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c_e1gpuc3suwlq | In mathematics, the secondary polynomials { q n ( x ) } {\displaystyle \{q_{n}(x)\}} associated with a sequence { p n ( x ) } {\displaystyle \{p_{n}(x)\}} of polynomials orthogonal with respect to a density ρ ( x ) {\displaystyle \rho (x)} are defined by q n ( x ) = ∫ R p n ( t ) − p n ( x ) t − x ρ ( t ) d t . {\displ... | Secondary polynomials |
c_tvubbbruo2y5 | {\displaystyle p_{0}(x)=x^{3}.} Then, q 0 ( x ) = ∫ R t 3 − x 3 t − x ρ ( t ) d t = ∫ R ( t − x ) ( t 2 + t x + x 2 ) t − x ρ ( t ) d t = ∫ R ( t 2 + t x + x 2 ) ρ ( t ) d t = ∫ R t 2 ρ ( t ) d t + x ∫ R t ρ ( t ) d t + x 2 ∫ R ρ ( t ) d t {\displaystyle {\begin{aligned}q_{0}(x)&{}=\int _{\mathbb {R} }\! {\frac {t^{3}-... | Secondary polynomials |
c_qt4u7o3jhrpi | In mathematics, the semi-implicit Euler method, also called symplectic Euler, semi-explicit Euler, Euler–Cromer, and Newton–Størmer–Verlet (NSV), is a modification of the Euler method for solving Hamilton's equations, a system of ordinary differential equations that arises in classical mechanics. It is a symplectic int... | Semi-implicit Euler method |
c_8pixkk881p7n | In mathematics, the set of positive real numbers, R > 0 = { x ∈ R ∣ x > 0 } , {\displaystyle \mathbb {R} _{>0}=\left\{x\in \mathbb {R} \mid x>0\right\},} is the subset of those real numbers that are greater than zero. The non-negative real numbers, R ≥ 0 = { x ∈ R ∣ x ≥ 0 } , {\displaystyle \mathbb {R} _{\geq 0}=\left\... | Logarithmic measure |
c_obfm2bsjw60g | This ray is used as reference in the polar form of a complex number. The real positive axis corresponds to complex numbers z = | z | e i φ , {\displaystyle z=|z|\mathrm {e} ^{\mathrm {i} \varphi },} with argument φ = 0. {\displaystyle \varphi =0.} | Logarithmic measure |
c_wdndcxymip95 | In mathematics, the seven-dimensional cross product is a bilinear operation on vectors in seven-dimensional Euclidean space. It assigns to any two vectors a, b in R 7 {\displaystyle \mathbb {R} ^{7}} a vector a × b also in R 7 {\displaystyle \mathbb {R} ^{7}} . Like the cross product in three dimensions, the seven-dime... | Seven-dimensional cross product |
c_i1gex3690ju4 | The seven-dimensional cross product has the same relationship to the octonions as the three-dimensional product does to the quaternions. The seven-dimensional cross product is one way of generalizing the cross product to other than three dimensions, and it is the only other bilinear product of two vectors that is vecto... | Seven-dimensional cross product |
c_v2qaoubd1qal | In mathematics, the sieve of Eratosthenes is an ancient algorithm for finding all prime numbers up to any given limit. It does so by iteratively marking as composite (i.e., not prime) the multiples of each prime, starting with the first prime number, 2. The multiples of a given prime are generated as a sequence of numb... | Sieve of Eratosthenes |
c_nv6fkcxmf0g2 | Once all the multiples of each discovered prime have been marked as composites, the remaining unmarked numbers are primes. The earliest known reference to the sieve (Ancient Greek: κόσκινον Ἐρατοσθένους, kóskinon Eratosthénous) is in Nicomachus of Gerasa's Introduction to Arithmetic, an early 2nd cent. CE book which at... | Sieve of Eratosthenes |
c_qcebgjdccs3z | In mathematics, the sieve of Pritchard is an algorithm for finding all prime numbers up to a specified bound. Like the ancient sieve of Eratosthenes, it has a simple conceptual basis in number theory. It is especially suited to quick hand computation for small bounds. | Sieve of Pritchard |
c_motn3xcl7fty | Whereas the sieve of Eratosthenes marks off each non-prime for each of its prime factors, the sieve of Pritchard avoids considering almost all non-prime numbers by building progressively larger wheels, which represent the pattern of numbers not divisible by any of the primes processed thus far. It thereby achieves a be... | Sieve of Pritchard |
c_w977vhgopmn1 | In mathematics, the sign function or signum function (from signum, Latin for "sign") is a function that returns the sign of a real number. In mathematical notation the sign function is often represented as sgn ( x ) {\displaystyle \operatorname {sgn}(x)} . | Sign function |
