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c_b455tfr6j2rm | In mathematics, a skeleton of a category is a subcategory that, roughly speaking, does not contain any extraneous isomorphisms. In a certain sense, the skeleton of a category is the "smallest" equivalent category, which captures all "categorical properties" of the original. In fact, two categories are equivalent if and... | Skeleton (category theory) |
c_8rrg1f78049i | In mathematics, a skew gradient of a harmonic function over a simply connected domain with two real dimensions is a vector field that is everywhere orthogonal to the gradient of the function and that has the same magnitude as the gradient. | Skew gradient |
c_z117kcn48li6 | In mathematics, a slender group is a torsion-free abelian group that is "small" in a sense that is made precise in the definition below. | Slender group |
c_ag427jtbi1gp | In mathematics, a smooth algebraic curve C {\displaystyle C} in the complex projective plane, of degree d {\displaystyle d} , has genus given by the genus–degree formula g = ( d − 1 ) ( d − 2 ) / 2 {\displaystyle g=(d-1)(d-2)/2} .The Thom conjecture, named after French mathematician René Thom, states that if Σ {\displa... | Thom conjecture |
c_c1q7npbap7w3 | There is at least one generalization of this conjecture, known as the symplectic Thom conjecture (which is now a theorem, as proved for example by Peter Ozsváth and Szabó in 2000). It states that a symplectic surface of a symplectic 4-manifold is genus minimizing within its homology class. This would imply the previous... | Thom conjecture |
c_5try9v8av5td | In mathematics, a smooth compact manifold M is called almost flat if for any ε > 0 {\displaystyle \varepsilon >0} there is a Riemannian metric g ε {\displaystyle g_{\varepsilon }} on M such that diam ( M , g ε ) ≤ 1 {\displaystyle {\mbox{diam}}(M,g_{\varepsilon })\leq 1} and g ε {\displaystyle g_{\varepsilon }} is ε {\... | Gromov-Ruh theorem |
c_7x8jw5n6det4 | In mathematics, a smooth maximum of an indexed family x1, ..., xn of numbers is a smooth approximation to the maximum function max ( x 1 , … , x n ) , {\displaystyle \max(x_{1},\ldots ,x_{n}),} meaning a parametric family of functions m α ( x 1 , … , x n ) {\displaystyle m_{\alpha }(x_{1},\ldots ,x_{n})} such that for ... | Smooth maximum |
c_7dsngk0f8lv6 | In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a smooth structure allows one to perform mathematical analysis on the manifold. | Smooth structure |
c_n6cw79a9ba0h | In mathematics, a sober space is a topological space X such that every (nonempty) irreducible closed subset of X is the closure of exactly one point of X: that is, every irreducible closed subset has a unique generic point. | Sober space |
c_xe9qstt45odm | In mathematics, a sofic group is a group whose Cayley graph is an initially subamenable graph, or equivalently a subgroup of an ultraproduct of finite-rank symmetric groups such that every two elements of the group have distance 1. They were introduced by Gromov (1999) as a common generalization of amenable and residua... | Sofic group |
c_wz55b3otccdp | A finitely generated group is sofic if it is the limit of a sequence of sofic groups. The limit of a sequence of amenable groups (that is, an initially subamenable group) is necessarily sofic, but there exist sofic groups that are not initially subamenable groups.As Gromov proved, Sofic groups are surjunctive. That is,... | Sofic group |
c_x8w751kccsag | In mathematics, a solid Klein bottle is a three-dimensional topological space (a 3-manifold) whose boundary is the Klein bottle.It is homeomorphic to the quotient space obtained by gluing the top disk of a cylinder D 2 × I {\displaystyle \scriptstyle D^{2}\times I} to the bottom disk by a reflection across a diameter o... | Solid Klein bottle |
c_z38gz4f1ns07 | In mathematics, a solid torus is the topological space formed by sweeping a disk around a circle. It is homeomorphic to the Cartesian product S 1 × D 2 {\displaystyle S^{1}\times D^{2}} of the disk and the circle, endowed with the product topology. A standard way to visualize a solid torus is as a toroid, embedded in 3... | Solid tori |
c_oyc4ke64vaej | However, it should be distinguished from a torus, which has the same visual appearance: the torus is the two-dimensional space on the boundary of a toroid, while the solid torus includes also the compact interior space enclosed by the torus. A solid torus is a torus plus the volume inside the torus. Real-world objects ... | Solid tori |
c_ww5gkf1jnodd | In mathematics, a solution set is the set of values that satisfy a given set of equations or inequalities. For example, for a set { f i } {\displaystyle \{f_{i}\}} of polynomials over a ring R {\displaystyle R} , the solution set is the subset of R {\displaystyle R} on which the polynomials all vanish (evaluate to 0), ... | Solution set |
