In this paper, we get a necessary and sufficient condition for the normalizers of higher dimensional Kleinian groups to be discrete. Also we obtain a necessary and sufficient condition for the isomorphisms between two higher dimensional Kleinian groups induced by quasiconformal mappings to be the same.

Let $n$ and $r$ be positive integers with $1\,{<}\,r\,{<}\,n$, and let $X_n\,{=}\break\{1,2,\ldots,n\}$. An $r$-set $A$ and a partition $\pi$ of $X_n$ are said to be orthogonal if every class of $\pi$ meets $A$ in exactly one element. We prove that if $A_{1},A_{2},\ldots, A_{\binom n r}$ is a list of the distinct $r$-sets of $X_ n$ with $|A_{i}\cap A_{i+1}|\,{=}\,r-1$ for $i=1,2,\ldots, \binom n r$ taken modulo $\binom n r$, then there exists a list of distinct partitions $\pi_{1},\pi_{2},\ldots, \pi_{\binom n r}$ such that $\pi_{i}$ is orthogonal to both $A_{i}$ and $A_{i+1}$. This result states that any constant weight Gray code admits a labeling by distinct orthogonal partitions. Using an algorithm from the literature on Gray codes, we provide a surprisingly efficient algorithm that on input $(n,r)$ outputs an orthogonally labeled constant weight Gray code. We also prove a two-fold Gray enumeration result, presenting an orthogonally labeled constant weight Gray code in which the partition labels form a cycle in the covering graph of the lattice of all partitions of $X_n$. This leads to a conjecture related to the Middle Levels Conjecture. Finally, we provide an application of our results to calculating minimal generating sets of idempotents for finite semigroups.

We present some results on locally (soluble-by-finite) groups with various restrictions on subgroups of infinite rank.

We investigate the radial behavior of holomorphic functions in the unit ball $B$ of $\mtc^n$. In particular, we prove the existence of universal holomorphic functions $f$ in the following sense : given any measurable function $\vphi$ on $\partial B$, there is a sequence $(r_n)_{n\geq 1}$, $0<r_n<1$, that converges to 1, such that $f(r_n\xi)$ converges to $\vphi(\xi)$ for almost every $\xi\,{\in}\,\partial B$.

In this paper we consider a generalization of supersolvability called groups of polycyclic breadth $n$ for $n\ge 1$, we see that a number of well known results for supersolvable groups generalize to groups of polycyclic breadth $n$. This generalization of supersolvability is especially strong for the groups of polycyclic breadth 2.

We give a spectral characterization of the norm closure of the class of all weakly hypercyclic operators on a Hilbert space. Analogous results are obtained for weakly supercyclic operators.

Let $p$ be a prime number, $\Q_p$ the field of $p$-adic numbers, $K$ a finite field extension of $\Q_p$, $\skew4\bar K$ a fixed algebraic closure of $K$, and $\C_p$ the completion of $\skew4\bar K$ with respect to the $p$-adic valuation. We discuss some properties of Lipschitzian elements, which are elements $T$ of $\C_p$ defined by a certain metric condition that allows one to integrate Lipschitzian functions along the Galois orbit of $T$ over $K$ with respect to the Haar distribution.

In this note, we show that results of U. Baumgartner and G. A. Willis concerning contraction groups of automorphisms of metrizable totally disconnected, locally compact groups remain valid also in the non-metrizable case, if one restricts attention to automorphisms for which small tidy subgroups exist.

We strengthen a characterization of the $p$-adic binomial series and a special case of a formal analogue of Hilbert's Theorem 90 for $p$-adic power series.

We show that there is a unique norm-preserving extension for norm-attaining 2-homogeneous polynomials on the predual $d_*(w,1)$ of a complex Lorentz sequence space $d(w,1)$ to $d^*(w,1)$, but there is no unique norm-preserving extension from $\mathcal{P}(^nd_*(w,1))$ to $\mathcal{P}(^nd^*(w,1))$ for $n\geq3$.

We show that the Fourier expansion in spherical $h$-harmonics (from Dunkl's theory) of a function $f$ on the sphere converges uniformly to $f$ if this function is sufficiently differentiable.

J.W.S. Cassels gave a solution to the problem of determining all instances of the sum of three consecutive cubes being a square. This amounts to finding all integer solutions to the Diophantine equation $y^2=3x(x^2+2)$. We describe an alternative approach to solving not only this equation, but any equation of the type $y^2=nx(x^2+2)$, with $n$ a natural number. Moreover, we provide an explicit upper bound for the number of solutions of such Diophantine equations. The method we present uses the ingenious work of Wilhelm Ljunggren, and a recent improvement by the authors.