c_vc6cnkhubfzr | In mathematics, the sign of a real number is its property of being either positive, negative, or zero. In some contexts, it makes sense to consider a signed zero (such as floating-point representations of real numbers within computers). Depending on local conventions, zero may be considered as being neither positive no... | Positive number |
c_d9jo97avlpo4 | Whenever not specifically mentioned, this article adheres to the first convention (zero having undefined sign). In mathematics and physics, the phrase "change of sign" is associated with the generation of the additive inverse (negation, or multiplication by −1) of any object that allows for this construction, and is no... | Positive number |
c_lafwcz9o9wqp | In mathematics, the signal magnitude area (abbreviated SMA or sma) is a statistical measure of the magnitude of a varying quantity. | Signal magnitude area |
c_9k2j0flf1loq | In mathematics, the signature (v, p, r) of a metric tensor g (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional vector space) is the number (counted with multiplicity) of positive, negative and zero eigenvalues of the real symmetric matrix gab of the metric tens... | Metric signature |
c_2negseqpjxuk | By Sylvester's law of inertia these numbers do not depend on the choice of basis and thus can be used to classify the metric. The signature is often denoted by a pair of integers (v, p) implying r= 0, or as an explicit list of signs of eigenvalues such as (+, −, −, −) or (−, +, +, +) for the signatures (1, 3, 0) and (3... | Metric signature |
c_rkoqgdffpzp8 | A Lorentzian metric is a metric with signature (p, 1), or (1, p). There is another notion of signature of a nondegenerate metric tensor given by a single number s defined as (v − p), where v and p are as above, which is equivalent to the above definition when the dimension n = v + p is given or implicit. For example, s... | Metric signature |
c_bvkrfscx63qu | In mathematics, the signature defect of a singularity measures the correction that a singularity contributes to the signature theorem. Hirzebruch (1973) introduced the signature defect for the cusp singularities of Hilbert modular surfaces. Michael Francis Atiyah, H. Donnelly, and I. M. Singer (1983) defined the signat... | Signature defect |
c_rp6qeuyiki5l | In mathematics, the signed area or oriented area of a region of an affine plane is its area with orientation specified by the ("plus" or "minus") sign. More generally, the signed area of an arbitrary surface region is its surface area with specified orientation. When the boundary of the region is a simple curve, the si... | Negative area |
c_rakh1r1hjpp2 | Negative area arises in the study of natural logarithm as signed area under the curve y = 1/x for x in the positive real numbers: Definition: ln x = ∫ 1 x d t t , x > 0. {\displaystyle \ln x=\int _{1}^{x}{\frac {dt}{t}},\quad x>0.} "For 0 < x < 1, ln x = ∫ 1 x d t t = − ∫ x 1 d t t < 0 {\displaystyle \ln x=\int _{1... | Negative area |
c_h6paiynqph3c | Since the wedge product has the anticommutative property, d y ∧ d x = − d x ∧ d y . {\displaystyle dy\wedge dx=-dx\wedge dy.} The density is associated with a planar orientation, something existing locally in a manifold but not necessarily globally. | Negative area |
c_yjpo3exxx4g3 | In the case of the natural logarithm, obtained by integrating area under the hyperbola xy=1, the density dx ∧ dy is positive for x>1, but since the logarithm is anchored to 1, the orientation of the x-axis is reversed in the unit interval. For this integration the (− dx) orientation yields the opposite density to the o... | Negative area |
c_nofj9otxd2ri | 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 ... | Parallelogram equality |
c_domdnd2sg7sp | In mathematics, the simplest real analytic Eisenstein series is a special function of two variables. It is used in the representation theory of SL(2,R) and in analytic number theory. It is closely related to the Epstein zeta function. There are many generalizations associated to more complicated groups. | Real analytic Eisenstein series |
c_9zsdiqvdyqfl | In mathematics, the simplex category (or simplicial category or nonempty finite ordinal category) is the category of non-empty finite ordinals and order-preserving maps. It is used to define simplicial and cosimplicial objects. | Simplex category |
