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https://en.wikipedia.org/wiki/Sigma%20function	In mathematics, by sigma function one can mean one of the following: The sum-of-divisors function σa(n), an arithmetic function Weierstrass sigma function, related to elliptic functions Rado's sigma function, see busy beaver See also sigmoid function.
https://en.wikipedia.org/wiki/Mathematics%20and%20God	Connections between mathematics and God include the use of mathematics in arguments about the existence of God and about whether belief in God is beneficial. Mathematical arguments for God's existence In the 1070s, Anselm of Canterbury, an Italian medieval philosopher and theologian, created an ontological argument which sought to use logic to prove the existence of God. A more elaborate version was given by Gottfried Leibniz in the early eighteenth century. Kurt Gödel created a formalization of Leibniz' version, known as Gödel's ontological proof. A more recent argument was made by Stephen D. Unwin in 2003, who suggested the use of Bayesian probability to estimate the probability of God's existence. Mathematical arguments for belief A common application of decision theory to the belief in God is Pascal's wager, published by Blaise Pascal in his 1669 work Pensées. The application was a defense of Christianity stating that "If God does not exist, the Atheist loses little by believing in him and gains little by not believing. If God does exist, the Atheist gains eternal life by believing and loses an infinite good by not believing". The atheist's wager has been proposed as a counterargument to Pascal's Wager. See also Existence of God Further reading Cohen, Daniel J., Equations from God: Pure Mathematics and Victorian Faith, Johns Hopkins University Press, 2007 . Livio, Mario, Is God a Mathematician?, Simon & Schuster, 2011 . Ransford, H. Chris, God and the Mathematics of Infinity: What Irreducible Mathematics Says about Godhood, Columbia University Press, 2017 . References Mathematics and culture God Arguments against the existence of God Arguments for the existence of God
https://en.wikipedia.org/wiki/Classical%20orthogonal%20polynomials	In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials: the Hermite polynomials, Laguerre polynomials, Jacobi polynomials (including as a special case the Gegenbauer polynomials, Chebyshev polynomials, and Legendre polynomials). They have many important applications in such areas as mathematical physics (in particular, the theory of random matrices), approximation theory, numerical analysis, and many others. Classical orthogonal polynomials appeared in the early 19th century in the works of Adrien-Marie Legendre, who introduced the Legendre polynomials. In the late 19th century, the study of continued fractions to solve the moment problem by P. L. Chebyshev and then A.A. Markov and T.J. Stieltjes led to the general notion of orthogonal polynomials. For given polynomials and the classical orthogonal polynomials are characterized by being solutions of the differential equation with to be determined constants . There are several more general definitions of orthogonal classical polynomials; for example, use the term for all polynomials in the Askey scheme. Definition In general, the orthogonal polynomials with respect to a weight satisfy The relations above define up to multiplication by a number. Various normalisations are used to fix the constant, e.g. The classical orthogonal polynomials correspond to the following three families of weights: The standard normalisation (also called standardization) is detailed below. Jacobi polynomials For the Jacobi polynomials are given by the formula They are normalised (standardized) by and satisfy the orthogonality condition The Jacobi polynomials are solutions to the differential equation Important special cases The Jacobi polynomials with are called the Gegenbauer polynomials (with parameter ) For , these are called the Legendre polynomials (for which the interval of orthogonality is [−1, 1] and the weight function is simply 1): For , one obtains the Chebyshev polynomials (of the second and first kind, respectively). Hermite polynomials The Hermite polynomials are defined by They satisfy the orthogonality condition and the differential equation Laguerre polynomials The generalised Laguerre polynomials are defined by (the classical Laguerre polynomials correspond to .) They satisfy the orthogonality relation and the differential equation Differential equation The classical orthogonal polynomials arise from a differential equation of the form where Q is a given quadratic (at most) polynomial, and L is a given linear polynomial. The function f, and the constant λ, are to be found. (Note that it makes sense for such an equation to have a polynomial solution. Each term in the equation is a polynomial, and the degrees are consistent.) This is a Sturm–Liouville type of equation. Such equations generally have singularities in their solution functions f except for particular values of λ. They can be thought of as eig
https://en.wikipedia.org/wiki/Girsanov%20theorem	In probability theory, the Girsanov theorem tells how stochastic processes change under changes in measure. The theorem is especially important in the theory of financial mathematics as it tells how to convert from the physical measure, which describes the probability that an underlying instrument (such as a share price or interest rate) will take a particular value or values, to the risk-neutral measure which is a very useful tool for evaluating the value of derivatives on the underlying. History Results of this type were first proved by Cameron-Martin in the 1940s and by Igor Girsanov in 1960. They have been subsequently extended to more general classes of process culminating in the general form of Lenglart (1977). Significance Girsanov's theorem is important in the general theory of stochastic processes since it enables the key result that if Q is a measure that is absolutely continuous with respect to P then every P-semimartingale is a Q-semimartingale. Statement of theorem We state the theorem first for the special case when the underlying stochastic process is a Wiener process. This special case is sufficient for risk-neutral pricing in the Black–Scholes model. Let be a Wiener process on the Wiener probability space . Let be a measurable process adapted to the natural filtration of the Wiener process ; we assume that the usual conditions have been satisfied. Given an adapted process define where is the stochastic exponential of X with respect to W, i.e. and denotes the quadratic variation of the process X. If is a martingale then a probability measure Q can be defined on such that Radon–Nikodym derivative Then for each t the measure Q restricted to the unaugmented sigma fields is equivalent to P restricted to Furthermore if is a local martingale under P then the process is a Q local martingale on the filtered probability space . Corollary If X is a continuous process and W is Brownian motion under measure P then is Brownian motion under Q. The fact that is continuous is trivial; by Girsanov's theorem it is a Q local martingale, and by computing it follows by Levy's characterization of Brownian motion that this is a Q Brownian motion. Comments In many common applications, the process X is defined by For X of this form then a necessary and sufficient condition for X to be a martingale is Novikov's condition which requires that The stochastic exponential is the process Z which solves the stochastic differential equation The measure Q constructed above is not equivalent to P on as this would only be the case if the Radon–Nikodym derivative were a uniformly integrable martingale, which the exponential martingale described above is not. On the other hand as long as Novikov's condition is satisfied the measures are equivalent on . Additionally, then combining this above observation in this case, we see that the process for is a Q Brownian motion. This was Igor Girsanov's original formulation of the above
https://en.wikipedia.org/wiki/Axial	Axial may refer to: one of the anatomical directions describing relationships in an animal body In geometry: a geometric term of location an axis of rotation In chemistry, referring to an axial bond a type of modal frame, in music axial-flow, a type of fan the Axial age in China, India, etc. Axial Seamount and submarine volcano off Oregon, USA Axial, Colorado, a ghost town See also Axiality (disambiguation) Axis (disambiguation)
https://en.wikipedia.org/wiki/PCC	PCC may refer to: Science and technology Pearson correlation coefficient (r), in statistics Periodic counter-current chromatography, a type of affinity chromatography Portable C Compiler, an early compiler for the C programming language Precipitated calcium carbonate, a chemical compound Proof-carrying code, a software mechanism that allows a host system to verify properties Pyridinium chlorochromate, a yellow-orange salt Pyrolytic chromium carbide coating, by vacuum deposition Medicine Pericardiocentesis, a procedure where fluid is aspirated from the pericardium Pheochromocytoma, a neuroendocrine tumor Posterior cingulate cortex, an anatomical brain region Prothrombin complex concentrate, a medication Propionyl-CoA carboxylase, catalyses the carboxylation reaction of propionyl CoA in the mitochondrial matrix 1-piperidinocyclohexanecarbonitrile, a precursor schedule II drug in the US Organizations C. Paul Phelps Correctional Center Clef Club of Jazz and Performing Arts Pacific Coast Conference, a defunct US college athletic conference Pacific Coffee Company Pacific Conference of Churches, the regional ecumenical organization in the Pacific region Pakistan Christian Congress Palmarian Catholic Church Panama Canal Commission PCC Community Markets, a food cooperative based in Seattle, Washington, US PCC SE, a German company Pentecost Convention Centre People's Computer Company Peoria Charter Coach Company, a bus company in Illinois, US Plains Conservation Center Plainfield Curling Club Polynesian Cultural Center Power Computing Corporation Power Corporation of Canada Pradesh Congress Committee, in India Precision Castparts Corp. in Portland, Oregon, US Presbyterian Church in Canada Press Complaints Commission, a voluntary regulatory body for British printed newspapers and magazines Primeiro Comando da Capital, a Brazilian prison gang-terrorist group Printed Circuit Corporation, US Education Pabna Cadet College Palmer College of Chiropractic Pasadena City College Penola Catholic College Pensacola Christian College Piedmont Community College Pima Community College Pitt Community College Pobalscoil Chloich Cheannfhaola Polk Community College, former name of Polk State College Portland Community College Presentation College, Chaguanas Pueblo Community College Punjab College of Commerce Government and politics Palestinian Central Council Palestinian Conciliation Commission Parochial church council in the Church of England Patents County Court, former name of the Intellectual Property Enterprise Court in the United Kingdom Philippine Competition Commission, independent quasi-judicial body of the Philippine government in charge of implementing the Philippine Competition Act Poison control center Police and crime commissioner, in England and Wales Porirua City Council, New Zealand Perth City Council, Australia Partido Comunista Colombiano, the Colombian Communist Party Partido Comunista d
https://en.wikipedia.org/wiki/Raoul%20Bott	Raoul Bott (September 24, 1923 – December 20, 2005) was a Hungarian-American mathematician known for numerous foundational contributions to geometry in its broad sense. He is best known for his Bott periodicity theorem, the Morse–Bott functions which he used in this context, and the Borel–Bott–Weil theorem. Early life Bott was born in Budapest, Hungary, the son of Margit Kovács and Rudolph Bott. His father was of Austrian descent, and his mother was of Hungarian Jewish descent; Bott was raised a Catholic by his mother and stepfather. Bott grew up in Czechoslovakia and spent his working life in the United States. His family emigrated to Canada in 1938, and subsequently he served in the Canadian Army in Europe during World War II. Career Bott later went to college at McGill University in Montreal, where he studied electrical engineering. He then earned a PhD in mathematics from Carnegie Mellon University in Pittsburgh in 1949. His thesis, titled Electrical Network Theory, was written under the direction of Richard Duffin. Afterward, he began teaching at the University of Michigan in Ann Arbor. Bott continued his study at the Institute for Advanced Study in Princeton. He was a professor at Harvard University from 1959 to 1999. In 2005 Bott died of cancer in San Diego. With Richard Duffin at Carnegie Mellon, Bott studied existence of electronic filters corresponding to given positive-real functions. In 1949 they proved a fundamental theorem of filter synthesis. Duffin and Bott extended earlier work by Otto Brune that requisite functions of complex frequency s could be realized by a passive network of inductors and capacitors. The proof relied on induction on the sum of the degrees of the polynomials in the numerator and denominator of the rational function. In his 2000 interview with Allyn Jackson of the American Mathematical Society, he explained that he sees "networks as discrete versions of harmonic theory", so his experience with network synthesis and electronic filter topology introduced him to algebraic topology. Bott met Arnold S. Shapiro at the IAS and they worked together. He studied the homotopy theory of Lie groups, using methods from Morse theory, leading to the Bott periodicity theorem (1957). In the course of this work, he introduced Morse–Bott functions, an important generalization of Morse functions. This led to his role as collaborator over many years with Michael Atiyah, initially via the part played by periodicity in K-theory. Bott made important contributions towards the index theorem, especially in formulating related fixed-point theorems, in particular the so-called 'Woods Hole fixed-point theorem', a combination of the Riemann–Roch theorem and Lefschetz fixed-point theorem (it is named after Woods Hole, Massachusetts, the site of a conference at which collective discussion formulated it). The major Atiyah–Bott papers on what is now the Atiyah–Bott fixed-point theorem were written in the years up to 1968; they collaborate
https://en.wikipedia.org/wiki/Frascati%20Manual	The Frascati Manual is a document setting forth the methodology for collecting statistics about research and development. The Manual was prepared and published by the Organisation for Economic Co-operation and Development. Contents The Frascati Manual classifies budgets according to what is done, what is studied, and who is studying it. For example, an oral history project conducted by a religious organization would be classified as being basic research, in the field of humanities (the sub-category of history), and performed by a non-governmental, non-profit organization. Three forms of research The manual gives definitions for: basic research, applied research, Research and development; research personnel: researchers, technicians, auxiliary personnel. The Frascati Manual classifies research into three categories: Basic research is experimental or theoretical work undertaken primarily to acquire new knowledge about observable phenomena and facts, not directed toward any particular use. Applied research is original investigation to acquire new knowledge directed primarily towards a specific practical aim or objective. Experimental development is systematic effort, based on existing knowledge from research or practical experience, directed toward creating novel or improved materials, products, devices, processes, systems, or services. These involve novelty, creativity, uncertainty, systematic, and reproducibility and transferability. Research areas It also organizes the fields of scholarly research endeavors, from mathematics to literature, into main and sub-categories. The 2002 Frascati Manual included a 'Field of Science' (FOS) classification. After several reviews, a Revised Fields of Science and Technology (FOS) classification was published in February 2007 consisting of the following high-level groupings: Natural sciences Engineering and technology Medical and Health sciences Agricultural sciences Social sciences Humanities Industry sectors The Frascati Manual deals primarily with measuring the expenditure and personnel resources devoted to R&D in the industry sectors performing it: higher education, government, business, and private non-profit organisations. History In June 1963, OECD experts met with the NESTI group (National Experts on Science and Technology Indicators) at the Villa Falconieri in Frascati, Italy. Based on a background document by Christopher Freeman they drafted the first version of Frascati Manual, which is officially known as The Proposed Standard Practice for Surveys of Research and Experimental Development. In 2002 the 6th edition was published. Use The definitions provided in the Frascati Manual have been adopted by many governments and serve as a common language for discussions of science and technology policy and economic development policy. Originally an OECD standard, it has become an acknowledged standard in R&D studies all over the world and is widely used by various organisations as
https://en.wikipedia.org/wiki/List%20of%20calculus%20topics	This is a list of calculus topics. Limits Limit (mathematics) Limit of a function One-sided limit Limit of a sequence Indeterminate form Orders of approximation (ε, δ)-definition of limit Continuous function Differential calculus Derivative Notation Newton's notation for differentiation Leibniz's notation for differentiation Simplest rules Derivative of a constant Sum rule in differentiation Constant factor rule in differentiation Linearity of differentiation Power rule Chain rule Local linearization Product rule Quotient rule Inverse functions and differentiation Implicit differentiation Stationary point Maxima and minima First derivative test Second derivative test Extreme value theorem Differential equation Differential operator Newton's method Taylor's theorem L'Hôpital's rule General Leibniz rule Mean value theorem Logarithmic derivative Differential (calculus) Related rates Regiomontanus' angle maximization problem Rolle's theorem Integral calculus Antiderivative/Indefinite integral Simplest rules Sum rule in integration Constant factor rule in integration Linearity of integration Arbitrary constant of integration Cavalieri's quadrature formula Fundamental theorem of calculus Integration by parts Inverse chain rule method Integration by substitution Tangent half-angle substitution Differentiation under the integral sign Trigonometric substitution Partial fractions in integration Quadratic integral Proof that 22/7 exceeds π Trapezium rule Integral of the secant function Integral of secant cubed Arclength Solid of revolution Shell integration Special functions and numbers Natural logarithm e (mathematical constant) Exponential function Hyperbolic angle Hyperbolic function Stirling's approximation Bernoulli numbers Absolute numerical See also list of numerical analysis topics Rectangle method Trapezoidal rule Simpson's rule Newton–Cotes formulas Gaussian quadrature Lists and tables Table of common limits Table of derivatives Table of integrals Table of mathematical symbols List of integrals List of integrals of rational functions List of integrals of irrational functions List of integrals of trigonometric functions List of integrals of inverse trigonometric functions List of integrals of hyperbolic functions List of integrals of exponential functions List of integrals of logarithmic functions List of integrals of area functions Multivariable Partial derivative Disk integration Gabriel's horn Jacobian matrix Hessian matrix Curvature Green's theorem Divergence theorem Stokes' theorem Series Infinite series Maclaurin series, Taylor series Fourier series Euler–Maclaurin formula History Adequality Infinitesimal Archimedes' use of infinitesimals Gottfried Leibniz Isaac Newton Method of Fluxions Infinitesimal calculus Brook Taylor Colin Maclaurin Leonhard Euler Gauss Joseph Fourier Law of continuity History of calculus Genera
https://en.wikipedia.org/wiki/Dehn%20invariant	In geometry, the Dehn invariant is a value used to determine whether one polyhedron can be cut into pieces and reassembled ("dissected") into another, and whether a polyhedron or its dissections can tile space. It is named after Max Dehn, who used it to solve Hilbert's third problem by proving that not all polyhedra with equal volume could be dissected into each other. Two polyhedra have a dissection into polyhedral pieces that can be reassembled into either one, if and only if their volumes and Dehn invariants are equal. Having Dehn invariant zero is a necessary (but not sufficient) condition for being a space-filling polyhedron, and a polyhedron can be cut up and reassembled into a space-filling polyhedron if and only if its Dehn invariant is zero. The Dehn invariant of a self-intersection-free flexible polyhedron is invariant as it flexes. Dehn invariants are also an invariant for dissection in higher dimensions, and (with volume) a complete invariant in four dimensions. The Dehn invariant is zero for the cube but nonzero for the other Platonic solids, implying that the other solids cannot tile space and that they cannot be dissected into a cube. All of the Archimedean solids have Dehn invariants that are rational combinations of the invariants for the Platonic solids. In particular, the truncated octahedron also tiles space and has Dehn invariant zero like the cube. The Dehn invariants of polyhedra are not numbers. Instead, they are elements of an infinite-dimensional tensor space. This space, viewed as an abelian group, is part of an exact sequence involving group homology. Similar invariants can also be defined for some other dissection puzzles, including the problem of dissecting rectilinear polygons into each other by axis-parallel cuts and translations. Background and history In two dimensions, the Wallace–Bolyai–Gerwien theorem from the early 19th century states that any two polygons of equal area can be cut up into polygonal pieces and reassembled into each other. In the late 19th century, David Hilbert became interested in this result. He used it as a way to axiomatize the area of two-dimensional polygons, in connection with Hilbert's axioms for Euclidean geometry. This was part of a program to make the foundations of geometry more rigorous, by treating explicitly notions like area that Euclid's Elements had handled more intuitively. Naturally, this raised the question of whether a similar axiomatic treatment could be extended to solid geometry. At the 1900 International Congress of Mathematicians, Hilbert formulated Hilbert's problems, a set of problems that became very influential in 20th-century mathematics. One of those, Hilbert's third problem, addressed this question on the axiomatization of solid volume. Hilbert's third problem asked, more specifically, whether every two polyhedra of equal volumes can always be cut into polyhedral pieces and reassembled into each other. If this were the case, then the volume of any polyhe
https://en.wikipedia.org/wiki/Plateau%27s%20problem	In mathematics, Plateau's problem is to show the existence of a minimal surface with a given boundary, a problem raised by Joseph-Louis Lagrange in 1760. However, it is named after Joseph Plateau who experimented with soap films. The problem is considered part of the calculus of variations. The existence and regularity problems are part of geometric measure theory. History Various specialized forms of the problem were solved, but it was only in 1930 that general solutions were found in the context of mappings (immersions) independently by Jesse Douglas and Tibor Radó. Their methods were quite different; Radó's work built on the previous work of René Garnier and held only for rectifiable simple closed curves, whereas Douglas used completely new ideas with his result holding for an arbitrary simple closed curve. Both relied on setting up minimization problems; Douglas minimized the now-named Douglas integral while Radó minimized the "energy". Douglas went on to be awarded the Fields Medal in 1936 for his efforts. In higher dimensions The extension of the problem to higher dimensions (that is, for -dimensional surfaces in -dimensional space) turns out to be much more difficult to study. Moreover, while the solutions to the original problem are always regular, it turns out that the solutions to the extended problem may have singularities if . In the hypersurface case where , singularities occur only for . An example of such singular solution of the Plateau problem is the Simons cone, a cone over in that was first described by Jim Simons and was shown to be an area minimizer by Bombieri, De Giorgi and Giusti. To solve the extended problem in certain special cases, the theory of perimeters (De Giorgi) for codimension 1 and the theory of rectifiable currents (Federer and Fleming) for higher codimension have been developed. The theory guarantees existence of codimension 1 solutions that are smooth away from a closed set of Hausdorff dimension . In the case of higher codimension Almgren proved existence of solutions with singular set of dimension at most in his regularity theorem. S. X. Chang, a student of Almgren, built upon Almgren’s work to show that the singularities of 2-dimensional area minimizing integral currents (in arbitrary codimension) form a finite discrete set. The axiomatic approach of Jenny Harrison and Harrison Pugh treats a wide variety of special cases. In particular, they solve the anisotropic Plateau problem in arbitrary dimension and codimension for any collection of rectifiable sets satisfying a combination of general homological, cohomological or homotopical spanning conditions. A different proof of Harrison-Pugh's results were obtained by Camillo De Lellis, Francesco Ghiraldin and Francesco Maggi. Physical applications Physical soap films are more accurately modeled by the -minimal sets of Frederick Almgren, but the lack of a compactness theorem makes it difficult to prove the existence of an area minimizer. In this co
https://en.wikipedia.org/wiki/Long%20line%20%28topology%29	In topology, the long line (or Alexandroff line) is a topological space somewhat similar to the real line, but in a certain way "longer". It behaves locally just like the real line, but has different large-scale properties (e.g., it is neither Lindelöf nor separable). Therefore, it serves as an important counterexample in topology. Intuitively, the usual real-number line consists of a countable number of line segments laid end-to-end, whereas the long line is constructed from an uncountable number of such segments. Definition The closed long ray is defined as the Cartesian product of the first uncountable ordinal with the half-open interval equipped with the order topology that arises from the lexicographical order on . The open long ray is obtained from the closed long ray by removing the smallest element The long line is obtained by "gluing" together two long rays, one in the positive direction and the other in the negative direction. More rigorously, it can be defined as the order topology on the disjoint union of the reversed open long ray (“reversed” means the order is reversed) (this is the negative half) and the (not reversed) closed long ray (the positive half), totally ordered by letting the points of the latter be greater than the points of the former. Alternatively, take two copies of the open long ray and identify the open interval of the one with the same interval of the other but reversing the interval, that is, identify the point (where is a real number such that ) of the one with the point of the other, and define the long line to be the topological space obtained by gluing the two open long rays along the open interval identified between the two. (The former construction is better in the sense that it defines the order on the long line and shows that the topology is the order topology; the latter is better in the sense that it uses gluing along an open set, which is clearer from the topological point of view.) Intuitively, the closed long ray is like a real (closed) half-line, except that it is much longer in one direction: we say that it is long at one end and closed at the other. The open long ray is like the real line (or equivalently an open half-line) except that it is much longer in one direction: we say that it is long at one end and short (open) at the other. The long line is longer than the real lines in both directions: we say that it is long in both directions. However, many authors speak of the “long line” where we have spoken of the (closed or open) long ray, and there is much confusion between the various long spaces. In many uses or counterexamples, however, the distinction is unessential, because the important part is the “long” end of the line, and it doesn't matter what happens at the other end (whether long, short, or closed). A related space, the (closed) extended long ray, is obtained as the one-point compactification of by adjoining an additional element to the right end of One can
https://en.wikipedia.org/wiki/Reinhard%20H%C3%B6ppner	Reinhard Höppner (2 December 1948 – 9 June 2014) was a German politician (SPD) and writer. Höppner held a Dr. rer. nat. in mathematics. In 1990, in the first (and last) free election in the assembly's history, he was elected a member of the East German People's Chamber (Volkskammer), becoming the assembly's vice president. He became the 4th Minister President of Saxony-Anhalt in July 1994 when, his SPD (party) having failed to secure an outright majority, entered into a minority governing coalition with the Green party. This was controversial at the time because most had expected that the SPD, if denied an overall majority, would govern in coalition with the PDS, successor to the old East German ruling party: together the SPD and PDS would have had an overall majority. The so-called Magdeburg model for a minority SPD/Green coalition that excluded the PDS but nevertheless was tolerated (not voted down) by them was subsequently followed in other regional assemblies. Höppner remained in office until 16 May 2002, when he was succeeded by Wolfgang Böhmer. References 1948 births People from Haldensleben Members of the 10th Volkskammer Members of the Landtag of Saxony-Anhalt Social Democratic Party of Germany politicians German Protestants Knights Commander of the Order of Merit of the Federal Republic of Germany 2014 deaths Ministers-President of Saxony-Anhalt
https://en.wikipedia.org/wiki/Radon%E2%80%93Nikodym%20theorem	In mathematics, the Radon–Nikodym theorem is a result in measure theory that expresses the relationship between two measures defined on the same measurable space. A measure is a set function that assigns a consistent magnitude to the measurable subsets of a measurable space. Examples of a measure include area and volume, where the subsets are sets of points; or the probability of an event, which is a subset of possible outcomes within a wider probability space. One way to derive a new measure from one already given is to assign a density to each point of the space, then integrate over the measurable subset of interest. This can be expressed as where is the new measure being defined for any measurable subset and the function is the density at a given point. The integral is with respect to an existing measure , which may often be the canonical Lebesgue measure on the real line or the n-dimensional Euclidean space (corresponding to our standard notions of length, area and volume). For example, if represented mass density and was the Lebesgue measure in three-dimensional space , then would equal the total mass in a spatial region . The Radon–Nikodym theorem essentially states that, under certain conditions, any measure can be expressed in this way with respect to another measure on the same space. The function is then called the Radon–Nikodym derivative and is denoted by . An important application is in probability theory, leading to the probability density function of a random variable. The theorem is named after Johann Radon, who proved the theorem for the special case where the underlying space is in 1913, and for Otto Nikodym who proved the general case in 1930. In 1936 Hans Freudenthal generalized the Radon–Nikodym theorem by proving the Freudenthal spectral theorem, a result in Riesz space theory; this contains the Radon–Nikodym theorem as a special case. A Banach space is said to have the Radon–Nikodym property if the generalization of the Radon–Nikodym theorem also holds, mutatis mutandis, for functions with values in . All Hilbert spaces have the Radon–Nikodym property. Formal description Radon–Nikodym theorem The Radon–Nikodym theorem involves a measurable space on which two σ-finite measures are defined, and It states that, if (that is, if is absolutely continuous with respect to ), then there exists a -measurable function such that for any measurable set Radon–Nikodym derivative The function satisfying the above equality is , that is, if is another function which satisfies the same property, then . The function is commonly written and is called the . The choice of notation and the name of the function reflects the fact that the function is analogous to a derivative in calculus in the sense that it describes the rate of change of density of one measure with respect to another (the way the Jacobian determinant is used in multivariable integration). Extension to signed or complex measures A similar theorem c
