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https://en.wikipedia.org/wiki/List%20of%20geometry%20topics	This is a list of geometry topics. Types, methodologies, and terminologies of geometry. Absolute geometry Affine geometry Algebraic geometry Analytic geometry Archimedes' use of infinitesimals Birational geometry Complex geometry Combinatorial geometry Computational geometry Conformal geometry Constructive solid geometry Contact geometry Convex geometry Descriptive geometry Differential geometry Digital geometry Discrete geometry Distance geometry Elliptic geometry Enumerative geometry Epipolar geometry Finite geometry Fractal geometry Geometry of numbers Hyperbolic geometry Incidence geometry Information geometry Integral geometry Inversive geometry Inversive ring geometry Klein geometry Lie sphere geometry Non-Euclidean geometry Noncommutative algebraic geometry Noncommutative geometry Numerical geometry Ordered geometry Parabolic geometry Plane geometry Projective geometry Quantum geometry Riemannian geometry Ruppeiner geometry Spherical geometry Symplectic geometry Synthetic geometry Systolic geometry Taxicab geometry Toric geometry Transformation geometry Tropical geometry Euclidean geometry foundations Hilbert's axioms Point Locus Line Parallel Angle Concurrent lines Adjacent angles Central angle Complementary angles Inscribed angle Internal angle Supplementary angles Angle trisection Congruence Reflection Rotation Coordinate rotations and reflections Translation Glide reflection Similarity Similarity transformation Homothety Shear mapping Euclidean plane geometry 2D computer graphics 2D geometric model Altitude Brahmagupta's formula Bretschneider's formula Compass and straightedge constructions Squaring the circle Complex geometry Conic section Focus Circle List of circle topics Thales' theorem Circumcircle Concyclic Incircle and excircles of a triangle Orthocentric system Monge's theorem Power center Nine-point circle Circle points segments proof Mrs. Miniver's problem Isoperimetric theorem Annulus Ptolemaios' theorem Steiner chain Eccentricity Ellipse Semi-major axis Hyperbola Parabola Matrix representation of conic sections Dandelin spheres Curve of constant width Reuleaux triangle Frieze group Golden angle Holditch's theorem Interactive geometry software Involutes Goat grazing problem Parallel postulate Polygon Star polygon Pick's theorem Shape dissection Bolyai–Gerwien theorem Poncelet–Steiner theorem Polygon triangulation Pons asinorum Quadrilateral Bicentric quadrilateral Cyclic quadrilateral Equidiagonal quadrilateral Kite (geometry) Orthodiagonal quadrilateral Rhombus Rectangle Square Tangential quadrilateral Trapezoid Isosceles trapezoid Sangaku Straightedge Symmedian Tessellation Prototile Aperiodic tiling Wang tile Penrose tiling Trapezoid (trapezium) Isosceles trapezoid Triangle Acute and obtuse triangles Equilateral triangle Euler's line Heron's formula Integer triangle H
https://en.wikipedia.org/wiki/Orthonormal%20frame	In Riemannian geometry and relativity theory, an orthonormal frame is a tool for studying the structure of a differentiable manifold equipped with a metric. If M is a manifold equipped with a metric g, then an orthonormal frame at a point P of M is an ordered basis of the tangent space at P consisting of vectors which are orthonormal with respect to the bilinear form gP. See also Frame (linear algebra) Frame bundle k-frame Moving frame Frame fields in general relativity References Riemannian geometry
https://en.wikipedia.org/wiki/Coherence%20%28philosophical%20gambling%20strategy%29	In a thought experiment proposed by the Italian probabilist Bruno de Finetti in order to justify Bayesian probability, an array of wagers is coherent precisely if it does not expose the wagerer to certain loss regardless of the outcomes of events on which they are wagering, even if their opponent makes the most judicious choices. Operational subjective probabilities as wagering odds One must set the price of a promise to pay $1 if John Smith wins tomorrow's election, and $0 otherwise. One knows that one's opponent will be able to choose either to buy such a promise from one at the price one has set, or require one to buy such a promise from them, still at the same price. In other words: Player A sets the odds, but Player B decides which side of the bet to take. The price one sets is the "operational subjective probability" that one assigns to the proposition on which one is betting. If one decides that John Smith is 12.5% likely to win—an arbitrary valuation—one might then set an odds of 7:1 against. This arbitrary valuation — the "operational subjective probability" — determines the payoff to a successful wager. $1 wagered at these odds will produce either a loss of $1 (if Smith loses) or a win of $7 (if Smith wins). If the $1 is placed in pledge as a condition of the bet, then the $1 will also be returned to the bettor, should the bettor win the bet. Dutch books A person who has set prices on an array of wagers, in such a way that he or she will make a net gain regardless of the outcome, is said to have made a Dutch book. When one has a Dutch book, one's opponent always loses. A person who sets prices in a way that gives his or her opponent a Dutch book is not behaving rationally. So the following Dutch book arguments show that rational agents must hold subjective probabilities that follow the common laws of probability. A very trivial Dutch book The rules do not forbid a set price higher than $1, but a prudent opponent may sell one a high-priced ticket, such that the opponent comes out ahead regardless of the outcome of the event on which the bet is made. The rules also do not forbid a negative price, but an opponent may extract a paid promise from the bettor to pay him or her later should a certain contingency arise. In either case, the price-setter loses. These lose-lose situations parallel the fact that a probability can neither exceed 1 (certainty) nor be less than 0 (no chance of winning). A more instructive Dutch book Now suppose one sets the price of a promise to pay $1 if the Boston Red Sox win next year's World Series, and also the price of a promise to pay $1 if the New York Yankees win, and finally the price of a promise to pay $1 if either the Red Sox or the Yankees win. One may set the prices in such a way that But if one sets the price of the third ticket lower than the sum of the first two tickets, a prudent opponent will buy that ticket and sell the other two tickets to the price-setter. By considering the thr
https://en.wikipedia.org/wiki/List%20of%20algebraic%20geometry%20topics	This is a list of algebraic geometry topics, by Wikipedia page. Classical topics in projective geometry Affine space Projective space Projective line, cross-ratio Projective plane Line at infinity Complex projective plane Complex projective space Plane at infinity, hyperplane at infinity Projective frame Projective transformation Fundamental theorem of projective geometry Duality (projective geometry) Real projective plane Real projective space Segre embedding of a product of projective spaces Rational normal curve Algebraic curves Conics, Pascal's theorem, Brianchon's theorem Twisted cubic Elliptic curve, cubic curve Elliptic function, Jacobi's elliptic functions, Weierstrass's elliptic functions Elliptic integral Complex multiplication Weil pairing Hyperelliptic curve Klein quartic Modular curve Modular equation Modular function Modular group Supersingular primes Fermat curve Bézout's theorem Brill–Noether theory Genus (mathematics) Riemann surface Riemann–Hurwitz formula Riemann–Roch theorem Abelian integral Differential of the first kind Jacobian variety Generalized Jacobian Moduli of algebraic curves Hurwitz's theorem on automorphisms of a curve Clifford's theorem on special divisors Gonality of an algebraic curve Weil reciprocity law Algebraic geometry codes Algebraic surfaces Enriques–Kodaira classification List of algebraic surfaces Ruled surface Cubic surface Veronese surface Del Pezzo surface Rational surface Enriques surface K3 surface Hodge index theorem Elliptic surface Surface of general type Zariski surface Algebraic geometry: classical approach Algebraic variety Hypersurface Quadric (algebraic geometry) Dimension of an algebraic variety Hilbert's Nullstellensatz Complete variety Elimination theory Gröbner basis Projective variety Quasiprojective variety Canonical bundle Complete intersection Serre duality Spaltenstein variety Arithmetic genus, geometric genus, irregularity Tangent space, Zariski tangent space Function field of an algebraic variety Ample line bundle Ample vector bundle Linear system of divisors Birational geometry Blowing up Resolution of singularities Rational variety Unirational variety Ruled variety Kodaira dimension Canonical ring Minimal model program Intersection theory Intersection number Chow ring Chern class Serre's multiplicity conjectures Albanese variety Picard group Modular form Moduli space Modular equation J-invariant Algebraic function Algebraic form Addition theorem Invariant theory Symbolic method of invariant theory Geometric invariant theory Toric variety Deformation theory Singular point, non-singular Singularity theory Newton polygon Weil conjectures Complex manifolds Kähler manifold Calabi–Yau manifold Stein manifold Hodge theory Hodge cycle Hodge conjecture Algebraic geometry and analytic geometry Mirror symmetry Algebraic groups Linear algebraic group Additive group Multiplicative group Algebraic torus Reductive group Borel subgroup Parabolic subgroup Radical of an algebraic group Uni
https://en.wikipedia.org/wiki/Almost%20surely	In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0. The concept is analogous to the concept of "almost everywhere" in measure theory. In probability experiments on a finite sample space with a non-zero probability for each outcome, there is no difference between almost surely and surely (since having a probability of 1 entails including all the sample points); however, this distinction becomes important when the sample space is an infinite set, because an infinite set can have non-empty subsets of probability 0. Some examples of the use of this concept include the strong and uniform versions of the law of large numbers, the continuity of the paths of Brownian motion, and the infinite monkey theorem. The terms almost certainly (a.c.) and almost always (a.a.) are also used. Almost never describes the opposite of almost surely: an event that happens with probability zero happens almost never. Formal definition Let be a probability space. An event happens almost surely if . Equivalently, happens almost surely if the probability of not occurring is zero: . More generally, any event (not necessarily in ) happens almost surely if is contained in a null set: a subset in such that The notion of almost sureness depends on the probability measure . If it is necessary to emphasize this dependence, it is customary to say that the event occurs P-almost surely, or almost surely . Illustrative examples In general, an event can happen "almost surely", even if the probability space in question includes outcomes which do not belong to the event—as the following examples illustrate. Throwing a dart Imagine throwing a dart at a unit square (a square with an area of 1) so that the dart always hits an exact point in the square, in such a way that each point in the square is equally likely to be hit. Since the square has area 1, the probability that the dart will hit any particular subregion of the square is equal to the area of that subregion. For example, the probability that the dart will hit the right half of the square is 0.5, since the right half has area 0.5. Next, consider the event that the dart hits exactly a point in the diagonals of the unit square. Since the area of the diagonals of the square is 0, the probability that the dart will land exactly on a diagonal is 0. That is, the dart will almost never land on a diagonal (equivalently, it will almost surely not land on a diagonal), even though the set of points on the diagonals is not empty, and a point on a diagonal is no less possible than any other point. Tossing a coin repeatedly Consider the case where a (possibly biased) coin is tossed, corresponding to the probability space , where the event occurs if a head is flipped, and if a tail is flipped. For this particular coin, it is assumed that the
https://en.wikipedia.org/wiki/List%20of%20abstract%20algebra%20topics	Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. The phrase abstract algebra was coined at the turn of the 20th century to distinguish this area from what was normally referred to as algebra, the study of the rules for manipulating formulae and algebraic expressions involving unknowns and real or complex numbers, often now called elementary algebra. The distinction is rarely made in more recent writings. Basic language Algebraic structures are defined primarily as sets with operations. Algebraic structure Subobjects: subgroup, subring, subalgebra, submodule etc. Binary operation Closure of an operation Associative property Distributive property Commutative property Unary operator Additive inverse, multiplicative inverse, inverse element Identity element Cancellation property Finitary operation Arity Structure preserving maps called homomorphisms are vital in the study of algebraic objects. Homomorphisms Kernels and cokernels Image and coimage Epimorphisms and monomorphisms Isomorphisms Isomorphism theorems There are several basic ways to combine algebraic objects of the same type to produce a third object of the same type. These constructions are used throughout algebra. Direct sum Direct limit Direct product Inverse limit Quotient objects: quotient group, quotient ring, quotient module etc. Tensor product Advanced concepts: Category theory Category of groups Category of abelian groups Category of rings Category of modules (over a fixed ring) Morita equivalence, Morita duality Category of vector spaces Homological algebra Filtration (algebra) Exact sequence Functor Zorn's lemma Semigroups and monoids Semigroup Subsemigroup Free semigroup Green's relations Inverse semigroup (or inversion semigroup, cf. ) Krohn–Rhodes theory Semigroup algebra Transformation semigroup Monoid Aperiodic monoid Free monoid Monoid (category theory) Monoid factorisation Syntactic monoid Group theory Structure Group (mathematics) Lagrange's theorem (group theory) Subgroup Coset Normal subgroup Characteristic subgroup Centralizer and normalizer subgroups Derived group Frattini subgroup Fitting subgroup Classification of finite simple groups Sylow theorems Local analysis Constructions Free group Presentation of a group Word problem for groups Quotient group Extension problem Direct sum, direct product Semidirect product Wreath product Types Simple group Finite group Abelian group Torsion subgroup Free abelian group Finitely generated abelian group Rank of an abelian group Cyclic group Locally cyclic group Solvable group Composition series Nilpotent group Divisible group Dedekind group, Hamiltonian group Examples Examples of groups Trivial group Additive group Permutation group Symmetric group Alternating group p-group List of small groups Klein four-group Quaternion group Dihedral group Dicyclic group Automorphism group Point group Circle group Linear group
https://en.wikipedia.org/wiki/Cantor%20space	In mathematics, a Cantor space, named for Georg Cantor, is a topological abstraction of the classical Cantor set: a topological space is a Cantor space if it is homeomorphic to the Cantor set. In set theory, the topological space 2ω is called "the" Cantor space. Examples The Cantor set itself is a Cantor space. But the canonical example of a Cantor space is the countably infinite topological product of the discrete 2-point space {0, 1}. This is usually written as or 2ω (where 2 denotes the 2-element set {0,1} with the discrete topology). A point in 2ω is an infinite binary sequence, that is a sequence that assumes only the values 0 or 1. Given such a sequence a0, a1, a2,..., one can map it to the real number This mapping gives a homeomorphism from 2ω onto the Cantor set, demonstrating that 2ω is indeed a Cantor space. Cantor spaces occur abundantly in real analysis. For example, they exist as subspaces in every perfect, complete metric space. (To see this, note that in such a space, any non-empty perfect set contains two disjoint non-empty perfect subsets of arbitrarily small diameter, and so one can imitate the construction of the usual Cantor set.) Also, every uncountable, separable, completely metrizable space contains Cantor spaces as subspaces. This includes most of the common spaces in real analysis. Characterization A topological characterization of Cantor spaces is given by Brouwer's theorem: The topological property of having a base consisting of clopen sets is sometimes known as "zero-dimensionality". Brouwer's theorem can be restated as: This theorem is also equivalent (via Stone's representation theorem for Boolean algebras) to the fact that any two countable atomless Boolean algebras are isomorphic. Properties As can be expected from Brouwer's theorem, Cantor spaces appear in several forms. But many properties of Cantor spaces can be established using 2ω, because its construction as a product makes it amenable to analysis. Cantor spaces have the following properties: The cardinality of any Cantor space is , that is, the cardinality of the continuum. The product of two (or even any finite or countable number of) Cantor spaces is a Cantor space. Along with the Cantor function, this fact can be used to construct space-filling curves. A (non-empty) Hausdorff topological space is compact metrizable if and only if it is a continuous image of a Cantor space. Let C(X) denote the space of all real-valued, bounded continuous functions on a topological space X. Let K denote a compact metric space, and Δ denote the Cantor set. Then the Cantor set has the following property: C(K) is isometric to a closed subspace of C(Δ). In general, this isometry is not unique, and thus is not properly a universal property in the categorical sense. The group of all homeomorphisms of the Cantor space is simple. See also Space (mathematics) Cantor set Cantor cube References Topological spaces Descriptive set theory Georg Cantor
https://en.wikipedia.org/wiki/1917%20in%20science	The year 1917 in science and technology involved some significant events, listed below. Biology D'Arcy Wentworth Thompson's On Growth and Form is published. Mathematics Paul Ehrenfest gives a conditional principle for a three-dimensional space. Medicine Shinobu Ishihara publishes his color perception test. Julius Wagner-Jauregg discovers malarial pyrotherapy for general paresis of the insane. Physics Albert Einstein introduces the idea of stimulated radiation emission. Nuclear fission: Ernest Rutherford (at the Victoria University of Manchester) achieves nuclear transmutation of nitrogen into oxygen, using alpha particles directed at nitrogen 14N + α → 17O + p, the first observation of a nuclear reaction, in which he also discovers and names the proton. Technology September 13 – Release in the United States of the first film made in Technicolor System 1, a two-color process, The Gulf Between. Alvin D. and Kelvin Keech introduce the "banjulele-banjo", an early form of the banjolele. Gilbert Vernam jointly reinvents the one-time pad encryption system. Awards Nobel Prize Physics – Charles Glover Barkla (announced 12 November 1918; presented 1 June 1920) Chemistry – not awarded Medicine – not awarded Births January 19 – Graham Higman (died 2008), English mathematician. January 25 – Ilya Prigogine (died 2003), Russian-born winner of the Nobel Prize in Chemistry. February 14 – Herbert A. Hauptman (died 2011), American mathematical biophysicist, winner of the Nobel Prize in Chemistry. March 23 – Howard McKern (died 2009), Australian analytical and organic chemist. March 24 – John Kendrew (died 1997), English molecular biologist, winner of the Nobel Prize in Chemistry. April 10 – Robert Burns Woodward (died 1979), American organic chemist, winner of the Nobel Prize in Chemistry. April 18 – Brian Harold Mason (died 2009), New Zealand born geochemist and mineralogist who was one of the pioneers in the study of meteorites. May 14 – W. T. Tutte (died 2002), English-born mathematician and cryptanalyst. June 1 – William S. Knowles (died 2012), American winner of the Nobel Prize in Chemistry. June 2 – Heinz Sielmann (died 2006), German zoological filmmaker. June 15 – John Fenn (died 2010), American analytical chemist, winner of the Nobel Prize in Chemistry. July 1 – Humphry Osmond (died 2004), English-born psychiatrist. July 15 – Walter S. Graf (died 2015), American cardiologist and pioneer of paramedic emergency medical services. July 22 – H. Boyd Woodruff (died 2017), American microbiologist. August 21 – Xu Shunshou (died 1968), Chinese aeronautical engineer. September 23 – Asima Chatterjee, née Mookerjee (died 2006), Indian organic chemist. October 2 – Christian de Duve (died 2013), English-born Belgian biologist, winner of the Nobel Prize in Physiology or Medicine October 8 – Rodney Porter (died 1985), English biochemist, winner of the Nobel Prize in Physiology or Medicine. November 22 – Andrew Huxley (died 2012), Engl
https://en.wikipedia.org/wiki/Germ%20%28mathematics%29	In mathematics, the notion of a germ of an object in/on a topological space is an equivalence class of that object and others of the same kind that captures their shared local properties. In particular, the objects in question are mostly functions (or maps) and subsets. In specific implementations of this idea, the functions or subsets in question will have some property, such as being analytic or smooth, but in general this is not needed (the functions in question need not even be continuous); it is however necessary that the space on/in which the object is defined is a topological space, in order that the word local has some meaning. Name The name is derived from cereal germ in a continuation of the sheaf metaphor, as a germ is (locally) the "heart" of a function, as it is for a grain. Formal definition Basic definition Given a point x of a topological space X, and two maps (where Y is any set), then and define the same germ at x if there is a neighbourhood U of x such that restricted to U, f and g are equal; meaning that for all u in U. Similarly, if S and T are any two subsets of X, then they define the same germ at x if there is again a neighbourhood U of x such that It is straightforward to see that defining the same germ at x is an equivalence relation (be it on maps or sets), and the equivalence classes are called germs (map-germs, or set-germs accordingly). The equivalence relation is usually written Given a map f on X, then its germ at x is usually denoted [f ]x. Similarly, the germ at x of a set S is written [S]x. Thus, A map germ at x in X that maps the point x in X to the point y in Y is denoted as When using this notation, f is then intended as an entire equivalence class of maps, using the same letter f for any representative map. Notice that two sets are germ-equivalent at x if and only if their characteristic functions are germ-equivalent at x: More generally Maps need not be defined on all of X, and in particular they don't need to have the same domain. However, if f has domain S and g has domain T, both subsets of X, then f and g are germ equivalent at x in X if first S and T are germ equivalent at x, say and then moreover , for some smaller neighbourhood V with . This is particularly relevant in two settings: f is defined on a subvariety V of X, and f has a pole of some sort at x, so is not even defined at x, as for example a rational function, which would be defined off a subvariety. Basic properties If f and g are germ equivalent at x, then they share all local properties, such as continuity, differentiability etc., so it makes sense to talk about a differentiable or analytic germ, etc. Similarly for subsets: if one representative of a germ is an analytic set then so are all representatives, at least on some neighbourhood of x. Algebraic structures on the target Y are inherited by the set of germs with values in Y. For instance, if the target Y is a group, then it makes sense to multiply germs: to def
https://en.wikipedia.org/wiki/Spin%E2%80%93statistics%20theorem	In quantum mechanics, the spin–statistics theorem relates the intrinsic spin of a particle (angular momentum not due to the orbital motion) to the particle statistics it obeys. In units of the reduced Planck constant ħ, all particles that move in 3 dimensions have either integer spin or half-integer spin. Background Quantum states and indistinguishable particles In a quantum system, a physical state is described by a state vector. A pair of distinct state vectors are physically equivalent if they differ only by an overall phase factor, ignoring other interactions. A pair of indistinguishable particles such as this have only one state. This means that if the positions of the particles are exchanged (i.e., they undergo a permutation), this does not identify a new physical state, but rather one matching the original physical state. In fact, one cannot tell which particle is in which position. While the physical state does not change under the exchange of the particles' positions, it is possible for the state vector to change sign as a result of an exchange. Since this sign change is just an overall phase, this does not affect the physical state. The essential ingredient in proving the spin-statistics relation is relativity, that the physical laws do not change under Lorentz transformations. The field operators transform under Lorentz transformations according to the spin of the particle that they create, by definition. Additionally, the assumption (known as microcausality) that spacelike-separated fields either commute or anticommute can be made only for relativistic theories with a time direction. Otherwise, the notion of being spacelike is meaningless. However, the proof involves looking at a Euclidean version of spacetime, in which the time direction is treated as a spatial one, as will be now explained. Lorentz transformations include 3-dimensional rotations and boosts. A boost transfers to a frame of reference with a different velocity and is mathematically like a rotation into time. By analytic continuation of the correlation functions of a quantum field theory, the time coordinate may become imaginary, and then boosts become rotations. The new "spacetime" has only spatial directions and is termed Euclidean. Exchange symmetry or permutation symmetry Bosons are particles whose wavefunction is symmetric under such an exchange or permutation, so if we swap the particles, the wavefunction does not change. Fermions are particles whose wavefunction is antisymmetric, so under such a swap the wavefunction gets a minus sign, meaning that the amplitude for two identical fermions to occupy the same state must be zero. This is the Pauli exclusion principle: two identical fermions cannot occupy the same state. This rule does not hold for bosons. In quantum field theory, a state or a wavefunction is described by field operators operating on some basic state called the vacuum. In order for the operators to project out the symmetric or antisymmetric
https://en.wikipedia.org/wiki/List%20of%20general%20topology%20topics	This is a list of general topology topics. Basic concepts Topological space Topological property Open set, closed set Clopen set Closure (topology) Boundary (topology) Dense (topology) G-delta set, F-sigma set closeness (mathematics) neighbourhood (mathematics) Continuity (topology) Homeomorphism Local homeomorphism Open and closed maps Germ (mathematics) Base (topology), subbase Open cover Covering space Atlas (topology) Limits Limit point Net (topology) Filter (topology) Ultrafilter Topological properties Baire category theorem Nowhere dense Baire space Banach–Mazur game Meagre set Comeagre set Compactness and countability Compact space Relatively compact subspace Heine–Borel theorem Tychonoff's theorem Finite intersection property Compactification Measure of non-compactness Paracompact space Locally compact space Compactly generated space Axiom of countability Sequential space First-countable space Second-countable space Separable space Lindelöf space Sigma-compact space Connectedness Connected space Separation axioms T0 space T1 space Hausdorff space Completely Hausdorff space Regular space Tychonoff space Normal space Urysohn's lemma Tietze extension theorem Paracompact Separated sets Topological constructions Direct sum and the dual construction product Subspace and the dual construction quotient Topological tensor product Examples Discrete space Locally constant function Trivial topology Cofinite topology Finer topology Product topology Restricted product Quotient space Unit interval Continuum (topology) Extended real number line Long line (topology) Sierpinski space Cantor set, Cantor space, Cantor cube Space-filling curve Topologist's sine curve Uniform norm Weak topology Strong topology Hilbert cube Lower limit topology Sorgenfrey plane Real tree Compact-open topology Zariski topology Kuratowski closure axioms Unicoherent Solenoid (mathematics) Uniform spaces Uniform continuity Lipschitz continuity Uniform isomorphism Uniform property Uniformly connected space Metric spaces Metric topology Manhattan distance Ultrametric space P-adic numbers, p-adic analysis Open ball Bounded subset Pointwise convergence Metrization theorems Complete space Cauchy sequence Banach fixed-point theorem Polish space Hausdorff distance Intrinsic metric Category of metric spaces Topology and order theory Stone duality Stone's representation theorem for Boolean algebras Specialization (pre)order Sober space Spectral space Alexandrov topology Upper topology Scott topology Scott continuity Lawson topology Descriptive set theory Polish Space Cantor space Dimension theory Inductive dimension Lebesgue covering dimension Lebesgue's number lemma Combinatorial topology Polytope Simplex Simplicial complex CW complex Manifold Triangulation Barycentric subdivision Sperner's lemma Simplicial approximation theorem Nerve of an open covering Foundations of algebraic topology Simply connected Semi-locally simply connected Path (topology) Homotopy
https://en.wikipedia.org/wiki/Chen%20Jingrun	Chen Jingrun (; 22 May 1933 – 19 March 1996), also known as Jing-Run Chen, was a Chinese mathematician who made significant contributions to number theory, including Chen's theorem and the Chen prime. Life and career Chen was the third son in a large family from Fuzhou, Fujian, China. His father was a postal worker. Chen Jingrun graduated from the Mathematics Department of Xiamen University in 1953. His advisor at the Chinese Academy of Sciences was Hua Luogeng. His work on the twin prime conjecture, Waring's problem, Goldbach's conjecture and Legendre's conjecture led to progress in analytic number theory. In a 1966 paper he proved what is now called Chen's theorem: every sufficiently large even number can be written as the sum of a prime and a semiprime (the product of two primes) – e.g., 100 = 23 + 7·11. Despite being persecuted during the Cultural Revolution, he expanded his proof in the 1970s. After the end of the Cultural Revolution, Xu Chi wrote a biography of Chen entitled Goldbach's Conjecture (). First published in People's Literature in January 1978, it was reprinted on the People's Daily a month later and became a national sensation. Chen became a household name in China and received a sackful of love letters from all over the country within two months. Chen died of complications of pneumonia on March 19, 1996, at the age of 63 years. Legacy The asteroid 7681 Chenjingrun, discovered in 1996, was named after him. In 1999, China issued an 80-cent postage stamp, titled The Best Result of Goldbach Conjecture, with a silhouette of Chen and the inequality: Several statues in China have been built in memory of Chen. At Xiamen University, the names of Chen and four other mathematicians — Peter Gustav Lejeune Dirichlet, Matti Jutila, Yuri Linnik, and Pan Chengdong — are inscribed in the marble slab behind Chen's statue (see image). Works J.-R. Chen, On the representation of a large even integer as the sum of a prime and a product of at most two primes, Sci. Sinica 16 (1973), 157–176. Chen, J.R, "On the representation of a large even integer as the sum of a prime and the product of at most two primes". [Chinese] J. Kexue Tongbao 17 (1966), 385–386. "Fundamental Number Theory" References Pan Chengdong and Wang Yuan, Chen Jingrun: a brief outline of his life and works, Acta Math. Sinica (NS) 12 (1996) 225–233. External links Chen's home page at the Chinese Institute of Mathematics. 1933 births 1996 deaths 20th-century Chinese mathematicians Members of the Chinese Academy of Sciences Number theorists Educators from Fujian Academic staff of Guizhou Nationalities University Academic staff of Henan University Academic staff of Xiamen University Academic staff of Qingdao University Academic staff of Huazhong University of Science and Technology Academic staff of Fujian Normal University People from Fuzhou Mathematicians from Fujian Xiamen University alumni Delegates to the 4th National People's Congress Delegates to the 5th National