c_5vr32bb7higi | In mathematics, the simplicial approximation theorem is a foundational result for algebraic topology, guaranteeing that continuous mappings can be (by a slight deformation) approximated by ones that are piecewise of the simplest kind. It applies to mappings between spaces that are built up from simplices—that is, finit... | Simplicial approximation theorem |
c_xge00iux0k17 | This theorem was first proved by L.E.J. Brouwer, by use of the Lebesgue covering theorem (a result based on compactness). It served to put the homology theory of the time—the first decade of the twentieth century—on a rigorous basis, since it showed that the topological effect (on homology groups) of continuous mapping... | Simplicial approximation theorem |
c_dlz4hfyao3g5 | This must be seen against the background of a realisation at the time that continuity was in general compatible with the pathological, in some other areas. This initiated, one could say, the era of combinatorial topology. There is a further simplicial approximation theorem for homotopies, stating that a homotopy betwee... | Simplicial approximation theorem |
c_rtpcs9p4f77n | In mathematics, the simultaneous uniformization theorem, proved by Bers (1960), states that it is possible to simultaneously uniformize two different Riemann surfaces of the same genus using a quasi-Fuchsian group of the first kind. The quasi-Fuchsian group is essentially uniquely determined by the two Riemann surfaces... | Bers's theorem |
c_6kadekhnyicp | In mathematics, the sinhc function appears frequently in papers about optical scattering, Heisenberg spacetime and hyperbolic geometry. For z ≠ 0 {\displaystyle z\neq 0} , it is defined as The sinhc function is the hyperbolic analogue of the sinc function, defined by sin x / x {\displaystyle \sin x/x} . It is a solut... | Sinhc function |
c_z5r7m9sndpm1 | In mathematics, the slice genus of a smooth knot K in S3 (sometimes called its Murasugi genus or 4-ball genus) is the least integer g such that K is the boundary of a connected, orientable 2-manifold S of genus g properly embedded in the 4-ball D4 bounded by S3. More precisely, if S is required to be smoothly embedded,... | Slice genus |
c_bike1w0vztdv | In mathematics, the slope or gradient of a line is a number that describes both the direction and the steepness of the line. Slope is often denoted by the letter m; there is no clear answer to the question why the letter m is used for slope, but its earliest use in English appears in O'Brien (1844) who wrote the equati... | Slope of a line |
c_bya1yqtinq8t | The line may be practical – as set by a road surveyor, or in a diagram that models a road or a roof either as a description or as a plan. The steepness, incline, or grade of a line is measured by the absolute value of the slope. A slope with a greater absolute value indicates a steeper line. | Slope of a line |
c_cipv2ynbskyv | The direction of a line is either increasing, decreasing, horizontal or vertical. A line is increasing if it goes up from left to right. The slope is positive, i.e. m > 0 {\displaystyle m>0} . | Slope of a line |
c_4qn3kxezwwlj | A line is decreasing if it goes down from left to right. The slope is negative, i.e. m < 0 {\displaystyle m<0} . If a line is horizontal the slope is zero. | Slope of a line |
c_eutvvf2q3hlq | This is a constant function. If a line is vertical the slope is undefined (see below).The rise of a road between two points is the difference between the altitude of the road at those two points, say y1 and y2, or in other words, the rise is (y2 − y1) = Δy. For relatively short distances, where the Earth's curvature ma... | Slope of a line |
c_hnxqnlfjofts | Here the slope of the road between the two points is simply described as the ratio of the altitude change to the horizontal distance between any two points on the line. In mathematical language, the slope m of the line is m = y 2 − y 1 x 2 − x 1 . {\displaystyle m={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}.} | Slope of a line |