https://en.wikipedia.org/wiki/Exponential%20family	In probability and statistics, an exponential family is a parametric set of probability distributions of a certain form, specified below. This special form is chosen for mathematical convenience, including the enabling of the user to calculate expectations, covariances using differentiation based on some useful algebraic properties, as well as for generality, as exponential families are in a sense very natural sets of distributions to consider. The term exponential class is sometimes used in place of "exponential family", or the older term Koopman–Darmois family. The terms "distribution" and "family" are often used loosely: specifically, an exponential family is a set of distributions, where the specific distribution varies with the parameter; however, a parametric family of distributions is often referred to as "a distribution" (like "the normal distribution", meaning "the family of normal distributions"), and the set of all exponential families is sometimes loosely referred to as "the" exponential family. They are distinct because they possess a variety of desirable properties, most importantly the existence of a sufficient statistic. The concept of exponential families is credited to E. J. G. Pitman, G. Darmois, and B. O. Koopman in 1935–1936. Exponential families of distributions provide a general framework for selecting a possible alternative parameterisation of a parametric family of distributions, in terms of natural parameters, and for defining useful sample statistics, called the natural sufficient statistics of the family. Definition Most of the commonly used distributions form an exponential family or subset of an exponential family, listed in the subsection below. The subsections following it are a sequence of increasingly more general mathematical definitions of an exponential family. A casual reader may wish to restrict attention to the first and simplest definition, which corresponds to a single-parameter family of discrete or continuous probability distributions. Examples of exponential family distributions Exponential families include many of the most common distributions. Among many others, exponential families includes the following: normal exponential gamma chi-squared beta Dirichlet Bernoulli categorical Poisson Wishart inverse Wishart geometric A number of common distributions are exponential families, but only when certain parameters are fixed and known. For example: binomial (with fixed number of trials) multinomial (with fixed number of trials) negative binomial (with fixed number of failures) Notice that in each case, the parameters which must be fixed determine a limit on the size of observation values. Examples of common distributions that are not exponential families are Student's t, most mixture distributions, and even the family of uniform distributions when the bounds are not fixed. See the section below on examples for more discussion. Scalar parameter A single-parameter exponential family is a set of pro
https://en.wikipedia.org/wiki/MOS	MOS or Mos may refer to: Technology MOSFET (metal–oxide–semiconductor field-effect transistor), also known as the MOS transistor Mathematical Optimization Society Model output statistics, a weather-forecasting technique MOS (filmmaking), term for a scene that is "motor only sync" or "motor only shot", or jokingly, “mit out sound” Mobile operating system, operating systems for mobile devices Computing Acorn MOS, an operating system used in the Acorn BBC computer range Media Object Server, a protocol used in newsroom computer systems Mean opinion score, a measure of the perceived quality of a signal MOS (operating system), a Soviet Unix clone My Oracle Support, a support site for the users of Oracle Corporation products, known until October 2010 as "MetaLink" macOS, an operating system for Macs Government and military Master of the Sword, the title for the head of physical education at the U.S. Military Academy at West Point Member of Service, any emergency responder (police officer, firefighter, emergency medical technician) that needs emergency help, usually over two-way radio Military occupation specialty code, used by the U.S. military to identify a specific job Ministry of Supply, former British government ministry that co-ordinated military supplies Places Ma On Shan (town), a town in the New Territories of Hong Kong Ma On Shan station, MTR station code Mos, Spain, a municipality in Galicia, Spain in the province of Pontevedra Museum of Science (Boston) (MoS), a Boston, Massachusetts landmark, located in Science Park, a plot of land spanning the Charles River, USA Companies and organizations MOS (brand), American brand of organizational tools MOS Technology, a defunct semiconductor company Mos, a startup tech company founded by Amira Yahyaoui MOS Burger, a fast-food restaurant chain that originated in Japan The Mosaic Company (NYSE: MOS), American fertilizer and mining company Other uses Mos, an uncommon singular form of mores, widely observed social norms (from Latin and ) Mos, a traditional dish of the Nivkh people Mos language, an aboriginal Mon–Khmer language of Malaya and Thailand Mannan oligosaccharide-based nutritional supplements Manual of style, also known a style guide or stylebook; a guide for writing and sometimes also for layout and typography Margin on services, a financial reporting method for Australian life insurance companies Moment of symmetry, in music, same as well formed generated collection MOS (gene), gene for a human protein expressed in testis during sperm formation MOS, German vehicle registration plate district code for Neckar-Odenwald-Kreis "Man on the street" () segments in broadcasting Mossi language ISO 639 alpha-2 language code Morvan Syndrome (MoS) though usually referred to as MVS MOS, minimum operating segment of a transportation system Mos, nickname of Thai singer Patiparn Patavekarn, also referred to as Mos Patiparn See also Mo's Restaurants, American r
https://en.wikipedia.org/wiki/Johann%20Radon	Johann Karl August Radon (; 16 December 1887 – 25 May 1956) was an Austrian mathematician. His doctoral dissertation was on the calculus of variations (in 1910, at the University of Vienna). Life Radon was born in Tetschen, Bohemia, Austria-Hungary, now Děčín, Czech Republic. He received his doctoral degree at the University of Vienna in 1910. He spent the winter semester 1910/11 at the University of Göttingen, then he was an assistant at the German Technical University in Brno, and from 1912 to 1919 at the Technical University of Vienna. In 1913/14, he passed his habilitation at the University of Vienna. Due to his near-sightedness, he was exempt from the draft during wartime. In 1919, he was called to become Professor extraordinarius at the newly founded University of Hamburg; in 1922, he became Professor ordinarius at the University of Greifswald, and in 1925 at the University of Erlangen. Then he was Ordinarius at the University of Breslau from 1928 to 1945. After a short stay at the University of Innsbruck he became Ordinarius at the Institute of Mathematics of the University of Vienna on 1 October 1946. In 1954/55, he was rector of the University of Vienna. In 1939, Radon became corresponding member of the Austrian Academy of Sciences, and in 1947, he became a member. From 1952 to 1956, he was Secretary of the Class of Mathematics and Science of this Academy. From 1948 to 1950, he was president of the Austrian Mathematical Society. Johann Radon married Maria Rigele, a secondary school teacher, in 1916. They had three sons who died young or very young. Their daughter Brigitte, born in 1924, obtained a Ph.D. in mathematics at the University of Innsbruck and married the Austrian mathematician Erich Bukovics in 1950. Brigitte lives in Vienna. Radon, as Curt C. Christian described him in 1987 at the occasion of the unveiling of his brass bust at the University of Vienna, was a friendly, good-natured man, highly esteemed by students and colleagues alike, a noble personality. He did make the impression of a quiet scholar, but he was also sociable and willing to celebrate. He loved music, and he played music with friends at home, being an excellent violinist himself, and a good singer. His love for classical literature lasted through all his life. In 2003, the Austrian Academy of Sciences founded an Institute for Computational and Applied Mathematics and named it after Johann Radon (see the external link below). Achievements Radon is known for a number of lasting contributions, including: his part in the Radon–Nikodym theorem; the Radon measure concept of measure as linear functional; the Radon transform, in integral geometry, based on integration over hyperplanes—with application to tomography for scanners (see tomographic reconstruction); Radon's theorem, that d + 2 points in d dimensions may always be partitioned into two subsets with intersecting convex hulls; the Radon–Hurwitz numbers. He is possibly the first to make use of
https://en.wikipedia.org/wiki/Semiprime	In mathematics, a semiprime is a natural number that is the product of exactly two prime numbers. The two primes in the product may equal each other, so the semiprimes include the squares of prime numbers. Because there are infinitely many prime numbers, there are also infinitely many semiprimes. Semiprimes are also called biprimes. Examples and variations The semiprimes less than 100 are: Semiprimes that are not square numbers are called discrete, distinct, or squarefree semiprimes: The semiprimes are the case of the -almost primes, numbers with exactly prime factors. However some sources use "semiprime" to refer to a larger set of numbers, the numbers with at most two prime factors (including unit (1), primes, and semiprimes). These are: Formula for number of semiprimes A semiprime counting formula was discovered by E. Noel and G. Panos in 2005. Let denote the number of semiprimes less than or equal to n. Then where is the prime-counting function and denotes the kth prime. Properties Semiprime numbers have no composite numbers as factors other than themselves. For example, the number 26 is semiprime and its only factors are 1, 2, 13, and 26, of which only 26 is composite. For a squarefree semiprime (with ) the value of Euler's totient function (the number of positive integers less than or equal to that are relatively prime to ) takes the simple form This calculation is an important part of the application of semiprimes in the RSA cryptosystem. For a square semiprime , the formula is again simple: Applications Semiprimes are highly useful in the area of cryptography and number theory, most notably in public key cryptography, where they are used by RSA and pseudorandom number generators such as Blum Blum Shub. These methods rely on the fact that finding two large primes and multiplying them together (resulting in a semiprime) is computationally simple, whereas finding the original factors appears to be difficult. In the RSA Factoring Challenge, RSA Security offered prizes for the factoring of specific large semiprimes and several prizes were awarded. The original RSA Factoring Challenge was issued in 1991, and was replaced in 2001 by the New RSA Factoring Challenge, which was later withdrawn in 2007. In 1974 the Arecibo message was sent with a radio signal aimed at a star cluster. It consisted of binary digits intended to be interpreted as a bitmap image. The number was chosen because it is a semiprime and therefore can be arranged into a rectangular image in only two distinct ways (23 rows and 73 columns, or 73 rows and 23 columns). See also Chen's theorem Sphenic number, a product of three distinct primes References External links Integer sequences Prime numbers Theory of cryptography
https://en.wikipedia.org/wiki/Almost%20prime	In number theory, a natural number is called k-almost prime if it has k prime factors. More formally, a number n is k-almost prime if and only if Ω(n) = k, where Ω(n) is the total number of primes in the prime factorization of n (can be also seen as the sum of all the primes' exponents): A natural number is thus prime if and only if it is 1-almost prime, and semiprime if and only if it is 2-almost prime. The set of k-almost primes is usually denoted by Pk. The smallest k-almost prime is 2k. The first few k-almost primes are: The number πk(n) of positive integers less than or equal to n with exactly k prime divisors (not necessarily distinct) is asymptotic to: a result of Landau. See also the Hardy–Ramanujan theorem. Properties The multiple of a -almost prime and a -almost prime is a -almost prime. A -almost prime cannot have a -almost prime as a factor for all . References External links Integer sequences Prime numbers
https://en.wikipedia.org/wiki/Pointwise%20convergence	In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function. It is weaker than uniform convergence, to which it is often compared. Definition Suppose that is a set and is a topological space, such as the real or complex numbers or a metric space, for example. A net or sequence of functions all having the same domain and codomain is said to converge pointwise to a given function often written as if (and only if) The function is said to be the pointwise limit function of the Sometimes, authors use the term bounded pointwise convergence when there is a constant such that . Properties This concept is often contrasted with uniform convergence. To say that means that where is the common domain of and , and stands for the supremum. That is a stronger statement than the assertion of pointwise convergence: every uniformly convergent sequence is pointwise convergent, to the same limiting function, but some pointwise convergent sequences are not uniformly convergent. For example, if is a sequence of functions defined by then pointwise on the interval but not uniformly. The pointwise limit of a sequence of continuous functions may be a discontinuous function, but only if the convergence is not uniform. For example, takes the value when is an integer and when is not an integer, and so is discontinuous at every integer. The values of the functions need not be real numbers, but may be in any topological space, in order that the concept of pointwise convergence make sense. Uniform convergence, on the other hand, does not make sense for functions taking values in topological spaces generally, but makes sense for functions taking values in metric spaces, and, more generally, in uniform spaces. Topology Let denote the set of all functions from some given set into some topological space As described in the article on characterizations of the category of topological spaces, if certain conditions are met then it is possible to define a unique topology on a set in terms of what nets do and do not converge. The definition of pointwise convergence meets these conditions and so it induces a topology, called the , on the set of all functions of the form A net in converges in this topology if and only if it converges pointwise. The topology of pointwise convergence is the same as convergence in the product topology on the space where is the domain and is the codomain. Explicitly, if is a set of functions from some set into some topological space then the topology of pointwise convergence on is equal to the subspace topology that it inherits from the product space when is identified as a subset of this Cartesian product via the canonical inclusion map defined by If the codomain is compact, then by Tychonoff's theorem, the space is also compact. Almost everywhere convergence In measure theory, one talks about almost everywhere conver
https://en.wikipedia.org/wiki/Partition%20of%20a%20set	In mathematics, a partition of a set is a grouping of its elements into non-empty subsets, in such a way that every element is included in exactly one subset. Every equivalence relation on a set defines a partition of this set, and every partition defines an equivalence relation. A set equipped with an equivalence relation or a partition is sometimes called a setoid, typically in type theory and proof theory. Definition and notation A partition of a set X is a set of non-empty subsets of X such that every element x in X is in exactly one of these subsets (i.e., the subsets are nonempty mutually disjoint sets). Equivalently, a family of sets P is a partition of X if and only if all of the following conditions hold: The family P does not contain the empty set (that is ). The union of the sets in P is equal to X (that is ). The sets in P are said to exhaust or cover X. See also collectively exhaustive events and cover (topology). The intersection of any two distinct sets in P is empty (that is ). The elements of P are said to be pairwise disjoint or mutually exclusive. See also mutual exclusivity. The sets in are called the blocks, parts, or cells, of the partition. If then we represent the cell containing by . That is to say, is notation for the cell in which contains . Every partition may be identified with an equivalence relation on , namely the relation such that for any we have if and only if (equivalently, if and only if ). The notation evokes the idea that the equivalence relation may be constructed from the partition. Conversely every equivalence relation may be identified with a partition. This is why it is sometimes said informally that "an equivalence relation is the same as a partition". If P is the partition identified with a given equivalence relation , then some authors write . This notation is suggestive of the idea that the partition is the set X divided in to cells. The notation also evokes the idea that, from the equivalence relation one may construct the partition. The rank of is , if is finite. Examples The empty set has exactly one partition, namely . (Note: this is the partition, not a member of the partition.) For any non-empty set X, P = is a partition of X, called the trivial partition. Particularly, every singleton set {x} has exactly one partition, namely . For any non-empty proper subset A of a set U, the set A together with its complement form a partition of U, namely, . The set has these five partitions (one partition per item): , sometimes written 1 \| 2 \| 3. , or 1 2 \| 3. , or 1 3 \| 2. , or 1 \| 2 3. , or 123 (in contexts where there will be no confusion with the number). The following are not partitions of : is not a partition (of any set) because one of its elements is the empty set. is not a partition (of any set) because the element 2 is contained in more than one block. is not a partition of because none of its blocks contains 3; however, it is a partition of . Par
https://en.wikipedia.org/wiki/Pentagonal%20number%20theorem	In mathematics, Euler's pentagonal number theorem relates the product and series representations of the Euler function. It states that In other words, The exponents 1, 2, 5, 7, 12, ... on the right hand side are given by the formula for k = 1, −1, 2, −2, 3, ... and are called (generalized) pentagonal numbers . (The constant term 1 corresponds to .) This holds as an identity of convergent power series for , and also as an identity of formal power series. A striking feature of this formula is the amount of cancellation in the expansion of the product. Relation with partitions The identity implies a recurrence for calculating , the number of partitions of n: or more formally, where the summation is over all nonzero integers k (positive and negative) and is the kth generalized pentagonal number. Since for all , the apparently infinite series on the right has only finitely many non-zero terms, enabling an efficient calculation of p(n). Franklin's bijective proof The theorem can be interpreted combinatorially in terms of partitions. In particular, the left hand side is a generating function for the number of partitions of n into an even number of distinct parts minus the number of partitions of n into an odd number of distinct parts. Each partition of n into an even number of distinct parts contributes +1 to the coefficient of xn; each partition into an odd number of distinct parts contributes −1. (The article on unrestricted partition functions discusses this type of generating function.) For example, the coefficient of x5 is +1 because there are two ways to split 5 into an even number of distinct parts (4+1 and 3+2), but only one way to do so for an odd number of distinct parts (the one-part partition 5). However, the coefficient of x12 is −1 because there are seven ways to partition 12 into an even number of distinct parts, but there are eight ways to partition 12 into an odd number of distinct parts, and 7 − 8 = −1. This interpretation leads to a proof of the identity by canceling pairs of matched terms (involution method). Consider the Ferrers diagram of any partition of n into distinct parts. For example, the diagram below shows n = 20 and the partition 20 = 7 + 6 + 4 + 3. Let m be the number of elements in the smallest row of the diagram (m = 3 in the above example). Let s be the number of elements in the rightmost 45 degree line of the diagram (s = 2 dots in red above, since 7−1 = 6, but 6−1 > 4). If m > s, take the rightmost 45-degree line and move it to form a new row, as in the matching diagram below. If m ≤ s (as in our newly formed diagram where m = 2, s = 5) we may reverse the process by moving the bottom row to form a new 45 degree line (adding 1 element to each of the first m rows), taking us back to the first diagram. A bit of thought shows that this process always changes the parity of the number of rows, and applying the process twice brings us back to the original diagram. This enables us to pair off Ferrers d
https://en.wikipedia.org/wiki/Ludwig%20Immanuel%20Magnus	Ludwig Immanuel Magnus (March 15, 1790 – September 25, 1861) was a German Jewish mathematician who, in 1831, published a paper about the inversion transformation, which leads to inversive geometry. His reputation as a mathematician was established by 1834 and an honorary doctorate conferred on him by the University of Bonn. His work appeared in Gergonne's Annales de mathématiques pures et appliquées vols. xi and xvi (1820–25); in Crelle's Journal, vols. v, vii, viii, and ix (1830–32); in the third part (1833) of Meier Hirsch's "Sammlung Geometrischer Aufgaben"; and in "Sammlung von Aufgaben und Lehrsätzen aus der Analytischen Geometrie des Raumes" (published in 1837, written earlier). He studied Euclid while working in his uncle's bank. From 1813 to 1815 he served as a gunner in the Napoleonic Wars. After the war he returned to banking and taught mathematics until 1834, when the founder of the academy at which he was teaching died. He then left teaching and spent nine years as the head revenue officer for the Berliner Kassenverein, retiring in 1843. References Allg. Deutsche Biographie, xx.91–92, Leipzig, 1884; H.S.M. Coxeter (1961) Introduction to Geometry, Chapter 6: Circles and Spheres (pp. 77–95), John Wiley & Sons. Poggendorff, Biog.-Literarisch Handwörterb. Leipzig, 1863, s.v. External links Jewish Encyclopedia biography Works by Ludwig Immanuel Magnus at Google Books 1790 births 1861 deaths 19th-century German mathematicians 19th-century German Jews Geometers
https://en.wikipedia.org/wiki/Economic%20statistics	Economic statistics is a topic in applied statistics and applied economics that concerns the collection, processing, compilation, dissemination, and analysis of economic data. It is closely related to business statistics and econometrics. It is also common to call the data themselves "economic statistics", but for this usage, "economic data" is the more common term. Overview The data of concern to economic statistics may include those of an economy within a region, country, or group of countries. Economic statistics may also refer to a subtopic of official statistics for data produced by official organizations (e.g. national statistical services, intergovernmental organizations such as United Nations, European Union or OECD, central banks, and ministries). Analyses within economic statistics both make use of and provide the empirical data needed in economic research, whether descriptive or econometric. They are a key input for decision making as to economic policy. The subject includes statistical analysis of topics and problems in microeconomics, macroeconomics, business, finance, forecasting, data quality, and policy evaluation. It also includes such considerations as what data to collect in order to quantify some particular aspect of an economy and of how best to collect in any given instance. See also Business statistics Econometrics Survey of production References Citations Sources Allen, R. G. D., 1956. "Official Economic Statistics," Economica, N.S., 23(92), pp. 360-365. Crum, W. L., 1925. An Introduction to the Methods of Economic Statistics, AW Shaw Co. Giovanini, Enrico, 2008. Understanding Economic Statistics. OECD Publishing. Fox, Karl A., 1968. Intermediate Economic Statistics, Wiley. Description. Kane, Edward J., 1968. Economic Statistics and Econometrics, Harper and Row. Morgenstern, Oskar, [1950] 1963. On the Accuracy of Economic Observations. 2nd rev. ed. ("The Accuracy of Economic Observation" ch. 16). Princeton University Press. Mirer, Thad W., 1995. Economic Statistics and Econometrics, 3rd ed. Prentice Hall. Description. Persons, Warren M., 1910. "The Correlation of Economic Statistics," Publications of the American Statistical Association, 12(92), pp. 287-322. Wonnacott, Thomas H., and Ronald J. Wonnacott, 1990. Introductory Statistics for Business and Economics, 4th ed., Wiley. Ullah, Aman, and David E. A. Giles, ed., 1998. Handbook of Applied Economic Statistics, Marcel Dekker. Description, preview, and back cover. Zellner, Arnold, ed. 1968. Readings in Economic Statistics and Econometrics, Little, Brown & Co. Journals Journal of Business and Economic Statistics Review of Economics and Statistics (from Review of Economic Statistics, 1919–47) External links Economic statistics section United Nations Economic Commission for Europe Statistics from UCB Libraries GovPubs Economic statistics: The White House pages on U.S. economic statistics Historical Financial Statistics: Center for Fina
https://en.wikipedia.org/wiki/List%20of%20complex%20analysis%20topics	Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics, including hydrodynamics, thermodynamics, and electrical engineering. Overview Complex numbers Complex plane Complex functions Complex derivative Holomorphic functions Harmonic functions Elementary functions Polynomial functions Exponential functions Trigonometric functions Hyperbolic functions Logarithmic functions Inverse trigonometric functions Inverse hyperbolic functions Residue theory Isometries in the complex plane Related fields Number theory Hydrodynamics Thermodynamics Electrical engineering Local theory Holomorphic function Antiholomorphic function Cauchy–Riemann equations Conformal mapping Conformal welding Power series Radius of convergence Laurent series Meromorphic function Entire function Pole (complex analysis) Zero (complex analysis) Residue (complex analysis) Isolated singularity Removable singularity Essential singularity Branch point Principal branch Weierstrass–Casorati theorem Landau's constants Holomorphic functions are analytic Schwarzian derivative Analytic capacity Disk algebra Growth and distribution of values Ahlfors theory Bieberbach conjecture Borel–Carathéodory theorem Corona theorem Hadamard three-circle theorem Hardy space Hardy's theorem Maximum modulus principle Nevanlinna theory Paley–Wiener theorem Progressive function Value distribution theory of holomorphic functions Contour integrals Line integral Cauchy's integral theorem Cauchy's integral formula Residue theorem Liouville's theorem (complex analysis) Examples of contour integration Fundamental theorem of algebra Simply connected Winding number Principle of the argument Rouché's theorem Bromwich integral Morera's theorem Mellin transform Kramers–Kronig relation, a. k. a. Hilbert transform Sokhotski–Plemelj theorem Special functions Exponential function Beta function Gamma function Riemann zeta function Riemann hypothesis Generalized Riemann hypothesis Elliptic function Half-period ratio Jacobi's elliptic functions Weierstrass's elliptic functions Theta function Elliptic modular function J-function Modular function Modular form Riemann surfaces Analytic continuation Riemann sphere Riemann surface Riemann mapping theorem Carathéodory's theorem (conformal mapping) Riemann–Roch theorem Other Amplitwist Antiderivative (complex analysis) Bôcher's theorem Cayley transform Harmonic conjugate Hilbert's inequality Method of steepest descent Montel's theorem Periodic points of complex quadratic mappings Pick matrix Runge approximation theorem Schwarz lemma Weierstrass factorization theorem Mittag-Leffler's theorem Sendov's conjecture Infinite compositions of analytic functions Several complex variables Biholomorphy Cartan's theorems A and B Cousin problem
https://en.wikipedia.org/wiki/Function%20space	In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vector space has a natural vector space structure given by pointwise addition and scalar multiplication. In other scenarios, the function space might inherit a topological or metric structure, hence the name function space. In linear algebra Let be a vector space over a field and let be any set. The functions → can be given the structure of a vector space over where the operations are defined pointwise, that is, for any , : → , any in , and any in , define When the domain has additional structure, one might consider instead the subset (or subspace) of all such functions which respect that structure. For example, if is also a vector space over , the set of linear maps → form a vector space over with pointwise operations (often denoted Hom(,)). One such space is the dual space of : the set of linear functionals → with addition and scalar multiplication defined pointwise. Examples Function spaces appear in various areas of mathematics: In set theory, the set of functions from X to Y may be denoted {X → Y} or YX. As a special case, the power set of a set X may be identified with the set of all functions from X to {0, 1}, denoted 2X. The set of bijections from X to Y is denoted . The factorial notation X! may be used for permutations of a single set X. In functional analysis, the same is seen for continuous linear transformations, including topologies on the vector spaces in the above, and many of the major examples are function spaces carrying a topology; the best known examples include Hilbert spaces and Banach spaces. In functional analysis, the set of all functions from the natural numbers to some set X is called a sequence space. It consists of the set of all possible sequences of elements of X. In topology, one may attempt to put a topology on the space of continuous functions from a topological space X to another one Y, with utility depending on the nature of the spaces. A commonly used example is the compact-open topology, e.g. loop space. Also available is the product topology on the space of set theoretic functions (i.e. not necessarily continuous functions) YX. In this context, this topology is also referred to as the topology of pointwise convergence. In algebraic topology, the study of homotopy theory is essentially that of discrete invariants of function spaces; In the theory of stochastic processes, the basic technical problem is how to construct a probability measure on a function space of paths of the process (functions of time); In category theory, the function space is called an exponential object or map object. It appears in one way as the representation canonical bifunctor; but as (single) functor, of type [X, -], it appears as an adjoint functor to a functor
