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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 . {\displaystyle q_{n}(x)=\int _{\mathbb {R} }\! {\frac {p_{n}(t)-p_{n}(x)}{t-x}}\rho (t)\,dt.} To see that the functions q n ( x ) {\displaystyle q_{n}(x)} are indeed polynomials, consider the simple example of p 0 ( x ) = x 3 .
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https://en.wikipedia.org/wiki/Secondary_polynomials
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{\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}-x^{3}}{t-x}}\rho (t)\,dt\\&{}=\int _{\mathbb {R} }\! {\frac {(t-x)(t^{2}+tx+x^{2})}{t-x}}\rho (t)\,dt\\&{}=\int _{\mathbb {R} }\! (t^{2}+tx+x^{2})\rho (t)\,dt\\&{}=\int _{\mathbb {R} }\!t^{2}\rho (t)\,dt+x\int _{\mathbb {R} }\!t\rho (t)\,dt+x^{2}\int _{\mathbb {R} }\!\rho (t)\,dt\end{aligned}}} which is a polynomial x {\displaystyle x} provided that the three integrals in t {\displaystyle t} (the moments of the density ρ {\displaystyle \rho } ) are convergent.
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https://en.wikipedia.org/wiki/Secondary_polynomials
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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 integrator and hence it yields better results than the standard Euler method.
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https://en.wikipedia.org/wiki/Semi-implicit_Euler_method
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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\{x\in \mathbb {R} \mid x\geq 0\right\},} also include zero. Although the symbols R + {\displaystyle \mathbb {R} _{+}} and R + {\displaystyle \mathbb {R} ^{+}} are ambiguously used for either of these, the notation R + {\displaystyle \mathbb {R} _{+}} or R + {\displaystyle \mathbb {R} ^{+}} for { x ∈ R ∣ x ≥ 0 } {\displaystyle \left\{x\in \mathbb {R} \mid x\geq 0\right\}} and R + ∗ {\displaystyle \mathbb {R} _{+}^{*}} or R ∗ + {\displaystyle \mathbb {R} _{*}^{+}} for { x ∈ R ∣ x > 0 } {\displaystyle \left\{x\in \mathbb {R} \mid x>0\right\}} has also been widely employed, is aligned with the practice in algebra of denoting the exclusion of the zero element with a star, and should be understandable to most practicing mathematicians.In a complex plane, R > 0 {\displaystyle \mathbb {R} _{>0}} is identified with the positive real axis, and is usually drawn as a horizontal ray.
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https://en.wikipedia.org/wiki/Logarithmic_measure
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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.}
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https://en.wikipedia.org/wiki/Logarithmic_measure
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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-dimensional product is anticommutative and a × b is orthogonal both to a and to b. Unlike in three dimensions, it does not satisfy the Jacobi identity, and while the three-dimensional cross product is unique up to a sign, there are many seven-dimensional cross products.
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https://en.wikipedia.org/wiki/Seven-dimensional_cross_product
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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 vector-valued, orthogonal, and has the same magnitude as in the 3D case. In other dimensions there are vector-valued products of three or more vectors that satisfy these conditions, and binary products with bivector results.
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https://en.wikipedia.org/wiki/Seven-dimensional_cross_product
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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 numbers starting from that prime, with constant difference between them that is equal to that prime. This is the sieve's key distinction from using trial division to sequentially test each candidate number for divisibility by each prime.
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https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes
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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 attributes it to Eratosthenes of Cyrene, a 3rd cent. BCE Greek mathematician, though describing the sieving by odd numbers instead of by primes.One of a number of prime number sieves, it is one of the most efficient ways to find all of the smaller primes. It may be used to find primes in arithmetic progressions.
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https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes
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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.
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https://en.wikipedia.org/wiki/Sieve_of_Pritchard
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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 better asymptotic complexity, and was the first sieve with a running time sublinear in the specified bound. Its asymptotic running-time has not been improved on, and it deletes fewer composites than any other known sieve. It was created in 1979 by Paul Pritchard.Since Pritchard has created a number of other sieve algorithms for finding prime numbers, the sieve of Pritchard is sometimes singled out by being called the wheel sieve (by Pritchard himself) or the dynamic wheel sieve.