https://en.wikipedia.org/wiki/CW%20complex	A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial nature that allows for computation (often with a much smaller complex). The C stands for "closure-finite", and the W for "weak" topology. Definition CW complex A CW complex is constructed by taking the union of a sequence of topological spaces such that each is obtained from by gluing copies of k-cells , each homeomorphic to , to by continuous gluing maps . The maps are also called attaching maps. Each is called the k-skeleton of the complex. The topology of is weak topology: a subset is open iff is open for each cell . In the language of category theory, the topology on is the direct limit of the diagram The name "CW" stands for "closure-finite weak topology", which is explained by the following theorem: This partition of X is also called a cellulation. The construction, in words The CW complex construction is a straightforward generalization of the following process: A 0-dimensional CW complex is just a set of zero or more discrete points (with the discrete topology). A 1-dimensional CW complex is constructed by taking the disjoint union of a 0-dimensional CW complex with one or more copies of the unit interval. For each copy, there is a map that "glues" its boundary (its two endpoints) to elements of the 0-dimensional complex (the points). The topology of the CW complex is the topology of the quotient space defined by these gluing maps. In general, an n-dimensional CW complex is constructed by taking the disjoint union of a k-dimensional CW complex (for some ) with one or more copies of the n-dimensional ball. For each copy, there is a map that "glues" its boundary (the -dimensional sphere) to elements of the -dimensional complex. The topology of the CW complex is the quotient topology defined by these gluing maps. An infinite-dimensional CW complex can be constructed by repeating the above process countably many times. Since the topology of the union is indeterminate, one takes the direct limit topology, since the diagram is highly suggestive of a direct limit. This turns out to have great technical benefits. Regular CW complexes A regular CW complex is a CW complex whose gluing maps are homeomorphisms. Accordingly, the partition of X is also called a regular cellulation. A loopless graph is represented by a regular 1-dimensional CW-complex. A closed 2-cell graph embedding on a surface is a regular 2-dimensional CW-complex. Finally, the 3-sphere regular cellulation conjecture claims that every 2-connected graph is the 1-skeleton of a regular CW-complex on the 3-dimensional sphere. Relative CW complexes Roughly speaking, a relative CW complex differs from
https://en.wikipedia.org/wiki/Vilnius%20County	Vilnius County () is the largest of the 10 counties of Lithuania, located in the east of the country around the city Vilnius and is also known as Capital Region () by the statistics department and Eurostat. On 1 July 2010, the county administration was abolished, and since that date, Vilnius County remains as the territorial and statistical unit. History Until the Partitions of the Polish–Lithuanian Commonwealth in late 18th century the area belonged to the Vilnius Voivodship and Trakai Voivodship of the Polish–Lithuanian Commonwealth. In the Russian Empire it belonged to the Northwestern Krai and approximately corresponded to its Vilna Governorate (as of 1843). During World War I, following the German offensive of 1915, it was occupied by the German army. After the war, some parts of the area was ruled by local Polish self-government established after the German Ober-Ost army withdrew from the area. Following the start of the Polish-Bolshevik War in 1919, it was occupied by the Red Army, which was pushed back by the Polish Army. In 1920, it was again occupied by the Red Army, but the Soviets officially recognized the sovereignty of the Lithuanian Republic over the city immediately after defeat during Battle of Warsaw. During their retreat, the Bolsheviks passed the sovereignty over the area to Lithuania. The Polish commander Józef Piłsudski ordered his subordinate general Lucjan Żeligowski to "rebel" his Lithuanian-Belarusian division and capture the city of Vilnius, without declaring war on Lithuania. The area of the future Vilnius County was seized by the Polish forces without significant opposition from Lithuanian forces and Gen. Żeligowski created a short-lived state called the Republic of Central Lithuania. Following the elections held there in 1922 the state was incorporated into Poland (see Vilnius region, Central Lithuania). As a result of the Nazi-Soviet Alliance and the Polish Defensive War of 1939, the area was captured by the Soviet Union, which transferred parts of what is now Vilnius County and Utena County to Lithuania, only to annex it the following year. In 1941, it was conquered by the Nazi Germany. During World War II, the area saw formation of many resistance units, most notably the Polish Home Army and, after 1943, Soviet partisans. After the war, Vilniaus Apskritis existed as a relic of the pre-war independent state in Lithuanian SSR between 1944 and 1950. In this period, a significant part of its population moved to Poland during the so-called repatriation. After 1990, when Lithuania became independent, Vilnius county was re-established differently in 1994. This entity has different boundaries than any previous entity and is not directly related to previous entities in this area. The mission of the county is different as well: its primary goal (as in that of Lithuania's other counties) is to oversee that municipalities in its area follow the laws of Lithuania. Municipalities The county is subdivided into six district
https://en.wikipedia.org/wiki/Outline%20of%20linear%20algebra	<noinclude>This is an outline of topics related to linear algebra, the branch of mathematics concerning linear equations and linear maps and their representations in vector spaces and through matrices. Linear equations Linear equation System of linear equations Determinant Minor Cauchy–Binet formula Cramer's rule Gaussian elimination Gauss–Jordan elimination Overcompleteness Strassen algorithm Matrices Matrix Matrix addition Matrix multiplication Basis transformation matrix Characteristic polynomial Trace Eigenvalue, eigenvector and eigenspace Cayley–Hamilton theorem Spread of a matrix Jordan normal form Weyr canonical form Rank Matrix inversion, invertible matrix Pseudoinverse Adjugate Transpose Dot product Symmetric matrix Orthogonal matrix Skew-symmetric matrix Conjugate transpose Unitary matrix Hermitian matrix, Antihermitian matrix Positive-definite, positive-semidefinite matrix Pfaffian Projection Spectral theorem Perron–Frobenius theorem List of matrices Diagonal matrix, main diagonal Diagonalizable matrix Triangular matrix Tridiagonal matrix Block matrix Sparse matrix Hessenberg matrix Hessian matrix Vandermonde matrix Stochastic matrix Toeplitz matrix Circulant matrix Hankel matrix (0,1)-matrix Matrix decompositions Matrix decomposition Cholesky decomposition LU decomposition QR decomposition Polar decomposition Reducing subspace Spectral theorem Singular value decomposition Higher-order singular value decomposition Schur decomposition Schur complement Haynsworth inertia additivity formula Relations Matrix equivalence Matrix congruence Matrix similarity Matrix consimilarity Row equivalence Computations Elementary row operations Householder transformation Least squares, linear least squares Gram–Schmidt process Woodbury matrix identity Vector spaces Vector space Linear combination Linear span Linear independence Scalar multiplication Basis Change of basis Hamel basis Cyclic decomposition theorem Dimension theorem for vector spaces Hamel dimension Examples of vector spaces Linear map Shear mapping or Galilean transformation Squeeze mapping or Lorentz transformation Linear subspace Row and column spaces Column space Row space Cyclic subspace Null space, nullity Rank–nullity theorem Nullity theorem Dual space Linear function Linear functional Category of vector spaces Structures Topological vector space Normed vector space Inner product space Euclidean space Orthogonality Orthogonal complement Orthogonal projection Orthogonal group Pseudo-Euclidean space Null vector Indefinite orthogonal group Orientation (geometry) Improper rotation Symplectic structure Multilinear algebra Multilinear algebra Tensor Classical treatment of tensors Component-free treatment of tensors Gamas's Theorem Outer product Tensor algebra Exterior algebra Symmetric algebra Clifford algebra Geometric algebra Topics related to affine spaces Affine space Affine transformation Affine group Affine geometry Affine coordinate system Flat (geometry) Cartesian coordin
https://en.wikipedia.org/wiki/Outline%20of%20category%20theory	The following outline is provided as an overview of and guide to category theory, the area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows (also called morphisms, although this term also has a specific, non category-theoretical sense), where these collections satisfy certain basic conditions. Many significant areas of mathematics can be formalised as categories, and the use of category theory allows many intricate and subtle mathematical results in these fields to be stated, and proved, in a much simpler way than without the use of categories. Essence of category theory Category Functor Natural transformation Branches of category theory Homological algebra Diagram chasing Topos theory Enriched category theory Higher category theory Categorical logic Specific categories Category of sets Concrete category Category of vector spaces Category of graded vector spaces Category of chain complexes Category of finite dimensional Hilbert spaces Category of sets and relations Category of topological spaces Category of metric spaces Category of preordered sets Category of groups Category of abelian groups Category of rings Category of magmas Category of medial magmas Objects Initial object Terminal object Zero object Subobject Group object Magma object Natural number object Exponential object Morphisms Epimorphism Monomorphism Zero morphism Normal morphism Dual (category theory) Groupoid Image (category theory) Coimage Commutative diagram Cartesian morphism Slice category Functors Isomorphism of categories Natural transformation Equivalence of categories Subcategory Faithful functor Full functor Forgetful functor Yoneda lemma Representable functor Functor category Adjoint functors Galois connection Pontryagin duality Affine scheme Monad (category theory) Comonad Combinatorial species Exact functor Derived functor Dominant functor Enriched functor Kan extension of a functor Hom functor Limits Product (category theory) Equaliser (mathematics) Kernel (category theory) Pullback (category theory)/fiber product Inverse limit Pro-finite group Colimit Coproduct Coequalizer Cokernel Pushout (category theory) Direct limit Biproduct Direct sum Additive structure Preadditive category Additive category Pre-Abelian category Abelian category Exact sequence Exact functor Snake lemma Nine lemma Five lemma Short five lemma Mitchell's embedding theorem Injective cogenerator Derived category Triangulated category Model category 2-category Dagger categories Dagger symmetric monoidal category Dagger compact category Strongly ribbon category Monoidal categories Closed monoidal category Braided monoidal category Cartesian closed category Topos Category of small categories Structure Semigroupoid Comma category Localization of a category Enri
https://en.wikipedia.org/wiki/Affine%20representation	In mathematics, an affine representation of a topological Lie group G on an affine space A is a continuous (smooth) group homomorphism from G to the automorphism group of A, the affine group Aff(A). Similarly, an affine representation of a Lie algebra g on A is a Lie algebra homomorphism from g to the Lie algebra aff(A) of the affine group of A. An example is the action of the Euclidean group E(n) on the Euclidean space En. Since the affine group in dimension n is a matrix group in dimension n + 1, an affine representation may be thought of as a particular kind of linear representation. We may ask whether a given affine representation has a fixed point in the given affine space A. If it does, we may take that as origin and regard A as a vector space; in that case, we actually have a linear representation in dimension n. This reduction depends on a group cohomology question, in general. See also Group action Projective representation References . Homological algebra Representation theory of Lie algebras Representation theory of Lie groups
https://en.wikipedia.org/wiki/Percentile	In statistics, a k-th percentile, also known as percentile score or centile, is a score a given percentage k of scores in its frequency distribution falls ("exclusive" definition) or a score a given percentage falls ("inclusive" definition). Percentiles are expressed in the same unit of measurement as the input scores, in percent; for example, if the scores refer to human weight, the corresponding percentiles will be expressed in kilograms or pounds. In the limit of an infinite sample size, the percentile approximates the percentile function, the inverse of the cumulative distribution function. Percentiles are a type of quantiles, obtained adopting a subdivision into 100 groups. The 25th percentile is also known as the first quartile (Q1), the 50th percentile as the median or second quartile (Q2), and the 75th percentile as the third quartile (Q3). For example, the 50th percentile (median) is the score (or , depending on the definition) which 50% of the scores in the distribution are found. A related quantity is the percentile rank of a score, expressed in percent, which represents the fraction of scores in its distribution that are less than it, an exclusive definition. Percentile scores and percentile ranks are often used in the reporting of test scores from norm-referenced tests, but, as just noted, they are not the same. For percentile ranks, a score is given and a percentage is computed. Percentile ranks are exclusive: if the percentile rank for a specified score is 90%, then 90% of the scores were lower. In contrast, for percentiles a percentage is given and a corresponding score is determined, which can be either exclusive or inclusive. The score for a specified percentage (e.g., 90th) indicates a score below which (exclusive definition) or at or below which (inclusive definition) other scores in the distribution fall. Definitions There is no standard definition of percentile, however all definitions yield similar results when the number of observations is very large and the probability distribution is continuous. In the limit, as the sample size approaches infinity, the 100pth percentile (0<p<1) approximates the inverse of the cumulative distribution function (CDF) thus formed, evaluated at p, as p approximates the CDF. This can be seen as a consequence of the Glivenko–Cantelli theorem. Some methods for calculating the percentiles are given below. The normal distribution and percentiles The methods given in the calculation methods section (below) are approximations for use in small-sample statistics. In general terms, for very large populations following a normal distribution, percentiles may often be represented by reference to a normal curve plot. The normal distribution is plotted along an axis scaled to standard deviations, or sigma () units. Mathematically, the normal distribution extends to negative infinity on the left and positive infinity on the right. Note, however, that only a very small proportion of individuals in
https://en.wikipedia.org/wiki/Quiver%20%28mathematics%29	In mathematics, especially representation theory, a quiver is another name for a multidigraph; that is, a directed graph where loops and multiple arrows between two vertices are allowed. Quivers are commonly used in representation theory: a representation of a quiver assigns a vector space to each vertex of the quiver and a linear map to each arrow . In category theory, a quiver can be understood to be the underlying structure of a category, but without composition or a designation of identity morphisms. That is, there is a forgetful functor from (the category of categories) to (the category of multidigraphs). Its left adjoint is a free functor which, from a quiver, makes the corresponding free category. Definition A quiver consists of: The set of vertices of The set of edges of Two functions: giving the start or source of the edge, and another function, giving the target of the edge. This definition is identical to that of a multidigraph. A morphism of quivers is defined as follows. If and are two quivers, then a morphism of quivers consists of two functions and such that the following diagrams commute: That is, and Category-theoretic definition The above definition is based in set theory; the category-theoretic definition generalizes this into a functor from the free quiver to the category of sets. The free quiver (also called the walking quiver, Kronecker quiver, 2-Kronecker quiver or Kronecker category) is a category with two objects, and four morphisms: The objects are and . The four morphisms are and the identity morphisms and That is, the free quiver is A quiver is then a functor More generally, a quiver in a category is a functor The category of quivers in is the functor category where: objects are functors morphisms are natural transformations between functors. Note that is the category of presheaves on the opposite category . Path algebra If is a quiver, then a path in is a sequence of arrows such that the head of is the tail of for , using the convention of concatenating paths from right to left. If is a field then the quiver algebra or path algebra is defined as a vector space having all the paths (of length ≥ 0) in the quiver as basis (including, for each vertex of the quiver , a trivial path of length 0; these paths are not assumed to be equal for different ), and multiplication given by concatenation of paths. If two paths cannot be concatenated because the end vertex of the first is not equal to the starting vertex of the second, their product is defined to be zero. This defines an associative algebra over . This algebra has a unit element if and only if the quiver has only finitely many vertices. In this case, the modules over are naturally identified with the representations of . If the quiver has infinitely many vertices, then has an approximate identity given by where ranges over finite subsets of the vertex set of . If the quiver has finitely many ver
https://en.wikipedia.org/wiki/Monad%20%28category%20theory%29	In category theory, a branch of mathematics, a monad (also triple, triad, standard construction and fundamental construction) is a monoid in the category of endofunctors of some fixed category. An endofunctor is a functor mapping a category to itself, and a monad is an endofunctor together with two natural transformations required to fulfill certain coherence conditions. Monads are used in the theory of pairs of adjoint functors, and they generalize closure operators on partially ordered sets to arbitrary categories. Monads are also useful in the theory of datatypes, the denotational semantics of imperative programming languages, and in functional programming languages, allowing languages with non-mutable states to do things such as simulate for-loops; see Monad (functional programming). Introduction and definition A monad is a certain type of endofunctor. For example, if and are a pair of adjoint functors, with left adjoint to , then the composition is a monad. If and are inverse functors, the corresponding monad is the identity functor. In general, adjunctions are not equivalences—they relate categories of different natures. The monad theory matters as part of the effort to capture what it is that adjunctions 'preserve'. The other half of the theory, of what can be learned likewise from consideration of , is discussed under the dual theory of comonads. Formal definition Throughout this article denotes a category. A monad on consists of an endofunctor together with two natural transformations: (where denotes the identity functor on ) and (where is the functor from to ). These are required to fulfill the following conditions (sometimes called coherence conditions): (as natural transformations ); here and are formed by "horizontal composition" (as natural transformations ; here denotes the identity transformation from to ). We can rewrite these conditions using the following commutative diagrams: See the article on natural transformations for the explanation of the notations and , or see below the commutative diagrams not using these notions: The first axiom is akin to the associativity in monoids if we think of as the monoid's binary operation, and the second axiom is akin to the existence of an identity element (which we think of as given by ). Indeed, a monad on can alternatively be defined as a monoid in the category whose objects are the endofunctors of and whose morphisms are the natural transformations between them, with the monoidal structure induced by the composition of endofunctors. The power set monad The power set monad is a monad on the category : For a set let be the power set of and for a function let be the function between the power sets induced by taking direct images under . For every set , we have a map , which assigns to every the singleton . The function takes a set of sets to its union. These data describe a monad. Remarks The axioms of a monad are formally similar to
https://en.wikipedia.org/wiki/Outline%20of%20discrete%20mathematics	Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not vary smoothly in this way, but have distinct, separated values. Discrete mathematics, therefore, excludes topics in "continuous mathematics" such as calculus and analysis. Included below are many of the standard terms used routinely in university-level courses and in research papers. This is not, however, intended as a complete list of mathematical terms; just a selection of typical terms of art that may be encountered. Subjects in discrete mathematics Logic – a study of reasoning Modal Logic: A type of logic for the study of necessity and probability Set theory – a study of collections of elements Number theory – study of integers and integer-valued functions Combinatorics – a study of Counting Finite mathematics – a course title Graph theory – a study of graphs Digital geometry and digital topology Algorithmics – a study of methods of calculation Information theory – a mathematical representation of the conditions and parameters affecting the transmission and processing of information Computability and complexity theories – deal with theoretical and practical limitations of algorithms Elementary probability theory and Markov chains Linear algebra – a study of related linear equations Functions – an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable) Partially ordered set – Probability – concerns with numerical descriptions of the chances of occurrence of an event Proofs – Relation – a collection of ordered pairs containing one object from each set Discrete mathematical disciplines For further reading in discrete mathematics, beyond a basic level, see these pages. Many of these disciplines are closely related to computer science. Automata theory – Coding theory – Combinatorics – Computational geometry – Digital geometry – Discrete geometry – Graph theory – a study of graphs Mathematical logic – Discrete optimization – Set theory – Number theory – Information theory – Game theory – Concepts in discrete mathematics Sets Set (mathematics) – Element (mathematics) – Venn diagram – Empty set – Subset – Union (set theory) – Disjoint union – Intersection (set theory) – Disjoint sets – Complement (set theory) – Symmetric difference – Ordered pair – Cartesian product – Power set – Simple theorems in the algebra of sets – Naive set theory – Multiset – Functions Function – Domain of a function – Codomain – Range of a function – Image (mathematics) – Injective function – Surjection – Bijection – Function composition – Partial function – Multivalued function – Binary function – Floor function – Sign function – I
https://en.wikipedia.org/wiki/Camille%20Jordan	Marie Ennemond Camille Jordan (; 5 January 1838 – 22 January 1922) was a French mathematician, known both for his foundational work in group theory and for his influential Cours d'analyse. Biography Jordan was born in Lyon and educated at the École polytechnique. He was an engineer by profession; later in life he taught at the École polytechnique and the Collège de France, where he had a reputation for eccentric choices of notation. He is remembered now by name in a number of results: The Jordan curve theorem, a topological result required in complex analysis The Jordan normal form and the Jordan matrix in linear algebra In mathematical analysis, Jordan measure (or Jordan content) is an area measure that predates measure theory In group theory, the Jordan–Hölder theorem on composition series is a basic result. Jordan's theorem on finite linear groups Jordan's work did much to bring Galois theory into the mainstream. He also investigated the Mathieu groups, the first examples of sporadic groups. His Traité des substitutions, on permutation groups, was published in 1870; this treatise won for Jordan the 1870 prix Poncelet. He was an Invited Speaker of the ICM in 1920 in Strasbourg. The asteroid 25593 Camillejordan and are named in his honour. Camille Jordan is not to be confused with the geodesist Wilhelm Jordan (Gauss–Jordan elimination) or the physicist Pascual Jordan (Jordan algebras). Bibliography Cours d'analyse de l'Ecole Polytechnique ; 1 Calcul différentiel (Gauthier-Villars, 1909) Cours d'analyse de l'Ecole Polytechnique ; 2 Calcul intégral (Gauthier-Villars, 1909) Cours d'analyse de l'Ecole Polytechnique ; 3 équations différentielles (Gauthier-Villars, 1909) Mémoire sur le nombre des valeurs des fonctions (1861–1869) Recherches sur les polyèdres (Gauthier-Villars, 1866) The collected works of Camille Jordan were published 1961–1964 in four volumes at Gauthier-Villars, Paris. See also Centered tree Frenet–Serret formulas Pochhammer contour References External links 1838 births 1922 deaths École Polytechnique alumni Mines Paris - PSL alumni Corps des mines Scientists from Lyon 19th-century French mathematicians Group theorists Linear algebraists Academic staff of the Collège de France Corresponding members of the Saint Petersburg Academy of Sciences Members of the French Academy of Sciences Foreign Members of the Royal Society Foreign associates of the National Academy of Sciences Members of the Ligue de la patrie française
https://en.wikipedia.org/wiki/Annulus%20%28mathematics%29	In mathematics, an annulus (: annuli or annuluses) is the region between two concentric circles. Informally, it is shaped like a ring or a hardware washer. The word "annulus" is borrowed from the Latin word anulus or annulus meaning 'little ring'. The adjectival form is annular (as in annular eclipse). The open annulus is topologically equivalent to both the open cylinder and the punctured plane. Area The area of an annulus is the difference in the areas of the larger circle of radius and the smaller one of radius : The area of an annulus is determined by the length of the longest line segment within the annulus, which is the chord tangent to the inner circle, in the accompanying diagram. That can be shown using the Pythagorean theorem since this line is tangent to the smaller circle and perpendicular to its radius at that point, so and are sides of a right-angled triangle with hypotenuse , and the area of the annulus is given by The area can also be obtained via calculus by dividing the annulus up into an infinite number of annuli of infinitesimal width and area and then integrating from to : The area of an annulus sector of angle , with measured in radians, is given by Complex structure In complex analysis an annulus in the complex plane is an open region defined as If is , the region is known as the punctured disk (a disk with a point hole in the center) of radius around the point . As a subset of the complex plane, an annulus can be considered as a Riemann surface. The complex structure of an annulus depends only on the ratio . Each annulus can be holomorphically mapped to a standard one centered at the origin and with outer radius 1 by the map The inner radius is then . The Hadamard three-circle theorem is a statement about the maximum value a holomorphic function may take inside an annulus. The Joukowsky transform conformally maps an annulus onto an ellipse with a slit cut between foci. See also References External links Annulus definition and properties With interactive animation Area of an annulus, formula With interactive animation Circles Elementary geometry Geometric shapes Planar surfaces
https://en.wikipedia.org/wiki/Submersion%20%28mathematics%29	In mathematics, a submersion is a differentiable map between differentiable manifolds whose differential is everywhere surjective. This is a basic concept in differential topology. The notion of a submersion is dual to the notion of an immersion. Definition Let M and N be differentiable manifolds and be a differentiable map between them. The map is a submersion at a point if its differential is a surjective linear map. In this case is called a regular point of the map , otherwise, is a critical point. A point is a regular value of if all points in the preimage are regular points. A differentiable map that is a submersion at each point is called a submersion. Equivalently, is a submersion if its differential has constant rank equal to the dimension of . A word of warning: some authors use the term critical point to describe a point where the rank of the Jacobian matrix of at is not maximal. Indeed, this is the more useful notion in singularity theory. If the dimension of is greater than or equal to the dimension of then these two notions of critical point coincide. But if the dimension of is less than the dimension of , all points are critical according to the definition above (the differential cannot be surjective) but the rank of the Jacobian may still be maximal (if it is equal to dim ). The definition given above is the more commonly used; e.g., in the formulation of Sard's theorem. Submersion theorem Given a submersion between smooth manifolds of dimensions and , for each there are surjective charts of around , and of around , such that restricts to a submersion which, when expressed in coordinates as , becomes an ordinary orthogonal projection. As an application, for each the corresponding fiber of , denoted can be equipped with the structure of a smooth submanifold of whose dimension is equal to the difference of the dimensions of and . The theorem is a consequence of the inverse function theorem (see Inverse function theorem#Giving a manifold structure). For example, consider given by The Jacobian matrix is This has maximal rank at every point except for . Also, the fibers are empty for , and equal to a point when . Hence we only have a smooth submersion and the subsets are two-dimensional smooth manifolds for . Examples Any projection Local diffeomorphisms Riemannian submersions The projection in a smooth vector bundle or a more general smooth fibration. The surjectivity of the differential is a necessary condition for the existence of a local trivialization. Maps between spheres One large class of examples of submersions are submersions between spheres of higher dimension, such as whose fibers have dimension . This is because the fibers (inverse images of elements ) are smooth manifolds of dimension . Then, if we take a path and take the pullback we get an example of a special kind of bordism, called a framed bordism. In fact, the framed cobordism groups are intimately related to
https://en.wikipedia.org/wiki/Yasha	Yasha may refer to: People with the name Gu Yasha (born 1990), Chinese footballer Nidhi Yasha (born 1983), Indian costume designer Yasha Asley (born 2003), British mathematics child prodigy Yasha Khalili (born 1988), Iranian footballer Yasha Levine (born 1981), Russian-American investigative journalist and author Yasha Malekzad (born 1984), English music video director and producer Yasha Manasherov (born 1980), Israeli Greco-Roman amateur wrestler Arts and entertainment Anime and manga Yasha (manga), a Japanese manga series by Akimi Yoshida Yasha Gozen, a Japanese one-shot manga by Ryoko Yamagishi Yashahime: Princess Half-Demon, a Japanese anime series and the sequel to Inuyasha Fictional characters Yasha, in the 1904 Russian play The Cherry Orchard by Anton Chekhov Yasha, in the 1990 Soviet adventure film Passport Yasha, in the 1993 Japanese anime film Yu Yu Hakusho: The Movie Yasha, in the 2012 Japanese action video game Asura's Wrath Yasha Nydoorin, in the American D&D web series Critical Role Yasha-ō, in the Japanese manga series RG Veda Other uses Yasha or yaksha, nature-spirits in Hindu and Buddhist mythology, sometimes depicted as demonic warriors See also Yash (disambiguation) Yaksha (disambiguation) Yeshua (disambiguation)