https://en.wikipedia.org/wiki/Quantum%20group	In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups (which are quasitriangular Hopf algebras), compact matrix quantum groups (which are structures on unital separable C*-algebras), and bicrossproduct quantum groups. Despite their name, they do not themselves have a natural group structure, though they are in some sense 'close' to a group. The term "quantum group" first appeared in the theory of quantum integrable systems, which was then formalized by Vladimir Drinfeld and Michio Jimbo as a particular class of Hopf algebra. The same term is also used for other Hopf algebras that deform or are close to classical Lie groups or Lie algebras, such as a "bicrossproduct" class of quantum groups introduced by Shahn Majid a little after the work of Drinfeld and Jimbo. In Drinfeld's approach, quantum groups arise as Hopf algebras depending on an auxiliary parameter q or h, which become universal enveloping algebras of a certain Lie algebra, frequently semisimple or affine, when q = 1 or h = 0. Closely related are certain dual objects, also Hopf algebras and also called quantum groups, deforming the algebra of functions on the corresponding semisimple algebraic group or a compact Lie group. Intuitive meaning The discovery of quantum groups was quite unexpected since it was known for a long time that compact groups and semisimple Lie algebras are "rigid" objects, in other words, they cannot be "deformed". One of the ideas behind quantum groups is that if we consider a structure that is in a sense equivalent but larger, namely a group algebra or a universal enveloping algebra, then a group or enveloping algebra can be "deformed", although the deformation will no longer remain a group or enveloping algebra. More precisely, deformation can be accomplished within the category of Hopf algebras that are not required to be either commutative or cocommutative. One can think of the deformed object as an algebra of functions on a "noncommutative space", in the spirit of the noncommutative geometry of Alain Connes. This intuition, however, came after particular classes of quantum groups had already proved their usefulness in the study of the quantum Yang–Baxter equation and quantum inverse scattering method developed by the Leningrad School (Ludwig Faddeev, Leon Takhtajan, Evgeny Sklyanin, Nicolai Reshetikhin and Vladimir Korepin) and related work by the Japanese School. The intuition behind the second, bicrossproduct, class of quantum groups was different and came from the search for self-dual objects as an approach to quantum gravity. Drinfeld–Jimbo type quantum groups One type of objects commonly called a "quantum group" appeared in the work of Vladimir Drinfeld and Michio Jimbo as a deformation of the universal enveloping algebra of a semisimple Lie algebra or, more generally, a Kac–Moody algebra, in the categ
https://en.wikipedia.org/wiki/Ferdinand%20Georg%20Frobenius	Ferdinand Georg Frobenius (26 October 1849 – 3 August 1917) was a German mathematician, best known for his contributions to the theory of elliptic functions, differential equations, number theory, and to group theory. He is known for the famous determinantal identities, known as Frobenius–Stickelberger formulae, governing elliptic functions, and for developing the theory of biquadratic forms. He was also the first to introduce the notion of rational approximations of functions (nowadays known as Padé approximants), and gave the first full proof for the Cayley–Hamilton theorem. He also lent his name to certain differential-geometric objects in modern mathematical physics, known as Frobenius manifolds. Biography Ferdinand Georg Frobenius was born on 26 October 1849 in Charlottenburg, a suburb of Berlin, from parents Christian Ferdinand Frobenius, a Protestant parson, and Christine Elizabeth Friedrich. He entered the Joachimsthal Gymnasium in 1860 when he was nearly eleven. In 1867, after graduating, he went to the University of Göttingen where he began his university studies but he only studied there for one semester before returning to Berlin, where he attended lectures by Kronecker, Kummer and Karl Weierstrass. He received his doctorate (awarded with distinction) in 1870 supervised by Weierstrass. His thesis was on the solution of differential equations. In 1874, after having taught at secondary school level first at the Joachimsthal Gymnasium then at the Sophienrealschule, he was appointed to the University of Berlin as an extraordinary professor of mathematics. Frobenius was only in Berlin a year before he went to Zürich to take up an appointment as an ordinary professor at the Eidgenössische Polytechnikum. For seventeen years, between 1875 and 1892, Frobenius worked in Zürich. It was there that he married, brought up his family, and did much important work in widely differing areas of mathematics. In the last days of December 1891 Kronecker died and, therefore, his chair in Berlin became vacant. Weierstrass, strongly believing that Frobenius was the right person to keep Berlin in the forefront of mathematics, used his considerable influence to have Frobenius appointed. In 1893 he returned to Berlin, where he was elected to the Prussian Academy of Sciences. Contributions to group theory Group theory was one of Frobenius' principal interests in the second half of his career. One of his first contributions was the proof of the Sylow theorems for abstract groups. Earlier proofs had been for permutation groups. His proof of the first Sylow theorem (on the existence of Sylow groups) is one of those frequently used today. Frobenius also has proved the following fundamental theorem: If a positive integer n divides the order \|G\| of a finite group G, then the number of solutions of the equation xn = 1 in G is equal to kn for some positive integer k. He also posed the following problem: If, in the above theorem, k = 1, then the solutions of the equat
https://en.wikipedia.org/wiki/Matrix%20similarity	In linear algebra, two n-by-n matrices and are called similar if there exists an invertible n-by-n matrix such that Similar matrices represent the same linear map under two (possibly) different bases, with being the change of basis matrix. A transformation is called a similarity transformation or conjugation of the matrix . In the general linear group, similarity is therefore the same as conjugacy, and similar matrices are also called conjugate; however, in a given subgroup of the general linear group, the notion of conjugacy may be more restrictive than similarity, since it requires that be chosen to lie in . Motivating example When defining a linear transformation, it can be the case that a change of basis can result in a simpler form of the same transformation. For example, the matrix representing a rotation in when the axis of rotation is not aligned with the coordinate axis can be complicated to compute. If the axis of rotation were aligned with the positive -axis, then it would simply be where is the angle of rotation. In the new coordinate system, the transformation would be written as where and are respectively the original and transformed vectors in a new basis containing a vector parallel to the axis of rotation. In the original basis, the transform would be written as where vectors and and the unknown transform matrix are in the original basis. To write in terms of the simpler matrix, we use the change-of-basis matrix that transforms and as and : Thus, the matrix in the original basis, , is given by . The transform in the original basis is found to be the product of three easy-to-derive matrices. In effect, the similarity transform operates in three steps: change to a new basis (), perform the simple transformation (), and change back to the old basis (). Properties Similarity is an equivalence relation on the space of square matrices. Because matrices are similar if and only if they represent the same linear operator with respect to (possibly) different bases, similar matrices share all properties of their shared underlying operator: Rank Characteristic polynomial, and attributes that can be derived from it: Determinant Trace Eigenvalues, and their algebraic multiplicities Geometric multiplicities of eigenvalues (but not the eigenspaces, which are transformed according to the base change matrix P used). Minimal polynomial Frobenius normal form Jordan normal form, up to a permutation of the Jordan blocks Index of nilpotence Elementary divisors, which form a complete set of invariants for similarity of matrices over a principal ideal domain Because of this, for a given matrix A, one is interested in finding a simple "normal form" B which is similar to A—the study of A then reduces to the study of the simpler matrix B. For example, A is called diagonalizable if it is similar to a diagonal matrix. Not all matrices are diagonalizable, but at least over the complex numbers (or any algebraically closed fie
https://en.wikipedia.org/wiki/Second-order%20logic	In logic and mathematics, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory. First-order logic quantifies only variables that range over individuals (elements of the domain of discourse); second-order logic, in addition, quantifies over relations. For example, the second-order sentence says that for every formula P, and every individual x, either Px is true or not(Px) is true (this is the law of excluded middle). Second-order logic also includes quantification over sets, functions, and other variables (see section below). Both first-order and second-order logic use the idea of a domain of discourse (often called simply the "domain" or the "universe"). The domain is a set over which individual elements may be quantified. Examples First-order logic can quantify over individuals, but not over properties. That is, we can take an atomic sentence like Cube(b) and obtain a quantified sentence by replacing the name with a variable and attaching a quantifier: However, we cannot do the same with the predicate. That is, the following expression is not a sentence of first-order logic, but this is a legitimate sentence of second-order logic. Here, P is a predicate variable and is semantically a set of individuals. As a result, second-order logic has greater expressive power than first-order logic. For example, there is no way in first-order logic to identify the set of all cubes and tetrahedrons. But the existence of this set can be asserted in second-order logic as We can then assert properties of this set. For instance, the following says that the set of all cubes and tetrahedrons does not contain any dodecahedrons: Second-order quantification is especially useful because it gives the ability to express reachability properties. For example, if Parent(x, y) denotes that x is a parent of y, then first-order logic cannot express the property that x is an ancestor of y. In second-order logic we can express this by saying that every set of people containing y and closed under the Parent relation contains x: It is notable that while we have variables for predicates in second-order-logic, we don't have variables for properties of predicates. We cannot say, for example, that there is a property Shape(P) that is true for the predicates P Cube, Tet, and Dodec. This would require third-order logic. Syntax and fragments The syntax of second-order logic tells which expressions are well formed formulas. In addition to the syntax of first-order logic, second-order logic includes many new sorts (sometimes called types) of variables. These are: A sort of variables that range over sets of individuals. If S is a variable of this sort and t is a first-order term then the expression t ∈ S (also written S(t), or St to save parentheses) is an atomic formula. Sets of individuals can also be viewed as unary relations on
https://en.wikipedia.org/wiki/Cantor%27s%20theorem	In mathematical set theory, Cantor's theorem is a fundamental result which states that, for any set , the set of all subsets of the power set of has a strictly greater cardinality than itself. For finite sets, Cantor's theorem can be seen to be true by simple enumeration of the number of subsets. Counting the empty set as a subset, a set with elements has a total of subsets, and the theorem holds because for all non-negative integers. Much more significant is Cantor's discovery of an argument that is applicable to any set, and shows that the theorem holds for infinite sets also. As a consequence, the cardinality of the real numbers, which is the same as that of the power set of the integers, is strictly larger than the cardinality of the integers; see Cardinality of the continuum for details. The theorem is named for German mathematician Georg Cantor, who first stated and proved it at the end of the 19th century. Cantor's theorem had immediate and important consequences for the philosophy of mathematics. For instance, by iteratively taking the power set of an infinite set and applying Cantor's theorem, we obtain an endless hierarchy of infinite cardinals, each strictly larger than the one before it. Consequently, the theorem implies that there is no largest cardinal number (colloquially, "there's no largest infinity"). Proof Cantor's argument is elegant and remarkably simple. The complete proof is presented below, with detailed explanations to follow. By definition of cardinality, we have for any two sets and if and only if there is an injective function but no bijective function from to It suffices to show that there is no surjection from to . This is the heart of Cantor's theorem: there is no surjective function from any set to its power set. To establish this, it is enough to show that no function that maps elements in to subsets of can reach every possible subset, i.e., we just need to demonstrate the existence of a subset of that is not equal to for any ∈ . (Recall that each is a subset of .) Such a subset is given by the following construction, sometimes called the Cantor diagonal set of : This means, by definition, that for all x ∈ A, x ∈ B if and only if x ∉ f(x). For all x the sets B and f(x) cannot be the same because B was constructed from elements of A whose images (under f) did not include themselves. More specifically, consider any x ∈ A, then either x ∈ f(x) or x ∉ f(x). In the former case, f(x) cannot equal B because x ∈ f(x) by assumption and x ∉ B by the construction of B. In the latter case, f(x) cannot equal B because x ∉ f(x) by assumption and x ∈ B by the construction of B. Equivalently, and slightly more formally, we just proved that the existence of ξ ∈ A such that f(ξ) = B implies the following contradiction: Therefore, by reductio ad absurdum, the assumption must be false. Thus there is no ξ ∈ A such that f(ξ) = B; in other words, B is not in the image of f and f does not map to every
https://en.wikipedia.org/wiki/30%20%28number%29	30 (thirty) is the natural number following 29 and preceding 31. In mathematics 30 is an even, composite, pronic number. With 2, 3, and 5 as its prime factors, it is a regular number and the first sphenic number, the smallest of the form , where is a prime greater than 3. It has an aliquot sum of 42, which is the second sphenic number. It is also: A semiperfect number, since adding some subsets of its divisors (e.g., 5, 10 and 15) equals 30. A primorial. A Harshad number in decimal. Divisible by the number of prime numbers (10) below it. The largest number such that all coprimes smaller than itself, except for 1, are prime. The sum of the first four squares, making it a square pyramidal number. The number of vertices in the Tutte–Coxeter graph. The measure of the central angle and exterior angle of a dodecagon, which is the petrie polygon of the 24-cell. The number of sides of a triacontagon, which in turn is the petrie polygon of the 120-cell and 600-cell. The number of edges of a dodecahedron and icosahedron, of vertices of an icosidodecahedron, and of faces of a rhombic dodecahedron. The sum of the number of elements of a 5-cell: 5 vertices, 10 edges, 10 faces, and 5 cells. The Coxeter number of E8. Furthermore, In a group , such that , where does not divide , and has a subgroup of order , 30 is the only number less than 60 that is neither a prime nor of the aforementioned form. Therefore, 30 is the only candidate for the order of a simple group less than 60, in which one needs other methods to specifically reject to eventually deduce said order. The SI prefix for 1030 is Quetta- (Q), and for 10−30 (i.e., the reciprocal of 1030) quecto (q). These numbers are the largest and smallest number to receive an SI prefix to date. In science The atomic number of zinc is 30. Astronomy Messier object M30, a magnitude 8.5 globular cluster in the constellation Capricornus The New General Catalogue object NGC 30, a double star in the constellation Pegasus Age 30 The minimum age for United States senators. In other fields Thirty is: Used (as –30–) to indicate the end of a newspaper (or broadcast) story, a copy editor's typographical notation The number of days in the months April, June, September and November (and in unusual circumstances February—see February 30) The total number of major and minor keys in Western tonal music, including enharmonic equivalents In years of marriage, the pearl wedding anniversary The duration in years of the Thirty Years' War - 1618 to 1648 The code for international direct dial phone calls to Greece The house number of 30 St Mary Axe (The Gherkin) The number of tracks on The Beatles' eponymous album, usually known as The White Album A stage in young adulthood Part of the name of: 30 Odd Foot of Grunts, the band fronted by actor Russell Crowe The movie title 13 Going on 30, starring Jennifer Garner The title of the Food Network show 30 Minute Meals 30 Days of Night, a comic book minise
https://en.wikipedia.org/wiki/Square%20root%20of%202	The square root of 2 (approximately 1.4142) is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as or . It is an algebraic number, and therefore not a transcendental number. Technically, it should be called the principal square root of 2, to distinguish it from the negative number with the same property. Geometrically, the square root of 2 is the length of a diagonal across a square with sides of one unit of length; this follows from the Pythagorean theorem. It was probably the first number known to be irrational. The fraction (≈ 1.4142857) is sometimes used as a good rational approximation with a reasonably small denominator. Sequence in the On-Line Encyclopedia of Integer Sequences consists of the digits in the decimal expansion of the square root of 2, here truncated to 65 decimal places: History The Babylonian clay tablet YBC 7289 (–1600 BC) gives an approximation of in four sexagesimal figures, , which is accurate to about six decimal digits, and is the closest possible three-place sexagesimal representation of : Another early approximation is given in ancient Indian mathematical texts, the Sulbasutras (–200 BC), as follows: Increase the length [of the side] by its third and this third by its own fourth less the thirty-fourth part of that fourth. That is, This approximation is the seventh in a sequence of increasingly accurate approximations based on the sequence of Pell numbers, which can be derived from the continued fraction expansion of . Despite having a smaller denominator, it is only slightly less accurate than the Babylonian approximation. Pythagoreans discovered that the diagonal of a square is incommensurable with its side, or in modern language, that the square root of two is irrational. Little is known with certainty about the time or circumstances of this discovery, but the name of Hippasus of Metapontum is often mentioned. For a while, the Pythagoreans treated as an official secret the discovery that the square root of two is irrational, and, according to legend, Hippasus was murdered for divulging it. The square root of two is occasionally called Pythagoras's number or Pythagoras's constant, for example by . Ancient Roman architecture In ancient Roman architecture, Vitruvius describes the use of the square root of 2 progression or ad quadratum technique. It consists basically in a geometric, rather than arithmetic, method to double a square, in which the diagonal of the original square is equal to the side of the resulting square. Vitruvius attributes the idea to Plato. The system was employed to build pavements by creating a square tangent to the corners of the original square at 45 degrees of it. The proportion was also used to design atria by giving them a length equal to a diagonal taken from a square, whose sides are equivalent to the intended atrium's width. Decimal value Computation algorithms There are many algorithms for approximating as a r
https://en.wikipedia.org/wiki/Diagonal%20argument	A diagonal argument, in mathematics, is a technique employed in the proofs of the following theorems: Cantor's diagonal argument (the earliest) Cantor's theorem Russell's paradox Diagonal lemma Gödel's first incompleteness theorem Tarski's undefinability theorem Halting problem Kleene's recursion theorem See also Diagonalization (disambiguation)
https://en.wikipedia.org/wiki/Central%20extension	Central extension may refer to: Central Extension (Long Island Rail Road), a rail line Central extension (mathematics), a type of group extension
https://en.wikipedia.org/wiki/Center%20%28algebra%29	The term center or centre is used in various contexts in abstract algebra to denote the set of all those elements that commute with all other elements. The center of a group G consists of all those elements x in G such that xg = gx for all g in G. This is a normal subgroup of G. The similarly named notion for a semigroup is defined likewise and it is a subsemigroup. The center of a ring (or an associative algebra) R is the subset of R consisting of all those elements x of R such that xr = rx for all r in R. The center is a commutative subring of R. The center of a Lie algebra L consists of all those elements x in L such that [x,a] = 0 for all a in L. This is an ideal of the Lie algebra L. See also Centralizer and normalizer Center (category theory) References Abstract algebra
https://en.wikipedia.org/wiki/Lehmann%E2%80%93Scheff%C3%A9%20theorem	In statistics, the Lehmann–Scheffé theorem is a prominent statement, tying together the ideas of completeness, sufficiency, uniqueness, and best unbiased estimation. The theorem states that any estimator which is unbiased for a given unknown quantity and that depends on the data only through a complete, sufficient statistic is the unique best unbiased estimator of that quantity. The Lehmann–Scheffé theorem is named after Erich Leo Lehmann and Henry Scheffé, given their two early papers. If T is a complete sufficient statistic for θ and E(g(T)) = τ(θ) then g(T) is the uniformly minimum-variance unbiased estimator (UMVUE) of τ(θ). Statement Let be a random sample from a distribution that has p.d.f (or p.m.f in the discrete case) where is a parameter in the parameter space. Suppose is a sufficient statistic for θ, and let be a complete family. If then is the unique MVUE of θ. Proof By the Rao–Blackwell theorem, if is an unbiased estimator of θ then defines an unbiased estimator of θ with the property that its variance is not greater than that of . Now we show that this function is unique. Suppose is another candidate MVUE estimator of θ. Then again defines an unbiased estimator of θ with the property that its variance is not greater than that of . Then Since is a complete family and therefore the function is the unique function of Y with variance not greater than that of any other unbiased estimator. We conclude that is the MVUE. Example for when using a non-complete minimal sufficient statistic An example of an improvable Rao–Blackwell improvement, when using a minimal sufficient statistic that is not complete, was provided by Galili and Meilijson in 2016. Let be a random sample from a scale-uniform distribution with unknown mean and known design parameter . In the search for "best" possible unbiased estimators for , it is natural to consider as an initial (crude) unbiased estimator for and then try to improve it. Since is not a function of , the minimal sufficient statistic for (where and ), it may be improved using the Rao–Blackwell theorem as follows: However, the following unbiased estimator can be shown to have lower variance: And in fact, it could be even further improved when using the following estimator: The model is a scale model. Optimal equivariant estimators can then be derived for loss functions that are invariant. See also Basu's theorem Complete class theorem Rao–Blackwell theorem References Theorems in statistics Estimation theory
https://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion%20principle	In combinatorics, a branch of mathematics, the inclusion–exclusion principle is a counting technique which generalizes the familiar method of obtaining the number of elements in the union of two finite sets; symbolically expressed as where A and B are two finite sets and \|S \| indicates the cardinality of a set S (which may be considered as the number of elements of the set, if the set is finite). The formula expresses the fact that the sum of the sizes of the two sets may be too large since some elements may be counted twice. The double-counted elements are those in the intersection of the two sets and the count is corrected by subtracting the size of the intersection. The inclusion-exclusion principle, being a generalization of the two-set case, is perhaps more clearly seen in the case of three sets, which for the sets A, B and C is given by This formula can be verified by counting how many times each region in the Venn diagram figure is included in the right-hand side of the formula. In this case, when removing the contributions of over-counted elements, the number of elements in the mutual intersection of the three sets has been subtracted too often, so must be added back in to get the correct total. Generalizing the results of these examples gives the principle of inclusion–exclusion. To find the cardinality of the union of sets: Include the cardinalities of the sets. Exclude the cardinalities of the pairwise intersections. Include the cardinalities of the triple-wise intersections. Exclude the cardinalities of the quadruple-wise intersections. Include the cardinalities of the quintuple-wise intersections. Continue, until the cardinality of the -tuple-wise intersection is included (if is odd) or excluded ( even). The name comes from the idea that the principle is based on over-generous inclusion, followed by compensating exclusion. This concept is attributed to Abraham de Moivre (1718), although it first appears in a paper of Daniel da Silva (1854) and later in a paper by J. J. Sylvester (1883). Sometimes the principle is referred to as the formula of Da Silva or Sylvester, due to these publications. The principle can be viewed as an example of the sieve method extensively used in number theory and is sometimes referred to as the sieve formula. As finite probabilities are computed as counts relative to the cardinality of the probability space, the formulas for the principle of inclusion–exclusion remain valid when the cardinalities of the sets are replaced by finite probabilities. More generally, both versions of the principle can be put under the common umbrella of measure theory. In a very abstract setting, the principle of inclusion–exclusion can be expressed as the calculation of the inverse of a certain matrix. This inverse has a special structure, making the principle an extremely valuable technique in combinatorics and related areas of mathematics. As Gian-Carlo Rota put it: "One of the most useful principles of enum
https://en.wikipedia.org/wiki/Image%20%28category%20theory%29	In category theory, a branch of mathematics, the image of a morphism is a generalization of the image of a function. General definition Given a category and a morphism in , the image of is a monomorphism satisfying the following universal property: There exists a morphism such that . For any object with a morphism and a monomorphism such that , there exists a unique morphism such that . Remarks: such a factorization does not necessarily exist. is unique by definition of monic. , therefore by monic. is monic. already implies that is unique. The image of is often denoted by or . Proposition: If has all equalizers then the in the factorization of (1) is an epimorphism. Second definition In a category with all finite limits and colimits, the image is defined as the equalizer of the so-called cokernel pair , which is the cocartesian of a morphism with itself over its domain, which will result in a pair of morphisms , on which the equalizer is taken, i.e. the first of the following diagrams is cocartesian, and the second equalizing. Remarks: Finite bicompleteness of the category ensures that pushouts and equalizers exist. can be called regular image as is a regular monomorphism, i.e. the equalizer of a pair of morphisms. (Recall also that an equalizer is automatically a monomorphism). In an abelian category, the cokernel pair property can be written and the equalizer condition . Moreover, all monomorphisms are regular. Examples In the category of sets the image of a morphism is the inclusion from the ordinary image to . In many concrete categories such as groups, abelian groups and (left- or right) modules, the image of a morphism is the image of the correspondent morphism in the category of sets. In any normal category with a zero object and kernels and cokernels for every morphism, the image of a morphism can be expressed as follows: im f = ker coker f In an abelian category (which is in particular binormal), if f is a monomorphism then f = ker coker f, and so f = im f. See also Subobject Coimage Image (mathematics) References Category theory
https://en.wikipedia.org/wiki/Cousin%20prime	In number theory, cousin primes are prime numbers that differ by four. Compare this with twin primes, pairs of prime numbers that differ by two, and sexy primes, pairs of prime numbers that differ by six. The cousin primes (sequences and in OEIS) below 1000 are: (3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47), (67, 71), (79, 83), (97, 101), (103, 107), (109, 113), (127, 131), (163, 167), (193, 197), (223, 227), (229, 233), (277, 281), (307, 311), (313, 317), (349, 353), (379, 383), (397, 401), (439, 443), (457, 461), (463,467), (487, 491), (499, 503), (613, 617), (643, 647), (673, 677), (739, 743), (757, 761), (769, 773), (823, 827), (853, 857), (859, 863), (877, 881), (883, 887), (907, 911), (937, 941), (967, 971) Properties The only prime belonging to two pairs of cousin primes is 7. One of the numbers will always be divisible by 3, so is the only case where all three are primes. An example of a large proven cousin prime pair is for which has 20008 digits. In fact, this is part of a prime triple since is also a twin prime (because is also a proven prime). , the largest-known pair of cousin primes was found by S. Batalov and has 51,934 digits. The primes are: It follows from the first Hardy–Littlewood conjecture that cousin primes have the same asymptotic density as twin primes. An analogue of Brun's constant for twin primes can be defined for cousin primes, called Brun's constant for cousin primes, with the initial term (3, 7) omitted, by the convergent sum: Using cousin primes up to 242, the value of was estimated by Marek Wolf in 1996 as This constant should not be confused with Brun's constant for prime quadruplets, which is also denoted . The Skewes number for cousin primes is 5206837 (). Notes References . Classes of prime numbers Unsolved problems in mathematics