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https://en.wikipedia.org/wiki/Sieve_of_Pritchard
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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)} .
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https://en.wikipedia.org/wiki/Sign_function
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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 nor negative (having no sign or a unique third sign), or it may be considered both positive and negative (having both signs).
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https://en.wikipedia.org/wiki/Positive_number
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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 not restricted to real numbers. It applies among other objects to vectors, matrices, and complex numbers, which are not prescribed to be only either positive, negative, or zero. The word "sign" is also often used to indicate other binary aspects of mathematical objects that resemble positivity and negativity, such as odd and even (sign of a permutation), sense of orientation or rotation (cw/ccw), one sided limits, and other concepts described in § Other meanings below.
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https://en.wikipedia.org/wiki/Positive_number
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In mathematics, the signal magnitude area (abbreviated SMA or sma) is a statistical measure of the magnitude of a varying quantity.
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https://en.wikipedia.org/wiki/Signal_magnitude_area
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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 tensor with respect to a basis. In relativistic physics, the v represents the time or virtual dimension, and the p for the space and physical dimension. Alternatively, it can be defined as the dimensions of a maximal positive and null subspace.
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https://en.wikipedia.org/wiki/Metric_signature
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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, 1, 0), respectively.The signature is said to be indefinite or mixed if both v and p are nonzero, and degenerate if r is nonzero. A Riemannian metric is a metric with a positive definite signature (v, 0).
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https://en.wikipedia.org/wiki/Metric_signature
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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 = 1 − 3 = −2 for (+, −, −, −) and its mirroring s' = −s = +2 for (−, +, +, +).
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https://en.wikipedia.org/wiki/Metric_signature
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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 signature defect of the boundary of a manifold as the eta invariant, the value as s = 0 of their eta function, and used this to show that Hirzebruch's signature defect of a cusp of a Hilbert modular surface can be expressed in terms of the value at s = 0 or 1 of a Shimizu L-function.
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https://en.wikipedia.org/wiki/Signature_defect
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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 signed area also indicates the orientation of the boundary. The integral of a real function can be imagined as the signed area between the line y = 0 {\displaystyle y=0} and the curve y = f ( x ) {\displaystyle y=f(x)} over an interval .
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https://en.wikipedia.org/wiki/Negative_area
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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}^{x}{\frac {dt}{t}}=-\int _{x}^{1}{\frac {dt}{t}}<0} and so ln x is the negative of the area,,,"In differential geometry, the sign of the area of a region of a surface is associated with the orientation of the surface: "In addition to the area ... one may consider also signed areas of portions of surfaces; in this case the area corresponding to one of the two possible orientations is defined by while the area corresponding to the other orientation is −A(H)"Area of a set A in differential geometry is obtained as an integration of a density: μ ( A ) = ∫ A d x ∧ d y {\displaystyle \mu (A)=\int _{A}dx\wedge dy} where dx and dy are differential 1-forms that make the density.
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https://en.wikipedia.org/wiki/Negative_area
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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.
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https://en.wikipedia.org/wiki/Negative_area
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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 one used for x>1. With this opposite density the area, under the hyperbola and above the unit interval, is taken as negative area, and the natural logarithm consequently is negative in this domain.
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https://en.wikipedia.org/wiki/Negative_area
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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 x {\displaystyle x} is the length of the line segment joining the midpoints of the diagonals. It can be seen from the diagram that x = 0 {\displaystyle x=0} for a parallelogram, and so the general formula simplifies to the parallelogram law.
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https://en.wikipedia.org/wiki/Parallelogram_equality
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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.
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https://en.wikipedia.org/wiki/Real_analytic_Eisenstein_series
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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.
