text stringlengths 14 4.79k | source stringlengths 13 304 | tokens float64 75 1.06k β | char_length float64 106 4.79k β | article_title stringlengths 16 300 β |
|---|---|---|---|---|
This is known as the theorema egregium, and was a major discovery of Carl Friedrich Gauss. It is particularly striking when one recalls the geometric definition of the Gaussian curvature of S as being defined by the maximum and minimum radii of osculating circles; they seem to be fundamentally defined by the geometry o... | Wikipedia - Differentiable surface | null | null | null |
As said by Marcel Berger: This theorem is baffling. It is the kind of theorem which could have waited dozens of years more before being discovered by another mathematician since, unlike so much of intellectual history, it was absolutely not in the air. To our knowledge there is no simple geometric proof of the theorema... | Wikipedia - Differentiable surface | null | null | null |
The Gauss-Codazzi equations can also be succinctly expressed and derived in the language of connection forms due to Γlie Cartan. In the language of tensor calculus, making use of natural metrics and connections on tensor bundles, the Gauss equation can be written as H2 β |h|2 = R and the two Codazzi equations can be wr... | Wikipedia - Differentiable surface | null | null | null |
Let S be a set of n points in the plane. If no more than n β k points lie on any line for some 0 β€ k < n β 2, then there exist Ξ©(nk) lines determined by the points of S. | Wikipedia - Beck's theorem (geometry) | null | null | null |
Let S be a statement of the form P implies Q (P β Q). Then the converse of S is the statement Q implies P (Q β P). In general, the truth of S says nothing about the truth of its converse, unless the antecedent P and the consequent Q are logically equivalent. For example, consider the true statement "If I am a human, th... | Wikipedia - Converse (logic) | null | null | null |
The converse of that statement is "If I am mortal, then I am a human," which is not necessarily true. On the other hand, the converse of a statement with mutually inclusive terms remains true, given the truth of the original proposition. This is equivalent to saying that the converse of a definition is true. | Wikipedia - Converse (logic) | null | null | null |
Thus, the statement "If I am a triangle, then I am a three-sided polygon" is logically equivalent to "If I am a three-sided polygon, then I am a triangle", because the definition of "triangle" is "three-sided polygon". A truth table makes it clear that S and the converse of S are not logically equivalent, unless both t... | Wikipedia - Converse (logic) | null | null | null |
Let S be a surface in three-dimensional Euclidean space that is parametrized by the position vector r(u, v). Let P = P(u, v) be a point on the surface. Then r u = β r β u , r v = β r β v {\displaystyle \mathbf {r} _{u}={\frac {\partial \mathbf {r} }{\partial u}},\quad \mathbf {r} _{v}={\frac {\partial \mathbf {r} }{\pa... | Wikipedia - Weingarten equations | null | null | null |
Let S be any finite set, f be any function from S to itself, and x0 be any element of S. For any i > 0, let xi = f(xi β 1). Let ΞΌ be the smallest index such that the value xΞΌ reappears infinitely often within the sequence of values xi, and let Ξ» (the loop length) be the smallest positive integer such that xΞΌ = xΞ» + ΞΌ. ... | Wikipedia - Cycle detection | null | null | null |
Let S be subjective time, R be real time, and define both to be zero at birth. One model proposes that the passage of subjective time relative to actual time is inversely proportional to real time: d S d R = K R {\displaystyle {\frac {dS}{dR}}={\frac {K}{R}}} When solved, S 2 β S 1 = K ( log β‘ R 2 β log β‘ R 1 ) = K log... | Wikipedia - Perception of time | null | null | null |
So a year would be experienced by a 55-year-old as passing approximately 5 times more quickly than a year experienced by an 11-year-old. If long-term time perception is based solely on the proportionality of a person's age, then the following four periods in life would appear to be quantitatively equal: ages 5β10 (1x),... | Wikipedia - Perception of time | null | null | null |
Let S be the subsemigroup of N generated by A. Then S is a numerical semigroup if and only if gcd (A) = 1. Moreover, every numerical semigroup arises in this way. | Wikipedia - Numerical semigroup | null | null | null |
Let S deg {\displaystyle S_{\text{deg}}} be a degenerate Sklyanin algebra. S deg {\displaystyle S_{\text{deg}}} contains non-zero zero divisors. The Hilbert series of S deg {\displaystyle S_{\text{deg}}} is H S deg = 1 + t 1 β 2 t {\displaystyle H_{S_{\text{deg}}}={\frac {1+t}{1-2t}}} . Degenerate Sklyanin algebras hav... | Wikipedia - Sklyanin algebra | null | null | null |