https://en.wikipedia.org/wiki/Sexy%20prime	In number theory, sexy primes are prime numbers that differ from each other by 6. For example, the numbers 5 and 11 are both sexy primes, because both are prime and . The term "sexy prime" is a pun stemming from the Latin word for six: . If or (where is the lower prime) is also prime, then the sexy prime is part of a prime triplet. In August 2014 the Polymath group, seeking the proof of the twin prime conjecture, showed that if the generalized Elliott–Halberstam conjecture is proven, one can show the existence of infinitely many pairs of consecutive primes that differ by at most 6 and as such they are either twin, cousin or sexy primes. Primorial n# notation As used in this article, # stands for the product 2 · 3 · 5 · 7 · … of all the primes ≤ . Types of groupings Sexy prime pairs The sexy primes (sequences and in OEIS) below 500 are: (5,11), (7,13), (11,17), (13,19), (17,23), (23,29), (31,37), (37,43), (41,47), (47,53), (53,59), (61,67), (67,73), (73,79), (83,89), (97,103), (101,107), (103,109), (107,113), (131,137), (151,157), (157,163), (167,173), (173,179), (191,197), (193,199), (223,229), (227,233), (233,239), (251,257), (257,263), (263,269), (271,277), (277,283), (307,313), (311,317), (331,337), (347,353), (353,359), (367,373), (373,379), (383,389), (433,439), (443,449), (457,463), (461,467). , the largest-known pair of sexy primes was found by S. Batalov and has 51,934 digits. The primes are: 11922002779 x (2172486 - 286243) + 286245 - 5 11922002779 x (2172486 - 286243) + 286245 + 1 Sexy prime triplets Sexy primes can be extended to larger constellations. Triplets of primes (, +6, +12) such that +18 is composite are called sexy prime triplets. Those below 1,000 are (, , ): (7,13,19), (17,23,29), (31,37,43), (47,53,59), (67,73,79), (97,103,109), (101,107,113), (151,157,163), (167,173,179), (227,233,239), (257,263,269), (271,277,283), (347,353,359), (367,373,379), (557,563,569), (587,593,599), (607,613,619), (647,653,659), (727,733,739), (941,947,953), (971,977,983). In January 2005 Ken Davis set a record for the largest-known sexy prime triplet with 5132 digits: (84055657369 · 205881 · 4001# · (205881 · 4001# + 1) + 210) · (205881 · 4001# - 1) / 35 + 1. In May 2019, Peter Kaiser improved this record to 6,031 digits: 10409207693×220000−1. Gerd Lamprecht improved the record to 6,116 digits in August 2019: 20730011943×14221#+344231. Ken Davis further improved the record with a 6,180 digit Brillhart-Lehmer-Selfridge provable triplet in October 2019: (728658978098574801#(8098574801#+1)+210)(8098574801#-1)/35+1 Norman Luhn & Gerd Lamprecht improved the record to 6,701 digits in October 2019: 22582235875×222224+1. Serge Batalov improved the record to 15,004 digits in April 2022: 2494779036241x249800+1. Sexy prime quadruplets Sexy prime quadruplets (, +6, +12, +18) can only begin with primes ending in a 1 in their decimal representation (except for the quadruplet with 5). The sexy prime quadruplets below
https://en.wikipedia.org/wiki/Schnirelmann%20density	In additive number theory, the Schnirelmann density of a sequence of numbers is a way to measure how "dense" the sequence is. It is named after Russian mathematician Lev Schnirelmann, who was the first to study it. Definition The Schnirelmann density of a set of natural numbers A is defined as where A(n) denotes the number of elements of A not exceeding n and inf is infimum. The Schnirelmann density is well-defined even if the limit of A(n)/n as fails to exist (see upper and lower asymptotic density). Properties By definition, and for all n, and therefore , and if and only if . Furthermore, Sensitivity The Schnirelmann density is sensitive to the first values of a set: . In particular, and Consequently, the Schnirelmann densities of the even numbers and the odd numbers, which one might expect to agree, are 0 and 1/2 respectively. Schnirelmann and Yuri Linnik exploited this sensitivity. Schnirelmann's theorems If we set , then Lagrange's four-square theorem can be restated as . (Here the symbol denotes the sumset of and .) It is clear that . In fact, we still have , and one might ask at what point the sumset attains Schnirelmann density 1 and how does it increase. It actually is the case that and one sees that sumsetting once again yields a more populous set, namely all of . Schnirelmann further succeeded in developing these ideas into the following theorems, aiming towards Additive Number Theory, and proving them to be a novel resource (if not greatly powerful) to attack important problems, such as Waring's problem and Goldbach's conjecture. Theorem. Let and be subsets of . Then Note that . Inductively, we have the following generalization. Corollary. Let be a finite family of subsets of . Then The theorem provides the first insights on how sumsets accumulate. It seems unfortunate that its conclusion stops short of showing being superadditive. Yet, Schnirelmann provided us with the following results, which sufficed for most of his purpose. Theorem. Let and be subsets of . If , then Theorem. (Schnirelmann) Let . If then there exists such that Additive bases A subset with the property that for a finite sum, is called an additive basis, and the least number of summands required is called the degree (sometimes order) of the basis. Thus, the last theorem states that any set with positive Schnirelmann density is an additive basis. In this terminology, the set of squares is an additive basis of degree 4. (About an open problem for additive bases, see Erdős–Turán conjecture on additive bases.) Mann's theorem Historically the theorems above were pointers to the following result, at one time known as the hypothesis. It was used by Edmund Landau and was finally proved by Henry Mann in 1942. Theorem. Let and be subsets of . In case that , we still have An analogue of this theorem for lower asymptotic density was obtained by Kneser. At a later date, E. Artin and P. Scherk simplified the proof of Mann's theorem. Warin
https://en.wikipedia.org/wiki/60%20%28number%29	60 (sixty) () is the natural number following 59 and preceding 61. Being three times 20, it is called threescore in older literature (kopa in Slavic, Schock in Germanic). In mathematics 60 is a highly composite number. Because it is the sum of its unitary divisors (excluding itself), it is a unitary perfect number, and it is an abundant number with an abundance of 48. Being ten times a perfect number, it is a semiperfect number. 60 is a Twin-prime sum of the fifth pair of twin-primes, 29 + 31. It is the smallest number divisible by the numbers 1 to 6: there is no smaller number divisible by the numbers 1 to 5 since any number divisible by 2 and 3 must also be divisible by 6. It is the smallest number with exactly 12 divisors. Having 12 as one of those divisors, 60 is also a refactorable number. It is one of seven integers that have more divisors than any number less than twice itself , one of six that are also lowest common multiple of a consecutive set of integers from 1, and one of six that are divisors of every highly composite number higher than itself. It is the smallest number that is the sum of two odd primes in six ways. The smallest nonsolvable group (A5) has order 60. There are four Archimedean solids with 60 vertices: the truncated icosahedron, the rhombicosidodecahedron, the snub dodecahedron, and the truncated dodecahedron. The skeletons of these polyhedra form 60-node vertex-transitive graphs. There are also two Archimedean solids with 60 edges: the snub cube and the icosidodecahedron. The skeleton of the icosidodecahedron forms a 60-edge symmetric graph. There are 60 one-sided hexominoes, the polyominoes made from six squares. In geometry, it is the number of seconds in a minute, and the number of minutes in a degree. In normal space, the three interior angles of an equilateral triangle each measure 60 degrees, adding up to 180 degrees. Because it is divisible by the sum of its digits in decimal, it is a Harshad number. A number system with base 60 is called sexagesimal (the original meaning of sexagesimal is sixtieth). It is the smallest positive integer that is written with only the smallest and the largest digit of base 2 (binary), base 3 (ternary), and base 4 (quaternary). 60 is also the product of the side lengths of the smallest whole number right triangle: 3, 4, 5, a type of Pythagorean triple. In science and technology The first fullerene to be discovered was buckminsterfullerene C60, an allotrope of carbon with 60 atoms in each molecule, arranged in a truncated icosahedron. This ball is known as a buckyball, and looks like a soccer ball. The atomic number of neodymium is 60, and cobalt-60 (60Co) is a radioactive isotope of cobalt. The electrical utility frequency in western Japan, South Korea, Taiwan, the Philippines, the United States, and several other countries in the Americas is 60 Hz. An exbibyte (sometimes called exabyte) is 260 bytes. Cultural number systems The Babylonian cuneiform numera
https://en.wikipedia.org/wiki/70%20%28number%29	70 (seventy) is the natural number following 69 and preceding 71. In mathematics 70 is: a sphenic number because its factors are 3 distinct primes. a Pell number. the seventh pentagonal number. the fourth tridecagonal number. the fifth pentatope number. the number of ways to choose 4 objects out of 8 if order does not matter. This makes it a central binomial coefficient. the smallest weird number, a natural number that is abundant but not semiperfect. a palindromic number in bases 9 (779), 13 (5513) and 34 (2234). a Harshad number in bases 6, 8, 9, 10, 11, 13, 14, 15 and 16. an Erdős–Woods number, since it is possible to find sequences of 70 consecutive integers such that each inner member shares a factor with either the first or the last member. The sum of the first 24 squares starting from 1 is 70 = 4900, i.e. a square pyramidal number. This is the only non trivial solution to the cannonball problem and relates 70 to the Leech lattice and thus string theory. In science 70 is the atomic number of ytterbium, a lanthanide Astronomy Messier object M70, a magnitude 9.0 globular cluster in the constellation Sagittarius The New General Catalogue object NGC 70, a magnitude 13.4 spiral galaxy in the constellation Andromeda In religion In Jewish tradition: Seventy souls went down to Egypt to begin the Hebrews' Egyptian exile (). There is a core of 70 nations and 70 world languages, paralleling the 70 names in the Table of Nations. There were 70 men in the Great Sanhedrin, the Supreme Court of ancient Israel. (Sanhedrin 1:4.) According to the Jewish Aggada, there are 70 perspectives ("faces") to the Torah (Numbers Rabbah 13:15). Seventy elders were assembled by Moses on God's command in the desert (). allots three score and ten (70 years) for a man's life, and the Mishnah attributes that age to "strength" (Avot 5:32), as one who survives that age is described by the verse as "the strong". Ptolemy II Philadelphus ordered 72 Jewish elders to translate the Torah into Greek; the result was the Septuagint (from the Latin for "seventy"). The Roman numeral seventy, LXX, is the scholarly symbol for the Septuagint. In Christianity: In , Jesus tells Peter to forgive people seventy times seven times. In , Jesus appoints Seventy Disciples and sends them out in pairs to preach the Gospel. Seventy is a priesthood office in the Latter Day Saint religion. In Islamic history and in Islamic interpretation the number 70 or 72 is most often and generally hyperbole for an infinite amount: There are 70 dead among the Prophet Muhammad's adversaries during the Battle of Badr. 70 of the Prophet Muhammad's followers are martyred at the Battle of Uhud. In Shia Islam, there are 70 martyrs among Imam Hussein's followers during the Battle of Karbala. In law In certain cases, copyrights expire after 70 years. In sports In Olympic archery, the targets are 70 meters from the archers. In college football, two teams scored 70 points in bowl games, the most in
https://en.wikipedia.org/wiki/Constructible%20universe	In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted by , is a particular class of sets that can be described entirely in terms of simpler sets. is the union of the constructible hierarchy . It was introduced by Kurt Gödel in his 1938 paper "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis". In this paper, he proved that the constructible universe is an inner model of ZF set theory (that is, of Zermelo–Fraenkel set theory with the axiom of choice excluded), 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. What L is can be thought of as being built in "stages" resembling the construction of von Neumann universe, . The stages are indexed by ordinals. In von Neumann's universe, at a successor stage, one takes to be the set of all subsets of the previous stage, . By contrast, in Gödel's constructible universe , one uses only those subsets of the previous stage that are: definable by a formula in the formal language of set theory, with parameters from the previous stage and, with the quantifiers interpreted to range over the previous stage. By limiting oneself to sets defined only in terms of what has already been constructed, one ensures that the resulting sets will be constructed in a way that is independent of the peculiarities of the surrounding model of set theory and contained in any such model. Define the Def operator: is defined by transfinite recursion as follows: If is a limit ordinal, then Here means precedes . Here Ord denotes the class of all ordinals. If is an element of , then . So is a subset of , which is a subset of the power set of . Consequently, this is a tower of nested transitive sets. But itself is a proper class. The elements of are called "constructible" sets; and itself is the "constructible universe". The "axiom of constructibility", aka "", says that every set (of ) is constructible, i.e. in . Additional facts about the sets An equivalent definition for is: For any finite ordinal , the sets and are the same (whether equals or not), and thus = : their elements are exactly the hereditarily finite sets. Equality beyond this point does not hold. Even in models of ZFC in which equals , is a proper subset of , and thereafter is a proper subset of the power set of for all . On the other hand, does imply that equals if , for example if is inaccessible. More generally, implies = for all infinite cardinals . If is an infinite ordinal then there is a bijection between and , and the bijection is constructible. So these sets are equinumerous in any model of set theory that includes them.
https://en.wikipedia.org/wiki/80%20%28number%29	80 (eighty) is the natural number following 79 and preceding 81. In mathematics 80 is: the sum of Euler's totient function φ(x) over the first sixteen integers. a semiperfect number, since adding up some subsets of its divisors (e.g., 1, 4, 5, 10, 20 and 40) gives 80. a ménage number. palindromic in bases 3 (22223), 6 (2126), 9 (889), 15 (5515), 19 (4419) and 39 (2239). a repdigit in bases 3, 9, 15, 19 and 39. the sum of the first 4 twin prime pairs ((3 + 5) + (5 + 7) + (11 + 13) + (17 + 19)). The Pareto principle (also known as the 80-20 rule) states that, for many events, roughly 80% of the effects come from 20% of the causes. Every solvable configuration of the 15 puzzle can be solved in no more than 80 single-tile moves. In science The atomic number of mercury In religion According to Exodus 7:7, Moses was 80 years old when he initially spoke to Pharaoh on behalf of his people. Today, 80 years of age is the upper age limit for cardinals to vote in papal elections. In other fields Eighty is also: used in the classic book title Around the World in Eighty Days the length of the Eighty Years' War or Dutch revolt (1568–1648) the standard TCP/IP port number for HTTP connections the 80A, 80B and 80C photographic filters correct for excessive redness under tungsten lighting The year AD 80, 80 BC, or 1980 Eighty shilling ale The older four-pin-base version of the 5Y3GT rectifier tube A common limit for the characters per line, in computing, derived from the number of columns in IBM cards American band Green Day has a song called "80" A fictional alien superhero named Ultraman 80 On the Réaumur scale, 80 degrees is the boiling temperature of pure water at sea level See also List of highways numbered 80 References External links wiktionary:eighty for 80 in other languages. Integers
https://en.wikipedia.org/wiki/Simple%20theorems%20in%20the%20algebra%20of%20sets	The simple theorems in the algebra of sets are some of the elementary properties of the algebra of union (infix operator: ∪), intersection (infix operator: ∩), and set complement (postfix ') of sets. These properties assume the existence of at least two sets: a given universal set, denoted U, and the empty set, denoted {}. The algebra of sets describes the properties of all possible subsets of U, called the power set of U and denoted P(U). P(U) is assumed closed under union, intersection, and set complement. The algebra of sets is an interpretation or model of Boolean algebra, with union, intersection, set complement, U, and {} interpreting Boolean sum, product, complement, 1, and 0, respectively. The properties below are stated without proof, but can be derived from a small number of properties taken as axioms. A "" follows the algebra of sets interpretation of Huntington's (1904) classic postulate set for Boolean algebra. These properties can be visualized with Venn diagrams. They also follow from the fact that P(U) is a Boolean lattice. The properties followed by "L" interpret the lattice axioms. Elementary discrete mathematics courses sometimes leave students with the impression that the subject matter of set theory is no more than these properties. For more about elementary set theory, see set, set theory, algebra of sets, and naive set theory. For an introduction to set theory at a higher level, see also axiomatic set theory, cardinal number, ordinal number, Cantor–Bernstein–Schroeder theorem, Cantor's diagonal argument, Cantor's first uncountability proof, Cantor's theorem, well-ordering theorem, axiom of choice, and Zorn's lemma. The properties below include a defined binary operation, relative complement, denoted by the infix operator "\". The "relative complement of A in B," denoted B \A, is defined as (A ∪) and as ∩B. PROPOSITION 1. For any U and any subset A of U: {} = U; = {}; A \ {} = A; {} \ A = {}; A ∩ {} = {}; A ∪ {} = A; A ∩ U = A; * A ∪ U = U; ∪ A = U; * ∩ A = {}; * A \ A = {}; U \ A = ; A \ U = {}; = A; A ∩ A = A; A ∪ A = A. PROPOSITION 2. For any sets A, B, and C: A ∩ B = B ∩ A; * L A ∪ B = B ∪ A; * L A ∪ (A ∩ B) = A; L A ∩ (A ∪ B) = A; L (A ∪ B) \ A = B \ A; A ∩ B = {} if and only if B \ A = B; ( ∪ B) ∪ ( ∪ ) = A; (A ∩ B) ∩ C = A ∩ (B ∩ C); L (A ∪ B) ∪ C = A ∪ (B ∪ C); L C \ (A ∩ B) = (C \ A) ∪ (C \ B); C \ (A ∪ B) = (C \ A) ∩ (C \ B); C \ (B \ A) = (C \ B) ∪(C ∩ A); (B \ A) ∩ C = (B ∩ C) \ A = B ∩ (C \ A); (B \ A) ∪ C = (B ∪ C) \ (A \ C). The distributive laws: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C); * A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C). * PROPOSITION 3. Some properties of ⊆: A ⊆ B if and only if A ∩ B = A; A ⊆ B if and only if A ∪ B = B; A ⊆ B if and only if ⊆ ; A ⊆ B if and only if A \ B = {}; A ∩ B ⊆ A ⊆ A ∪ B. See also References Edward Huntington (1904) "Sets of independent postulates for the algebra of logic," Transactions of the American Mathematical Society 5: 288-309. Whitesitt, J. E. (1961) Boolean
https://en.wikipedia.org/wiki/Set-theoretic%20limit	In mathematics, the limit of a sequence of sets (subsets of a common set ) is a set whose elements are determined by the sequence in either of two equivalent ways: (1) by upper and lower bounds on the sequence that converge monotonically to the same set (analogous to convergence of real-valued sequences) and (2) by convergence of a sequence of indicator functions which are themselves real-valued. As is the case with sequences of other objects, convergence is not necessary or even usual. More generally, again analogous to real-valued sequences, the less restrictive limit infimum and limit supremum of a set sequence always exist and can be used to determine convergence: the limit exists if the limit infimum and limit supremum are identical. (See below). Such set limits are essential in measure theory and probability. It is a common misconception that the limits infimum and supremum described here involve sets of accumulation points, that is, sets of where each is in some This is only true if convergence is determined by the discrete metric (that is, if there is such that for all ). This article is restricted to that situation as it is the only one relevant for measure theory and probability. See the examples below. (On the other hand, there are more general topological notions of set convergence that do involve accumulation points under different metrics or topologies.) Definitions The two definitions Suppose that is a sequence of sets. The two equivalent definitions are as follows. Using union and intersection: define and If these two sets are equal, then the set-theoretic limit of the sequence exists and is equal to that common set. Either set as described above can be used to get the limit, and there may be other means to get the limit as well. Using indicator functions: let equal if and otherwise. Define and where the expressions inside the brackets on the right are, respectively, the limit infimum and limit supremum of the real-valued sequence Again, if these two sets are equal, then the set-theoretic limit of the sequence exists and is equal to that common set, and either set as described above can be used to get the limit. To see the equivalence of the definitions, consider the limit infimum. The use of De Morgan's law below explains why this suffices for the limit supremum. Since indicator functions take only values and if and only if takes value only finitely many times. Equivalently, if and only if there exists such that the element is in for every which is to say if and only if for only finitely many Therefore, is in the if and only if is in all but finitely many For this reason, a shorthand phrase for the limit infimum is " is in all but finitely often", typically expressed by writing " a.b.f.o.". Similarly, an element is in the limit supremum if, no matter how large is, there exists such that the element is in That is, is in the limit supremum if and only if is in infinitely many F
https://en.wikipedia.org/wiki/Korteweg%E2%80%93De%20Vries%20equation	In mathematics, the Korteweg–De Vries (KdV) equation is a partial differential equation (PDE) which serves as a mathematical model of waves on shallow water surfaces. It is particularly notable as the prototypical example of an integrable PDE and exhibits many of the expected behaviors for an integrable PDE, such as a large number of explicit solutions, in particular soliton solutions, and an infinite number of conserved quantities, despite the nonlinearity which typically renders PDEs intractable. The KdV can be solved by the inverse scattering method (ISM). In fact, Gardner, Greene, Kruskal and Miura developed the classical inverse scattering method to solve the KdV equation. The KdV equation was first introduced by and rediscovered by , who found the simplest solution, the one-soliton solution. Understanding of the equation and behavior of solutions was greatly advanced by the computer simulations of Zabusky and Kruskal in 1965 and then the development of the inverse scattering transform in 1967. Definition The KdV equation is a nonlinear, dispersive partial differential equation for a function of two dimensionless real variables, and which are proportional to space and time respectively: with and denoting partial derivatives with respect to and . For modelling shallow water waves, is the height displacement of the water surface from its equilibrium height. The constant in front of the last term is conventional but of no great significance: multiplying , , and by constants can be used to make the coefficients of any of the three terms equal to any given non-zero constants. The introduces dispersion while is an advection term. Soliton solutions One-soliton solution Consider solutions in which a fixed wave form (given by ) maintains its shape as it travels to the right at phase speed . Such a solution is given by . Substituting it into the KdV equation gives the ordinary differential equation or, integrating with respect to , where is a constant of integration. Interpreting the independent variable above as a virtual time variable, this means satisfies Newton's equation of motion of a particle of unit mass in a cubic potential If then the potential function has local maximum at , there is a solution in which starts at this point at 'virtual time' , eventually slides down to the local minimum, then back up the other side, reaching an equal height, then reverses direction, ending up at the local maximum again at time . In other words, approaches as . This is the characteristic shape of the solitary wave solution. More precisely, the solution is where stands for the hyperbolic secant and is an arbitrary constant. This describes a right-moving soliton with velocity . N-soliton solution There is a known expression for a solution which is an -soliton solution, which at late times resolves into separate single solitons. The solution depends on an decreasing positive set of parameters and a non-zero set of p
https://en.wikipedia.org/wiki/Mean%20deviation	Mean deviation may refer to: Statistics Mean signed deviation, a measure of central tendency Mean absolute deviation, a measure of statistical dispersion Mean squared deviation, another measure of statistical dispersion Other Mean Deviation (book), a 2010 non-fiction book by former Metal Maniacs magazine editor Jeff Wagner See also Deviation (statistics) Mean difference (disambiguation)
https://en.wikipedia.org/wiki/Multiplicative%20order	In number theory, given a positive integer n and an integer a coprime to n, the multiplicative order of a modulo n is the smallest positive integer k such that . In other words, the multiplicative order of a modulo n is the order of a in the multiplicative group of the units in the ring of the integers modulo n. The order of a modulo n is sometimes written as . Example The powers of 4 modulo 7 are as follows: The smallest positive integer k such that 4k ≡ 1 (mod 7) is 3, so the order of 4 (mod 7) is 3. Properties Even without knowledge that we are working in the multiplicative group of integers modulo n, we can show that a actually has an order by noting that the powers of a can only take a finite number of different values modulo n, so according to the pigeonhole principle there must be two powers, say s and t and without loss of generality s > t, such that as ≡ at (mod n). Since a and n are coprime, a has an inverse element a−1 and we can multiply both sides of the congruence with a−t, yielding as−t ≡ 1 (mod n). The concept of multiplicative order is a special case of the order of group elements. The multiplicative order of a number a modulo n is the order of a in the multiplicative group whose elements are the residues modulo n of the numbers coprime to n, and whose group operation is multiplication modulo n. This is the group of units of the ring Zn; it has φ(n) elements, φ being Euler's totient function, and is denoted as U(n) or U(Zn). As a consequence of Lagrange's theorem, the order of a (mod n) always divides φ(n). If the order of a is actually equal to φ(n), and therefore as large as possible, then a is called a primitive root modulo n. This means that the group U(n) is cyclic and the residue class of a generates it. The order of a (mod n) also divides λ(n), a value of the Carmichael function, which is an even stronger statement than the divisibility of φ(n). Programming languages Maxima CAS : zn_order (a, n) Rosetta Code - examples of multiplicative order in various languages See also Discrete logarithm Modular arithmetic References External links Modular arithmetic
https://en.wikipedia.org/wiki/Pierre-Simon%20Laplace	Pierre-Simon, Marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized and extended the work of his predecessors in his five-volume Mécanique céleste (Celestial Mechanics) (1799–1825). This work translated the geometric study of classical mechanics to one based on calculus, opening up a broader range of problems. In statistics, the Bayesian interpretation of probability was developed mainly by Laplace. Laplace formulated Laplace's equation, and pioneered the Laplace transform which appears in many branches of mathematical physics, a field that he took a leading role in forming. The Laplacian differential operator, widely used in mathematics, is also named after him. He restated and developed the nebular hypothesis of the origin of the Solar System and was one of the first scientists to suggest an idea similar to that of a black hole, with Stephen Hawking stating that "Laplace essentially predicted the existence of black holes". Laplace is regarded as one of the greatest scientists of all time. Sometimes referred to as the French Newton or Newton of France, he has been described as possessing a phenomenal natural mathematical faculty superior to that of almost all of his contemporaries. He was Napoleon's examiner when Napoleon graduated from the École Militaire in Paris in 1785. Laplace became a count of the Empire in 1806 and was named a marquis in 1817, after the Bourbon Restoration. Early years Some details of Laplace's life are not known, as records of it were burned in 1925 with the family château in Saint Julien de Mailloc, near Lisieux, the home of his great-great-grandson the Comte de Colbert-Laplace. Others had been destroyed earlier, when his house at Arcueil near Paris was looted in 1871. Laplace was born in Beaumont-en-Auge, Normandy on 23 March 1749, a village four miles west of Pont l'Évêque. According to W. W. Rouse Ball, his father, Pierre de Laplace, owned and farmed the small estates of Maarquis. His great-uncle, Maitre Oliver de Laplace, had held the title of Chirurgien Royal. It would seem that from a pupil he became an usher in the school at Beaumont; but, having procured a letter of introduction to d'Alembert, he went to Paris to advance his fortune. However, Karl Pearson is scathing about the inaccuracies in Rouse Ball's account and states: His parents, Pierre Laplace and Marie-Anne Sochon, were from comfortable families. The Laplace family was involved in agriculture until at least 1750, but Pierre Laplace senior was also a cider merchant and syndic of the town of Beaumont. Pierre Simon Laplace attended a school in the village run at a Benedictine priory, his father intending that he be ordained in the Roman Catholic Church. At sixteen, to further his father's intention, he was sent to the University of Caen to read theology. At the university, he was
https://en.wikipedia.org/wiki/Boole%27s%20inequality	In probability theory, Boole's inequality, also known as the union bound, says that for any finite or countable set of events, the probability that at least one of the events happens is no greater than the sum of the probabilities of the individual events. This inequality provides an upper bound on the probability of occurrence of at least one of a countable number of events in terms of the individual probabilities of the events. Boole's inequality is named for its discoverer, George Boole. Formally, for a countable set of events A1, A2, A3, ..., we have In measure-theoretic terms, Boole's inequality follows from the fact that a measure (and certainly any probability measure) is σ-sub-additive. Proof Proof using induction Boole's inequality may be proved for finite collections of events using the method of induction. For the case, it follows that For the case , we have Since and because the union operation is associative, we have Since by the first axiom of probability, we have and therefore Proof without using induction For any events in in our probability space we have One of the axioms of a probability space is that if are disjoint subsets of the probability space then this is called countable additivity. If we modify the sets , so they become disjoint, we can show that by proving both directions of inclusion. Suppose . Then for some minimum such that . Therefore . So the first inclusion is true: . Next suppose that . It follows that for some . And so , and we have the other inclusion: . By construction of each , . For it is the case that So, we can conclude that the desired inequality is true: Bonferroni inequalities Boole's inequality may be generalized to find upper and lower bounds on the probability of finite unions of events. These bounds are known as Bonferroni inequalities, after Carlo Emilio Bonferroni; see . Let for all integers k in {1, ..., n}. When n is odd, the sequence of inequalities: and both hold. When n is even, then: and both hold. The chains of inequalities among the partial sums follow from the observation that the events Sk form a decreasing sequence of sets. The equalities follow from the inclusion–exclusion principle, and Boole's inequality is the special case of the extremal upper bounds. Example Suppose that you are estimating 5 parameters based on a random sample, and you can control each parameter separately. If you want your estimations of all five parameters to be good with a chance 95%, what should you do to each parameter? Tuning each parameter's chance to be good to within 95% is not enough because "all are good" is a subset of each event "Estimate i is good". We can use Boole's Inequality to solve this problem. By finding the complement of event "all fives are good", we can change this question into another condition: P( at least one estimation is bad) = 0.05 ≤ P( A1 is bad) + P( A2 is bad) + P( A3 is bad) + P( A4 is bad) + P( A5 is bad) One way is to make each