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https://en.wikipedia.org/wiki/Simplex_category
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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, finite simplicial complexes. The general continuous mapping between such spaces can be represented approximately by the type of mapping that is (affine-) linear on each simplex into another simplex, at the cost (i) of sufficient barycentric subdivision of the simplices of the domain, and (ii) replacement of the actual mapping by a homotopic one.
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https://en.wikipedia.org/wiki/Simplicial_approximation_theorem
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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 mappings could in a given case be expressed in a finitary way.
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https://en.wikipedia.org/wiki/Simplicial_approximation_theorem
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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 between continuous mappings can likewise be approximated by a combinatorial version.
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https://en.wikipedia.org/wiki/Simplicial_approximation_theorem
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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, so the space of marked quasi-Fuchsian group of the first kind of some fixed genus g can be identified with the product of two copies of Teichmüller space of the same genus.
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https://en.wikipedia.org/wiki/Bers's_theorem
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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 solution of the following differential equation:
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https://en.wikipedia.org/wiki/Sinhc_function
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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, then this integer g is the smooth slice genus of K and is often denoted gs(K) or g4(K), whereas if S is required only to be topologically locally flatly embedded then g is the topologically locally flat slice genus of K. (There is no point considering g if S is required only to be a topological embedding, since the cone on K is a 2-disk with genus 0.) There can be an arbitrarily great difference between the smooth and the topologically locally flat slice genus of a knot; a theorem of Michael Freedman says that if the Alexander polynomial of K is 1, then the topologically locally flat slice genus of K is 0, but it can be proved in many ways (originally with gauge theory) that for every g there exist knots K such that the Alexander polynomial of K is 1 while the genus and the smooth slice genus of K both equal g. The (smooth) slice genus of a knot K is bounded below by a quantity involving the Thurston–Bennequin invariant of K: g s ( K ) ≥ ( T B ( K ) + 1 ) / 2. {\displaystyle g_{s}(K)\geq ({\rm {TB}}(K)+1)/2.\,} The (smooth) slice genus is zero if and only if the knot is concordant to the unknot.
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https://en.wikipedia.org/wiki/Slice_genus
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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 equation of a straight line as "y = mx + b" and it can also be found in Todhunter (1888) who wrote it as "y = mx + c".Slope is calculated by finding the ratio of the "vertical change" to the "horizontal change" between (any) two distinct points on a line. Sometimes the ratio is expressed as a quotient ("rise over run"), giving the same number for every two distinct points on the same line. A line that is decreasing has a negative "rise".
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https://en.wikipedia.org/wiki/Slope_of_a_line
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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.
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https://en.wikipedia.org/wiki/Slope_of_a_line
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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} .
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https://en.wikipedia.org/wiki/Slope_of_a_line
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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.
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https://en.wikipedia.org/wiki/Slope_of_a_line
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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 may be neglected, the run is the difference in distance from a fixed point measured along a level, horizontal line, or in other words, the run is (x2 − x1) = Δx.
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https://en.wikipedia.org/wiki/Slope_of_a_line
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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}}}.}
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https://en.wikipedia.org/wiki/Slope_of_a_line
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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 has a slope of −1. As a generalization of this practical description, the mathematics of differential calculus defines the slope of a curve at a point as the slope of the tangent line at that point.
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https://en.wikipedia.org/wiki/Slope_of_a_line
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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 formula for the slope of the curve at any point in the middle of the curve. This generalization of the concept of slope allows very complex constructions to be planned and built that go well beyond static structures that are either horizontals or verticals, but can change in time, move in curves, and change depending on the rate of change of other factors. Thereby, the simple idea of slope becomes one of the main basis of the modern world in terms of both technology and the built environment.
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https://en.wikipedia.org/wiki/Slope_of_a_line
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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 modelling. For example, some global and regional models of the atmosphere or oceans resolve the so-called quasi-geostrophic flow dynamics on the slow manifold of the atmosphere/oceanic dynamics, and is thus crucial to forecasting with a climate model. In some cases, a slow manifold is defined to be the invariant manifold on which the dynamics are slow compared to the dynamics off the manifold.
