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In mathematics, the quadratic bottleneck assignment problem (QBAP) is one of the fundamental combinatorial optimization problems in the branch of optimization or operations research, from the category of the facilities location problems.It is related to the quadratic assignment problem in the same way as the linear bottleneck assignment problem is related to the linear assignment problem, the "sum" is replaced with "max" in the objective function. The problem models the following real-life problem: There are a set of n facilities and a set of n locations. For each pair of locations, a distance is specified and for each pair of facilities a weight or flow is specified (e.g., the amount of supplies transported between the two facilities). The problem is to assign all facilities to different locations with the goal of minimizing the maximum of the distances multiplied by the corresponding flows.
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https://en.wikipedia.org/wiki/Quadratic_bottleneck_assignment_problem
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In mathematics, the quadratic eigenvalue problem (QEP), is to find scalar eigenvalues λ {\displaystyle \lambda } , left eigenvectors y {\displaystyle y} and right eigenvectors x {\displaystyle x} such that Q ( λ ) x = 0 and y ∗ Q ( λ ) = 0 , {\displaystyle Q(\lambda )x=0~{\text{ and }}~y^{\ast }Q(\lambda )=0,} where Q ( λ ) = λ 2 M + λ C + K {\displaystyle Q(\lambda )=\lambda ^{2}M+\lambda C+K} , with matrix coefficients M , C , K ∈ C n × n {\displaystyle M,\,C,K\in \mathbb {C} ^{n\times n}} and we require that M ≠ 0 {\displaystyle M\,\neq 0} , (so that we have a nonzero leading coefficient). There are 2 n {\displaystyle 2n} eigenvalues that may be infinite or finite, and possibly zero. This is a special case of a nonlinear eigenproblem. Q ( λ ) {\displaystyle Q(\lambda )} is also known as a quadratic polynomial matrix.
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https://en.wikipedia.org/wiki/Quadratic_eigenvalue_problem
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In mathematics, the quadruple product is a product of four vectors in three-dimensional Euclidean space. The name "quadruple product" is used for two different products, the scalar-valued scalar quadruple product and the vector-valued vector quadruple product or vector product of four vectors.
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https://en.wikipedia.org/wiki/Quadruple_product
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In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f ( x ) {\displaystyle f(x)} of some function f . {\displaystyle f.} An important class of pointwise concepts are the pointwise operations, that is, operations defined on functions by applying the operations to function values separately for each point in the domain of definition. Important relations can also be defined pointwise.
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https://en.wikipedia.org/wiki/Pointwise
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In mathematics, the qualitative theory of differential equations studies the behavior of differential equations by means other than finding their solutions. It originated from the works of Henri Poincaré and Aleksandr Lyapunov. There are relatively few differential equations that can be solved explicitly, but using tools from analysis and topology, one can "solve" them in the qualitative sense, obtaining information about their properties.
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https://en.wikipedia.org/wiki/Qualitative_theory_of_differential_equations
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In mathematics, the quantum Markov chain is a reformulation of the ideas of a classical Markov chain, replacing the classical definitions of probability with quantum probability.
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https://en.wikipedia.org/wiki/Quantum_Markov_chain
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In mathematics, the quantum dilogarithm is a special function defined by the formula ϕ ( x ) ≡ ( x ; q ) ∞ = ∏ n = 0 ∞ ( 1 − x q n ) , | q | < 1 {\displaystyle \phi (x)\equiv (x;q)_{\infty }=\prod _{n=0}^{\infty }(1-xq^{n}),\quad |q|<1} It is the same as the q-exponential function E q ( x ) {\displaystyle E_{q}(x)} . Let u , v {\displaystyle u,v} be "q-commuting variables", that is elements of a suitable noncommutative algebra satisfying Weyl's relation u v = q v u {\displaystyle uv=qvu} . Then, the quantum dilogarithm satisfies Schützenberger's identity ϕ ( u ) ϕ ( v ) = ϕ ( u + v ) , {\displaystyle \phi (u)\phi (v)=\phi (u+v),} Faddeev-Volkov's identity ϕ ( v ) ϕ ( u ) = ϕ ( u + v − v u ) , {\displaystyle \phi (v)\phi (u)=\phi (u+v-vu),} and Faddeev-Kashaev's identity ϕ ( v ) ϕ ( u ) = ϕ ( u ) ϕ ( − v u ) ϕ ( v ) . {\displaystyle \phi (v)\phi (u)=\phi (u)\phi (-vu)\phi (v).}
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https://en.wikipedia.org/wiki/Quantum_dilogarithm
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The latter is known to be a quantum generalization of Rogers' five term dilogarithm identity. Faddeev's quantum dilogarithm Φ b ( w ) {\displaystyle \Phi _{b}(w)} is defined by the following formula: Φ b ( z ) = exp ( 1 4 ∫ C e − 2 i z w sinh ( w b ) sinh ( w / b ) d w w ) , {\displaystyle \Phi _{b}(z)=\exp \left({\frac {1}{4}}\int _{C}{\frac {e^{-2izw}}{\sinh(wb)\sinh(w/b)}}{\frac {dw}{w}}\right),} where the contour of integration C {\displaystyle C} goes along the real axis outside a small neighborhood of the origin and deviates into the upper half-plane near the origin. The same function can be described by the integral formula of Woronowicz: Φ b ( x ) = exp ( i 2 π ∫ R log ( 1 + e t b 2 + 2 π b x ) 1 + e t d t ) .
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https://en.wikipedia.org/wiki/Quantum_dilogarithm
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In mathematics, the quantum q-Krawtchouk polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties.
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https://en.wikipedia.org/wiki/Quantum_q-Krawtchouk_polynomials
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In mathematics, the quasi-commutative property is an extension or generalization of the general commutative property. This property is used in specific applications with various definitions.
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https://en.wikipedia.org/wiki/Quasi-commutative_property
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In mathematics, the quasi-derivative is one of several generalizations of the derivative of a function between two Banach spaces. The quasi-derivative is a slightly stronger version of the Gateaux derivative, though weaker than the Fréchet derivative. Let f: A → F be a continuous function from an open set A in a Banach space E to another Banach space F. Then the quasi-derivative of f at x0 ∈ A is a linear transformation u: E → F with the following property: for every continuous function g: → A with g(0)=x0 such that g′(0) ∈ E exists, lim t → 0 + f ( g ( t ) ) − f ( x 0 ) t = u ( g ′ ( 0 ) ) . {\displaystyle \lim _{t\to 0^{+}}{\frac {f(g(t))-f(x_{0})}{t}}=u(g'(0)).}
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https://en.wikipedia.org/wiki/Quasi-derivative
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If such a linear map u exists, then f is said to be quasi-differentiable at x0. Continuity of u need not be assumed, but it follows instead from the definition of the quasi-derivative. If f is Fréchet differentiable at x0, then by the chain rule, f is also quasi-differentiable and its quasi-derivative is equal to its Fréchet derivative at x0. The converse is true provided E is finite-dimensional. Finally, if f is quasi-differentiable, then it is Gateaux differentiable and its Gateaux derivative is equal to its quasi-derivative.
