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 |
|---|---|---|---|---|
Let C β P 2 {\displaystyle C\subset \mathbf {P} ^{2}} be a smooth plane curve cut out by a degree d {\displaystyle d} homogeneous polynomial F ( X , Y , Z ) {\displaystyle F(X,Y,Z)} . We claim that the canonical divisor is K = ( d β 3 ) {\displaystyle K=(d-3)} where H {\displaystyle H} is the hyperplane divisor. First... | Wikipedia - Adjunction formula | null | null | null |
We will explicitly compute the divisor of the differential Ο := d x β f / β y = β d y β f / β x . {\displaystyle \omega :={\frac {dx}{\partial f/\partial y}}={\frac {-dy}{\partial f/\partial x}}.} At any point ( x 0 , y 0 ) {\displaystyle (x_{0},y_{0})} either β f / β y β 0 {\displaystyle \partial f/\partial y\neq 0} s... | Wikipedia - Adjunction formula | null | null | null |
In both cases the order of vanishing of Ο {\displaystyle \omega } at the point is zero. Thus all contributions to the divisor div ( Ο ) {\displaystyle {\text{div}}(\omega )} are at the line at infinity, Z = 0 {\displaystyle Z=0} . Now look on the line Z = 0 {\displaystyle {Z=0}} . | Wikipedia - Adjunction formula | null | null | null |
Assume that β C {\displaystyle \not \in C} so it suffices to look in the chart Y β 0 {\displaystyle Y\neq 0} with coordinates u = 1 / y {\displaystyle u=1/y} and v = x / y {\displaystyle v=x/y} . The equation of the curve becomes g ( u , v ) = F ( v , 1 , u ) = F ( x / y , 1 , 1 / y ) = y β d F ( x , y , 1 ) = y β d f... | Wikipedia - Adjunction formula | null | null | null |
{\displaystyle g(u,v)=F(v,1,u)=F(x/y,1,1/y)=y^{-d}F(x,y,1)=y^{-d}f(x,y).} Hence β f / β x = y d β g β v β v β x = y d β 1 β g β v {\displaystyle \partial f/\partial x=y^{d}{\frac {\partial g}{\partial v}}{\frac {\partial v}{\partial x}}=y^{d-1}{\frac {\partial g}{\partial v}}} so Ο = β d y β f / β x = 1 u 2 d u y d β 1... | Wikipedia - Adjunction formula | null | null | null |
Let CG be the complex group algebra over G. Then the center of CG, denoted Z(CG), is defined by Z β‘ ( C G ) = { f β C G β£ β g β C G , f g = g f } {\displaystyle \operatorname {Z} (\mathbf {C} G)=\{f\in \mathbf {C} G\mid \forall g\in \mathbf {C} G,fg=gf\}} .This is equal to the set of all class functions (functions whic... | Wikipedia - Conjugacy class sum | null | null | null |
Let Ci (for i between 1 and k) be the sum of subset i in a given partition. Instead of minimizing the objective function max(Ci), one can minimize the objective function max(f(Ci)), where f is any fixed function. Similarly, one can minimize the objective function sum(f(Ci)), or maximize min(f(Ci)), or maximize sum(f(Ci... | Wikipedia - Multiway number partitioning | null | null | null |
Alon, Azar, Woeginger and Yadid presented general PTAS-s (generalizing the PTAS-s of Sanhi, Hochbaum and Shmoys, and Woeginger) for these four problems. Their algorithm works for any f which satisfies the following two conditions: A strong continuity condition called Condition F*: for every Ξ΅>0 there exists Ξ΄>0 such th... | Wikipedia - Multiway number partitioning | null | null | null |
The PTAS for minimizing sum(f(Ci)) is based on some combinatorial observations: Let L := the average sum in a single subset (1/k the sum of all inputs). If some input x is at least L, then there is an optimal partition in which one part contains only x. This follows from the convexity of f. Therefore, the input can be ... | Wikipedia - Multiway number partitioning | null | null | null |
Particularly, the partition minimizing the sum of squares Ci2, among all optimal partitions, satisfies these inequalities.The PTAS uses an input rounding technique. Given the input sequence S = (v1,...,vn) and a positive integer d, the rounded sequence S#(d) is defined as follows: For any vj > L/d, the sequence S#(d) c... | Wikipedia - Multiway number partitioning | null | null | null |
In addition, the sequence S#(d) contains some inputs equal to L/d. The number of these inputs is determined such that the sum of all these new inputs equals the sum of all inputs in S#(d) that are at most L/d, rounded up to the next integer multiple of L/d (for example, if the sum of all "short" inputs in S is 51.3L/d,... | Wikipedia - Multiway number partitioning | null | null | null |
By construction, L# is at least L. Since L itself is an integer multiple of L/d2, the rounding-up of inputs smaller than L cannot make them larger than L. Therefore, all inputs in S#(d) are smaller than L, and hence smaller than L#. There is an optimal partition of S#(d) in which all subset sums are strictly between L#... | Wikipedia - Multiway number partitioning | null | null | null |