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https://en.wikipedia.org/wiki/Slow_manifold
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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 definition described in the first paragraph. Furthermore, one might define the slow manifold to be tangent to more than one eigenspace by choosing a cut-off point in an ordering of the real part eigenvalues in magnitude from least to greatest. In practice, one should be careful to see what definition the literature is suggesting.
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https://en.wikipedia.org/wiki/Slow_manifold
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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 Feferman–Schütte ordinal Γ 0 {\displaystyle \Gamma _{0}} . Most systems of notation use symbols such as ψ ( α ) {\displaystyle \psi (\alpha )} , θ ( α ) {\displaystyle \theta (\alpha )} , ψ α ( β ) {\displaystyle \psi _{\alpha }(\beta )} , some of which are modifications of the Veblen functions to produce countable ordinals even for uncountable arguments, and some of which are "collapsing functions".
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https://en.wikipedia.org/wiki/Small_Veblen_ordinal
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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 theorem. It is also the ordinal type of a certain ordering of rooted trees (Jervell 2005).
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https://en.wikipedia.org/wiki/Small_Veblen_ordinal
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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.
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https://en.wikipedia.org/wiki/Small_boundary_property
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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 given to the incorrect identity ( x + y ) n = x n + y n {\textstyle (x+y)^{n}=x^{n}+y^{n}} . The sophomore's dream has a similar too-good-to-be-true feel, but is true.
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https://en.wikipedia.org/wiki/Bernoulli's_identity
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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 soul conjecture, formulated by Cheeger and Gromoll at that time, was proved twenty years later by Grigori Perelman.
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https://en.wikipedia.org/wiki/Soul_conjecture
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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} . This algebra is well studied and understood, and is often used as a model for the study of other Lie algebras. The Lie group that it generates is the special linear group.
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https://en.wikipedia.org/wiki/Special_linear_Lie_algebra
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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 }}ad-bc=1\right\}.} It is a connected non-compact simple real Lie group of dimension 3 with applications in geometry, topology, representation theory, and physics.
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https://en.wikipedia.org/wiki/SL2(R)
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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.
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https://en.wikipedia.org/wiki/SL2(R)
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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, however.
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https://en.wikipedia.org/wiki/SL2(R)
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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 ) → F × . {\displaystyle \det \colon \operatorname {GL} (n,F)\to F^{\times }.}
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https://en.wikipedia.org/wiki/Special_linear_group
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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 sometimes used.
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https://en.wikipedia.org/wiki/Special_linear_group
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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 are three degrees of freedom, so that the dimension of SO(3) is three. In numerous applications one or other coordinate system is used, and the question arises how to convert from a given system to another.
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https://en.wikipedia.org/wiki/Hypersphere_of_rotations
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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. The special unitary group is a normal subgroup of the unitary group U(n), consisting of all n×n unitary matrices.
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https://en.wikipedia.org/wiki/Special_unitary_group
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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,\mathbb {C} )} . The SU(n) groups find wide application in the Standard Model of particle physics, especially SU(2) in the electroweak interaction and SU(3) in quantum chromodynamics.
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https://en.wikipedia.org/wiki/Special_unitary_group
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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 multiplying by a global phase factor e i φ {\displaystyle e^{i\varphi }} does not change the expectation values of a quantum operator.) The simplest case, SU(1), is the trivial group, having only a single element.
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https://en.wikipedia.org/wiki/Special_unitary_group
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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 identical to one of the symmetry groups of spinors, Spin(3), that enables a spinor presentation of rotations.
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https://en.wikipedia.org/wiki/Special_unitary_group
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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 spectral abscissa maps a square matrix onto its largest real eigenvalue.
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https://en.wikipedia.org/wiki/Spectral_abscissa
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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.
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https://en.wikipedia.org/wiki/Spectral_gap
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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(·).
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https://en.wikipedia.org/wiki/Spectral_radius
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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 on a finite closed interval to second order differential operators with singularities at the endpoints of the interval, possibly semi-infinite or infinite. Unlike the classical case, the spectrum may no longer consist of just a countable set of eigenvalues, but may also contain a continuous part.