https://en.wikipedia.org/wiki/Leech%20lattice	In mathematics, the Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space, which is one of the best models for the kissing number problem. It was discovered by . It may also have been discovered (but not published) by Ernst Witt in 1940. Characterization The Leech lattice Λ24 is the unique lattice in 24-dimensional Euclidean space, E24, with the following list of properties: It is unimodular; i.e., it can be generated by the columns of a certain 24×24 matrix with determinant 1. It is even; i.e., the square of the length of each vector in Λ24 is an even integer. The length of every non-zero vector in Λ24 is at least 2. The last condition is equivalent to the condition that unit balls centered at the points of Λ24 do not overlap. Each is tangent to 196,560 neighbors, and this is known to be the largest number of non-overlapping 24-dimensional unit balls that can simultaneously touch a single unit ball. This arrangement of 196,560 unit balls centred about another unit ball is so efficient that there is no room to move any of the balls; this configuration, together with its mirror-image, is the only 24-dimensional arrangement where 196,560 unit balls simultaneously touch another. This property is also true in 1, 2 and 8 dimensions, with 2, 6 and 240 unit balls, respectively, based on the integer lattice, hexagonal tiling and E8 lattice, respectively. It has no root system and in fact is the first unimodular lattice with no roots (vectors of norm less than 4), and therefore has a centre density of 1. By multiplying this value by the volume of a unit ball in 24 dimensions, , one can derive its absolute density. showed that the Leech lattice is isometric to the set of simple roots (or the Dynkin diagram) of the reflection group of the 26-dimensional even Lorentzian unimodular lattice II25,1. By comparison, the Dynkin diagrams of II9,1 and II17,1 are finite. Applications The binary Golay code, independently developed in 1949, is an application in coding theory. More specifically, it is an error-correcting code capable of correcting up to three errors in each 24-bit word, and detecting up to seven. It was used to communicate with the Voyager probes, as it is much more compact than the previously-used Hadamard code. Quantizers, or analog-to-digital converters, can use lattices to minimise the average root-mean-square error. Most quantizers are based on the one-dimensional integer lattice, but using multi-dimensional lattices reduces the RMS error. The Leech lattice is a good solution to this problem, as the Voronoi cells have a low second moment. The vertex algebra of the two-dimensional conformal field theory describing bosonic string theory, compactified on the 24-dimensional quotient torus R24/Λ24 and orbifolded by a two-element reflection group, provides an explicit construction of the Griess algebra that has the monster group as its automorphism group. This monster vertex algebra was also used to prove the mons
https://en.wikipedia.org/wiki/Binary%20Golay%20code	In mathematics and electronics engineering, a binary Golay code is a type of linear error-correcting code used in digital communications. The binary Golay code, along with the ternary Golay code, has a particularly deep and interesting connection to the theory of finite sporadic groups in mathematics. These codes are named in honor of Marcel J. E. Golay whose 1949 paper introducing them has been called, by E. R. Berlekamp, the "best single published page" in coding theory. There are two closely related binary Golay codes. The extended binary Golay code, G24 (sometimes just called the "Golay code" in finite group theory) encodes 12 bits of data in a 24-bit word in such a way that any 3-bit errors can be corrected or any 7-bit errors can be detected. The other, the perfect binary Golay code, G23, has codewords of length 23 and is obtained from the extended binary Golay code by deleting one coordinate position (conversely, the extended binary Golay code is obtained from the perfect binary Golay code by adding a parity bit). In standard coding notation, the codes have parameters [24, 12, 8] and [23, 12, 7], corresponding to the length of the codewords, the dimension of the code, and the minimum Hamming distance between two codewords, respectively. Mathematical definition In mathematical terms, the extended binary Golay code G24 consists of a 12-dimensional linear subspace W of the space of 24-bit words such that any two distinct elements of W differ in at least 8 coordinates. W is called a linear code because it is a vector space. In all, W comprises elements. The elements of W are called code words. They can also be described as subsets of a set of 24 elements, where addition is defined as taking the symmetric difference of the subsets. In the extended binary Golay code, all code words have Hamming weights of 0, 8, 12, 16, or 24. Code words of weight 8 are called octads and code words of weight 12 are called dodecads. Octads of the code G24 are elements of the S(5,8,24) Steiner system. There are octads and 759 complements thereof. It follows that there are dodecads. Two octads intersect (have 1's in common) in 0, 2, or 4 coordinates in the binary vector representation (these are the possible intersection sizes in the subset representation). An octad and a dodecad intersect at 2, 4, or 6 coordinates. Up to relabeling coordinates, W is unique. The binary Golay code, G23 is a perfect code. That is, the spheres of radius three around code words form a partition of the vector space. G23 is a 12-dimensional subspace of the space F. The automorphism group of the perfect binary Golay code G23 (meaning the subgroup of the group S23 of permutations of the coordinates of F which leave G23 invariant), is the Mathieu group . The automorphism group of the extended binary Golay code is the Mathieu group , of order . is transitive on octads and on dodecads. The other Mathieu groups occur as stabilizers of one or several elements of W. Construct
https://en.wikipedia.org/wiki/Finite%20volume%20method	The finite volume method (FVM) is a method for representing and evaluating partial differential equations in the form of algebraic equations. In the finite volume method, volume integrals in a partial differential equation that contain a divergence term are converted to surface integrals, using the divergence theorem. These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods are conservative. Another advantage of the finite volume method is that it is easily formulated to allow for unstructured meshes. The method is used in many computational fluid dynamics packages. "Finite volume" refers to the small volume surrounding each node point on a mesh. Finite volume methods can be compared and contrasted with the finite difference methods, which approximate derivatives using nodal values, or finite element methods, which create local approximations of a solution using local data, and construct a global approximation by stitching them together. In contrast a finite volume method evaluates exact expressions for the average value of the solution over some volume, and uses this data to construct approximations of the solution within cells. Example Consider a simple 1D advection problem: Here, represents the state variable and represents the flux or flow of . Conventionally, positive represents flow to the right while negative represents flow to the left. If we assume that equation () represents a flowing medium of constant area, we can sub-divide the spatial domain, , into finite volumes or cells with cell centers indexed as . For a particular cell, , we can define the volume average value of at time and , as and at time as, where and represent locations of the upstream and downstream faces or edges respectively of the cell. Integrating equation () in time, we have: where . To obtain the volume average of at time , we integrate over the cell volume, and divide the result by , i.e. We assume that is well behaved and that we can reverse the order of integration. Also, recall that flow is normal to the unit area of the cell. Now, since in one dimension , we can apply the divergence theorem, i.e. , and substitute for the volume integral of the divergence with the values of evaluated at the cell surface (edges and ) of the finite volume as follows: where . We can therefore derive a semi-discrete numerical scheme for the above problem with cell centers indexed as , and with cell edge fluxes indexed as , by differentiating () with respect to time to obtain: where values for the edge fluxes, , can be reconstructed by interpolation or extrapolation of the cell averages. Equation () is exact for the volume averages; i.e., no approximations have been made during its derivation. This method can also be applied to a 2D situation by considering the north and south faces along with the east and west faces around a
https://en.wikipedia.org/wiki/Intuitionistic%20type%20theory	Intuitionistic type theory (also known as constructive type theory, or Martin-Löf type theory) is a type theory and an alternative foundation of mathematics. Intuitionistic type theory was created by Per Martin-Löf, a Swedish mathematician and philosopher, who first published it in 1972. There are multiple versions of the type theory: Martin-Löf proposed both intensional and extensional variants of the theory and early impredicative versions, shown to be inconsistent by Girard's paradox, gave way to predicative versions. However, all versions keep the core design of constructive logic using dependent types. Design Martin-Löf designed the type theory on the principles of mathematical constructivism. Constructivism requires any existence proof to contain a "witness". So, any proof of "there exists a prime greater than 1000" must identify a specific number that is both prime and greater than 1000. Intuitionistic type theory accomplished this design goal by internalizing the BHK interpretation. An interesting consequence is that proofs become mathematical objects that can be examined, compared, and manipulated. Intuitionistic type theory's type constructors were built to follow a one-to-one correspondence with logical connectives. For example, the logical connective called implication () corresponds to the type of a function (). This correspondence is called the Curry–Howard isomorphism. Previous type theories had also followed this isomorphism, but Martin-Löf's was the first to extend it to predicate logic by introducing dependent types. Type theory Intuitionistic type theory has 3 finite types, which are then composed using 5 different type constructors. Unlike set theories, type theories are not built on top of a logic like Frege's. So, each feature of the type theory does double duty as a feature of both math and logic. If you are unfamiliar with type theory and know set theory, a quick summary is: Types contain terms just like sets contain elements. Terms belong to one and only one type. Terms like and compute ("reduce") down to canonical terms like 4. For more, see the article on Type theory. 0 type, 1 type and 2 type There are 3 finite types: The 0 type contains 0 terms. The 1 type contains 1 canonical term. And the 2 type contains 2 canonical terms. Because the 0 type contains 0 terms, it is also called the empty type. It is used to represent anything that cannot exist. It is also written and represents anything unprovable. (That is, a proof of it cannot exist.) As a result, negation is defined as a function to it: . Likewise, the 1 type contains 1 canonical term and represents existence. It also is called the unit type. It often represents propositions that can be proven and is, therefore, sometimes written . Finally, the 2 type contains 2 canonical terms. It represents a definite choice between two values. It is used for Boolean values but not propositions. Propositions are considered the 1 type and ma
https://en.wikipedia.org/wiki/Hell%2C%20Michigan	Hell is an unincorporated community in Livingston County in the U.S. state of Michigan. As an unincorporated community, Hell has no defined boundaries or population statistics of its own. Located within Putnam Township, the community is centered along Patterson Lake Road about northwest of Ann Arbor and southwest of Pinckney. The community is served by the Pinckney post office with the 48169 ZIP Code. History Hell developed around a sawmill, gristmill, distillery and tavern. All four were operated by George Reeves, who moved to the area in the 1830s from the Catskill Mountains in New York. He purchased a sawmill on what is now known as Hell Creek in 1841. In addition to the sawmill, Reeves purchased of land surrounding the mill. Reeves then built a gristmill on Hell Creek which was powered by water that was impounded by a small dam across the creek. Farmers in the area were quite successful in growing wheat and had an abundance of grain. Reeves opened a distillery to process the excess grain into whiskey. Reeves also opened a general store/tavern on his property. The tavern and distillery soon became a thriving business for Reeves. He built a ballroom on the second floor of the establishment and a sulky racetrack around his millpond. Reeves also sold his alcohol to nearby roadhouses and stores for as little as ten cents a gallon. His operation came under the scrutiny of the U.S. government in the years after the American Civil War. When tax collectors came to Hell to assess his operation, Reeves and his customers conspired to hide the whiskey by filling barrels and sinking them to the bottom of the millpond. When the government agents left the area, the barrels were hauled to the surface with ropes. As Reeves aged, he slowed his business ventures, closing the distillery and witnessing the burning of the gristmill. He died in 1877. Reeves' family sold the land to a group of investors from Detroit in 1924. The investors increased the size of the millpond by raising the level of the dam, creating what is now Hiland Lake. The area soon became a summer resort area, attracting visitors for swimming and fishing. Henry Ford considered building some manufacturing facilities in the area but decided against it. Etymology Hell has been noted on a list of unusual place names. There are a number of theories for the origin of Hell's name. The first is that a pair of German travelers stepped out of a stagecoach one sunny afternoon in the 1830s, and one said to the other, "So schön hell!" (translated as, "So beautifully bright!") Their comments were overheard by some locals and the name stuck. The second theory is tied to the "hell-like" conditions encountered by early explorers including mosquitos, thick forest cover, and extensive wetlands. The third is that George’s habit of paying the local farmers for their grain with home distilled whiskey led many wives to comment “He’s gone to Hell again” when questioned about their husband’s whereabouts during har
https://en.wikipedia.org/wiki/VEGAS%20algorithm	The VEGAS algorithm, due to G. Peter Lepage, is a method for reducing error in Monte Carlo simulations by using a known or approximate probability distribution function to concentrate the search in those areas of the integrand that make the greatest contribution to the final integral. The VEGAS algorithm is based on importance sampling. It samples points from the probability distribution described by the function so that the points are concentrated in the regions that make the largest contribution to the integral. The GNU Scientific Library (GSL) provides a VEGAS routine. Sampling method In general, if the Monte Carlo integral of over a volume is sampled with points distributed according to a probability distribution described by the function we obtain an estimate The variance of the new estimate is then where is the variance of the original estimate, If the probability distribution is chosen as then it can be shown that the variance vanishes, and the error in the estimate will be zero. In practice it is not possible to sample from the exact distribution g for an arbitrary function, so importance sampling algorithms aim to produce efficient approximations to the desired distribution. Approximation of probability distribution The VEGAS algorithm approximates the exact distribution by making a number of passes over the integration region while histogramming the function f. Each histogram is used to define a sampling distribution for the next pass. Asymptotically this procedure converges to the desired distribution. In order to avoid the number of histogram bins growing like with dimension d the probability distribution is approximated by a separable function: so that the number of bins required is only Kd. This is equivalent to locating the peaks of the function from the projections of the integrand onto the coordinate axes. The efficiency of VEGAS depends on the validity of this assumption. It is most efficient when the peaks of the integrand are well-localized. If an integrand can be rewritten in a form which is approximately separable this will increase the efficiency of integration with VEGAS. See also Las Vegas algorithm Monte Carlo integration Importance sampling References Monte Carlo methods Computational physics Statistical algorithms Variance reduction
https://en.wikipedia.org/wiki/List%20of%20differential%20geometry%20topics	This is a list of differential geometry topics. See also glossary of differential and metric geometry and list of Lie group topics. Differential geometry of curves and surfaces Differential geometry of curves List of curves topics Frenet–Serret formulas Curves in differential geometry Line element Curvature Radius of curvature Osculating circle Curve Fenchel's theorem Differential geometry of surfaces Theorema egregium Gauss–Bonnet theorem First fundamental form Second fundamental form Gauss–Codazzi–Mainardi equations Dupin indicatrix Asymptotic curve Curvature Principal curvatures Mean curvature Gauss curvature Elliptic point Types of surfaces Minimal surface Ruled surface Conical surface Developable surface Nadirashvili surface Foundations Calculus on manifolds See also multivariable calculus, list of multivariable calculus topics Manifold Differentiable manifold Smooth manifold Banach manifold Fréchet manifold Tensor analysis Tangent vector Tangent space Tangent bundle Cotangent space Cotangent bundle Tensor Tensor bundle Vector field Tensor field Differential form Exterior derivative Lie derivative pullback (differential geometry) pushforward (differential) jet (mathematics) Contact (mathematics) jet bundle Frobenius theorem (differential topology) Integral curve Differential topology Diffeomorphism Large diffeomorphism Orientability characteristic class Chern class Pontrjagin class spin structure differentiable map submersion immersion Embedding Whitney embedding theorem Critical value Sard's theorem Saddle point Morse theory Lie derivative Hairy ball theorem Poincaré–Hopf theorem Stokes' theorem De Rham cohomology Sphere eversion Frobenius theorem (differential topology) Distribution (differential geometry) integral curve foliation integrability conditions for differential systems Fiber bundles Fiber bundle Principal bundle Frame bundle Hopf bundle Associated bundle Vector bundle Tangent bundle Cotangent bundle Line bundle Jet bundle Fundamental structures Sheaf (mathematics) Pseudogroup G-structure synthetic differential geometry Riemannian geometry Fundamental notions Metric tensor Riemannian manifold Pseudo-Riemannian manifold Levi-Civita connection Non-Euclidean geometry Non-Euclidean geometry Elliptic geometry Spherical geometry Sphere-world Angle excess hyperbolic geometry hyperbolic space hyperboloid model Poincaré disc model Poincaré half-plane model Poincaré metric Angle of parallelism Geodesic Prime geodesic Geodesic flow Exponential map (Lie theory) Exponential map (Riemannian geometry) Injectivity radius Geodesic deviation equation Jacobi field Symmetric spaces (and related topics) Riemannian symmetric space Margulis lemma Space form Constant curvature taut submanifold Uniformization theorem Myers theorem Gromov's compactness theorem Riemannian submanifolds Gauss–Codazzi equations Darboux frame Hypersurface Induced metric Nash embedding theorem minimal surface Helicoid Catenoid Costa's minimal surface Hsi
https://en.wikipedia.org/wiki/Ein	Ein or EIN may refer to: Science and technology Ein function, in mathematics Endometrial intraepithelial neoplasia, a lesion of the uterine lining Equivalent input noise, of a microphone European Informatics Network, a 1970s computer network Fictional characters Ein, a character in the anime series Cowboy Bebop Ein, a character in the video game series Dead or Alive Ein, the protagonist of the Game Boy Advance game Riviera: The Promised Land Other uses Aer Lingus (ICAO code), the flag carrier airline of Ireland Eindhoven Airport (IATA code), in the Netherlands Employer Identification Number, assigned by the US Internal Revenue Service EPODE International Network, a Belgian obesity organization
https://en.wikipedia.org/wiki/Conway%20group	In the area of modern algebra known as group theory, the Conway groups are the three sporadic simple groups Co1, Co2 and Co3 along with the related finite group Co0 introduced by . The largest of the Conway groups, Co0, is the group of automorphisms of the Leech lattice Λ with respect to addition and inner product. It has order but it is not a simple group. The simple group Co1 of order = 221395472111323 is defined as the quotient of Co0 by its center, which consists of the scalar matrices ±1. The groups Co2 of order = 218365371123 and Co3 of order = 210375371123 consist of the automorphisms of Λ fixing a lattice vector of type 2 and type 3, respectively. As the scalar −1 fixes no non-zero vector, these two groups are isomorphic to subgroups of Co1. The inner product on the Leech lattice is defined as 1/8 the sum of the products of respective co-ordinates of the two multiplicand vectors; it is an integer. The square norm of a vector is its inner product with itself, always an even integer. It is common to speak of the type of a Leech lattice vector: half the square norm. Subgroups are often named in reference to the types of relevant fixed points. This lattice has no vectors of type 1. History relates how, in about 1964, John Leech investigated close packings of spheres in Euclidean spaces of large dimension. One of Leech's discoveries was a lattice packing in 24-space, based on what came to be called the Leech lattice Λ. He wondered whether his lattice's symmetry group contained an interesting simple group, but felt he needed the help of someone better acquainted with group theory. He had to do much asking around because the mathematicians were pre-occupied with agendas of their own. John Conway agreed to look at the problem. John G. Thompson said he would be interested if he were given the order of the group. Conway expected to spend months or years on the problem, but found results in just a few sessions. stated that he found the Leech lattice in 1940 and hinted that he calculated the order of its automorphism group Co0. Monomial subgroup N of Co0 Conway started his investigation of Co0 with a subgroup he called N, a holomorph of the (extended) binary Golay code (as diagonal matrices with 1 or −1 as diagonal elements) by the Mathieu group M24 (as permutation matrices). . A standard representation, used throughout this article, of the binary Golay code arranges the 24 co-ordinates so that 6 consecutive blocks (tetrads) of 4 constitute a sextet. The matrices of Co0 are orthogonal; i. e., they leave the inner product invariant. The inverse is the transpose. Co0 has no matrices of determinant −1. The Leech lattice can easily be defined as the Z-module generated by the set Λ2 of all vectors of type 2, consisting of (4, 4, 022) (28, 016) (−3, 123) and their images under N. Λ2 under N falls into 3 orbits of sizes 1104, 97152, and 98304. Then . Conway strongly suspected that Co0 was transitive on Λ2, and indeed he
https://en.wikipedia.org/wiki/The%20Dot%20and%20the%20Line	The Dot and the Line: A Romance in Lower Mathematics is a 1965 animated short film directed by Chuck Jones and co-directed by Maurice Noble, based on the 1963 book of the same name written and illustrated by Norton Juster, who also provided the film's script. The film was narrated by Robert Morley and produced by Metro-Goldwyn-Mayer. It won the 1965 Academy Award for Animated Short Film and was entered into the Short Film Palme d'Or competition at the 1966 Cannes Film Festival. Story The story details a straight blue line who is hopelessly in love with a red dot. The dot, finding the line to be stiff, dull, and conventional, turns her affections toward a wild and unkempt squiggle. Taking advantage of the line's stiffness, the squiggle rubs it in that he is a lot more fun for the dot. The depressed line's friends try to get him to settle down with a female line, but he refuses. He tries to dream of greatness until he finally understands what the squiggle means, and decides to be more unconventional. Willing to do whatever it takes to win the dot's affection, the line manages to bend himself and form angle after angle until he is nothing more than a mess of sides, bends and angles. After he straightens himself out, he settles down and focuses more responsibly on this new ability, creating shapes so complex that he has to label his sides and angles in order to keep his place. When competing again, the squiggle claims that the line still has nothing to show to the dot. The line proves his rival wrong and is able to show the dot what she is really worth to him. When she sees this, the dot is overwhelmed by the line's responsibility and unconventionality. She then faces the now nervous squiggle, whom she gives a chance to make his case to win her love. He makes an effort to reclaim the dot's heart by trying to copy what the line did, but to no avail. No matter how hard he tries to re-shape himself, the squiggle still remains the same tangled, chaotic mess of lines and curves. He tries to tell the dot a joke, but she has realized the flatness of it, and he's forced to retreat. She realized how much her relationship with the squiggle had been a mistake. What she thought was freedom and joy was nothing more than sloth, chaos and anarchy. Fed up, the dot tells the squiggle off how she really feels about him, denouncing him as meaningless, undisciplined, insignificant, and out of luck. She leaves with the line, having accepted that he has much more to offer, and the punning moral is presented: "To the vector belong the spoils." Authorship Though listed as being directed by Chuck Jones, the true acting director was Maurice Noble according to his own recollection, Noble being the long-term background artist and eventually credited co-director on numerous projects with Jones. Chuck Jones was one of the originators for the adaptation and did the first treatment for the short. However, the results did not please the producers, who asked Maurice Noble to take
https://en.wikipedia.org/wiki/Poisson%20algebra	In mathematics, a Poisson algebra is an associative algebra together with a Lie bracket that also satisfies Leibniz's law; that is, the bracket is also a derivation. Poisson algebras appear naturally in Hamiltonian mechanics, and are also central in the study of quantum groups. Manifolds with a Poisson algebra structure are known as Poisson manifolds, of which the symplectic manifolds and the Poisson–Lie groups are a special case. The algebra is named in honour of Siméon Denis Poisson. Definition A Poisson algebra is a vector space over a field K equipped with two bilinear products, ⋅ and {, }, having the following properties: The product ⋅ forms an associative K-algebra. The product {, }, called the Poisson bracket, forms a Lie algebra, and so it is anti-symmetric, and obeys the Jacobi identity. The Poisson bracket acts as a derivation of the associative product ⋅, so that for any three elements x, y and z in the algebra, one has {x, y ⋅ z} = {x, y} ⋅ z + y ⋅ {x, z}. The last property often allows a variety of different formulations of the algebra to be given, as noted in the examples below. Examples Poisson algebras occur in various settings. Symplectic manifolds The space of real-valued smooth functions over a symplectic manifold forms a Poisson algebra. On a symplectic manifold, every real-valued function H on the manifold induces a vector field XH, the Hamiltonian vector field. Then, given any two smooth functions F and G over the symplectic manifold, the Poisson bracket may be defined as: . This definition is consistent in part because the Poisson bracket acts as a derivation. Equivalently, one may define the bracket {,} as where [,] is the Lie derivative. When the symplectic manifold is R2n with the standard symplectic structure, then the Poisson bracket takes on the well-known form Similar considerations apply for Poisson manifolds, which generalize symplectic manifolds by allowing the symplectic bivector to be rank deficient. Lie algebras The tensor algebra of a Lie algebra has a Poisson algebra structure. A very explicit construction of this is given in the article on universal enveloping algebras. The construction proceeds by first building the tensor algebra of the underlying vector space of the Lie algebra. The tensor algebra is simply the disjoint union (direct sum ⊕) of all tensor products of this vector space. One can then show that the Lie bracket can be consistently lifted to the entire tensor algebra: it obeys both the product rule, and the Jacobi identity of the Poisson bracket, and thus is the Poisson bracket, when lifted. The pair of products {,} and ⊗ then form a Poisson algebra. Observe that ⊗ is neither commutative nor is it anti-commutative: it is merely associative. Thus, one has the general statement that the tensor algebra of any Lie algebra is a Poisson algebra. The universal enveloping algebra is obtained by modding out the Poisson algebra structure. Associative algebras If A is an associat
https://en.wikipedia.org/wiki/Group%20extension	In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If and are two groups, then is an extension of by if there is a short exact sequence If is an extension of by , then is a group, is a normal subgroup of and the quotient group is isomorphic to the group . Group extensions arise in the context of the extension problem, where the groups and are known and the properties of are to be determined. Note that the phrasing " is an extension of by " is also used by some. Since any finite group possesses a maximal normal subgroup with simple factor group , all finite groups may be constructed as a series of extensions with finite simple groups. This fact was a motivation for completing the classification of finite simple groups. An extension is called a central extension if the subgroup lies in the center of . Extensions in general One extension, the direct product, is immediately obvious. If one requires and to be abelian groups, then the set of isomorphism classes of extensions of by a given (abelian) group is in fact a group, which is isomorphic to cf. the Ext functor. Several other general classes of extensions are known but no theory exists that treats all the possible extensions at one time. Group extension is usually described as a hard problem; it is termed the extension problem. To consider some examples, if , then is an extension of both and . More generally, if is a semidirect product of and , written as , then is an extension of by , so such products as the wreath product provide further examples of extensions. Extension problem The question of what groups are extensions of by is called the extension problem, and has been studied heavily since the late nineteenth century. As to its motivation, consider that the composition series of a finite group is a finite sequence of subgroups , where each is an extension of by some simple group. The classification of finite simple groups gives us a complete list of finite simple groups; so the solution to the extension problem would give us enough information to construct and classify all finite groups in general. Classifying extensions Solving the extension problem amounts to classifying all extensions of H by K; or more practically, by expressing all such extensions in terms of mathematical objects that are easier to understand and compute. In general, this problem is very hard, and all the most useful results classify extensions that satisfy some additional condition. It is important to know when two extensions are equivalent or congruent. We say that the extensions and are equivalent (or congruent) if there exists a group isomorphism making commutative the diagram of Figure 1. In fact it is sufficient to have a group homomorphism; due to the assumed commutativity of the diagram, the map is forced to be an isomorphism by the short five lemma. Warning It may ha