S deg {\displaystyle S_{\text{deg}}} is neither left nor right Noetherian. S deg {\displaystyle S_{\text{deg}}} is a Koszul algebra. Degenerate Sklyanin algebras have infinite global dimension. | Wikipedia - Sklyanin algebra | null | null | null |
Let S {\displaystyle S} be a non-degenerate Sklyanin algebra. S {\displaystyle S} contains no non-zero zero divisors. The hilbert series of S {\displaystyle S} is H S = 1 ( 1 β t ) 3 {\displaystyle H_{S}={\frac {1}{(1-t)^{3}}}} . Non-degenerate Sklyanin algebras are Noetherian. | Wikipedia - Sklyanin algebra | null | null | null |
S {\displaystyle S} is Koszul. Non-degenerate Sklyanin algebras are Artin-Schelter regular. Therefore, they have global dimension 3 and GelfandβKirillov dimension 3. There exists a normal central element in every non-degenerate Sklyanin algebra. | Wikipedia - Sklyanin algebra | null | null | null |
Let S {\displaystyle S} be a set of symbols. Let Ξ¦ {\displaystyle \Phi } be a maximally consistent set of S {\displaystyle S} -formulas containing witnesses. Define an equivalence relation βΌ {\displaystyle \sim } on the set of S {\displaystyle S} -terms by t 0 βΌ t 1 {\displaystyle t_{0}\sim t_{1}} if t 0 β‘ t 1 β Ξ¦ {\di... | Wikipedia - Consistent theory | null | null | null |
Define the S {\displaystyle S} -structure T Ξ¦ {\displaystyle {\mathfrak {T}}_{\Phi }} over T Ξ¦ {\displaystyle T_{\Phi }} , also called the term-structure corresponding to Ξ¦ {\displaystyle \Phi } , by: for each n {\displaystyle n} -ary relation symbol R β S {\displaystyle R\in S} , define R T Ξ¦ t 0 Β― β¦ t n β 1 Β― {\displ... | Wikipedia - Consistent theory | null | null | null |
Define a variable assignment Ξ² Ξ¦ {\displaystyle \beta _{\Phi }} by Ξ² Ξ¦ ( x ) := x Β― {\displaystyle \beta _{\Phi }(x):={\bar {x}}} for each variable x {\displaystyle x} . Let I Ξ¦ := ( T Ξ¦ , Ξ² Ξ¦ ) {\displaystyle {\mathfrak {I}}_{\Phi }:=({\mathfrak {T}}_{\Phi },\beta _{\Phi })} be the term interpretation associated with ... | Wikipedia - Consistent theory | null | null | null |
Let S {\displaystyle S} be a subset of V {\displaystyle V} . The annihilator of S {\displaystyle S} in V β {\displaystyle V^{*}} , denoted here S 0 {\displaystyle S^{0}} , is the collection of linear functionals f β V β {\displaystyle f\in V^{*}} such that = 0 {\displaystyle =0} for all s β S {\displaystyle s\in S} . ... | Wikipedia - Dual space | null | null | null |
The annihilator of a subset is itself a vector space. The annihilator of the zero vector is the whole dual space: { 0 } 0 = V β {\displaystyle \{0\}^{0}=V^{*}} , and the annihilator of the whole space is just the zero covector: V 0 = { 0 } β V β {\displaystyle V^{0}=\{0\}\subseteq V^{*}} . Furthermore, the assignment o... | Wikipedia - Dual space | null | null | null |
{\displaystyle \{0\}\subseteq T^{0}\subseteq S^{0}\subseteq V^{*}.} If A {\displaystyle A} and B {\displaystyle B} are two subsets of V {\displaystyle V} then A 0 + B 0 β ( A β© B ) 0 , {\displaystyle A^{0}+B^{0}\subseteq (A\cap B)^{0},} and equality holds provided V {\displaystyle V} is finite-dimensional. If ( A i ) i... | Wikipedia - Dual space | null | null | null |
{\displaystyle \left(\bigcup _{i\in I}A_{i}\right)^{0}=\bigcap _{i\in I}A_{i}^{0}.} In particular if A {\displaystyle A} and B {\displaystyle B} are subspaces of V {\displaystyle V} then ( A + B ) 0 = A 0 β© B 0 . | Wikipedia - Dual space | null | null | null |
{\displaystyle (A+B)^{0}=A^{0}\cap B^{0}.} If V {\displaystyle V} is finite-dimensional and W {\displaystyle W} is a vector subspace, then W 00 = W {\displaystyle W^{00}=W} after identifying W {\displaystyle W} with its image in the second dual space under the double duality isomorphism V β V β β {\displaystyle V\appro... | Wikipedia - Dual space | null | null | null |
If W {\displaystyle W} is a subspace of V {\displaystyle V} then the quotient space V / W {\displaystyle V/W} is a vector space in its own right, and so has a dual. By the first isomorphism theorem, a functional f: V β F {\displaystyle f:V\to F} factors through V / W {\displaystyle V/W} if and only if W {\displaystyle ... | Wikipedia - Dual space | null | null | null |
Since G {\displaystyle G} is orthogonal, S {\displaystyle S} and S β² {\displaystyle S^{\prime }} have the same Frobenius norm | | β
| | F {\displaystyle ||\cdot ||_{F}} (the square-root sum of squares of all components), however we can choose ΞΈ {\displaystyle \theta } such that S i j β² = 0 {\displaystyle S_{ij}^{\prime... | Wikipedia - Jacobi eigenvalue algorithm | null | null | null |