Therefore, the input can be represented as an integer vector n = ( n d , n d + 1 , β¦ , n d 2 ) {\displaystyle \mathbf {n} =(n_{d},n_{d+1},\ldots ,n_{d^{2}})} , where n h {\displaystyle n_{h}} is the number of hL/d2 inputs in S#(d). Moreover, each subset can be represented as an integer vector t = ( t d , t d + 1 , β¦ , ... | Wikipedia - Multiway number partitioning | null | null | null |
Let D ( A ) {\displaystyle D(A)} be the module of derivations of a ring A {\displaystyle A} . A connection on an A-module M {\displaystyle M} is defined as an A-module morphism β: D ( A ) β D i f f 1 ( M , M ) ; u β¦ β u {\displaystyle \nabla :D(A)\to \mathrm {Diff} _{1}(M,M);u\mapsto \nabla _{u}} such that the first or... | Wikipedia - Connection (algebraic framework) | null | null | null |
The curvature of the connection β {\displaystyle \nabla } is defined as the zero-order differential operator R ( u , u β² ) = β β {\displaystyle R(u,u')=-\nabla _{}\,} on the module M {\displaystyle M} for all u , u β² β D ( A ) {\displaystyle u,u'\in D(A)} . If E β X {\displaystyle E\to X} is a vector bundle, there is... | Wikipedia - Connection (algebraic framework) | null | null | null |
Let D ( Ο S 2 k ) {\displaystyle D(\tau _{S^{2k}})} denote the disk bundle associated to the tangent bundle of the 2k-sphere. If we plumb eight copies of D ( Ο S 2 k ) {\displaystyle D(\tau _{S^{2k}})} according to the diagram E 8 {\displaystyle E_{8}} , we obtain a 4k-dimensional manifold which certain authors call th... | Wikipedia - Plumbing (mathematics) | null | null | null |
Let D = (V, E) be a directed graph, and 0 an element not in V. The graph algebra associated with D has underlying set V βͺ { 0 } {\displaystyle V\cup \{0\}} , and is equipped with a multiplication defined by the rules xy = x if x , y β V {\displaystyle x,y\in V} and ( x , y ) β E {\displaystyle (x,y)\in E} , xy = 0 if x... | Wikipedia - Graph algebra | null | null | null |
Let D = (X, B) be a t-(v,k,Ξ») design and p a point of X. The derived design Dp has point set X β {p} and as block set all the blocks of D which contain p with p removed. It is a (t β 1)-(v β 1, k β 1, Ξ») design. Note that derived designs with respect to different points may not be isomorphic. A design E is called an ex... | Wikipedia - Biplane geometries | null | null | null |
Theorem: If a t-(v,k,Ξ») design has an extension, then k + 1 divides b(v + 1). The only extendable projective planes (symmetric 2-(n2 + n + 1, n + 1, 1) designs) are those of orders 2 and 4.Every Hadamard 2-design is extendable (to an Hadamard 3-design).Theorem:. If D, a symmetric 2-(v,k,Ξ») design, is extendable, then o... | Wikipedia - Biplane geometries | null | null | null |
Let D = k be the ring of dual numbers over a field k and Y a scheme of finite type over k. Given a closed subscheme X of Y, by definition, an embedded first-order infinitesimal deformation of X is a closed subscheme X' of Y ΓSpec(k) Spec(D) such that the projection X' β Spec D is flat and has X as the special fiber. If... | Wikipedia - Flat degeneration | null | null | null |
Let D be the division algebra in question. Let n be the dimension of D. We identify the real multiples of 1 with R. When we write a β€ 0 for an element a of D, we imply that a is contained in R. We can consider D as a finite-dimensional R-vector space. Any element d of D defines an endomorphism of D by left-multiplicati... | Wikipedia - Frobenius theorem (real division algebras) | null | null | null |
For any z in C define the following real quadratic polynomial: Q ( z ; x ) = x 2 β 2 Re β‘ ( z ) x + | z | 2 = ( x β z ) ( x β z Β― ) β R . {\displaystyle Q(z;x)=x^{2}-2\operatorname {Re} (z)x+|z|^{2}=(x-z)(x-{\overline {z}})\in \mathbf {R} .} Note that if z β C β R then Q(z; x) is irreducible over R. | Wikipedia - Frobenius theorem (real division algebras) | null | null | null |
Let D {\displaystyle D} such that P β = D Γ r β = ( p 1 , p 2 , β¦ , p n ) T {\displaystyle {\vec {P}}=D\times {\vec {r}}=(p_{1},p_{2},\dots ,p_{n})^{T}} Where D = A Γ B β C = ( d i j ) {\displaystyle D=A\times B-C=(d_{ij})} .Since A Γ B β C {\displaystyle A\times B\neq C} , we have that some element of D {\displaystyle... | Wikipedia - Freivalds' algorithm | null | null | null |
Using Bayes' theorem, we can partition over y {\displaystyle y}: We use that: Pr = Pr = 1 2 {\displaystyle \Pr=\Pr={\frac {1}{2}}} Pr = Pr β€ Pr = 1 2 {\displaystyle \Pr=\Pr\leq \Pr={\frac {1}{2}}} Plugging these in the equation (1), we get: Pr β€ 1 2 β
Pr + 1 2 β
Pr = 1 2 β
Pr + 1 2 β