https://en.wikipedia.org/wiki/Betti%20number	In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of n-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicial complexes or CW complexes), the sequence of Betti numbers is 0 from some point onward (Betti numbers vanish above the dimension of a space), and they are all finite. The nth Betti number represents the rank of the nth homology group, denoted Hn, which tells us the maximum number of cuts that can be made before separating a surface into two pieces or 0-cycles, 1-cycles, etc. For example, if then , if then , if then , if then , etc. Note that only the ranks of infinite groups are considered, so for example if , where is the finite cyclic group of order 2, then . These finite components of the homology groups are their torsion subgroups, and they are denoted by torsion coefficients. The term "Betti numbers" was coined by Henri Poincaré after Enrico Betti. The modern formulation is due to Emmy Noether. Betti numbers are used today in fields such as simplicial homology, computer science and digital images. Geometric interpretation Informally, the kth Betti number refers to the number of k-dimensional holes on a topological surface. A "k-dimensional hole" is a k-dimensional cycle that is not a boundary of a (k+1)-dimensional object. The first few Betti numbers have the following definitions for 0-dimensional, 1-dimensional, and 2-dimensional simplicial complexes: b0 is the number of connected components; b1 is the number of one-dimensional or "circular" holes; b2 is the number of two-dimensional "voids" or "cavities". Thus, for example, a torus has one connected surface component so b0 = 1, two "circular" holes (one equatorial and one meridional) so b1 = 2, and a single cavity enclosed within the surface so b2 = 1. Another interpretation of bk is the maximum number of k-dimensional curves that can be removed while the object remains connected. For example, the torus remains connected after removing two 1-dimensional curves (equatorial and meridional) so b1 = 2. The two-dimensional Betti numbers are easier to understand because we can see the world in 0, 1, 2, and 3-dimensions. Formal definition For a non-negative integer k, the kth Betti number bk(X) of the space X is defined as the rank (number of linearly independent generators) of the abelian group Hk(X), the kth homology group of X. The kth homology group is , the s are the boundary maps of the simplicial complex and the rank of Hk is the kth Betti number. Equivalently, one can define it as the vector space dimension of Hk(X; Q) since the homology group in this case is a vector space over Q. The universal coefficient theorem, in a very simple torsion-free case, shows that these definitions are the same. More generally, given a field F one can define bk(X, F), the kth Betti number with coefficients in F, as the vector space dimension o
https://en.wikipedia.org/wiki/Algebraic%20quantum%20field%20theory	Algebraic quantum field theory (AQFT) is an application to local quantum physics of C-algebra theory. Also referred to as the Haag–Kastler axiomatic framework for quantum field theory, because it was introduced by . The axioms are stated in terms of an algebra given for every open set in Minkowski space, and mappings between those. Haag–Kastler axioms Let be the set of all open and bounded subsets of Minkowski space. An algebraic quantum field theory is defined via a net of von Neumann algebras on a common Hilbert space satisfying the following axioms: Isotony: implies . Causality: If is space-like separated from , then . Poincaré covariance: A strongly continuous unitary representation of the Poincaré group on exists such that Spectrum condition: The joint spectrum of the energy-momentum operator (i.e. the generator of space-time translations) is contained in the closed forward lightcone. Existence of a vacuum vector: A cyclic and Poincaré-invariant vector exists. The net algebras are called local algebras and the C algebra is called the quasilocal algebra. Category-theoretic formulation Let Mink be the category of open subsets of Minkowski space M with inclusion maps as morphisms. We are given a covariant functor from Mink to uCalg, the category of unital C algebras, such that every morphism in Mink maps to a monomorphism in uCalg (isotony). The Poincaré group acts continuously on Mink. There exists a pullback of this action, which is continuous in the norm topology of (Poincaré covariance). Minkowski space has a causal structure. If an open set V lies in the causal complement of an open set U, then the image of the maps and commute (spacelike commutativity). If is the causal completion of an open set U, then is an isomorphism (primitive causality). A state with respect to a C-algebra is a positive linear functional over it with unit norm. If we have a state over , we can take the "partial trace" to get states associated with for each open set via the net monomorphism. The states over the open sets form a presheaf structure. According to the GNS construction, for each state, we can associate a Hilbert space representation of Pure states correspond to irreducible representations and mixed states correspond to reducible representations. Each irreducible representation (up to equivalence) is called a superselection sector. We assume there is a pure state called the vacuum such that the Hilbert space associated with it is a unitary representation of the Poincaré group compatible with the Poincaré covariance of the net such that if we look at the Poincaré algebra, the spectrum with respect to energy-momentum (corresponding to spacetime translations) lies on and in the positive light cone. This is the vacuum sector. QFT in curved spacetime More recently, the approach has been further implemented to include an algebraic version of quantum field theory in curved spacetime. Indeed, the viewpoint of local qu
https://en.wikipedia.org/wiki/Axiom%20of%20countability	In mathematics, an axiom of countability is a property of certain mathematical objects that asserts the existence of a countable set with certain properties. Without such an axiom, such a set might not provably exist. Important examples Important countability axioms for topological spaces include: sequential space: a set is open if every sequence convergent to a point in the set is eventually in the set first-countable space: every point has a countable neighbourhood basis (local base) second-countable space: the topology has a countable base separable space: there exists a countable dense subset Lindelöf space: every open cover has a countable subcover σ-compact space: there exists a countable cover by compact spaces Relationships with each other These axioms are related to each other in the following ways: Every first-countable space is sequential. Every second-countable space is first countable, separable, and Lindelöf. Every σ-compact space is Lindelöf. Every metric space is first countable. For metric spaces, second-countability, separability, and the Lindelöf property are all equivalent. Related concepts Other examples of mathematical objects obeying axioms of countability include sigma-finite measure spaces, and lattices of countable type. References General topology Mathematical axioms
https://en.wikipedia.org/wiki/%CE%A3-compact%20space	In mathematics, a topological space is said to be σ-compact if it is the union of countably many compact subspaces. A space is said to be σ-locally compact if it is both σ-compact and (weakly) locally compact. That terminology can be somewhat confusing as it does not fit the usual pattern of σ-(property) meaning a countable union of spaces satisfying (property); that's why such spaces are more commonly referred to explicitly as σ-compact (weakly) locally compact, which is also equivalent to being exhaustible by compact sets. Properties and examples Every compact space is σ-compact, and every σ-compact space is Lindelöf (i.e. every open cover has a countable subcover). The reverse implications do not hold, for example, standard Euclidean space (Rn) is σ-compact but not compact, and the lower limit topology on the real line is Lindelöf but not σ-compact. In fact, the countable complement topology on any uncountable set is Lindelöf but neither σ-compact nor locally compact. However, it is true that any locally compact Lindelöf space is σ-compact. (The irrational numbers) is not σ-compact. A Hausdorff, Baire space that is also σ-compact, must be locally compact at at least one point. If G is a topological group and G is locally compact at one point, then G is locally compact everywhere. Therefore, the previous property tells us that if G is a σ-compact, Hausdorff topological group that is also a Baire space, then G is locally compact. This shows that for Hausdorff topological groups that are also Baire spaces, σ-compactness implies local compactness. The previous property implies for instance that Rω is not σ-compact: if it were σ-compact, it would necessarily be locally compact since Rω is a topological group that is also a Baire space. Every hemicompact space is σ-compact. The converse, however, is not true; for example, the space of rationals, with the usual topology, is σ-compact but not hemicompact. The product of a finite number of σ-compact spaces is σ-compact. However the product of an infinite number of σ-compact spaces may fail to be σ-compact. A σ-compact space X is second category (respectively Baire) if and only if the set of points at which is X is locally compact is nonempty (respectively dense) in X. See also Notes References Steen, Lynn A. and Seebach, J. Arthur Jr.; Counterexamples in Topology, Holt, Rinehart and Winston (1970). . Compactness (mathematics) General topology Properties of topological spaces
https://en.wikipedia.org/wiki/List%20of%20number%20theory%20topics	This is a list of number theory topics. See also: List of recreational number theory topics Topics in cryptography Divisibility Composite number Highly composite number Even and odd numbers Parity Divisor, aliquot part Greatest common divisor Least common multiple Euclidean algorithm Coprime Euclid's lemma Bézout's identity, Bézout's lemma Extended Euclidean algorithm Table of divisors Prime number, prime power Bonse's inequality Prime factor Table of prime factors Formula for primes Factorization RSA number Fundamental theorem of arithmetic Square-free Square-free integer Square-free polynomial Square number Power of two Integer-valued polynomial Fractions Rational number Unit fraction Irreducible fraction = in lowest terms Dyadic fraction Recurring decimal Cyclic number Farey sequence Ford circle Stern–Brocot tree Dedekind sum Egyptian fraction Modular arithmetic Montgomery reduction Modular exponentiation Linear congruence theorem Method of successive substitution Chinese remainder theorem Fermat's little theorem Proofs of Fermat's little theorem Fermat quotient Euler's totient function Noncototient Nontotient Euler's theorem Wilson's theorem Primitive root modulo n Multiplicative order Discrete logarithm Quadratic residue Euler's criterion Legendre symbol Gauss's lemma (number theory) Congruence of squares Luhn formula Mod n cryptanalysis Arithmetic functions Multiplicative function Additive function Dirichlet convolution Erdős–Kac theorem Möbius function Möbius inversion formula Divisor function Liouville function Partition function (number theory) Integer partition Bell numbers Landau's function Pentagonal number theorem Bell series Lambert series Analytic number theory: additive problems Twin prime Brun's constant Cousin prime Prime triplet Prime quadruplet Sexy prime Sophie Germain prime Cunningham chain Goldbach's conjecture Goldbach's weak conjecture Second Hardy–Littlewood conjecture Hardy–Littlewood circle method Schinzel's hypothesis H Bateman–Horn conjecture Waring's problem Brahmagupta–Fibonacci identity Euler's four-square identity Lagrange's four-square theorem Taxicab number Generalized taxicab number Cabtaxi number Schnirelmann density Sumset Landau–Ramanujan constant Sierpinski number Seventeen or Bust Niven's constant Algebraic number theory See list of algebraic number theory topics Quadratic forms Unimodular lattice Fermat's theorem on sums of two squares Proofs of Fermat's theorem on sums of two squares L-functions Riemann zeta function Basel problem on ζ(2) Hurwitz zeta function Bernoulli number Agoh–Giuga conjecture Von Staudt–Clausen theorem Dirichlet series Euler product Prime number theorem Prime-counting function Meissel–Lehmer algorithm Offset logarithmic integral Legendre's constant Skewes' number Bertrand's postulate Proof of Bertrand's postulate Proof that the sum of the reciprocals of the primes diverges Cramér's conjecture Riemann hypothesis Critical line theorem Hilbert–Pólya conjecture General
https://en.wikipedia.org/wiki/Dimension%20theorem%20for%20vector%20spaces	In mathematics, the dimension theorem for vector spaces states that all bases of a vector space have equally many elements. This number of elements may be finite or infinite (in the latter case, it is a cardinal number), and defines the dimension of the vector space. Formally, the dimension theorem for vector spaces states that: As a basis is a generating set that is linearly independent, the theorem is a consequence of the following theorem, which is also useful: In particular if is finitely generated, then all its bases are finite and have the same number of elements. While the proof of the existence of a basis for any vector space in the general case requires Zorn's lemma and is in fact equivalent to the axiom of choice, the uniqueness of the cardinality of the basis requires only the ultrafilter lemma, which is strictly weaker (the proof given below, however, assumes trichotomy, i.e., that all cardinal numbers are comparable, a statement which is also equivalent to the axiom of choice). The theorem can be generalized to arbitrary -modules for rings having invariant basis number. In the finitely generated case the proof uses only elementary arguments of algebra, and does not require the axiom of choice nor its weaker variants. Proof Let be a vector space, be a linearly independent set of elements of , and be a generating set. One has to prove that the cardinality of is not larger than that of . If is finite, this results from the Steinitz exchange lemma. (Indeed, the Steinitz exchange lemma implies every finite subset of has cardinality not larger than that of , hence is finite with cardinality not larger than that of .) If is finite, a proof based on matrix theory is also possible. Assume that is infinite. If is finite, there is nothing to prove. Thus, we may assume that is also infinite. Let us suppose that the cardinality of is larger than that of . We have to prove that this leads to a contradiction. By Zorn's lemma, every linearly independent set is contained in a maximal linearly independent set . This maximality implies that spans and is therefore a basis (the maximality implies that every element of is linearly dependent from the elements of , and therefore is a linear combination of elements of ). As the cardinality of is greater than or equal to the cardinality of , one may replace with , that is, one may suppose, without loss of generality, that is a basis. Thus, every can be written as a finite sum where is a finite subset of As is infinite, has the same cardinality as . Therefore has cardinality smaller than that of . So there is some which does not appear in any . The corresponding can be expressed as a finite linear combination of s, which in turn can be expressed as finite linear combination of s, not involving . Hence is linearly dependent on the other s, which provides the desired contradiction. Kernel extension theorem for vector spaces This application of the dimension theorem i
https://en.wikipedia.org/wiki/Valuation%20%28algebra%29	In algebra (in particular in algebraic geometry or algebraic number theory), a valuation is a function on a field that provides a measure of the size or multiplicity of elements of the field. It generalizes to commutative algebra the notion of size inherent in consideration of the degree of a pole or multiplicity of a zero in complex analysis, the degree of divisibility of a number by a prime number in number theory, and the geometrical concept of contact between two algebraic or analytic varieties in algebraic geometry. A field with a valuation on it is called a valued field. Definition One starts with the following objects: a field and its multiplicative group K×, an abelian totally ordered group . The ordering and group law on are extended to the set } by the rules for all ∈ , for all ∈ . Then a valuation of is any map which satisfies the following properties for all a, b in K: if and only if , , , with equality if v(a) ≠ v(b). A valuation v is trivial if v(a) = 0 for all a in K×, otherwise it is non-trivial. The second property asserts that any valuation is a group homomorphism. The third property is a version of the triangle inequality on metric spaces adapted to an arbitrary Γ (see Multiplicative notation below). For valuations used in geometric applications, the first property implies that any non-empty germ of an analytic variety near a point contains that point. The valuation can be interpreted as the order of the leading-order term. The third property then corresponds to the order of a sum being the order of the larger term, unless the two terms have the same order, in which case they may cancel, in which case the sum may have larger order. For many applications, is an additive subgroup of the real numbers in which case ∞ can be interpreted as +∞ in the extended real numbers; note that for any real number a, and thus +∞ is the unit under the binary operation of minimum. The real numbers (extended by +∞) with the operations of minimum and addition form a semiring, called the min tropical semiring, and a valuation v is almost a semiring homomorphism from K to the tropical semiring, except that the homomorphism property can fail when two elements with the same valuation are added together. Multiplicative notation and absolute values The concept was developed by Emil Artin in his book Geometric Algebra writing the group in multiplicative notation as : Instead of ∞, we adjoin a formal symbol O to Γ, with the ordering and group law extended by the rules for all ∈ , for all ∈ . Then a valuation of is any map satisfying the following properties for all a, b ∈ K: if and only if , , , with equality if . (Note that the directions of the inequalities are reversed from those in the additive notation.) If is a subgroup of the positive real numbers under multiplication, the last condition is the ultrametric inequality, a stronger form of the triangle inequality , and is an absolute value. In this case, we may pa
https://en.wikipedia.org/wiki/List%20of%20algebraic%20topology%20topics	This is a list of algebraic topology topics. Homology (mathematics) Simplex Simplicial complex Polytope Triangulation Barycentric subdivision Simplicial approximation theorem Abstract simplicial complex Simplicial set Simplicial category Chain (algebraic topology) Betti number Euler characteristic Genus Riemann–Hurwitz formula Singular homology Cellular homology Relative homology Mayer–Vietoris sequence Excision theorem Universal coefficient theorem Cohomology List of cohomology theories Cocycle class Cup product Cohomology ring De Rham cohomology Čech cohomology Alexander–Spanier cohomology Intersection cohomology Lusternik–Schnirelmann category Poincaré duality Fundamental class Applications Jordan curve theorem Brouwer fixed point theorem Invariance of domain Lefschetz fixed-point theorem Hairy ball theorem Degree of a continuous mapping Borsuk–Ulam theorem Ham sandwich theorem Homology sphere Homotopy theory Homotopy Path (topology) Fundamental group Homotopy group Seifert–van Kampen theorem Pointed space Winding number Simply connected Universal cover Monodromy Homotopy lifting property Mapping cylinder Mapping cone (topology) Wedge sum Smash product Adjunction space Cohomotopy Cohomotopy group Brown's representability theorem Eilenberg–MacLane space Fibre bundle Möbius strip Line bundle Canonical line bundle Vector bundle Associated bundle Fibration Hopf bundle Classifying space Cofibration Homotopy groups of spheres Plus construction Whitehead theorem Weak equivalence Hurewicz theorem H-space Further developments Künneth theorem De Rham cohomology Obstruction theory Characteristic class Chern class Chern–Simons form Pontryagin class Pontryagin number Stiefel–Whitney class Poincaré conjecture Cohomology operation Steenrod algebra Bott periodicity theorem K-theory Topological K-theory Adams operation Algebraic K-theory Whitehead torsion Twisted K-theory Cobordism Thom space Suspension functor Stable homotopy theory Spectrum (homotopy theory) Morava K-theory Hodge conjecture Weil conjectures Directed algebraic topology Applied topology Example: DE-9IM Homological algebra Chain complex Commutative diagram Exact sequence Five lemma Short five lemma Snake lemma Splitting lemma Extension problem Spectral sequence Abelian category Group cohomology Sheaf Sheaf cohomology Grothendieck topology Derived category History Combinatorial topology See also Glossary of algebraic topology topology glossary List of topology topics List of general topology topics List of geometric topology topics Publications in topology Topological property Mathematics-related lists Outlines of mathematics and logic Outlines
https://en.wikipedia.org/wiki/List%20of%20polynomial%20topics	This is a list of polynomial topics, by Wikipedia page. See also trigonometric polynomial, list of algebraic geometry topics. Terminology Degree: The maximum exponents among the monomials. Factor: An expression being multiplied. Linear factor: A factor of degree one. Coefficient: An expression multiplying one of the monomials of the polynomial. Root (or zero) of a polynomial: Given a polynomial p(x), the x values that satisfy p(x) = 0 are called roots (or zeroes) of the polynomial p. Graphing End behaviour – Concavity – Orientation – Tangency point – Inflection point – Point where concavity changes. Basics Polynomial Coefficient Monomial Polynomial long division Synthetic division Polynomial factorization Rational function Partial fraction Partial fraction decomposition over R Vieta's formulas Integer-valued polynomial Algebraic equation Factor theorem Polynomial remainder theorem Elementary abstract algebra See also Theory of equations below. Polynomial ring Greatest common divisior of two polynomials Symmetric function Homogeneous polynomial Polynomial SOS (sum of squares) Theory of equations Polynomial family Quadratic function Cubic function Quartic function Quintic function Sextic function Septic function Octic function Completing the square Abel–Ruffini theorem Bring radical Binomial theorem Blossom (functional) Root of a function nth root (radical) Surd Square root Methods of computing square roots Cube root Root of unity Constructible number Complex conjugate root theorem Algebraic element Horner scheme Rational root theorem Gauss's lemma (polynomial) Irreducible polynomial Eisenstein's criterion Primitive polynomial Fundamental theorem of algebra Hurwitz polynomial Polynomial transformation Tschirnhaus transformation Galois theory Discriminant of a polynomial Resultant Elimination theory Gröbner basis Regular chain Triangular decomposition Sturm's theorem Descartes' rule of signs Carlitz–Wan conjecture Polynomial decomposition, factorization under functional composition Calculus with polynomials Delta operator Bernstein–Sato polynomial Polynomial interpolation Lagrange polynomial Runge's phenomenon Spline (mathematics) Weierstrass approximation theorem Bernstein polynomial Linear algebra Characteristic polynomial Minimal polynomial Invariant polynomial Named polynomials and polynomial sequences Abel polynomials Actuarial polynomials Additive polynomials All one polynomials Appell sequence Askey–Wilson polynomials Bell polynomials Bernoulli polynomials Bernstein polynomial Bessel polynomials Binomial type Caloric polynomial Charlier polynomials Chebyshev polynomials Chihara–Ismail polynomials Cyclotomic polynomials Dickson polynomial Ehrhart polynomial Exponential polynomials Favard's theorem Fibonacci polynomials Gegenbauer polynomials Hahn polynomials Hall–Littlewood polynomials Heat polynomial — see caloric polynomial Heckman–Opdam polynomials Hermite polynomials Hurwitz po
https://en.wikipedia.org/wiki/Arg%20max	In mathematics, the arguments of the maxima (abbreviated arg max or argmax) are the points, or elements, of the domain of some function at which the function values are maximized. In contrast to global maxima, which refers to the largest outputs of a function, arg max refers to the inputs, or arguments, at which the function outputs are as large as possible. Definition Given an arbitrary set a totally ordered set and a function, the over some subset of is defined by If or is clear from the context, then is often left out, as in In other words, is the set of points for which attains the function's largest value (if it exists). may be the empty set, a singleton, or contain multiple elements. In the fields of convex analysis and variational analysis, a slightly different definition is used in the special case where are the extended real numbers. In this case, if is identically equal to on then (that is, ) and otherwise is defined as above, where in this case can also be written as: where it is emphasized that this equality involving holds when is not identically on Arg min The notion of (or ), which stands for argument of the minimum, is defined analogously. For instance, are points for which attains its smallest value. It is the complementary operator of In the special case where are the extended real numbers, if is identically equal to on then (that is, ) and otherwise is defined as above and moreover, in this case (of not identically equal to ) it also satisfies: Examples and properties For example, if is then attains its maximum value of only at the point Thus The operator is different from the operator. The operator, when given the same function, returns the of the function instead of the that cause that function to reach that value; in other words is the element in Like max may be the empty set (in which case the maximum is undefined) or a singleton, but unlike may not contain multiple elements: for example, if is then but because the function attains the same value at every element of Equivalently, if is the maximum of then the is the level set of the maximum: We can rearrange to give the simple identity If the maximum is reached at a single point then this point is often referred to as and is considered a point, not a set of points. So, for example, (rather than the singleton set ), since the maximum value of is which occurs for However, in case the maximum is reached at many points, needs to be considered a of points. For example because the maximum value of is which occurs on this interval for or On the whole real line so an infinite set. Functions need not in general attain a maximum value, and hence the is sometimes the empty set; for example, since is unbounded on the real line. As another example, although is bounded by However, by the extreme value theorem, a continuous real-valued function on a closed interval has a maximum, and
https://en.wikipedia.org/wiki/Virasoro%20algebra	In mathematics, the Virasoro algebra (named after the physicist Miguel Ángel Virasoro) is a complex Lie algebra and the unique central extension of the Witt algebra. It is widely used in two-dimensional conformal field theory and in string theory. Definition The Virasoro algebra is spanned by generators for and the central charge . These generators satisfy and The factor of is merely a matter of convention. For a derivation of the algebra as the unique central extension of the Witt algebra, see derivation of the Virasoro algebra. The Virasoro algebra has a presentation in terms of two generators (e.g. 3 and −2) and six relations. Representation theory Highest weight representations A highest weight representation of the Virasoro algebra is a representation generated by a primary state: a vector such that where the number is called the conformal dimension or conformal weight of . A highest weight representation is spanned by eigenstates of . The eigenvalues take the form , where the integer is called the level of the corresponding eigenstate. More precisely, a highest weight representation is spanned by -eigenstates of the type with and , whose levels are . Any state whose level is not zero is called a descendant state of . For any pair of complex numbers and , the Verma module is the largest possible highest weight representation. (The same letter is used for both the element of the Virasoro algebra and its eigenvalue in a representation.) The states with and form a basis of the Verma module. The Verma module is indecomposable, and for generic values of and it is also irreducible. When it is reducible, there exist other highest weight representations with these values of and , called degenerate representations, which are cosets of the Verma module. In particular, the unique irreducible highest weight representation with these values of and is the quotient of the Verma module by its maximal submodule. A Verma module is irreducible if and only if it has no singular vectors. Singular vectors A singular vector or null vector of a highest weight representation is a state that is both descendant and primary. A sufficient condition for the Verma module to have a singular vector at the level is for some positive integers such that , with In particular, , and the reducible Verma module has a singular vector at the level . Then , and the corresponding reducible Verma module has a singular vector at the level . This condition for the existence of a singular vector at the level is not necessary. In particular, there is a singular vector at the level if with and . This singular vector is now a descendant of another singular vector at the level . This type of singular vectors can however only exist if the central charge is of the type . (For coprime, these are the central charges of the minimal models.) Hermitian form and unitarity A highest weight representation with a real value of has a unique Hermit
https://en.wikipedia.org/wiki/Spinor%20bundle	In differential geometry, given a spin structure on an -dimensional orientable Riemannian manifold one defines the spinor bundle to be the complex vector bundle associated to the corresponding principal bundle of spin frames over and the spin representation of its structure group on the space of spinors . A section of the spinor bundle is called a spinor field. Formal definition Let be a spin structure on a Riemannian manifold that is, an equivariant lift of the oriented orthonormal frame bundle with respect to the double covering of the special orthogonal group by the spin group. The spinor bundle is defined to be the complex vector bundle associated to the spin structure via the spin representation where denotes the group of unitary operators acting on a Hilbert space It is worth noting that the spin representation is a faithful and unitary representation of the group See also Clifford bundle Clifford module bundle Orthonormal frame bundle Spin geometry Spinor Spinor representation Notes Further reading \| Algebraic topology Riemannian geometry Structures on manifolds
https://en.wikipedia.org/wiki/Double%20cover	In mathematics, a double cover or double covering may refer to: Double cover (topology), a two-to-one mapping from one topological space to another. Frequently occurring special cases include The orientable double cover of a non-orientable manifold The bipartite double cover of an undirected graph G, formed by the graph tensor product G ×K2 A double covering group of a topological group such as a Lie group, a group extension of index two formed by a topological double cover. A double cover may also be used to refer to non-topological group extensions of index two, for instance extensions of finite groups. Doubling space, a possible property of a metric space. Cycle double cover, a collection of cycles in a graph that together include each edge twice. The cycle double cover conjecture is the unproven assertion that every bridgeless graph has a cycle double cover. Double coverage may also refer to: Double coverage, a defensive strategy in American football, basketball, and other sports Being covered by more than one health insurance plan A type of Asphalt roll roofing.