c_2eae8lmdbzf3 | In mathematics, a solvmanifold is a homogeneous space of a connected solvable Lie group. It may also be characterized as a quotient of a connected solvable Lie group by a closed subgroup. (Some authors also require that the Lie group be simply-connected, or that the quotient be compact.) A special class of solvmanifold... | Sol manifold |
c_4z0tpgtu8vnw | In mathematics, a source for the representation theory of the group of diffeomorphisms of a smooth manifold M is the initial observation that (for M connected) that group acts transitively on M. | Representation theory of the diffeomorphism group |
c_cnwfzv8f47xf | In mathematics, a space form is a complete Riemannian manifold M of constant sectional curvature K. The three most fundamental examples are Euclidean n-space, the n-dimensional sphere, and hyperbolic space, although a space form need not be simply connected. | Space form |
c_ju8e7vpo1i6s | In mathematics, a space is a set (sometimes called a universe) with some added structure. While modern mathematics uses many types of spaces, such as Euclidean spaces, linear spaces, topological spaces, Hilbert spaces, or probability spaces, it does not define the notion of "space" itself. A space consists of selected ... | Space (mathematics) |
c_s5sf924udre5 | It is the relationships that define the nature of the space. More precisely, isomorphic spaces are considered identical, where an isomorphism between two spaces is a one-to-one correspondence between their points that preserves the relationships. For example, the relationships between the points of a three-dimensional ... | Space (mathematics) |
c_j7agdcdpehyt | Topological notions such as continuity have natural definitions in every Euclidean space. However, topology does not distinguish straight lines from curved lines, and the relation between Euclidean and topological spaces is thus "forgetful". | Space (mathematics) |
c_817xkz6nbjzd | Relations of this kind are treated in more detail in the Section "Types of spaces". It is not always clear whether a given mathematical object should be considered as a geometric "space", or an algebraic "structure". A general definition of "structure", proposed by Bourbaki, embraces all common types of spaces, provide... | Space (mathematics) |
c_tqpon6lljdtx | In mathematics, a space of convolution quotients is a field of fractions of a convolution ring of functions: a convolution quotient is to the operation of convolution as a quotient of integers is to multiplication. The construction of convolution quotients allows easy algebraic representation of the Dirac delta functio... | Convolution quotient |
c_2hi8qfhfmpws | The kind of convolution ( f , g ) ↦ f ∗ g {\textstyle (f,g)\mapsto f*g} with which this theory is concerned is defined by ( f ∗ g ) ( x ) = ∫ 0 x f ( u ) g ( x − u ) d u . {\displaystyle (f*g)(x)=\int _{0}^{x}f(u)g(x-u)\,du.} It follows from the Titchmarsh convolution theorem that if the convolution f ∗ g {\textstyle f... | Convolution quotient |
c_2ofejpoz5mtx | A consequence is that if f , g , h {\textstyle f,g,h} are continuous on [ 0 , + ∞ ) {\textstyle [0,+\infty )} then h ∗ f = h ∗ g {\textstyle h*f=h*g} only if f = g . {\textstyle f=g.} This fact makes it possible to define convolution quotients by saying that for two functions ƒ, g, the pair (ƒ, g) has the same convolut... | Convolution quotient |
c_m6bs5veodeyi | As with the construction of the rational numbers from the integers, the field of convolution quotients is a direct extension of the convolution ring from which it was built. Every "ordinary" function f {\displaystyle f} in the original space embeds canonically into the space of convolution quotients as the (equivalence... | Convolution quotient |
c_hvcbvoq5icm0 | In mathematics, a sparse polynomial (also lacunary polynomial or fewnomial) is a polynomial that has far fewer terms than its degree and number of variables would suggest. For example, x10 + 3x3 - 1 is a sparse polynomial as it is a trinomial with a degree of 10. The motivation for studying sparse polynomials is to con... | Sparse polynomial |
c_dkeul8s95n0o | Sparse polynomials have also been used in pure mathematics, especially in the study of Galois groups, because it has been easier to determine the Galois groups of certain families of sparse polynomials than it is for other polynomials.The algebraic varieties determined by sparse polynomials have a simple structure, whi... | Sparse polynomial |
c_q5pgkn0j99l5 | It states that the non-negativity of a polynomial can be certified by sos polynomials whose degree only depends on the number of monomials of the polynomial.Sparse polynomials oftentimes come up in sum or difference of powers equations. The sum of two cubes states that (a + b)(a2 - 2ab + b2) = a3 + b3. a3 + b3, here, i... | Sparse polynomial |