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https://en.wikipedia.org/wiki/Quasi-derivative
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In mathematics, the quasi-dihedral groups, also called semi-dihedral groups, are certain non-abelian groups of order a power of 2. For every positive integer n greater than or equal to 4, there are exactly four isomorphism classes of non-abelian groups of order 2n which have a cyclic subgroup of index 2. Two are well known, the generalized quaternion group and the dihedral group.
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https://en.wikipedia.org/wiki/Quasi-dihedral_group
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One of the remaining two groups is often considered particularly important, since it is an example of a 2-group of maximal nilpotency class. In Bertram Huppert's text Endliche Gruppen, this group is called a "Quasidiedergruppe". In Daniel Gorenstein's text, Finite Groups, this group is called the "semidihedral group".
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https://en.wikipedia.org/wiki/Quasi-dihedral_group
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Dummit and Foote refer to it as the "quasidihedral group"; we adopt that name in this article. All give the same presentation for this group: ⟨ r , s ∣ r 2 n − 1 = s 2 = 1 , s r s = r 2 n − 2 − 1 ⟩ {\displaystyle \langle r,s\mid r^{2^{n-1}}=s^{2}=1,\ srs=r^{2^{n-2}-1}\rangle \,\!} .The other non-abelian 2-group with cyclic subgroup of index 2 is not given a special name in either text, but referred to as just G or Mm(2).
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https://en.wikipedia.org/wiki/Quasi-dihedral_group
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When this group has order 16, Dummit and Foote refer to this group as the "modular group of order 16", as its lattice of subgroups is modular. In this article this group will be called the modular maximal-cyclic group of order 2 n {\displaystyle 2^{n}} . Its presentation is: ⟨ r , s ∣ r 2 n − 1 = s 2 = 1 , s r s = r 2 n − 2 + 1 ⟩ {\displaystyle \langle r,s\mid r^{2^{n-1}}=s^{2}=1,\ srs=r^{2^{n-2}+1}\rangle \,\!}
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https://en.wikipedia.org/wiki/Quasi-dihedral_group
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.Both these two groups and the dihedral group are semidirect products of a cyclic group of order 2n−1 with a cyclic group of order 2. Such a non-abelian semidirect product is uniquely determined by an element of order 2 in the group of units of the ring Z / 2 n − 1 Z {\displaystyle \mathbb {Z} /2^{n-1}\mathbb {Z} } and there are precisely three such elements, 2 n − 1 − 1 {\displaystyle 2^{n-1}-1} , 2 n − 2 − 1 {\displaystyle 2^{n-2}-1} , and 2 n − 2 + 1 {\displaystyle 2^{n-2}+1} , corresponding to the dihedral group, the quasidihedral, and the modular maximal-cyclic group. The generalized quaternion group, the dihedral group, and the quasidihedral group of order 2n all have nilpotency class n − 1, and are the only isomorphism classes of groups of order 2n with nilpotency class n − 1.
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https://en.wikipedia.org/wiki/Quasi-dihedral_group
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The groups of order pn and nilpotency class n − 1 were the beginning of the classification of all p-groups via coclass. The modular maximal-cyclic group of order 2n always has nilpotency class 2. This makes the modular maximal-cyclic group less interesting, since most groups of order pn for large n have nilpotency class 2 and have proven difficult to understand directly. The generalized quaternion, the dihedral, and the quasidihedral group are the only 2-groups whose derived subgroup has index 4. The Alperin–Brauer–Gorenstein theorem classifies the simple groups, and to a degree the finite groups, with quasidihedral Sylow 2-subgroups.
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https://en.wikipedia.org/wiki/Quasi-dihedral_group
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In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space, or, equivalently, as the quotient of two vectors. Multiplication of quaternions is noncommutative.
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https://en.wikipedia.org/wiki/Norm_of_a_quaternion
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Quaternions are generally represented in the form where a, b, c, and d are real numbers; and 1, i, j, and k are the basis vectors or basis elements.Quaternions are used in pure mathematics, but also have practical uses in applied mathematics, particularly for calculations involving three-dimensional rotations, such as in three-dimensional computer graphics, computer vision, and crystallographic texture analysis. They can be used alongside other methods of rotation, such as Euler angles and rotation matrices, or as an alternative to them, depending on the application.
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https://en.wikipedia.org/wiki/Norm_of_a_quaternion
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In modern mathematical language, quaternions form a four-dimensional associative normed division algebra over the real numbers, and therefore a ring, being both a division ring and a domain. The algebra of quaternions is often denoted by H (for Hamilton), or in blackboard bold by H . {\displaystyle \mathbb {H} .}
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https://en.wikipedia.org/wiki/Norm_of_a_quaternion
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It can also be given by the Clifford algebra classifications Cl 0 , 2 ( R ) ≅ Cl 3 , 0 + ( R ) . {\displaystyle \operatorname {Cl} _{0,2}(\mathbb {R} )\cong \operatorname {Cl} _{3,0}^{+}(\mathbb {R} ).} In fact, it was the first noncommutative division algebra to be discovered.
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https://en.wikipedia.org/wiki/Norm_of_a_quaternion
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According to the Frobenius theorem, the algebra H {\displaystyle \mathbb {H} } is one of only two finite-dimensional division rings containing a proper subring isomorphic to the real numbers; the other being the complex numbers. These rings are also Euclidean Hurwitz algebras, of which the quaternions are the largest associative algebra (and hence the largest ring). Further extending the quaternions yields the non-associative octonions, which is the last normed division algebra over the real numbers. (The sedenions, the extension of the octonions, have zero divisors and so cannot be a normed division algebra. )The unit quaternions can be thought of as a choice of a group structure on the 3-sphere S3 that gives the group Spin(3), which is isomorphic to SU(2) and also to the universal cover of SO(3).
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https://en.wikipedia.org/wiki/Norm_of_a_quaternion
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In mathematics, the question of whether the Fourier series of a periodic function converges to a given function is researched by a field known as classical harmonic analysis, a branch of pure mathematics. Convergence is not necessarily given in the general case, and certain criteria must be met for convergence to occur. Determination of convergence requires the comprehension of pointwise convergence, uniform convergence, absolute convergence, Lp spaces, summability methods and the Cesàro mean.
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https://en.wikipedia.org/wiki/Convergence_of_Fourier_series
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In mathematics, the quotient (also called Serre quotient or Gabriel quotient) of an abelian category A {\displaystyle {\mathcal {A}}} by a Serre subcategory B {\displaystyle {\mathcal {B}}} is the abelian category A / B {\displaystyle {\mathcal {A}}/{\mathcal {B}}} which, intuitively, is obtained from A {\displaystyle {\mathcal {A}}} by ignoring (i.e. treating as zero) all objects from B {\displaystyle {\mathcal {B}}} . There is a canonical exact functor Q: A → A / B {\displaystyle Q\colon {\mathcal {A}}\to {\mathcal {A}}/{\mathcal {B}}} whose kernel is B {\displaystyle {\mathcal {B}}} , and A / B {\displaystyle {\mathcal {A}}/{\mathcal {B}}} is in a certain sense the most general abelian category with this property. Forming Serre quotients of abelian categories is thus formally akin to forming quotients of groups. Serre quotients are somewhat similar to quotient categories, the difference being that with Serre quotients all involved categories are abelian and all functors are exact. Serre quotients also often have the character of localizations of categories, especially if the Serre subcategory is localizing.