Let S {\displaystyle S} be any set, and let A {\displaystyle \mathbf {A} } be an algebraic structure of type Ο {\displaystyle \rho } generated by S {\displaystyle S} . Let the underlying set of this algebraic structure A {\displaystyle \mathbf {A} } , sometimes called its universe, be A {\displaystyle A} , and let Ο: S... | Wikipedia - Free functor | null | null | null |
Let S2 and T2 be two distinct projective planes in a projective 3-space R3. With O and O* being points of R3 in neither plane, use the construction of the last section to project S2 onto T2 by the perspectivity with center O followed by the projection of T2 back onto S2 with the perspectivity with center O*. This compo... | Wikipedia - Perspectivity | null | null | null |
Each point of the line of intersection of S2 and T2 will be fixed by Ο and this line is called the axis of Ο. Let point P be the intersection of line OO* with the plane S2. P is also fixed by Ο and every line of S2 that passes through P is stabilized by Ο (fixed, but not necessarily pointwise fixed). P is called the ce... | Wikipedia - Perspectivity | null | null | null |
Let Spc be the category of all topological spaces. Given any family of functions {uΞ±: VΞ± β X}, we say that it is a surjective family or that the morphisms uΞ± are jointly surjective if βͺ {\displaystyle \cup } uΞ±(VΞ±) equals X. We define a pretopology on Spc by taking the covering families to be surjective families all of... | Wikipedia - Grothendieck topologies | null | null | null |
The covering sieves and covering families are almost exactly the same; the only difference is that now all the maps involved commute with the fixed maps to X. This is the big site associated to a topological space X . Notice that Spc is the big site associated to the one point space. This site was first considered by J... | Wikipedia - Grothendieck topologies | null | null | null |
Let T = { 1 , β¦ , n } {\displaystyle T=\{1,\ldots ,n\}\,\!} . The domain of decision criterion, V d {\displaystyle V_{d}\,\!} consist of n {\displaystyle n\,\!} | Wikipedia - Dominance-based rough set approach | null | null | null |
elements (without loss of generality we assume V d = T {\displaystyle V_{d}=T\,\!} ) and induces a partition of U {\displaystyle U\,\!} into n {\displaystyle n\,\!} | Wikipedia - Dominance-based rough set approach | null | null | null |
classes Cl = { C l t , t β T } {\displaystyle {\textbf {Cl}}=\{Cl_{t},t\in T\}} , where C l t = { x β U: f ( x , d ) = t } {\displaystyle Cl_{t}=\{x\in U\colon f(x,d)=t\}} . Each object x β U {\displaystyle x\in U} is assigned to one and only one class C l t , t β T {\displaystyle Cl_{t},t\in T} . The classes are prefe... | Wikipedia - Dominance-based rough set approach | null | null | null |
, the objects from C l r {\displaystyle Cl_{r}\,\!} are strictly preferred to the objects from C l s {\displaystyle Cl_{s}\,\!} . For this reason, we can consider the upward and downward unions of classes, defined respectively, as: C l t β₯ = β s β₯ t C l s C l t β€ = β s β€ t C l s t β T {\displaystyle Cl_{t}^{\geq }=\big... | Wikipedia - Dominance-based rough set approach | null | null | null |
Let T be a theory. A complete type p(x1, ..., xn) is called principal or atomic (relative to T) if it is axiomatized relative to T by a single formula Ο(x1, ..., xn) β p(x1, ..., xn). A formula Ο is called complete in T if for every formula Ο(x1, ..., xn), the theory T βͺ {Ο} entails exactly one of Ο and Β¬Ο. It follows ... | Wikipedia - Atomic model (mathematical logic) | null | null | null |
Let T be totally non-unitary contraction on H. Then the minimal unitary dilation U of T on K β H is unitarily equivalent to a direct sum of copies the bilateral shift operator, i.e. multiplication by z on L2(S1).If P is the orthogonal projection onto H then for f in Lβ = Lβ(S1) it follows that the operator f(T) can be ... | Wikipedia - Contraction (operator theory) | null | null | null |
In fact if f ( z ) = β n β₯ 0 a n z n {\displaystyle \displaystyle {f(z)=\sum _{n\geq 0}a_{n}z^{n}}} for |z| < 1, then for r < 1 f r ( z ) ) = β n β₯ 0 r n a n z n {\displaystyle \displaystyle {f_{r}(z))=\sum _{n\geq 0}r^{n}a_{n}z^{n}}} is holomorphic on |z| < 1/r. In that case fr(T) is defined by the holomorphic functio... | Wikipedia - Contraction (operator theory) | null | null | null |