c_7sex79xxkx1y | In mathematics, a sparsely totient number is a certain kind of natural number. A natural number, n, is sparsely totient if for all m > n, φ ( m ) > φ ( n ) {\displaystyle \varphi (m)>\varphi (n)} where φ {\displaystyle \varphi } is Euler's totient function. The first few sparsely totient numbers are: 2, 6, 12, 18, 30, ... | Sparsely totient number |
c_t1bj3mekydb7 | In mathematics, a spectral space is a topological space that is homeomorphic to the spectrum of a commutative ring. It is sometimes also called a coherent space because of the connection to coherent topos. | Spectral spaces |
c_6hezz787s6w4 | In mathematics, a spherical 3-manifold M is a 3-manifold of the form M = S 3 / Γ {\displaystyle M=S^{3}/\Gamma } where Γ {\displaystyle \Gamma } is a finite subgroup of SO(4) acting freely by rotations on the 3-sphere S 3 {\displaystyle S^{3}} . All such manifolds are prime, orientable, and closed. Spherical 3-manifold... | Spherical 3-manifold |
c_tdd7f1veql5d | In mathematics, a spherical conic or sphero-conic is a curve on the sphere, the intersection of the sphere with a concentric elliptic cone. It is the spherical analog of a conic section (ellipse, parabola, or hyperbola) in the plane, and as in the planar case, a spherical conic can be defined as the locus of points the... | Spherical conic |
c_ceoz783pbjwv | Like planar conics, spherical conics also satisfy a "reflection property": the great-circle arcs from the two foci to any point on the conic have the tangent and normal to the conic at that point as their angle bisectors. Many theorems about conics in the plane extend to spherical conics. For example, Graves's theorem ... | Spherical conic |
c_5jttiekz1xta | When restricted to the surface of a sphere, the remaining coordinates are confocal spherical conics. Sometimes this is called an elliptic coordinate system on the sphere, by analogy to a planar elliptic coordinate system. Such coordinates can be used in the computation of conformal maps from the sphere to the plane. | Spherical conic |
c_2npazjkv1dma | In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin; its polar angle measured from a fixed polar axis or zenith direction; and the azimuthal angle of its orthog... | Spherical coordinates |
c_nj0j4omfl0mw | The polar angle is often replaced by the elevation angle measured from the reference plane towards the positive Z axis; the depression angle is the negative of the elevation angle. The use of symbols and the order of the coordinates differs among sources and disciplines. This article will use the ISO convention frequen... | Spherical coordinates |
c_bd2prozp4967 | By contrast, in many mathematics books, ( ρ , θ , φ ) {\displaystyle (\rho ,\theta ,\varphi )} or ( r , θ , φ ) {\displaystyle (r,\theta ,\varphi )} gives the radial distance, azimuthal angle, and polar angle, switching the meanings of θ and φ. Other conventions are also used, such as r for radius from the z-axis, so g... | Spherical coordinates |
c_rqmw7turencn | There are a number of celestial coordinate systems based on different fundamental planes and with different terms for the various coordinates. The spherical coordinate systems used in mathematics normally use radians rather than degrees and measure the azimuthal angle counterclockwise from the x-axis to the y-axis rath... | Spherical coordinates |
c_r9guzlyuek3x | In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point. It is a subtype of whorled patterns, a broad group that also includes concentric objects. | Spherical spiral |
c_8ukftbzt9sxn | In mathematics, a spline is a special function defined piecewise by polynomials. In interpolating problems, spline interpolation is often preferred to polynomial interpolation because it yields similar results, even when using low degree polynomials, while avoiding Runge's phenomenon for higher degrees. In the computer... | Spline curve |
c_zrwju31cq30w | In mathematics, a split exact sequence is a short exact sequence in which the middle term is built out of the two outer terms in the simplest possible way. | Split exact sequence |
c_xg4kdlmgpc18 | In mathematics, a split-biquaternion is a hypercomplex number of the form q = w + x i + y j + z k {\displaystyle q=w+xi+yj+zk} where w, x, y, and z are split-complex numbers and i, j, and k multiply as in the quaternion group. Since each coefficient w, x, y, z spans two real dimensions, the split-biquaternion is an ele... | Split-biquaternion |