c_io2uxbe2b2t5 | The concept of slope applies directly to grades or gradients in geography and civil engineering. Through trigonometry, the slope m of a line is related to its angle of inclination θ by the tangent function m = tan ( θ ) {\displaystyle m=\tan(\theta )} Thus, a 45° rising line has a slope of +1 and a 45° falling line h... | Slope of a line |
c_3c9havv96bf0 | When the curve is given by a series of points in a diagram or in a list of the coordinates of points, the slope may be calculated not at a point but between any two given points. When the curve is given as a continuous function, perhaps as an algebraic expression, then the differential calculus provides rules giving a ... | Slope of a line |
c_pn798zyha4ex | In mathematics, the slow manifold of an equilibrium point of a dynamical system occurs as the most common example of a center manifold. One of the main methods of simplifying dynamical systems, is to reduce the dimension of the system to that of the slow manifold—center manifold theory rigorously justifies the modellin... | Slow manifold |
c_bmg8dpn06b5e | The slow manifold in a particular problem would be a sub-manifold of either the stable, unstable, or center manifold, exclusively, that has the same dimension of, and is tangent to, the eigenspace with an associated eigenvalue (or eigenvalue pair) that has the smallest real part in magnitude. This generalizes the defin... | Slow manifold |
c_hm2qwa4o5rak | In mathematics, the small Veblen ordinal is a certain large countable ordinal, named after Oswald Veblen. It is occasionally called the Ackermann ordinal, though the Ackermann ordinal described by Ackermann (1951) is somewhat smaller than the small Veblen ordinal. There is no standard notation for ordinals beyond the F... | Small Veblen ordinal |
c_uwci3fb7ga9s | The small Veblen ordinal θ Ω ω ( 0 ) {\displaystyle \theta _{\Omega ^{\omega }}(0)} or ψ ( Ω Ω ω ) {\displaystyle \psi (\Omega ^{\Omega ^{\omega }})} is the limit of ordinals that can be described using a version of Veblen functions with finitely many arguments. It is the ordinal that measures the strength of Kruskal's... | Small Veblen ordinal |
c_0i557k5cfgka | In mathematics, the small boundary property is a property of certain topological dynamical systems. It is dynamical analog of the inductive definition of Lebesgue covering dimension zero. | Small boundary property |
c_ibp51rw0q3fz | In mathematics, the sophomore's dream is the pair of identities (especially the first) discovered in 1697 by Johann Bernoulli. The numerical values of these constants are approximately 1.291285997... and 0.7834305107..., respectively. The name "sophomore's dream" is in contrast to the name "freshman's dream" which is g... | Bernoulli's identity |
c_4niqw81xs7v9 | In mathematics, the soul theorem is a theorem of Riemannian geometry that largely reduces the study of complete manifolds of non-negative sectional curvature to that of the compact case. Jeff Cheeger and Detlef Gromoll proved the theorem in 1972 by generalizing a 1969 result of Gromoll and Wolfgang Meyer. The related s... | Soul conjecture |
c_2kapr87a1ge0 | In mathematics, the special linear Lie algebra of order n (denoted s l n ( F ) {\displaystyle {\mathfrak {sl}}_{n}(F)} or s l ( n , F ) {\displaystyle {\mathfrak {sl}}(n,F)} ) is the Lie algebra of n × n {\displaystyle n\times n} matrices with trace zero and with the Lie bracket := X Y − Y X {\displaystyle :=XY-YX} . ... | Special linear Lie algebra |
c_z0v37725g1m3 | In mathematics, the special linear group SL(2, R) or SL2(R) is the group of 2 × 2 real matrices with determinant one: SL ( 2 , R ) = { ( a b c d ): a , b , c , d ∈ R and a d − b c = 1 } . {\displaystyle {\mbox{SL}}(2,\mathbf {R} )=\left\{{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\colon a,b,c,d\in \mathbf {R} {\mbox{ and }}... | SL2(R) |
c_5x6u9i7vc804 | SL(2, R) acts on the complex upper half-plane by fractional linear transformations. The group action factors through the quotient PSL(2, R) (the 2 × 2 projective special linear group over R). More specifically, PSL(2, R) = SL(2, R) / {±I},where I denotes the 2 × 2 identity matrix. | SL2(R) |
c_gj3orbt83pfv | It contains the modular group PSL(2, Z). Also closely related is the 2-fold covering group, Mp(2, R), a metaplectic group (thinking of SL(2, R) as a symplectic group). Another related group is SL±(2, R), the group of real 2 × 2 matrices with determinant ±1; this is more commonly used in the context of the modular group... | SL2(R) |