The map sending f to f(T) defines an algebra homomorphism of Hβ into bounded operators on H. Moreover, if f βΌ ( z ) = β n β₯ 0 a n z Β― n , {\displaystyle \displaystyle {f^{\sim }(z)=\sum _{n\geq 0}a_{n}{\overline {z}}^{n},}} then f βΌ ( T ) = f ( T β ) β . {\displaystyle \displaystyle {f^{\sim }(T)=f(T^{*})^{*}.}} This m... | Wikipedia - Contraction (operator theory) | null | null | null |
For t β₯ 0, let et be the inner function e t ( z ) = exp β‘ t z + 1 z β 1 . {\displaystyle \displaystyle {e_{t}(z)=\exp t{z+1 \over z-1}.}} If T is the cogenerator of a one-parameter semigroup of completely non-unitary contractions T(t), then T ( t ) = e t ( T ) {\displaystyle \displaystyle {T(t)=e_{t}(T)}} and T = 1 2 I... | Wikipedia - Contraction (operator theory) | null | null | null |
Let T {\displaystyle T} and Ξ³ {\displaystyle \gamma } as above and let t β T {\displaystyle t\in T} some fixed time. We are now going to explain what it means that a MTL formula Ο {\displaystyle \phi } holds at time t {\displaystyle t} , which is denoted Ξ³ , t β¨ Ο {\displaystyle \gamma ,t\models \phi } . Let I β R + {\... | Wikipedia - Metric Temporal Logic | null | null | null |
Let T {\displaystyle T} be a regular chain of k {\displaystyle \mathbf {k} } for some ordering of the variables x = x 1 , β¦ , x n {\displaystyle \mathbf {x} =x_{1},\ldots ,x_{n}} and a real closed field k {\displaystyle \mathbf {k} } . Let u = u 1 , β¦ , u d {\displaystyle \mathbf {u} =u_{1},\ldots ,u_{d}} and y = y 1 ... | Wikipedia - Regular semi-algebraic system | null | null | null |
Define P > := { p > 0 β£ p β P } {\displaystyle P_{>}:=\{p>0\mid p\in P\}} . Let Q {\displaystyle {\mathcal {Q}}} be a quantifier-free formula of k {\displaystyle \mathbf {k} } involving only the variables of u {\displaystyle \mathbf {u} } . We say that R := {\displaystyle R:=} is a regular semi-algebraic system if th... | Wikipedia - Regular semi-algebraic system | null | null | null |
Let T {\displaystyle T} be the initial solution with root r {\displaystyle r} . The neighborhood consists of any combination of a single node or subtree (general subtrees, not as in the introduction of this article) displacing one in a different component of T β r {\displaystyle T\setminus r} such that the displaced st... | Wikipedia - Capacitated minimum spanning tree | null | null | null |
Let T {\displaystyle \mathbb {T} } be the free vector space spanned by all rooted trees. One can introduce a bilinear product βΆ {\displaystyle \curvearrowleft } on T {\displaystyle \mathbb {T} } as follows. Let Ο 1 {\displaystyle \tau _{1}} and Ο 2 {\displaystyle \tau _{2}} be two rooted trees. | Wikipedia - Pre-Lie algebra | null | null | null |
Ο 1 βΆ Ο 2 = β s β V e r t i c e s ( Ο 1 ) Ο 1 β s Ο 2 {\displaystyle \tau _{1}\curvearrowleft \tau _{2}=\sum _{s\in \mathrm {Vertices} (\tau _{1})}\tau _{1}\circ _{s}\tau _{2}} where Ο 1 β s Ο 2 {\displaystyle \tau _{1}\circ _{s}\tau _{2}} is the rooted tree obtained by adding to the disjoint union of Ο 1 {\displaystyl... | Wikipedia - Pre-Lie algebra | null | null | null |
Let T β R + {\displaystyle T\subseteq \mathbb {R} _{+}} , which intuitively represents a set of times. Let Ξ³: T β P ( A P ) {\displaystyle \gamma :T\to {\mathcal {P}}(AP)} a function that associates to each moment t β T {\displaystyle t\in T} a set of propositions from AP. A model of a TPTL formula is such a function Ξ³... | Wikipedia - Timed propositional temporal logic | null | null | null |
Let T β R + {\displaystyle T\subseteq \mathbb {R} _{+}} intuitively represent a set of points in time. Let Ξ³: T β A {\displaystyle \gamma :T\to A} a function which associates a letter to each moment t β T {\displaystyle t\in T} . A model of a MTL formula is such a function Ξ³ {\displaystyle \gamma } . Usually, Ξ³ {\displ... | Wikipedia - Metric Temporal Logic | null | null | null |
Let TN consist of the double-negation translations of the formulas in T. The fundamental soundness theorem (Avigad and Feferman 1998, p. 342; Buss 1998 p. 66) states: If T is a set of axioms and Ο is a formula, then T proves Ο using classical logic if and only if TN proves ΟN using intuitionistic logic. | Wikipedia - Double-negation translation | null | null | null |
Let U be an open set in Rn. A differential 0-form ("zero-form") is defined to be a smooth function f on U β the set of which is denoted Cβ(U). If v is any vector in Rn, then f has a directional derivative βv f, which is another function on U whose value at a point p β U is the rate of change (at p) of f in the v direct... | Wikipedia - Differential forms | null | null | null |
{\frac {d}{dt}}f(p+t\mathbf {v} )\right|_{t=0}.} (This notion can be extended pointwise to the case that v is a vector field on U by evaluating v at the point p in the definition.) In particular, if v = ej is the jth coordinate vector then βv f is the partial derivative of f with respect to the jth coordinate vector, i... | Wikipedia - Differential forms | null | null | null |