c_8aurdt7wcp62 | This algebra was introduced by William Kingdon Clifford in an 1873 article for the London Mathematical Society. It has been repeatedly noted in mathematical literature since then, variously as a deviation in terminology, an illustration of the tensor product of algebras, and as an illustration of the direct sum of alge... | Split-biquaternion |
c_f9taferon1zc | In mathematics, a sporadic group is one of the 26 exceptional groups found in the classification of finite simple groups. A simple group is a group G that does not have any normal subgroups except for the trivial group and G itself. The classification theorem states that the list of finite simple groups consists of 18 ... | Sporadic groups |
c_bbpieldbw1cq | These 26 exceptions are the sporadic groups. They are also known as the sporadic simple groups, or the sporadic finite groups. Because it is not strictly a group of Lie type, the Tits group is sometimes regarded as a sporadic group, in which case there would be 27 sporadic groups. The monster group, or friendly giant, ... | Sporadic groups |
c_4dyvq32tz9me | In mathematics, a square is the result of multiplying a number by itself. The verb "to square" is used to denote this operation. Squaring is the same as raising to the power 2, and is denoted by a superscript 2; for instance, the square of 3 may be written as 32, which is the number 9. In some cases when superscripts a... | Modulus squared |
c_ybdpjt7qglp6 | The adjective which corresponds to squaring is quadratic. The square of an integer may also be called a square number or a perfect square. In algebra, the operation of squaring is often generalized to polynomials, other expressions, or values in systems of mathematical values other than the numbers. | Modulus squared |
c_mdctzkyhsz5z | For instance, the square of the linear polynomial x + 1 is the quadratic polynomial (x + 1)2 = x2 + 2x + 1. One of the important properties of squaring, for numbers as well as in many other mathematical systems, is that (for all numbers x), the square of x is the same as the square of its additive inverse −x. That is, ... | Modulus squared |
c_upgwyvqhvojw | In mathematics, a square matrix is a matrix with the same number of rows and columns. An n-by-n matrix is known as a square matrix of order n {\displaystyle n} . Any two square matrices of the same order can be added and multiplied. | Square matrices |
c_da6xsznp8xbc | Square matrices are often used to represent simple linear transformations, such as shearing or rotation. For example, if R {\displaystyle R} is a square matrix representing a rotation (rotation matrix) and v {\displaystyle \mathbf {v} } is a column vector describing the position of a point in space, the product R v {\d... | Square matrices |
c_4zybjtknkv3g | In mathematics, a square matrix is said to be diagonally dominant if, for every row of the matrix, the magnitude of the diagonal entry in a row is larger than or equal to the sum of the magnitudes of all the other (non-diagonal) entries in that row. More precisely, the matrix A is diagonally dominant if | a i i | ≥ ∑ j... | Diagonally dominant matrix |
c_jr03pjjyjg88 | In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals 32 and can be written as 3 × 3. The usual notation for the square of a number n is not the product n × n, bu... | Square number |
c_y8hcxz16fdew | The unit of area is defined as the area of a unit square (1 × 1). Hence, a square with side length n has area n2. If a square number is represented by n points, the points can be arranged in rows as a square each side of which has the same number of points as the square root of n; thus, square numbers are a type of fig... | Square number |
c_ls0eb9mn5jic | In the real number system, square numbers are non-negative. A non-negative integer is a square number when its square root is again an integer. For example, 9 = 3 , {\displaystyle {\sqrt {9}}=3,} so 9 is a square number. | Square number |
c_fs3xry7obl9q | A positive integer that has no square divisors except 1 is called square-free. For a non-negative integer n, the nth square number is n2, with 02 = 0 being the zeroth one. | Square number |
c_lvfh406z0eub | The concept of square can be extended to some other number systems. If rational numbers are included, then a square is the ratio of two square integers, and, conversely, the ratio of two square integers is a square, for example, 4 9 = ( 2 3 ) 2 {\displaystyle \textstyle {\frac {4}{9}}=\left({\frac {2}{3}}\right)^{2}} .... | Square number |
c_0v4b3gi1nw6x | In mathematics, a square root of a number x is a number y such that y 2 = x {\displaystyle y^{2}=x} ; in other words, a number y whose square (the result of multiplying the number by itself, or y ⋅ y {\displaystyle y\cdot y} ) is x. For example, 4 and −4 are square roots of 16 because 4 2 = ( − 4 ) 2 = 16 {\displaystyl... | Square root |