c_9ordvmisxljg | In mathematics, the special linear group SL(n, F) of degree n over a field F is the set of n × n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the general linear group given by the kernel of the determinant det: GL ( n , F... | Special linear group |
c_hetd6z49jf3y | where F× is the multiplicative group of F (that is, F excluding 0). These elements are "special" in that they form an algebraic subvariety of the general linear group – they satisfy a polynomial equation (since the determinant is polynomial in the entries). When F is a finite field of order q, the notation SL(n, q) is ... | Special linear group |
c_ef0dx6n0tnat | In mathematics, the special orthogonal group in three dimensions, otherwise known as the rotation group SO(3), is a naturally occurring example of a manifold. The various charts on SO(3) set up rival coordinate systems: in this case there cannot be said to be a preferred set of parameters describing a rotation. There a... | Hypersphere of rotations |
c_vf7dhra9ztcl | In mathematics, the special unitary group of degree n, denoted SU(n), is the Lie group of n × n unitary matrices with determinant 1. The matrices of the more general unitary group may have complex determinants with absolute value 1, rather than real 1 in the special case. The group operation is matrix multiplication. T... | Special unitary group |
c_8pfpmy1bm5jn | As a compact classical group, U(n) is the group that preserves the standard inner product on C n {\displaystyle \mathbb {C} ^{n}} . It is itself a subgroup of the general linear group, SU ( n ) ⊂ U ( n ) ⊂ GL ( n , C ) {\displaystyle \operatorname {SU} (n)\subset \operatorname {U} (n)\subset \operatorname {GL} (n... | Special unitary group |
c_f6r7gz4wy07n | The groups SU(2n) are important in quantum computing, as they represent the possible quantum logic gate operations in a quantum circuit with n {\displaystyle n} qubits and thus 2 n {\displaystyle 2^{n}} basis states. (Alternatively, the more general unitary group U ( 2 n ) {\displaystyle U(2^{n})} can be used, since mu... | Special unitary group |
c_ipkk9rgtf3g0 | The group SU(2) is isomorphic to the group of quaternions of norm 1, and is thus diffeomorphic to the 3-sphere. Since unit quaternions can be used to represent rotations in 3-dimensional space (up to sign), there is a surjective homomorphism from SU(2) to the rotation group SO(3) whose kernel is {+I, −I}. SU(2) is also... | Special unitary group |
c_8lkclq7iaduj | In mathematics, the spectral abscissa of a matrix or a bounded linear operator is the greatest real part of the matrix's spectrum (its set of eigenvalues). It is sometimes denoted α ( A ) {\displaystyle \alpha (A)} . As a transformation α: M n → R {\displaystyle \alpha :\mathrm {M} ^{n}\rightarrow \mathbb {R} } , the s... | Spectral abscissa |
c_y4hqs6zwofuh | In mathematics, the spectral gap is the difference between the moduli of the two largest eigenvalues of a matrix or operator; alternately, it is sometimes taken as the smallest non-zero eigenvalue. Various theorems relate this difference to other properties of the system. | Spectral gap |
c_79hvkqequ6m1 | In mathematics, the spectral radius of a square matrix is the maximum of the absolute values of its eigenvalues. More generally, the spectral radius of a bounded linear operator is the supremum of the absolute values of the elements of its spectrum. The spectral radius is often denoted by ρ(·). | Spectral radius |
c_23q5izp2di04 | In mathematics, the spectral theory of ordinary differential equations is the part of spectral theory concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation. In his dissertation, Hermann Weyl generalized the classical Sturm–Liouville theory o... | Spectral measure |
c_60mkdfjri2r0 | In this case the eigenfunction expansion involves an integral over the continuous part with respect to a spectral measure, given by the Titchmarsh–Kodaira formula. The theory was put in its final simplified form for singular differential equations of even degree by Kodaira and others, using von Neumann's spectral theor... | Spectral measure |