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https://en.wikipedia.org/wiki/Quotient_of_an_abelian_category
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In mathematics, the quotient of subspace theorem is an important property of finite-dimensional normed spaces, discovered by Vitali Milman.Let (X, ||·||) be an N-dimensional normed space. There exist subspaces Z ⊂ Y ⊂ X such that the following holds: The quotient space E = Y / Z is of dimension dim E ≥ c N, where c > 0 is a universal constant. The induced norm || · || on E, defined by ‖ e ‖ = min y ∈ e ‖ y ‖ , e ∈ E , {\displaystyle \|e\|=\min _{y\in e}\|y\|,\quad e\in E,} is uniformly isomorphic to Euclidean. That is, there exists a positive quadratic form ("Euclidean structure") Q on E, such that Q ( e ) K ≤ ‖ e ‖ ≤ K Q ( e ) {\displaystyle {\frac {\sqrt {Q(e)}}{K}}\leq \|e\|\leq K{\sqrt {Q(e)}}} for e ∈ E , {\displaystyle e\in E,} with K > 1 a universal constant.
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https://en.wikipedia.org/wiki/Quotient_of_subspace_theorem
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The statement is relative easy to prove by induction on the dimension of Z (even for Y=Z, X=0, c=1) with a K that depends only on N; the point of the theorem is that K is independent of N. In fact, the constant c can be made arbitrarily close to 1, at the expense of the constant K becoming large. The original proof allowed c ( K ) ≈ 1 − const / log log K . {\displaystyle c(K)\approx 1-{\text{const}}/\log \log K.}
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https://en.wikipedia.org/wiki/Quotient_of_subspace_theorem
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In mathematics, the radical symbol, radical sign, root symbol, radix, or surd is a symbol for the square root or higher-order root of a number. The square root of a number x is written as 11 , {\displaystyle {\sqrt {11}},} while the nth root of x is written as x n . {\displaystyle {\sqrt{x}}.} It is also used for other meanings in more advanced mathematics, such as the radical of an ideal. In linguistics, the symbol is used to denote a root word.
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https://en.wikipedia.org/wiki/Radical_symbol
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In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or ∞ {\displaystyle \infty } . When it is positive, the power series converges absolutely and uniformly on compact sets inside the open disk of radius equal to the radius of convergence, and it is the Taylor series of the analytic function to which it converges. In case of multiple singularities of a function (singularities are those values of the argument for which the function is not defined), the radius of convergence is the shortest or minimum of all the respective distances (which are all non-negative numbers) calculated from the center of the disk of convergence to the respective singularities of the function.
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https://en.wikipedia.org/wiki/Disc_of_convergence
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In mathematics, the ramification theory of valuations studies the set of extensions of a valuation v of a field K to an extension L of K. It is a generalization of the ramification theory of Dedekind domains.The structure of the set of extensions is known better when L/K is Galois.
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https://en.wikipedia.org/wiki/Inertia_group
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In mathematics, the random Fibonacci sequence is a stochastic analogue of the Fibonacci sequence defined by the recurrence relation f n = f n − 1 ± f n − 2 {\displaystyle f_{n}=f_{n-1}\pm f_{n-2}} , where the signs + or − are chosen at random with equal probability 1 2 {\displaystyle {\tfrac {1}{2}}} , independently for different n {\displaystyle n} . By a theorem of Harry Kesten and Hillel Furstenberg, random recurrent sequences of this kind grow at a certain exponential rate, but it is difficult to compute the rate explicitly. In 1999, Divakar Viswanath showed that the growth rate of the random Fibonacci sequence is equal to 1.1319882487943... (sequence A078416 in the OEIS), a mathematical constant that was later named Viswanath's constant.
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https://en.wikipedia.org/wiki/Embree–Trefethen_constant
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In mathematics, the range of a function may refer to either of two closely related concepts: The codomain of the function The image of the functionGiven two sets X and Y, a binary relation f between X and Y is a (total) function (from X to Y) if for every x in X there is exactly one y in Y such that f relates x to y. The sets X and Y are called domain and codomain of f, respectively. The image of f is then the subset of Y consisting of only those elements y of Y such that there is at least one x in X with f(x) = y.
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https://en.wikipedia.org/wiki/Range_of_a_function
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In mathematics, the rank of a differentiable map f: M → N {\displaystyle f:M\to N} between differentiable manifolds at a point p ∈ M {\displaystyle p\in M} is the rank of the derivative of f {\displaystyle f} at p {\displaystyle p} . Recall that the derivative of f {\displaystyle f} at p {\displaystyle p} is a linear map d p f: T p M → T f ( p ) N {\displaystyle d_{p}f:T_{p}M\to T_{f(p)}N\,} from the tangent space at p to the tangent space at f(p). As a linear map between vector spaces it has a well-defined rank, which is just the dimension of the image in Tf(p)N: rank ( f ) p = dim ( im ( d p f ) ) . {\displaystyle \operatorname {rank} (f)_{p}=\dim(\operatorname {im} (d_{p}f)).}
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https://en.wikipedia.org/wiki/Rank_(differential_topology)
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In mathematics, the rank of an elliptic curve is the rational Mordell–Weil rank of an elliptic curve E {\displaystyle E} defined over the field of rational numbers. Mordell's theorem says the group of rational points on an elliptic curve has a finite basis. This means that for any elliptic curve there is a finite subset of the rational points on the curve, from which all further rational points may be generated. If the number of rational points on a curve is infinite then some point in a finite basis must have infinite order.
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https://en.wikipedia.org/wiki/Average_rank_of_elliptic_curves
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The number of independent basis points with infinite order is the rank of the curve. The rank is related to several outstanding problems in number theory, most notably the Birch–Swinnerton-Dyer conjecture.
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https://en.wikipedia.org/wiki/Average_rank_of_elliptic_curves
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It is widely believed that there is no maximum rank for an elliptic curve, and it has been shown that there exist curves with rank as large as 28, but it is widely believed that such curves are rare. Indeed, Goldfeld and later Katz–Sarnak conjectured that in a suitable asymptotic sense (see below), the rank of elliptic curves should be 1/2 on average. In other words, half of all elliptic curves should have rank 0 (meaning that the infinite part of its Mordell–Weil group is trivial) and the other half should have rank 1; all remaining ranks consist of a total of 0% of all elliptic curves.