c_iuusu6qzkarb | The radicand is the number or expression underneath the radical sign, in this case, 9. For non-negative x, the principal square root can also be written in exponent notation, as x 1 / 2 {\displaystyle x^{1/2}} . Every positive number x has two square roots: x {\displaystyle {\sqrt {x}}} (which is positive) and − x {\di... | Square root |
c_puimpwqdi9i3 | The two roots can be written more concisely using the ± sign as ± x {\displaystyle \pm {\sqrt {x}}} . Although the principal square root of a positive number is only one of its two square roots, the designation "the square root" is often used to refer to the principal square root.Square roots of negative numbers can be... | Square root |
c_774tv68wc6m5 | In mathematics, a square triangular number (or triangular square number) is a number which is both a triangular number and a square number. There are infinitely many square triangular numbers; the first few are: 0, 1, 36, 1225, 41616, 1413721, 48024900, 1631432881, 55420693056, 1882672131025 (sequence A001110 in the OE... | Square triangular number |
c_7kmwc2wkso59 | In mathematics, a square-difference-free set is a set of natural numbers, no two of which differ by a square number. Hillel Furstenberg and András Sárközy proved in the late 1970s the Furstenberg–Sárközy theorem of additive number theory showing that, in a certain sense, these sets cannot be very large. In the game of ... | Square-difference-free set |
c_mkd7geew6sdl | Another square-difference-free set is obtained by doubling the Moser–de Bruijn sequence. The best known upper bound on the size of a square-difference-free set of numbers up to n {\displaystyle n} is only slightly sublinear, but the largest known sets of this form are significantly smaller, of size ≈ n 0.733412 {\displ... | Square-difference-free set |
c_fxnfzkx2gcbf | In mathematics, a square-free element is an element r of a unique factorization domain R that is not divisible by a non-trivial square. This means that every s such that s 2 ∣ r {\displaystyle s^{2}\mid r} is a unit of R. | Square-free element |
c_kbu296nr5jxs | In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, 10 = 2 ⋅ 5 is square-free, but 18 = 2 ⋅ 3 ⋅ 3 is not, because 18 is divisible by 9 = 32... | Squarefree integer |
c_ydyoynub4s71 | In mathematics, a square-free polynomial is a polynomial defined over a field (or more generally, an integral domain) that does not have as a divisor any square of a non-constant polynomial. A univariate polynomial is square free if and only if it has no multiple root in an algebraically closed field containing its coe... | Square-free factorization |
c_8i966qfnylz9 | In the case of univariate polynomials, the product rule implies that, if p2 divides f, then p divides the formal derivative f ' of f. The converse is also true and hence, f {\displaystyle f} is square-free if and only if 1 {\displaystyle 1} is a greatest common divisor of the polynomial and its derivative.A square-free... | Square-free factorization |
c_ayr4ht8jrefl | Square-free factorization is the first step of the polynomial factorization algorithms that are implemented in computer algebra systems. Therefore, the algorithm of square-free factorization is basic in computer algebra. Over a field of characteristic 0, the quotient of f {\displaystyle f} by its GCD with its derivativ... | Square-free factorization |
c_klt9prtqqogl | Over a perfect field of non-zero characteristic p, this quotient is the product of the a i {\displaystyle a_{i}} such that i is not a multiple of p. Further GCD computations and exact divisions allow computing the square-free factorization (see square-free factorization over a finite field). In characteristic zero, a b... | Square-free factorization |
c_a2f0mdsebqf0 | In mathematics, a square-integrable function, also called a quadratically integrable function or L 2 {\displaystyle L^{2}} function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite. Thus, square-integrability on the real l... | Square integrable |
c_proiakmcsidj | For this to be true, the integrals of the positive and negative portions of the real part must both be finite, as well as those for the imaginary part. The vector space of (equivalence classes of) square integrable functions (with respect to Lebesgue measure) forms the L p {\displaystyle L^{p}} space with p = 2. {\disp... | Square integrable |
c_y2ap2m68f09s | Among the L p {\displaystyle L^{p}} spaces, the class of square integrable functions is unique in being compatible with an inner product, which allows notions like angle and orthogonality to be defined. Along with this inner product, the square integrable functions form a Hilbert space, since all of the L p {\displayst... | Square integrable |