c_28zgwdprkfy7 | In mathematics, the spectrum of a C*-algebra or dual of a C*-algebra A, denoted Â, is the set of unitary equivalence classes of irreducible *-representations of A. A *-representation π of A on a Hilbert space H is irreducible if, and only if, there is no closed subspace K different from H and {0} which is invariant und... | Hull-kernel topology |
c_vnikx4uwkbsg | In mathematics, the spectrum of a matrix is the multiset of the eigenvalues of the matrix. In functional analysis, the concept of the spectrum of a bounded operator is a generalization of the eigenvalue concept for matrices. In algebraic topology, a spectrum is an object representing a generalized cohomology theory. | Energy spectrum |
c_nkkm6q0ae1uf | In mathematics, the spectrum of a matrix is the set of its eigenvalues. More generally, if T: V → V {\displaystyle T\colon V\to V} is a linear operator on any finite-dimensional vector space, its spectrum is the set of scalars λ {\displaystyle \lambda } such that T − λ I {\displaystyle T-\lambda I} is not invertible. T... | Spectrum of a matrix |
c_njapv0a99ueb | Similarly, the trace of the matrix equals the sum of its eigenvalues. From this point of view, we can define the pseudo-determinant for a singular matrix to be the product of its nonzero eigenvalues (the density of multivariate normal distribution will need this quantity). In many applications, such as PageRank, one is... | Spectrum of a matrix |
c_cgetfx68jxm4 | In mathematics, the spherical mean of a function around a point is the average of all values of that function on a sphere of given radius centered at that point. | Spherical mean |
c_3j9b7ndgsppm | In mathematics, the spheroidal wave equation is given by ( 1 − t 2 ) d 2 y d t 2 − 2 ( b + 1 ) t d y d t + ( c − 4 q t 2 ) y = 0 {\displaystyle (1-t^{2}){\frac {d^{2}y}{dt^{2}}}-2(b+1)t\,{\frac {dy}{dt}}+(c-4qt^{2})\,y=0} It is a generalization of the Mathieu differential equation. If y ( t ) {\displaystyle y(t)} is a ... | Spheroidal wave equation |
c_pu0igygy75vv | In mathematics, the spin representations are particular projective representations of the orthogonal or special orthogonal groups in arbitrary dimension and signature (i.e., including indefinite orthogonal groups). More precisely, they are two equivalent representations of the spin groups, which are double covers of th... | Spin representation |
c_t890f7b21kuu | Elements of a spin representation are called spinors. They play an important role in the physical description of fermions such as the electron. The spin representations may be constructed in several ways, but typically the construction involves (perhaps only implicitly) the choice of a maximal isotropic subspace in the... | Spin representation |
c_4lc3vcadl0oj | Over the real numbers, this usually requires using a complexification of the vector representation. For this reason, it is convenient to define the spin representations over the complex numbers first, and derive real representations by introducing real structures. The properties of the spin representations depend, in a... | Spin representation |
c_cbg9cdsunhjr | In mathematics, the spinor concept as specialised to three dimensions can be treated by means of the traditional notions of dot product and cross product. This is part of the detailed algebraic discussion of the rotation group SO(3). | Spinors in three dimensions |
c_mssvmobf89gj | In mathematics, the spinor genus is a classification of quadratic forms and lattices over the ring of integers, introduced by Martin Eichler. It refines the genus but may be coarser than proper equivalence. | Spinor genus |
c_0b6tuwad73m1 | In mathematics, the spiral optimization (SPO) algorithm is a metaheuristic inspired by spiral phenomena in nature. The first SPO algorithm was proposed for two-dimensional unconstrained optimization based on two-dimensional spiral models. This was extended to n-dimensional problems by generalizing the two-dimensional s... | Spiral optimization algorithm |
c_z7tfs076dpb5 | In mathematics, the splitting circle method is a numerical algorithm for the numerical factorization of a polynomial and, ultimately, for finding its complex roots. It was introduced by Arnold Schönhage in his 1982 paper The fundamental theorem of algebra in terms of computational complexity (Technical report, Mathemat... | Splitting circle method |
