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.

An infinite family of perfect, non-extremal Riemann surfaces is constructed, the first examples of this type of surfaces. The examples are based on normal subgroups of the modular group $\text{PSL}\left( 2,\,\mathbb{Z} \right)$ of level 6. They provide non-Euclidean analogues to the existence of perfect, non-extremal positive definite quadratic forms. The analogy uses the function syst which associates to every Riemann surface $M$ the length of a systole, which is a shortest closed geodesic of $M$.

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.

Let R be a ring with 1. All modules considered are to be unital left R-modules unless otherwise noted.

Definition. A σ-set for R is a nonempty set Σ of left ideals of R satisfying the following conditions:

1. (σ1) If I ∊ Σ, J is a left ideal of I, and J ⊇ I, then J ∊ Σ.

2. (σ2) If I ∊ Σ and r ∊ R, then Ir-1 = {s ∊ R | sr ∊ I} ∊ Σ.

3. (σ3) If I is a left ideal of R, J ∊ Σ, and It-1 ∊ Σ for each t ∊ J, then I ∊ Σ.

It is known that a unital simple ${{C}^{*}}$-algebra $A$ with tracial topological rank zero has real rank zero. We show in this note that, in general, there are unital ${{C}^{*}}$-algebras with tracial topological rank zero that have real rank other than zero.

Let $0\,\to \,J\,\to \,E\,\to A\,\to \,0$ be a short exact sequence of ${{C}^{*}}$-algebras. Suppose that $J$ and $A$ have tracial topological rank zero. It is known that $E$ has tracial topological rank zero as a ${{C}^{*}}$-algebra if and only if $E$ is tracially quasidiagonal as an extension. We present an example of a tracially quasidiagonal extension which is not quasidiagonal.

We characterize those linear projections represented as a convex combination of two surjective isometries on standard Banach spaces of continuous functions with values in a strictly convex Banach space.

Since 1963 several authors ([13], [2], [6], [14], [10], [11], [12], [9]) have considered integral equations each one of which is contained as a special case in one of the two equations

1.1

1.2

for Re b > 0 and x ∊ [α, β].

A vector measure (countable additive set function with values in a Banach space) on a field may be extended to a vector measure on the generated σ- field, under certain hypotheses. For example, the extension is established for the bounded variation case [2, 5, 8], and there are more general conditions under which the extension exists [ 1 ]. The above results have as hypotheses fairly strong boundedness conditions on the n o rm of the measure to be extended. In this paper we prove an extension theorem of the same type with a restriction on the range, supposing further that the measure is merely bounded. In fact a vector measure on a σ- field is bounded (III. 4. 5 of [3]) but it is conceivable that a vector measure on a field could be unbounded.

The non-separable Banach space of right continuous functions with left hand limits and the supremum norm is investigated to find the isomorphic types of complemented subspaces. It is shown that every isometric isomorph of c0 is complemented in this space which may be identified as a non-separable C(K) space. Sufficient conditions are given for other isomorphs of C0 to be complemented in the space and the complement of a C0 subspace is characterized isomorphically.

In 1993, N. Danikas and A. G. Siskakis showed that the Cesàro operator ${\mathcal{C}}$ is not bounded on $H^{\infty }$; that is, ${\mathcal{C}}(H^{\infty })\nsubseteq H^{\infty }$, but ${\mathcal{C}}(H^{\infty })$ is a subset of $BMOA$. In 1997, M. Essén and J. Xiao gave that ${\mathcal{C}}(H^{\infty })\subsetneq {\mathcal{Q}}_{p}$ for every $0<p<1$. In this paper, we characterize positive Borel measures $\unicode[STIX]{x1D707}$ such that ${\mathcal{C}}(H^{\infty })\subseteq M({\mathcal{D}}_{\unicode[STIX]{x1D707}})$ and show that ${\mathcal{C}}(H^{\infty })\subsetneq M({\mathcal{D}}_{\unicode[STIX]{x1D707}_{0}})\subsetneq \bigcap _{0<p<\infty }{\mathcal{Q}}_{p}$ by constructing some measures $\unicode[STIX]{x1D707}_{0}$. Here, $M({\mathcal{D}}_{\unicode[STIX]{x1D707}})$ denotes the Möbius invariant function space generated by ${\mathcal{D}}_{\unicode[STIX]{x1D707}}$, where ${\mathcal{D}}_{\unicode[STIX]{x1D707}}$ is a Dirichlet space with superharmonic weight induced by a positive Borel measure $\unicode[STIX]{x1D707}$ on the open unit disk. Our conclusions improve results mentioned above.

A family of sets is said to have property B if there exists a set S such that S∩F≠ ϕ and SF for all F . S is called a B-set for . Let n≥2 and N≥2n-1. Let V = { 1, 2,≠, N} and let = {G:G⊂ V, |G| = rc}. Erdös [3] defined mN(n) to be the size of a smallest subfamily of which does not have property B and proved the following results:

Let Un be an n-dimensional unitary space with inner product In Un let Sn-1 denote the unit sphere:

Let A be an arbitrary linear transformation of Un.

This paper uses both the maximum principle and coupling method to study gradient estimates of positive solutions to Lu = 0 on Rd, where

with (aij) uniformly positive definite and aij,bi € C1(Rd). We obtain some upper bounds of |∇u|/u and ∥∇u∥∞/∥u∥∞, which imply a Harnack inequality and improve the corresponding results proved in Cranston [4]. Besides, two examples show that our estimates can be sharp.

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.

The purpose of this note is to give a new characterization of Prüfer domains using the concept of ring epimorphism, and to indicate some connections with well-known properties of Prüfer domains. All rings are commutative and have a unit element.

The normalized eigenvalues ${{\Lambda }_{i}}\left( M,\,g \right)$ of the Laplace–Beltrami operator can be considered as functionals on the space of all Riemannian metrics $g$ on a fixed surface $M$. In recent papers several explicit examples of extremal metrics were provided. These metrics are induced by minimal immersions of surfaces in ${{\mathbb{S}}^{3}}$ or ${{\mathbb{S}}^{4}}$. In this paper a family of extremal metrics induced by minimal immersions in ${{\mathbb{S}}^{5}}$ is investigated.

For α ∈ [0, 1] the operator is the operator formally defined on the Hardy space H2 by

If α = 1, then the usual identification of H2 with l2 takes A1 onto the discrete Cesàro operator. Here we see that {Aα: α ∈ [0, 1]} is not arcwise connected, that Re Aα ≥ 0, that Aα is a Hilbert-Schmidt operator if α ∈[0, 1), and that Aα is neither normaloid nor spectraloid if α ∈(0, 1).

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.

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.

Associated with any near-ring R, and any set S of distributive elements of R, one has a natural notion of S-distributive R-modules, analogous to that of modules over rings and including the latter as special case (Frohlich [6]; we recall the details in Section 1). Since near-rings can be viewed as slightly deficient rings, it makes sense to enquire whether such near-ring modules share with modules over rings the familiar and important property of having injective hulls.