{\displaystyle {\frac {\partial f}{\partial x^{j}}}=\sum _{i=1}^{n}{\frac {\partial y^{i}}{\partial x^{j}}}{\frac {\partial f}{\partial y^{i}}}.} The first idea leading to differential forms is the observation that βv f (p) is a linear function of v: ( β v + w f ) ( p ) = ( β v f ) ( p ) + ( β w f ) ( p ) ( β c v f ) (... | Wikipedia - Differential forms | null | null | null |
Since any vector v is a linear combination Ξ£ vjej of its components, df is uniquely determined by dfp(ej) for each j and each p β U, which are just the partial derivatives of f on U. Thus df provides a way of encoding the partial derivatives of f. It can be decoded by noticing that the coordinates x1, x2, ..., xn are t... | Wikipedia - Differential forms | null | null | null |
{\displaystyle df_{p}=\sum _{i=1}^{n}{\frac {\partial f}{\partial x^{i}}}(p)(dx^{i})_{p}.} Applying both sides to ej, the result on each side is the jth partial derivative of f at p. Since p and j were arbitrary, this proves the formula (*). More generally, for any smooth functions gi and hi on U, we define the differe... | Wikipedia - Differential forms | null | null | null |
The above expansion reduces this question to the search for a function f whose partial derivatives βf / βxi are equal to n given functions fi. For n > 1, such a function does not always exist: any smooth function f satisfies β 2 f β x i β x j = β 2 f β x j β x i , {\displaystyle {\frac {\partial ^{2}f}{\partial x^{i}\,... | Wikipedia - Differential forms | null | null | null |
This is an example of a differential 2-form. This 2-form is called the exterior derivative dΞ± of Ξ± = Ξ£nj=1 fj dxj. | Wikipedia - Differential forms | null | null | null |
It is given by d Ξ± = β j = 1 n d f j β§ d x j = β i , j = 1 n β f j β x i d x i β§ d x j . {\displaystyle d\alpha =\sum _{j=1}^{n}df_{j}\wedge dx^{j}=\sum _{i,j=1}^{n}{\frac {\partial f_{j}}{\partial x^{i}}}\,dx^{i}\wedge dx^{j}.} | Wikipedia - Differential forms | null | null | null |
To summarize: dΞ± = 0 is a necessary condition for the existence of a function f with Ξ± = df. Differential 0-forms, 1-forms, and 2-forms are special cases of differential forms. For each k, there is a space of differential k-forms, which can be expressed in terms of the coordinates as β i 1 , i 2 β¦ i k = 1 n f i 1 i 2 β¦... | Wikipedia - Differential forms | null | null | null |
Antisymmetry, which was already present for 2-forms, makes it possible to restrict the sum to those sets of indices for which i1 < i2 < ... < ikβ1 < ik. Differential forms can be multiplied together using the exterior product, and for any differential k-form Ξ±, there is a differential (k + 1)-form dΞ± called the exterio... | Wikipedia - Differential forms | null | null | null |
Let U be an open subset of Rn for some n. A semialgebraic function on U is defined to be a continuous real-valued function on U whose restriction to any semialgebraic set contained in U has a graph which is a semialgebraic subset of the product space RnΓR. This endows Rn with a sheaf O R n {\displaystyle {\mathcal {O}}... | Wikipedia - Semialgebraic space | null | null | null |
Let U be an open subset of βm and let V be an open subset of βn. For each i and j between 1 and n, let gij be a smooth real-valued function on U, such that for each p in U, one has that the m Γ m matrix is symmetric and positive-definite. For each Ξ± and Ξ² between 1 and m, let hΞ±Ξ² be a smooth real-valued function on V,... | Wikipedia - Harmonic map | null | null | null |
Let U be the universal enveloping algebra of a Lie algebra g {\displaystyle {\mathfrak {g}}} over a field k; it is filtered by degree. The PoincarΓ©βBirkhoffβWitt theorem implies that gr β‘ U {\displaystyle \operatorname {gr} U} is a polynomial ring; in fact, it is the coordinate ring k {\displaystyle k} . The associate... | Wikipedia - Associated graded ring | null | null | null |
Let U {\displaystyle U} be a unitary operator acting on an m {\displaystyle m} -qubit register. Unitarity implies that all the eigenvalues of U {\displaystyle U} have unit modulus, and can therefore be characterized by their phase. Thus if | Ο β© {\displaystyle |\psi \rangle } is an eigenvector of U {\displaystyle U} , ... | Wikipedia - Quantum phase estimation | null | null | null |