https://en.wikipedia.org/wiki/Decision%20mathematics	Decision mathematics may refer to: Discrete mathematics Decision theory, identifying the values, uncertainties and other issues relevant in a decision
https://en.wikipedia.org/wiki/Gr%C3%B6bner%20basis	In mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating set of an ideal in a polynomial ring over a field . A Gröbner basis allows many important properties of the ideal and the associated algebraic variety to be deduced easily, such as the dimension and the number of zeros when it is finite. Gröbner basis computation is one of the main practical tools for solving systems of polynomial equations and computing the images of algebraic varieties under projections or rational maps. Gröbner basis computation can be seen as a multivariate, non-linear generalization of both Euclid's algorithm for computing polynomial greatest common divisors, and Gaussian elimination for linear systems. Gröbner bases were introduced by Bruno Buchberger in his 1965 Ph.D. thesis, which also included an algorithm to compute them (Buchberger's algorithm). He named them after his advisor Wolfgang Gröbner. In 2007, Buchberger received the Association for Computing Machinery's Paris Kanellakis Theory and Practice Award for this work. However, the Russian mathematician Nikolai Günther had introduced a similar notion in 1913, published in various Russian mathematical journals. These papers were largely ignored by the mathematical community until their rediscovery in 1987 by Bodo Renschuch et al. An analogous concept for multivariate power series was developed independently by Heisuke Hironaka in 1964, who named them standard bases. This term has been used by some authors to also denote Gröbner bases. The theory of Gröbner bases has been extended by many authors in various directions. It has been generalized to other structures such as polynomials over principal ideal rings or polynomial rings, and also some classes of non-commutative rings and algebras, like Ore algebras. Tools Polynomial ring Gröbner bases are primarily defined for ideals in a polynomial ring over a field . Although the theory works for any field, most Gröbner basis computations are done either when is the field of rationals or the integers modulo a prime number. In the context of Gröbner bases, a nonzero polynomial in is commonly represented as a sum where the are nonzero elements of , called coefficients, and the are monomials (called power products by Buchberger and some of his followers) of the form where the are nonnegative integers. The vector is called the exponent vector of the monomial. When the list of the variables is fixed, the notation of monomials is often abbreviated as Monomials are uniquely defined by their exponent vectors, and, when a monomial ordering (see below) is fixed, a polynomial is uniquely represented by the ordered list of the ordered pairs formed by an exponent vector and the corresponding coefficient. This representation of polynomials is especially efficient for Gröbner basis computation in computers, although it is less convenient fo
https://en.wikipedia.org/wiki/Cauchy%E2%80%93Binet%20formula	In mathematics, specifically linear algebra, the Cauchy–Binet formula, named after Augustin-Louis Cauchy and Jacques Philippe Marie Binet, is an identity for the determinant of the product of two rectangular matrices of transpose shapes (so that the product is well-defined and square). It generalizes the statement that the determinant of a product of square matrices is equal to the product of their determinants. The formula is valid for matrices with the entries from any commutative ring. Statement Let A be an m×n matrix and B an n×m matrix. Write [n] for the set {1, ..., n}, and for the set of m-combinations of [n] (i.e., subsets of [n] of size m; there are of them). For , write A[m],S for the m×m matrix whose columns are the columns of A at indices from S, and BS,[m] for the m×m matrix whose rows are the rows of B at indices from S. The Cauchy–Binet formula then states Example: Taking m = 2 and n = 3, and matrices and , the Cauchy–Binet formula gives the determinant Indeed , and its determinant is which equals from the right hand side of the formula. Special cases If n < m then is the empty set, and the formula says that det(AB) = 0 (its right hand side is an empty sum); indeed in this case the rank of the m×m matrix AB is at most n, which implies that its determinant is zero. If n = m, the case where A and B are square matrices, (a singleton set), so the sum only involves S = [n], and the formula states that det(AB) = det(A)det(B). For m = 0, A and B are empty matrices (but of different shapes if n > 0), as is their product AB; the summation involves a single term S = Ø, and the formula states 1 = 1, with both sides given by the determinant of the 0×0 matrix. For m = 1, the summation ranges over the collection of the n different singletons taken from [n], and both sides of the formula give , the dot product of the pair of vectors represented by the matrices. The smallest value of m for which the formula states a non-trivial equality is m = 2; it is discussed in the article on the Binet–Cauchy identity. In the case n = 3 Let be three-dimensional vectors. In the case m > 3, the right-hand side always equals 0. A simple proof The following simple proof relies on two facts that can be proven in several different ways: For any the coefficient of in the polynomial is the sum of the principal minors of . If and is an matrix and an matrix, then . Now, if we compare the coefficient of in the equation , the left hand side will give the sum of the principal minors of while the right hand side will give the constant term of , which is simply , which is what the Cauchy–Binet formula states, i.e. Proof There are various kinds of proofs that can be given for the Cauchy−Binet formula. The proof below is based on formal manipulations only, and avoids using any particular interpretation of determinants, which may be taken to be defined by the Leibniz formula. Only their multilinearity with respect to rows and co
https://en.wikipedia.org/wiki/Dickson%27s%20lemma	In mathematics, Dickson's lemma states that every set of -tuples of natural numbers has finitely many minimal elements. This simple fact from combinatorics has become attributed to the American algebraist L. E. Dickson, who used it to prove a result in number theory about perfect numbers. However, the lemma was certainly known earlier, for example to Paul Gordan in his research on invariant theory. Example Let be a fixed number, and let be the set of pairs of numbers whose product is at least . When defined over the positive real numbers, has infinitely many minimal elements of the form , one for each positive number ; this set of points forms one of the branches of a hyperbola. The pairs on this hyperbola are minimal, because it is not possible for a different pair that belongs to to be less than or equal to in both of its coordinates. However, Dickson's lemma concerns only tuples of natural numbers, and over the natural numbers there are only finitely many minimal pairs. Every minimal pair of natural numbers has and , for if x were greater than K then (x − 1, y) would also belong to S, contradicting the minimality of (x, y), and symmetrically if y were greater than K then (x, y − 1) would also belong to S. Therefore, over the natural numbers, has at most minimal elements, a finite number. Formal statement Let be the set of non-negative integers (natural numbers), let n be any fixed constant, and let be the set of -tuples of natural numbers. These tuples may be given a pointwise partial order, the product order, in which if and only if for every . The set of tuples that are greater than or equal to some particular tuple forms a positive orthant with its apex at the given tuple. With this notation, Dickson's lemma may be stated in several equivalent forms: In every non-empty subset of there is at least one but no more than a finite number of elements that are minimal elements of for the pointwise partial order. For every infinite sequence of -tuples of natural numbers, there exist two indices such that holds with respect to the pointwise order. The partially ordered set does not contain infinite antichains nor infinite (strictly) descending sequences of -tuples. The partially ordered set is a well partial order. Every subset of may be covered by a finite set of positive orthants, whose apexes all belong to . Generalizations and applications Dickson used his lemma to prove that, for any given number , there can exist only a finite number of odd perfect numbers that have at most prime factors. However, it remains open whether there exist any odd perfect numbers at all. The divisibility relation among the P-smooth numbers, natural numbers whose prime factors all belong to the finite set P, gives these numbers the structure of a partially ordered set isomorphic to . Thus, for any set S of P-smooth numbers, there is a finite subset of S such that every element of S is divisible by one of the numbers in this subset. This
https://en.wikipedia.org/wiki/Monomial	In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: A monomial, also called power product, is a product of powers of variables with nonnegative integer exponents, or, in other words, a product of variables, possibly with repetitions. For example, is a monomial. The constant is a monomial, being equal to the empty product and to for any variable . If only a single variable is considered, this means that a monomial is either or a power of , with a positive integer. If several variables are considered, say, then each can be given an exponent, so that any monomial is of the form with non-negative integers (taking note that any exponent makes the corresponding factor equal to ). A monomial is a monomial in the first sense multiplied by a nonzero constant, called the coefficient of the monomial. A monomial in the first sense is a special case of a monomial in the second sense, where the coefficient is . For example, in this interpretation and are monomials (in the second example, the variables are and the coefficient is a complex number). In the context of Laurent polynomials and Laurent series, the exponents of a monomial may be negative, and in the context of Puiseux series, the exponents may be rational numbers. Since the word "monomial", as well as the word "polynomial", comes from the late Latin word "binomium" (binomial), by changing the prefix "bi-" (two in Latin), a monomial should theoretically be called a "mononomial". "Monomial" is a syncope by haplology of "mononomial". Comparison of the two definitions With either definition, the set of monomials is a subset of all polynomials that is closed under multiplication. Both uses of this notion can be found, and in many cases the distinction is simply ignored, see for instance examples for the first and second meaning. In informal discussions the distinction is seldom important, and tendency is towards the broader second meaning. When studying the structure of polynomials however, one often definitely needs a notion with the first meaning. This is for instance the case when considering a monomial basis of a polynomial ring, or a monomial ordering of that basis. An argument in favor of the first meaning is also that no obvious other notion is available to designate these values (the term power product is in use, in particular when monomial is used with the first meaning, but it does not make the absence of constants clear either), while the notion term of a polynomial unambiguously coincides with the second meaning of monomial. The remainder of this article assumes the first meaning of "monomial". Monomial basis The most obvious fact about monomials (first meaning) is that any polynomial is a linear combination of them, so they form a basis of the vector space of all polynomials, called the monomial basis - a fact of constant implicit use in mathematics. Number The number of monomials of
https://en.wikipedia.org/wiki/Induced%20representation	In group theory, the induced representation is a representation of a group, , which is constructed using a known representation of a subgroup . Given a representation of , the induced representation is, in a sense, the "most general" representation of that extends the given one. Since it is often easier to find representations of the smaller group than of , the operation of forming induced representations is an important tool to construct new representations. Induced representations were initially defined by Frobenius, for linear representations of finite groups. The idea is by no means limited to the case of finite groups, but the theory in that case is particularly well-behaved. Constructions Algebraic Let be a finite group and any subgroup of . Furthermore let be a representation of . Let be the index of in and let be a full set of representatives in of the left cosets in . The induced representation can be thought of as acting on the following space: Here each is an isomorphic copy of the vector space V whose elements are written as with . For each g in and each gi there is an hi in and j(i) in {1, ..., n} such that . (This is just another way of saying that is a full set of representatives.) Via the induced representation acts on as follows: where for each i. Alternatively, one can construct induced representations by extension of scalars: any K-linear representation of the group H can be viewed as a module V over the group ring K[H]. We can then define This latter formula can also be used to define for any group and subgroup , without requiring any finiteness. Examples For any group, the induced representation of the trivial representation of the trivial subgroup is the right regular representation. More generally the induced representation of the trivial representation of any subgroup is the permutation representation on the cosets of that subgroup. An induced representation of a one dimensional representation is called a monomial representation, because it can be represented as monomial matrices. Some groups have the property that all of their irreducible representations are monomial, the so-called monomial groups. Properties If is a subgroup of the group , then every -linear representation of can be viewed as a -linear representation of ; this is known as the restriction of to and denoted by . In the case of finite groups and finite-dimensional representations, the Frobenius reciprocity theorem states that, given representations of and of , the space of -equivariant linear maps from to has the same dimension over K as that of -equivariant linear maps from to . The universal property of the induced representation, which is also valid for infinite groups, is equivalent to the adjunction asserted in the reciprocity theorem. If is a representation of H and is the representation of G induced by , then there exists a -equivariant linear map with the following property: given any representa
https://en.wikipedia.org/wiki/Posterior%20probability	The posterior probability is a type of conditional probability that results from updating the prior probability with information summarized by the likelihood via an application of Bayes' rule. From an epistemological perspective, the posterior probability contains everything there is to know about an uncertain proposition (such as a scientific hypothesis, or parameter values), given prior knowledge and a mathematical model describing the observations available at a particular time. After the arrival of new information, the current posterior probability may serve as the prior in another round of Bayesian updating. In the context of Bayesian statistics, the posterior probability distribution usually describes the epistemic uncertainty about statistical parameters conditional on a collection of observed data. From a given posterior distribution, various point and interval estimates can be derived, such as the maximum a posteriori (MAP) or the highest posterior density interval (HPDI). But while conceptually simple, the posterior distribution is generally not tractable and therefore needs to be either analytically or numerically approximated. Definition in the distributional case In variational Bayesian methods, the posterior probability is the probability of the parameters given the evidence , and is denoted . It contrasts with the likelihood function, which is the probability of the evidence given the parameters: . The two are related as follows: Given a prior belief that a probability distribution function is and that the observations have a likelihood , then the posterior probability is defined as , where is the normalizing constant and is calculated as for continuous , or by summing over all possible values of for discrete . The posterior probability is therefore proportional to the product Likelihood · Prior probability. Example Suppose there is a school with 60% boys and 40% girls as students. The girls wear trousers or skirts in equal numbers; all boys wear trousers. An observer sees a (random) student from a distance; all the observer can see is that this student is wearing trousers. What is the probability this student is a girl? The correct answer can be computed using Bayes' theorem. The event is that the student observed is a girl, and the event is that the student observed is wearing trousers. To compute the posterior probability , we first need to know: , or the probability that the student is a girl regardless of any other information. Since the observer sees a random student, meaning that all students have the same probability of being observed, and the percentage of girls among the students is 40%, this probability equals 0.4. , or the probability that the student is not a girl (i.e. a boy) regardless of any other information ( is the complementary event to ). This is 60%, or 0.6. , or the probability of the student wearing trousers given that the student is a girl. As they are as likely to wear skirts as tro
https://en.wikipedia.org/wiki/Proof%20by%20infinite%20descent	In mathematics, a proof by infinite descent, also known as Fermat's method of descent, is a particular kind of proof by contradiction used to show that a statement cannot possibly hold for any number, by showing that if the statement were to hold for a number, then the same would be true for a smaller number, leading to an infinite descent and ultimately a contradiction. It is a method which relies on the well-ordering principle, and is often used to show that a given equation, such as a Diophantine equation, has no solutions. Typically, one shows that if a solution to a problem existed, which in some sense was related to one or more natural numbers, it would necessarily imply that a second solution existed, which was related to one or more 'smaller' natural numbers. This in turn would imply a third solution related to smaller natural numbers, implying a fourth solution, therefore a fifth solution, and so on. However, there cannot be an infinity of ever-smaller natural numbers, and therefore by mathematical induction, the original premise—that any solution exists—is incorrect: its correctness produces a contradiction. An alternative way to express this is to assume one or more solutions or examples exists, from which a smallest solution or example—a minimal counterexample—can then be inferred. Once there, one would try to prove that if a smallest solution exists, then it must imply the existence of a smaller solution (in some sense), which again proves that the existence of any solution would lead to a contradiction. The earliest uses of the method of infinite descent appear in Euclid's Elements. A typical example is Proposition 31 of Book 7, in which Euclid proves that every composite integer is divided (in Euclid's terminology "measured") by some prime number. The method was much later developed by Fermat, who coined the term and often used it for Diophantine equations. Two typical examples are showing the non-solvability of the Diophantine equation and proving Fermat's theorem on sums of two squares, which states that an odd prime p can be expressed as a sum of two squares when (see Modular arithmetic and proof by infinite descent). In this way Fermat was able to show the non-existence of solutions in many cases of Diophantine equations of classical interest (for example, the problem of four perfect squares in arithmetic progression). In some cases, to the modern eye, his "method of infinite descent" is an exploitation of the inversion of the doubling function for rational points on an elliptic curve E. The context is of a hypothetical non-trivial rational point on E. Doubling a point on E roughly doubles the length of the numbers required to write it (as number of digits), so that a "halving" a point gives a rational with smaller terms. Since the terms are positive, they cannot decrease forever. Number theory In the number theory of the twentieth century, the infinite descent method was taken up again, and pushed to a point where it
https://en.wikipedia.org/wiki/Cayley%20graph	In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group, is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem (named after Arthur Cayley), and uses a specified set of generators for the group. It is a central tool in combinatorial and geometric group theory. The structure and symmetry of Cayley graphs makes them particularly good candidates for constructing expander graphs. Definition Let be a group and be a generating set of . The Cayley graph is an edge-colored directed graph constructed as follows: Each element of is assigned a vertex: the vertex set of is identified with Each element of is assigned a color . For every and , there is a directed edge of color from the vertex corresponding to to the one corresponding to . Not every convention requires that generate the group. If is not a generating set for , then is disconnected and each connected component represents a coset of the subgroup generated by . If an element of is its own inverse, then it is typically represented by an undirected edge. The set is often assumed to be finite, especially in geometric group theory, which corresponds to being locally finite and being finitely generated. The set is sometimes assumed to be symmetric () and not containing the group identity element. In this case, the uncolored Cayley graph can be represented as a simple undirected graph. Examples Suppose that is the infinite cyclic group and the set consists of the standard generator 1 and its inverse (−1 in the additive notation); then the Cayley graph is an infinite path. Similarly, if is the finite cyclic group of order and the set consists of two elements, the standard generator of and its inverse, then the Cayley graph is the cycle . More generally, the Cayley graphs of finite cyclic groups are exactly the circulant graphs. The Cayley graph of the direct product of groups (with the cartesian product of generating sets as a generating set) is the cartesian product of the corresponding Cayley graphs. Thus the Cayley graph of the abelian group with the set of generators consisting of four elements is the infinite grid on the plane , while for the direct product with similar generators the Cayley graph is the finite grid on a torus. A Cayley graph of the dihedral group on two generators and is depicted to the left. Red arrows represent composition with . Since is self-inverse, the blue lines, which represent composition with , are undirected. Therefore the graph is mixed: it has eight vertices, eight arrows, and four edges. The Cayley table of the group can be derived from the group presentation A different Cayley graph of is shown on the right. is still the horizontal reflection and is represented by blue lines, and is a diagonal reflection and is represented by pink lines. As both reflections are self-inverse the Cayley graph on the rig
https://en.wikipedia.org/wiki/Closed%20form	Closed form may refer to: Mathematics Closed-form expression, a finitary expression Closed differential form, a differential form whose exterior derivative is the zero form , meaning . Poetry In poetry analysis, a type of poetry that exhibits regular structure, such as meter or a rhyming pattern Trobar clus, an allusive and obscure style adopted by some 12th-century troubadours Mathematics disambiguation pages
https://en.wikipedia.org/wiki/Peano%20curve	In geometry, the Peano curve is the first example of a space-filling curve to be discovered, by Giuseppe Peano in 1890. Peano's curve is a surjective, continuous function from the unit interval onto the unit square, however it is not injective. Peano was motivated by an earlier result of Georg Cantor that these two sets have the same cardinality. Because of this example, some authors use the phrase "Peano curve" to refer more generally to any space-filling curve. Construction Peano's curve may be constructed by a sequence of steps, where the ith step constructs a set Si of squares, and a sequence Pi of the centers of the squares, from the set and sequence constructed in the previous step. As a base case, S0 consists of the single unit square, and P0 is the one-element sequence consisting of its center point. In step i, each square s of Si − 1 is partitioned into nine smaller equal squares, and its center point c is replaced by a contiguous subsequence of the centers of these nine smaller squares. This subsequence is formed by grouping the nine smaller squares into three columns, ordering the centers contiguously within each column, and then ordering the columns from one side of the square to the other, in such a way that the distance between each consecutive pair of points in the subsequence equals the side length of the small squares. There are four such orderings possible: Left three centers bottom to top, middle three centers top to bottom, and right three centers bottom to top Right three centers bottom to top, middle three centers top to bottom, and left three centers bottom to top Left three centers top to bottom, middle three centers bottom to top, and right three centers top to bottom Right three centers top to bottom, middle three centers bottom to top, and left three centers top to bottom Among these four orderings, the one for s is chosen in such a way that the distance between the first point of the ordering and its predecessor in Pi also equals the side length of the small squares. If c was the first point in its ordering, then the first of these four orderings is chosen for the nine centers that replace c. The Peano curve itself is the limit of the curves through the sequences of square centers, as i goes to infinity. Variants In the definition of the Peano curve, it is possible to perform some or all of the steps by making the centers of each row of three squares be contiguous, rather than the centers of each column of squares. These choices lead to many different variants of the Peano curve. A "multiple radix" variant of this curve with different numbers of subdivisions in different directions can be used to fill rectangles of arbitrary shapes. The Hilbert curve is a simpler variant of the same idea, based on subdividing squares into four equal smaller squares instead of into nine equal smaller squares. References Theory of continuous functions Fractal curves
https://en.wikipedia.org/wiki/Random%20graph	In mathematics, random graph is the general term to refer to probability distributions over graphs. Random graphs may be described simply by a probability distribution, or by a random process which generates them. The theory of random graphs lies at the intersection between graph theory and probability theory. From a mathematical perspective, random graphs are used to answer questions about the properties of typical graphs. Its practical applications are found in all areas in which complex networks need to be modeled – many random graph models are thus known, mirroring the diverse types of complex networks encountered in different areas. In a mathematical context, random graph refers almost exclusively to the Erdős–Rényi random graph model. In other contexts, any graph model may be referred to as a random graph. Models A random graph is obtained by starting with a set of n isolated vertices and adding successive edges between them at random. The aim of the study in this field is to determine at what stage a particular property of the graph is likely to arise. Different random graph models produce different probability distributions on graphs. Most commonly studied is the one proposed by Edgar Gilbert, denoted G(n,p), in which every possible edge occurs independently with probability 0 < p < 1. The probability of obtaining any one particular random graph with m edges is with the notation . A closely related model, the Erdős–Rényi model denoted G(n,M), assigns equal probability to all graphs with exactly M edges. With 0 ≤ M ≤ N, G(n,M) has elements and every element occurs with probability . The latter model can be viewed as a snapshot at a particular time (M) of the random graph process , which is a stochastic process that starts with n vertices and no edges, and at each step adds one new edge chosen uniformly from the set of missing edges. If instead we start with an infinite set of vertices, and again let every possible edge occur independently with probability 0 < p < 1, then we get an object G called an infinite random graph. Except in the trivial cases when p is 0 or 1, such a G almost surely has the following property: Given any n + m elements , there is a vertex c in V that is adjacent to each of and is not adjacent to any of . It turns out that if the vertex set is countable then there is, up to isomorphism, only a single graph with this property, namely the Rado graph. Thus any countably infinite random graph is almost surely the Rado graph, which for this reason is sometimes called simply the random graph. However, the analogous result is not true for uncountable graphs, of which there are many (nonisomorphic) graphs satisfying the above property. Another model, which generalizes Gilbert's random graph model, is the random dot-product model. A random dot-product graph associates with each vertex a real vector. The probability of an edge uv between any vertices u and v is some function of the dot product u • v of their respec
https://en.wikipedia.org/wiki/Girard%20Desargues	Girard Desargues (; 21 February 1591September 1661) was a French mathematician and engineer, who is considered one of the founders of projective geometry. Desargues' theorem, the Desargues graph, and the crater Desargues on the Moon are named in his honour. Biography Born in Lyon, Desargues came from a family devoted to service to the French crown. His father was a royal notary, an investigating commissioner of the Seneschal's court in Lyon (1574), the collector of the tithes on ecclesiastical revenues for the city of Lyon (1583) and for the diocese of Lyon. Girard Desargues worked as an architect from 1645. Prior to that, he had worked as a tutor and may have served as an engineer and technical consultant in the entourage of Richelieu. As an architect, Desargues planned several private and public buildings in Paris and Lyon. As an engineer, he designed a system for raising water that he installed near Paris. It was based on the use of the epicycloidal wheel, the principle of which was unrecognized at the time. His research on perspective and geometrical projections can be seen as a culmination of centuries of scientific inquiry across the classical epoch in optics that stretched from al-Hasan Ibn al-Haytham (Alhazen) to Johannes Kepler, and going beyond a mere synthesis of these traditions with Renaissance perspective theories and practices. His work was rediscovered and republished in 1864. A collection of his works was published in 1951, and the 1864 compilation remains in print. One notable work, often cited by others in mathematics, is "Rough draft for an essay on the results of taking plane sections of a cone" (1639). Late in his life, Desargues published a paper with the cryptic title of DALG. The most common theory about what this stands for is Des Argues, Lyonnais, Géometre (proposed by Henri Brocard). He died in Lyon. See also Desarguesian plane, non-Desarguesian plane Desargues' theorem Desargues graph Desargues configuration Desargues (crater) Perspective (graphical) / Perspective (visual) Optics References J. V. Field & J. J. Gray (1987) The Geometrical Work of Girard Desargues, Springer-Verlag, . René Taton (1962) Sur la naissance de Girard Desargues., Revue d'histoire des sciences et de leurs applications Tome 15 n°2. pp. 165–166. External links Richard Westfall, Gerard Desargues, The Galileo Project Gerard Desargues, Brouillon Project d'une Atteinte aux Evenemens des Rencontres du Cone avec un Plan 1591 births 1661 deaths 17th-century French mathematicians 17th-century French architects French engineers French geometers Architects from Lyon
