Every continuous Volterra right inverse to the derivative in the space of complex-valued infinitely differentiable functions has the form of an integral.

An exponential polynomial is a finite linear combination of terms unea: t→tneat where n is any non-negative integer and a is any complex number. The set X of exponetial polynomials is clearly a vector space over the field of complex numbers C and this set is identical with the set of solutions to all homogeneous linear ordinary differential equations with constant coefficients.

It is well known that if M is a module with finite spanning dimension, then one can talk of Sd(K), the spanning dimension of K only when K is a supplement submodule in M. In this paper we extend this concept to general submodules and obtained some important results. We characterize the set of all supplement submodules of the module R/(x) over R where R is a Euclidean domain and x ∈ R. Moreover, it is proved that the number of distinct supplements in R/(x) is 2k and Sd(R/(x)) = k where k is the number of distinct nonassociate prime factors of x.

In this paper we study commutative topological semigroups S admitting an absolutely continuous measure. When S is cancellative we show that S admits a weaker topology J with respect to which (S, J) is embeddable as a subsemigroup with non-empty interior in some locally compact topological group. As a consequence, we deduce certain results related to the existence of invariant measures on S and for a large class of locally compact topological semigroups S, we associate S with some useful topological subsemigroup of a locally compact group.

By using methods of subordinacy theory, we study packing continuity properties of spectral measures of discrete one-dimensional Schrödinger operators acting on the whole line. Then we apply these methods to Sturmian operators with rotation numbers of quasibounded density to show that they have purely $\unicode[STIX]{x1D6FC}$-packing continuous spectrum. A dimensional stability result is also mentioned.

It is shown that if a1,…,am are relatively prime integers then for every integer n the equation has infinitely many solutions in pairwise relatively prime integers x1,…,xm.

A subgroup A of a group G is said to be complemented in G if G contains a subgroup C such that . Every subgroup C with its property is called a complement of A in G. Various results have been obtained about groups in which each member of a given set of subgroups is complemented. Some of these results state, roughly speaking, that the existence of complements of all members of a given set of subgroups implies that all members of larger set are also complements. In this paper we derive another theorem of this kind.

A recursive construction for orthogonal diagonal latin squares, using group divisible designs, is presented. In consequence the numbers of orders for which the existence of such squares is in question is reduced to 72.

The inverse semigroup TE of isomorphisms of principal ideals of E onto principal ideals of E, where E is a semilattice, has been introduced and studied by Munn (1966, 1970). He showed that, for any inverse semigroup S with semilattice E, there is a representation of S by an inverse subsemigroup of TE. The Munn representation, however, is not always faithful. In this paper, the possibility is considered of enlarging the carrier set E of the Munn representation in order to obtain a faithful representation of S as an inverse subsemigroup of a structure resembling TE in many ways. A structure X is obtained by replacing each element of E by a set. Then X = ∪{Xe: e ∈ E}, where Xe, denotes some set, has a natural pre-order relation ≤ (where x ≤ y if and only if x ∈ Xe, y ∈ Xf and e ≦ f ) inherited from E such that if T = {(x, y)∈X × X;x ≤ y and y ≤ x} then X/T is isomorphic to E. Such a set X is referred to as a pre-semilattice with semilattice E. If Tx denotes the set of all isomorphisms of principal ideals of X onto principal ideals of X then Tx is an inverse semigroup. Basic properties of Tx are considered. It is shown that when X is locally uniform, that is, when |Xe| = |Xf|, for all e, f ∈ E, Tx may be described as a wreath product of a permutation group with TE.

The set s itself is a presemilattice with semilattice E with respect to the pre-order ≤ defined by a ≤ b if and only if a−1a ≦ b−1b. It is then shown that the Vagner-Preston representation embeds S as a full inverse subsemigroup of Ts. As an application of these concepts the following result is established. Let R and S be inverse semigroups and let θ1(θ2) be an isomorphism of a semilattice E onto the semilattice of R(S). Then there exists a locally uniform presemilattice W and embeddings ϕ1, ϕ2 of R and S, respectively, as full inverse subsemigroups of Tw such that (1) θ1ϕ1 = θ2ϕ2 and (2) (eθ1ϕ1, eθ2ϕ2) ∈ if and only if Ee is isomorphic to Ef.

Under certain general conditions an n x n stochastic matrix P = (pij), where

is known to possess the “ergodic” property. This means that there exists a finite matrix E such that

In this paper we obtain some improved $L^p$ -Hardy and $L^p$ -Rellich inequalities on bounded domains of Riemannian manifolds. For Cartan–Hadamard manifolds we prove the inequalities with sharp constants and with weights being hyperbolic functions of the Riemannian distance.