Our goal is to find a good approximation to ΞΈ {\displaystyle \theta } with a small number of gates and with high probability. The quantum phase estimation algorithm achieves this under the assumptions of having oracular access to U {\displaystyle U} , and having | Ο β© {\displaystyle |\psi \rangle } available as a quant... | Wikipedia - Quantum phase estimation | null | null | null |
Let U β ( E n + 1 , β β
β E ) {\displaystyle U\subset (\mathbb {E} ^{n+1},\|\cdot \|_{\mathbb {E} })} be a bounded open convex set with the boundary of class C2 and positive normal curvatures. Similarly to the Lobachevsky space, the hypersurface β U {\displaystyle \partial U} is called the absolute of Hilbert's geometr... | Wikipedia - Hilbert's fourth problem | null | null | null |
{\displaystyle F_{U}(x,y)={\frac {1}{2}}\|y\|_{\mathbb {E} }\left({\frac {1}{\|x-x_{+}\|_{\mathbb {E} }}}+{\frac {1}{\|x-x_{-}\|_{\mathbb {E} }}}\right).} The metric is symmetric and flat. In 1895, Hilbert introduced this metric as a generalization of the Lobachevsky geometry. If the hypersurface β U {\displaystyle \pa... | Wikipedia - Hilbert's fourth problem | null | null | null |
Let U β R {\displaystyle U\subseteq \mathbb {R} } be open (for example, an interval ( a , b ) {\displaystyle (a,b)} ), and consider a differentiable function f: U β R , {\displaystyle f:U\to \mathbb {R} ,} with derivative f β² . {\displaystyle f'.} The differential d f {\displaystyle df} of f , {\displaystyle f,} at a p... | Wikipedia - One-form (differential geometry) | null | null | null |
Specifically, d f ( x 0 , β
): d x β¦ f β² ( x 0 ) d x . {\displaystyle df(x_{0},\cdot ):dx\mapsto f'(x_{0})dx.} | Wikipedia - One-form (differential geometry) | null | null | null |
(The meaning of the symbol d x {\displaystyle dx} is thus revealed: it is simply an argument, or independent variable, of the linear function d f ( x 0 , β
) . {\displaystyle df(x_{0},\cdot ).} ) Hence the map x β¦ d f ( x ) {\displaystyle x\mapsto df(x)} sends each point x {\displaystyle x} to a linear functional d f (... | Wikipedia - One-form (differential geometry) | null | null | null |
{\displaystyle df(x,\cdot ).} This is the simplest example of a differential (one-)form. In terms of the de Rham cochain complex, one has an assignment from zero-forms (scalar functions) to one-forms; that is, f β¦ d f . {\displaystyle f\mapsto df.} | Wikipedia - One-form (differential geometry) | null | null | null |
Let Un be the group of upper-triangular matrices in GL(n) with diagonal entries equal to 1, over a field k. A group scheme over a field k (for example, a linear algebraic group) is called unipotent if it is isomorphic to a closed subgroup scheme of Un for some n. It is straightforward to check that the group Un is nilp... | Wikipedia - Affine algebraic group | null | null | null |
Let V , W {\displaystyle V,W} be finite-dimensional real vector spaces. There exists a bilinear map, called the exterior product, which is uniquely characterized by the following two properties: it is continuous with respect to the usual topologies on Val {\displaystyle \operatorname {Val} } and Val β . {\displaystyle ... | Wikipedia - Valuation (geometry) | null | null | null |
Let V , W {\displaystyle V,W} be g {\displaystyle {\mathfrak {g}}} -modules, g {\displaystyle {\mathfrak {g}}} a Lie algebra. Then Hom β‘ ( V , W ) {\displaystyle \operatorname {Hom} (V,W)} becomes a g {\displaystyle {\mathfrak {g}}} -module by setting ( X β
f ) ( v ) = X f ( v ) β f ( X v ) {\displaystyle (X\cdot f)(v)... | Wikipedia - Representation of Lie algebras | null | null | null |
Let V , W {\displaystyle V,W} be vector spaces over some field F , {\displaystyle F,} and T {\displaystyle T} defined as in the statement of the theorem with dim β‘ V = n {\displaystyle \dim V=n} . As Ker β‘ T β V {\displaystyle \operatorname {Ker} T\subset V} is a subspace, there exists a basis for it. Suppose dim β‘ Ker... | Wikipedia - Fundamental theorem of linear algebra | null | null | null |
We may now, by the Steinitz exchange lemma, extend K {\displaystyle {\mathcal {K}}} with n β k {\displaystyle n-k} linearly independent vectors w 1 , β¦ , w n β k {\displaystyle w_{1},\ldots ,w_{n-k}} to form a full basis of V {\displaystyle V} . Let such that is a basis for V {\displaystyle V} . From this, we know that... | Wikipedia - Fundamental theorem of linear algebra | null | null | null |
{\displaystyle =\operatorname {Span} \{T(w_{1}),\ldots ,T(w_{n-k})\}=\operatorname {Span} T({\mathcal {S}}).} We now claim that T ( S ) {\displaystyle T({\mathcal {S}})} is a basis for Im β‘ T {\displaystyle \operatorname {Im} T} . The above equality already states that T ( S ) {\displaystyle T({\mathcal {S}})} is a gen... | Wikipedia - Fundamental theorem of linear algebra | null | null | null |