https://en.wikipedia.org/wiki/Desargues%27s%20theorem	In projective geometry, Desargues's theorem, named after Girard Desargues, states: Two triangles are in perspective axially if and only if they are in perspective centrally. Denote the three vertices of one triangle by and , and those of the other by and . Axial perspectivity means that lines and meet in a point, lines and meet in a second point, and lines and meet in a third point, and that these three points all lie on a common line called the axis of perspectivity. Central perspectivity means that the three lines and are concurrent, at a point called the center of perspectivity. This intersection theorem is true in the usual Euclidean plane but special care needs to be taken in exceptional cases, as when a pair of sides are parallel, so that their "point of intersection" recedes to infinity. Commonly, to remove these exceptions, mathematicians "complete" the Euclidean plane by adding points at infinity, following Jean-Victor Poncelet. This results in a projective plane. Desargues's theorem is true for the real projective plane and for any projective space defined arithmetically from a field or division ring; that includes any projective space of dimension greater than two or in which Pappus's theorem holds. However, there are many "non-Desarguesian planes", in which Desargues's theorem is false. History Desargues never published this theorem, but it appeared in an appendix entitled Universal Method of M. Desargues for Using Perspective (Manière universelle de M. Desargues pour practiquer la perspective) to a practical book on the use of perspective published in 1648. by his friend and pupil Abraham Bosse (1602–1676). Coordinatization The importance of Desargues's theorem in abstract projective geometry is due especially to the fact that a projective space satisfies that theorem if and only if it is isomorphic to a projective space defined over a field or division ring. Projective versus affine spaces In an affine space such as the Euclidean plane a similar statement is true, but only if one lists various exceptions involving parallel lines. Desargues's theorem is therefore one of the simplest geometric theorems whose natural home is in projective rather than affine space. Self-duality By definition, two triangles are perspective if and only if they are in perspective centrally (or, equivalently according to this theorem, in perspective axially). Note that perspective triangles need not be similar. Under the standard duality of plane projective geometry (where points correspond to lines and collinearity of points corresponds to concurrency of lines), the statement of Desargues's theorem is self-dual: axial perspectivity is translated into central perspectivity and vice versa. The Desargues configuration (below) is a self-dual configuration. This self-duality in the statement is due to the usual modern way of writing the theorem. Historically, the theorem only read, "In a projective space, a pair of centrally perspective
https://en.wikipedia.org/wiki/Method%20of%20complements	In mathematics and computing, the method of complements is a technique to encode a symmetric range of positive and negative integers in a way that they can use the same algorithm (or mechanism) for addition throughout the whole range. For a given number of places half of the possible representations of numbers encode the positive numbers, the other half represents their respective additive inverses. The pairs of mutually additive inverse numbers are called complements. Thus subtraction of any number is implemented by adding its complement. Changing the sign of any number is encoded by generating its complement, which can be done by a very simple and efficient algorithm. This method was commonly used in mechanical calculators and is still used in modern computers. The generalized concept of the radix complement (as described below) is also valuable in number theory, such as in Midy's theorem. The nines' complement of a number given in decimal representation is formed by replacing each digit with nine minus that digit. To subtract a decimal number y (the subtrahend) from another number x (the minuend) two methods may be used: In the first method the nines' complement of x is added to y. Then the nines' complement of the result obtained is formed to produce the desired result. In the second method the nines' complement of y is added to x and one is added to the sum. The leftmost digit '1' of the result is then discarded. Discarding the leftmost '1' is especially convenient on calculators or computers that use a fixed number of digits: there is nowhere for it to go so it is simply lost during the calculation. The nines' complement plus one is known as the ten's complement. The method of complements can be extended to other number bases (radices); in particular, it is used on most digital computers to perform subtraction, represent negative numbers in base 2 or binary arithmetic and test underflow and overflow in calculation. Numeric complements The radix complement of an digit number in radix is defined as . In practice, the radix complement is more easily obtained by adding 1 to the diminished radix complement, which is . While this seems equally difficult to calculate as the radix complement, it is actually simpler since is simply the digit repeated times. This is because (see also Geometric series Formula). Knowing this, the diminished radix complement of a number can be found by complementing each digit with respect to , i.e. subtracting each digit in from . The subtraction of from using diminished radix complements may be performed as follows. Add the diminished radix complement of to to obtain or equivalently , which is the diminished radix complement of . Further taking the diminished radix complement of results in the desired answer of . Alternatively using the radix complement, may be obtained by adding the radix complement of to to obtain or . Assuming , the result will be greater or equal to and dropping the
https://en.wikipedia.org/wiki/Lucas%20primality%20test	In computational number theory, the Lucas test is a primality test for a natural number n; it requires that the prime factors of n − 1 be already known. It is the basis of the Pratt certificate that gives a concise verification that n is prime. Concepts Let n be a positive integer. If there exists an integer a, 1 < a < n, such that and for every prime factor q of n − 1 then n is prime. If no such number a exists, then n is either 1, 2, or composite. The reason for the correctness of this claim is as follows: if the first equivalence holds for a, we can deduce that a and n are coprime. If a also survives the second step, then the order of a in the group (Z/nZ)* is equal to n−1, which means that the order of that group is n−1 (because the order of every element of a group divides the order of the group), implying that n is prime. Conversely, if n is prime, then there exists a primitive root modulo n, or generator of the group (Z/nZ). Such a generator has order \|(Z/nZ)\| = n−1 and both equivalences will hold for any such primitive root. Note that if there exists an a < n such that the first equivalence fails, a is called a Fermat witness for the compositeness of n. Example For example, take n = 71. Then n − 1 = 70 and the prime factors of 70 are 2, 5 and 7. We randomly select an a=17 < n. Now we compute: For all integers a it is known that Therefore, the multiplicative order of 17 (mod 71) is not necessarily 70 because some factor of 70 may also work above. So check 70 divided by its prime factors: Unfortunately, we get that 1710≡1 (mod 71). So we still don't know if 71 is prime or not. We try another random a, this time choosing a = 11. Now we compute: Again, this does not show that the multiplicative order of 11 (mod 71) is 70 because some factor of 70 may also work. So check 70 divided by its prime factors: So the multiplicative order of 11 (mod 71) is 70, and thus 71 is prime. (To carry out these modular exponentiations, one could use a fast exponentiation algorithm like binary or addition-chain exponentiation). Algorithm The algorithm can be written in pseudocode as follows: algorithm lucas_primality_test is input: n > 2, an odd integer to be tested for primality. k, a parameter that determines the accuracy of the test. output: prime if n is prime, otherwise composite or possibly composite. determine the prime factors of n−1. LOOP1: repeat k times: pick a randomly in the range [2, n − 1] return composite else LOOP2: for all prime factors q of n−1: if we checked this equality for all prime factors of n−1 then return prime else continue LOOP2 else continue LOOP1 return possibly composite. See also Édouard Lucas, for whom this test is name
https://en.wikipedia.org/wiki/Serre%20duality	In algebraic geometry, a branch of mathematics, Serre duality is a duality for the coherent sheaf cohomology of algebraic varieties, proved by Jean-Pierre Serre. The basic version applies to vector bundles on a smooth projective variety, but Alexander Grothendieck found wide generalizations, for example to singular varieties. On an n-dimensional variety, the theorem says that a cohomology group is the dual space of another one, . Serre duality is the analog for coherent sheaf cohomology of Poincaré duality in topology, with the canonical line bundle replacing the orientation sheaf. The Serre duality theorem is also true in complex geometry more generally, for compact complex manifolds that are not necessarily projective complex algebraic varieties. In this setting, the Serre duality theorem is an application of Hodge theory for Dolbeault cohomology, and may be seen as a result in the theory of elliptic operators. These two different interpretations of Serre duality coincide for non-singular projective complex algebraic varieties, by an application of Dolbeault's theorem relating sheaf cohomology to Dolbeault cohomology. Serre duality for vector bundles Algebraic theorem Let X be a smooth variety of dimension n over a field k. Define the canonical line bundle to be the bundle of n-forms on X, the top exterior power of the cotangent bundle: Suppose in addition that X is proper (for example, projective) over k. Then Serre duality says: for an algebraic vector bundle E on X and an integer i, there is a natural isomorphism: of finite-dimensional k-vector spaces. Here denotes the tensor product of vector bundles. It follows that the dimensions of the two cohomology groups are equal: As in Poincaré duality, the isomorphism in Serre duality comes from the cup product in sheaf cohomology. Namely, the composition of the cup product with a natural trace map on is a perfect pairing: The trace map is the analog for coherent sheaf cohomology of integration in de Rham cohomology. Differential-geometric theorem Serre also proved the same duality statement for X a compact complex manifold and E a holomorphic vector bundle. Here, the Serre duality theorem is a consequence of Hodge theory. Namely, on a compact complex manifold equipped with a Riemannian metric, there is a Hodge star operator: where . Additionally, since is complex, there is a splitting of the complex differential forms into forms of type . The Hodge star operator (extended complex-linearly to complex-valued differential forms) interacts with this grading as: Notice that the holomorphic and anti-holomorphic indices have switched places. There is a conjugation on complex differential forms which interchanges forms of type and , and if one defines the conjugate-linear Hodge star operator by then we have: Using the conjugate-linear Hodge star, one may define a Hermitian -inner product on complex differential forms, by: where now is an -form, and in particular a complex-valued -for
https://en.wikipedia.org/wiki/Oscar%20Zariski	Oscar Zariski (April 24, 1899 – July 4, 1986) was a Russian-born American mathematician and one of the most influential algebraic geometers of the 20th century. Education Zariski was born Oscher (also transliterated as Ascher or Osher) Zaritsky to a Jewish family (his parents were Bezalel Zaritsky and Hanna Tennenbaum) and in 1918 studied at the University of Kyiv. He left Kyiv in 1920 to study at the University of Rome where he became a disciple of the Italian school of algebraic geometry, studying with Guido Castelnuovo, Federigo Enriques and Francesco Severi. Zariski wrote a doctoral dissertation in 1924 on a topic in Galois theory, which was proposed to him by Castelnuovo. At the time of his dissertation publication, he changed his name to Oscar Zariski. Johns Hopkins University years Zariski emigrated to the United States in 1927 supported by Solomon Lefschetz. He had a position at Johns Hopkins University where he became professor in 1937. During this period, he wrote Algebraic Surfaces as a summation of the work of the Italian school. The book was published in 1935 and reissued 36 years later, with detailed notes by Zariski's students that illustrated how the field of algebraic geometry had changed. It is still an important reference. It seems to have been this work that set the seal of Zariski's discontent with the approach of the Italians to birational geometry. He addressed the question of rigour by recourse to commutative algebra. The Zariski topology, as it was later known, is adequate for biregular geometry, where varieties are mapped by polynomial functions. That theory is too limited for algebraic surfaces, and even for curves with singular points. A rational map is to a regular map as a rational function is to a polynomial: it may be indeterminate at some points. In geometric terms, one has to work with functions defined on some open, dense set of a given variety. The description of the behaviour on the complement may require infinitely near points to be introduced to account for limiting behaviour along different directions. This introduces a need, in the surface case, to use also valuation theory to describe the phenomena such as blowing up (balloon-style, rather than explosively). Harvard University years After spending a year 1946–1947 at the University of Illinois at Urbana–Champaign, Zariski became professor at Harvard University in 1947 where he remained until his retirement in 1969. In 1945, he fruitfully discussed foundational matters for algebraic geometry with André Weil. Weil's interest was in putting an abstract variety theory in place, to support the use of the Jacobian variety in his proof of the Riemann hypothesis for curves over finite fields, a direction rather oblique to Zariski's interests. The two sets of foundations weren't reconciled at that point. At Harvard, Zariski's students included Shreeram Abhyankar, Heisuke Hironaka, David Mumford, Michael Artin and Steven Kleiman—thus spanning the main areas o
https://en.wikipedia.org/wiki/List%20of%20mathematical%20examples	This page will attempt to list examples in mathematics. To qualify for inclusion, an article should be about a mathematical object with a fair amount of concreteness. Usually a definition of an abstract concept, a theorem, or a proof would not be an "example" as the term should be understood here (an elegant proof of an isolated but particularly striking fact, as opposed to a proof of a general theorem, could perhaps be considered an "example"). The discussion page for list of mathematical topics has some comments on this. Eventually this page may have its own discussion page. This page links to itself in order that edits to this page will be included among related changes when the user clicks on that button. The concrete example within the article titled Rao-Blackwell theorem is perhaps one of the best ways for a probabilist ignorant of statistical inference to get a quick impression of the flavor of that subject. Uncategorized examples, alphabetized Alexander horned sphere All horses are the same color Cantor function Cantor set Checking if a coin is biased Concrete illustration of the central limit theorem Differential equations of mathematical physics Dirichlet function Discontinuous linear map Efron's non-transitive dice Example of a game without a value Examples of contour integration Examples of differential equations Examples of generating functions Examples of groups List of the 230 crystallographic 3D space groups Examples of Markov chains Examples of vector spaces Fano plane Frieze group Gray graph Hall–Janko graph Higman–Sims graph Hilbert matrix Illustration of a low-discrepancy sequence Illustration of the central limit theorem An infinitely differentiable function that is not analytic Leech lattice Lewy's example on PDEs List of finite simple groups Long line Normally distributed and uncorrelated does not imply independent Pairwise independence of random variables need not imply mutual independence. Petersen graph Sierpinski space Simple example of Azuma's inequality for coin flips Proof that 22/7 exceeds π Solenoid (mathematics) Sorgenfrey plane Stein's example Three cards and a top hat Topologist's sine curve Tsirelson space Tutte eight cage Weierstrass function Wilkinson's polynomial Wallpaper group Uses of trigonometry (The "examples" in that article are not mathematical objects, i.e., numbers, functions, equations, sets, etc., but applications of trigonometry or scientific fields to which trigonometry is applied.) Specialized lists of mathematical examples List of algebraic surfaces List of curves List of complexity classes List of examples in general topology List of finite simple groups List of Fourier-related transforms List of mathematical functions List of knots List of mathematical knots and links List of manifolds List of mathematical shapes List of matrices List of numbers List of polygons, polyhedra and polytopes List of prime numbers —not merely a numerical table, but a list of va
https://en.wikipedia.org/wiki/Cumulant	In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the moments of the distribution. Any two probability distributions whose moments are identical will have identical cumulants as well, and vice versa. The first cumulant is the mean, the second cumulant is the variance, and the third cumulant is the same as the third central moment. But fourth and higher-order cumulants are not equal to central moments. In some cases theoretical treatments of problems in terms of cumulants are simpler than those using moments. In particular, when two or more random variables are statistically independent, the -th-order cumulant of their sum is equal to the sum of their -th-order cumulants. As well, the third and higher-order cumulants of a normal distribution are zero, and it is the only distribution with this property. Just as for moments, where joint moments are used for collections of random variables, it is possible to define joint cumulants. Definition The cumulants of a random variable are defined using the cumulant-generating function , which is the natural logarithm of the moment-generating function: The cumulants are obtained from a power series expansion of the cumulant generating function: This expansion is a Maclaurin series, so the -th cumulant can be obtained by differentiating the above expansion times and evaluating the result at zero: If the moment-generating function does not exist, the cumulants can be defined in terms of the relationship between cumulants and moments discussed later. Alternative definition of the cumulant generating function Some writers prefer to define the cumulant-generating function as the natural logarithm of the characteristic function, which is sometimes also called the second characteristic function, An advantage of —in some sense the function evaluated for purely imaginary arguments—is that is well defined for all real values of even when is not well defined for all real values of , such as can occur when there is "too much" probability that has a large magnitude. Although the function will be well defined, it will nonetheless mimic in terms of the length of its Maclaurin series, which may not extend beyond (or, rarely, even to) linear order in the argument , and in particular the number of cumulants that are well defined will not change. Nevertheless, even when does not have a long Maclaurin series, it can be used directly in analyzing and, particularly, adding random variables. Both the Cauchy distribution (also called the Lorentzian) and more generally, stable distributions (related to the Lévy distribution) are examples of distributions for which the power-series expansions of the generating functions have only finitely many well-defined terms. Some basic properties The -th cumulant of (the distribution of) a random variable enjoys the following properties: If and is constant (i.e. not random) then
https://en.wikipedia.org/wiki/Unicoherent%20space	In mathematics, a unicoherent space is a topological space that is connected and in which the following property holds: For any closed, connected with , the intersection is connected. For example, any closed interval on the real line is unicoherent, but a circle is not. If a unicoherent space is more strongly hereditarily unicoherent (meaning that every subcontinuum is unicoherent) and arcwise connected, then it is called a dendroid. If in addition it is locally connected then it is called a dendrite. The Phragmen–Brouwer theorem states that, for locally connected spaces, unicoherence is equivalent to a separation property of the closed sets of the space. References External links General topology Trees (topology)
https://en.wikipedia.org/wiki/Regular%20representation	In mathematics, and in particular the theory of group representations, the regular representation of a group G is the linear representation afforded by the group action of G on itself by translation. One distinguishes the left regular representation λ given by left translation and the right regular representation ρ given by the inverse of right translation. Finite groups For a finite group G, the left regular representation λ (over a field K) is a linear representation on the K-vector space V freely generated by the elements of G, i. e. they can be identified with a basis of V. Given g ∈ G, λg is the linear map determined by its action on the basis by left translation by g, i.e. For the right regular representation ρ, an inversion must occur in order to satisfy the axioms of a representation. Specifically, given g ∈ G, ρg is the linear map on V determined by its action on the basis by right translation by g−1, i.e. Alternatively, these representations can be defined on the K-vector space W of all functions . It is in this form that the regular representation is generalized to topological groups such as Lie groups. The specific definition in terms of W is as follows. Given a function and an element g ∈ G, and Significance of the regular representation of a group Every group G acts on itself by translations. If we consider this action as a permutation representation it is characterised as having a single orbit and stabilizer the identity subgroup {e} of G. The regular representation of G, for a given field K, is the linear representation made by taking this permutation representation as a set of basis vectors of a vector space over K. The significance is that while the permutation representation doesn't decompose – it is transitive – the regular representation in general breaks up into smaller representations. For example, if G is a finite group and K is the complex number field, the regular representation decomposes as a direct sum of irreducible representations, with each irreducible representation appearing in the decomposition with multiplicity its dimension. The number of these irreducibles is equal to the number of conjugacy classes of G. The above fact can be explained by character theory. Recall that the character of the regular representation χ(g) is the number of fixed points of g acting on the regular representation V. It means the number of fixed points χ(g) is zero when g is not id and \|G\| otherwise. Let V have the decomposition ⊕aiVi where Vi's are irreducible representations of G and ai's are the corresponding multiplicities. By character theory, the multiplicity ai can be computed as which means the multiplicity of each irreducible representation is its dimension. The article on group rings articulates the regular representation for finite groups, as well as showing how the regular representation can be taken to be a module. Module theory point of view To put the construction more abstractly, the group ring K[G] is c
https://en.wikipedia.org/wiki/Free%20product	In mathematics, specifically group theory, the free product is an operation that takes two groups G and H and constructs a new The result contains both G and H as subgroups, is generated by the elements of these subgroups, and is the “universal” group having these properties, in the sense that any two homomorphisms from G and H into a group K factor uniquely through a homomorphism from to K. Unless one of the groups G and H is trivial, the free product is always infinite. The construction of a free product is similar in spirit to the construction of a free group (the universal group with a given set of generators). The free product is the coproduct in the category of groups. That is, the free product plays the same role in group theory that disjoint union plays in set theory, or that the direct sum plays in module theory. Even if the groups are commutative, their free product is not, unless one of the two groups is the trivial group. Therefore, the free product is not the coproduct in the category of abelian groups. The free product is important in algebraic topology because of van Kampen's theorem, which states that the fundamental group of the union of two path-connected topological spaces whose intersection is also path-connected is always an amalgamated free product of the fundamental groups of the spaces. In particular, the fundamental group of the wedge sum of two spaces (i.e. the space obtained by joining two spaces together at a single point) is, under certain conditions given in the Seifert van-Kampen theorem, the free product of the fundamental groups of the spaces. Free products are also important in Bass–Serre theory, the study of groups acting by automorphisms on trees. Specifically, any group acting with finite vertex stabilizers on a tree may be constructed from finite groups using amalgamated free products and HNN extensions. Using the action of the modular group on a certain tessellation of the hyperbolic plane, it follows from this theory that the modular group is isomorphic to the free product of cyclic groups of orders 4 and 6 amalgamated over a cyclic group of order 2. Construction If G and H are groups, a word in G and H is a product of the form where each si is either an element of G or an element of H. Such a word may be reduced using the following operations: Remove an instance of the identity element (of either G or H). Replace a pair of the form g1g2 by its product in G, or a pair h1h2 by its product in H. Every reduced word is an alternating product of elements of G and elements of H, e.g. The free product G ∗ H is the group whose elements are the reduced words in G and H, under the operation of concatenation followed by reduction. For example, if G is the infinite cyclic group , and H is the infinite cyclic group , then every element of G ∗ H is an alternating product of powers of x with powers of y. In this case, G ∗ H is isomorphic to the free group generated by x and y. Presentation Suppose that
https://en.wikipedia.org/wiki/Constructive%20proof	In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof (also known as an existence proof or pure existence theorem), which proves the existence of a particular kind of object without providing an example. For avoiding confusion with the stronger concept that follows, such a constructive proof is sometimes called an effective proof. A constructive proof may also refer to the stronger concept of a proof that is valid in constructive mathematics. Constructivism is a mathematical philosophy that rejects all proof methods that involve the existence of objects that are not explicitly built. This excludes, in particular, the use of the law of the excluded middle, the axiom of infinity, and the axiom of choice, and induces a different meaning for some terminology (for example, the term "or" has a stronger meaning in constructive mathematics than in classical). Some non-constructive proofs show that if a certain proposition is false, a contradiction ensues; consequently the proposition must be true (proof by contradiction). However, the principle of explosion (ex falso quodlibet) has been accepted in some varieties of constructive mathematics, including intuitionism. Constructive proofs can be seen as defining certified mathematical algorithms: this idea is explored in the Brouwer–Heyting–Kolmogorov interpretation of constructive logic, the Curry–Howard correspondence between proofs and programs, and such logical systems as Per Martin-Löf's intuitionistic type theory, and Thierry Coquand and Gérard Huet's calculus of constructions. A historical example Until the end of 19th century, all mathematical proofs were essentially constructive. The first non-constructive constructions appeared with Georg Cantor’s theory of infinite sets, and the formal definition of real numbers. The first use of non-constructive proofs for solving previously considered problems seems to be Hilbert's Nullstellensatz and Hilbert's basis theorem. From a philosophical point of view, the former is especially interesting, as implying the existence of a well specified object. The Nullstellensatz may be stated as follows: If are polynomials in indeterminates with complex coefficients, which have no common complex zeros, then there are polynomials such that Such a non-constructive existence theorem was such a surprise for mathematicians of that time that one of them, Paul Gordan, wrote: "this is not mathematics, it is theology". Twenty five years later, Grete Hermann provided an algorithm for computing which is not a constructive proof in the strong sense, as she used Hilbert's result. She proved that, if exist, they can be found with degrees less than . This provides an algorithm, as the problem is reduced to solving a system of linear equations, by considering as unknowns the finite number of coefficients
https://en.wikipedia.org/wiki/Function%20of%20several%20complex%20variables	The theory of functions of several complex variables is the branch of mathematics dealing with functions defined on the complex coordinate space , that is, -tuples of complex numbers. The name of the field dealing with the properties of these functions is called several complex variables (and analytic space), which the Mathematics Subject Classification has as a top-level heading. As in complex analysis of functions of one variable, which is the case , the functions studied are holomorphic or complex analytic so that, locally, they are power series in the variables . Equivalently, they are locally uniform limits of polynomials; or locally square-integrable solutions to the -dimensional Cauchy–Riemann equations. For one complex variable, every domain(), is the domain of holomorphy of some function, in other words every domain has a function for which it is the domain of holomorphy. For several complex variables, this is not the case; there exist domains () that are not the domain of holomorphy of any function, and so is not always the domain of holomorphy, so the domain of holomorphy is one of the themes in this field. Patching the local data of meromorphic functions, i.e. the problem of creating a global meromorphic function from zeros and poles, is called the Cousin problem. Also, the interesting phenomena that occur in several complex variables are fundamentally important to the study of compact complex manifolds and complex projective varieties () and has a different flavour to complex analytic geometry in or on Stein manifolds, these are much similar to study of algebraic varieties that is study of the algebraic geometry than complex analytic geometry. Historical perspective Many examples of such functions were familiar in nineteenth-century mathematics; abelian functions, theta functions, and some hypergeometric series, and also, as an example of an inverse problem; the Jacobi inversion problem. Naturally also same function of one variable that depends on some complex parameter is a candidate. The theory, however, for many years didn't become a full-fledged field in mathematical analysis, since its characteristic phenomena weren't uncovered. The Weierstrass preparation theorem would now be classed as commutative algebra; it did justify the local picture, ramification, that addresses the generalization of the branch points of Riemann surface theory. With work of Friedrich Hartogs, , E. E. Levi, and of Kiyoshi Oka in the 1930s, a general theory began to emerge; others working in the area at the time were Heinrich Behnke, Peter Thullen, Karl Stein, Wilhelm Wirtinger and Francesco Severi. Hartogs proved some basic results, such as every isolated singularity is removable, for every analytic function whenever . Naturally the analogues of contour integrals will be harder to handle; when an integral surrounding a point should be over a three-dimensional manifold (since we are in four real dimensions), while iterating contour (line) integral