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https://en.wikipedia.org/wiki/Average_rank_of_elliptic_curves
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In mathematics, the rank, Prüfer rank, or torsion-free rank of an abelian group A is the cardinality of a maximal linearly independent subset. The rank of A determines the size of the largest free abelian group contained in A. If A is torsion-free then it embeds into a vector space over the rational numbers of dimension rank A. For finitely generated abelian groups, rank is a strong invariant and every such group is determined up to isomorphism by its rank and torsion subgroup. Torsion-free abelian groups of rank 1 have been completely classified. However, the theory of abelian groups of higher rank is more involved. The term rank has a different meaning in the context of elementary abelian groups.
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https://en.wikipedia.org/wiki/Torsion-free_rank
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In mathematics, the ratio test is a test (or "criterion") for the convergence of a series ∑ n = 1 ∞ a n , {\displaystyle \sum _{n=1}^{\infty }a_{n},} where each term is a real or complex number and an is nonzero when n is large. The test was first published by Jean le Rond d'Alembert and is sometimes known as d'Alembert's ratio test or as the Cauchy ratio test.
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https://en.wikipedia.org/wiki/Ratio_test
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In mathematics, the rational normal curve is a smooth, rational curve C of degree n in projective n-space Pn. It is a simple example of a projective variety; formally, it is the Veronese variety when the domain is the projective line. For n = 2 it is the plane conic Z0Z2 = Z21, and for n = 3 it is the twisted cubic. The term "normal" refers to projective normality, not normal schemes. The intersection of the rational normal curve with an affine space is called the moment curve.
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https://en.wikipedia.org/wiki/Rational_normal_curve
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In mathematics, the rational points on the unit circle are those points (x, y) such that both x and y are rational numbers ("fractions") and satisfy x2 + y2 = 1. The set of such points turns out to be closely related to primitive Pythagorean triples. Consider a primitive right triangle, that is, with integer side lengths a, b, c, with c the hypotenuse, such that the sides have no common factor larger than 1. Then on the unit circle there exists the rational point (a/c, b/c), which, in the complex plane, is just a/c + ib/c, where i is the imaginary unit. Conversely, if (x, y) is a rational point on the unit circle in the 1st quadrant of the coordinate system (i.e. x > 0, y > 0), then there exists a primitive right triangle with sides xc, yc, c, with c being the least common multiple of the denominators of x and y. There is a correspondence between points (a, b) in the x-y plane and points a + ib in the complex plane which is used below.
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https://en.wikipedia.org/wiki/Group_of_rational_points_on_the_unit_circle
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In mathematics, the rational sieve is a general algorithm for factoring integers into prime factors. It is a special case of the general number field sieve. While it is less efficient than the general algorithm, it is conceptually simpler. It serves as a helpful first step in understanding how the general number field sieve works.
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https://en.wikipedia.org/wiki/Rational_sieve
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In mathematics, the real coordinate space of dimension n, denoted Rn or R n {\displaystyle \mathbb {R} ^{n}} , is the set of the n-tuples of real numbers, that is the set of all sequences of n real numbers. Special cases are called the real line R1 and the real coordinate plane R2. With component-wise addition and scalar multiplication, it is a real vector space, and its elements are called coordinate vectors. The coordinates over any basis of the elements of a real vector space form a real coordinate space of the same dimension as that of the vector space.
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https://en.wikipedia.org/wiki/Real_coordinate_plane
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Similarly, the Cartesian coordinates of the points of a Euclidean space of dimension n form a real coordinate space of dimension n. These one to one correspondences between vectors, points and coordinate vectors explain the names of coordinate space and coordinate vector. It allows using geometric terms and methods for studying real coordinate spaces, and, conversely, to use methods of calculus in geometry. This approach of geometry was introduced by René Descartes in the 17th century. It is widely used, as it allows locating points in Euclidean spaces, and computing with them.
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https://en.wikipedia.org/wiki/Real_coordinate_plane
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In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has basic applications to geometry, since the common construction of the real projective plane is as the space of lines in R 3 {\displaystyle \mathbb {R} ^{3}} passing through the origin. The plane is also often described topologically, in terms of a construction based on the Möbius strip: if one could glue the (single) edge of the Möbius strip to itself in the correct direction, one would obtain the projective plane.
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https://en.wikipedia.org/wiki/Real_projective_plane
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(This cannot be done in three-dimensional space without the surface intersecting itself.) Equivalently, gluing a disk along the boundary of the Möbius strip gives the projective plane. Topologically, it has Euler characteristic 1, hence a demigenus (non-orientable genus, Euler genus) of 1. Since the Möbius strip, in turn, can be constructed from a square by gluing two of its sides together with a half-twist, the real projective plane can thus be represented as a unit square (that is, × ) with its sides identified by the following equivalence relations: (0, y) ~ (1, 1 − y) for 0 ≤ y ≤ 1and (x, 0) ~ (1 − x, 1) for 0 ≤ x ≤ 1,as in the leftmost diagram shown here.
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https://en.wikipedia.org/wiki/Real_projective_plane
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In mathematics, the real rank of a C*-algebra is a noncommutative analogue of Lebesgue covering dimension. The notion was first introduced by Lawrence G. Brown and Gert K. Pedersen.
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https://en.wikipedia.org/wiki/Real_rank_(C*-algebras)
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In mathematics, the rearrangement inequality states that for every choice of real numbers and every permutation y σ ( 1 ) , … , y σ ( n ) {\displaystyle y_{\sigma (1)},\ldots ,y_{\sigma (n)}} of y 1 , … , y n . {\displaystyle y_{1},\ldots ,y_{n}.} If the numbers x 1 , … , x n {\displaystyle x_{1},\ldots ,x_{n}} are different, meaning that x 1 < ⋯ < x n , {\displaystyle x_{1}<\cdots
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https://en.wikipedia.org/wiki/Rearrangement_inequality
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In mathematics, the reciprocal gamma function is the function f ( z ) = 1 Γ ( z ) , {\displaystyle f(z)={\frac {1}{\Gamma (z)}},} where Γ(z) denotes the gamma function. Since the gamma function is meromorphic and nonzero everywhere in the complex plane, its reciprocal is an entire function. As an entire function, it is of order 1 (meaning that log log |1/Γ(z)| grows no faster than log |z|), but of infinite type (meaning that log |1/Γ(z)| grows faster than any multiple of |z|, since its growth is approximately proportional to |z| log |z| in the left-half plane). The reciprocal is sometimes used as a starting point for numerical computation of the gamma function, and a few software libraries provide it separately from the regular gamma function. Karl Weierstrass called the reciprocal gamma function the "factorielle" and used it in his development of the Weierstrass factorization theorem.
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https://en.wikipedia.org/wiki/Reciprocal_Gamma_function
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In mathematics, the reduced derivative is a generalization of the notion of derivative that is well-suited to the study of functions of bounded variation. Although functions of bounded variation have derivatives in the sense of Radon measures, it is desirable to have a derivative that takes values in the same space as the functions themselves. Although the precise definition of the reduced derivative is quite involved, its key properties are quite easy to remember: it is a multiple of the usual derivative wherever it exists; at jump points, it is a multiple of the jump vector.The notion of reduced derivative appears to have been introduced by Alexander Mielke and Florian Theil in 2004.