( 1 β Pr ) = 1 2 {\displa... | Wikipedia - Freivalds' algorithm | null | null | null |
Let D βͺ X {\displaystyle D\hookrightarrow X} be an effective Cartier divisor. Then the normal bundle to it (or equivalently the normal cone to it) is | Wikipedia - Conormal sheaf | null | null | null |
Let E ( Ο , t 0 , y ( t 0 ) ) {\displaystyle E(\ \tau ,t_{0},y(t_{0})\ )} be the exact solution operator so that: y ( t 0 + Ο ) = E ( Ο , t 0 , y ( t 0 ) ) y ( t 0 ) {\displaystyle y(t_{0}+\tau )=E(\tau ,t_{0},y(t_{0}))\ y(t_{0})} with t 0 {\displaystyle t_{0}} denoting the initial time and y ( t ) {\displaystyle y(t)}... | Wikipedia - Lady Windermere's Fan (mathematics) | null | null | null |
Let E be a Euclidean Jordan algebra with complexification A = EC = E + iE. The unit ball or disk D in A is just the convex bounded open set of elements a such the ||a|| < 1, i.e. the unit ball for the spectral norm. The tube domain T in A is the unbounded convex open set T = E + iC, where C is the open positive cone in... | Wikipedia - Jordan frame (Jordan algebra) | null | null | null |
Let E be a Euclidean Jordan algebra with the inner product given by the trace form. Let (ei) be a fixed Jordan frame in E. For given a in EC choose u in Ξu(EC) such that ua = Ξ£ Ξ±i ei with Ξ±i β₯ 0. Then the spectral norm ||a|| = max Ξ±i is independent of all choices. It is a norm on EC with β a β β = β a β , β { a , a β ,... | Wikipedia - Jordan frame (Jordan algebra) | null | null | null |
{\displaystyle \displaystyle {\|a^{*}\|=\|a\|,\,\,\,\|\{a,a^{*},a\}\|=\|a\|^{3}.}} In addition ||a||2 is given by the operator norm of Q(a) on the inner product space EC. | Wikipedia - Jordan frame (Jordan algebra) | null | null | null |
The fundamental identity for the quadratic representation implies that ||Q(a)b|| β€ ||a||2||b||. The spectral norm of an element a is defined in terms of C so depends only on a and not the particular Euclidean Jordan algebra in which it is calculated.The compact set S is the set of extreme points of the closed unit ball... | Wikipedia - Jordan frame (Jordan algebra) | null | null | null |
Each u in S has norm one. Moreover, if u = eia and v = eib, then ||uv|| β€ 1. Indeed, by the CohnβShirshov theorem the unital Jordan subalgebra of E generated by a and b is special. | Wikipedia - Jordan frame (Jordan algebra) | null | null | null |
The inequality is easy to establish in non-exceptional simple Euclidean Jordan algebras, since each such Jordan algebra and its complexification can be realized as a subalgebra of some Hn(R) and its complexification Hn(C) β Mn(C). The spectral norm in Hn(C) is the usual operator norm. In that case, for unitary matrices... | Wikipedia - Jordan frame (Jordan algebra) | null | null | null |
The inequality therefore follows in any special Euclidean Jordan algebra and hence in general.On the other hand, by the KreinβMilman theorem, the closed unit ball is the (closed) convex span of S. It follows that ||L(u)|| = 1, in the operator norm corresponding to either the inner product norm or spectral norm. Hence |... | Wikipedia - Jordan frame (Jordan algebra) | null | null | null |
Let E be a Euclidean Jordan algebra. The complexification EC = E β iE has a natural conjugation operation (a + ib)* = a β ib and a natural complex inner product and norm. The Jordan product on E extends bilinearly to EC, so that (a + ib)(c + id) = (ac β bd) + i(ad + bc). If multiplication is defined by L(a)b = ab then ... | Wikipedia - Symmetric cone | null | null | null |
Indeed, the identity above holds when a is replaced by a + tb for t real; and since the left side is then a polynomial with values in End EC vanishing for real t, it vanishes also t complex. Analytic continuation also shows that all for the formulas involving power-associativity for a single element a in E, including r... | Wikipedia - Symmetric cone | null | null | null |
Similarly the symmetric bilinear form Ξ²(a,b) = (a,b*) satisfies Ξ²(ab,c) = Ξ²(b,ac). If the inner product comes from the trace form, then Ξ²(a,b) = Tr L(ab). For a in EC, the quadratic representation is defined as before by Q(a)=2L(a)2 β L(a2). | Wikipedia - Symmetric cone | null | null | null |
By analytic continuation the fundamental identity still holds: Q ( Q ( a ) b ) = Q ( a ) Q ( b ) Q ( a ) , Q ( a m ) = Q ( a ) m ( m β₯ 0 ) . {\displaystyle \displaystyle {Q(Q(a)b)=Q(a)Q(b)Q(a),\,\,\,Q(a^{m})=Q(a)^{m}\,\,(m\geq 0).}} An element a in E is called invertible if it is invertible in C. Power associativity sh... | Wikipedia - Symmetric cone | null | null | null |