https://en.wikipedia.org/wiki/Gimel%20function	In axiomatic set theory, the gimel function is the following function mapping cardinal numbers to cardinal numbers: where cf denotes the cofinality function; the gimel function is used for studying the continuum function and the cardinal exponentiation function. The symbol is a serif form of the Hebrew letter gimel. Values of the gimel function The gimel function has the property for all infinite cardinals by König's theorem. For regular cardinals , , and Easton's theorem says we don't know much about the values of this function. For singular , upper bounds for can be found from Shelah's PCF theory. The gimel hypothesis The gimel hypothesis states that . In essence, this means that for singular is the smallest value allowed by the axioms of Zermelo–Fraenkel set theory (assuming consistency). Under this hypothesis cardinal exponentiation is simplified, though not to the extent of the continuum hypothesis (which implies the gimel hypothesis). Reducing the exponentiation function to the gimel function showed that all cardinal exponentiation is determined (recursively) by the gimel function as follows. If is an infinite regular cardinal (in particular any infinite successor) then If is infinite and singular and the continuum function is eventually constant below then If is a limit and the continuum function is not eventually constant below then The remaining rules hold whenever and are both infinite: If then If for some then If and for all and then If and for all and then See also Aleph number Beth number References Thomas Jech, Set Theory, 3rd millennium ed., 2003, Springer Monographs in Mathematics, Springer, . Cardinal numbers
https://en.wikipedia.org/wiki/Amalgamation	Amalgamation is the process of combining or uniting multiple entities into one form. Amalgamation, amalgam, and other derivatives may refer to: Mathematics and science Amalgam (chemistry), the combination of mercury with another metal Pan amalgamation, another extraction method with additional compound Patio process, the use of mercury amalgamation to extract silver Amalgamation (geology), the creation of a stable continent or craton by the union of two terranes; see Tectonic evolution of the Barberton greenstone belt Amalgamation paradox in probability and statistics, also known as Simpson's paradox Amalgamation property in model theory Free product with amalgamation, in mathematics, especially group theory, an important construction Arts, entertainment, and media Amalgamated Broadcasting System, a short-lived American radio network during the 1930s Amalgamation (fiction), the concept of creating an element in a work of fiction by combining existing things Amalgamation, a 1994 EP by the band Pop Will Eat Itself Amalgamation, the debut studio album by the band Trapt Other uses Amalgamated (1917 automobile), car manufactured by the Amalgamated Machinery Corp. Amalgamated (organization name) Amalgamation (business), the merge or consolidation of companies Amalgamation (land), the formal combination of adjoining plots; in some jurisdictions distinct from a merger Amalgamation (names), the strategy of naming something after a combination of existing names Amalgamation (race), a now largely archaic term for the merger of people of different ethnicities and "races" Amalgamation, another name for a trade union, chiefly used in the UK Amalgamation, in C (programming language) (C) and C++ programming, merging all the source codes of a library into a single header file Conflation, also known as "idiom amalgamation", the combination of two expressions Merger (politics), consolidation or amalgamation, in geopolitics, joining two or more political or administrative entities, such as municipalities, cities, towns, counties, districts etc. into a single entity See also Amalgam (disambiguation)
https://en.wikipedia.org/wiki/Weil%20restriction	In mathematics, restriction of scalars (also known as "Weil restriction") is a functor which, for any finite extension of fields L/k and any algebraic variety X over L, produces another variety ResL/kX, defined over k. It is useful for reducing questions about varieties over large fields to questions about more complicated varieties over smaller fields. Definition Let L/k be a finite extension of fields, and X a variety defined over L. The functor from k-schemesop to sets is defined by (In particular, the k-rational points of are the L-rational points of X.) The variety that represents this functor is called the restriction of scalars, and is unique up to unique isomorphism if it exists. From the standpoint of sheaves of sets, restriction of scalars is just a pushforward along the morphism and is right adjoint to fiber product of schemes, so the above definition can be rephrased in much more generality. In particular, one can replace the extension of fields by any morphism of ringed topoi, and the hypotheses on X can be weakened to e.g. stacks. This comes at the cost of having less control over the behavior of the restriction of scalars. Alternative definition Let be a morphism of schemes. For a -scheme , if the contravariant functor is representable, then we call the corresponding -scheme, which we also denote with , the Weil restriction of with respect to . Where denotes the dual of the category of schemes over a fixed scheme . Properties For any finite extension of fields, the restriction of scalars takes quasiprojective varieties to quasiprojective varieties. The dimension of the resulting variety is multiplied by the degree of the extension. Under appropriate hypotheses (e.g., flat, proper, finitely presented), any morphism of algebraic spaces yields a restriction of scalars functor that takes algebraic stacks to algebraic stacks, preserving properties such as Artin, Deligne-Mumford, and representability. Examples and applications Simple examples are the following: Let L be a finite extension of k of degree s. Then and is an s-dimensional affine space over Spec k. If X is an affine L-variety, defined by we can write as Spec , where () are new variables, and () are polynomials in given by taking a k-basis of L and setting and . If a scheme is a group scheme then any Weil restriction of it will be as well. This is frequently used in number theory, for instance: The torus where denotes the multiplicative group, plays a significant role in Hodge theory, since the Tannakian category of real Hodge structures is equivalent to the category of representations of The real points have a Lie group structure isomorphic to . See Mumford–Tate group. The Weil restriction of a (commutative) group variety is again a (commutative) group variety of dimension if L is separable over k. Aleksander Momot applied Weil restrictions of commutative group varieties with and in order to derive new results in transcendence th
https://en.wikipedia.org/wiki/Generalized%20hypergeometric%20function	In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by n is a rational function of n. The series, if convergent, defines a generalized hypergeometric function, which may then be defined over a wider domain of the argument by analytic continuation. The generalized hypergeometric series is sometimes just called the hypergeometric series, though this term also sometimes just refers to the Gaussian hypergeometric series. Generalized hypergeometric functions include the (Gaussian) hypergeometric function and the confluent hypergeometric function as special cases, which in turn have many particular special functions as special cases, such as elementary functions, Bessel functions, and the classical orthogonal polynomials. Notation A hypergeometric series is formally defined as a power series in which the ratio of successive coefficients is a rational function of n. That is, where A(n) and B(n) are polynomials in n. For example, in the case of the series for the exponential function, we have: So this satisfies the definition with and . It is customary to factor out the leading term, so β0 is assumed to be 1. The polynomials can be factored into linear factors of the form (aj + n) and (bk + n) respectively, where the aj and bk are complex numbers. For historical reasons, it is assumed that (1 + n) is a factor of B. If this is not already the case then both A and B can be multiplied by this factor; the factor cancels so the terms are unchanged and there is no loss of generality. The ratio between consecutive coefficients now has the form , where c and d are the leading coefficients of A and B. The series then has the form , or, by scaling z by the appropriate factor and rearranging, . This has the form of an exponential generating function. This series is usually denoted by or Using the rising factorial or Pochhammer symbol this can be written (Note that this use of the Pochhammer symbol is not standard; however it is the standard usage in this context.) Terminology When all the terms of the series are defined and it has a non-zero radius of convergence, then the series defines an analytic function. Such a function, and its analytic continuations, is called the hypergeometric function. The case when the radius of convergence is 0 yields many interesting series in mathematics, for example the incomplete gamma function has the asymptotic expansion which could be written za−1e−z 2F0(1−a,1;;−z−1). However, the use of the term hypergeometric series is usually restricted to the case where the series defines an actual analytic function. The ordinary hypergeometric series should not be confused with the basic hypergeometric series, which, despite its name, is a rather more complicated and recondite series. The "basic" series is the q-analog of the ordinary hypergeometric series. There are several such generalizations of the ordinary hypergeometric series, including th
https://en.wikipedia.org/wiki/Rational%20function	In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rational numbers; they may be taken in any field K. In this case, one speaks of a rational function and a rational fraction over K. The values of the variables may be taken in any field L containing K. Then the domain of the function is the set of the values of the variables for which the denominator is not zero, and the codomain is L. The set of rational functions over a field K is a field, the field of fractions of the ring of the polynomial functions over K. Definitions A function is called a rational function if and only if it can be written in the form where and are polynomial functions of and is not the zero function. The domain of is the set of all values of for which the denominator is not zero. However, if and have a non-constant polynomial greatest common divisor , then setting and produces a rational function which may have a larger domain than , and is equal to on the domain of It is a common usage to identify and , that is to extend "by continuity" the domain of to that of Indeed, one can define a rational fraction as an equivalence class of fractions of polynomials, where two fractions and are considered equivalent if . In this case is equivalent to . A proper rational function is a rational function in which the degree of is less than the degree of and both are real polynomials, named by analogy to a proper fraction in . Degree There are several non equivalent definitions of the degree of a rational function. Most commonly, the degree of a rational function is the maximum of the degrees of its constituent polynomials and , when the fraction is reduced to lowest terms. If the degree of is , then the equation has distinct solutions in except for certain values of , called critical values, where two or more solutions coincide or where some solution is rejected at infinity (that is, when the degree of the equation decrease after having cleared the denominator). In the case of complex coefficients, a rational function with degree one is a Möbius transformation. The degree of the graph of a rational function is not the degree as defined above: it is the maximum of the degree of the numerator and one plus the degree of the denominator. In some contexts, such as in asymptotic analysis, the degree of a rational function is the difference between the degrees of the numerator and the denominator. In network synthesis and network analysis, a rational function of degree two (that is, the ratio of two polynomials of degree at most two) is often called a . Examples The rational function is not defined at It is asymptotic to as The rational function is defined for all real numbers, but not for all complex numbers, since if x were a square root of (i.e. the imagi
https://en.wikipedia.org/wiki/Negentropy	In information theory and statistics, negentropy is used as a measure of distance to normality. The concept and phrase "negative entropy" was introduced by Erwin Schrödinger in his 1944 popular-science book What is Life? Later, French physicist Léon Brillouin shortened the phrase to néguentropie (negentropy). In 1974, Albert Szent-Györgyi proposed replacing the term negentropy with syntropy. That term may have originated in the 1940s with the Italian mathematician Luigi Fantappiè, who tried to construct a unified theory of biology and physics. Buckminster Fuller tried to popularize this usage, but negentropy remains common. In a note to What is Life? Schrödinger explained his use of this phrase. Information theory In information theory and statistics, negentropy is used as a measure of distance to normality. Out of all distributions with a given mean and variance, the normal or Gaussian distribution is the one with the highest entropy. Negentropy measures the difference in entropy between a given distribution and the Gaussian distribution with the same mean and variance. Thus, negentropy is always nonnegative, is invariant by any linear invertible change of coordinates, and vanishes if and only if the signal is Gaussian. Negentropy is defined as where is the differential entropy of the Gaussian density with the same mean and variance as and is the differential entropy of : Negentropy is used in statistics and signal processing. It is related to network entropy, which is used in independent component analysis. The negentropy of a distribution is equal to the Kullback–Leibler divergence between and a Gaussian distribution with the same mean and variance as (see for a proof). In particular, it is always nonnegative. Correlation between statistical negentropy and Gibbs' free energy There is a physical quantity closely linked to free energy (free enthalpy), with a unit of entropy and isomorphic to negentropy known in statistics and information theory. In 1873, Willard Gibbs created a diagram illustrating the concept of free energy corresponding to free enthalpy. On the diagram one can see the quantity called capacity for entropy. This quantity is the amount of entropy that may be increased without changing an internal energy or increasing its volume. In other words, it is a difference between maximum possible, under assumed conditions, entropy and its actual entropy. It corresponds exactly to the definition of negentropy adopted in statistics and information theory. A similar physical quantity was introduced in 1869 by Massieu for the isothermal process (both quantities differs just with a figure sign) and then Planck for the isothermal-isobaric process. More recently, the Massieu–Planck thermodynamic potential, known also as free entropy, has been shown to play a great role in the so-called entropic formulation of statistical mechanics, applied among the others in molecular biology and thermodynamic non-equilibrium processes. where
https://en.wikipedia.org/wiki/Line%20bundle	In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the tangent bundle is a way of organising these. More formally, in algebraic topology and differential topology, a line bundle is defined as a vector bundle of rank 1. Line bundles are specified by choosing a one-dimensional vector space for each point of the space in a continuous manner. In topological applications, this vector space is usually real or complex. The two cases display fundamentally different behavior because of the different topological properties of real and complex vector spaces: If the origin is removed from the real line, then the result is the set of 1×1 invertible real matrices, which is homotopy-equivalent to a discrete two-point space by contracting the positive and negative reals each to a point; whereas removing the origin from the complex plane yields the 1×1 invertible complex matrices, which have the homotopy type of a circle. From the perspective of homotopy theory, a real line bundle therefore behaves much the same as a fiber bundle with a two-point fiber, that is, like a double cover. A special case of this is the orientable double cover of a differentiable manifold, where the corresponding line bundle is the determinant bundle of the tangent bundle (see below). The Möbius strip corresponds to a double cover of the circle (the θ → 2θ mapping) and by changing the fiber, can also be viewed as having a two-point fiber, the unit interval as a fiber, or the real line. Complex line bundles are closely related to circle bundles. There are some celebrated ones, for example the Hopf fibrations of spheres to spheres. In algebraic geometry, an invertible sheaf (i.e., locally free sheaf of rank one) is often called a line bundle. Every line bundle arises from a divisor with the following conditions (I) If X is reduced and irreducible scheme, then every line bundle comes from a divisor. (II) If X is projective scheme then the same statement holds. The tautological bundle on projective space One of the most important line bundles in algebraic geometry is the tautological line bundle on projective space. The projectivization P(V) of a vector space V over a field k is defined to be the quotient of by the action of the multiplicative group k×. Each point of P(V) therefore corresponds to a copy of k×, and these copies of k× can be assembled into a k×-bundle over P(V). k× differs from k only by a single point, and by adjoining that point to each fiber, we get a line bundle on P(V). This line bundle is called the tautological line bundle. This line bundle is sometimes denoted since it corresponds to the dual of the Serre twisting sheaf . Maps to projective space Suppose that X is a space and that L is a line bundle on X. A global section of L is a function such that if is the natural projection, then = idX. In a sma
https://en.wikipedia.org/wiki/Descent%20%28mathematics%29	In mathematics, the idea of descent extends the intuitive idea of 'gluing' in topology. Since the topologists' glue is the use of equivalence relations on topological spaces, the theory starts with some ideas on identification. Descent of vector bundles The case of the construction of vector bundles from data on a disjoint union of topological spaces is a straightforward place to start. Suppose X is a topological space covered by open sets Xi. Let Y be the disjoint union of the Xi, so that there is a natural mapping We think of Y as 'above' X, with the Xi projection 'down' onto X. With this language, descent implies a vector bundle on Y (so, a bundle given on each Xi), and our concern is to 'glue' those bundles Vi, to make a single bundle V on X. What we mean is that V should, when restricted to Xi, give back Vi, up to a bundle isomorphism. The data needed is then this: on each overlap intersection of Xi and Xj, we'll require mappings to use to identify Vi and Vj there, fiber by fiber. Further the fij must satisfy conditions based on the reflexive, symmetric and transitive properties of an equivalence relation (gluing conditions). For example, the composition for transitivity (and choosing apt notation). The fii should be identity maps and hence symmetry becomes (so that it is fiberwise an isomorphism). These are indeed standard conditions in fiber bundle theory (see transition map). One important application to note is change of fiber: if the fij are all you need to make a bundle, then there are many ways to make an associated bundle. That is, we can take essentially same fij, acting on various fibers. Another major point is the relation with the chain rule: the discussion of the way there of constructing tensor fields can be summed up as 'once you learn to descend the tangent bundle, for which transitivity is the Jacobian chain rule, the rest is just 'naturality of tensor constructions'. To move closer towards the abstract theory we need to interpret the disjoint union of the now as the fiber product (here an equalizer) of two copies of the projection p. The bundles on the Xij that we must control are V′ and V", the pullbacks to the fiber of V via the two different projection maps to X. Therefore, by going to a more abstract level one can eliminate the combinatorial side (that is, leave out the indices) and get something that makes sense for p not of the special form of covering with which we began. This then allows a category theory approach: what remains to do is to re-express the gluing conditions. History The ideas were developed in the period 1955–1965 (which was roughly the time at which the requirements of algebraic topology were met but those of algebraic geometry were not). From the point of view of abstract category theory the work of comonads of Beck was a summation of those ideas; see Beck's monadicity theorem. The difficulties of algebraic geometry with passage to the quotient are acute. The urgency (to put it tha
https://en.wikipedia.org/wiki/Second%20Hardy%E2%80%93Littlewood%20conjecture	In number theory, the second Hardy–Littlewood conjecture concerns the number of primes in intervals. Along with the first Hardy–Littlewood conjecture, the second Hardy–Littlewood conjecture was proposed by G. H. Hardy and John Edensor Littlewood in 1923. Statement The conjecture states that for integers , where denotes the prime-counting function, giving the number of prime numbers up to and including . Connection to the first Hardy–Littlewood conjecture The statement of the second Hardy–Littlewood conjecture is equivalent to the statement that the number of primes from to is always less than or equal to the number of primes from 1 to . This was proved to be inconsistent with the first Hardy–Littlewood conjecture on prime -tuples, and the first violation is expected to likely occur for very large values of . For example, an admissible k-tuple (or prime constellation) of 447 primes can be found in an interval of integers, while . If the first Hardy–Littlewood conjecture holds, then the first such -tuple is expected for greater than but less than . References External links Analytic number theory Conjectures about prime numbers Unsolved problems in number theory
https://en.wikipedia.org/wiki/Moduli%20space	In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spaces frequently arise as solutions to classification problems: If one can show that a collection of interesting objects (e.g., the smooth algebraic curves of a fixed genus) can be given the structure of a geometric space, then one can parametrize such objects by introducing coordinates on the resulting space. In this context, the term "modulus" is used synonymously with "parameter"; moduli spaces were first understood as spaces of parameters rather than as spaces of objects. A variant of moduli spaces is formal moduli. Bernhard Riemann first used the term "moduli" in 1857. Motivation Moduli spaces are spaces of solutions of geometric classification problems. That is, the points of a moduli space correspond to solutions of geometric problems. Here different solutions are identified if they are isomorphic (that is, geometrically the same). Moduli spaces can be thought of as giving a universal space of parameters for the problem. For example, consider the problem of finding all circles in the Euclidean plane up to congruence. Any circle can be described uniquely by giving three points, but many different sets of three points give the same circle: the correspondence is many-to-one. However, circles are uniquely parameterized by giving their center and radius: this is two real parameters and one positive real parameter. Since we are only interested in circles "up to congruence", we identify circles having different centers but the same radius, and so the radius alone suffices to parameterize the set of interest. The moduli space is, therefore, the positive real numbers. Moduli spaces often carry natural geometric and topological structures as well. In the example of circles, for instance, the moduli space is not just an abstract set, but the absolute value of the difference of the radii defines a metric for determining when two circles are "close". The geometric structure of moduli spaces locally tells us when two solutions of a geometric classification problem are "close", but generally moduli spaces also have a complicated global structure as well. For example, consider how to describe the collection of lines in R2 which intersect the origin. We want to assign to each line L of this family a quantity that can uniquely identify it—a modulus. An example of such a quantity is the positive angle θ(L) with 0 ≤ θ < π radians. The set of lines L so parametrized is known as P1(R) and is called the real projective line. We can also describe the collection of lines in R2 that intersect the origin by means of a topological construction. To wit: consider S1 ⊂ R2 and notice that every point s ∈ S1 gives a line L(s) in the collection (which joins the origin and s). However, this map is two-to-one, so we wan
https://en.wikipedia.org/wiki/Order%20theory	Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and provides basic definitions. A list of order-theoretic terms can be found in the order theory glossary. Background and motivation Orders are everywhere in mathematics and related fields like computer science. The first order often discussed in primary school is the standard order on the natural numbers e.g. "2 is less than 3", "10 is greater than 5", or "Does Tom have fewer cookies than Sally?". This intuitive concept can be extended to orders on other sets of numbers, such as the integers and the reals. The idea of being greater than or less than another number is one of the basic intuitions of number systems (compare with numeral systems) in general (although one usually is also interested in the actual difference of two numbers, which is not given by the order). Other familiar examples of orderings are the alphabetical order of words in a dictionary and the genealogical property of lineal descent within a group of people. The notion of order is very general, extending beyond contexts that have an immediate, intuitive feel of sequence or relative quantity. In other contexts orders may capture notions of containment or specialization. Abstractly, this type of order amounts to the subset relation, e.g., "Pediatricians are physicians," and "Circles are merely special-case ellipses." Some orders, like "less-than" on the natural numbers and alphabetical order on words, have a special property: each element can be compared to any other element, i.e. it is smaller (earlier) than, larger (later) than, or identical to. However, many other orders do not. Consider for example the subset order on a collection of sets: though the set of birds and the set of dogs are both subsets of the set of animals, neither the birds nor the dogs constitutes a subset of the other. Those orders like the "subset-of" relation for which there exist incomparable elements are called partial orders; orders for which every pair of elements is comparable are total orders. Order theory captures the intuition of orders that arises from such examples in a general setting. This is achieved by specifying properties that a relation ≤ must have to be a mathematical order. This more abstract approach makes much sense, because one can derive numerous theorems in the general setting, without focusing on the details of any particular order. These insights can then be readily transferred to many less abstract applications. Driven by the wide practical usage of orders, numerous special kinds of ordered sets have been defined, some of which have grown into mathematical fields of their own. In addition, order theory does not restrict itself to the various classes of ordering relations, but also considers appropriate functions betw
https://en.wikipedia.org/wiki/Idempotent%20%28ring%20theory%29	In ring theory, a branch of mathematics, an idempotent element or simply idempotent of a ring is an element such that . That is, the element is idempotent under the ring's multiplication. Inductively then, one can also conclude that for any positive integer . For example, an idempotent element of a matrix ring is precisely an idempotent matrix. For general rings, elements idempotent under multiplication are involved in decompositions of modules, and connected to homological properties of the ring. In Boolean algebra, the main objects of study are rings in which all elements are idempotent under both addition and multiplication. Examples Quotients of Z One may consider the ring of integers modulo , where is squarefree. By the Chinese remainder theorem, this ring factors into the product of rings of integers modulo , where is prime. Now each of these factors is a field, so it is clear that the factors' only idempotents will be and . That is, each factor has two idempotents. So if there are factors, there will be idempotents. We can check this for the integers , . Since has two prime factors ( and ) it should have idempotents. From these computations, , , , and are idempotents of this ring, while and are not. This also demonstrates the decomposition properties described below: because , there is a ring decomposition . In the identity is and in the identity is . Quotient of polynomial ring Given a ring and an element such that , then the quotient ring has the idempotent . For example, this could be applied to , or any polynomial . Idempotents in split-quaternion rings There is a hyperboloid of idempotents in the split-quaternion ring. Types of ring idempotents A partial list of important types of idempotents includes: Two idempotents and are called orthogonal if . If is idempotent in the ring (with unity), then so is ; moreover, and are orthogonal. An idempotent in is called a central idempotent if for all in , that is, if is in the centre of . A trivial idempotent refers to either of the elements and , which are always idempotent. A primitive idempotent of a ring is a nonzero idempotent such that is indecomposable as a right -module; that is, such that is not a direct sum of two nonzero submodules. Equivalently, is a primitive idempotent if it cannot be written as , where and are nonzero orthogonal idempotents in . A local idempotent is an idempotent such that is a local ring. This implies that is directly indecomposable, so local idempotents are also primitive. A right irreducible idempotent is an idempotent for which is a simple module. By Schur's lemma, is a division ring, and hence is a local ring, so right (and left) irreducible idempotents are local. A centrally primitive idempotent is a central idempotent that cannot be written as the sum of two nonzero orthogonal central idempotents. An idempotent in the quotient ring is said to lift modulo if there is an ide
https://en.wikipedia.org/wiki/SCFG	SCFG may refer to Stochastic context-free grammar, generative probability model that takes the shape of a context-free grammar Synchronous context-free grammar, in machine translation
https://en.wikipedia.org/wiki/22%20%28number%29	22 (twenty-two) is the natural number following 21 and preceding 23. In mathematics 22 is a palindromic number. It is the second Smith number, the second Erdős–Woods number, and the fourth large Schröder number. It is also a Perrin number, from a sum of 10 and 12. 22 is the sixth distinct semiprime, and the fouth of the form where is a higher prime. It is the second member of the second cluster of discrete biprimes (21, 22), where the next such cluster is (38, 39). It contains an aliquot sum of 14 (itself semiprime), within an aliquot sequence of four composite numbers (22, 14, 10, 8, 7, 1, 0) that are rooted in the prime 7-aliquot tree. The maximum number of regions into which five intersecting circles divide the plane is 22. 22 is also the quantity of pieces in a disc that can be created with six straight cuts, which makes 22 the seventh central polygonal number. 22 is the fourth pentagonal number, the third hexagonal pyramidal number, and the third centered heptagonal number. is a commonly used approximation of the irrational number , the ratio of the circumference of a circle to its diameter, where in particular 22 and 7 are consecutive hexagonal pyramidal numbers. Also, from an approximate construction of the squaring of the circle by Srinivasa Ramanujan, correct to eight decimal places. Natural logarithms of integers in binary are known to have Bailey–Borwein–Plouffe type formulae for for all integers . 22 is the number of partitions of 8, as well as the sum of the totient function over the first eight integers, with for 22 returning 10. 22 can read as "two twos", which is the only fixed point of John Conway's look-and-say function. In other words, "22" generates the infinite repeating sequence "22, 22, 22, ..." All regular polygons with < edges can be constructed with an angle trisector, with the exception of the 11-sided hendecagon. There is an elementary set of twenty-two single-orbit convex tilings that tessellate two-dimensional space with face-transitive, edge-transitive, and/or vertex-transitive properties: eleven of these are regular and semiregular Archimedean tilings, while the other eleven are their dual Laves tilings. Twenty-two edge-to-edge star polygon tilings exist in the second dimension that incorporate regular convex polygons: eighteen involve specific angles, while four involve angles that are adjustable. Finally, there are also twenty-two regular complex apeirohedra of the form p{a}q{b}r: eight are self-dual, while fourteen exist as dual polytope pairs; twenty-one belong in while one belongs in . There are twenty-two different subgroups that describe full icosahedral symmetry. Three groups are generated by particular inversions, five groups by reflections, and nine groups by rotations, alongside three mixed groups, the pyritohedral group, and the full icosahedral group. There are 22 finite semiregular polytopes through the eighth dimension, aside from the infinite families of prisms and antip