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https://en.wikipedia.org/wiki/Reduced_derivative
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In mathematics, the reflexive closure of a binary relation R {\displaystyle R} on a set X {\displaystyle X} is the smallest reflexive relation on X {\displaystyle X} that contains R . {\displaystyle R.} A relation is called reflexive if it relates every element of X {\displaystyle X} to itself. For example, if X {\displaystyle X} is a set of distinct numbers and x R y {\displaystyle xRy} means " x {\displaystyle x} is less than y {\displaystyle y} ", then the reflexive closure of R {\displaystyle R} is the relation " x {\displaystyle x} is less than or equal to y {\displaystyle y} ".
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https://en.wikipedia.org/wiki/Reflexive_closure
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In mathematics, the regula falsi, method of false position, or false position method is a very old method for solving an equation with one unknown; this method, in modified form, is still in use. In simple terms, the method is the trial and error technique of using test ("false") values for the variable and then adjusting the test value according to the outcome. This is sometimes also referred to as "guess and check". Versions of the method predate the advent of algebra and the use of equations.
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https://en.wikipedia.org/wiki/False_position
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As an example, consider problem 26 in the Rhind papyrus, which asks for a solution of (written in modern notation) the equation x + x/4 = 15. This is solved by false position. First, guess that x = 4 to obtain, on the left, 4 + 4/4 = 5.
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https://en.wikipedia.org/wiki/False_position
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This guess is a good choice since it produces an integer value. However, 4 is not the solution of the original equation, as it gives a value which is three times too small. To compensate, multiply x (currently set to 4) by 3 and substitute again to get 12 + 12/4 = 15, verifying that the solution is x = 12. Modern versions of the technique employ systematic ways of choosing new test values and are concerned with the questions of whether or not an approximation to a solution can be obtained, and if it can, how fast can the approximation be found.
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https://en.wikipedia.org/wiki/False_position
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In mathematics, the regular part of a Laurent series consists of the series of terms with positive powers. That is, if f ( z ) = ∑ n = − ∞ ∞ a n ( z − c ) n , {\displaystyle f(z)=\sum _{n=-\infty }^{\infty }a_{n}(z-c)^{n},} then the regular part of this Laurent series is ∑ n = 0 ∞ a n ( z − c ) n . {\displaystyle \sum _{n=0}^{\infty }a_{n}(z-c)^{n}.} In contrast, the series of terms with negative powers is the principal part. == References ==
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https://en.wikipedia.org/wiki/Regular_part
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In mathematics, the regulated integral is a definition of integration for regulated functions, which are defined to be uniform limits of step functions. The use of the regulated integral instead of the Riemann integral has been advocated by Nicolas Bourbaki and Jean Dieudonné.
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https://en.wikipedia.org/wiki/Regulated_integral
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In mathematics, the relative interior of a set is a refinement of the concept of the interior, which is often more useful when dealing with low-dimensional sets placed in higher-dimensional spaces. Formally, the relative interior of a set S {\displaystyle S} (denoted relint ( S ) {\displaystyle \operatorname {relint} (S)} ) is defined as its interior within the affine hull of S . {\displaystyle S.} In other words, where aff ( S ) {\displaystyle \operatorname {aff} (S)} is the affine hull of S , {\displaystyle S,} and N ϵ ( x ) {\displaystyle N_{\epsilon }(x)} is a ball of radius ϵ {\displaystyle \epsilon } centered on x {\displaystyle x} .
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https://en.wikipedia.org/wiki/Relative_interior
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Any metric can be used for the construction of the ball; all metrics define the same set as the relative interior. A set is relatively open iff it is equal to its relative interior. Note that when aff ( S ) {\displaystyle \operatorname {aff} (S)} is a closed subspace of the full vector space (always the case when the full vector space is finite dimensional) then being relatively closed is equivalent to being closed. For any convex set C ⊆ R n {\displaystyle C\subseteq \mathbb {R} ^{n}} the relative interior is equivalently defined as where x ∈ ( y , z ) {\displaystyle x\in (y,z)} means that there exists some 0 < λ < 1 {\displaystyle 0<\lambda <1} such that x = λ z + ( 1 − λ ) y {\displaystyle x=\lambda z+(1-\lambda )y} .
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https://en.wikipedia.org/wiki/Relative_interior
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In mathematics, the relaxation of a (mixed) integer linear program is the problem that arises by removing the integrality constraint of each variable. For example, in a 0–1 integer program, all constraints are of the form x i ∈ { 0 , 1 } {\displaystyle x_{i}\in \{0,1\}} .The relaxation of the original integer program instead uses a collection of linear constraints 0 ≤ x i ≤ 1. {\displaystyle 0\leq x_{i}\leq 1.} The resulting relaxation is a linear program, hence the name. This relaxation technique transforms an NP-hard optimization problem (integer programming) into a related problem that is solvable in polynomial time (linear programming); the solution to the relaxed linear program can be used to gain information about the solution to the original integer program.
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https://en.wikipedia.org/wiki/Linear_programming_relaxation
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In mathematics, the remainder is the amount "left over" after performing some computation. In arithmetic, the remainder is the integer "left over" after dividing one integer by another to produce an integer quotient (integer division). In algebra of polynomials, the remainder is the polynomial "left over" after dividing one polynomial by another.
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https://en.wikipedia.org/wiki/Remainder
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The modulo operation is the operation that produces such a remainder when given a dividend and divisor. Alternatively, a remainder is also what is left after subtracting one number from another, although this is more precisely called the difference. This usage can be found in some elementary textbooks; colloquially it is replaced by the expression "the rest" as in "Give me two dollars back and keep the rest." However, the term "remainder" is still used in this sense when a function is approximated by a series expansion, where the error expression ("the rest") is referred to as the remainder term.
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https://en.wikipedia.org/wiki/Remainder
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In mathematics, the replicator equation is a deterministic monotone non-linear and non-innovative game dynamic used in evolutionary game theory. The replicator equation differs from other equations used to model replication, such as the quasispecies equation, in that it allows the fitness function to incorporate the distribution of the population types rather than setting the fitness of a particular type constant. This important property allows the replicator equation to capture the essence of selection. Unlike the quasispecies equation, the replicator equation does not incorporate mutation and so is not able to innovate new types or pure strategies.
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https://en.wikipedia.org/wiki/Replicator_equation
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In mathematics, the representation theory of semisimple Lie algebras is one of the crowning achievements of the theory of Lie groups and Lie algebras. The theory was worked out mainly by E. Cartan and H. Weyl and because of that, the theory is also known as the Cartan–Weyl theory. The theory gives the structural description and classification of a finite-dimensional representation of a semisimple Lie algebra (over C {\displaystyle \mathbb {C} } ); in particular, it gives a way to parametrize (or classify) irreducible finite-dimensional representations of a semisimple Lie algebra, the result known as the theorem of the highest weight.