Suppose T ( S ) {\displaystyle T({\mathcal {S}})} is not linearly independent, and let for some Ξ± j β F {\displaystyle \alpha _{j}\in F} . Thus, owing to the linearity of T {\displaystyle T} , it follows that This is a contradiction to B {\displaystyle {\mathcal {B}}} being a basis, unless all Ξ± j {\displaystyle \alpha... | Wikipedia - Fundamental theorem of linear algebra | null | null | null |
To summarize, we have K {\displaystyle {\mathcal {K}}} , a basis for Ker β‘ T {\displaystyle \operatorname {Ker} T} , and T ( S ) {\displaystyle T({\mathcal {S}})} , a basis for Im β‘ T {\displaystyle \operatorname {Im} T} . Finally we may state that = | T ( S ) | + | K | = ( n β k ) + k = n = dim β‘ V . {\displaystyle =|... | Wikipedia - Fundamental theorem of linear algebra | null | null | null |
Let V = V(n + 1, q) denote the vector space of (algebraic) dimension n + 1 defined over the finite field GF(q). The projective space PG(n, q) consists of all the positive (algebraic) dimensional vector subspaces of V. An alternate way to view the construction is to define the points of PG(n, q) as the equivalence class... | Wikipedia - Galois geometry | null | null | null |
Let V be a finite-dimensional vector space over a field F, g l ( V ) {\displaystyle {\mathfrak {gl}}(V)} the Lie algebra of linear transformations and g β g l ( V ) {\displaystyle {\mathfrak {g}}\subseteq {\mathfrak {gl}}(V)} a Lie subalgebra. Then g {\displaystyle {\mathfrak {g}}} is said to be split if the roots of t... | Wikipedia - Lie algebra | null | null | null |
Let V be a finite-dimensional vector space over a field k. The scheme over k defined by Proj(k) is called projectivization of V. The projective n-space on k is the projectivization of the vector space A k n + 1 {\displaystyle \mathbb {A} _{k}^{n+1}} . The definition of the sheaf is done on the base of open sets of prin... | Wikipedia - Algebraic geometry of projective spaces | null | null | null |
It can be noted that the ring of global sections of this scheme is a field, which implies that the scheme is not affine. Any two open sets intersect non-trivially: ie the scheme is irreducible. When the field k is algebraically closed, P ( V ) {\displaystyle \mathbb {P} (V)} is in fact an abstract variety, that further... | Wikipedia - Algebraic geometry of projective spaces | null | null | null |
Let V be a finite-dimensional vector space. The Grassmannian variety Gn(V) is the set of all n-dimensional subspaces of V. It is a projective variety: it is embedded into a projective space via the PlΓΌcker embedding: { G n ( V ) βͺ P ( β§ n V ) β¨ b 1 , β¦ , b n β© β¦ {\displaystyle {\begin{cases}G_{n}(V)\hookrightarrow \ma... | Wikipedia - Affine curve | null | null | null |
Let V be a projective algebraic set defined as the set of the common zeros of a homogeneous ideal I in a polynomial ring R = K {\displaystyle R=K} over a field K, and let A=R/I be the graded algebra of the polynomials over V. All the definitions of the previous section apply, with the change that, when A or I appear e... | Wikipedia - Dimension of an algebraic variety | null | null | null |
Let V be a representation of a Lie algebra g {\displaystyle {\mathfrak {g}}} . Then V is said to be completely reducible (or semisimple) if it is isomorphic to a direct sum of irreducible representations (cf. semisimple module). If V is finite-dimensional, then V is completely reducible if and only if every invariant s... | Wikipedia - Representation theory of Lie algebras | null | null | null |
If g {\displaystyle {\mathfrak {g}}} is a finite-dimensional semisimple Lie algebra over a field of characteristic zero and V is finite-dimensional, then V is semisimple; this is Weyl's complete reducibility theorem. Thus, for semisimple Lie algebras, a classification of irreducible (i.e. simple) representations leads ... | Wikipedia - Representation theory of Lie algebras | null | null | null |
A Lie algebra is said to be reductive if the adjoint representation is semisimple. Certainly, every (finite-dimensional) semisimple Lie algebra g {\displaystyle {\mathfrak {g}}} is reductive, since every representation of g {\displaystyle {\mathfrak {g}}} is completely reducible, as we have just noted. In the other dir... | Wikipedia - Representation theory of Lie algebras | null | null | null |
Let V be a set of vertices. Let C be an abstract simplicial complex on V. Let Vy (for y in Y) be subsets of V. A C-V-transversal is a set in C (an element of C) whose intersection with each Vy contains exactly one vertex. For every subset Y0 of Y, let V Y 0 := β y β Y 0 V y . {\displaystyle V_{Y_{0}}:=\bigcup _{y\in Y_... | Wikipedia - Hall-type theorems for hypergraphs | null | null | null |
