S+ + Extensionality cannot prove those theorems of the conventional set theory ZFC whose proofs require Choice. Inf guarantees the existence of stages ω, and of ω + n for finite n, but not of stage ω + ω.

The pioneer of computer science, Alan Turing Multiple new fields of mathematics were developed in the 20th century. In the first part of the 20th century, measure theory, functional analysis, and topology were established, and significant developments were made in fields such as abstract algebra and probability. The development of set theory and formal logic led to Gödel's incompleteness theorems. Later in the 20th century, the development of computers led to the establishment of a theory of computation.

In mathematics, in particular in functional analysis and nonlinear analysis, it is possible to define the derivative of a function between two Fréchet spaces. This notion of differentiation, as it is Gateaux derivative between Fréchet spaces, is significantly weaker than the derivative in a Banach space, even between general topological vector spaces. Nevertheless, it is the weakest notion of differentiation for which many of the familiar theorems from calculus hold. In particular, the chain rule is true.

Griffin identifies the complexity of the AUM theory as a weakness arguing, "hypothetically, the 47 axioms could spawn over a thousand theorems.". Potential expansion of the axioms as a result of incorporation of more cultural variability indicates the possibility of causing greater confusion and complication. Ting-Toomey explores the content of AUM theory as a potential weakness demanding further revision of the theories. She points out five conceptual issues in relation to URT and the social penetration theory.

Amal Kumar Raychaudhuri (; 14 September 1923 – 18 June 2005) was an Indian physicist, known for his research in general relativity and cosmology. His most significant contribution is the eponymous Raychaudhuri equation, which demonstrates that singularities arise inevitably in general relativity and is a key ingredient in the proofs of the Penrose–Hawking singularity theorems. Raychaudhuri was also revered as a teacher during his tenure at Presidency College, Kolkata. Many of his students have gone on to become established scientists.

Class field theory is used to prove Artin-Verdier duality.Milne, J. S. Arithmetic duality theorems. Charleston, SC: BookSurge, LLC 2006 Very explicit class field theory is used in many subareas of algebraic number theory such as Iwasawa theory and Galois modules theory. Most main achievements in the Langlands correspondence for number fields, the BSD conjecture for number fields, and Iwasawa theory for number fields are using very explicit but narrow class field theory methods or their generalizations.

"The Space-time Manifold of Relativity. The Non-Euclidean Geometry of Mechanics and Electromagnetics", Proceedings of the American Academy of Arts and Sciences 48:387–507Synthetic Spacetime, a digest of the axioms used, and theorems proved, by Wilson and Lewis. Archived by WebCite to express the special theory of relativity. In 1918, Hermann Weyl referred to affine geometry for his text Space, Time, Matter. He used affine geometry to introduce vector addition and subtractionHermann Weyl (1918)Raum, Zeit, Materie.

Many theorems about geometric properties of holomorphic functions of one complex variable have been extended to quasiregular maps. These extensions are usually highly non-trivial. Perhaps the most famous result of this sort is the extension of Picard's theorem which is due to Seppo Rickman: :A K-quasiregular map Rn -> Rn can omit at most a finite set. When n = 2, this omitted set can contain at most two points (this is a simple extension of Picard's theorem).

Charles A. Weibel, Robert W. Thomason (1952–1995). Specifically, he proved equivariant analogs of fundamental theorems such as the localization theorem. Equivalently, K_i^G(X) may be defined as the K_i of the category of coherent sheaves on the quotient stack [X/G]. (Hence, the equivariant K-theory is a specific case of the K-theory of a stack.) A version of the Lefschetz fixed point theorem holds in the setting of equivariant (algebraic) K-theory.

Andreu Mas-Colell, Michael Whinston and Jerry Green wrote a textbookMas-Colell et al., ‘Microeconomic Theory’ (1995 and subsequent editions). which is nowadays the standard reference for the fundamental theorems. In it they recast the Bergson–Samuelson second theorem into the Arrow–Debreu formalism, introducing a requirement that the desired end state should be a ‘competitive equilibrium’ – a condition previously absent which excludes some Pareto optima in the case that indifference curves may be non-convex.

Together with F. Fahroo, W. Kang and Q. Gong, Ross proved a series of results on the convergence of pseudospectral discretizations of optimal control problems.W. Kang, I. M. Ross, Q. Gong, Pseudospectral optimal control and its convergence theorems, Analysis and Design of Nonlinear Control Systems, Springer, pp. 109–124, 2008. Ross and his coworkers showed that the Legendre and Chebyshev pseudospectral discretizations converge to an optimal solution of a problem under the mild condition of boundedness of variations.

Many authors impose some connectivity conditions on the spaces X and C in the definition of a covering map. In particular, many authors require both spaces to be path-connected and locally path-connected. This can prove helpful because many theorems hold only if the spaces in question have these properties. Some authors omit the assumption of surjectivity, for if X is connected and C is nonempty then surjectivity of the covering map actually follows from the other axioms.

Cinderella was initially developed by Jürgen Richter-Gebert and Henry Crapo and was used to input incidence theorems and conjectures for automatic theorem proving using the binomial proving method by Richter-Gebert. The initial software was created in Objective-C on the NeXT platform. In 1996, the software was rewritten in Java from scratch by Jürgen Richter-Gebert and Ulrich Kortenkamp. It still included the binomial prover, but was not suitable for classroom teaching as it still was prototypical.

These > thinkers seem to have maintained a modified observational standpoint for the > introduction of natural numbers, for the principle of complete induction > [...] For these, even for such theorems as were deduced by means of > classical logic, they postulated an existence and exactness independent of > language and logic and regarded its non-contradictority as certain, even > without logical proof. For the continuum, however, they seem not to have > sought an origin strictly extraneous to language and logic.