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https://en.wikipedia.org/wiki/Representation_theory_of_semisimple_Lie_algebras
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There is a natural one-to-one correspondence between the finite-dimensional representations of a simply connected compact Lie group K and the finite-dimensional representations of the complex semisimple Lie algebra g {\displaystyle {\mathfrak {g}}} that is the complexification of the Lie algebra of K (this fact is essentially a special case of the Lie group–Lie algebra correspondence). Also, finite-dimensional representations of a connected compact Lie group can be studied through finite-dimensional representations of the universal cover of such a group. Hence, the representation theory of semisimple Lie algebras marks the starting point for the general theory of representations of connected compact Lie groups. The theory is a basis for the later works of Harish-Chandra that concern (infinite-dimensional) representation theory of real reductive groups.
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https://en.wikipedia.org/wiki/Representation_theory_of_semisimple_Lie_algebras
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In mathematics, the representation theory of the Poincaré group is an example of the representation theory of a Lie group that is neither a compact group nor a semisimple group. It is fundamental in theoretical physics. In a physical theory having Minkowski space as the underlying spacetime, the space of physical states is typically a representation of the Poincaré group. (More generally, it may be a projective representation, which amounts to a representation of the double cover of the group.)
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https://en.wikipedia.org/wiki/Representation_theory_of_the_Poincaré_group
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In a classical field theory, the physical states are sections of a Poincaré-equivariant vector bundle over Minkowski space. The equivariance condition means that the group acts on the total space of the vector bundle, and the projection to Minkowski space is an equivariant map. Therefore, the Poincaré group also acts on the space of sections.
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https://en.wikipedia.org/wiki/Representation_theory_of_the_Poincaré_group
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Representations arising in this way (and their subquotients) are called covariant field representations, and are not usually unitary. For a discussion of such unitary representations, see Wigner's classification. In quantum mechanics, the state of the system is determined by the Schrödinger equation, which is invariant under Galilean transformations.
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https://en.wikipedia.org/wiki/Representation_theory_of_the_Poincaré_group
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Quantum field theory is the relativistic extension of quantum mechanics, where relativistic (Lorentz/Poincaré invariant) wave equations are solved, "quantized", and act on a Hilbert space composed of Fock states. There are no finite unitary representations of the full Lorentz (and thus Poincaré) transformations due to the non-compact nature of Lorentz boosts (rotations in Minkowski space along a space and time axis). However, there are finite non-unitary indecomposable representations of the Poincaré algebra, which may be used for modelling of unstable particles.In case of spin 1/2 particles, it is possible to find a construction that includes both a finite-dimensional representation and a scalar product preserved by this representation by associating a 4-component Dirac spinor ψ {\displaystyle \psi } with each particle.
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https://en.wikipedia.org/wiki/Representation_theory_of_the_Poincaré_group
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These spinors transform under Lorentz transformations generated by the gamma matrices ( γ μ {\displaystyle \gamma _{\mu }} ). It can be shown that the scalar product ⟨ ψ | ϕ ⟩ = ψ ¯ ϕ = ψ † γ 0 ϕ {\displaystyle \langle \psi |\phi \rangle ={\bar {\psi }}\phi =\psi ^{\dagger }\gamma _{0}\phi } is preserved. It is not, however, positive definite, so the representation is not unitary.
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https://en.wikipedia.org/wiki/Representation_theory_of_the_Poincaré_group
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In mathematics, the representation theory of the symmetric group is a particular case of the representation theory of finite groups, for which a concrete and detailed theory can be obtained. This has a large area of potential applications, from symmetric function theory to quantum chemistry studies of atoms, molecules and solids.The symmetric group Sn has order n!. Its conjugacy classes are labeled by partitions of n. Therefore according to the representation theory of a finite group, the number of inequivalent irreducible representations, over the complex numbers, is equal to the number of partitions of n. Unlike the general situation for finite groups, there is in fact a natural way to parametrize irreducible representations by the same set that parametrizes conjugacy classes, namely by partitions of n or equivalently Young diagrams of size n. Each such irreducible representation can in fact be realized over the integers (every permutation acting by a matrix with integer coefficients); it can be explicitly constructed by computing the Young symmetrizers acting on a space generated by the Young tableaux of shape given by the Young diagram. The dimension d λ {\displaystyle d_{\lambda }} of the representation that corresponds to the Young diagram λ {\displaystyle \lambda } is given by the hook length formula.
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https://en.wikipedia.org/wiki/Representation_theory_of_symmetric_groups
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To each irreducible representation ρ we can associate an irreducible character, χρ. To compute χρ(π) where π is a permutation, one can use the combinatorial Murnaghan–Nakayama rule . Note that χρ is constant on conjugacy classes, that is, χρ(π) = χρ(σ−1πσ) for all permutations σ. Over other fields the situation can become much more complicated.
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https://en.wikipedia.org/wiki/Representation_theory_of_symmetric_groups
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If the field K has characteristic equal to zero or greater than n then by Maschke's theorem the group algebra KSn is semisimple. In these cases the irreducible representations defined over the integers give the complete set of irreducible representations (after reduction modulo the characteristic if necessary). However, the irreducible representations of the symmetric group are not known in arbitrary characteristic.
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https://en.wikipedia.org/wiki/Representation_theory_of_symmetric_groups
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In this context it is more usual to use the language of modules rather than representations. The representation obtained from an irreducible representation defined over the integers by reducing modulo the characteristic will not in general be irreducible.
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https://en.wikipedia.org/wiki/Representation_theory_of_symmetric_groups
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The modules so constructed are called Specht modules, and every irreducible does arise inside some such module. There are now fewer irreducibles, and although they can be classified they are very poorly understood. For example, even their dimensions are not known in general. The determination of the irreducible modules for the symmetric group over an arbitrary field is widely regarded as one of the most important open problems in representation theory.
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https://en.wikipedia.org/wiki/Representation_theory_of_symmetric_groups
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In mathematics, the residue field is a basic construction in commutative algebra. If R is a commutative ring and m is a maximal ideal, then the residue field is the quotient ring k = R/m, which is a field. Frequently, R is a local ring and m is then its unique maximal ideal. This construction is applied in algebraic geometry, where to every point x of a scheme X one associates its residue field k(x). One can say a little loosely that the residue field of a point of an abstract algebraic variety is the 'natural domain' for the coordinates of the point.
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https://en.wikipedia.org/wiki/Residue_field
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In mathematics, the residue formula says that the sum of the residues of a meromorphic differential form on a smooth proper algebraic curve vanishes.
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https://en.wikipedia.org/wiki/Sum_of_residues_formula
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In mathematics, the resolvent formalism is a technique for applying concepts from complex analysis to the study of the spectrum of operators on Banach spaces and more general spaces. Formal justification for the manipulations can be found in the framework of holomorphic functional calculus. The resolvent captures the spectral properties of an operator in the analytic structure of the functional.