https://en.wikipedia.org/wiki/24%20%28number%29	24 (twenty-four) is the natural number following 23 and preceding 25. In mathematics 24 is an even composite number, with 2 and 3 as its distinct prime factors. It is the first number of the form 2q, where q is an odd prime. It is the smallest number with at least eight positive divisors: 1, 2, 3, 4, 6, 8, 12, and 24; thus, it is a highly composite number, having more divisors than any smaller number. Furthermore, it is an abundant number, since the sum of its proper divisors (36) is greater than itself, as well as a superabundant number. In number theory and algebra 24 is the smallest 5-hemiperfect number, as it has a half-integer abundancy index: 1 + 2 + 3 + 4 + 6 + 8 + 12 + 24 = 60 = × 24 24 is a semiperfect number, since adding up all the proper divisors of 24 except 4 and 8 gives 24. 24 is a practical number, since all smaller positive integers than 24 can be represented as sums of distinct divisors of 24. 24 is a Harshad number, since it is divisible by the sum of its digits in decimal. 24 is a refactorable number, as it has exactly eight positive divisors, and 8 is one of them. 24 is a twin-prime sum, specifically the sum of the third pair of twin primes . 24 is a highly totient number, as there are 10 solutions to the equation φ(x) = 24, which is more than any integer below 24. 144 (the square of 12) and 576 (the square of 24) are also highly totient. 24 is a polite number, an amenable number, an idoneal number, and a tribonacci number. 24 forms a Ruth-Aaron pair with 25, since the sums of distinct prime factors of each are equal (5). 24 is a compositorial, as it is the product of composite numbers up to 6. 24 is a pernicious number, since its Hamming weight in its binary representation (11000) is prime (2). 24 is the third nonagonal number. 24's digits in decimal can be manipulated to form two of its factors, as 2 * 4 is 8 and 2 + 4 is 6. In turn 6 * 8 is 48, which is twice 24, and 4 + 8 is 12, which is half 24. 24 is a congruent number, as 24 is the area of a right triangle with a rational number of sides. 24 is a semi-meandric number, where an order-6 semi-meander intersects an oriented ray in R2 at 24 points. 24 is the number of digits of the fifth and largest known unitary perfect number, when written in decimal: 146361946186458562560000. Subtracting 1 from any of its divisors (except 1 and 2 but including itself) yields a prime number; 24 is the largest number with this property. 24 is the largest integer that is divisible by all natural numbers no larger than its square root. The product of any four consecutive numbers is divisible by 24. This is because, among any four consecutive numbers, there must be two even numbers, one of which is a multiple of four, and there must be at least one multiple of three. 24 = 4!, the factorial of 4. It is the largest factorial that does not contain a trailing zero at the end of its digits (since factorial of any integer greater than 4 is divisible by both 2 and 5), and represents the num
https://en.wikipedia.org/wiki/23%20%28number%29	23 (twenty-three) is the natural number following 22 and preceding 24. In mathematics Twenty-three is the ninth prime number, the smallest odd prime that is not a twin prime. It is, however, a cousin prime with 19, and a sexy prime with 17 and 29; while also being the largest member of the first prime sextuplet (7, 11, 13, 17, 19, 23). Twenty-three is also the fifth factorial prime, the second Woodall prime, and a happy number in decimal. It is an Eisenstein prime with no imaginary part and real part of the form It is also the fifth Sophie Germain prime and the fourth safe prime, and the next to last member of the first Cunningham chain of the first kind to have five terms (2, 5, 11, 23, 47). Since 14! + 1 is a multiple of 23, but 23 is not one more than a multiple of 14, 23 is the first Pillai prime. 23 is the smallest odd prime to be a highly cototient number, as the solution to for the integers 95, 119, 143, and 529. The third decimal repunit prime after R2 and R19 is R23, followed by R1031. 23 is the second Smarandache–Wellin prime in base ten, as it is the concatenation of the decimal representations of the first two primes (2 and 3) and is itself also prime. The sum of the first nine primes up to 23 is a square: and the sum of the first 23 primes is 874, which is divisible by 23, a property shared by few other numbers. In the list of fortunate numbers, 23 occurs twice, since adding 23 to either the fifth or eighth primorial gives a prime number (namely 2333 and 9699713). 23 has the distinction of being one of two integers that cannot be expressed as the sum of fewer than 9 cubes of positive integers (the other is 239). See Waring's problem. The twenty-third highly composite number 20,160 is one less than the last number (the 339th super-prime 20,161) that cannot be expressed as the sum of two abundant numbers. 23 is the number of trees on 8 unlabeled nodes. It is also a Wedderburn–Etherington number, which are numbers that can be used to count certain binary trees. The natural logarithms of all positive integers lower than 23 are known to have binary BBP-type formulae. 23 is the first prime p for which unique factorization of cyclotomic integers based on the pth root of unity breaks down. 23 is the smallest positive solution to Sunzi's original formulation of the Chinese remainder theorem. 23 is the smallest prime such that the largest consecutive pair of smooth numbers (11859210, 11859211) is the same as the largest consecutive pair of smooth numbers. According to the birthday paradox, in a group of 23 or more randomly chosen people, the probability is more than 50% that some pair of them will have the same birthday. A related coincidence is that 365 times the natural logarithm of 2, approximately 252.999, is very close to the number of pairs of 23 items and 22nd triangular number, 253. The first twenty-three odd prime numbers (between 3 and 89 inclusive), are all cluster primes such that every even positive
https://en.wikipedia.org/wiki/25%20%28number%29	25 (twenty-five) is the natural number following 24 and preceding 26. In mathematics It is a square number, being 52 = 5 × 5, and hence the third non-unitary square prime of the form p2. It is one of two two-digit numbers whose square and higher powers of the number also ends in the same last two digits, e.g., 252 = 625; the other is 76. Twenty five has an even aliquot sum of 6, which is itself the first even and perfect number root of an aliquot sequence; not ending in (1 and 0). It is the smallest square that is also a sum of two (non-zero) squares: 25 = 32 + 42. Hence, it often appears in illustrations of the Pythagorean theorem. 25 is the sum of the five consecutive single-digit odd natural numbers 1, 3, 5, 7, and 9. 25 is a centered octagonal number, a centered square number, a centered octahedral number, and an automorphic number. 25 percent (%) is equal to . It is the smallest decimal Friedman number as it can be expressed by its own digits: 52. It is also a Cullen number and a vertically symmetrical number. 25 is the smallest pseudoprime satisfying the congruence 7n = 7 mod n. 25 is the smallest aspiring number — a composite non-sociable number whose aliquot sequence does not terminate. According to the Shapiro inequality, 25 is the smallest odd integer n such that there exist x, x, ..., x such that where x = x, x = x. Within decimal, one can readily test for divisibility by 25 by seeing if the last two digits of the number match 00, 25, 50, or 75. There are 25 primes under 100. In science The Standard Model of physics features a total of 25 elementary particles: 12 fermions (made of 6 quarks and 6 leptons) and 13 bosons (made of 12 gauge bosons and 1 scalar boson). The atomic number of manganese. The average percentage DNA overlap of an individual with their half-sibling, grandparent, grandchild, aunt, uncle, niece, nephew, identical twin cousin (offspring of identical twins), or double cousin. In religion In Ezekiel's vision of a new temple: The number twenty-five is of cardinal importance in Ezekiel's Temple Vision (in the Bible, Ezekiel chapters 40–48). In Islam, there are 25 prophets mentioned in the Quran. In sports Before 2020, the size of the full roster on a Major League Baseball team for most of the season, except for regular-season games on or after September 1, when teams expanded their roster to 40 players. The size of the playing roster on a Nippon Professional Baseball team for a particular game. Active NPB rosters consist of 28 players, but prior to each game, managers must designate three players who will be ineligible for that game. In baseball, the number 25 is typically reserved for the best slugger on the team. Examples include Mark McGwire, Barry Bonds, Jim Thome, and Mark Teixeira. The number of points needed to win a set in volleyball under rally scoring rules (except for the fifth set), so long as the losing team's score is two less than the winning team's score (i.e., if the winning team sco
https://en.wikipedia.org/wiki/26%20%28number%29	26 (twenty-six) is the natural number following 25 and preceding 27. In mathematics 26 is the seventh discrete semiprime () and the fifth with 2 as the lowest non-unitary factor thus of the form (2.q), where q is a higher prime. with an aliquot sum of 16, within an aliquot sequence of five composite numbers (26,16,15,9,4,3,1,0) to the Prime in the 3-aliquot tree. 26 is the only integer that is one greater than a square (5 + 1) and one less than a cube (3 − 1). 26 is a telephone number, specifically, the number of ways of connecting 5 points with pairwise connections. There are 26 sporadic groups. The 26-dimensional Lorentzian unimodular lattice II25,1 plays a significant role in sphere packing problems and the classification of finite simple groups. In particular, the Leech lattice is obtained in a simple way as a subquotient. 26 is the smallest number that is both a nontotient and a noncototient number. There are 26 faces of a rhombicuboctahedron. When a 3 × 3 × 3 cube is made of 27 unit cubes, 26 of them are viewable as the exterior layer. Also a 26 sided polygon is called an icosihexagon. φ(26) = φ(σ(26)). Properties of its positional representation in certain radixes Twenty-six is a repdigit in base three (2223) and in base 12 (2212). In base ten, 26 is the smallest positive integer that is not a palindrome to have a square (262 = 676) that is a palindrome. In science The atomic number of iron. The number of spacetime dimensions in bosonic string theory. Astronomy Messier object M26, a magnitude 9.5 open cluster in the constellation Scutum. The New General Catalogue object NGC 26, a spiral galaxy in the constellation Pegasus. In religion 26 is the gematric number, being the sum of the Hebrew characters () being the name of the god of Israel – YHWH (Yahweh). GOD=26=G7+O15+D4 in Simple6,74 English7,74 Gematria8,74 ('The Key': A=1, B2, C3, ..., Z26). The Greek Strongs number G26 is "Agape", which means "Love". The expression "For His mercy endures forever" is found verbatim in English and the original Hebrew 26 times in Psalm 136. The expression is found once in each of the 26 verses. According to Jewish chronology, God gave the Torah in the 26th generation since Creation. In sports The number of complete miles in a marathon (26 miles and 385 yards). Effective in 2020, the size of the full roster on a Major League Baseball team, except for regular-season games on or after September 1, when teams expand their rosters to 28 players. The number of holes in a floorball ball. The maximum capacity of cars allowed to compete in an official Formula One Grand Prix race. In other fields Twenty-six is: A 2003 novel by Leo McKay, Jr. The title refers to the number of miners killed in the Westray Mine explosion. 26 is: The sixth track of After Laughter, the fifth studio album by American alternative band Paramore. The eighth track of The Balcony, the first studio album by British modern rock band Catfish and the Bottlemen. In
https://en.wikipedia.org/wiki/29%20%28number%29	29 (twenty-nine) is the natural number following 28 and preceding 30. Mathematics 29 is the tenth prime number, and the fifth primorial prime. 29 forms a twin prime pair with thirty-one, which is also a primorial prime. Twenty-nine is also the sixth Sophie Germain prime. 29 is the sum of three consecutive squares, 22 + 32 + 42. 29 is a Lucas prime, a Pell prime, and a tetranacci number. 29 is an Eisenstein prime with no imaginary part and real part of the form 3n − 1. 29 is also the 10th supersingular prime. None of the first 29 natural numbers have more than two different prime factors. This is the longest such consecutive sequence. 29 is a Markov number, appearing in the solutions to x + y + z = 3xyz: {2, 5, 29}, {2, 29, 169}, {5, 29, 433}, {29, 169, 14701}, etc. 29 is a Perrin number, preceded in the sequence by 12, 17, 22. 29 is the smallest positive whole number that cannot be made from the numbers {1, 2, 3, 4}, using each exactly once and using only addition, subtraction, multiplication, and division. 29 is the number of pentacubes if reflections are considered distinct. The 29th dimension is the highest dimension for compact hyperbolic Coxeter polytopes that are bounded by a fundamental polyhedron, and the highest dimension that holds arithmetic discrete groups of reflections with noncompact unbounded fundamental polyhedra. Religion The Bishnois community follows 29 principles. Guru Jambheshwar had laid down 29 principles to be followed by the sect in 1485 A.D. In Hindi, Bish means 20 and noi means 9; thus, Bishnoi translates as Twenty-niners. The number of suras in the Qur'an that begin with muqatta'at. Science and astronomy The atomic number of copper. Messier object M29, a magnitude 6.6 open cluster in the constellation Cygnus. The New General Catalogue object NGC 29, a spiral galaxy in the constellation Andromeda. Saturn requires over 29 years to orbit the Sun. The number of days February has in leap years. Language and literature The number of letters in the Turkish, Finnish, Swedish, Danish and Norwegian alphabets The number of Knuts in one Sickle in the fictional currency in the Harry Potter novels Geography In the name of the town Twentynine Palms, California The number of the French department of Finistère Military 29th Regiment of Foot, a former regiment in the British Army Marine Corps Air Ground Combat Center Twentynine Palms, affectionately referred to by Marines as "Twentynine Stumps". Boeing B-29, a large bomber Music and entertainment "$29.00" is a song on the album Blue Valentine by Tom Waits. 29, an album by Ryan Adams. "No. 29", a song about a washed-up high school football star from the album Exit 0 by Steve Earle. The track from which the Chattanooga Choo Choo train departs in the Glenn Miller song. The number of attributes existing according to The Strokes in "You Only Live Once". A track from Bon Iver's album, 22, A Million, all tracks being numerically themed. A 2022 song by
https://en.wikipedia.org/wiki/28%20%28number%29	28 (twenty-eight) is the natural number following 27 and preceding 29. In mathematics It is a composite number; a square-prime, of the form (p2,q) where q is a higher prime. It is the third of this form and of the specific form (22.q), with proper divisors being 1, 2, 4, 7, and 14. Twenty-eight is the second perfect number - it is the sum of its proper divisors: . As a perfect number, it is related to the Mersenne prime 7, since . The next perfect number is 496, the previous being 6. Though perfect, 28 is not the aliquot sum of any other number other than itself, and so; unusually, is not part of a multi-number aliquot sequence. The next perfect number, 496, has the single Aliquot sum, 652 which leads to multiple aliquot sequencing. Twenty-eight is the sum of the totient function for the first nine integers. Since the greatest prime factor of is 157, which is more than 28 twice, 28 is a Størmer number. Twenty-eight is a harmonic divisor number, a happy number, a triangular number, a hexagonal number, a Leyland number of the second kind and a centered nonagonal number. It appears in the Padovan sequence, preceded by the terms 12, 16, 21 (it is the sum of the first two of these). It is also a Keith number, because it recurs in a Fibonacci-like sequence started from its decimal digits: 2, 8, 10, 18, 28... Twenty-eight is the ninth and last number in early Indian magic square of order 3. There are twenty-eight convex uniform honeycombs. Twenty-eight is the only positive integer that has a unique Kayles nim-value. Twenty-eight is the only known number that can be expressed as a sum of the first nonnegative (or positive) integers (), a sum of the first primes () and a sum of the first nonprimes (), and it is unlikely that any other number has this property. There are twenty-eight oriented diffeomorphism classes of manifolds homeomorphic to the 7-sphere. There are 28 elements of the cuboid: 8 vertices, 12 edges, 6 faces, 2 3-dimensional elements (interior and exterior). There are 28 non-equivalent ways of expressing 1000 as the sum of two prime numbers In science The atomic mass of silicon. The atomic number of nickel. The fourth magic number in physics. The curing time of concrete is classically considered 28 days. The average human menstrual cycle is 28 days, although no link has been established with the nightlighting and the Moon. Astronomy The rotation time of the surface of the Sun varies due to it being gas and plasma. At 45°, it rotates every 28 days. The Sun's surface gravity is 28 times that of Earth. Messier 28, a magnitude 8.5 globular cluster in the constellation Sagittarius. The New General Catalogue object NGC 28, an elliptical galaxy in the constellation Phoenix. In sports The number of players on the active roster of teams in Nippon Professional Baseball. However, each team is limited to using 25 players in a given game; before every game, the manager must designate three players who will be ineligible for
https://en.wikipedia.org/wiki/Walks%20plus%20hits%20per%20inning%20pitched	In baseball statistics, walks plus hits per inning pitched (WHIP) is a sabermetric measurement of the number of baserunners a pitcher has allowed per inning pitched. WHIP is calculated by adding the number of walks and hits allowed and dividing this sum by the number of innings pitched. WHIP reflects a pitcher's propensity for allowing batters to reach base, therefore a lower WHIP indicates better performance. While earned run average (ERA) measures the runs a pitcher gives up, WHIP more directly measures a pitcher's effectiveness against batters. Like ERA, WHIP accounts for pitcher performance regardless of errors and unearned runs. History The stat was invented in 1979 by writer Daniel Okrent, who called the metric "innings pitched ratio" at the time. Okrent excluded hit batsmen from the numerator of baserunners allowed since Sunday newspapers did not include hit batsmen in their agate box scores. WHIP is one of the few sabermetric statistics to enter mainstream baseball usage. In addition to its use in live games, the WHIP is one of the most commonly used statistics in fantasy baseball, and is standard in fantasy leagues that employ 4×4, 5×5, and 6×6 formats. Leaders WHIP near 1.00 or lower over the course of a season will often rank among the league leaders in Major League Baseball (MLB). The lowest single-season WHIP in MLB history through 2018 is 0.7373 from Pedro Martínez pitching for the Boston Red Sox in 2000, which broke the previous record of 0.7692 of Guy Hecker of the Louisville Eclipse in 1882. Walter Johnson, with a 0.7803 WHIP in 1913, has the third-lowest single-season WHIP. Cleveland Guardians (then called the Bronchos and Naps during his career) right-handed pitcher Addie Joss held the MLB record for the lowest career WHIP as of 2018, with a 0.9678 WHIP in 2,327 innings. Chicago White Sox spitballer Ed Walsh was second, with a 0.9996 WHIP in 2,964 innings, the lowest career WHIP for a qualified pitcher with 10 or more seasons pitched. Reliever Mariano Rivera ranked third among qualified pitchers with a career WHIP of 1.0003 in 1,283 innings. Los Angeles Dodgers left-handed pitcher Clayton Kershaw was fourth with a WHIP of 1.0046. See also List of Major League Baseball career WHIP leaders References Notes External links Baseball's single-season WHIP leaders Baseball's all-time WHIP leaders Pitching statistics
https://en.wikipedia.org/wiki/Separable%20polynomial	In mathematics, a polynomial P(X) over a given field K is separable if its roots are distinct in an algebraic closure of K, that is, the number of distinct roots is equal to the degree of the polynomial. This concept is closely related to square-free polynomial. If K is a perfect field then the two concepts coincide. In general, P(X) is separable if and only if it is square-free over any field that contains K, which holds if and only if P(X) is coprime to its formal derivative D P(X). Older definition In an older definition, P(X) was considered separable if each of its irreducible factors in K[X] is separable in the modern definition. In this definition, separability depended on the field K; for example, any polynomial over a perfect field would have been considered separable. This definition, although it can be convenient for Galois theory, is no longer in use. Separable field extensions Separable polynomials are used to define separable extensions: A field extension is a separable extension if and only if for every in which is algebraic over , the minimal polynomial of over is a separable polynomial. Inseparable extensions (that is, extensions which are not separable) may occur only in positive characteristic. The criterion above leads to the quick conclusion that if P is irreducible and not separable, then D P(X) = 0. Thus we must have P(X) = Q(X&hairsp;p) for some polynomial Q over K, where the prime number p is the characteristic. With this clue we can construct an example: P(X) = X&hairsp;p − T with K the field of rational functions in the indeterminate T over the finite field with p elements. Here one can prove directly that P(X) is irreducible and not separable. This is actually a typical example of why inseparability matters; in geometric terms P represents the mapping on the projective line over the finite field, taking co-ordinates to their pth power. Such mappings are fundamental to the algebraic geometry of finite fields. Put another way, there are coverings in that setting that cannot be 'seen' by Galois theory. (See Radical morphism for a higher-level discussion.) If L is the field extension K(T&hairsp;1/p), in other words the splitting field of P, then L/K is an example of a purely inseparable field extension. It is of degree p, but has no automorphism fixing K, other than the identity, because T&hairsp;1/p is the unique root of P. This shows directly that Galois theory must here break down. A field such that there are no such extensions is called perfect. That finite fields are perfect follows a posteriori from their known structure. One can show that the tensor product of fields of L with itself over K for this example has nilpotent elements that are non-zero. This is another manifestation of inseparability: that is, the tensor product operation on fields need not produce a ring that is a product of fields (so, not a commutative semisimple ring). If P(x) is separable, and its roots form a group (a subgroup of
https://en.wikipedia.org/wiki/Karol%20Borsuk	Karol Borsuk (8 May 1905 – 24 January 1982) was a Polish mathematician. His main interest was topology, while he obtained significant results also in functional analysis. Borsuk introduced the theory of absolute retracts (ARs) and absolute neighborhood retracts (ANRs), and the cohomotopy groups, later called Borsuk–Spanier cohomotopy groups. He also founded shape theory. He has constructed various beautiful examples of topological spaces, e.g. an acyclic, 3-dimensional continuum which admits a fixed point free homeomorphism onto itself; also 2-dimensional, contractible polyhedra which have no free edge. His topological and geometric conjectures and themes stimulated research for more than half a century; in particular, his open problems stimulated the infinite-dimensional topology. Borsuk received his master's degree and doctorate from Warsaw University in 1927 and 1930, respectively; his PhD thesis advisor was Stefan Mazurkiewicz. He was a member of the Polish Academy of Sciences from 1952. Borsuk's students included Samuel Eilenberg, Włodzimierz Holsztyński, Jan Jaworowski, Krystyna Kuperberg, Włodzimierz Kuperberg, Hanna Patkowska, and Andrzej Trybulec. Works Geometria analityczna w n wymiarach (1950) (translated to English as Multidimensional Analytic Geometry, Polish Scientific Publishers, 1969) Podstawy geometrii (1955) Foundations of Geometry (1960) with Wanda Szmielew, North Holland publisher Theory of Retracts (1967), PWN, Warszawa. Theory of Shape (1975) Collected papers vol. I, (1983), PWN, Warszawa. See also Zygmunt Janiszewski Stanislaw Ulam Scottish Café Animal Husbandry, an educational dice game published by Borsuk at his own expense in 1943 during the German occupation of Warsaw. The original game was lost during the Warsaw uprising in August 1944. Very few copies survived outside Warsaw and one was given back to the Borsuk family. The game is now published by Granna under the name of "Super Farmer". References External links Warsaw School of Mathematics Topologists Members of the Polish Academy of Sciences University of Warsaw alumni Academic staff of the University of Warsaw People from Warsaw Governorate 1905 births 1982 deaths
https://en.wikipedia.org/wiki/Samuel%20Eilenberg	Samuel Eilenberg (September 30, 1913 – January 30, 1998) was a Polish-American mathematician who co-founded category theory (with Saunders Mac Lane) and homological algebra. Early life and education He was born in Warsaw, Kingdom of Poland to a Jewish family. He spent much of his career as a professor at Columbia University. He earned his Ph.D. from University of Warsaw in 1936, with thesis On the Topological Applications of Maps onto a Circle; his thesis advisors were Kazimierz Kuratowski and Karol Borsuk. He died in New York City in January 1998. Career Eilenberg's main body of work was in algebraic topology. He worked on the axiomatic treatment of homology theory with Norman Steenrod (and the Eilenberg–Steenrod axioms are named for the pair), and on homological algebra with Saunders Mac Lane. In the process, Eilenberg and Mac Lane created category theory. Eilenberg was a member of Bourbaki and, with Henri Cartan, wrote the 1956 book Homological Algebra. Later in life he worked mainly in pure category theory, being one of the founders of the field. The Eilenberg swindle (or telescope) is a construction applying the telescoping cancellation idea to projective modules. Eilenberg contributed to automata theory and algebraic automata theory. In particular, he introduced a model of computation called X-machine and a new prime decomposition algorithm for finite state machines in the vein of Krohn–Rhodes theory. Art collection Eilenberg was also a prominent collector of Asian art. His collection mainly consisted of small sculptures and other artifacts from India, Indonesia, Nepal, Thailand, Cambodia, Sri Lanka and Central Asia. In 1991–1992, the Metropolitan Museum of Art in New York staged an exhibition from more than 400 items that Eilenberg had donated to the museum, entitled The Lotus Transcendent: Indian and Southeast Asian Art From the Samuel Eilenberg Collection. In reciprocity, the Metropolitan Museum of Art donated substantially to the endowment of the Samuel Eilenberg Visiting Professorship in Mathematics at Columbia University. Selected publications See also Stefan Banach Stanislaw Ulam Eilenberg–Montgomery fixed point theorem Footnotes External links Eilenberg's biography − from the National Academies Press, by Hyman Bass, Henri Cartan, Peter Freyd, Alex Heller and Saunders Mac Lane. 1913 births 1998 deaths 20th-century American mathematicians Category theorists Columbia University faculty Nicolas Bourbaki Scientists from New York City Warsaw School of Mathematics People from Warsaw Governorate Polish emigrants to the United States 20th-century Polish Jews Topologists University of Warsaw alumni Wolf Prize in Mathematics laureates Members of the United States National Academy of Sciences Mathematicians from New York (state)
https://en.wikipedia.org/wiki/Henri%20Cartan	Henri Paul Cartan (; 8 July 1904 – 13 August 2008) was a French mathematician who made substantial contributions to algebraic topology. He was the son of the mathematician Élie Cartan, nephew of mathematician Anna Cartan, oldest brother of composer , physicist and mathematician , and the son-in-law of physicist Pierre Weiss. Life According to his own words, Henri Cartan was interested in mathematics at a very young age, without being influenced by his family. He moved to Paris with his family after his father's appointment at Sorbonne in 1909 and he attended secondary school at Lycée Hoche in Versailles. In 1923 he started studying mathematics at École Normale Supérieure, receiving an agrégation in 1926 and a doctorate in 1928. His PhD thesis, entitled Sur les systèmes de fonctions holomorphes a variétés linéaires lacunaires et leurs applications, was supervised by Paul Montel. Cartan taught at Lycée Malherbe in Caen from 1928 to 1929, at University of Lille from 1929 to 1931 and at University of Strasbourg from 1931 to 1939. After German invasion of France the university staff was moved to Clermont Ferrand, but in 1940 he returned to Paris to work at Université de Paris and École Normale Supérieure. From 1969 until his retirement in 1975 he was professor at Paris-Sud University. Cartan died on 13 August 2008 at the age of 104. His funeral took place the following Wednesday on 20 August in Die, Drome. Honours and awards In 1932 Cartan was invited to give a Cours Peccot at the Collège de France. In 1950 he was elected president of the Société mathématique de France and from 1967 to 1970 he was president of the International Mathematics Union. He was awarded the Émile Picard Medal in 1959, the CNRS Gold Medal in 1976, and the Wolf Prize in 1980. He was an invited Speaker at the International Congress of Mathematics in 1932 in Zürich and a Plenary Speaker at the ICM in 1950 in Cambridge, Massachusetts and in 1958 in Edinburgh. From 1974 until his death he had been a member of the French Academy of Sciences. He was elected a foreign member of many academies and societies, including the American Academy of Arts and Sciences (1950), London Mathematical Society (1959), Royal Danish Academy of Sciences and Letters (1962), (1967), Royal Society of London (1971), Göttingen Academy of Sciences and Humanities (1971), Spanish Royal Academy of Sciences (1971), United States National Academy of Sciences (1972), Bavarian Academy of Science (1974), Royal Academy of Belgium (1978), Japan Academy (1979), Finnish Academy of Science and Letters (1979), Royal Swedish Academy of Sciences (1981), Polish Academy of Sciences (1985) and Russian Academy of Sciences (1999). He was awarded Honorary Doctorates from Münster (1952), ETH Zürich (1955), Oslo (1961), Sussex (1969), Cambridge (1969), Stockholm (1978), Oxford University (1980), Zaragoza (1985) and Athens (1992). The French government named him Commandeur des Palmes Académiques in 1964, Officier de la Légi
