Script 2013W 104.271 Discrete Mathematics VO (Gittenberger)
... Kruskal’s and Prim’s algorithm are greedy algorithms. These algorithms only work on a local view of the graph and they use these local values to solve the maximization (or minimization) problem. Since these algorithms are greedy, they generally don’t produce optimal maximal or minimal spanning trees ...
... Kruskal’s and Prim’s algorithm are greedy algorithms. These algorithms only work on a local view of the graph and they use these local values to solve the maximization (or minimization) problem. Since these algorithms are greedy, they generally don’t produce optimal maximal or minimal spanning trees ...
skew-primitive elements of quantum groups and braided lie algebras
... The central idea leading to the structure of braided Lie algebras is the concept of symmetrization. For any module P in the category of Yetter-Drinfel'd modules the n-th tensor power P of P has a natural braid structure. We construct submodules P ( ) P for any nonzero in the base eld k , that ...
... The central idea leading to the structure of braided Lie algebras is the concept of symmetrization. For any module P in the category of Yetter-Drinfel'd modules the n-th tensor power P of P has a natural braid structure. We construct submodules P ( ) P for any nonzero in the base eld k , that ...
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... G is a finite group. But by [EG1, Theorem 2.1], there exist a finite group G and a twist J ∈ k[G] ⊗ k[G] such that H ∼ = k[G]J as Hopf algebras. Hence the result follows from Theorem 3.3. Theorem 4.3. Let H be a semisimple and cosemisimple Hopf algebra over k. Then exp(H) divides dim(H)3 . Theorem 4.3 ...
... G is a finite group. But by [EG1, Theorem 2.1], there exist a finite group G and a twist J ∈ k[G] ⊗ k[G] such that H ∼ = k[G]J as Hopf algebras. Hence the result follows from Theorem 3.3. Theorem 4.3. Let H be a semisimple and cosemisimple Hopf algebra over k. Then exp(H) divides dim(H)3 . Theorem 4.3 ...
STRONGLY PRIME ALGEBRAIC LIE PI-ALGEBRAS
... Theorem L. Let L be a prime nondegenerate Lie algebra over a field F of characteristic 0. If L has an algebraic adjoint representation and satisfies a polynomial identity, then L is simple and finite dimensional over its centroid, this being an algebraic extension of F. This theorem is proved in [8] ...
... Theorem L. Let L be a prime nondegenerate Lie algebra over a field F of characteristic 0. If L has an algebraic adjoint representation and satisfies a polynomial identity, then L is simple and finite dimensional over its centroid, this being an algebraic extension of F. This theorem is proved in [8] ...
The Knot Quandle
... Knots that the quandle does allow us to distinguish are, for example 5-1 and the unknot, and 6-3 and 5-1. We couldn’t distinguish these knots using the 3-coloring invariant. Def. The 3-coloring invariant is the number of ways to color a knot diagram with three colors. To three color a diagram, each ...
... Knots that the quandle does allow us to distinguish are, for example 5-1 and the unknot, and 6-3 and 5-1. We couldn’t distinguish these knots using the 3-coloring invariant. Def. The 3-coloring invariant is the number of ways to color a knot diagram with three colors. To three color a diagram, each ...
Semisimplicity - UC Davis Mathematics
... algebra over a division ring. The following corollary is closer to the original version of Wedderburn’s theorem: Corollary 3.11. Let k be a field, and R a k-algebra which is finite-dimensional as a k-vector space, and has no two-sided ideals other than 0 and R. Then there is a division ring D contai ...
... algebra over a division ring. The following corollary is closer to the original version of Wedderburn’s theorem: Corollary 3.11. Let k be a field, and R a k-algebra which is finite-dimensional as a k-vector space, and has no two-sided ideals other than 0 and R. Then there is a division ring D contai ...
A non-archimedean Ax-Lindemann theorem - IMJ-PRG
... 2.1. Non-archimedean analytic spaces. — Given a complete non-archimedean valued field F , we shall consider in this paper F -analytic spaces in the sense of Berkovich [3]. However, the statements, and essentially the proofs, can be carried on mutatis mutandis in the rigid analytic setting. In this c ...
