In this paper we investigate some subclasses of strongly regular congruences on an $E$-inversive semigroup $S$. We describe the minimum and the maximum strongly orthodox congruences on $S$ whose characteristic trace coincides with the characteristic trace of given congruences and, in each case, we present an alternative characterization for them. A description of all strongly orthodox congruences on $S$ with characteristic trace $\tau $ is given. Further, we investigate the kernel relation of strongly orthodox congruences on an $E$-inversive semigroup and give the least and the greatest element in the class of the same kernel with a given congruence.

The notion of duality of functors is used to study and characterize spaces satisfying the Radon-Nikodym property. A theorem of equivalences concerning the Radon-Nikodym property is proved by categorical means; the classical Dunford-Pettis theorem is then deduced using an adjointness argument. The functorial properties of integral operators, compact operators, and weakly compact operators are discussed. It is shown that as an instance of Kan extension the weakly compact operators can be expressed as a certain direct limit of ordinary hom functors. Characterizations of spaces satisfying the Radon-Nikodym property are then given in terms of the agreement of dual functors of the functors mentioned above.

Let R be a ring and M a left R-module. We investigate when the functors HomR(M, —), HomR( —, M), and — ⊗RM are exact for certain restricted subcategories of modules.

A Fitting class of finite soluble groups is one closed under the formation of normal subgroups and products of normal subgroups. It is shown that the Fitting classes of metanilpotent groups which are quotient group closed as well are primitive saturated formuations.

The Laplace-Stieltjes Transform m(s) of the matrix renewal function M(t) of a Markov Renewal process is expanded in powers of the argument s, in this paper, by using a generalized inverse of the matrix I–P0, where P0 is the transition probability matrix of the imbedded Markov chain. This helps in obtaining the values of moments of any order of the number of renewals and also of the moments of the first passage times, for large values of t, the time. All the results of renewal theory are hidden under the Laplacian curtain and this expansion helps to lift this curtain at least for large values of t and is thus useful in predicting the number of renewals.

An (n + 1, n2 + n + 1)-packing is a collection of blocks, each of size n + 1, chosen from a set of size n2 + n + 1, such that no pair of points is contained in more than one block. If any two blocks contain a common point, then the packing can be extended to a projective plane of order n, provided the number of blocks is sufficiently large. We study packings which have a pair of disjoint blocks (such a packing clearly cannot be extended to a projective plane of order n). No such packing can contain more than n2 + n/2 blocks. Also, if n is the order of a projective plane, then we can construct such a packing with n2 + 1 blocks.

We establish existence of weighted Hardy and Rellich inequalities on the spaces $L_{p}(\unicode[STIX]{x1D6FA})$, where $\unicode[STIX]{x1D6FA}=\mathbf{R}^{d}\backslash K$ with $K$ a closed convex subset of $\mathbf{R}^{d}$. Let $\unicode[STIX]{x1D6E4}=\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ denote the boundary of $\unicode[STIX]{x1D6FA}$ and $d_{\unicode[STIX]{x1D6E4}}$ the Euclidean distance to $\unicode[STIX]{x1D6E4}$. We consider weighting functions $c_{\unicode[STIX]{x1D6FA}}=c\circ d_{\unicode[STIX]{x1D6E4}}$ with $c(s)=s^{\unicode[STIX]{x1D6FF}}(1+s)^{\unicode[STIX]{x1D6FF}^{\prime }-\unicode[STIX]{x1D6FF}}$ and $\unicode[STIX]{x1D6FF},\unicode[STIX]{x1D6FF}^{\prime }\geq 0$. Then the Hardy inequalities take the form

$$\begin{eqnarray}\int _{\unicode[STIX]{x1D6FA}}c_{\unicode[STIX]{x1D6FA}}|\unicode[STIX]{x1D6FB}\unicode[STIX]{x1D711}|^{p}\geq b_{p}\int _{\unicode[STIX]{x1D6FA}}c_{\unicode[STIX]{x1D6FA}}\,d_{\unicode[STIX]{x1D6E4}}^{-p}|\unicode[STIX]{x1D711}|^{p}\end{eqnarray}$$