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https://en.wikipedia.org/wiki/Spectral_measure
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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 theorem. It has had important applications in quantum mechanics, operator theory and harmonic analysis on semisimple Lie groups.
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https://en.wikipedia.org/wiki/Spectral_measure
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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 under all operators π(x) with x ∈ A. We implicitly assume that irreducible representation means non-null irreducible representation, thus excluding trivial (i.e. identically 0) representations on one-dimensional spaces. As explained below, the spectrum  is also naturally a topological space; this is similar to the notion of the spectrum of a ring. One of the most important applications of this concept is to provide a notion of dual object for any locally compact group. This dual object is suitable for formulating a Fourier transform and a Plancherel theorem for unimodular separable locally compact groups of type I and a decomposition theorem for arbitrary representations of separable locally compact groups of type I. The resulting duality theory for locally compact groups is however much weaker than the Tannaka–Krein duality theory for compact topological groups or Pontryagin duality for locally compact abelian groups, both of which are complete invariants. That the dual is not a complete invariant is easily seen as the dual of any finite-dimensional full matrix algebra Mn(C) consists of a single point.
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https://en.wikipedia.org/wiki/Hull-kernel_topology
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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.
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https://en.wikipedia.org/wiki/Energy_spectrum
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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. The determinant of the matrix equals the product of its eigenvalues.
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https://en.wikipedia.org/wiki/Spectrum_of_a_matrix
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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 interested in the dominant eigenvalue, i.e. that which is largest in absolute value. In other applications, the smallest eigenvalue is important, but in general, the whole spectrum provides valuable information about a matrix.
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https://en.wikipedia.org/wiki/Spectrum_of_a_matrix
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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.
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https://en.wikipedia.org/wiki/Spherical_mean
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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 solution to this equation and we define S ( t ) := ( 1 − t 2 ) b / 2 y ( t ) {\displaystyle S(t):=(1-t^{2})^{b/2}y(t)} , then S ( t ) {\displaystyle S(t)} is a prolate spheroidal wave function in the sense that it satisfies the equation ( 1 − t 2 ) d 2 S d t 2 − 2 t d S d t + ( c − 4 q + b + b 2 + 4 q ( 1 − t 2 ) − b 2 1 − t 2 ) S = 0 {\displaystyle (1-t^{2}){\frac {d^{2}S}{dt^{2}}}-2t\,{\frac {dS}{dt}}+(c-4q+b+b^{2}+4q(1-t^{2})-{\frac {b^{2}}{1-t^{2}}})\,S=0}
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https://en.wikipedia.org/wiki/Spheroidal_wave_equation
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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 the special orthogonal groups. They are usually studied over the real or complex numbers, but they can be defined over other fields.
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https://en.wikipedia.org/wiki/Spin_representation
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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 vector representation of the group.
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https://en.wikipedia.org/wiki/Spin_representation
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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 subtle way, on the dimension and signature of the orthogonal group. In particular, spin representations often admit invariant bilinear forms, which can be used to embed the spin groups into classical Lie groups. In low dimensions, these embeddings are surjective and determine special isomorphisms between the spin groups and more familiar Lie groups; this elucidates the properties of spinors in these dimensions.
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https://en.wikipedia.org/wiki/Spin_representation
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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).
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https://en.wikipedia.org/wiki/Spinors_in_three_dimensions
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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.
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https://en.wikipedia.org/wiki/Spinor_genus
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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 spiral model to an n-dimensional spiral model. There are effective settings for the SPO algorithm: the periodic descent direction setting and the convergence setting.
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https://en.wikipedia.org/wiki/Spiral_optimization_algorithm
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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, Mathematisches Institut der Universität Tübingen). A revised algorithm was presented by Victor Pan in 1998. An implementation was provided by Xavier Gourdon in 1996 for the Magma and PARI/GP computer algebra systems.
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https://en.wikipedia.org/wiki/Splitting_circle_method
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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. In this case the splitting principle can be quite useful.
