We define a family of generalizations of the two-variable quandle polynomial. These polynomial invariants generalize in a natural way to eight-variable polynomial invariants of finite biquandles. We use these polynomials to define a family of link invariants that further generalize the quandle counting invariant.

A family of sets is said to possess property if there exists a set such that and F ⊄ B for each In [1], P. Erdos and A. Hajnal ask the following question: Does there exist for every positive integer k a finite family of finite sets satisfying

1. (i) |F|=k for each

1. (ii) | F∩ G| ≤ 1 for each F, , F ≠ G

1. (iii) does not possess property ?

In a previous work, we associated a complete differential graded Lie algebra to any finite simplicial complex in a functorial way. Similarly, we also have a realization functor fromthe category of complete differential graded Lie algebras to the category of simplicial sets. We have already interpreted the homology of a Lie algebra in terms of homotopy groups of its realization. In this paper, we begin a dictionary between models and simplicial complexes by establishing a correspondence between the Deligne groupoid of the model and the connected components of the finite simplicial complex.

A discrete group $G$ is called identity excluding if the only irreducible unitary representation of $G$ which weakly contains the 1-dimensional identity representation is the 1-dimensional identity representation itself. Given a unitary representation $\pi $ of $G$ and a probability measure $\mu $ on $G$, let ${{P}_{\mu }}$ denote the $\mu $-average $\int{\pi (g)\mu (dg)}$. The goal of this article is twofold: (1) to study the asymptotic behaviour of the powers $P_{\mu }^{n}$, and (2) to provide a characterization of countable amenable identity excluding groups. We prove that for every adapted probability measure $\mu $ on an identity excluding group and every unitary representation $\pi $ there exists and orthogonal projection ${{E}_{\mu }}$ onto a $\pi $-invariant subspace such that $s-{{\lim }_{n\to \infty }}\,(P_{\mu }^{n}\,-\,\pi {{(a)}^{n}}\,{{E}_{\mu }})\,\,=\,0$ for every $a\,\in $ supp $\mu $. This also remains true for suitably defined identity excluding locally compact groups. We show that the class of countable amenable identity excluding groups coincides with the class of $\text{FC}$-hypercentral groups; in the finitely generated case this is precisely the class of groups of polynomial growth. We also establish that every adapted random walk on a countable amenable identity excluding group is ergodic.

All rings R considered here are commutative with identity and all the modules are unital right modules. As defined by Mehdi [6] a module MR is said to be a multiplication module if for every pair of submodules K and N of M, K ⊂ N implies K=NA for some ideal A of R. This concept generalizes the well known concept of a multiplication ring.

A complex number λ will be said to be free if the multiplicative group Fλ generated by the two matrices

is a free group, and non-free, otherwise. Very little is known about the distribution of free and non-free numbers [1]. It is, for instance, unknown whether the domain

contains an open set which consists of only free points.

In this paper, we consolidate into one two separate problems - dependent random variables with independent subsets and construction of a joint distribution with given marginals. Let N = {1,2,3,...} and X = {Xn; n ∊ N} be a sequence of random variables with nondegenerate one-dimensional marginal distributions {Fn; n ∊ N}. An example is constructed to show that there exists a sequence of random variables Y = {Yn; n ∊ N} such that the components of a subset of Y are independent if and only if its size is ≦ k, where k ≧ 2 is a prefixed integer. Furthermore, the one-dimensional marginal distributions of Y are those of X.

The purpose of this note is to prove the following theorem conjectured by P. Erdös.

Theorem. Let a0, a1, …, ak be integers satisfying 1 ≤ a0 a1 … k, and let [ai-1 ai] denote the least common multiple of ai-1 and ai Then

1

with equality occurring if and only if ai = 2i for 1 ≤ i ≤ k.

Let G be a periodic group and ZG its integral group ring. The elements ±g(g∈G) are called the trivial units of ZG. In [1], S. D. Berman has shown that if G is finite, then every unit of finite order is trivial if and only if G is abelian or the direct product of a quaternion group of order 8 and an elementary abelin 2-group. By comparison, Losey in [7] has shown that if ZG contains one non-trivial unit of finite order, then it contains infinitely many.

Let $G$ be a connected graph with vertex set $V\left( G \right)$.The degree Kirchhoff index of $G$ is defined as ${{S}^{\prime }}\left( G \right)\,=\,\sum{_{\left\{ u,v \right\}\,\subseteq \,V\left( G \right)}d\left( u \right)d\left( v \right)R\left( u,\,v \right)}$, where $d\left( u \right)$ is the degree of vertex $u$, and $R\left( u,\,v \right)$ denotes the resistance distance between vertices $u$ and $v$. In this paper, we characterize the graphs having maximum and minimum degree Kirchhoff index among all $n$-vertex bicyclic graphs with exactly two cycles.

Rings (all of which are assumed to be associative) with no non-zero nilpotent elements will be called reduced rings; R is a reduced ring if and only if x2=0 implies x=0, for all x∈R. In 2. we prove that the following conditions on an annihilator ideal I of a reduced ring are equivalent: I is a maximal annihilator, I is prime, I is a minimal prime, I is completely prime. A characterization of reduced rings with the maximum condition on annihilators is given in 3.

We show that, if a Banach algebra $\mathfrak{A}$ is a left ideal in its second dual algebra and has a left bounded approximate identity, then the weak amenability of $\mathfrak{A}$ implies the $\left( 2m+1 \right)$-weak amenability of $\mathfrak{A}$ for all $m\,\ge \,1$.