$$\begin{eqnarray}\int _{\unicode[STIX]{x1D6FA}}|H\unicode[STIX]{x1D711}|^{p}\geq d_{p}\int _{\unicode[STIX]{x1D6FA}}|c_{\unicode[STIX]{x1D6FA}}\,d_{\unicode[STIX]{x1D6E4}}^{-2}\unicode[STIX]{x1D711}|^{p}\end{eqnarray}$$

$H=-\text{div}(c_{\unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D6FB})$

$b_{p},d_{p}$

$\unicode[STIX]{x1D6FF},\unicode[STIX]{x1D6FF}^{\prime }\geq 0$

For any positive integer q≧2, let Fq be a finite field with q elements, Fq ((z-1)) be the field of all formal Laurent series in an inderminate z, I denote the valuation ideal z-1Fq [[z-1]] in the ring of formal power series Fq ((z-1)) normalized by P(l) = 1. For any x ∈ I, let the series be the Engel expansin of Laurent series of x. Grabner and Knopfmacher have shown that the P-measure of the set A(α) = {x ∞ I: limn→∞ deg an(x)/n = ά} is l when α = q/(q -l), where deg an(x) is the degree of polynomial an(x). In this paper, we prove that for any α ≧ l, A(α) has Hausdorff dimension l. Among other thing we also show that for any integer m, the following set B(m) = {x ∈ l: deg an+1(x) - deg an(x) = m for any n ≧ l} has Hausdorff dimension 1.

A graph G is divisible by t if its edge set can be partitioned into t subsets, such that the subgraphs (called factors) induced by the subsets are all isomorphic. Such an edge partition is an isomorphic factorization. It is proved that a 2k-regular graph with an even number of vertices is divisble by 2k provided it contains either no 3-cycles or no 5-cycles. It is also shown that any 4-regular graph with an even number of vertices is divisible by 4. In both cases the components of the factors found are paths of length 1 and 2, and the factorizations can be constructed in polynomial time.

A Banach algebra $A$ is said to be a zero Jordan product determined Banach algebra if, for every Banach space $X$, every bilinear map $\unicode[STIX]{x1D711}:A\times A\rightarrow X$ satisfying $\unicode[STIX]{x1D711}(a,b)=0$ whenever $a$, $b\in A$ are such that $ab+ba=0$, is of the form $\unicode[STIX]{x1D711}(a,b)=\unicode[STIX]{x1D70E}(ab+ba)$ for some continuous linear map $\unicode[STIX]{x1D70E}$. We show that all $C^{\ast }$-algebras and all group algebras $L^{1}(G)$ of amenable locally compact groups have this property and also discuss some applications.

The object of this note is to extend Dini's theorem about (monotonic) sequences of continuous functions on a compact topological space to the case where the underlying domain is an abstract group which is free from topological restrictions. Continuous functions are replaced by almost periodic real-valued functions and the main result may be stated as follows: If a monotonically increasing sequence (fn) of almost periodic real-valued functions on a group G converges pointwise to an almost periodic function f on G, then the sequence converges to f uniformly. The basic idea of the present (elementary) proof is due to v. Kampen [2] and A. Weil [4], i.e., every almost periodic function on a group induces a kind of compact topology in it, relative to which the function is continuous. We modify this idea with the aid of the mean-value of an almost periodic function and obtain a pseudometric topology. This topology facilitates convergence proofs greatly. Moreover, it turns out to be equivalent with the previous one (Lemma 1). No use will be made of the theory of bounded matrix representations. This is significant as any use of the ‘Approximation Theorem’ [3, p. 66, see also p. 226] would violate the claim of an elementary proof.