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https://en.wikipedia.org/wiki/Splitting_principle
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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 ∗ ( Y ) {\displaystyle p^{*}\colon H^{*}(X)\rightarrow H^{*}(Y)} is injective means that any equation which holds in H ∗ ( Y ) {\displaystyle H^{*}(Y)} (say between various Chern classes) also holds in H ∗ ( X ) {\displaystyle H^{*}(X)} .
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https://en.wikipedia.org/wiki/Splitting_principle
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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 ) {\displaystyle K(X)} , it is important to note that p ∗: K ( X ) → K ( Y ) {\displaystyle p^{*}\colon K(X)\rightarrow K(Y)} is also injective for the map p {\displaystyle p} in the above theorem.The splitting principle admits many variations. The following, in particular, concerns real vector bundles and their complexifications:
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https://en.wikipedia.org/wiki/Splitting_principle
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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 IUC notation as p4m, Coxeter notation as , and orbifold notation as *442.Two orientations of an image of the lattice are by far the most common. They can conveniently be referred to as the upright square lattice and diagonal square lattice; the latter is also called the centered square lattice. They differ by an angle of 45°. This is related to the fact that a square lattice can be partitioned into two square sub-lattices, as is evident in the colouring of a checkerboard.
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https://en.wikipedia.org/wiki/Square_lattice
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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 unique matrix B that is positive semidefinite and such that BB = BTB = A (for real-valued matrices, where BT is the transpose of B). Less frequently, the name square root may be used for any factorization of a positive semidefinite matrix A as BTB = A, as in the Cholesky factorization, even if BB ≠ A. This distinct meaning is discussed in Positive definite matrix § Decomposition.
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https://en.wikipedia.org/wiki/Square_root_of_a_matrix
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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 {\hat {p}}} denotes the nominal point, P {\displaystyle P} denotes the space of all possible values of the object p {\displaystyle p} , and the shaded area, P ( s ) {\displaystyle P(s)} , represents the set of points that satisfy the stability conditions. The radius of the blue circle, shown in red, is the stability radius.
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https://en.wikipedia.org/wiki/Stability_radius
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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 Euclidean plane R 2 {\displaystyle \mathbb {R} ^{2}} formed by the pairs (x, y) of real numbers, the standard basis is formed by the vectors e x = ( 1 , 0 ) , e y = ( 0 , 1 ) . {\displaystyle \mathbf {e} _{x}=(1,0),\quad \mathbf {e} _{y}=(0,1).} Similarly, the standard basis for the three-dimensional space R 3 {\displaystyle \mathbb {R} ^{3}} is formed by vectors e x = ( 1 , 0 , 0 ) , e y = ( 0 , 1 , 0 ) , e z = ( 0 , 0 , 1 ) .
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https://en.wikipedia.org/wiki/Standard_basis
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{\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.
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https://en.wikipedia.org/wiki/Standard_basis
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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 uniquely as a linear combination of these.
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https://en.wikipedia.org/wiki/Standard_basis
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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 components of the vector v. In the n-dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} , the standard basis consists of n distinct vectors { e i: 1 ≤ i ≤ n } , {\displaystyle \{\mathbf {e} _{i}:1\leq i\leq n\},} where ei denotes the vector with a 1 in the ith coordinate and 0's elsewhere. Standard bases can be defined for other vector spaces, whose definition involves coefficients, such as polynomials and matrices. In both cases, the standard basis consists of the elements of the space such that all coefficients but one are 0 and the non-zero one is 1.