Moreover, aβ1 is invertible with inverse a. As in E, a is invertible if and only if Q(a) is invertible. In that case Q ( a ) β 1 a = a β 1 , Q ( a β 1 ) = Q ( a ) β 1 . {\displaystyle \displaystyle {Q(a)^{-1}a=a^{-1},\,\,\,Q(a^{-1})=Q(a)^{-1}.}} | Wikipedia - Symmetric cone | null | null | null |
Indeed, as for E, if Q(a) is invertible it carries C onto itself, while Q(a)1 = a2, so ( Q ( a ) β 1 a ) a = a Q ( a ) β 1 a = L ( a ) Q ( a ) β 1 a = Q ( a ) β 1 a 2 = 1 , {\displaystyle \displaystyle {(Q(a)^{-1}a)a=aQ(a)^{-1}a=L(a)Q(a)^{-1}a=Q(a)^{-1}a^{2}=1,}} so a is invertible. Conversely if a is invertible, takin... | Wikipedia - Symmetric cone | null | null | null |
Finally if a and b are invertible then so is c = Q(a)b and it satisfies the inverse identity: ( Q ( a ) b ) β 1 = Q ( a β 1 ) b β 1 . {\displaystyle \displaystyle {(Q(a)b)^{-1}=Q(a^{-1})b^{-1}.}} Invertibility of c follows from the fundamental formula which gives Q(c) = Q(a)Q(b)Q(a). | Wikipedia - Symmetric cone | null | null | null |
Hence c β 1 = Q ( c ) β 1 c = Q ( a ) β 1 Q ( b ) β 1 b = Q ( a ) β 1 b β 1 . {\displaystyle \displaystyle {c^{-1}=Q(c)^{-1}c=Q(a)^{-1}Q(b)^{-1}b=Q(a)^{-1}b^{-1}.}} The formula Q ( e a ) = e 2 L ( a ) {\displaystyle \displaystyle {Q(e^{a})=e^{2L(a)}}} also follows by analytic continuation. | Wikipedia - Symmetric cone | null | null | null |
Let E be a finite-dimensional real vector space with a symmetric bilinear product operation E Γ E β E , a , b β¦ a b = b a , {\displaystyle \displaystyle {E\times E\rightarrow E,\,\,\,a,b\mapsto ab=ba,}} with an identity element 1 such that a1 = a for a in A and a real inner product (a,b) for which the multiplication op... | Wikipedia - Structure group (Jordan algebra) | null | null | null |
The trace form has the advantage of being manifestly invariant under automorphisms of the Jordan algebra, which is thus a closed subgroup of O(E) and thus a compact Lie group. In practical examples, however, it is often easier to produce an inner product for which the L(a) are self-adjoint than verify directly positive... | Wikipedia - Structure group (Jordan algebra) | null | null | null |
Let E be a simple Euclidean Jordan algebra with inner product given by the trace form Ο(a)= Tr L(a). The proof that E has the above form rests on constructing an analogue of matrix units for a Jordan frame in E. The following properties of idempotents hold in E. An idempotent e is minimal in E if and only if E1(e) has ... | Wikipedia - Symmetric cone | null | null | null |
If e and f are non-orthogonal minimal idempotents, then there is a period 2 automorphism Ο of E such that Οe=f, so that e and f have the same trace. If e and f are orthogonal minimal idempotents then E1/2(e) β© E1/2(f) β (0). Moreover, there is a period 2 automorphism Ο of E such that Οe=f, so that e and f have the same... | Wikipedia - Symmetric cone | null | null | null |
All minimal idempotents in E are in the same orbit of the automorphism group so have the same trace Ο0. If e, f, g are three minimal orthogonal idempotents, then for a in E1/2(e) β© E1/2(f) and b in E1/2(f) β© E1/2(g), L(a)2 b = 1/8 Ο0 |a|2 b and |ab|2 = 1/8 Ο0 |a|2|b|2. Moreover, E1/2(e) β© E1/2(f) β© E1/2(g) = (0). | Wikipedia - Symmetric cone | null | null | null |
If e1, ..., er and f1, ..., fr are Jordan frames in E, then there is an automorphism Ξ± such that Ξ±ei = fi. If (ei) is a Jordan frame and Eii = E1(ei) and Eij = E1/2(ei) β© E1/2(ej), then E is the orthogonal direct sum the Eii's and Eij's. Since E is simple, the Eii's are one-dimensional and the subspaces Eij are all non... | Wikipedia - Symmetric cone | null | null | null |
Let E be a simple Euclidean Jordan algebra. From the properties of the Peirce decomposition it follows that: If E has rank 2, then it has the form V β R for some inner product space V with Jordan product as described above. If E has rank r > 2, then there is a non-associative unital algebra A, associative if r > 3, equ... | Wikipedia - Jordan frame (Jordan algebra) | null | null | null |
)Such an algebra A is called a Euclidean Hurwitz algebra. In A if Ξ»(a)b = ab and Ο(a)b = ba, then: the involution is an antiautomorphism, i.e. (a b)*=b* a* a a* = β a β2 1 = a* a Ξ»(a*) = Ξ»(a)*, Ο(a*) = Ο(a)*, so that the involution on the algebra corresponds to taking adjoints Re(a b) = Re(b a) if Re x = (x + x*)/2 = (... | Wikipedia - Jordan frame (Jordan algebra) | null | null | null |
Because A is associative when A = R, C or H, it is immediate that Hr(A) is a Jordan algebra for r β₯ 3. A separate argument, given originally by Albert (1934), is required to show that H3(O) with Jordan product aβb = 1/2(ab + ba) satisfies the Jordan identity = 0. There is a later more direct proof using the Freudentha... | Wikipedia - Jordan frame (Jordan algebra) | null | null | null |