Let G be a group of order pn where p is some prime. Denote by nr (G) the number of subgroups of G of order pr. If H is the elementary abelian group of order pn, then.

We investigate the problem of the existence of filters with some properties. This leads to a solution of two problems of Ulam concerning σ-fields on the real line.

Let ρ and σ be two congruences on a completely 0-simple semigroup. Suppose that there is a maximal chain of congruences from ρ to σ which is of finite length. Then, as we shall show, any maximal chain of congruences from ρ to σ finite and of the same length.

A λ-graph system is a labeled Bratteli diagram with shift transformation. It is a generalization of finite labeled graphs and presents a subshift. In Doc. Math. 7 (2002) 1–30, the author constructed a C*-algebra O£ associated with a λ-graph system £ from a graph theoretic view-point. If a λ-graph system comes from a finite labeled graph, the algebra becomes a Cuntz-Krieger algebra. In this paper, we prove that there is a bijective correspondence between the lattice of all saturated hereditary subsets of £ and the lattice of all ideals of the algebra O£, under a certain condition on £ called (II). As a result, the class of the C*-algebras associated with λ-graph systems under condition (II) is closed under quotients by its ideals.

Certain semigroups of continuous selfmaps of the closed unit interval are shown to have the property that all their automorphisms are inner. Contrary to expectation, certain other such semigroups do have outer automorphisms.

1980 Mathematics subject classification (Amer. Math. Soc.): primary 20 M 20; secondary 54 C 40.

We consider a multi-parameter family of canonical coordinates and mirror maps originally introduced by Zudilin. This family includes many of the known one-variable mirror maps as special cases, in particular many of modular origin and the celebrated ‘quintic’ example of Candelas, de la Ossa, Green and Parkes. In a previous paper, we proved that all coefficients in the Taylor expansions at 0 of these canonical coordinates (and, hence, of the corresponding mirror maps) are integers. Here we prove that all coefficients in their Taylor expansions at 0 are positive. Furthermore, we provide several results about the behaviour of the canonical coordinates and mirror maps as complex functions. In particular, we address their analytic continuation, points of singularity, and radius of covergence. We present several very precise conjectures on the radius of covergence of the mirror maps and the sign pattern of the coefficients in their Taylor expansions at 0.

In this paper, we consider contact metric three-manifolds $(M;\unicode[STIX]{x1D702},g,\unicode[STIX]{x1D711},\unicode[STIX]{x1D709})$ which satisfy the condition $\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x1D709}}h=\unicode[STIX]{x1D707}h\unicode[STIX]{x1D711}+\unicode[STIX]{x1D708}h$ for some smooth functions $\unicode[STIX]{x1D707}$ and $\unicode[STIX]{x1D708}$, where $2h=\unicode[STIX]{x00A3}_{\unicode[STIX]{x1D709}}\unicode[STIX]{x1D711}$. We prove that if $M$ is conformally flat and if $\unicode[STIX]{x1D707}$ is constant, then $M$ is either a flat manifold or a Sasakian manifold of constant curvature $+1$. We cannot extend this result for a smooth function $\unicode[STIX]{x1D707}$. Indeed, we give such an example of a conformally flat contact three-manifold which is not of constant curvature.

In this paper we describe the moduli spaces of degree d branched superminimal immersions of a compact Riemann surface of genus g into S4. We prove that when d ≥ max {2g, g + 2}, such spaces have the structure of projectivzed fibre products and are path-connected quasi projective varieties of dimension 2d − g + 4. This generalizes known results for spaces of harmonic 2-spheres in S4.

Given two C1 -functions g: R → R, u: [0,1] → R such that u(0) = u(1) = 0, g(0) = 0, we prove that there exists c, with 0 < c < 1, such that u′(c) = g(u(c)). This result implies the classical Rolle's Theorem when g ≡ 0. Next we apply our result to prove the existence of solutions of the Dirichlet problem for the equation x′ = f(t, x, x′).