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https://en.wikipedia.org/wiki/Resolvent_formalism
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Given an operator A, the resolvent may be defined as R ( z ; A ) = ( A − z I ) − 1 . {\displaystyle R(z;A)=(A-zI)^{-1}~.} Among other uses, the resolvent may be used to solve the inhomogeneous Fredholm integral equations; a commonly used approach is a series solution, the Liouville–Neumann series. The resolvent of A can be used to directly obtain information about the spectral decomposition of A. For example, suppose λ is an isolated eigenvalue in the spectrum of A. That is, suppose there exists a simple closed curve C λ {\displaystyle C_{\lambda }} in the complex plane that separates λ from the rest of the spectrum of A. Then the residue − 1 2 π i ∮ C λ ( A − z I ) − 1 d z {\displaystyle -{\frac {1}{2\pi i}}\oint _{C_{\lambda }}(A-zI)^{-1}~dz} defines a projection operator onto the λ eigenspace of A. The Hille–Yosida theorem relates the resolvent through a Laplace transform to an integral over the one-parameter group of transformations generated by A. Thus, for example, if A is a Hermitian, then U(t) = exp(tA) is a one-parameter group of unitary operators. Whenever | z | > ‖ A ‖ {\displaystyle |z|>\|A\|} , the resolvent of A at z can be expressed as the Laplace transform R ( z ; A ) = ∫ 0 ∞ e − z t U ( t ) d t , {\displaystyle R(z;A)=\int _{0}^{\infty }e^{-zt}U(t)~dt,} where the integral is taken along the ray arg t = − arg λ {\displaystyle \arg t=-\arg \lambda } .
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https://en.wikipedia.org/wiki/Resolvent_formalism
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In mathematics, the restricted product is a construction in the theory of topological groups. Let I {\displaystyle I} be an index set; S {\displaystyle S} a finite subset of I {\displaystyle I} . If G i {\displaystyle G_{i}} is a locally compact group for each i ∈ I {\displaystyle i\in I} , and K i ⊂ G i {\displaystyle K_{i}\subset G_{i}} is an open compact subgroup for each i ∈ I ∖ S {\displaystyle i\in I\setminus S} , then the restricted product ∏ i ′ G i {\displaystyle \prod _{i}\nolimits 'G_{i}\,} is the subset of the product of the G i {\displaystyle G_{i}} 's consisting of all elements ( g i ) i ∈ I {\displaystyle (g_{i})_{i\in I}} such that g i ∈ K i {\displaystyle g_{i}\in K_{i}} for all but finitely many i ∈ I ∖ S {\displaystyle i\in I\setminus S} .
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https://en.wikipedia.org/wiki/Restricted_product
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This group is given the topology whose basis of open sets are those of the form ∏ i A i , {\displaystyle \prod _{i}A_{i}\,,} where A i {\displaystyle A_{i}} is open in G i {\displaystyle G_{i}} and A i = K i {\displaystyle A_{i}=K_{i}} for all but finitely many i {\displaystyle i} . One can easily prove that the restricted product is itself a locally compact group. The best known example of this construction is that of the adele ring and idele group of a global field.
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https://en.wikipedia.org/wiki/Restricted_product
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In mathematics, the restriction of a function f {\displaystyle f} is a new function, denoted f | A {\displaystyle f\vert _{A}} or f ↾ A , {\displaystyle f{\upharpoonright _{A}},} obtained by choosing a smaller domain A {\displaystyle A} for the original function f . {\displaystyle f.} The function f {\displaystyle f} is then said to extend f | A . {\displaystyle f\vert _{A}.}
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https://en.wikipedia.org/wiki/Restricted_function
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In mathematics, the result of the modulo operation is an equivalence class, and any member of the class may be chosen as representative; however, the usual representative is the least positive residue, the smallest non-negative integer that belongs to that class (i.e., the remainder of the Euclidean division). However, other conventions are possible. Computers and calculators have various ways of storing and representing numbers; thus their definition of the modulo operation depends on the programming language or the underlying hardware. In nearly all computing systems, the quotient q and the remainder r of a divided by n satisfy the following conditions: This still leaves a sign ambiguity if the remainder is non-zero: two possible choices for the remainder occur, one negative and the other positive, and two possible choices for the quotient occur.
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https://en.wikipedia.org/wiki/Modulo_operation
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In number theory, the positive remainder is always chosen, but in computing, programming languages choose depending on the language and the signs of a or n. Standard Pascal and ALGOL 68, for example, give a positive remainder (or 0) even for negative divisors, and some programming languages, such as C90, leave it to the implementation when either of n or a is negative (see the table under § In programming languages for details). a modulo 0 is undefined in most systems, although some do define it as a. As described by Leijen, Boute argues that Euclidean division is superior to the other ones in terms of regularity and useful mathematical properties, although floored division, promoted by Knuth, is also a good definition. Despite its widespread use, truncated division is shown to be inferior to the other definitions. However, truncated division satisfies the identity ( − a ) / b = − ( a / b ) = a / ( − b ) {\displaystyle ({-a})/b={-(a/b)}=a/({-b})} .
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https://en.wikipedia.org/wiki/Modulo_operation
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In mathematics, the resultant of two polynomials is a polynomial expression of their coefficients that is equal to zero if and only if the polynomials have a common root (possibly in a field extension), or, equivalently, a common factor (over their field of coefficients). In some older texts, the resultant is also called the eliminant.The resultant is widely used in number theory, either directly or through the discriminant, which is essentially the resultant of a polynomial and its derivative. The resultant of two polynomials with rational or polynomial coefficients may be computed efficiently on a computer. It is a basic tool of computer algebra, and is a built-in function of most computer algebra systems.
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https://en.wikipedia.org/wiki/Multivariate_resultant
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It is used, among others, for cylindrical algebraic decomposition, integration of rational functions and drawing of curves defined by a bivariate polynomial equation. The resultant of n homogeneous polynomials in n variables (also called multivariate resultant, or Macaulay's resultant for distinguishing it from the usual resultant) is a generalization, introduced by Macaulay, of the usual resultant. It is, with Gröbner bases, one of the main tools of elimination theory.
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https://en.wikipedia.org/wiki/Multivariate_resultant
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In mathematics, the ring of integers of an algebraic number field K {\displaystyle K} is the ring of all algebraic integers contained in K {\displaystyle K} . An algebraic integer is a root of a monic polynomial with integer coefficients: x n + c n − 1 x n − 1 + ⋯ + c 0 {\displaystyle x^{n}+c_{n-1}x^{n-1}+\cdots +c_{0}} . This ring is often denoted by O K {\displaystyle O_{K}} or O K {\displaystyle {\mathcal {O}}_{K}} .
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https://en.wikipedia.org/wiki/Number_ring
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Since any integer belongs to K {\displaystyle K} and is an integral element of K {\displaystyle K} , the ring Z {\displaystyle \mathbb {Z} } is always a subring of O K {\displaystyle O_{K}} . The ring of integers Z {\displaystyle \mathbb {Z} } is the simplest possible ring of integers. Namely, Z = O Q {\displaystyle \mathbb {Z} =O_{\mathbb {Q} }} where Q {\displaystyle \mathbb {Q} } is the field of rational numbers.