https://en.wikipedia.org/wiki/List%20of%20geometric%20topology%20topics	This is a list of geometric topology topics. Low-dimensional topology Knot theory Knot (mathematics) Link (knot theory) Wild knots Examples of knots Unknot Trefoil knot Figure-eight knot (mathematics) Borromean rings Types of knots Torus knot Prime knot Alternating knot Hyperbolic link Knot invariants Crossing number Linking number Skein relation Knot polynomials Alexander polynomial Jones polynomial Knot group Writhe Quandle Seifert surface Braids Braid theory Braid group Kirby calculus Surfaces Genus (mathematics) Examples Positive Euler characteristic 2-disk Sphere Real projective plane Zero Euler characteristic Annulus Möbius strip Torus Klein bottle Negative Euler characteristic The boundary of the pretzel is a genus three surface Embedded/Immersed in Euclidean space Cross-cap Boy's surface Roman surface Steiner surface Alexander horned sphere Klein bottle Mapping class group Dehn twist Nielsen–Thurston classification Three-manifolds Moise's Theorem (see also Hauptvermutung) Poincaré conjecture Thurston elliptization conjecture Thurston's geometrization conjecture Hyperbolic 3-manifolds Spherical 3-manifolds Euclidean 3-manifolds, Bieberbach Theorem, Flat manifolds, Crystallographic groups Seifert fiber space Heegaard splitting Waldhausen conjecture Compression body Handlebody Incompressible surface Dehn's lemma Loop theorem (aka the Disk theorem) Sphere theorem Haken manifold JSJ decomposition Branched surface Lamination Examples 3-sphere Torus bundles Surface bundles over the circle Graph manifolds Knot complements Whitehead manifold Invariants Fundamental group Heegaard genus tri-genus Analytic torsion Manifolds in general Orientable manifold Connected sum Jordan-Schönflies theorem Signature (topology) Handle decomposition Handlebody h-cobordism theorem s-cobordism theorem Manifold decomposition Hilbert-Smith conjecture Mapping class group Orbifolds Examples Exotic sphere Homology sphere Lens space I-bundle See also topology glossary List of topology topics List of general topology topics List of algebraic topology topics Publications in topology Mathematics-related lists Outlines of mathematics and logic Outlines
https://en.wikipedia.org/wiki/Morse%20theory	In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiable function on a manifold will reflect the topology quite directly. Morse theory allows one to find CW structures and handle decompositions on manifolds and to obtain substantial information about their homology. Before Morse, Arthur Cayley and James Clerk Maxwell had developed some of the ideas of Morse theory in the context of topography. Morse originally applied his theory to geodesics (critical points of the energy functional on the space of paths). These techniques were used in Raoul Bott's proof of his periodicity theorem. The analogue of Morse theory for complex manifolds is Picard–Lefschetz theory. Basic concepts To illustrate, consider a mountainous landscape surface (more generally, a manifold). If is the function giving the elevation of each point, then the inverse image of a point in is a contour line (more generally, a level set). Each connected component of a contour line is either a point, a simple closed curve, or a closed curve with a double point. Contour lines may also have points of higher order (triple points, etc.), but these are unstable and may be removed by a slight deformation of the landscape. Double points in contour lines occur at saddle points, or passes, where the surrounding landscape curves up in one direction and down in the other. Imagine flooding this landscape with water. When the water reaches elevation , the underwater surface is , the points with elevation or below. Consider how the topology of this surface changes as the water rises. It appears unchanged except when passes the height of a critical point, where the gradient of is (more generally, the Jacobian matrix acting as a linear map between tangent spaces does not have maximal rank). In other words, the topology of does not change except when the water either (1) starts filling a basin, (2) covers a saddle (a mountain pass), or (3) submerges a peak. To these three types of critical pointsbasins, passes, and peaks (i.e. minima, saddles, and maxima)one associates a number called the index, the number of independent directions in which decreases from the point. More precisely, the index of a non-degenerate critical point of is the dimension of the largest subspace of the tangent space to at on which the Hessian of is negative definite. The indices of basins, passes, and peaks are and respectively. Considering a more general surface, let be a torus oriented as in the picture, with again taking a point to its height above the plane. One can again analyze how the topology of the underwater surface changes as the water level rises. Starting from the bottom of the torus, let and be the four critical points of index and corresponding to the basin, two saddles, and peak, respectively.
https://en.wikipedia.org/wiki/List%20of%20order%20theory%20topics	Order theory is a branch of mathematics that studies various kinds of objects (often binary relations) that capture the intuitive notion of ordering, providing a framework for saying when one thing is "less than" or "precedes" another. An alphabetical list of many notions of order theory can be found in the order theory glossary. See also inequality, extreme value and mathematical optimization. Overview Partially ordered set Preorder Totally ordered set Total preorder Chain Trichotomy Extended real number line Antichain Strict order Hasse diagram Directed acyclic graph Duality (order theory) Product order Distinguished elements of partial orders Greatest element (maximum, top, unit), Least element (minimum, bottom, zero) Maximal element, minimal element Upper bound Least upper bound (supremum, join) Greatest lower bound (infimum, meet) Limit superior and limit inferior Irreducible element Prime element Compact element Subsets of partial orders Cofinal and coinitial set, sometimes also called dense Meet-dense set and join-dense set Linked set (upwards and downwards) Directed set (upwards and downwards) centered and σ-centered set Net (mathematics) Upper set and lower set Ideal and filter Ultrafilter Special types of partial orders Completeness (order theory) Dense order Distributivity (order theory) modular lattice distributive lattice completely distributive lattice Ascending chain condition Infinite descending chain Countable chain condition, often abbreviated as ccc Knaster's condition, sometimes denoted property (K) Well-orders Well-founded relation Ordinal number Well-quasi-ordering Completeness properties Semilattice Lattice (Directed) complete partial order, (d)cpo Bounded complete Complete lattice Knaster–Tarski theorem Infinite divisibility Orders with further algebraic operations Heyting algebra Relatively complemented lattice Complete Heyting algebra Pointless topology MV-algebra Ockham algebras: Stone algebra De Morgan algebra Kleene algebra (with involution) Łukasiewicz–Moisil algebra Boolean algebra (structure) Boolean ring Complete Boolean algebra Orthocomplemented lattice Quantale Orders in algebra Partially ordered monoid Ordered group Archimedean property Ordered ring Ordered field Artinian ring Noetherian Linearly ordered group Monomial order Weak order of permutations Bruhat order on a Coxeter group Incidence algebra Functions between partial orders Monotonic Pointwise order of functions Galois connection Order embedding Order isomorphism Closure operator Functions that preserve suprema/infima Completions and free constructions Dedekind completion Ideal completion Domain theory Way-below relation Continuous poset Continuous lattice Algebraic poset Scott domain Algebraic lattice Scott information system Powerdomain Scott topology Scott continuity Orders in mathematical logic Lindenbaum algebra Zorn's lemma Hausdorff maximality theorem Boolean prime ideal theorem Ultrafilter Ultrafilter lemma Tree (set the
https://en.wikipedia.org/wiki/Vandermonde%20matrix	In linear algebra, a Vandermonde matrix, named after Alexandre-Théophile Vandermonde, is a matrix with the terms of a geometric progression in each row: an matrix with entries , the jth power of the number , for all zero-based indices and . Most authors define the Vandermonde matrix as the transpose of the above matrix. The determinant of a square Vandermonde matrix (when ) is called a Vandermonde determinant or Vandermonde polynomial. Its value is: This is non-zero if and only if all are distinct (no two are equal), making the Vandermonde matrix invertible. Applications The polynomial interpolation problem is to find a polynomial which satisfies for given data points . This problem can be reformulated in terms of linear algebra by means of the Vandermonde matrix, as follows. computes the values of at the points via a matrix multiplication , where is the vector of coefficients and is the vector of values (both written as column vectors): If and are distinct, then V is a square matrix with non-zero determinant, i.e. an invertible matrix. Thus, given V and y, one can find the required by solving for its coefficients in the equation : . That is, the map from coefficients to values of polynomials is a bijective linear mapping with matrix V, and the interpolation problem has a unique solution. This result is called the unisolvence theorem, and is a special case of the Chinese remainder theorem for polynomials. In statistics, the equation means that the Vandermonde matrix is the design matrix of polynomial regression. In numerical analysis, solving the equation naïvely by Gaussian elimination results in an algorithm with time complexity O(n3). Exploiting the structure of the Vandermonde matrix, one can use Newton's divided differences method (or the Lagrange interpolation formula) to solve the equation in O(n2) time, which also gives the UL factorization of . The resulting algorithm produces extremely accurate solutions, even if is ill-conditioned. (See polynomial interpolation.) The Vandermonde determinant is used in the representation theory of the symmetric group. When the values belong to a finite field, the Vandermonde determinant is also called the Moore determinant, and has properties which are important in the theory of BCH codes and Reed–Solomon error correction codes. The discrete Fourier transform is defined by a specific Vandermonde matrix, the DFT matrix, where the are chosen to be nth roots of unity. The Fast Fourier transform computes the product of this matrix with a vector in O(n log2n) time. In the physical theory of the quantum Hall effect, the Vandermonde determinant shows that the Laughlin wavefunction with filling factor 1 is equal to a Slater determinant. This is no longer true for filling factors different from 1 in the fractional quantum Hall effect. In the geometry of polyhedra, the Vandermonde matrix gives the normalized volume of arbitrary -faces of cyclic polytopes. Specifically, if is a -fa
https://en.wikipedia.org/wiki/Mandelbrot	Mandelbrot may refer to: Benoit Mandelbrot (1924–2010), a mathematician associated with fractal geometry Mandelbrot set, a fractal popularized by Benoit Mandelbrot Mandelbrot Competition, a mathematics competition Mandelbrot (cookie), dessert associated with Eastern European Jews Szolem Mandelbrojt, a Polish-French mathematician Surnames of Jewish origin Yiddish-language surnames
https://en.wikipedia.org/wiki/Torsion-free	In mathematics, torsion-free may refer to: Abstract algebra Torsion-free group, a group whose only element of finite order is the identity Torsion-free module, module over an integral domain where zero is the only torsion element Torsion-free abelian group, an abelian group which is a torsion-free group Torsion-free rank, the cardinality of a maximal linearly independent subset of an abelian group or of a module over an integral domain Differential geometry Torsion-free affine connection, an affine connection whose torsion tensor vanishes Torsion-free metric connection or Levi-Civita connection, a unique symmetric connection on the tangent bundle of a manifold compatible with the metric See also Torsion (disambiguation)
https://en.wikipedia.org/wiki/Projective%20representation	In the field of representation theory in mathematics, a projective representation of a group G on a vector space V over a field F is a group homomorphism from G to the projective linear group where GL(V) is the general linear group of invertible linear transformations of V over F, and F∗ is the normal subgroup consisting of nonzero scalar multiples of the identity transformation (see Scalar transformation). In more concrete terms, a projective representation of is a collection of operators satisfying the homomorphism property up to a constant: for some constant . Equivalently, a projective representation of is a collection of operators , such that . Note that, in this notation, is a set of linear operators related by multiplication with some nonzero scalar. If it is possible to choose a particular representative in each family of operators in such a way that the homomorphism property is satisfied on the nose, rather than just up to a constant, then we say that can be "de-projectivized", or that can be "lifted to an ordinary representation". More concretely, we thus say that can be de-projectivized if there are for each such that . This possibility is discussed further below. Linear representations and projective representations One way in which a projective representation can arise is by taking a linear group representation of on and applying the quotient map which is the quotient by the subgroup of scalar transformations (diagonal matrices with all diagonal entries equal). The interest for algebra is in the process in the other direction: given a projective representation, try to 'lift' it to an ordinary linear representation. A general projective representation cannot be lifted to a linear representation , and the obstruction to this lifting can be understood via group cohomology, as described below. However, one can lift a projective representation of to a linear representation of a different group , which will be a central extension of . The group is the subgroup of defined as follows: , where is the quotient map of onto . Since is a homomorphism, it is easy to check that is, indeed, a subgroup of . If the original projective representation is faithful, then is isomorphic to the preimage in of . We can define a homomorphism by setting . The kernel of is: , which is contained in the center of . It is clear also that is surjective, so that is a central extension of . We can also define an ordinary representation of by setting . The ordinary representation of is a lift of the projective representation of in the sense that: . If is a perfect group there is a single universal perfect central extension of that can be used. Group cohomology The analysis of the lifting question involves group cohomology. Indeed, if one fixes for each in a lifted element in lifting from back to , the lifts then satisfy for some scalar in . It follows that the 2-cocycle or Schur multiplier satisfies the cocycle equ
https://en.wikipedia.org/wiki/Antichain	In mathematics, in the area of order theory, an antichain is a subset of a partially ordered set such that any two distinct elements in the subset are incomparable. The size of the largest antichain in a partially ordered set is known as its width. By Dilworth's theorem, this also equals the minimum number of chains (totally ordered subsets) into which the set can be partitioned. Dually, the height of the partially ordered set (the length of its longest chain) equals by Mirsky's theorem the minimum number of antichains into which the set can be partitioned. The family of all antichains in a finite partially ordered set can be given join and meet operations, making them into a distributive lattice. For the partially ordered system of all subsets of a finite set, ordered by set inclusion, the antichains are called Sperner families and their lattice is a free distributive lattice, with a Dedekind number of elements. More generally, counting the number of antichains of a finite partially ordered set is #P-complete. Definitions Let be a partially ordered set. Two elements and of a partially ordered set are called comparable if If two elements are not comparable, they are called incomparable; that is, and are incomparable if neither A chain in is a subset in which each pair of elements is comparable; that is, is totally ordered. An antichain in is a subset of in which each pair of different elements is incomparable; that is, there is no order relation between any two different elements in (However, some authors use the term "antichain" to mean strong antichain, a subset such that there is no element of the poset smaller than two distinct elements of the antichain.) Height and width A maximal antichain is an antichain that is not a proper subset of any other antichain. A maximum antichain is an antichain that has cardinality at least as large as every other antichain. The of a partially ordered set is the cardinality of a maximum antichain. Any antichain can intersect any chain in at most one element, so, if we can partition the elements of an order into chains then the width of the order must be at most (if the antichain has more than elements, by the pigeonhole principle, there would be 2 of its elements belonging to the same chain, a contradiction). Dilworth's theorem states that this bound can always be reached: there always exists an antichain, and a partition of the elements into chains, such that the number of chains equals the number of elements in the antichain, which must therefore also equal the width. Similarly, one can define the of a partial order to be the maximum cardinality of a chain. Mirsky's theorem states that in any partial order of finite height, the height equals the smallest number of antichains into which the order may be partitioned. Sperner families An antichain in the inclusion ordering of subsets of an -element set is known as a Sperner family. The number of different Sperner families is counte
https://en.wikipedia.org/wiki/Lindemann%E2%80%93Weierstrass%20theorem	In transcendental number theory, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers. It states the following: In other words, the extension field has transcendence degree over . An equivalent formulation , is the following: This equivalence transforms a linear relation over the algebraic numbers into an algebraic relation over by using the fact that a symmetric polynomial whose arguments are all conjugates of one another gives a rational number. The theorem is named for Ferdinand von Lindemann and Karl Weierstrass. Lindemann proved in 1882 that is transcendental for every non-zero algebraic number thereby establishing that is transcendental (see below). Weierstrass proved the above more general statement in 1885. The theorem, along with the Gelfond–Schneider theorem, is extended by Baker's theorem, and all of these would be further generalized by Schanuel's conjecture. Naming convention The theorem is also known variously as the Hermite–Lindemann theorem and the Hermite–Lindemann–Weierstrass theorem. Charles Hermite first proved the simpler theorem where the exponents are required to be rational integers and linear independence is only assured over the rational integers, a result sometimes referred to as Hermite's theorem. Although that appears to be a special case of the above theorem, the general result can be reduced to this simpler case. Lindemann was the first to allow algebraic numbers into Hermite's work in 1882. Shortly afterwards Weierstrass obtained the full result, and further simplifications have been made by several mathematicians, most notably by David Hilbert and Paul Gordan. Transcendence of and The transcendence of and are direct corollaries of this theorem. Suppose is a non-zero algebraic number; then is a linearly independent set over the rationals, and therefore by the first formulation of the theorem is an algebraically independent set; or in other words is transcendental. In particular, is transcendental. (A more elementary proof that is transcendental is outlined in the article on transcendental numbers.) Alternatively, by the second formulation of the theorem, if is a non-zero algebraic number, then is a set of distinct algebraic numbers, and so the set is linearly independent over the algebraic numbers and in particular cannot be algebraic and so it is transcendental. To prove that is transcendental, we prove that it is not algebraic. If were algebraic, i would be algebraic as well, and then by the Lindemann–Weierstrass theorem (see Euler's identity) would be transcendental, a contradiction. Therefore is not algebraic, which means that it is transcendental. A slight variant on the same proof will show that if is a non-zero algebraic number then and their hyperbolic counterparts are also transcendental. -adic conjecture Modular conjecture An analogue of the theorem involving the modular function was conjectured by Dan
https://en.wikipedia.org/wiki/Linearly%20ordered%20group	In mathematics, specifically abstract algebra, a linearly ordered or totally ordered group is a group G equipped with a total order "≤" that is translation-invariant. This may have different meanings. We say that (G, ≤) is a: left-ordered group if ≤ is left-invariant, that is a ≤ b implies ca ≤ cb for all a, b, c in G, right-ordered group if ≤ is right-invariant, that is a ≤ b implies ac ≤ bc for all a, b, c in G, bi-ordered group if ≤ is bi-invariant, that is it is both left- and right-invariant. A group G is said to be left-orderable (or right-orderable, or bi-orderable) if there exists a left- (or right-, or bi-) invariant order on G. A simple necessary condition for a group to be left-orderable is to have no elements of finite order; however this is not a sufficient condition. It is equivalent for a group to be left- or right-orderable; however there exist left-orderable groups which are not bi-orderable. Further definitions In this section is a left-invariant order on a group with identity element . All that is said applies to right-invariant orders with the obvious modifications. Note that being left-invariant is equivalent to the order defined by if and only if being right-invariant. In particular a group being left-orderable is the same as it being right-orderable. In analogy with ordinary numbers we call an element of an ordered group positive if . The set of positive elements in an ordered group is called the positive cone, it is often denoted with ; the slightly different notation is used for the positive cone together with the identity element. The positive cone characterises the order ; indeed, by left-invariance we see that if and only if . In fact a left-ordered group can be defined as a group together with a subset satisfying the two conditions that: for we have also ; let , then is the disjoint union of and . The order associated with is defined by ; the first condition amounts to left-invariance and the second to the order being well-defined and total. The positive cone of is . The left-invariant order is bi-invariant if and only if it is conjugacy invariant, that is if then for any we have as well. This is equivalent to the positive cone being stable under inner automorphisms. If , then the absolute value of , denoted by , is defined to be: If in addition the group is abelian, then for any a triangle inequality is satisfied: . Examples Any left- or right-orderable group is torsion-free, that is it contains no elements of finite order besides the identity. Conversely, F. W. Levi showed that a torsion-free abelian group is bi-orderable; this is still true for nilpotent groups but there exist torsion-free, finitely presented groups which are not left-orderable. Archimedean ordered groups Otto Hölder showed that every Archimedean group (a bi-ordered group satisfying an Archimedean property) is isomorphic to a subgroup of the additive group of real numbers, . If we write the Archimedean
https://en.wikipedia.org/wiki/Guido%20Fubini	Guido Fubini (19 January 1879 – 6 June 1943) was an Italian mathematician, known for Fubini's theorem and the Fubini–Study metric. Life Born in Venice, he was steered towards mathematics at an early age by his teachers and his father, who was himself a teacher of mathematics. In 1896 he entered the Scuola Normale Superiore di Pisa, where he studied differential geometry under Ulisse Dini and Luigi Bianchi. His 1900 doctoral thesis was about Clifford's parallelism in elliptic spaces. After earning his doctorate, he took up a series of professorships. In 1901 he began teaching at the University of Catania in Sicily; shortly afterwards he moved to the University of Genoa; and in 1908 he moved to the Politecnico in Turin and then the University of Turin, where he stayed for a few decades. During this time his research focused primarily on topics in mathematical analysis, especially differential equations, functional analysis, and complex analysis; but he also studied the calculus of variations, group theory, non-Euclidean geometry, and projective geometry, among other topics. With the outbreak of World War I, he shifted his work towards more applied topics, studying the accuracy of artillery fire; after the war, he continued in an applied direction, applying results from this work to problems in electrical circuits and acoustics. In 1938, when Fubini at the age of 59 was nearing retirement, Benito Mussolini's Fascists adopted the anti-Jewish policies advocated for several years by Adolf Hitler's Nazis. As a Jew, Fubini feared for the safety of his family, and so accepted an invitation by Princeton University to teach there; he died in New York City four years later. Legacy A main-belt asteroid, 22495 Fubini, was named in his honour. Publications 1920: Lezioni di analisi matematica (Società Tipografico-Editrice Nazionale, Torino) References . The "Proceedings of the mathematical conference for the celebration of the centenary of the birth of Guido Fubini and Francesco Severi", including several research as well as historical papers describing the contributions of Guido Fubini and Francesco Severi to various branches of pure and applied mathematics: the conference was held on 8–10 October 1979 at the Accademia delle Scienze di Torino. . In this paper Gaetano Fichera describes the main contributions of the two scientists to the Cauchy and the Dirichlet problem for holomorphic functions of several complex variables, as well as the impact of their work on subsequent researches. . In this paper Dionigi Galletto describes the main contributions of the two scientists to the theory of special and general relativity. External links 1879 births 1943 deaths 19th-century Italian mathematicians 20th-century Italian Jews 20th-century Italian mathematicians Differential geometers Mathematical analysts Scientists from Venice Italian emigrants to the United States Princeton University faculty