Let E be a vector space over a field k equipped with a linear mapping d: E β E β§ E {\displaystyle d\colon E\to E\wedge E} from E to the exterior product of E with itself. It is possible to extend d uniquely to a graded derivation (this means that, for any a, b β E which are homogeneous elements, d ( a β§ b ) = ( d a ) β§... | Wikipedia - Lie coalgebra | null | null | null |
Let E {\displaystyle E} be a finite dimensional vector-space over K {\displaystyle K} and { Ο 1 , β¦ , Ο n } {\displaystyle \{\omega _{1},\ldots ,\omega _{n}\}} a basis for E {\displaystyle E} over K . {\displaystyle K.} For each place v {\displaystyle v} of K {\displaystyle K}: E v := E β K K v β
K v Ο 1 β β― β K v Ο n ... | Wikipedia - Adele ring | null | null | null |
This definition is based on the alternative description of the adele ring as a tensor product equipped with the same topology that was defined when giving an alternate definition of adele ring for number fields. Next, A E {\displaystyle \mathbb {A} _{E}} is equipped with the restricted product topology. Then A E = E β ... | Wikipedia - Adele ring | null | null | null |
{\displaystyle e\mapsto e\otimes 1.} An alternative definition of the topology on A E {\displaystyle \mathbb {A} _{E}} can be provided. Consider all linear maps: E β K . | Wikipedia - Adele ring | null | null | null |
{\displaystyle E\to K.} Using the natural embeddings E β A E {\displaystyle E\to \mathbb {A} _{E}} and K β A K , {\displaystyle K\to \mathbb {A} _{K},} extend these linear maps to: A E β A K . {\displaystyle \mathbb {A} _{E}\to \mathbb {A} _{K}.} | Wikipedia - Adele ring | null | null | null |
The topology on A E {\displaystyle \mathbb {A} _{E}} is the coarsest topology for which all these extensions are continuous. The topology can be defined in a different way. Fixing a basis for E {\displaystyle E} over K {\displaystyle K} results in an isomorphism E β
K n . | Wikipedia - Adele ring | null | null | null |
{\displaystyle E\cong K^{n}.} Therefore fixing a basis induces an isomorphism ( A K ) n β
A E . {\displaystyle (\mathbb {A} _{K})^{n}\cong \mathbb {A} _{E}.} | Wikipedia - Adele ring | null | null | null |
The left-hand side is supplied with the product topology and transport this topology with the isomorphism onto the right-hand side. The topology doesn't depend on the choice of the basis, because another basis defines a second isomorphism. By composing both isomorphisms, a linear homeomorphism which transfers the two t... | Wikipedia - Adele ring | null | null | null |
More formally A E = E β K A K β
( K β K A K ) β β― β ( K β K A K ) β
A K β β― β A K {\displaystyle {\begin{aligned}\mathbb {A} _{E}&=E\otimes _{K}\mathbb {A} _{K}\\&\cong (K\otimes _{K}\mathbb {A} _{K})\oplus \cdots \oplus (K\otimes _{K}\mathbb {A} _{K})\\&\cong \mathbb {A} _{K}\oplus \cdots \oplus \mathbb {A} _{K}\end{a... | Wikipedia - Adele ring | null | null | null |
Let E {\displaystyle E} be a finite extension of a field k {\displaystyle k} of inseparable degree p ΞΌ , {\displaystyle p^{\mu },} and let k {\displaystyle k} have algebraic closure K . {\displaystyle K.} If the distinct embeddings of E {\displaystyle E} are { Ο j } j , {\displaystyle \left\{\sigma _{j}\right\}_{j},} t... | Wikipedia - Equivalent norm | null | null | null |
Let E {\displaystyle E} be an elliptic curve defined over the finite field F q {\displaystyle \mathbb {F} _{q}} , where q = p n {\displaystyle q=p^{n}} for p {\displaystyle p} a prime and n {\displaystyle n} an integer β₯ 1 {\displaystyle \geq 1} . Over a field of characteristic β 2 , 3 {\displaystyle \neq 2,3} an ellip... | Wikipedia - Schoof's algorithm | null | null | null |
Using the group law on elliptic curves restricted to this set one can see that this set E ( F q ) {\displaystyle E(\mathbb {F} _{q})} forms an abelian group, with O {\displaystyle O} acting as the zero element. In order to count points on an elliptic curve, we compute the cardinality of E ( F q ) {\displaystyle E(\math... | Wikipedia - Schoof's algorithm | null | null | null |
Let E β R 3 {\displaystyle {\mathcal {E}}\in \mathbb {R} ^{3}} be a three-dimensional Euclidean space. Let B 0 , B a β E {\displaystyle {\mathcal {B}}_{0},{\mathcal {B}}_{a}\in {\mathcal {E}}} be two regions occupied by the material in two different instants of time. Let Ο {\displaystyle {\bf {\chi }}} be the deformati... | Wikipedia - Incremental deformations | null | null | null |