... 2.1. Non-archimedean analytic spaces. — Given a complete non-archimedean valued field F , we shall consider in this paper F -analytic spaces in the sense of Berkovich [3]. However, the statements, and essentially the proofs, can be carried on mutatis mutandis in the rigid analytic setting. In this c ...
Commutative Algebra Notes Introduction to Commutative Algebra
... such that t ≤ x for every t ∈ T . Finally, a maximal element in S is an element x ∈ S so that for all y such that x ≤ y, we have x = y. Theorem 1.2 (Zorn’s Lemma). If every chain T of S has an upper bound in S then S has at least one maximal element. Zorn’s Lemma is equivalent to the axiom of choice ...
... such that t ≤ x for every t ∈ T . Finally, a maximal element in S is an element x ∈ S so that for all y such that x ≤ y, we have x = y. Theorem 1.2 (Zorn’s Lemma). If every chain T of S has an upper bound in S then S has at least one maximal element. Zorn’s Lemma is equivalent to the axiom of choice ...
a set of postulates for ordinary complex algebra
... concepts—(the class of complex numbers, with the operations of addition and of multiplication, and the subclass of real numbers, with the relation of order)— and deducible from a small number of fundamental propositions, or hypotheses. The object of the present paper is to analyze these fundamental ...
... concepts—(the class of complex numbers, with the operations of addition and of multiplication, and the subclass of real numbers, with the relation of order)— and deducible from a small number of fundamental propositions, or hypotheses. The object of the present paper is to analyze these fundamental ...
decompositions of groups of invertible elements in a ring
... correspondence between subgroups of R∗ and R◦ . We will determine when a group of invertible elements in R can be written as a product of its subgroups. This is to say, given a group G and subgroups A, B we will write G = A · B if every g ∈ G can be uniquely factorized as g = ab where a ∈ A and b ∈ ...
... correspondence between subgroups of R∗ and R◦ . We will determine when a group of invertible elements in R can be written as a product of its subgroups. This is to say, given a group G and subgroups A, B we will write G = A · B if every g ∈ G can be uniquely factorized as g = ab where a ∈ A and b ∈ ...
categories - Andrew.cmu.edu
... any set X, the powerset P(X) is a poset under the usual inclusion relation U ⊆ V between the subsets U, V of X. What is a functor F : P → Q between poset categories P and Q? It must satisfy the identity and composition laws . . . . Clearly, these are just the monotone functions already considered ab ...
... any set X, the powerset P(X) is a poset under the usual inclusion relation U ⊆ V between the subsets U, V of X. What is a functor F : P → Q between poset categories P and Q? It must satisfy the identity and composition laws . . . . Clearly, these are just the monotone functions already considered ab ...
A convenient category for directed homotopy
... X if and only if fi (a) ≤ fi (b) for each i ∈ I. (2) An ordered set is a preordered set (A, ≤) where ≤ is also antisymmetric, i.e., if it satisfies (∀x, y)(x ≤ y ∧ y ≤ x → x = y). The category of ordered sets is not topological because the underlying functor to sets does not preserve colimits. All t ...
... X if and only if fi (a) ≤ fi (b) for each i ∈ I. (2) An ordered set is a preordered set (A, ≤) where ≤ is also antisymmetric, i.e., if it satisfies (∀x, y)(x ≤ y ∧ y ≤ x → x = y). The category of ordered sets is not topological because the underlying functor to sets does not preserve colimits. All t ...
Moduli Spaces of K3 Surfaces with Large Picard Number
... single lattice isomorphic to E82 ⊕ U 3 , which we will call the K3 lattice and denote by ΛK3 . Let X be a K3 surface, then the K3 lattice is just the lattice given by the cup product form on H 2 (X, Z). This is a rank 22 lattice, and hence it is by no means simple, however, it is this complexity tha ...
... single lattice isomorphic to E82 ⊕ U 3 , which we will call the K3 lattice and denote by ΛK3 . Let X be a K3 surface, then the K3 lattice is just the lattice given by the cup product form on H 2 (X, Z). This is a rank 22 lattice, and hence it is by no means simple, however, it is this complexity tha ...
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.