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https://en.wikipedia.org/wiki/Number_ring
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And indeed, in algebraic number theory the elements of Z {\displaystyle \mathbb {Z} } are often called the "rational integers" because of this. The next simplest example is the ring of Gaussian integers Z {\displaystyle \mathbb {Z} } , consisting of complex numbers whose real and imaginary parts are integers. It is the ring of integers in the number field Q ( i ) {\displaystyle \mathbb {Q} (i)} of Gaussian rationals, consisting of complex numbers whose real and imaginary parts are rational numbers.
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https://en.wikipedia.org/wiki/Number_ring
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Like the rational integers, Z {\displaystyle \mathbb {Z} } is a Euclidean domain. The ring of integers of an algebraic number field is the unique maximal order in the field. It is always a Dedekind domain.
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https://en.wikipedia.org/wiki/Number_ring
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In mathematics, the ring of modular forms associated to a subgroup Γ of the special linear group SL(2, Z) is the graded ring generated by the modular forms of Γ. The study of rings of modular forms describes the algebraic structure of the space of modular forms.
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https://en.wikipedia.org/wiki/Ring_of_modular_forms
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In mathematics, the ring of polynomial functions on a vector space V over a field k gives a coordinate-free analog of a polynomial ring. It is denoted by k. If V is finite dimensional and is viewed as an algebraic variety, then k is precisely the coordinate ring of V. The explicit definition of the ring can be given as follows. If k {\displaystyle k} is a polynomial ring, then we can view t i {\displaystyle t_{i}} as coordinate functions on k n {\displaystyle k^{n}} ; i.e., t i ( x ) = x i {\displaystyle t_{i}(x)=x_{i}} when x = ( x 1 , … , x n ) . {\displaystyle x=(x_{1},\dots ,x_{n}).}
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https://en.wikipedia.org/wiki/Polynomials_on_vector_spaces
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This suggests the following: given a vector space V, let k be the commutative k-algebra generated by the dual space V ∗ {\displaystyle V^{*}} , which is a subring of the ring of all functions V → k {\displaystyle V\to k} . If we fix a basis for V and write t i {\displaystyle t_{i}} for its dual basis, then k consists of polynomials in t i {\displaystyle t_{i}} . If k is infinite, then k is the symmetric algebra of the dual space V ∗ {\displaystyle V^{*}} .
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https://en.wikipedia.org/wiki/Polynomials_on_vector_spaces
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In applications, one also defines k when V is defined over some subfield of k (e.g., k is the complex field and V is a real vector space.) The same definition still applies. Throughout the article, for simplicity, the base field k is assumed to be infinite.
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https://en.wikipedia.org/wiki/Polynomials_on_vector_spaces
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In mathematics, the root system for the simple Lie group G2 is in the form of a hexagram, with six long roots and six short roots.
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https://en.wikipedia.org/wiki/Hexagram
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In mathematics, the root test is a criterion for the convergence (a convergence test) of an infinite series. It depends on the quantity lim sup n → ∞ | a n | n , {\displaystyle \limsup _{n\rightarrow \infty }{\sqrt{|a_{n}|}},} where a n {\displaystyle a_{n}} are the terms of the series, and states that the series converges absolutely if this quantity is less than one, but diverges if it is greater than one. It is particularly useful in connection with power series.
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https://en.wikipedia.org/wiki/Cauchy's_radical_test
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In mathematics, the rotation number is an invariant of homeomorphisms of the circle.
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https://en.wikipedia.org/wiki/Map_winding_number
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In mathematics, the scalar projection of a vector a {\displaystyle \mathbf {a} } on (or onto) a vector b , {\displaystyle \mathbf {b} ,} also known as the scalar resolute of a {\displaystyle \mathbf {a} } in the direction of b , {\displaystyle \mathbf {b} ,} is given by: s = ‖ a ‖ cos θ = a ⋅ b ^ , {\displaystyle s=\left\|\mathbf {a} \right\|\cos \theta =\mathbf {a} \cdot \mathbf {\hat {b}} ,} where the operator ⋅ {\displaystyle \cdot } denotes a dot product, b ^ {\displaystyle {\hat {\mathbf {b} }}} is the unit vector in the direction of b , {\displaystyle \mathbf {b} ,} ‖ a ‖ {\displaystyle \left\|\mathbf {a} \right\|} is the length of a , {\displaystyle \mathbf {a} ,} and θ {\displaystyle \theta } is the angle between a {\displaystyle \mathbf {a} } and b {\displaystyle \mathbf {b} } . The term scalar component refers sometimes to scalar projection, as, in Cartesian coordinates, the components of a vector are the scalar projections in the directions of the coordinate axes. The scalar projection is a scalar, equal to the length of the orthogonal projection of a {\displaystyle \mathbf {a} } on b {\displaystyle \mathbf {b} } , with a negative sign if the projection has an opposite direction with respect to b {\displaystyle \mathbf {b} } . Multiplying the scalar projection of a {\displaystyle \mathbf {a} } on b {\displaystyle \mathbf {b} } by b ^ {\displaystyle \mathbf {\hat {b}} } converts it into the above-mentioned orthogonal projection, also called vector projection of a {\displaystyle \mathbf {a} } on b {\displaystyle \mathbf {b} } .
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https://en.wikipedia.org/wiki/Scalar_projection
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In mathematics, the scale convolution of two functions s ( t ) {\displaystyle s(t)} and r ( t ) {\displaystyle r(t)} , also known as their logarithmic convolution is defined as the function s ∗ l r ( t ) = r ∗ l s ( t ) = ∫ 0 ∞ s ( t a ) r ( a ) d a a {\displaystyle s*_{l}r(t)=r*_{l}s(t)=\int _{0}^{\infty }s\left({\frac {t}{a}}\right)r(a)\,{\frac {da}{a}}} when this quantity exists.
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https://en.wikipedia.org/wiki/Logarithmic_convolution
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In mathematics, the second moment method is a technique used in probability theory and analysis to show that a random variable has positive probability of being positive. More generally, the "moment method" consists of bounding the probability that a random variable fluctuates far from its mean, by using its moments.The method is often quantitative, in that one can often deduce a lower bound on the probability that the random variable is larger than some constant times its expectation. The method involves comparing the second moment of random variables to the square of the first moment.
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https://en.wikipedia.org/wiki/Second_moment_method
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In mathematics, the second partial derivative test is a method in multivariable calculus used to determine if a critical point of a function is a local minimum, maximum or saddle point.
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https://en.wikipedia.org/wiki/Second_partial_derivative_test
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In mathematics, the secondary measure associated with a measure of positive density ρ when there is one, is a measure of positive density μ, turning the secondary polynomials associated with the orthogonal polynomials for ρ into an orthogonal system.
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https://en.wikipedia.org/wiki/Secondary_measure
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