https://en.wikipedia.org/wiki/J.%20A.%20Todd	John Arthur Todd (23 August 1908 – 22 December 1994) was an English mathematician who specialised in geometry. Biography He was born in Liverpool, and went up to Trinity College, Cambridge in 1925. He did research under H.F. Baker, and in 1931 took a position at the University of Manchester. He became a lecturer at Cambridge in 1937. He remained at Cambridge for the rest of his working life. Work The Todd class in the theory of the higher-dimensional Riemann–Roch theorem is an example of a characteristic class (or, more accurately, a reciprocal of one) that was discovered by Todd in work published in 1937. It used the methods of the Italian school of algebraic geometry. The Todd–Coxeter process for coset enumeration is a major method of computational algebra, and dates from a collaboration with H.S.M. Coxeter in 1936. In 1953 he and Coxeter discovered the Coxeter–Todd lattice. In 1954 he and G. C. Shephard classified the finite complex reflection groups. Honours In March 1948 he was elected a Fellow of the Royal Society. Selected publications 1936: "A practical method for enumerating cosets of a finite abstract group", Proc. Edin. Math. Soc. 5(1), 26-34 (with Harold Scott MacDonald Coxeter) 1937: "Rational quartic primals and associated Cremona transformations of four-dimensional space", Proc. London Math. Soc. s2-42, 324-339 (with Dennis Babbage), "The geometrical invariants of algebraic varieties", Proc. London Math. Soc. 43(2), 127-138, "The arithmetical invariants of algebraic loci", Proc. London Math. Soc. 43(2), 190-225 1939: "The geometrical invariants of algebraic loci", Proc. London Math. Soc. 45, 410-424 1953: "An extreme duodenary form", Can. J. Math. 5, 384-392 (with Harold Scott MacDonald Coxeter) 1954: "Finite unitary reflection groups", Canadian Journal of Mathematics 6, 274-304 (with Geoffrey Colin Shephard) 1960: "On complex Stiefel manifolds", Mathematical Proc. Camb. Phil. Soc. 56, 342-353 (with Michael Atiyah) 1966: "A representation of the Mathieu group M24 as a collineation group", Ann. Mat. Pura Appl. 71(4), 199-238 References External links Todd's Mactutor biography 1908 births 1994 deaths Alumni of Trinity College, Cambridge 20th-century English mathematicians Scientists from Liverpool Fellows of Downing College, Cambridge Fellows of the Royal Society
https://en.wikipedia.org/wiki/Kazimierz%20Zarankiewicz	Kazimierz Zarankiewicz (2 May 1902 – 5 September 1959) was a Polish mathematician and Professor at the Warsaw University of Technology who was interested primarily in topology and graph theory. Biography Zarankiewicz was born in Częstochowa to father Stanisław and mother Józefa (née Borowska). He studied at the University of Warsaw, together with Zygmunt Janiszewski, Stefan Mazurkiewicz, Wacław Sierpiński, Kazimierz Kuratowski, and Stanisław Saks. During World War II, Zarankiewicz took part in illegal teaching, forbidden by the German authorities, and eventually was sent to a concentration camp. He survived and became a teacher at Warsaw University of Technology (Polish: Politechnika Warszawska). He visited universities in Tomsk, Harvard, London, and Vienna. He served as president of the Warsaw section of the Polish Mathematical Society and the International Astronautical Federation. He died in London, England. Research contributions Zarankiewicz wrote works on cut-points in connected spaces, on conformal mappings, on complex functions and number theory, and triangular numbers. The Zarankiewicz problem is named after Zarankiewicz. This problem asks, for a given size of (0,1)-matrix, how many matrix entries must be set equal to 1 in order to guarantee that the matrix contains at least one a × b submatrix is made up only of 1's. An equivalent formulation in extremal graph theory asks for the maximum number of edges in a bipartite graph with no complete bipartite subgraph Ka,b. The Zarankiewicz crossing number conjecture in the mathematical field of graph theory is also named after Zarankiewicz. The conjecture states that the crossing number of a complete bipartite graph equals Zarankiewicz proved that this formula is an upper bound for the actual crossing number. The problem of determining the number was suggested by Paul Turán and became known as Turán's brick factory problem. See also List of Polish mathematicians References External links 20th-century Polish mathematicians Topologists University of Warsaw alumni Academic staff of the Warsaw University of Technology Nazi concentration camp survivors People from Częstochowa 1902 births 1959 deaths
https://en.wikipedia.org/wiki/Stanis%C5%82aw%20Saks	Stanisław Saks (30 December 1897 – 23 November 1942) was a Polish mathematician and university tutor, a member of the Lwów School of Mathematics, known primarily for his membership in the Scottish Café circle, an extensive monograph on the theory of integrals, his works on measure theory and the Vitali–Hahn–Saks theorem. Life and work Stanisław Saks was born on 30 December 1897 in Kalisz, Congress Poland, to an assimilated Polish-Jewish family. In 1915 he graduated from a local gymnasium and joined the newly recreated Warsaw University. In 1922 he received a doctorate of his alma mater with a prestigious distinction maxima cum laude. Soon afterwards he also passed his habilitation and received the Rockefeller fellowship, which allowed him to travel to the United States. Around that time he started publishing articles in various mathematical journals, mostly the Fundamenta Mathematicae, but also in the Transactions of the American Mathematical Society. He participated in the Silesian Uprisings and was awarded the Cross of the Valorous and the Medal of Independence for his bravery. Following the end of the uprising he returned to Warsaw and resumed his academic career. For most of it he studied the theories of functions and functionals in particular. In 1930 he published his most notable book, the Zarys teorii całki (Sketch on the Theory of the Integral), which later got expanded and translated into several languages, including English (Theory of the Integral), French (Théorie de l'Intégrale) and Russian (Teoriya Integrala). Despite his successes, Saks was never awarded the title of professor and remained an ordinary tutor, initially at his alma mater and the Warsaw University of Technology, and later at the Lwów University and Wilno University. He was also an active socialist and a journalist at the Robotnik weekly (1919–1926) and later a collaborator of the Association of Socialist Youth. Saks wrote a mathematics book with Antoni Zygmund, Analytic Functions, in 1933. It was translated into English in 1952 by E. J. Scott. In the preface to the English edition, Zygmund writes: Stanislaw Saks was a man of moral as well as physical courage, of rare intelligence and wit. To his colleagues and pupils he was an inspiration not only as a mathematician but as a human being. In the period between the two world wars he exerted great influence upon a whole generation of Polish mathematicians in Warsaw and Lwów. In November 1942, at the age of 45, Saks died in a Warsaw prison, victim of a policy of extermination. After the outbreak of World War II and the occupation of Poland by Germany, Saks joined the Polish underground. Arrested in November 1942, he was executed on 23 November 1942 by the German Gestapo in Warsaw. Publications . English translation by Laurence Chisholm Young, with two additional notes by Stefan Banach. See also Lwów School of Mathematics Notes References Functional analysts Measure theorists 1897 births 1942 deaths Lwów
https://en.wikipedia.org/wiki/Scottish%20Caf%C3%A9	The Scottish Café () was a café in Lwów, Poland (now Lviv, Ukraine) where, in the 1930s and 1940s, mathematicians from the Lwów School of Mathematics collaboratively discussed research problems, particularly in functional analysis and topology. Stanisław Ulam recounts that the tables of the café had marble tops, so they could write in pencil, directly on the table, during their discussions. To keep the results from being lost, and after becoming annoyed with their writing directly on the table tops, Stefan Banach's wife provided the mathematicians with a large notebook, which was used for writing the problems and answers and eventually became known as the Scottish Book. The book—a collection of solved, unsolved, and even probably unsolvable problems—could be borrowed by any of the guests of the café. Solving any of the problems was rewarded with prizes, with the most difficult and challenging problems having expensive prizes (during the Great Depression and on the eve of World War II), such as a bottle of fine brandy. For problem 153, which was later recognized as being closely related to Stefan Banach's "basis problem", Stanisław Mazur offered the prize of a live goose. This problem was solved only in 1972 by Per Enflo, who was presented with the live goose in a ceremony that was broadcast throughout Poland. The café building now houses the Szkocka Restaurant & Bar (named for the original Scottish Café) and the Atlas Deluxe hotel at the street address of 27 Taras Shevchenko Prospekt. See also The following mathematicians were associated with the Lwów School of Mathematics or contributed to The Scottish Book: Stefan Banach Karol Borsuk Mark Kac Stefan Kaczmarz Bronisław Knaster Kazimierz Kuratowski Stanisław Saks Juliusz Schauder Hugo Steinhaus Stanisław Ulam Gus Ward References External links Scottish book Scottish book Web page at Home Page of Stefan Banach at Adam Mickiewicz University in Poznań website Manuscript of Scottish book (PDF) Typescript of English version of Scottish book (PDF) Kawiarnia Szkocka at the MacTutor archive "The Life of Stefan Banach" review by Sheldon Axler History of Lviv History of mathematics History of education in Poland Coffeehouses and cafés in Poland Buildings and structures in Lviv
https://en.wikipedia.org/wiki/W%C5%82adys%C5%82aw%20Orlicz	Władysław Roman Orlicz (May 24, 1903 – August 9, 1990) was a Polish mathematician of Lwów School of Mathematics. His main interests were functional analysis and topology: Orlicz spaces are named after him. Education and career Orlicz was the third of Franciszek and Maria Orlicz's five children. His youngest brother died in the Polish-Soviet War, the eldest perished in the Stutthof concentration camp. The other brothers also became professors. The family moved several times. Orlicz attended school in Tarnów, Znojmo, and Lviv, where he finished school in 1920 and began studying mathematics at the Lviv Polytechnic University. He studied with Hugo Steinhaus, Antoni Łomnicki and Stanisław Ruziewicz, among others. As early as 1923 he took on small tasks at the Faculty of Mathematics. On August 1, 1925 he became a junior assistant at the Jan Kazimierz University in Lemberg (now University of Lviv). He published his first scientific work in 1926 at the age of 23. In 1928 he completed his dissertation on the theory of orthogonal sequences. In 1929 he went to University of Göttingen on a scholarship and returned to Lemberg in 1930 as a senior assistant. In 1934 he presented his habilitation thesis The investigations of orthogonal systems. The following year he became an assistant professor at the Lviv Polytechnic University and received his teaching license at the University of Lviv. In 1937, he became an associate professor at the University of Poznań. The outbreak of the Second World War surprised him while he was on vacation in Lemberg. Since he could not return to Poznań, he was appointed professor in Lemberg. Officially he worked as a teacher. When it became clear in early 1945 that Lemberg would no longer belong to Poland, Orlicz went back to Poznań. In 1948 he was appointed full professor at the University of Poznań, where he remained until his retirement in 1970. Orlicz was awarded the Stefan Banach Prize by the Polish Mathematical Society in 1948. See also Orlicz space Orlicz–Pettis theorem List of Polish mathematicians External links References 1903 births 1990 deaths Lwów School of Mathematics Functional analysts Topologists Recipients of the State Award Badge (Poland) People from Brzesko County Lviv Polytechnic alumni Academic staff of Adam Mickiewicz University in Poznań Academic staff of Lviv Polytechnic Academic staff of the University of Lviv 20th-century Polish mathematicians
https://en.wikipedia.org/wiki/Fantasy%20hockey	Fantasy hockey is a form of fantasy sport where players build a team that competes with other players who do the same, based on the statistics generated by professional hockey players or teams. The majority of fantasy hockey pools are based on the teams and players of the ice hockey National Hockey League (NHL). A typical fantasy hockey league or hockey pool, has 8 - 12 teams but often have as many as 20. Other types of pools may have a greater number of teams, which may dilute the average talent making it more or less fun depending on the league, but also represents more closely the actual NHL, which currently has 32 teams. Other forms of fantasy hockey may allow an unlimited number of teams, whereby any number of owners may draft the same player(s). These typically have a restricted number of "trades" where one player may simply be exchanged for any other in the player pool, typically of the same position. NHL.com/Fantasy has recently integrated year-round fantasy coverage, featuring rankings, news and advice. Rules The most common way for choosing NHL players or teams to comprise a fantasy team is via a draft, either online or in person. However, the method ranges from basic (i.e., the draft has a predetermined number of rounds, and every team has a pick in each round, unless they have traded their picks) to complicated (i.e. 'auction' style). Some leagues require an entry fee at the start of the season, with the league champion at the end of the year collecting some or all of the money. In other leagues, money is not involved at all and the league is simply for fun. Team structure Most office hockey pools keep the teams simple - merely choose 12 or 15 or 20 skaters from any position, most points win. However, fantasy hockey leagues usually use more complicated formats. One common format of a fantasy team (rotisserie style) is: 2 Centers (Forward) 2 Left Wings (Forward) 2 Right Wings (Forward) 4 Defensemen 2 Goaltenders 5 Bench Players (who can play any position) Point scoring Common categories in which fantasy owners collect points include: Points Goals Assists Plus/Minus (+/-) Penalty Minutes Power Play Points Game-Winning Goals Faceoffs Won Shots On Goal Hits Blocked Shots Wins (Goalie) Goals Against Average (Goalie) Save Percentage (Goalie) Shutouts (Goalie) Hat Tricks "Points" (which is goals plus assists) is the most common measure of a fantasy hockey team's performance. Some pools offer additional scoring based on the player's position (such as points for a goaltender victory) or skill level (such as points for penalties earned by an enforcer). Other pools have much higher levels of complexity, taking into account defensive statistics and +/-. When management of the participant's roster is required (i.e., activating certain players), this type of pool is often referred to as fantasy hockey. Example The most common scoring scheme is simply: 1 point per goal 1 point per assist 2 points per win 0 poi
https://en.wikipedia.org/wiki/Johnny%20Ball	Johnny Ball (born Graham Thalben Ball; 23 May 1938) is an English television personality and a populariser of mathematics. He is also the father of BBC Radio 2 DJ Zoe Ball. Early life Ball was born in Bristol and attended Kingswood Primary School on the eastern edge of the city. Later in his childhood the family moved to Bolton, Lancashire, where he attended Bolton County Grammar School. He left formal education with two "O" Levels, one in mathematics and one in geography. He was called up for national service and spent three years in the Royal Air Force. He was posted to Wales as a radar operator and was later sent to Germany to monitor the Hamburg-Berlin air corridor. Ball began his entertainment career by working as a Butlin's Redcoat, and was an entertainer in northern clubs and cabaret. He was nicknamed Johnny after John Ball, who played for Bolton Wanderers from 1950 to 1958 and the name stuck. Television and radio career Ball was a regular fixture on children's television from the mid 1970s and throughout the 1980s, presenting several series of science and technology programmes intended for children (including Think of a Number; Think Again; Think Backwards; Think...This Way and Johnny Ball Reveals All). He was also one of the hosts of pre-school programme Play School beginning in 1967 and continuing throughout the 1970s and beyond. As well as appearing on screen Ball wrote jokes for some shows including Crackerjack. All of these shows (except the ITV programme ...Reveals All) appeared on the BBC. Ball's shows were known for presenting scientific and technological principles in an entertaining and accessible way for young people. In 2003, he appeared on The Terry and Gaby Show in which he answered viewers' questions. In July 2004, he was named in the Radio Times list of the top 40 most eccentric TV presenters of all time. In July 2012, he presented a Horizon special on ageing on BBC Four. He has starred in ITV and Channel 4 television adverts as well as radio adverts for the Yorkshire-based firm Help-Link. In 2012, Ball took part in the Strictly Come Dancing TV show, where he was paired with Aliona Vilani. A training accident in the three-week interval resulted in torn ligaments for Vilani, causing her to retire temporarily from the show. She was replaced by Iveta Lukošiūtė who, with Ball, was eliminated in the first week. Vilani returned in the final group dance alongside Ball. In a TV interview in October 2017, Ball claimed Vilani faked the injury, with Vilani denying the allegation and saying she would take legal advice over Ball's comments. There are no reports that she subsequently took any form of legal action. Personal life Ball's daughter Zoe by his first wife, Julia née Anderson, previously presented Strictly Come Dancing: It Takes Two for BBC TV and currently presents the breakfast show on BBC Radio 2. The couple divorced when Zoe was two. Ball lives with his current wife Diane in Buckinghamshire. Series guide Think of
https://en.wikipedia.org/wiki/Parallelogram%20law	In mathematics, the simplest form of the parallelogram law (also called the parallelogram identity) belongs to elementary geometry. It states that the sum of the squares of the lengths of the four sides of a parallelogram equals the sum of the squares of the lengths of the two diagonals. We use these notations for the sides: AB, BC, CD, DA. But since in Euclidean geometry a parallelogram necessarily has opposite sides equal, that is, AB = CD and BC = DA, the law can be stated as If the parallelogram is a rectangle, the two diagonals are of equal lengths AC = BD, so and the statement reduces to the Pythagorean theorem. For the general quadrilateral with four sides not necessarily equal, where is the length of the line segment joining the midpoints of the diagonals. It can be seen from the diagram that for a parallelogram, and so the general formula simplifies to the parallelogram law. Proof In the parallelogram on the right, let AD = BC = a, AB = DC = b, By using the law of cosines in triangle we get: In a parallelogram, adjacent angles are supplementary, therefore Using the law of cosines in triangle produces: By applying the trigonometric identity to the former result proves: Now the sum of squares can be expressed as: Simplifying this expression, it becomes: The parallelogram law in inner product spaces In a normed space, the statement of the parallelogram law is an equation relating norms: The parallelogram law is equivalent to the seemingly weaker statement: because the reverse inequality can be obtained from it by substituting for and for and then simplifying. With the same proof, the parallelogram law is also equivalent to: In an inner product space, the norm is determined using the inner product: As a consequence of this definition, in an inner product space the parallelogram law is an algebraic identity, readily established using the properties of the inner product: Adding these two expressions: as required. If is orthogonal to meaning and the above equation for the norm of a sum becomes: which is Pythagoras' theorem. Normed vector spaces satisfying the parallelogram law Most real and complex normed vector spaces do not have inner products, but all normed vector spaces have norms (by definition). For example, a commonly used norm for a vector in the real coordinate space is the -norm: Given a norm, one can evaluate both sides of the parallelogram law above. A remarkable fact is that if the parallelogram law holds, then the norm must arise in the usual way from some inner product. In particular, it holds for the -norm if and only if the so-called norm or norm. For any norm satisfying the parallelogram law (which necessarily is an inner product norm), the inner product generating the norm is unique as a consequence of the polarization identity. In the real case, the polarization identity is given by: or equivalently by In the complex case it is given by: For example, using the -norm with
https://en.wikipedia.org/wiki/SPSD	SPSD can refer to: Southfield Public School District Saskatoon Public School Division Symmetric Positive Semi-Definite matrix, in linear algebra
https://en.wikipedia.org/wiki/38%20%28number%29	38 (thirty-eight) is the natural number following 37 and preceding 39. In mathematics specifically, the 11th discrete Semiprime, it being the 7th of the form (2.q). the first member of the third cluster of two discrete semiprimes 38, 39 the next such cluster is 57, 58. with an aliquot sum of 22 in an aliquot sequence of five composite numbers (38,22,14,10,8,7,1,0) to the Prime in the 7-aliquot tree. 34 is the first semiprime within a chain of 4 semiprimes in its aliquot sequence (38,22,14,10). The next semiprime with a four semiprime chain is 166. 38! − 1 yields which is the 16th factorial prime. There is no answer to the equation φ(x) = 38, making 38 a nontotient. 38 is the sum of the squares of the first three primes. 37 and 38 are the first pair of consecutive positive integers not divisible by any of their digits. 38 is the largest even number which cannot be written as the sum of two odd composite numbers. The sum of each row of the only non-trivial (order 3) magic hexagon is 38. In science The atomic number of strontium Astronomy The Messier object M38, a magnitude 7.0 open cluster in the constellation Auriga The New General Catalogue object NGC 38, a spiral galaxy in the constellation Pisces In other fields Thirty-eight is also: The 38th parallel north is the pre-Korean War boundary between North Korea and South Korea. The number of slots on an American roulette wheel (0, 00, and 1 through 36; European roulette does not use the 00 slot and has only 37 slots) The Ishihara test is a color vision test consisting of 38 pseudoisochromatic plates. A "38" is often the name for a snub nose .38 caliber revolver. The 38 class is the most famous class of steam locomotive used in New South Wales Gerald Ford, 38th President of the United States Arnold Schwarzenegger, 38th Governor of California, most recent Republican governor of California, and the second governor to be born outside of the United States Cats have a total of 38 chromosomes in their genome. Number used by Zane Smith and Front Row Motorsports to win the 2022 NASCAR Camping World Truck Series championship. See also List of highways numbered 38 References Integers
https://en.wikipedia.org/wiki/Homogeneous%20space	In mathematics, a homogeneous space is, very informally, a space that looks the same everywhere, as you move through it, with movement given by the action of a group. Homogeneous spaces occur in the theories of Lie groups, algebraic groups and topological groups. More precisely, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts transitively. The elements of G are called the symmetries of X. A special case of this is when the group G in question is the automorphism group of the space X – here "automorphism group" can mean isometry group, diffeomorphism group, or homeomorphism group. In this case, X is homogeneous if intuitively X looks locally the same at each point, either in the sense of isometry (rigid geometry), diffeomorphism (differential geometry), or homeomorphism (topology). Some authors insist that the action of G be faithful (non-identity elements act non-trivially), although the present article does not. Thus there is a group action of G on X which can be thought of as preserving some "geometric structure" on X, and making X into a single G-orbit. Formal definition Let X be a non-empty set and G a group. Then X is called a G-space if it is equipped with an action of G on X. Note that automatically G acts by automorphisms (bijections) on the set. If X in addition belongs to some category, then the elements of G are assumed to act as automorphisms in the same category. That is, the maps on X coming from elements of G preserve the structure associated with the category (for example, if X is an object in Diff then the action is required to be by diffeomorphisms). A homogeneous space is a G-space on which G acts transitively. Succinctly, if X is an object of the category C, then the structure of a G-space is a homomorphism: into the group of automorphisms of the object X in the category C. The pair (X, ρ) defines a homogeneous space provided ρ(G) is a transitive group of symmetries of the underlying set of X. Examples For example, if X is a topological space, then group elements are assumed to act as homeomorphisms on X. The structure of a G-space is a group homomorphism ρ : G → Homeo(X) into the homeomorphism group of X. Similarly, if X is a differentiable manifold, then the group elements are diffeomorphisms. The structure of a G-space is a group homomorphism ρ : G → Diffeo(X) into the diffeomorphism group of X. Riemannian symmetric spaces are an important class of homogeneous spaces, and include many of the examples listed below. Concrete examples include: Isometry groups Positive curvature: Sphere (orthogonal group): . This is true because of the following observations: First, is the set of vectors in with norm . If we consider one of these vectors as a base vector, then any other vector can be constructed using an orthogonal transformation. If we consider the span of this vector as a one dimensional subspace of , then the complement is an -dimensional vector space which is inva
https://en.wikipedia.org/wiki/Separable%20extension	In field theory, a branch of algebra, an algebraic field extension is called a separable extension if for every , the minimal polynomial of over is a separable polynomial (i.e., its formal derivative is not the zero polynomial, or equivalently it has no repeated roots in any extension field). There is also a more general definition that applies when is not necessarily algebraic over . An extension that is not separable is said to be inseparable. Every algebraic extension of a field of characteristic zero is separable, and every algebraic extension of a finite field is separable. It follows that most extensions that are considered in mathematics are separable. Nevertheless, the concept of separability is important, as the existence of inseparable extensions is the main obstacle for extending many theorems proved in characteristic zero to non-zero characteristic. For example, the fundamental theorem of Galois theory is a theorem about normal extensions, which remains true in non-zero characteristic only if the extensions are also assumed to be separable. The opposite concept, a purely inseparable extension, also occurs naturally, as every algebraic extension may be decomposed uniquely as a purely inseparable extension of a separable extension. An algebraic extension of fields of non-zero characteristics is a purely inseparable extension if and only if for every , the minimal polynomial of over is not a separable polynomial, or, equivalently, for every element of , there is a positive integer such that . The simplest example of a (purely) inseparable extension is , fields of rational functions in the indeterminate x with coefficients in the finite field . The element has minimal polynomial , having and a p-fold multiple root, as . This is a simple algebraic extension of degree p, as , but it is not a normal extension since the Galois group is trivial. Informal discussion An arbitrary polynomial with coefficients in some field is said to have distinct roots or to be square-free if it has roots in some extension field . For instance, the polynomial has precisely roots in the complex plane; namely and , and hence does have distinct roots. On the other hand, the polynomial , which is the square of a non-constant polynomial does not have distinct roots, as its degree is two, and is its only root. Every polynomial may be factored in linear factors over an algebraic closure of the field of its coefficients. Therefore, the polynomial does not have distinct roots if and only if it is divisible by the square of a polynomial of positive degree. This is the case if and only if the greatest common divisor of the polynomial and its derivative is not a constant. Thus for testing if a polynomial is square-free, it is not necessary to consider explicitly any field extension nor to compute the roots. In this context, the case of irreducible polynomials requires some care. A priori, it may seem that being divisible by a square is impossibl
https://en.wikipedia.org/wiki/Fredholm%20integral%20equation	In mathematics, the Fredholm integral equation is an integral equation whose solution gives rise to Fredholm theory, the study of Fredholm kernels and Fredholm operators. The integral equation was studied by Ivar Fredholm. A useful method to solve such equations, the Adomian decomposition method, is due to George Adomian. Equation of the first kind A Fredholm equation is an integral equation in which the term containing the kernel function (defined below) has constants as integration limits. A closely related form is the Volterra integral equation which has variable integral limits. An inhomogeneous Fredholm equation of the first kind is written as and the problem is, given the continuous kernel function and the function , to find the function . An important case of these types of equation is the case when the kernel is a function only of the difference of its arguments, namely , and the limits of integration are ±∞, then the right hand side of the equation can be rewritten as a convolution of the functions and and therefore, formally, the solution is given by where and are the direct and inverse Fourier transforms, respectively. This case would not be typically included under the umbrella of Fredholm integral equations, a name that is usually reserved for when the integral operator defines a compact operator (convolution operators on non-compact groups are non-compact, since, in general, the spectrum of the operator of convolution with contains the range of , which is usually a non-countable set, whereas compact operators have discrete countable spectra). Equation of the second kind An inhomogeneous Fredholm equation of the second kind is given as Given the kernel , and the function , the problem is typically to find the function . A standard approach to solving this is to use iteration, amounting to the resolvent formalism; written as a series, the solution is known as the Liouville–Neumann series. General theory The general theory underlying the Fredholm equations is known as Fredholm theory. One of the principal results is that the kernel yields a compact operator. Compactness may be shown by invoking equicontinuity. As an operator, it has a spectral theory that can be understood in terms of a discrete spectrum of eigenvalues that tend to 0. Applications Fredholm equations arise naturally in the theory of signal processing, for example as the famous spectral concentration problem popularized by David Slepian. The operators involved are the same as linear filters. They also commonly arise in linear forward modeling and inverse problems. In physics, the solution of such integral equations allows for experimental spectra to be related to various underlying distributions, for instance the mass distribution of polymers in a polymeric melt, or the distribution of relaxation times in the system. In addition, Fredholm integral equations also arise in fluid mechanics problems involving hydrodynamic interactions near fini
https://en.wikipedia.org/wiki/Combinatorial%20topology	In mathematics, combinatorial topology was an older name for algebraic topology, dating from the time when topological invariants of spaces (for example the Betti numbers) were regarded as derived from combinatorial decompositions of spaces, such as decomposition into simplicial complexes. After the proof of the simplicial approximation theorem this approach provided rigour. The change of name reflected the move to organise topological classes such as cycles-modulo-boundaries explicitly into abelian groups. This point of view is often attributed to Emmy Noether, and so the change of title may reflect her influence. The transition is also attributed to the work of Heinz Hopf, who was influenced by Noether, and to Leopold Vietoris and Walther Mayer, who independently defined homology. A fairly precise date can be supplied in the internal notes of the Bourbaki group. While topology was still combinatorial in 1942, it had become algebraic by 1944. This corresponds also to the period where homological algebra and category theory were introduced for the study of topological spaces, and largely supplanted combinatorial methods. Azriel Rosenfeld (1973) proposed digital topology for a type of image processing that can be considered as a new development of combinatorial topology. The digital forms of the Euler characteristic theorem and the Gauss–Bonnet theorem were obtained by Li Chen and Yongwu Rong. A 2D grid cell topology already appeared in the Alexandrov–Hopf book Topologie I (1935). See also Hauptvermutung Topological combinatorics Topological graph theory Notes References Algebraic topology Combinatorics es:Topología combinatoria