The deformation gradient is given by Considering a hyperelastic material with an elastic strain energy density W ( F ) {\displaystyle W({\bf {F}})} , the Piola-Kirchhoff stress tensor S {\displaystyle {\bf {S}}} is given by S = β W β F {\displaystyle {\bf {S}}={\frac {\partial W}{\partial \,{\bf {F}}}}} . For a quasi-s... | Wikipedia - Incremental deformations | null | null | null |
Let E β F {\displaystyle E\supseteq F} be an algebraic extension of fields of characteristic p. The separable closure of F in E is S = { Ξ± β E β£ Ξ± is separable over F } . {\displaystyle S=\{\alpha \in E\mid \alpha {\text{ is separable over }}F\}.} For every element x β E β S {\displaystyle x\in E\setminus S} there exis... | Wikipedia - Separably closed | null | null | null |
The former, often denoted sep, is referred to as the separable part of , or as the separable degree of E/F; the latter is referred to as the inseparable part of the degree or the inseparable degree. The inseparable degree is 1 in characteristic zero and a power of p in characteristic p > 0.On the other hand, an arbitra... | Wikipedia - Separably closed | null | null | null |
Let E: x 2 a 2 + y 2 b 2 = 1 {\displaystyle E:\;{\tfrac {x^{2}}{a^{2}}}+{\tfrac {y^{2}}{b^{2}}}=1} be the ellipse of consideration. The tangents to the ellipse E {\displaystyle E} at the vertices and co-vertices intersect at the 4 points ( Β± a , Β± b ) {\displaystyle (\pm a,\pm b)} , which lie on the desired orthoptic c... | Wikipedia - Orthoptic (geometry) | null | null | null |
{\displaystyle y=-{\tfrac {b^{2}u}{a^{2}v}}\;x\;+\;{\tfrac {b^{2}}{v}}\,.} Using the abbreviations and the equation u 2 a 2 = 1 β v 2 b 2 = 1 β b 2 n 2 {\displaystyle {\color {blue}{\tfrac {u^{2}}{a^{2}}}=1-{\tfrac {v^{2}}{b^{2}}}=1-{\tfrac {b^{2}}{n^{2}}}}} one gets: Hence and the equation of a non vertical tangent is... | Wikipedia - Orthoptic (geometry) | null | null | null |
Eliminating the square root leads to which has two solutions m 1 , m 2 {\displaystyle m_{1},m_{2}} corresponding to the two tangents passing through ( x 0 , y 0 ) {\displaystyle (x_{0},y_{0})} . The constant term of a monic quadratic equation is always the product of its solutions. Hence, if the tangents meet at ( x 0 ... | Wikipedia - Orthoptic (geometry) | null | null | null |
Let F ( X , R ) {\displaystyle {\mathcal {F}}(X,{\mathbb {R} })} be the set of all functions from a set X to real numbers R {\displaystyle \mathbb {R} } . Because R {\displaystyle \mathbb {R} } is a field, F ( X , R ) {\displaystyle {\mathcal {F}}(X,{\mathbb {R} })} may be turned into a vector space and a commutative a... | Wikipedia - Real-valued function | null | null | null |
Let F ( x , y ) = 0 {\displaystyle F(x,y)=0} be a well-posed problem, i.e. F: X Γ Y β R {\displaystyle F:X\times Y\rightarrow \mathbb {R} } is a real or complex functional relationship, defined on the cross-product of an input data set X {\displaystyle X} and an output data set Y {\displaystyle Y} , such that exists a ... | Wikipedia - Numerical method | null | null | null |
Let F / K {\displaystyle F/K} be a Galois extension of algebraic number fields with Galois group G = G a l ( F / K ) {\displaystyle G=\mathrm {Gal} (F/K)} and let P K , P F {\displaystyle \mathbb {P} _{K},\mathbb {P} _{F}} denote the set of prime ideals of the fields K , F {\displaystyle K,F} respectively. Suppose that... | Wikipedia - Principalization (algebra) | null | null | null |
Let F = GF(q) = Fq, the finite field of order q and V the 2n-dimensional vector space over F represented as: V = { ( x , y ): x , y β F n } . {\displaystyle V=\{(x,y)\colon x,y\in F^{n}\}.} Let M0, M1, ..., Mqn - 1 be n Γ n matrices over F with the property that Mi β Mj is nonsingular whenever i β j. For i = 0, 1, ...,... | Wikipedia - Translation plane | null | null | null |
Let F be a field and let X be the simplicial complex with vertices the non-trivial vector subspaces of V = Fn. Two subspaces U1 and U2 are connected if one of them is a subset of the other. The k-simplices of X are formed by sets of k + 1 mutually connected subspaces. Maximal connectivity is obtained by taking n β 1 pr... | Wikipedia - Building (mathematics) | null | null | null |
To define the apartments in X, it is convenient to define a frame in V as a basis (vi) determined up to scalar multiplication of each of its vectors vi; in other words a frame is a set of one-dimensional subspaces Li = FΒ·vi such that any k of them generate a k-dimensional subspace. Now an ordered frame L1, ..., Ln defi... | Wikipedia - Building (mathematics) | null | null | null |
