Factoring in Skew-Polynomial Rings over Finite Fields
... ring F[x; σ, δ] is then defined such that xa = σ(a)x + δ(a) for any a ∈ F. In this paper we only consider the case when δ = 0 and F is finite. Assume throughout this paper that F has size pω , where p is a prime and ω ≥ 1. For any f, g ∈ F[x; σ] we find that deg(fg) = deg f + deg g, where deg: F[x; ...
... ring F[x; σ, δ] is then defined such that xa = σ(a)x + δ(a) for any a ∈ F. In this paper we only consider the case when δ = 0 and F is finite. Assume throughout this paper that F has size pω , where p is a prime and ω ≥ 1. For any f, g ∈ F[x; σ] we find that deg(fg) = deg f + deg g, where deg: F[x; ...
diagram algebras, hecke algebras and decomposition numbers at
... applies when R is an algebraically closed field whose characteristic is either 0 or p > 0 such that pe > n, where e is the multiplicative order of q 2 . na (q) be the extended affine Hecke algebra of type A ...
... applies when R is an algebraically closed field whose characteristic is either 0 or p > 0 such that pe > n, where e is the multiplicative order of q 2 . na (q) be the extended affine Hecke algebra of type A ...
Automorphisms of 2--dimensional right
... Proposition 3.2 Let 2 Aut0 .A / be a pure automorphism of A and let J D U W be a maximal join in . Then maps AJ D F hU i F hW i to a conjugate of itself. Moreover, if U contains no leaves, then preserves the factor F hU i up to conjugacy. Proof It suffices to verify the proposition for ...
... Proposition 3.2 Let 2 Aut0 .A / be a pure automorphism of A and let J D U W be a maximal join in . Then maps AJ D F hU i F hW i to a conjugate of itself. Moreover, if U contains no leaves, then preserves the factor F hU i up to conjugacy. Proof It suffices to verify the proposition for ...
On the continuity of the inverses of strictly monotonic
... and subspace topology of A do not coincide whenever R \ A possesses a bounded component that is neither closed nor open. The converse is also true, see Corollary 3.5 below in the next section. 3. Strictly monotonic functions on subsets of connected linearly ordered spaces In this section we want to ...
... and subspace topology of A do not coincide whenever R \ A possesses a bounded component that is neither closed nor open. The converse is also true, see Corollary 3.5 below in the next section. 3. Strictly monotonic functions on subsets of connected linearly ordered spaces In this section we want to ...
Sample pages 2 PDF
... nontrivial examples. An extreme case is when the ring R itself is nilpotent. Example 2.4 Take any additive group R, and equip it with trivial product: xy = 0 for all x, y ∈ R. Then R2 = 0. Example 2.5 A nilpotent element lying in the center Z(R) of the ring R clearly generates a nilpotent ideal. A s ...
... nontrivial examples. An extreme case is when the ring R itself is nilpotent. Example 2.4 Take any additive group R, and equip it with trivial product: xy = 0 for all x, y ∈ R. Then R2 = 0. Example 2.5 A nilpotent element lying in the center Z(R) of the ring R clearly generates a nilpotent ideal. A s ...
24 pp. pdf
... rings, the algorithm retains the title of number field sieve. In this case, if q is prime, one of the rings can be taken to be Z as is done for factoring. However, it is often advantageous to use two non-trivial extensions of Z instead (see Section 2.2). When the chosen rings are polynomial rings, t ...
... rings, the algorithm retains the title of number field sieve. In this case, if q is prime, one of the rings can be taken to be Z as is done for factoring. However, it is often advantageous to use two non-trivial extensions of Z instead (see Section 2.2). When the chosen rings are polynomial rings, t ...
PDF - Cryptology ePrint Archive
... the number of elements. When #E(Fp ) = p, which means E has trace 1, E is called anomalous. If E 0 is a twist of E of degree 1, then #E(Fp ) = #E 0 (Fp ). If E 0 is a twist of E of degree ≥ 2, then #E(Fp ) 6= #E 0 (Fp ) in general. Let E be an elliptic curve over a finite field Fp (p ≥ 5) given by E ...