https://en.wikipedia.org/wiki/Principal%20homogeneous%20space	In mathematics, a principal homogeneous space, or torsor, for a group G is a homogeneous space X for G in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a group G is a non-empty set X on which G acts freely and transitively (meaning that, for any x, y in X, there exists a unique g in G such that , where · denotes the (right) action of G on X). An analogous definition holds in other categories, where, for example, G is a topological group, X is a topological space and the action is continuous, G is a Lie group, X is a smooth manifold and the action is smooth, G is an algebraic group, X is an algebraic variety and the action is regular. Definition If G is nonabelian then one must distinguish between left and right torsors according to whether the action is on the left or right. In this article, we will use right actions. To state the definition more explicitly, X is a G-torsor or G-principal homogeneous space if X is nonempty and is equipped with a map (in the appropriate category) such that x·1 = x x·(gh) = (x·g)·h for all and all and such that the map given by is an isomorphism (of sets, or topological spaces or ..., as appropriate, i.e. in the category in question). Note that this means that X and G are isomorphic (in the category in question; not as groups: see the following). However—and this is the essential point—there is no preferred 'identity' point in X. That is, X looks exactly like G except that which point is the identity has been forgotten. (This concept is often used in mathematics as a way of passing to a more intrinsic point of view, under the heading 'throw away the origin'.) Since X is not a group, we cannot multiply elements; we can, however, take their "quotient". That is, there is a map that sends to the unique element such that . The composition of the latter operation with the right group action, however, yields a ternary operation , which serves as an affine generalization of group multiplication and which is sufficient to both characterize a principal homogeneous space algebraically and intrinsically characterize the group it is associated with. If we denote the result of this ternary operation, then the following identities will suffice to define a principal homogeneous space, while the additional property identifies those spaces that are associated with abelian groups. The group may be defined as formal quotients subject to the equivalence relation , with the group product, identity and inverse defined, respectively, by , , and the group action by Examples Every group G can itself be thought of as a left or right G-torsor under the natural action of left or right multiplication. Another example is the affine space concept: the idea of the affine space A underlying a vector space V can be said succinctly by saying that A is a principal homogeneous space for V acting as the additive group of translations. The flags of any regular polytope fo
https://en.wikipedia.org/wiki/Colombeau%20algebra	In mathematics, a Colombeau algebra is an algebra of a certain kind containing the space of Schwartz distributions. While in classical distribution theory a general multiplication of distributions is not possible, Colombeau algebras provide a rigorous framework for this. Such a multiplication of distributions has long been believed to be impossible because of L. Schwartz' impossibility result, which basically states that there cannot be a differential algebra containing the space of distributions and preserving the product of continuous functions. However, if one only wants to preserve the product of smooth functions instead such a construction becomes possible, as demonstrated first by Colombeau. As a mathematical tool, Colombeau algebras can be said to combine a treatment of singularities, differentiation and nonlinear operations in one framework, lifting the limitations of distribution theory. These algebras have found numerous applications in the fields of partial differential equations, geophysics, microlocal analysis and general relativity so far. Colombeau algebras are named after French mathematician Jean François Colombeau. Schwartz' impossibility result Attempting to embed the space of distributions on into an associative algebra , the following requirements seem to be natural: is linearly embedded into such that the constant function becomes the unity in , There is a partial derivative operator on which is linear and satisfies the Leibniz rule, the restriction of to coincides with the usual partial derivative, the restriction of to coincides with the pointwise product. However, L. Schwartz' result implies that these requirements cannot hold simultaneously. The same is true even if, in 4., one replaces by , the space of times continuously differentiable functions. While this result has often been interpreted as saying that a general multiplication of distributions is not possible, in fact it only states that one cannot unrestrictedly combine differentiation, multiplication of continuous functions and the presence of singular objects like the Dirac delta. Colombeau algebras are constructed to satisfy conditions 1.–3. and a condition like 4., but with replaced by , i.e., they preserve the product of smooth (infinitely differentiable) functions only. Basic idea The Colombeau Algebra is defined as the quotient algebra Here the algebra of moderate functions on is the algebra of families of smooth regularisations (fε) of smooth functions on (where R+ = (0,∞) is the "regularization" parameter ε), such that for all compact subsets K of and all multiindices α, there is an N > 0 such that The ideal of negligible functions is defined in the same way but with the partial derivatives instead bounded by O(ε+N) for all N > 0. Embedding of distributions The space(s) of Schwartz distributions can be embedded into the simplified algebra by (component-wise) convolution with any element of the algebra having as repre
https://en.wikipedia.org/wiki/Characteristic%20class	In mathematics, a characteristic class is a way of associating to each principal bundle of X a cohomology class of X. The cohomology class measures the extent the bundle is "twisted" and whether it possesses sections. Characteristic classes are global invariants that measure the deviation of a local product structure from a global product structure. They are one of the unifying geometric concepts in algebraic topology, differential geometry, and algebraic geometry. The notion of characteristic class arose in 1935 in the work of Eduard Stiefel and Hassler Whitney about vector fields on manifolds. Definition Let G be a topological group, and for a topological space , write for the set of isomorphism classes of principal G-bundles over . This is a contravariant functor from Top (the category of topological spaces and continuous functions) to Set (the category of sets and functions), sending a map to the pullback operation . A characteristic class c of principal G-bundles is then a natural transformation from to a cohomology functor , regarded also as a functor to Set. In other words, a characteristic class associates to each principal G-bundle in an element c(P) in H(X) such that, if f : Y → X is a continuous map, then c(fP) = f*c(P). On the left is the class of the pullback of P to Y; on the right is the image of the class of P under the induced map in cohomology. Characteristic numbers Characteristic classes are elements of cohomology groups; one can obtain integers from characteristic classes, called characteristic numbers. Some important examples of characteristic numbers are Stiefel–Whitney numbers, Chern numbers, Pontryagin numbers, and the Euler characteristic. Given an oriented manifold M of dimension n with fundamental class , and a G-bundle with characteristic classes , one can pair a product of characteristic classes of total degree n with the fundamental class. The number of distinct characteristic numbers is the number of monomials of degree n in the characteristic classes, or equivalently the partitions of n into . Formally, given such that , the corresponding characteristic number is: where denotes the cup product of cohomology classes. These are notated variously as either the product of characteristic classes, such as , or by some alternative notation, such as for the Pontryagin number corresponding to , or for the Euler characteristic. From the point of view of de Rham cohomology, one can take differential forms representing the characteristic classes, take a wedge product so that one obtains a top dimensional form, then integrate over the manifold; this is analogous to taking the product in cohomology and pairing with the fundamental class. This also works for non-orientable manifolds, which have a -orientation, in which case one obtains -valued characteristic numbers, such as the Stiefel-Whitney numbers. Characteristic numbers solve the oriented and unoriented bordism questions: two manifolds are (respect
https://en.wikipedia.org/wiki/Serre%E2%80%93Swan%20theorem	In the mathematical fields of topology and K-theory, the Serre–Swan theorem, also called Swan's theorem, relates the geometric notion of vector bundles to the algebraic concept of projective modules and gives rise to a common intuition throughout mathematics: "projective modules over commutative rings are like vector bundles on compact spaces". The two precise formulations of the theorems differ somewhat. The original theorem, as stated by Jean-Pierre Serre in 1955, is more algebraic in nature, and concerns vector bundles on an algebraic variety over an algebraically closed field (of any characteristic). The complementary variant stated by Richard Swan in 1962 is more analytic, and concerns (real, complex, or quaternionic) vector bundles on a smooth manifold or Hausdorff space. Differential geometry Suppose M is a smooth manifold (not necessarily compact), and E is a smooth vector bundle over M. Then Γ(E), the space of smooth sections of E, is a module over C∞(M) (the commutative algebra of smooth real-valued functions on M). Swan's theorem states that this module is finitely generated and projective over C∞(M). In other words, every vector bundle is a direct summand of some trivial bundle: for some k. The theorem can be proved by constructing a bundle epimorphism from a trivial bundle This can be done by, for instance, exhibiting sections s1...sk with the property that for each point p, {si(p)} span the fiber over p. When M is connected, the converse is also true: every finitely generated projective module over C∞(M) arises in this way from some smooth vector bundle on M. Such a module can be viewed as a smooth function f on M with values in the n × n idempotent matrices for some n. The fiber of the corresponding vector bundle over x is then the range of f(x). If M is not connected, the converse does not hold unless one allows for vector bundles of non-constant rank (which means admitting manifolds of non-constant dimension). For example, if M is a zero-dimensional 2-point manifold, the module is finitely-generated and projective over but is not free, and so cannot correspond to the sections of any (constant-rank) vector bundle over M (all of which are trivial). Another way of stating the above is that for any connected smooth manifold M, the section functor Γ from the category of smooth vector bundles over M to the category of finitely generated, projective C∞(M)-modules is full, faithful, and essentially surjective. Therefore the category of smooth vector bundles on M is equivalent to the category of finitely generated, projective C∞(M)-modules. Details may be found in . Topology Suppose X is a compact Hausdorff space, and C(X) is the ring of continuous real-valued functions on X. Analogous to the result above, the category of real vector bundles on X is equivalent to the category of finitely generated projective modules over C(X). The same result holds if one replaces "real-valued" by "complex-valued" and "real vector bundle" by "
https://en.wikipedia.org/wiki/Universal%20enveloping%20algebra	In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra. Universal enveloping algebras are used in the representation theory of Lie groups and Lie algebras. For example, Verma modules can be constructed as quotients of the universal enveloping algebra. In addition, the enveloping algebra gives a precise definition for the Casimir operators. Because Casimir operators commute with all elements of a Lie algebra, they can be used to classify representations. The precise definition also allows the importation of Casimir operators into other areas of mathematics, specifically, those that have a differential algebra. They also play a central role in some recent developments in mathematics. In particular, their dual provides a commutative example of the objects studied in non-commutative geometry, the quantum groups. This dual can be shown, by the Gelfand–Naimark theorem, to contain the C* algebra of the corresponding Lie group. This relationship generalizes to the idea of Tannaka–Krein duality between compact topological groups and their representations. From an analytic viewpoint, the universal enveloping algebra of the Lie algebra of a Lie group may be identified with the algebra of left-invariant differential operators on the group. Informal construction The idea of the universal enveloping algebra is to embed a Lie algebra into an associative algebra with identity in such a way that the abstract bracket operation in corresponds to the commutator in and the algebra is generated by the elements of . There may be many ways to make such an embedding, but there is a unique "largest" such , called the universal enveloping algebra of . Generators and relations Let be a Lie algebra, assumed finite-dimensional for simplicity, with basis . Let be the structure constants for this basis, so that Then the universal enveloping algebra is the associative algebra (with identity) generated by elements subject to the relations and no other relations. Below we will make this "generators and relations" construction more precise by constructing the universal enveloping algebra as a quotient of the tensor algebra over . Consider, for example, the Lie algebra sl(2,C), spanned by the matrices which satisfy the commutation relations , , and . The universal enveloping algebra of sl(2,C) is then the algebra generated by three elements subject to the relations and no other relations. We emphasize that the universal enveloping algebra is not the same as (or contained in) the algebra of matrices. For example, the matrix satisfies , as is easily verified. But in the universal enveloping algebra, the element does not satisfy —because we do not impose this relation in the construction of the enveloping algebra. Indeed, it follows from the Poincaré–Birkhoff–Witt theorem (discussed below) that the elements are all linearly indepen
https://en.wikipedia.org/wiki/Tensor%20algebra	In mathematics, the tensor algebra of a vector space V, denoted T(V) or T(V), is the algebra of tensors on V (of any rank) with multiplication being the tensor product. It is the free algebra on V, in the sense of being left adjoint to the forgetful functor from algebras to vector spaces: it is the "most general" algebra containing V, in the sense of the corresponding universal property (see below). The tensor algebra is important because many other algebras arise as quotient algebras of T(V). These include the exterior algebra, the symmetric algebra, Clifford algebras, the Weyl algebra and universal enveloping algebras. The tensor algebra also has two coalgebra structures; one simple one, which does not make it a bialgebra, but does lead to the concept of a cofree coalgebra, and a more complicated one, which yields a bialgebra, and can be extended by giving an antipode to create a Hopf algebra structure. Note: In this article, all algebras are assumed to be unital and associative. The unit is explicitly required to define the coproduct. Construction Let V be a vector space over a field K. For any nonnegative integer k, we define the kth tensor power of V to be the tensor product of V with itself k times: That is, TkV consists of all tensors on V of order k. By convention T0V is the ground field K (as a one-dimensional vector space over itself). We then construct T(V) as the direct sum of TkV for k = 0,1,2,… The multiplication in T(V) is determined by the canonical isomorphism given by the tensor product, which is then extended by linearity to all of T(V). This multiplication rule implies that the tensor algebra T(V) is naturally a graded algebra with TkV serving as the grade-k subspace. This grading can be extended to a Z-grading by appending subspaces for negative integers k. The construction generalizes in a straightforward manner to the tensor algebra of any module M over a commutative ring. If R is a non-commutative ring, one can still perform the construction for any R-R bimodule M. (It does not work for ordinary R-modules because the iterated tensor products cannot be formed.) Adjunction and universal property The tensor algebra is also called the free algebra on the vector space , and is functorial; this means that the map extends to linear maps for forming a functor from the category of -vector spaces to the category of associative algebras. Similarly with other free constructions, the functor is left adjoint to the forgetful functor that sends each associative -algebra to its underlying vector space. Explicitly, the tensor algebra satisfies the following universal property, which formally expresses the statement that it is the most general algebra containing V: Any linear map from to an associative algebra over can be uniquely extended to an algebra homomorphism from to as indicated by the following commutative diagram: Here is the canonical inclusion of into . As for other universal properties, the tensor alg
https://en.wikipedia.org/wiki/Invariant%20theory	Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit description of polynomial functions that do not change, or are invariant, under the transformations from a given linear group. For example, if we consider the action of the special linear group SLn on the space of n by n matrices by left multiplication, then the determinant is an invariant of this action because the determinant of A X equals the determinant of X, when A is in SLn. Introduction Let be a group, and a finite-dimensional vector space over a field (which in classical invariant theory was usually assumed to be the complex numbers). A representation of in is a group homomorphism , which induces a group action of on . If is the space of polynomial functions on , then the group action of on produces an action on by the following formula: With this action it is natural to consider the subspace of all polynomial functions which are invariant under this group action, in other words the set of polynomials such that for all . This space of invariant polynomials is denoted . First problem of invariant theory: Is a finitely generated algebra over ? For example, if and the space of square matrices, and the action of on is given by left multiplication, then is isomorphic to a polynomial algebra in one variable, generated by the determinant. In other words, in this case, every invariant polynomial is a linear combination of powers of the determinant polynomial. So in this case, is finitely generated over . If the answer is yes, then the next question is to find a minimal basis, and ask whether the module of polynomial relations between the basis elements (known as the syzygies) is finitely generated over . Invariant theory of finite groups has intimate connections with Galois theory. One of the first major results was the main theorem on the symmetric functions that described the invariants of the symmetric group acting on the polynomial ring ] by permutations of the variables. More generally, the Chevalley–Shephard–Todd theorem characterizes finite groups whose algebra of invariants is a polynomial ring. Modern research in invariant theory of finite groups emphasizes "effective" results, such as explicit bounds on the degrees of the generators. The case of positive characteristic, ideologically close to modular representation theory, is an area of active study, with links to algebraic topology. Invariant theory of infinite groups is inextricably linked with the development of linear algebra, especially, the theories of quadratic forms and determinants. Another subject with strong mutual influence was projective geometry, where invariant theory was expected to play a major role in organizing the material. One of the highlights of this relationship is the symbolic method. Representatio
https://en.wikipedia.org/wiki/Canonical%20correlation	In statistics, canonical-correlation analysis (CCA), also called canonical variates analysis, is a way of inferring information from cross-covariance matrices. If we have two vectors X = (X1, ..., Xn) and Y = (Y1, ..., Ym) of random variables, and there are correlations among the variables, then canonical-correlation analysis will find linear combinations of X and Y which have maximum correlation with each other. T. R. Knapp notes that "virtually all of the commonly encountered parametric tests of significance can be treated as special cases of canonical-correlation analysis, which is the general procedure for investigating the relationships between two sets of variables." The method was first introduced by Harold Hotelling in 1936, although in the context of angles between flats the mathematical concept was published by Jordan in 1875. CCA is now a cornerstone of multivariate statistics and multi-view learning; and a great number of interpretations and extensions have been proposed (e.g. probabilistic CCA, sparse CCA, multi-view CCA, Deep CCA - all of which are implemented in the CCA-Zoo python package). Unfortunately, perhaps because of its popularity, the literature can be inconsistent with notation, we attempt to highlight such inconsistencies in this article to help the reader make best use of the existing literature and techniques available. Like its sister method PCA, CCA can be viewed in population form (corresponding to random vectors and their covariance matrices) or in sample form (corresponding to datasets and their sample covariance matrices). These two forms are almost exact analogues of each other, which is why their distinction is often overlooked, but they can behave very differently in high dimensional settings. We next give explicit mathematical definitions for the population problem and highlight the different objects in the so-called canonical decomposition - understanding the differences between this objects is crucial for interpretation of the technique. Population CCA definition via correlations Given two column vectors and of random variables with finite second moments, one may define the cross-covariance to be the matrix whose entry is the covariance . In practice, we would estimate the covariance matrix based on sampled data from and (i.e. from a pair of data matrices). Canonical-correlation analysis seeks a sequence of vectors () and () such that the random variables and maximize the correlation . The (scalar) random variables and are the first pair of canonical variables. Then one seeks vectors maximizing the same correlation subject to the constraint that they are to be uncorrelated with the first pair of canonical variables; this gives the second pair of canonical variables. This procedure may be continued up to times. The sets of vectors are called canonical directions or weight vectors or simply weights. The 'dual' sets of vectors are called canonical loading vectors or simply loadings; t
https://en.wikipedia.org/wiki/Benjamin%20Peirce	Benjamin Peirce (; April 4, 1809 – October 6, 1880) was an American mathematician who taught at Harvard University for approximately 50 years. He made contributions to celestial mechanics, statistics, number theory, algebra, and the philosophy of mathematics. Early life He was born in Salem, Massachusetts, the son of first cousins Benjamin Peirce (1778–1831), later librarian of Harvard, and Lydia Ropes Nichols Peirce (1781–1868). After graduating from Harvard University in 1829, he taught mathematics for two years at the Round Hill School in Northampton, and in 1831 was appointed professor of mathematics at Harvard. He added astronomy to his portfolio in 1842, and remained as Harvard professor until his death. In addition, he was instrumental in the development of Harvard's science curriculum, served as the college librarian, and was director of the U.S. Coast Survey from 1867 to 1874. In 1842, he was elected as a member of the American Philosophical Society. He was elected a Foreign Member of the Royal Society of London in 1852. Research Benjamin Peirce is often regarded as the earliest American scientist whose research was recognized as world class. He was an apologist for slavery, opining that it should be condoned if it was used to allow an elite to pursue scientific enquiry. Mathematics In number theory, he proved there is no odd perfect number with fewer than four prime factors. In algebra, he was notable for the study of associative algebras. He first introduced the terms idempotent and nilpotent in 1870 to describe elements of these algebras, and he also introduced the Peirce decomposition. In the philosophy of mathematics, he became known for the statement that "Mathematics is the science that draws necessary conclusions". Peirce's definition of mathematics was credited by his son, Charles Sanders Peirce, as helping to initiate the consequence-oriented philosophy of pragmatism. Like George Boole, Peirce believed that mathematics could be used to study logic. These ideas were further developed by his son Charles , who noted that logic also includes the study of faulty reasoning. In contrast, the later logicist program of Gottlob Frege and Bertrand Russell attempted to base mathematics on logic. Statistics Peirce proposed what came to be known as Peirce's Criterion for the statistical treatment of outliers, that is, of apparently extreme observations. His ideas were further developed by his son Charles. Peirce was an expert witness in the Howland will forgery trial, where he was assisted by his son Charles. Their analysis of the questioned signature showed that it resembled another particular handwriting example so closely that the chance of such a match occurring at random, i.e. by pure coincidence, was extremely small. Private life He was devoutly religious, though he seldom published his theological thoughts. Peirce credited God as shaping nature in ways that account for the efficacy of pure mathematics in describing empir
https://en.wikipedia.org/wiki/Gelfand%20representation	In mathematics, the Gelfand representation in functional analysis (named after I. M. Gelfand) is either of two things: a way of representing commutative Banach algebras as algebras of continuous functions; the fact that for commutative C-algebras, this representation is an isometric isomorphism. In the former case, one may regard the Gelfand representation as a far-reaching generalization of the Fourier transform of an integrable function. In the latter case, the Gelfand–Naimark representation theorem is one avenue in the development of spectral theory for normal operators, and generalizes the notion of diagonalizing a normal matrix. Historical remarks One of Gelfand's original applications (and one which historically motivated much of the study of Banach algebras) was to give a much shorter and more conceptual proof of a celebrated lemma of Norbert Wiener (see the citation below), characterizing the elements of the group algebras L1(R) and whose translates span dense subspaces in the respective algebras. The model algebra For any locally compact Hausdorff topological space X, the space C0(X) of continuous complex-valued functions on X which vanish at infinity is in a natural way a commutative C-algebra: The structure of algebra over the complex numbers is obtained by considering the pointwise operations of addition and multiplication. The involution is pointwise complex conjugation. The norm is the uniform norm on functions. The importance of X being locally compact and Hausdorff is that this turns X into a completely regular space. In such a space every closed subset of X is the common zero set of a family of continuous complex-valued functions on X, allowing one to recover the topology of X from C0(X). Note that C0(X) is unital if and only if X is compact, in which case C0(X) is equal to C(X), the algebra of all continuous complex-valued functions on X. Gelfand representation of a commutative Banach algebra Let be a commutative Banach algebra, defined over the field of complex numbers. A non-zero algebra homomorphism (a multiplicative linear functional) is called a character of ; the set of all characters of is denoted by . It can be shown that every character on is automatically continuous, and hence is a subset of the space of continuous linear functionals on ; moreover, when equipped with the relative weak-* topology, turns out to be locally compact and Hausdorff. (This follows from the Banach–Alaoglu theorem.) The space is compact (in the topology just defined) if and only if the algebra has an identity element. Given , one defines the function by . The definition of and the topology on it ensure that is continuous and vanishes at infinity, and that the map defines a norm-decreasing, unit-preserving algebra homomorphism from to . This homomorphism is the Gelfand representation of , and is the Gelfand transform of the element . In general, the representation is neither injective nor surjective. In the
https://en.wikipedia.org/wiki/Functional%20equation	In mathematics, a functional equation is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning is often used, where a functional equation is an equation that relates several values of the same function. For example, the logarithm functions are essentially characterized by the logarithmic functional equation If the domain of the unknown function is supposed to be the natural numbers, the function is generally viewed as a sequence, and, in this case, a functional equation (in the narrower meaning) is called a recurrence relation. Thus the term functional equation is used mainly for real functions and complex functions. Moreover a smoothness condition is often assumed for the solutions, since without such a condition, most functional equations have very irregular solutions. For example, the gamma function is a function that satisfies the functional equation and the initial value There are many functions that satisfy these conditions, but the gamma function is the unique one that is meromorphic in the whole complex plane, and logarithmically convex for real and positive (Bohr–Mollerup theorem). Examples Recurrence relations can be seen as functional equations in functions over the integers or natural numbers, in which the differences between terms' indexes can be seen as an application of the shift operator. For example, the recurrence relation defining the Fibonacci numbers, , where and , which characterizes the periodic functions , which characterizes the even functions, and likewise , which characterizes the odd functions , which characterizes the functional square roots of the function g (Cauchy's functional equation), satisfied by linear maps. The equation may, contingent on the axiom of choice, also have other pathological nonlinear solutions, whose existence can be proven with a Hamel basis for the real numbers satisfied by all exponential functions. Like Cauchy's additive functional equation, this too may have pathological, discontinuous solutions , satisfied by all logarithmic functions and, over coprime integer arguments, additive functions , satisfied by all power functions and, over coprime integer arguments, multiplicative functions (quadratic equation or parallelogram law) (Jensen's functional equation) (d'Alembert's functional equation) (Abel equation) (Schröder's equation). (Böttcher's equation). (Julia's equation). (Levi-Civita), (sine addition formula and hyperbolic sine addition formula), (cosine addition formula), (hyperbolic cosine addition formula). The commutative and associative laws are functional equations. In its familiar form, the associative law is expressed by writing the binary operation in infix notation, but if we write f(a, b) instead of then the associative law looks more like a conventional functional equation, The functional equation

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