Let F be a field of characteristic not equal to 2. A biquaternion algebra over F is a tensor product of two quaternion algebras.A biquaternion algebra is a central simple algebra of dimension 16 and degree 4 over the base field: it has exponent (order of its Brauer class in the Brauer group of F) equal to 1 or 2. | Wikipedia - Biquaternion algebra | null | null | null |
Let F be a field of characteristic not equal to 2. Let A = (a1,a2) and B = (b1,b2) be quaternion algebras over F. The algebras A and B are linked quaternion algebras over F if there is x in F such that A is equivalent to (x,y) and B is equivalent to (x,z). : 69 The Albert form for A, B is q = β¨ β a 1 , β a 2 , a 1 a 2 ... | Wikipedia - Linked field | null | null | null |
{\displaystyle q=\left\langle {-a_{1},-a_{2},a_{1}a_{2},b_{1},b_{2},-b_{1}b_{2}}\right\rangle \ .} It can be regarded as the difference in the Witt ring of the ternary forms attached to the imaginary subspaces of A and B. The quaternion algebras are linked if and only if the Albert form is isotropic. : 70 | Wikipedia - Linked field | null | null | null |
Let F be a field of characteristic zero, G a finite group and H a subgroup of G. Let F {\displaystyle F} denote the group algebra of G: the space of F-valued functions on G with the multiplication given by convolution. We write F {\displaystyle F} for the space of F-valued functions on G / H {\displaystyle G/H} . An ... | Wikipedia - Hecke algebra of a finite group | null | null | null |
{\displaystyle F=F^{H}.} Similarly, there is the identification R := End G β‘ ( F ) = F H Γ H {\displaystyle R:=\operatorname {End} _{G}(F)=F^{H\times H}} given by sending a G-linear map f to the value of f evaluated at the characteristic function of H. For each double coset H g H {\displaystyle HgH} , let T g {\displ... | Wikipedia - Hecke algebra of a finite group | null | null | null |
Let F n = F ( x 1 , β¦ , x n ) {\displaystyle F_{n}=F(x_{1},\dots ,x_{n})} be a free group of rank n β₯ 2 {\displaystyle n\geq 2} with a free basis X = { x 1 , β¦ , x n } {\displaystyle X=\{x_{1},\dots ,x_{n}\}} . The automorphism problem, or the automorphic equivalence problem for F n {\displaystyle F_{n}} asks, given tw... | Wikipedia - Primitive element (free group) | null | null | null |
Therefore, the automorphism problem for F n {\displaystyle F_{n}} is often formulated in terms of Out β‘ ( F n ) {\displaystyle \operatorname {Out} (F_{n})} -equivalence of conjugacy classes of elements of F n {\displaystyle F_{n}} . For an element w β F n {\displaystyle w\in F_{n}} , | w | X {\displaystyle |w|_{X}} den... | Wikipedia - Primitive element (free group) | null | null | null |
Let F {\displaystyle F} and F β² {\displaystyle F'} be the two fragments that need to be combined. There are two ways to do this: Merge: This operation occurs if both F {\displaystyle F} and F β² {\displaystyle F'} share a common minimum weight outgoing edge, and L e v e l ( F ) = L e v e l ( F β² ) {\displaystyle {\mathi... | Wikipedia - Distributed minimum spanning tree | null | null | null |
The combined fragment will have the same level as F β² {\displaystyle F'} .Furthermore, when an "Absorb" operation occurs, F {\displaystyle F} must be in the stage of changing the core, while F β² {\displaystyle F'} can be in an arbitrary stage. Therefore, "Absorb" operations may be done differently depending on the stat... | Wikipedia - Distributed minimum spanning tree | null | null | null |
There are two cases to consider: Node n β² {\displaystyle n'} has received broadcast message but it has not sent a convergecast message back to the core. In this case, fragment F {\displaystyle F} can simply join the broadcast process of F β² {\displaystyle F'} . Specifically, we image F {\displaystyle F} and F β² {\displ... | Wikipedia - Distributed minimum spanning tree | null | null | null |
In order to do that, node n β² {\displaystyle n'} can initiate a broadcast to F {\displaystyle F} to update the fragment ID of each node in F {\displaystyle F} and collect minimum weight outgoing edge in F {\displaystyle F} . Node n β² {\displaystyle n'} has already sent a convergecast message back to the core. Before no... | Wikipedia - Distributed minimum spanning tree | null | null | null |
As we discussed above, n β² {\displaystyle n'} does that by choosing its minimum weight basic edge, sending a test message to the other side of the chosen edge, and waiting for the response. Suppose e β² {\displaystyle e'} is the chosen edge, we can conclude the following: e β² β e {\displaystyle e'\neq e} w e i g h t ( e... | Wikipedia - Distributed minimum spanning tree | null | null | null |