Suppose that, for every subset Y0 of Y, the homological connectivity plus 2 of the sub-complex induced by V Y 0 {\displaystyle V_{Y_{0}}} is at least |Y0|, that is: Ξ· H ( C ) β₯ | Y 0 | . {\displaystyle \eta _{H}(C)\geq |Y_{0}|.} | Wikipedia - Hall-type theorems for hypergraphs | null | null | null |
Then there exists a C-V-transversal. That is: there is a set in C that intersects each Vy by exactly one element. This theorem has a deficiency version. If, for every subset Y0 of Y: Ξ· H ( C ) β₯ | Y 0 | β d , {\displaystyle \eta _{H}(C)\geq |Y_{0}|-d,} then there exists a partial C-transversal, that intersects some |Y... | Wikipedia - Hall-type theorems for hypergraphs | null | null | null |
Let V be a vector space and U, W two finite-dimensional subspaces of V with the following spanning sets: U = β¨ u 1 , β¦ , u n β© {\displaystyle U=\langle u_{1},\ldots ,u_{n}\rangle } and W = β¨ w 1 , β¦ , w k β© . {\displaystyle W=\langle w_{1},\ldots ,w_{k}\rangle .} Finally, let B 1 , β¦ , B m {\displaystyle B_{1},\ldots ,... | Wikipedia - Zassenhaus algorithm | null | null | null |
Let V be a vector space over a field F and let X be any set. The functions X β F can be given the structure of a vector space over F where the operations are defined pointwise, that is, for any f, g: X β F, any x in X, and any c in F, define When the domain X has additional structure, one might consider instead the sub... | Wikipedia - Function spaces | null | null | null |
Let V be a vector space over a field K, and let Q: V β K be a quadratic form on V. In most cases of interest the field K is either the field of real numbers R, or the field of complex numbers C, or a finite field. A Clifford algebra Cl(V, Q) is a pair (A, i), where A is a unital associative algebra over K and i is a li... | Wikipedia - Clifford Algebra | null | null | null |
It is then straightforward to show that Cl(V, Q) contains V and satisfies the above universal property, so that Cl is unique up to a unique isomorphism; thus one speaks of "the" Clifford algebra Cl(V, Q). It also follows from this construction that i is injective. One usually drops the i and considers V as a linear sub... | Wikipedia - Clifford Algebra | null | null | null |
The universal characterization of the Clifford algebra shows that the construction of Cl(V, Q) is functorial in nature. Namely, Cl can be considered as a functor from the category of vector spaces with quadratic forms (whose morphisms are linear maps preserving the quadratic form) to the category of associative algebra... | Wikipedia - Clifford Algebra | null | null | null |
Let V be a vector space over a field K. For any nonnegative integer k, we define the kth tensor power of V to be the tensor product of V with itself k times: T k V = V β k = V β V β β― β V . {\displaystyle T^{k}V=V^{\otimes k}=V\otimes V\otimes \cdots \otimes V.} That is, TkV consists of all tensors on V of order k. By ... | Wikipedia - Tensor-algebra bundle | null | null | null |
{\displaystyle T(V)=\bigoplus _{k=0}^{\infty }T^{k}V=K\oplus V\oplus (V\otimes V)\oplus (V\otimes V\otimes V)\oplus \cdots .} The multiplication in T(V) is determined by the canonical isomorphism T k V β T β V β T k + β V {\displaystyle T^{k}V\otimes T^{\ell }V\to T^{k+\ell }V} given by the tensor product, which is the... | Wikipedia - Tensor-algebra bundle | null | null | null |
This grading can be extended to a Z grading by appending subspaces T k V = { 0 } {\displaystyle T^{k}V=\{0\}} for negative integers k. The construction generalizes in a straightforward manner to the tensor algebra of any module M over a commutative ring. If R is a non-commutative ring, one can still perform the constru... | Wikipedia - Tensor-algebra bundle | null | null | null |
Let V be a vector space over the field K. Informally, multiplication in β ( V ) {\textstyle \bigwedge (V)} is performed by manipulating symbols and imposing a distributive law, an associative law, and using the identity v β§ v = 0 {\displaystyle v\wedge v=0} for v β V. Formally, β ( V ) {\textstyle \bigwedge (V)} is the... | Wikipedia - Alternating form | null | null | null |
It is then straightforward to show that β ( V ) {\textstyle \bigwedge (V)} contains V and satisfies the above universal property. As a consequence of this construction, the operation of assigning to a vector space V its exterior algebra β ( V ) {\textstyle \bigwedge (V)} is a functor from the category of vector spaces ... | Wikipedia - Alternating form | null | null | null |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.