While arXiv does contain some dubious e-prints, such as those claiming to refute famous theorems or proving famous conjectures such as Fermat's Last Theorem using only high-school mathematics, a 2002 article which appeared in Notices of the American Mathematical Society described those as "surprisingly rare". arXiv generally re-classifies these works, e.g. in "General mathematics", rather than deleting them; however, some authors have voiced concern over the lack of transparency in the arXiv screening process.

Having begun to apply these theorems to organizations, by 1954 Simon determined that the best way to study problem-solving was to simulate it with computer programs, which led to his interest in computer simulation of human cognition. Founded during the 1950s, he was among the first members of the Society for General Systems Research. Simon had a keen interest in the arts, as he was a pianist. He was a friend of Robert Lepper and Richard Rappaport.

In Parfit's original formulation, the repugnant conclusion states that Parfit arrives at this conclusion by showing that there is a series of steps, each of which intuitively makes the overall state of the world better, that leads from an "A" world—one with a large population with high average wellbeing—to a "Z" world—one with an extremely large population but just barely positive average wellbeing. Totalism leads to the repugnant conclusion because it holds that the Z world is better than the A world, as the total wellbeing is higher in the Z world for a sufficiently large population. Greaves writes that Parfit searched for a way to avoid the repugnant conclusion, but that he The impossibility theorems in population ethics highlight the difficulty of avoiding the repugnant conclusion without giving up even more fundamental axioms in ethics and rationality. In light of this, several prominent academics have come to accept and even defend the repugnant conclusion, including philosophers Torbjörn Tannsjö and Michael Huemer, because this strategy avoids all the impossibility theorems.

Zorn's lemma, also known as the Kuratowski–Zorn lemma, after mathematicians Max Zorn and Kazimierz Kuratowski, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (that is, every totally ordered subset) necessarily contains at least one maximal element. Proved by Kuratowski in 1922 and independently by Zorn in 1935, this lemma occurs in the proofs of several theorems of crucial importance, for instance the Hahn–Banach theorem in functional analysis, the theorem that every vector space has a basis, Tychonoff's theorem in topology stating that every product of compact spaces is compact, and the theorems in abstract algebra that in a ring with identity every proper ideal is contained in a maximal ideal and that every field has an algebraic closure. Zorn's lemma is equivalent to the well-ordering theorem and also to the axiom of choice, in the sense that any one of the three, together with the Zermelo–Fraenkel axioms of set theory, is sufficient to prove the other two.

This is perhaps best seen in his work on topological methods in nonlinear analysis which he developed into a universal method for finding answers to such qualitative problems such as evaluating the number of solutions, describing the structure of a solution set and conditions for the connectedness of this set, convergence of Galerkin type approximations, the bifurcation of solutions in nonlinear systems, and so on. Krasnosel'skii also presented many new general principles on solvability of a large variety of nonlinear equations, including one-sided estimates, cone stretching and contractions, fixed-point theorems for monotone operators and a combination of the Schauder fixed point and contraction mapping theorems that was the genesis of condensing operators. He suggested a new general method for investigating degenerate extremals in variational problems and developed qualitative methods for studying critical and bifurcation parameter values based on restricted information of nonlinear equations. such as the properties of equations linearized at zero or at infinity, which have been very useful in determining the existence of bounded or periodic solutions.

In computational complexity theory, a branch of computer science, the Max/min CSP/Ones classification theorems state necessary and sufficient conditions that determine the complexity classes of problems about satisfying a subset S of boolean relations. They are similar to Schaefer's dichotomy theorem, which classifies the complexity of satisfying finite sets of relations; however, the Max/min CSP/Ones classification theorems give information about the complexity of approximating an optimal solution to a problem defined by S. Given a set S of clauses, the Max constraint satisfaction problem (CSP) is to find the maximum number (in the weighted case: the maximal sum of weights) of satisfiable clauses in S. Similarly, the Min CSP problem is to minimize the number of unsatisfied clauses. The Max Ones problem is to maximize the number of boolean variables in S that are set to 1 under the restriction that all clauses are satisfied, and the Min Ones problem is to minimize this number. When using the classifications below, the problem's complexity class is determined by the topmost classification that it satisfies.

Today, the bulk of extant mathematics is believed to be derivable logically from a small number of extralogical axioms, such as the axioms of Zermelo–Fraenkel set theory (or its extension ZFC), from which no inconsistencies have as yet been derived. Thus, elements of the logicist programmes have proved viable, but in the process theories of classes, sets and mappings, and higher-order logics other than with Henkin semantics, have come to be regarded as extralogical in nature, in part under the influence of Quine's later thought. Kurt Gödel's incompleteness theorems show that no formal system from which the Peano axioms for the natural numbers may be derived — such as Russell's systems in PM — can decide all the well-formed sentences of that system."On the philosophical relevance of Gödel's incompleteness theorems" This result damaged Hilbert's programme for foundations of mathematics whereby 'infinitary' theories — such as that of PM — were to be proved consistent from finitary theories, with the aim that those uneasy about 'infinitary methods' could be reassurred that their use should provably not result in the derivation of a contradiction.

The whole of Iamblichus's complex theory is ruled by a mathematical formalism of triad, hebdomad, etc., while the first principle is identified with the monad, dyad and triad; symbolic meanings being also assigned to the other numbers. The theorems of mathematics, he says, apply absolutely to all things, from things divine to original matter. But though he subjects all things to number, he holds elsewhere that numbers are independent existences, and occupy a middle place between the limited and unlimited.