Then, node n {\displaystyle n} will delay the response (according to case 3 of "Finding the minimum weight incident outgoing edge"). Then, it is impossible that n β² {\displaystyle n'} has already sent its convergecast message. By the aforementioned conclusions 1 and 2, we can conclude it is safe to absorb F {\displayst... | Wikipedia - Distributed minimum spanning tree | null | null | null |
Let F {\displaystyle F} be a field (a division ring actually suffices) and let V {\displaystyle V} be a vector space over F {\displaystyle F} . Then V {\displaystyle V} is a pregeometry where closures of sets are defined to be their span. The closed sets are the linear subspaces of V {\displaystyle V} and the notion of... | Wikipedia - Pregeometry (model theory) | null | null | null |
Vector spaces are considered to be the prototypical example of modularity. V {\displaystyle V} is locally finite if and only if F {\displaystyle F} is finite. V {\displaystyle V} is not a geometry, as the closure of any nontrivial vector is a subspace of size at least 2 {\displaystyle 2} . The associated geometry of a ... | Wikipedia - Pregeometry (model theory) | null | null | null |
Let F {\displaystyle F} be a field with algebraic closure F Β― {\displaystyle {\overline {F}}} . Then a F {\displaystyle F} -torus is an algebraic group defined over F {\displaystyle F} which is isomorphic over F Β― {\displaystyle {\overline {F}}} to a finite product of copies of the multiplicative group. In other words,... | Wikipedia - Algebraic torus | null | null | null |
The integer r {\displaystyle r} is called the rank or absolute rank of the torus T {\displaystyle \mathrm {T} } . The torus is said to be split over a field extension E / F {\displaystyle E/F} if T ( E ) β
( E Γ ) r {\displaystyle \mathbf {T} (E)\cong (E^{\times })^{r}} . There is a unique minimal finite extension of F... | Wikipedia - Algebraic torus | null | null | null |
The F {\displaystyle F} -rank of T {\displaystyle \mathbf {T} } is the maximal rank of a split sub-torus of T {\displaystyle \mathbf {T} } . A torus is split if and only if its F {\displaystyle F} -rank equals its absolute rank. A torus is said to be anisotropic if its F {\displaystyle F} -rank is zero. | Wikipedia - Algebraic torus | null | null | null |
Let F {\displaystyle F} be an arbitrary family of subsets of X . {\displaystyle X.} Then there exists a unique smallest Ο-algebra which contains every set in F {\displaystyle F} (even though F {\displaystyle F} may or may not itself be a Ο-algebra). It is, in fact, the intersection of all Ο-algebras containing F . | Wikipedia - Product sigma-algebra | null | null | null |
{\displaystyle F.} (See intersections of Ο-algebras above.) This Ο-algebra is denoted Ο ( F ) {\displaystyle \sigma (F)} and is called the Ο-algebra generated by F . | Wikipedia - Product sigma-algebra | null | null | null |
{\displaystyle F.} If F {\displaystyle F} is empty, then Ο ( β
) = { β
, X } . {\displaystyle \sigma (\varnothing )=\{\varnothing ,X\}.} | Wikipedia - Product sigma-algebra | null | null | null |
Otherwise Ο ( F ) {\displaystyle \sigma (F)} consists of all the subsets of X {\displaystyle X} that can be made from elements of F {\displaystyle F} by a countable number of complement, union and intersection operations. For a simple example, consider the set X = { 1 , 2 , 3 } . {\displaystyle X=\{1,2,3\}.} | Wikipedia - Product sigma-algebra | null | null | null |
Then the Ο-algebra generated by the single subset { 1 } {\displaystyle \{1\}} is Ο ( { 1 } ) = { β
, { 1 } , { 2 , 3 } , { 1 , 2 , 3 } } . {\displaystyle \sigma (\{1\})=\{\varnothing ,\{1\},\{2,3\},\{1,2,3\}\}.} By an abuse of notation, when a collection of subsets contains only one element, A , {\displaystyle A,} Ο ( ... | Wikipedia - Product sigma-algebra | null | null | null |
{\displaystyle \sigma (\{\{1\}\}).} Indeed, using Ο ( A 1 , A 2 , β¦ ) {\displaystyle \sigma \left(A_{1},A_{2},\ldots \right)} to mean Ο ( { A 1 , A 2 , β¦ } ) {\displaystyle \sigma \left(\left\{A_{1},A_{2},\ldots \right\}\right)} is also quite common. There are many families of subsets that generate useful Ο-algebras. S... | Wikipedia - Product sigma-algebra | null | null | null |
Let F {\displaystyle F} be an r {\displaystyle r} -grade multivector. Then we can define an additional pair of operators, the interior and exterior derivatives, β β
F = β¨ β F β© r β 1 = e i β
β i F , {\displaystyle \nabla \cdot F=\langle \nabla F\rangle _{r-1}=e^{i}\cdot \partial _{i}F,} β β§ F = β¨ β F β© r + 1 = e i β§ β ... | Wikipedia - Geometric calculus | null | null | null |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.