... the number of elements. When #E(Fp ) = p, which means E has trace 1, E is called anomalous. If E 0 is a twist of E of degree 1, then #E(Fp ) = #E 0 (Fp ). If E 0 is a twist of E of degree ≥ 2, then #E(Fp ) 6= #E 0 (Fp ) in general. Let E be an elliptic curve over a finite field Fp (p ≥ 5) given by E ...
NOTES FOR MATH 4510, FALL 2010 1. Metric Spaces The
... the exponents may now be negative. Fix a prime number p as before, and define n(x, y) in the same way, and use the same formula for the distance. For instance, if p = 5 we have, in addition to the examples given above, ...
... the exponents may now be negative. Fix a prime number p as before, and define n(x, y) in the same way, and use the same formula for the distance. For instance, if p = 5 we have, in addition to the examples given above, ...
Factorization in Integral Domains II
... is irreducible in Q[x], since it is a polynomial with integer coefficients whose reduction mod 2 is irreducible. (2) To see why we need to make some assumptions about the leading coefficient of f , or equivalently that deg f¯ = deg f , consider the polynomial f = (2x + 1)(3x + 1) = 6x2 + 5x + 1. Tak ...
... is irreducible in Q[x], since it is a polynomial with integer coefficients whose reduction mod 2 is irreducible. (2) To see why we need to make some assumptions about the leading coefficient of f , or equivalently that deg f¯ = deg f , consider the polynomial f = (2x + 1)(3x + 1) = 6x2 + 5x + 1. Tak ...
Let us assume that Y is a non-empty set. A function ψ : Y × Y → C is
... for any ξ, η ∈ H (note that this result also follows as a corollary to Proposition 1.1). U. Haagerup has shown that on the non-abelian free groups there are Herz–Schur multipliers which cannot be realized as coefficients of uniformly bounded representations. The proof by Haagerup has remained unpubl ...
... for any ξ, η ∈ H (note that this result also follows as a corollary to Proposition 1.1). U. Haagerup has shown that on the non-abelian free groups there are Herz–Schur multipliers which cannot be realized as coefficients of uniformly bounded representations. The proof by Haagerup has remained unpubl ...
Rewriting Systems for Coxeter Groups
... complete rewriting systems; a simplified version is as follows. To begin the KnuthBendix procedure, one must start with a finite set S of generators and a finite set E of equations sufficient to present the group or monoid involved. Put a partial well-founded ordering on S ∗ , which is compatible wi ...
... complete rewriting systems; a simplified version is as follows. To begin the KnuthBendix procedure, one must start with a finite set S of generators and a finite set E of equations sufficient to present the group or monoid involved. Put a partial well-founded ordering on S ∗ , which is compatible wi ...
Birkhoff's representation theorem
This is about lattice theory. For other similarly named results, see Birkhoff's theorem (disambiguation).In mathematics, Birkhoff's representation theorem for distributive lattices states that the elements of any finite distributive lattice can be represented as finite sets, in such a way that the lattice operations correspond to unions and intersections of sets. The theorem can be interpreted as providing a one-to-one correspondence between distributive lattices and partial orders, between quasi-ordinal knowledge spaces and preorders, or between finite topological spaces and preorders. It is named after Garrett Birkhoff, who published a proof of it in 1937.The name “Birkhoff's representation theorem” has also been applied to two other results of Birkhoff, one from 1935 on the representation of Boolean algebras as families of sets closed under union, intersection, and complement (so-called fields of sets, closely related to the rings of sets used by Birkhoff to represent distributive lattices), and Birkhoff's HSP theorem representing algebras as products of irreducible algebras. Birkhoff's representation theorem has also been called the fundamental theorem for finite distributive lattices.