The primary aim of this paper is to extend Barnes [1], Marshall-Olkin [6], and Nehari [8] inequalities as applications of some results introduced in [10] by the author.

Since several results from various sources are adopted here, a unified notation is required in order to simplify our subsequent arguments. To this end, let Lp = Lp(S, ∑, μ), p>0 (unless otherwise stated), be the space of all pth power non-negative integrable functions over a given finite measure space (S, ∑, μ) (where S may be regarded as a bounded subset of real numbers).

In this paper we study the topology of the space of harmonic maps from S2 to ℂℙ2.We prove that the subspaces consisting of maps of a fixed degree and energy are path connected. By a result of Guest and Ohnita it follows that the same is true for the space of harmonic maps to ℂℙn for n ≥ 2. We show that the components of maps to ℂℙ2 are complex manifolds.

It is known that if K is a field, then the ring of formal power series in one or more variables, with coefficients in K, is Henselian at its maximal ideal. In this note we show that if R is a ring (commutative and with identity element) which is Henselian at the maximal ideals M1, M2, … then R[[x]] - the ring of formal power series in x with coefficients from R - is also Henselian at the maximal ideals M1 ⋅ R[[x]] + x⋅ R[[x]], etc.

We provide necessary and sufficient conditions for a skew polynomial ring of derivation type to be semiprimitive when the base ring has no nonzero nil ideals. This extends existing results on the Jacobson radical of skew polynomial rings of derivation type.

Let $R$ be a prime ring with extended centroid $\text{C,Q}$ maximal right ring of quotients of $R$, $RC$ central closure of $R$ such that ${{\dim}_{C}}(RC)>4,f({{X}_{1}},...,{{X}_{n}})$ a multilinear polynomial over $C$ that is not central-valued on $R$, and $f(R)$ the set of all evaluations of the multilinear polynomial $f({{X}_{1}},...,{{X}_{n}})$ in $R$. Suppose that $G$ is a nonzero generalized derivation of $R$ such that ${{G}^{2}}(u)u\in C$ for all $u\in f(R)$. Then one of the following conditions holds:

• (i) there exists $a\in \text{Q}$ such that ${{a}^{2}}=0$ and either $G(x)=ax$ for all $x\in R$or $G(x)=xa$ for all $x\in R$;

• (ii) there exists $a\in \text{Q}$ such that $0\ne {{a}^{2}}\in C$ and either $G(x)=ax$ for all $x\in R$ or $G(x)=xa$ for all $x\in R$ and $f{{({{X}_{1}},...,{{X}_{n}})}^{2}}$is central-valued on $R$;

• (iii) char $(R)=2$ and one of the following holds:

• (a) there exist $a,b,\in \text{Q}$ such that $G(x)=ax+xb$ for all $x\in R$ and ${{a}^{2}}={{b}^{2}}\in C$;

• (b) there exist $a,b,\in \text{Q}$ such that $G(x)=ax+xb$ for all $x\in R,\,{{a}^{2}},{{b}^{2}}\in C$ and $f{{({{X}_{1}},...,{{X}_{n}})}^{2}}$ is central-valued on $R$;

• (c) there exist $a\in \text{Q}$ and an $X$-outer derivation $d$ of $R$ such that $G(x)=ax+d(x)$ for all $x\in R,{{d}^{2}}=0$ and ${{a}^{2}}+d(a)=0$;

• (d) there exist $a\in \text{Q}$ and an $X$-outer derivation $d$ of $R$ such that $G(x)=ax+d(x)$ for all $x\in R,\,{{d}^{2}}=0,\,{{a}^{2}}+d(a)\in C$ and $f{{({{X}_{1}},...,{{X}_{n}})}^{2}}$ is central-valued on $R$.

Moreover, we characterize the form of nonzero generalized derivations $G$ of $R$ satisfying ${{G}^{2}}(x)=\lambda x$ for all $x\in R$, where $\lambda \in C$.

About a hundred years ago, the author of Through the Looking-Glass wrote another book called Euclid and his Modern Rivals. Were these rivals Lobachevsky and Bolyai, Riemann and Schlafli? No, they were merely the authors of dull school textbooks that would soon be forgotten. The sad truth is that, in 1873, hardly anyone in England knew of the breakthrough that had occurred on the Continent some fifty years before: only Cayley in Cambridge, Clifford in London, and a few students. Even if Cayley or Clifford had visited Oxford and given a lecture there, it is doubtful that he would have succeeded in convincing the conservative Dodgson that Euclid's postulates could be modified to yield two new worlds, surpassing in strangeness the worlds of the two Alice books and yet just as logically consistent as Euclid.

Let (Ω, , P) be a probability space, and A1, A2… be a sequence of members of . The classical Borel-Cantelli problem is to determine the probability that infinitely many events Ak occur. The classical results may be found in Feller [2, p. 188]; while related work may be found in Spitzer [3, p. 317], and Dawson and Sankoff [1]. The latter works are generalizations of the Borel-Cantelli lemmas, taken in different directions.

In this paper, necessary and sufficient conditions will be given for infinitely many events Ak to occur, with probability 1. A lower bound for the probability that only finitely many Ak occur, is developed. In addition, necessary and sufficient conditions that only finitely many Ak occur, with probability 1, are given.

We prove an equivalence between weighted Poincaré inequalities and the existence of weak solutions to a Neumann problem related to a degenerate $p$-Laplacian. The Poincaré inequalities are formulated in the context of degenerate Sobolev spaces defined in terms of a quadratic form, and the associated matrix is the source of the degeneracy in the $p$-Laplacian.