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https://en.wikipedia.org/wiki/Standard_basis
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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 formed by the 4 matrices e 11 = ( 1 0 0 0 ) , e 12 = ( 0 1 0 0 ) , e 21 = ( 0 0 1 0 ) , e 22 = ( 0 0 0 1 ) . {\displaystyle \mathbf {e} _{11}={\begin{pmatrix}1&0\\0&0\end{pmatrix}},\quad \mathbf {e} _{12}={\begin{pmatrix}0&1\\0&0\end{pmatrix}},\quad \mathbf {e} _{21}={\begin{pmatrix}0&0\\1&0\end{pmatrix}},\quad \mathbf {e} _{22}={\begin{pmatrix}0&0\\0&1\end{pmatrix}}.}
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https://en.wikipedia.org/wiki/Standard_basis
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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 Cartan and Eilenberg (1956, IX.6) and has since been generalized in many ways. The name "bar complex" comes from the fact that Eilenberg & Mac Lane (1953) used a vertical bar | as a shortened form of the tensor product ⊗ {\displaystyle \otimes } in their notation for the complex.
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https://en.wikipedia.org/wiki/Bar_resolution
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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 abelian category that is semisimple. Moreover, as he pointed out, the standard conjectures also imply the hardest part of the Weil conjectures, namely the "Riemann hypothesis" conjecture that remained open at the end of the 1960s and was proved later by Pierre Deligne; for details on the link between Weil and standard conjectures, see Kleiman (1968). The standard conjectures remain open problems, so that their application gives only conditional proofs of results.
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https://en.wikipedia.org/wiki/Standard_conjectures
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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 the cohomology of a smooth projective variety H ∗(X) → H ∗(X)induced by an algebraic cycle with rational coefficients on the product X × X via the cycle class map, which is part of the structure of a Weil cohomology theory. Conjecture A is equivalent to Conjecture B (see Grothendieck (1969), p. 196), and so is not listed.
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https://en.wikipedia.org/wiki/Standard_conjectures
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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.
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https://en.wikipedia.org/wiki/Star_product
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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 method and the method of steepest descent, but Laplace's contribution precedes the others.
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https://en.wikipedia.org/wiki/Stationary_phase_approximation
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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 vectors allows to compute the product of any elements (just like a matrix allows to compute the action of the linear operator on any vector by providing the action of the operator on basis vectors). Therefore, the structure constants can be used to specify the product operation of the algebra (just like a matrix defines a linear operator).
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https://en.wikipedia.org/wiki/Structure_constant
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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 when discussing Lie algebras in physics, as the basis vectors indicate specific directions in physical space, or correspond to specific particles (recall that Lie algebras are algebras over a field, with the bilinear product being given by the Lie bracket, usually defined via the commutator).
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https://en.wikipedia.org/wiki/Structure_constant
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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 often used in image processing and computer vision.
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https://en.wikipedia.org/wiki/Structure_Tensor
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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 result due to H. Satô (1969) which proves that "any Gaussian measure on a separable Banach space is an abstract Wiener measure in the sense of L. Gross". The result by Dudley et al. generalizes this result to the setting of Gaussian measures on a general topological vector space.
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https://en.wikipedia.org/wiki/Structure_theorem_for_Gaussian_measures
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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 series.
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https://en.wikipedia.org/wiki/Interchange_of_limiting_operations
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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-function for the field of Gaussian rational numbers. This formula is a special case of the analytic class number formula, and in those terms reads that the Gaussian field has class number 1. The factor 1 4 {\displaystyle {\tfrac {1}{4}}} on the right hand side of the formula corresponds to the fact that this field contains four roots of unity.
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https://en.wikipedia.org/wiki/Bloch-Kato_conjecture_on_Tamagawa_numbers
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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} } be a real-valued convex function defined on an open interval of the real line. Such a function need not be differentiable at all points: For example, the absolute value function f ( x ) = | x | {\displaystyle f(x)=|x|} is non-differentiable when x = 0 {\displaystyle x=0} . However, as seen in the graph on the right (where f ( x ) {\displaystyle f(x)} in blue has non-differentiable kinks similar to the absolute value function), for any x 0 {\displaystyle x_{0}} in the domain of the function one can draw a line which goes through the point ( x 0 , f ( x 0 ) ) {\displaystyle (x_{0},f(x_{0}))} and which is everywhere either touching or below the graph of f. The slope of such a line is called a subderivative.
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https://en.wikipedia.org/wiki/Subderivative
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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).
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https://en.wikipedia.org/wiki/Subspace_theorem
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