c_lfhqoqr4l9z4 | In mathematics, the splitting principle is a technique used to reduce questions about vector bundles to the case of line bundles. In the theory of vector bundles, one often wishes to simplify computations, say of Chern classes. Often computations are well understood for line bundles and for direct sums of line bundles.... | Splitting principle |
c_5r3funonr9x2 | The theorem above holds for complex vector bundles and integer coefficients or for real vector bundles with Z 2 {\displaystyle \mathbb {Z} _{2}} coefficients. In the complex case, the line bundles L i {\displaystyle L_{i}} or their first characteristic classes are called Chern roots. The fact that p ∗: H ∗ ( X ) → H ∗ ... | Splitting principle |
c_lvgpizf456ko | The point is that these equations are easier to understand for direct sums of line bundles than for arbitrary vector bundles, so equations should be understood in Y {\displaystyle Y} and then pushed down to X {\displaystyle X} . Since vector bundles on X {\displaystyle X} are used to define the K-theory group K ( X ) {... | Splitting principle |
c_f4jl3899g3on | In mathematics, the square lattice is a type of lattice in a two-dimensional Euclidean space. It is the two-dimensional version of the integer lattice, denoted as Z 2 {\displaystyle \mathbb {Z} ^{2}} . It is one of the five types of two-dimensional lattices as classified by their symmetry groups; its symmetry group in ... | Square lattice |
c_lhvq48452ybl | In mathematics, the square root of a matrix extends the notion of square root from numbers to matrices. A matrix B is said to be a square root of A if the matrix product BB is equal to A.Some authors use the name square root or the notation A1/2 only for the specific case when A is positive semidefinite, to denote the ... | Square root of a matrix |
c_75zk8xoejupl | In mathematics, the stability radius of an object (system, function, matrix, parameter) at a given nominal point is the radius of the largest ball, centered at the nominal point, all of whose elements satisfy pre-determined stability conditions. The picture of this intuitive notion is this: where p ^ {\displaystyle {\h... | Stability radius |
c_ep6w5z8lis3a | In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as R n {\displaystyle \mathbb {R} ^{n}} or C n {\displaystyle \mathbb {C} ^{n}} ) is the set of vectors, each of whose components are all zero, except one that equals 1. For example, in the case of the E... | Standard basis |
c_cpdc1i1mm03e | {\displaystyle \mathbf {e} _{x}=(1,0,0),\quad \mathbf {e} _{y}=(0,1,0),\quad \mathbf {e} _{z}=(0,0,1).} Here the vector ex points in the x direction, the vector ey points in the y direction, and the vector ez points in the z direction. | Standard basis |
c_1nmhk3i43dnh | There are several common notations for standard-basis vectors, including {ex, ey, ez}, {e1, e2, e3}, {i, j, k}, and {x, y, z}. These vectors are sometimes written with a hat to emphasize their status as unit vectors (standard unit vectors). These vectors are a basis in the sense that any other vector can be expressed u... | Standard basis |
c_pj5wqijguhx6 | For example, every vector v in three-dimensional space can be written uniquely as v x e x + v y e y + v z e z , {\displaystyle v_{x}\,\mathbf {e} _{x}+v_{y}\,\mathbf {e} _{y}+v_{z}\,\mathbf {e} _{z},} the scalars v x {\displaystyle v_{x}} , v y {\displaystyle v_{y}} , v z {\displaystyle v_{z}} being the scalar componen... | Standard basis |
c_5x82ni4iimvx | For polynomials, the standard basis thus consists of the monomials and is commonly called monomial basis. For matrices M m × n {\displaystyle {\mathcal {M}}_{m\times n}} , the standard basis consists of the m×n-matrices with exactly one non-zero entry, which is 1. For example, the standard basis for 2×2 matrices is for... | Standard basis |
c_yuatvunsivqi | In mathematics, the standard complex, also called standard resolution, bar resolution, bar complex, bar construction, is a way of constructing resolutions in homological algebra. It was first introduced for the special case of algebras over a commutative ring by Samuel Eilenberg and Saunders Mac Lane (1953) and Henri C... | Bar resolution |