c_seumfbwd2a4d | In mathematics, a stable vector bundle is a (holomorphic or algebraic) vector bundle that is stable in the sense of geometric invariant theory. Any holomorphic vector bundle may be built from stable ones using Harder–Narasimhan filtration. Stable bundles were defined by David Mumford in Mumford (1963) and later built u... | Stable holomorphic vector bundle |
c_r9g353yx17xs | In mathematics, a stably free module is a module which is close to being free. | Stably free module |
c_7kkva8hi1d9n | In mathematics, a stacky curve is an object in algebraic geometry that is roughly an algebraic curve with potentially "fractional points" called stacky points. A stacky curve is a type of stack used in studying Gromov–Witten theory, enumerative geometry, and rings of modular forms. Stacky curves are deeply related to 1... | Stacky curve |
c_61pveaizjum6 | In mathematics, a standard Borel space is the Borel space associated to a Polish space. Discounting Borel spaces of discrete Polish spaces, there is, up to isomorphism of measurable spaces, only one standard Borel space. | Standard Borel space |
c_ijdat669kgpw | In mathematics, a statistical manifold is a Riemannian manifold, each of whose points is a probability distribution. Statistical manifolds provide a setting for the field of information geometry. The Fisher information metric provides a metric on these manifolds. Following this definition, the log-likelihood function i... | Statistical manifold |
c_63c8j89vx4u1 | In mathematics, a stella octangula number is a figurate number based on the stella octangula, of the form n(2n2 − 1).The sequence of stella octangula numbers is 0, 1, 14, 51, 124, 245, 426, 679, 1016, 1449, 1990, ... (sequence A007588 in the OEIS)Only two of these numbers are square. | Stella octangula number |
c_z27fjcp3ts4i | In mathematics, a stiff equation is a differential equation for which certain numerical methods for solving the equation are numerically unstable, unless the step size is taken to be extremely small. It has proven difficult to formulate a precise definition of stiffness, but the main idea is that the equation includes ... | Stiff equation |
c_dgl9iqr5se1a | In order for a numerical method to give a reliable solution to the differential system sometimes the step size is required to be at an unacceptably small level in a region where the solution curve is very smooth. The phenomenon is known as stiffness. | Stiff equation |
c_lris0bdcbt6y | In some cases there may be two different problems with the same solution, yet one is not stiff and the other is. The phenomenon cannot therefore be a property of the exact solution, since this is the same for both problems, and must be a property of the differential system itself. Such systems are thus known as stiff s... | Stiff equation |
c_5malr7ts5kn9 | In mathematics, a stochastic matrix is a square matrix used to describe the transitions of a Markov chain. Each of its entries is a nonnegative real number representing a probability. : 9–11 It is also called a probability matrix, transition matrix, substitution matrix, or Markov matrix. : 9–11 The stochastic matrix wa... | Right stochastic matrix |
c_o4hf6zd09n94 | : 1–8 There are several different definitions and types of stochastic matrices:: 9–11 A right stochastic matrix is a real square matrix, with each row summing to 1. A left stochastic matrix is a real square matrix, with each column summing to 1. A doubly stochastic matrix is a square matrix of nonnegative real numbers ... | Right stochastic matrix |
c_8zvrtmoadsx8 | Thus, each row of a right stochastic matrix (or column of a left stochastic matrix) is a stochastic vector. : 9–11 A common convention in English language mathematics literature is to use row vectors of probabilities and right stochastic matrices rather than column vectors of probabilities and left stochastic matrices;... | Right stochastic matrix |
c_qjkse1wt5umq | In mathematics, a strange nonchaotic attractor (SNA) is a form of attractor which, while converging to a limit, is strange, because it is not piecewise differentiable, and also non-chaotic, in that its Lyapunov exponents are non-positive. SNAs were introduced as a topic of study by Grebogi et al. in 1984. SNAs can be d... | Strange nonchaotic attractor |
c_d6et0bmqerwu | In addition to periodic or quasiperiodic motion, they can exhibit chaotic or nonchaotic motion on strange attractors. Although quasiperiodic forcing is not necessary for strange nonchaotic dynamics (e.g., the period doubling accumulation point of a period doubling cascade), if quasiperiodic driving is not present, stra... | Strange nonchaotic attractor |
