Algorithms for Factoring Square-Free Polynomials over
... not x divides either x(q−1)/2 − 1 or x(q−1)/2 + 1. If the factors are distributed in such a way that at least one divides each, then we can split m(x) by taking GCD(m(x), x(q−1)/2 − 1). However, it may be that the factors are not distributed in this way and therefore the GCD is trivial. Rabin showed ...
... not x divides either x(q−1)/2 − 1 or x(q−1)/2 + 1. If the factors are distributed in such a way that at least one divides each, then we can split m(x) by taking GCD(m(x), x(q−1)/2 − 1). However, it may be that the factors are not distributed in this way and therefore the GCD is trivial. Rabin showed ...
![arXiv:math/0105237v3 [math.DG] 8 Nov 2002](http://s1.studyres.com/store/data/017056594_1-99fcf3e85def2a6659feb2cb8a9b60c3-300x300.png)
arXiv:math/0105237v3 [math.DG] 8 Nov 2002
... as original Q̂), and certain natural properties hold. The space DM = T ∗ M with such a structure is called the double of M . The double DM so defined inherits half the original structure of M , a homological field. Using a linear connection on M , it is possible to define on DM an “almost” Schouten ...
... as original Q̂), and certain natural properties hold. The space DM = T ∗ M with such a structure is called the double of M . The double DM so defined inherits half the original structure of M , a homological field. Using a linear connection on M , it is possible to define on DM an “almost” Schouten ...
EXERCISES IN MA 510 : COMMUTATIVE ALGEBRA
... (ii) Show that C ∗ and H ∗ are projectively equivalent to P ∗ . (43) Let φ : P1 → P2 be given by φ((x0 : x1 )) = (x20 : x0 x1 : x21 ). Show that C = φ(P1 ) and P1 are isomorphic as projective varieties but their homogeneous coordinate rings are not. (44) The variety defined by a linear form is calle ...
... (ii) Show that C ∗ and H ∗ are projectively equivalent to P ∗ . (43) Let φ : P1 → P2 be given by φ((x0 : x1 )) = (x20 : x0 x1 : x21 ). Show that C = φ(P1 ) and P1 are isomorphic as projective varieties but their homogeneous coordinate rings are not. (44) The variety defined by a linear form is calle ...
Connectedness and local connectedness of topological groups and
... It is easy to verify (see Section 6 of [6]) that both F Γ(X) and AΓ(X), respectively the free topological and the free Abelian topological groups on a space X in the sense of Graev are connected iff X is connected. The situation in the case of the free (Abelian) topological group F (X) (resp., A(X)) ...
... It is easy to verify (see Section 6 of [6]) that both F Γ(X) and AΓ(X), respectively the free topological and the free Abelian topological groups on a space X in the sense of Graev are connected iff X is connected. The situation in the case of the free (Abelian) topological group F (X) (resp., A(X)) ...
13-2004 - Institut für Mathematik
... of a weak metric, introduced in [10], means that the topology induced by µ(k) on any tangent space TΦ Vir, Φ ∈ Vir, is weaker than the Fréchet topology on TΦ Vir. The aim of this paper is to show that results of classical Riemannian geometry concerning the geodesic exponential map induced by the me ...
... of a weak metric, introduced in [10], means that the topology induced by µ(k) on any tangent space TΦ Vir, Φ ∈ Vir, is weaker than the Fréchet topology on TΦ Vir. The aim of this paper is to show that results of classical Riemannian geometry concerning the geodesic exponential map induced by the me ...
Neighborly Polytopes and Sparse Solution of Underdetermined
... |I|-neighborly. In consequence, most sets I of size k are sets of local equivalence if and only if most k-dimensional ‘intrinsic’ sections are k-neighborly. The other notion - individual equivalence - asks that, for a fraction 1 − of vectors x with k nonzeros, x is both the sparsest solution to y ...
... |I|-neighborly. In consequence, most sets I of size k are sets of local equivalence if and only if most k-dimensional ‘intrinsic’ sections are k-neighborly. The other notion - individual equivalence - asks that, for a fraction 1 − of vectors x with k nonzeros, x is both the sparsest solution to y ...