https://en.wikipedia.org/wiki/Iterated%20integral	In multivariable calculus, an iterated integral is the result of applying integrals to a function of more than one variable (for example or ) in such a way that each of the integrals considers some of the variables as given constants. For example, the function , if is considered a given parameter, can be integrated with respect to , . The result is a function of and therefore its integral can be considered. If this is done, the result is the iterated integral It is key for the notion of iterated integrals that this is different, in principle, from the multiple integral In general, although these two can be different, Fubini's theorem states that under specific conditions, they are equivalent. The alternative notation for iterated integrals is also used. In the notation that uses parentheses, iterated integrals are computed following the operational order indicated by the parentheses starting from the most inner integral outside. In the alternative notation, writing , the innermost integrand is computed first. Examples A simple computation For the iterated integral the integral is computed first and then the result is used to compute the integral with respect to y. This example omits the constants of integration. After the first integration with respect to x, we would rigorously need to introduce a "constant" function of y. That is, If we were to differentiate this function with respect to x, any terms containing only y would vanish, leaving the original integrand. Similarly for the second integral, we would introduce a "constant" function of x, because we have integrated with respect to y. In this way, indefinite integration does not make very much sense for functions of several variables. The order is important The order in which the integrals are computed is important in iterated integrals, particularly when the integrand is not continuous on the domain of integration. Examples in which the different orders lead to different results are usually for complicated functions as the one that follows. Define the sequence such that . Let be a sequence of continuous functions not vanishing in the interval and zero elsewhere, such that for every . Define In the previous sum, at each specific , at most one term is different from zero. For this function it happens that See also References Integrals
https://en.wikipedia.org/wiki/1907%20in%20science	The year 1907 in science and technology involved some significant events, listed below. Mathematics Paul Koebe conjectures the result of the Koebe quarter theorem. Physics The Ehrenfest model of diffusion is proposed by Tatiana and Paul Ehrenfest to explain the second law of thermodynamics. Albert Einstein introduces the principle of equivalence of gravitation and inertia and uses it to predict the gravitational redshift. Chemistry June 6 – Persil laundry detergent is first marketed by Henkel of Düsseldorf, Germany, the first to combine a bleaching agent (sodium perborate) with a base washing agent (sodium silicate) commercially. Emil Fischer artificially synthesizes peptide amino acid chains and thereby shows that amino acids in proteins are connected by amino group-acid group bonds. Hermann Staudinger prepares the first synthetic β-lactam. Georges Urbain discovers Lutetium (from Lutetia, the ancient name of Paris). Geology January 14 – 1907 Kingston earthquake: Earthquake in Kingston, Jamaica. c. March 28 – Volcanic eruption of Ksudach in the Kamchatka Peninsula. Bertram Boltwood proposes that the amount of lead in uranium and thorium ores might be used to determine the Earth's age and crudely dates some rocks to have ages between 410–2200 million years. The Moine Thrust Belt in Scotland is identified by Ben Peach and John Horne, one of the first to be discovered. The rare phosphate mineral tarbuttite is first discovered at Broken Hill, Barotziland-North-Western Rhodesia. Ludovic Mrazek describes and names diapirs. Medicine Paul Ehrlich develops a chemotherapeutic cure for sleeping sickness. George Soper identifies "Typhoid Mary" Mallon as an asymptomatic carrier of typhoid in New York. Dengue fever becomes the second disease shown to be caused by a virus. Indiana, in the United States, becomes the world's first legislature to place laws permitting compulsory sterilization for eugenic purposes on the statute book. Paleontology October 21 – Jaw of Homo heidelbergensis (Mauer 1) found. Astronomy Alfred Russel Wallace publishes the book Is Mars Habitable?, a refutation of Percival Lowell's theory of mars canals, and the first work in the emerging field of astrobiology Psychology Ivan Pavlov demonstrates conditioned responses with salivating dogs. Vladimir Bekhterev begins publication of Objective Psychology. Technology August 10 – Peking to Paris motor race concludes after 2 months, won by Prince Scipione Borghese driving a 7-litre 35/45 hp Itala. August 29 – The partially completed Quebec Bridge collapses. October 17 – Guglielmo Marconi initiates commercial transatlantic radio communications between his high power longwave wireless telegraphy stations in Clifden, Ireland, and Glace Bay, Nova Scotia. Lee de Forest invents the triode thermionic amplifier, starting the development of electronics as a practical technology. Furuholmen Lighthouse in Sweden is the world's first to be equipped with AGA's Dalén light inc
https://en.wikipedia.org/wiki/Fotini%20Markopoulou-Kalamara	Fotini G. Markopoulou-Kalamara (; born April 3, 1971) is a Greek theoretical physicist interested in quantum gravity, foundational mathematics, quantum mechanics and a design engineer working on embodied cognition technologies. Markopoulou is co-founder and CEO of Empathic Technologies. She was a founding faculty member at Perimeter Institute for Theoretical Physics and was an adjunct professor at the University of Waterloo. Quantum gravity Markopoulou received her PhD from Imperial College London in 1998 and held postdoctoral positions at the Max Planck Institute for Gravitational Physics, Imperial College London, and Pennsylvania State University. She shared First Prize in the Young Researchers competition at the Ultimate Reality Symposium in Princeton, New Jersey. She has been influenced by researchers such as Christopher Isham who call attention to the unstated assumption in most modern physics that physical properties are most naturally calibrated by a real-number continuum. She, and others, attempt to make explicit some of the implicit mathematical assumptions underpinning modern theoretical physics and cosmology. In her interdisciplinary paper "The Internal Description of a Causal Set: What the Universe Looks Like from the Inside", Markopoulou instantiates some abstract terms from mathematical category theory to develop straightforward models of space-time. It proposes simple quantum models of space-time based on category-theoretic notions of a topos and its subobject classifier (which has a Heyting algebra structure, but not necessarily a Boolean algebra structure). For example, hard-to-picture category-theoretic "presheaves" from topos theory become easy-to-picture "evolving (or varying) sets" in her discussions of quantum spacetime. The diagrams in Markopoulou's papers (including hand-drawn diagrams in one of the earlier versions of "The Internal Description of a Causal Set") are straightforward presentations of possible models of space-time. They are intended as meaningful and provocative, not just for specialists but also for newcomers. In May 2006, Markopoulou published a paper with Lee Smolin that further popularized this Causal dynamical triangulation (CDT) theory by explaining the time-slicing of the Ambjorn–Loll CDT model as a result of gauge fixing. Their approach relaxed the definition of the Ambjorn–Loll CDT model in 1 + 1 dimensions to allow for a varying lapse. Quantum graphity In 2008, Markopoulou, Tomasz Konopka, Mohammad H. Ansari, and Simone Severini initiated the study of a new background independent model of evolutionary space called quantum graphity. In the quantum graphity model, points in spacetime are represented by nodes on a graph connected by links that can be on or off. This indicates whether or not the two points are directly connected as if they are next to each other in spacetime. When they are on the links have additional state variables which are used to define the random dynamics of the graph unde
https://en.wikipedia.org/wiki/Group%20ring	In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the given group. As a ring, its addition law is that of the free module and its multiplication extends "by linearity" the given group law on the basis. Less formally, a group ring is a generalization of a given group, by attaching to each element of the group a "weighting factor" from a given ring. If the ring is commutative then the group ring is also referred to as a group algebra, for it is indeed an algebra over the given ring. A group algebra over a field has a further structure of a Hopf algebra; in this case, it is thus called a group Hopf algebra. The apparatus of group rings is especially useful in the theory of group representations. Definition Let be a group, written multiplicatively, and let be a ring. The group ring of over , which we will denote by , or simply , is the set of mappings of finite support ( is nonzero for only finitely many elements ), where the module scalar product of a scalar in and a mapping is defined as the mapping , and the module group sum of two mappings and is defined as the mapping . To turn the additive group into a ring, we define the product of and to be the mapping The summation is legitimate because and are of finite support, and the ring axioms are readily verified. Some variations in the notation and terminology are in use. In particular, the mappings such as are sometimes written as what are called "formal linear combinations of elements of with coefficients in ": or simply Note that if the ring is in fact a field , then the module structure of the group ring is in fact a vector space over . Examples 1. Let , the cyclic group of order 3, with generator and identity element 1G. An element r of C[G] can be written as where z0, z1 and z2 are in C, the complex numbers. This is the same thing as a polynomial ring in variable such that i.e. C[G] is isomorphic to the ring C[]/. Writing a different element s as , their sum is and their product is Notice that the identity element 1G of G induces a canonical embedding of the coefficient ring (in this case C) into C[G]; however strictly speaking the multiplicative identity element of C[G] is 1⋅1G where the first 1 comes from C and the second from G. The additive identity element is zero. When G is a non-commutative group, one must be careful to preserve the order of the group elements (and not accidentally commute them) when multiplying the terms. 2. A different example is that of the Laurent polynomials over a ring R: these are nothing more or less than the group ring of the infinite cyclic group Z over R. 3. Let Q be the quaternion group with elements . Consider the group ring RQ, where R is the set of real numbers. An arbitrary element of this group ring is of the form where is a real
https://en.wikipedia.org/wiki/Diagonal	In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word diagonal derives from the ancient Greek διαγώνιος diagonios, "from angle to angle" (from διά- dia-, "through", "across" and γωνία gonia, "angle", related to gony "knee"); it was used by both Strabo and Euclid to refer to a line connecting two vertices of a rhombus or cuboid, and later adopted into Latin as diagonus ("slanting line"). In matrix algebra, the diagonal of a square matrix consists of the entries on the line from the top left corner to the bottom right corner. There are also many other non-mathematical uses. Non-mathematical uses In engineering, a diagonal brace is a beam used to brace a rectangular structure (such as scaffolding) to withstand strong forces pushing into it; although called a diagonal, due to practical considerations diagonal braces are often not connected to the corners of the rectangle. Diagonal pliers are wire-cutting pliers defined by the cutting edges of the jaws intersects the joint rivet at an angle or "on a diagonal", hence the name. A diagonal lashing is a type of lashing used to bind spars or poles together applied so that the lashings cross over the poles at an angle. In association football, the diagonal system of control is the method referees and assistant referees use to position themselves in one of the four quadrants of the pitch. Polygons As applied to a polygon, a diagonal is a line segment joining any two non-consecutive vertices. Therefore, a quadrilateral has two diagonals, joining opposite pairs of vertices. For any convex polygon, all the diagonals are inside the polygon, but for re-entrant polygons, some diagonals are outside of the polygon. Any n-sided polygon (n ≥ 3), convex or concave, has total diagonals, as each vertex has diagonals to all other vertices except itself and the two adjacent vertices, or n − 3 diagonals, and each diagonal is shared by two vertices. In general, a regular n-sided polygon has distinct diagonals in length, which follows the pattern 1,1,2,2,3,3... starting from a square. Regions formed by diagonals In a convex polygon, if no three diagonals are concurrent at a single point in the interior, the number of regions that the diagonals divide the interior into is given by For n-gons with n=3, 4, ... the number of regions is 1, 4, 11, 25, 50, 91, 154, 246... This is OEIS sequence A006522. Intersections of diagonals If no three diagonals of a convex polygon are concurrent at a point in the interior, the number of interior intersections of diagonals is given by . This holds, for example, for any regular polygon with an odd number of sides. The formula follows from the fact that each intersection is uniquely determined by the four endpoints of the two intersecting diagonals: the number of intersections is thus the number of combinations of the n vertices four a
https://en.wikipedia.org/wiki/Integral%20geometry	In mathematics, integral geometry is the theory of measures on a geometrical space invariant under the symmetry group of that space. In more recent times, the meaning has been broadened to include a view of invariant (or equivariant) transformations from the space of functions on one geometrical space to the space of functions on another geometrical space. Such transformations often take the form of integral transforms such as the Radon transform and its generalizations. Classical context Integral geometry as such first emerged as an attempt to refine certain statements of geometric probability theory. The early work of Luis Santaló and Wilhelm Blaschke was in this connection. It follows from the classic theorem of Crofton expressing the length of a plane curve as an expectation of the number of intersections with a random line. Here the word 'random' must be interpreted as subject to correct symmetry considerations. There is a sample space of lines, one on which the affine group of the plane acts. A probability measure is sought on this space, invariant under the symmetry group. If, as in this case, we can find a unique such invariant measure, then that solves the problem of formulating accurately what 'random line' means and expectations become integrals with respect to that measure. (Note for example that the phrase 'random chord of a circle' can be used to construct some paradoxes—for example Bertrand's paradox.) We can therefore say that integral geometry in this sense is the application of probability theory (as axiomatized by Kolmogorov) in the context of the Erlangen programme of Klein. The content of the theory is effectively that of invariant (smooth) measures on (preferably compact) homogeneous spaces of Lie groups; and the evaluation of integrals of the differential forms. A very celebrated case is the problem of Buffon's needle: drop a needle on a floor made of planks and calculate the probability the needle lies across a crack. Generalising, this theory is applied to various stochastic processes concerned with geometric and incidence questions. See stochastic geometry. 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 Helgason and Israel Gelfand. It deals more specifically with integral transforms, modeled on the Radon transform. Here the underlying geometrical incidence relation (points lying on lines, in Crofton's case) is seen in a freer light, as the site for an integral transform composed as pullback onto the incidence graph and then push forward. Notes Further reading Sors, Luis Antonio Santaló, and Luis A. Santaló. Integral geometry and geometric probability. Cambridge university press, 2004. A systematic exposition of the theory and
https://en.wikipedia.org/wiki/Spherical%20circle	In spherical geometry, a spherical circle (often shortened to circle) is the locus of points on a sphere at constant spherical distance (the spherical radius) from a given point on the sphere (the pole or spherical center). It is a curve of constant geodesic curvature relative to the sphere, analogous to a line or circle in the Euclidean plane; the curves analogous to straight lines are called great circles, and the curves analogous to planar circles are called small circles or lesser circles. Fundamental concepts Intrinsic characterization A spherical circle with zero geodesic curvature is called a great circle, and is a geodesic analogous to a straight line in the plane. A great circle separates the sphere into two equal hemispheres, each with the great circle as its boundary. If a great circle passes through a point on the sphere, it also passes through the antipodal point (the unique furthest other point on the sphere). For any pair of distinct non-antipodal points, a unique great circle passes through both. Any two points on a great circle separate it into two arcs analogous to line segments in the plane; the shorter is called the minor arc and is the shortest path between the points, and the longer is called the major arc. A circle with non-zero geodesic curvature is called a small circle, and is analogous to a circle in the plane. A small circle separates the sphere into two spherical disks or spherical caps, each with the circle as its boundary. For any triple of distinct non-antipodal points a unique small circle passes through all three. Any two points on the small circle separate it into two arcs, analogous to circular arcs in the plane. Every circle has two antipodal poles (or centers) intrinsic to the sphere. A great circle is equidistant to its poles, while a small circle is closer to one pole than the other. Concentric circles are sometimes called parallels, because they each have constant distance to each-other, and in particular to their concentric great circle, and are in that sense analogous to parallel lines in the plane. Extrinsic characterization If the sphere is isometrically embedded in Euclidean space, the sphere's intersection with a plane is a circle, which can be interpreted extrinsically to the sphere as a Euclidean circle: a locus of points in the plane at a constant Euclidean distance (the extrinsic radius) from a point in the plane (the extrinsic center). A great circle lies on a plane passing through the center of the sphere, so its extrinsic radius is equal to the radius of the sphere itself, and its extrinsic center is the sphere's center. A small circle lies on a plane not passing through the sphere's center, so its extrinsic radius is smaller than that of the sphere and its extrinsic center is an arbitrary point in the interior of the sphere. Parallel planes cut the sphere into parallel (concentric) small circles; the pair of parallel planes tangent to the sphere are tangent at the poles of these circ
https://en.wikipedia.org/wiki/Farey%20sequence	In mathematics, the Farey sequence of order n is the sequence of completely reduced fractions, either between 0 and 1, or without this restriction, which when in lowest terms have denominators less than or equal to n, arranged in order of increasing size. With the restricted definition, each Farey sequence starts with the value 0, denoted by the fraction , and ends with the value 1, denoted by the fraction (although some authors omit these terms). A Farey sequence is sometimes called a Farey series, which is not strictly correct, because the terms are not summed. Examples The Farey sequences of orders 1 to 8 are : F1 = { , } F2 = { , , } F3 = { , , , , } F4 = { , , , , , , } F5 = { , , , , , , , , , , } F6 = { , , , , , , , , , , , , } F7 = { , , , , , , , , , , , , , , , , , , } F8 = { , , , , , , , , , , , , , , , , , , , , , , } Farey sunburst Plotting the numerators versus the denominators of a Farey sequence gives a shape like the one to the right, shown for 6. Reflecting this shape around the diagonal and main axes generates the Farey sunburst, shown below. The Farey sunburst of order connects the visible integer grid points from the origin in the square of side 2, centered at the origin. Using Pick's theorem, the area of the sunburst is 4(\|n\|−1), where \|n\| is the number of fractions in n. History The history of 'Farey series' is very curious — Hardy & Wright (1979) ... once again the man whose name was given to a mathematical relation was not the original discoverer so far as the records go. — Beiler (1964) Farey sequences are named after the British geologist John Farey, Sr., whose letter about these sequences was published in the Philosophical Magazine in 1816. Farey conjectured, without offering proof, that each new term in a Farey sequence expansion is the mediant of its neighbours. Farey's letter was read by Cauchy, who provided a proof in his Exercices de mathématique, and attributed this result to Farey. In fact, another mathematician, Charles Haros, had published similar results in 1802 which were not known either to Farey or to Cauchy. Thus it was a historical accident that linked Farey's name with these sequences. This is an example of Stigler's law of eponymy. Properties Sequence length and index of a fraction The Farey sequence of order n contains all of the members of the Farey sequences of lower orders. In particular Fn contains all of the members of Fn−1 and also contains an additional fraction for each number that is less than n and coprime to n. Thus F6 consists of F5 together with the fractions and . The middle term of a Farey sequence Fn is always , for n > 1. From this, we can relate the lengths of Fn and Fn−1 using Euler's totient function : Using the fact that \|F1\| = 2, we can derive an expression for the length of Fn: where is the summatory totient. We also have : and by a Möbius inversion formula : where µ(d) is the number-theoretic Möbius function, and is the floor function. The a
https://en.wikipedia.org/wiki/Outline%20of%20combinatorics	Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Essence of combinatorics Matroid Greedoid Ramsey theory Van der Waerden's theorem Hales–Jewett theorem Umbral calculus, binomial type polynomial sequences Combinatorial species Branches of combinatorics Algebraic combinatorics Analytic combinatorics Arithmetic combinatorics Combinatorics on words Combinatorial design theory Enumerative combinatorics Extremal combinatorics Geometric combinatorics Graph theory Infinitary combinatorics Matroid theory Order theory Partition theory Probabilistic combinatorics Topological combinatorics Multi-disciplinary fields that include combinatorics Coding theory Combinatorial optimization Combinatorics and dynamical systems Combinatorics and physics Discrete geometry Finite geometry Phylogenetics History of combinatorics History of combinatorics General combinatorial principles and methods Combinatorial principles Trial and error, brute-force search, bogosort, British Museum algorithm Pigeonhole principle Method of distinguished element Mathematical induction Recurrence relation, telescoping series Generating functions as an application of formal power series Cyclic sieving Schrödinger method Exponential generating function Stanley's reciprocity theorem Binomial coefficients and their properties Combinatorial proof Double counting (proof technique) Bijective proof Inclusion–exclusion principle Möbius inversion formula Parity, even and odd permutations Combinatorial Nullstellensatz Incidence algebra Greedy algorithm Divide and conquer algorithm Akra–Bazzi method Dynamic programming Branch and bound Birthday attack, birthday paradox Floyd's cycle-finding algorithm Reduction to linear algebra Sparsity Weight function Minimax algorithm Alpha–beta pruning Probabilistic method Sieve methods Analytic combinatorics Symbolic combinatorics Combinatorial class Exponential formula Twelvefold way MacMahon Master theorem Data structure concepts Data structure Data type Abstract data type Algebraic data type Composite type Array Associative array Deque List Linked list Queue Priority queue Skip list Stack Tree data structure Automatic garbage collection Problem solving as an art Heuristic Inductive reasoning How to Solve It Creative problem solving Morphological analysis (problem-solving) Living with large numbers Names of large numbers, long scale History of large numbers Graham's number Moser's number Skewes' number Large number notations Conway chained arrow notation Hyper4 Knuth's up-arrow notation Moser polygon notation Steinhaus polygon notation Large number effects Exponential growth Combinatorial explosion Branching factor Granularity Curse of dimensionality Concentration of measure Persons influential in the field of combinatorics Noga Alon George Andrews József Beck Eric Tem
https://en.wikipedia.org/wiki/Logicism	In the philosophy of mathematics, logicism is a programme comprising one or more of the theses that — for some coherent meaning of 'logic' — mathematics is an extension of logic, some or all of mathematics is reducible to logic, or some or all of mathematics may be modelled in logic. Bertrand Russell and Alfred North Whitehead championed this programme, initiated by Gottlob Frege and subsequently developed by Richard Dedekind and Giuseppe Peano. Overview Dedekind's path to logicism had a turning point when he was able to construct a model satisfying the axioms characterizing the real numbers using certain sets of rational numbers. This and related ideas convinced him that arithmetic, algebra and analysis were reducible to the natural numbers plus a "logic" of classes. Furthermore by 1872 he had concluded that the naturals themselves were reducible to sets and mappings. It is likely that other logicists, most importantly Frege, were also guided by the new theories of the real numbers published in the year 1872. The philosophical impetus behind Frege's logicist programme from the Grundlagen der Arithmetik onwards was in part his dissatisfaction with the epistemological and ontological commitments of then-extant accounts of the natural numbers, and his conviction that Kant's use of truths about the natural numbers as examples of synthetic a priori truth was incorrect. This started a period of expansion for logicism, with Dedekind and Frege as its main exponents. However, this initial phase of the logicist programme was brought into crisis with the discovery of the classical paradoxes of set theory (Cantor 1896, Zermelo and Russell 1900–1901). Frege gave up on the project after Russell recognized and communicated his paradox identifying an inconsistency in Frege's system set out in the Grundgesetze der Arithmetik. Note that naive set theory also suffers from this difficulty. On the other hand, Russell wrote The Principles of Mathematics in 1903 using the paradox and developments of Giuseppe Peano's school of geometry. Since he treated the subject of primitive notions in geometry and set theory, this text is a watershed in the development of logicism. Evidence of the assertion of logicism was collected by Russell and Whitehead in their Principia Mathematica. 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
https://en.wikipedia.org/wiki/Laplace%20transform%20applied%20to%20differential%20equations	In mathematics, the Laplace transform is a powerful integral transform used to switch a function from the time domain to the s-domain. The Laplace transform can be used in some cases to solve linear differential equations with given initial conditions. First consider the following property of the Laplace transform: One can prove by induction that Now we consider the following differential equation: with given initial conditions Using the linearity of the Laplace transform it is equivalent to rewrite the equation as obtaining Solving the equation for and substituting with one obtains The solution for f(t) is obtained by applying the inverse Laplace transform to Note that if the initial conditions are all zero, i.e. then the formula simplifies to An example We want to solve with initial conditions f(0) = 0 and f′(0)=0. We note that and we get The equation is then equivalent to We deduce Now we apply the Laplace inverse transform to get Bibliography A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, Boca Raton, 2002. Integral transforms Differential equations Differential calculus Ordinary differential equations

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