In this article, we derive a partition result and employ its special case to offer a new proof for an identity of N. J. Fine.

We prove that if the order-one differential operator $S=\partial_1 + \sum_{i=2}^{n} \beta_i\partial_i + \gamma$, with $\beta_i,\gamma \in K[x_1,\ldots,x_n]$, generates a maximal left ideal of the Weyl algebra $A_n(K)$, then $S$ does not admit any Darboux differential operator in $K[x_1,\ldots,x_n]\langle \partial_2,\ldots,\partial_n\rangle $; hence in particular, the derivation $\partial_1 + \sum_{i=2}^{n} \beta_i\partial_i$ does not admit any Darboux polynomial in $K[x_1,\ldots,x_n]$. We show that the converse is true when $\beta_i \in K[x_1,x_i]$, for every $i=2,\ldots,n$. Then, we generalize to $K[x_1,\ldots,x_n]$ the classical result of Shamsuddin that characterizes the simple linear derivations of $K[x_1,x_2]$. Finally, we establish a criterion for the left ideal generated by $S$ in $A_n(K)$ to be maximal in terms of the existence of polynomial solutions of a finite system of differential polynomial equations.

A generalization due to Gessel [3] of Miki's identity between Bernoulli numbers is shown to be a direct consequence of a functional equation for the generating function.

We study the analysis of a probability density $K$ on a Lie group $G$, where $G$ is a semidirect product of a compact group $M$ with a nilpotent group $N$. To approximate analysis on $G$ with analysis on $N$, it is natural to consider certain maps (“realizations”) of $G$ onto $N$. In this paper, we prove the existence of a realization of $G$ in $N$ which is $K$-harmonic (modulo the commutator subgroup of $N$). By utilizing this result and extending some ideas of Alexopoulos, we can prove the boundedness in $L^p$ spaces of some new Riesz transforms associated with $K$, and obtain new regularity estimates for the convolution powers of $K$.

This paper is mainly dedicated to describing the congruences on certain monoids of transformations on a finite chain $X_n$ with $n$ elements. Namely, we consider the monoids $\od_n$ and $\mpod_n$ of all full, respectively partial, transformations on $X_n$ that preserve or reverse the order, as well as the submonoid $\po_n$ of $\mpod_n$ of all its order-preserving elements. The inverse monoid $\podi_n$ of all injective elements of $\mpod_n$ is also considered.

We show that in $\po_n$ any congruence is a Rees congruence, but this may not happen in the monoids $\od_n$, $\podi_n$ and $\mpod_n$. However in all these cases the congruences form a chain.

We prove under quite general assumptions the global existence of classical solutions for quasilinear parabolic equations in bounded domains with homogeneous Neumann boundary conditions.

It is shown that, if $E$ and $F$ are Banach spaces containing complemented copies of $\ell_1$, then the space of integral operators ${\mathcal I}(E,F^*)\equiv (E\otimes_\eps F)^*$ contains a complemented copy of $\ell_2$. This answers a question of Félix Cabello and Ricardo García.

The present paper deals with two graph parameters related to cover graphs and acyclic orientations of graphs.

The parameter $c(G)$ of a graph $G$, introduced by B. Bollobás, G. Brightwell and J. Nešetřil [Order 3 245–255], is defined as the minimum number of edges one needs to delete from $G$ in order to obtain a cover graph. Extending their results, we prove that, for $\delta >0$, $(1-\delta) \frac{1}{l} \frac{n^2p}{2} \leq c({\mathcal G}_{n,p}) \leq (1+\delta) \frac{1}{l} \frac{n^2p}{2}$ asymptotically almost surely as long as $C n^{-1 + \frac{1}{l}} \leq p(n) \leq c n^{-1 + \frac{1}{ l-1} }$ for some positive constants $c$ and $C$. Here, as usual, ${\mathcal G}_{n,p}$ is the random graph.

Given an acyclic orientation of a graph $G$, an arc is called dependent if its reversal creates an oriented cycle. Let $d_{\min}(G)$ be the minimum number of dependent arcs in any acyclic orientation of $G$. We determine the supremum, denoted by $r_{\chi,g}$, of $d_{\min}(G)/e(G)$ in the class of graphs $G$ with chromatic number $\chi$ and girth $g$. Namely, we show that $r_{\chi,g} = {(\scriptsize\begin{array}{@{}c@{}}{\chi}-g+2\\ 2\end{array})} / {(\scriptsize\begin{array}{@{}c@{}}{\chi}\\ 2\end{array})}$. This extends results of D. C. Fisher, K. Fraughnaugh, L. Langley and D. B. West [J. Combin. Theory Ser. B 71 73–78].