1.1. Khovanov’s Homology and Conjecture. In his 1999 paper [K1] in Duke Mathematical Journal, Mikhail Khovanov showed that the Jones link polynomial is the graded Euler characteristic of a bigraded homology module H (D) over Z[c], associated to a diagram D of the link. H (D) is the homology of a bigraded chain complex C(D) with a (1, 0)-bigraded differential. He also explained how each link cob...

We show that there exists a natural non-degenerate pairing of the homomorphism space between two neighbor standard modules over a quasi-hereditary algebra with the first extension space between the corresponding costandard modules and vise versa. Investigation of this phenomenon leads to a family of pairings involving standard, costandard and tilting modules. In the graded case, under some ”Kos...

Let Fn be a free group of rank n and F n the quotient group of Fn by the subgroup [Γn(3),Γn(3)][[Γn(2),Γn(2)],Γn(2)] where Γn(k) denotes the k-th subgroup of the lower central series of the free group Fn. In this paper, we determine the group structure of the graded quotients of the lower central series of the group F n by using a generalized Chen’s integration in free groups. Then we apply it ...

208 NOTICES OF THE AMS VOLUME 51, NUMBER 2 As far as I know, the notion of a quasi-morphism does not have much to do with category theory. This very natural idea underlies several interesting recent developments at the crossroads of algebra, topology, geometry, and dynamics. Other, perhaps better, names currently in use for the same concept are quasi-homomorphism and pseudo-character. Let G be ...

Resolution reasoners, when applied to set theory problems, typically suffer from “lack of focus.” MARS is a program that attempts to rectify this difficulty by exploiting the definition-like character of the set theory axioms. As in the case of its predecessor, SLIM, it employs a tableau proof procedure based on binary resolution, but MARS IS enhanced by an equality substitution rule and a devi...

10. Dirichlet characters and Dirichlet L-functions Definition 10.1. Let m ∈ N. A Dirichlet character modulo m is a function χ : N→ C that satisfies the following three conditions. (1) χ is periodic with period m, i.e. if a ≡ b (mod m) then χ(a) = χ(b). (2) χ is completely multiplicative, i.e. χ(ab) = χ(a)χ(b) for all a, b ∈ N. (3) χ(a) = 0 if and only if gcd(a,m) > 1. For m ∈ N we let (Z/mZ)× d...

The character formula of any finite dimensional irreducible module Lλ for Lie superalgebra osp(n|2) is computed. As a by-product, the decomposition of tensor module Lλ ⊗ C n|2, where Cn|2 is the natural representation, is obtained.

Let (R, m) and (S, n) be commutative Noetherian local rings, and let φ : R→ S be a flat local homomorphism such that mS = n and the induced map on residue fields R/m→ S/n is an isomorphism. Given a finitely generated R-module M , we show that M has an S-module structure compatible with the given R-module structure if and only if Ext R (S,M) = 0 for each i ≥ 1. We say that an S-module N is exten...

The Unified Medical Language System (UMLS) is an extensive source of biomedical knowledge developed and maintained by the U.S. National Library of Medicine (NLM). The UMLS began to include biomedical terms in other languages a few years ago. However, providing foreign terms for existing concepts is only the first step for the UMLS to become international. The current limits of the use of the UM...

Let $nin mathbb{N}$. An additive map $h:Ato B$ between algebras $A$ and $B$ is called $n$-Jordan homomorphism if $h(a^n)=(h(a))^n$ for all $ain A$. We show that every $n$-Jordan homomorphism between commutative Banach algebras is a $n$-ring homomorphism when $n < 8$. For these cases, every involutive $n$-Jordan homomorphism between commutative C-algebras is norm continuous.