Let p be a class of finite soluble groups which is closed under epimorphic images and let g be a saturated formation. Then if G is a group of minimal order belonging to p but not to g, F(G), the Fitting subgroup of G, is the unique minimal normal subgroup of G. It is to groups with this property that the following proposition is applicable.

The aim of this paper is to study the heat kernel and the jump kernel of the Dirichlet form associated to the ultrametric Cantor set $\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$ that is the infinite path space of the stationary $k$-Bratteli diagram ${\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$, where $\unicode[STIX]{x1D6EC}$ is a finite strongly connected $k$-graph. The Dirichlet form which we are interested in is induced by an even spectral triple $(C_{\operatorname{Lip}}(\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}),\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D719}},{\mathcal{H}},D,\unicode[STIX]{x1D6E4})$ and is given by

$$\begin{eqnarray}Q_{s}(f,g)=\frac{1}{2}\int _{\unicode[STIX]{x1D6EF}}\operatorname{Tr}(|D|^{-s}[D,\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D719}}(f)]^{\ast }[D,\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D719}}(g)])\,d\unicode[STIX]{x1D708}(\unicode[STIX]{x1D719}),\end{eqnarray}$$

$\unicode[STIX]{x1D6EF}$

$\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}\times \unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$

$d^{(s)}$

$d_{w_{\unicode[STIX]{x1D6FF}}}$

$\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$

$\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$

$d^{(s)}$

$\unicode[STIX]{x1D6E5}_{s}$

$Q_{s}$

$d_{w_{\unicode[STIX]{x1D6FF}}}$

$w_{\unicode[STIX]{x1D6FF}}$

${\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$

$\unicode[STIX]{x1D6FF}\in (0,1)$

$\unicode[STIX]{x1D707}$

$\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$

$d^{(s)}$

$d_{w_{\unicode[STIX]{x1D6FF}}}$

$Q_{s}$

$Q_{s}$

${\mathcal{Q}}_{J_{s},\unicode[STIX]{x1D707}}$

$J_{s}$

$\unicode[STIX]{x1D707}$

$\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$

A group G is called a c-group if each of its subnormal subgroups is characteristic in G. It is the object of this note to give a characterization of finite solvable c-groups.

We use the exterior and composition products of double forms together with the alternating operator to reformulate Pontrjagin classes and all Pontrjagin numbers in terms of the Riemannian curvature. We show that the alternating operator is obtained by a succession of applications of the first Bianchi sum and we prove some useful identities relating the previous four operations on double forms. As an application, we prove that for a $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}k$-conformally flat manifold of dimension $n\geq 4k$, the Pontrjagin classes $P_i$ vanish for any $i\geq k$. Finally, we study the equality case in an inequality of Thorpe between the Euler–Poincaré characteristic and the $k{\rm th}$ Pontrjagin number of a $4k$-dimensional Thorpe manifold.

Using a result on arithmetic progressions, we describe a method for finding the rational h–tuples ρ = (ρl,…,ρh) such that all the multiples mρ (for m coprime to a denominator of ρ) lie in a linear variety modulo Z. We give an application to hypergeometric functions.

A two-term Edgeworth expansion for the distribution of an M-estimator of a simple linear regression parameter is obtained without assuming any Cramér-type conditions. As an application, it is shown that certain modification of the naive bootstrap procedure is second order correct even when the error variables have a lattice distribution. This is in marked contrast with the results of Singh on the sample mean of independent and identically distributed random variables.

A subset S of an associative ring R is a uniform insulator for R provided a S b ≠ 0 for any nonzero a, b ∈ R. The ring R is called uniformly strongly prime of bound m if R has uniform insulators and the smallest of those has cardinality m. Here we compute these bounds for matrix rings over fields and obtain refinements of some results of van den Berg in this context.

More precisely, for a field F and a positive integer k, let m be the bound of the matrix ring Mk(F), and let n be dimF(V), where V is a subspace of Mk(F) of maximal dimension with respect to not containing rank one matrices. We show that m + n = k2. This implies, for example, that n = k2 − k if and only if there exists a (nonassociative) division algebra over F of dimension k.

Taking residual finiteness as a starting point, we consider three related finiteness properties: weak subsemigroup separability, strong subsemigroup separability and complete separability. We investigate whether each of these properties is inherited by Schützenberger groups. The main result of this paper states that for a finitely generated commutative semigroup S, these three separability conditions coincide and are equivalent to every $\mathcal {H}$ -class of S being finite. We also provide examples to show that these properties in general differ for commutative semigroups and finitely generated semigroups. For a semigroup with finitely many $\mathcal {H}$ -classes, we investigate whether it has one of these properties if and only if all its Schützenberger groups have the property.

We study Frobenius groups acting on curves.