Lecture 1: Lattice ideals and lattice basis ideals
... Proof. Let K (A) be the quotient field of K [A]. Then the Krull dimension of K [A] is equal to the transcendence degree trdeg(K (A)/K ) of K (A) over K . Let V ⊂ Qd be the Q-subspace of Qd generated by the column vectors of A. Then rank A = dimQ V . Let b1 , . . . , bm be a Q-basis of integer vecto ...
... Proof. Let K (A) be the quotient field of K [A]. Then the Krull dimension of K [A] is equal to the transcendence degree trdeg(K (A)/K ) of K (A) over K . Let V ⊂ Qd be the Q-subspace of Qd generated by the column vectors of A. Then rank A = dimQ V . Let b1 , . . . , bm be a Q-basis of integer vecto ...
"The Sieve Re-Imagined: Integer Factorization Methods"
... The QS is a method to find x and y with the property that x2 ≡ y 2 (mod n) and x 6≡ ±y (mod n) by sieving for smooth numbers over evaluations of quadratic polynomials. The Number Field Sieve (NFS) uses this same idea, but goes about finding x and y a bit differently from the Quadratic Sieve. An earl ...
... The QS is a method to find x and y with the property that x2 ≡ y 2 (mod n) and x 6≡ ±y (mod n) by sieving for smooth numbers over evaluations of quadratic polynomials. The Number Field Sieve (NFS) uses this same idea, but goes about finding x and y a bit differently from the Quadratic Sieve. An earl ...
FORMALIZATION OF A PLAUSIBLE INFERENCE
... pr1 (a1 ) ∈ X or (b) there exists r ∈ R(Z) such that (a1 ) ∈ r. If (a) holds then A∗ (1) = ∅ and moreover, Z(X ∪ A1 (1)) = Z(X) for A1 (1) = {pr1 (a1 )}. If (b) holds then from the definition of the set R(Z) applied for the set X and p-inference (a1 ), it follows that (pr2 (a1 ) = ∗ ⇒ pr1 (a1 ) ∈ Z( ...
... pr1 (a1 ) ∈ X or (b) there exists r ∈ R(Z) such that (a1 ) ∈ r. If (a) holds then A∗ (1) = ∅ and moreover, Z(X ∪ A1 (1)) = Z(X) for A1 (1) = {pr1 (a1 )}. If (b) holds then from the definition of the set R(Z) applied for the set X and p-inference (a1 ), it follows that (pr2 (a1 ) = ∗ ⇒ pr1 (a1 ) ∈ Z( ...
ON THE WEAK LEFSCHETZ PROPERTY FOR POWERS OF
... shown the following: Let I = h`t1 , . . . , `tn i ⊂ k[x1 , . . . , x4 ] with `i general linear forms. If n ∈ {5, 6, 7, 8} then the WLP fails, respectively, for t ≥ {3, 27, 140, 704}. A famous conjecture of Fröberg gives the expected Hilbert function for an ideal of s general forms of prescribed deg ...
... shown the following: Let I = h`t1 , . . . , `tn i ⊂ k[x1 , . . . , x4 ] with `i general linear forms. If n ∈ {5, 6, 7, 8} then the WLP fails, respectively, for t ≥ {3, 27, 140, 704}. A famous conjecture of Fröberg gives the expected Hilbert function for an ideal of s general forms of prescribed deg ...
B Linear Algebra: Matrices
... in which x1 = x11 , x2 = x22 and x3 = x33 . Are x and X the same thing? If so we could treat column vectors as one-column matrices and dispense with the distinction. Indeed in many contexts a column vector of order n may be treated as a matrix with a single column, i.e., as a matrix of order n × 1. ...
... in which x1 = x11 , x2 = x22 and x3 = x33 . Are x and X the same thing? If so we could treat column vectors as one-column matrices and dispense with the distinction. Indeed in many contexts a column vector of order n may be treated as a matrix with a single column, i.e., as a matrix of order n × 1. ...
Basis (linear algebra)
Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.A set of vectors in a vector space V is called a basis, or a set of basis vectors, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set. In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space V, every element of V can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector coordinates or components. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.