c_0a482v01yh8c | The first experiment to demonstrate a robust strange nonchaotic attractor involved the buckling of a magnetoelastic ribbon driven quasiperiodically by two incommensurate frequencies in the golden ratio. Strange nonchaotic attractors have been robustly observed in laboratory experiments involving magnetoelastic ribbons,... | Strange nonchaotic attractor |
c_1pe2kwyl804o | In mathematics, a strictly convex space is a normed vector space (X, || ||) for which the closed unit ball is a strictly convex set. Put another way, a strictly convex space is one for which, given any two distinct points x and y on the unit sphere ∂B (i.e. the boundary of the unit ball B of X), the segment joining x a... | Strictly convex space |
c_2ukcumwd2jf8 | In mathematics, a strong prime is a prime number with certain special properties. The definitions of strong primes are different in cryptography and number theory. | Strong prime |
c_yv2989tv8ja3 | In mathematics, a strong topology is a topology which is stronger than some other "default" topology. This term is used to describe different topologies depending on context, and it may refer to: the final topology on the disjoint union the topology arising from a norm the strong operator topology the strong topology (... | Strong topology |
c_l5tgpyv6xg5g | In mathematics, a structure is a set endowed with some additional features on the set (e.g. an operation, relation, metric, or topology). Often, the additional features are attached or related to the set, so as to provide it with some additional meaning or significance. A partial list of possible structures are measure... | Mathematical structure |
c_tbwlt55kdqrs | Sometimes, a set is endowed with more than one feature simultaneously, which allows mathematicians to study the interaction between the different structures more richly. For example, an ordering imposes a rigid form, shape, or topology on the set, and if a set has both a topology feature and a group feature, such that ... | Mathematical structure |
c_kfwluhk1u0ak | In mathematics, a stuck unknot is a closed polygonal chain in three-dimensional space (a skew polygon) that is topologically equal to the unknot but cannot be deformed to a simple polygon when interpreted as a mechanical linkage, by rigid length-preserving and non-self-intersecting motions of its segments. Similarly a ... | Stuck unknot |
c_4zdpaar4u6of | In mathematics, a stunted projective space is a construction on a projective space of importance in homotopy theory, introduced by James (1959). Part of a conventional projective space is collapsed down to a point. More concretely, in a real projective space, complex projective space or quaternionic projective space KP... | Stunted projective space |
c_86y7yuzaxp2q | This makes a topological space that is no longer a manifold. The importance of this construction was realised when it was shown that real stunted projective spaces arose as Spanier–Whitehead duals of spaces of Ioan James, so-called quasi-projective spaces, constructed from Stiefel manifolds. Their properties were there... | Stunted projective space |
c_r5vsx3puxpwu | In this way the vector fields on spheres question was reduced to a question on stunted projective spaces: for RPn,m, is there a degree one mapping on the 'next cell up' (of the first dimension not collapsed in the 'stunting') that extends to the whole space? Frank Adams showed that this could not happen, completing the... | Stunted projective space |
c_n4urm9m1sb3q | In mathematics, a sub-Riemannian manifold is a certain type of generalization of a Riemannian manifold. Roughly speaking, to measure distances in a sub-Riemannian manifold, you are allowed to go only along curves tangent to so-called horizontal subspaces. Sub-Riemannian manifolds (and so, a fortiori, Riemannian manifol... | Sub-Riemannian manifold |
c_jyotld3z4u68 | The Hausdorff dimension of such metric spaces is always an integer and larger than its topological dimension (unless it is actually a Riemannian manifold). Sub-Riemannian manifolds often occur in the study of constrained systems in classical mechanics, such as the motion of vehicles on a surface, the motion of robot ar... | Sub-Riemannian manifold |
c_kqhu8vb77kiu | In mathematics, a subadditive set function is a set function whose value, informally, has the property that the value of function on the union of two sets is at most the sum of values of the function on each of the sets. This is thematically related to the subadditivity property of real-valued functions. | Subadditive utility |
c_g1i217v2ege0 | In mathematics, a subalgebra is a subset of an algebra, closed under all its operations, and carrying the induced operations. "Algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear operation. Algebras in universal algebra are far more general: they are a comm... | Subalgebra |
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