c_fhubb178rebo | In mathematics, the standard conjectures about algebraic cycles are several conjectures describing the relationship of algebraic cycles and Weil cohomology theories. One of the original applications of these conjectures, envisaged by Alexander Grothendieck, was to prove that his construction of pure motives gave an abe... | Standard conjectures |
c_b9rhbev4g1ms | In quite a few cases, including that of the Weil conjectures, other methods have been found to prove such results unconditionally. The classical formulations of the standard conjectures involve a fixed Weil cohomology theory H. All of the conjectures deal with "algebraic" cohomology classes, which means a morphism on t... | Standard conjectures |
c_hp3huizl0kbq | In mathematics, the star product is a method of combining graded posets with unique minimal and maximal elements, preserving the property that the posets are Eulerian. | Star product |
c_59gezbw5e0dm | In mathematics, the stationary phase approximation is a basic principle of asymptotic analysis, applying to functions given by integration against a rapidly-varying complex exponential. This method originates from the 19th century, and is due to George Gabriel Stokes and Lord Kelvin. It is closely related to Laplace's ... | Stationary phase approximation |
c_p4dttca77qqp | In mathematics, the structure constants or structure coefficients of an algebra over a field are the coefficients of the basis expansion (into linear combination of basis vectors) of the products of basis vectors. Because the product operation in the algebra is bilinear, by linearity knowing the product of basis vector... | Structure constant |
c_j255gpddstv4 | Given the structure constants, the resulting product is obtained by bilinearity and can be uniquely extended to all vectors in the vector space, thus uniquely determining the product for the algebra. Structure constants are used whenever an explicit form for the algebra must be given. Thus, they are frequently used whe... | Structure constant |
c_u7tunj8e8f5f | In mathematics, the structure tensor, also referred to as the second-moment matrix, is a matrix derived from the gradient of a function. It describes the distribution of the gradient in a specified neighborhood around a point and makes the information invariant respect the observing coordinates. The structure tensor is... | Structure Tensor |
c_krk1a5r7baji | In mathematics, the structure theorem for Gaussian measures shows that the abstract Wiener space construction is essentially the only way to obtain a strictly positive Gaussian measure on a separable Banach space. It was proved in the 1970s by Kallianpur–Sato–Stefan and Dudley–Feldman–le Cam. There is the earlier resul... | Structure theorem for Gaussian measures |
c_8fg77u7ovb5n | In mathematics, the study of interchange of limiting operations is one of the major concerns of mathematical analysis, in that two given limiting operations, say L and M, cannot be assumed to give the same result when applied in either order. One of the historical sources for this theory is the study of trigonometric s... | Interchange of limiting operations |
c_wzyh3agluut1 | In mathematics, the study of special values of L-functions is a subfield of number theory devoted to generalising formulae such as the Leibniz formula for pi, namely by the recognition that expression on the left-hand side is also L ( 1 ) {\displaystyle L(1)} where L ( s ) {\displaystyle L(s)} is the Dirichlet L-functi... | Bloch-Kato conjecture on Tamagawa numbers |
c_63zk9xubtru7 | In mathematics, the subderivative, subgradient, and subdifferential generalize the derivative to convex functions which are not necessarily differentiable. Subderivatives arise in convex analysis, the study of convex functions, often in connection to convex optimization. Let f: I → R {\displaystyle f:I\to \mathbb {R} }... | Subderivative |
c_0elwoibz3nq4 | In mathematics, the subspace theorem says that points of small height in projective space lie in a finite number of hyperplanes. It is a result obtained by Wolfgang M. Schmidt (1972). | Subspace theorem |
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