Several properties of some quadratic elements of a unitial Banach algebra are studied. Deddens subspaces are also introduced and discussed.

In this note we prove that a locally graded group $G$ in which all proper subgroups are (nilpotent of class not exceeding $n$)-by-Černikov, is itself (nilpotent of class not exceeding $n$)-by-Černikov.

As a preparatory result that is used for the proof of the former statement in the case of a periodic group, we also prove that a group $G$, containing a nilpotent of class $n$ subgroup of finite index, also contains a characteristic subgroup of finite index that is nilpotent of class not exceeding $n$.

Strata in manifold stratified spaces are shown to have neighborhoods that are teardrops of manifold stratified approximate fibrations (under dimension and compactness assumptions). This is the best possible version of the tubular neighborhood theorem for strata in the topological setting. Applications are given to replacement of singularities, to the structure of neighborhoods of points in manifold stratified spaces, and to spaces of manifold stratified approximate fibrations.

Let $(X,\mathcal{F},\mu)$ be a complete probability space and let $\mathcal{B}$ be a sub-$\sigma$-algebra of $\mathcal{F}$. We consider the extreme points of the closed unit ball $\mathbb{B}(\mathcal{A})$ of the normed space $\mathcal{A}$ whose points are the elements of $L^\infty(X,\mathcal{F},\mu)$ with the norm$\Vert\, f \Vert = \Vert\Phi(\vert\, f \vert)\Vert_\infty$, where $\Phi$ is the probabilistic conditional expectation operator determined by $\mathcal{B}$. No $\mathcal{B}$- measurable function is an extreme point of the closed unit ball of $\mathcal{A}$, and in certain cases there are no extreme points of $\mathbb{B}(\mathcal{A})$.

For an interesting family of examples, depending on a parameter $n$, we characterize the extreme points of the unit ball and show that every element of the open unit ball is a convex combination of extreme points. Although in these examples every element of the open ball of radius $\frac{1}{n}$ can be shown to be a convex combination of at most $2n$ extreme points by elementary arguments, our proof for the open unit ball requires use of the $\lambda$-function of Aron and Lohman. In the case of the open unit ball, we only obtain estimates for the number of extreme points required in very special cases, e.g. the $\mathcal{B}$-measurable functions, where $2n$ extreme points suffice.

In this paper we investigate the most general dichotomy concept of evolutionary processes. This dichotomy concept includes many interesting situations, among them we note the nonuniform dichotomy. We characterize the $(a,b)$-dichotomy in terms of the admissibility of the pair $(L^1_a, L^{\infty}_b)$. Also, generalizations of the results of [20], [23] are obtained.

Face 2-colourable triangulations of complete tripartite graphs $K_{n,n,n}$ correspond to biembeddings of Latin squares. Up to isomorphism, we give all such embeddings for $n=3,4,5$ and 6, and we summarize the corresponding results for $n=7$. Closely related to these are Hamiltonian decompositions of complete bipartite directed graphs $K^*_{n,n}$, and we also give computational results for these in the cases $n=3,4,5$ and 6.

An associative ring with unity is called clean if every element is the sum of an idempotent and a unit; if this representation is unique for every element, we call the ring uniquely clean. These rings represent a natural generalization of the Boolean rings in that a ring is uniquely clean if and only if it is Boolean modulo the Jacobson radical and idempotents lift uniquely modulo the radical. We also show that every image of a uniquely clean ring is uniquely clean, and construct several noncommutative examples.

We characterize and classify the “regular classes of heaps” introduced by the author using ideas of Fan and of Stembridge. The irreducible objects fall into five infinite families with one exceptional case.

In this paper, we discuss the structure of the global attractor of a positively bounded system. In particular, we are concerned with the existence of connecting orbits and the relation between maximal elliptic sectors and connecting orbits. For the systems with two singular points a necessary and sufficient condition for the existence of connecting orbits is given.

In 1989, Alladi, Erdös and Vaaler confirmed their own conjecture about a class of multiplicative functions by means of a deep result of Baranyai on hypergraphs. In this note we give a simple direct proof of the result which is derived in their proof as a consequence of the above mentioned graph theoretic result.

Let $X$ be a smooth complex projective curve, $S^{b}X$ the b-symmetric product of $X$. Assume that $X$ has an automorphism h. Our aim is to compute the characteristic classes of the fixed point set of h in $S^{b}X$.

With the help of some $p$-adic formal series over $p$-adic number fields and the estimates of character sums over Galois rings, we prove that there is a constant $C(n)$ such that there exists a primitive polynomial $f(x)\,{=}\,x^{n}-a_{1}x^{n-1}+\cdots +(-1)^{n}a_{n}$ of degree $n$ over $F_{q}$ with the first $m=\lfloor\frac{n-1}{2}\rfloor$ coefficients $a_{1},\ldots ,a_{m}$ prescribed in advance if $q\,{>}\,C(n)$.

A free boundary problem for a parabolic system arising from the mathematical theory of combustion will be considered in the one dimensional case. The existence and uniqueness of the classical solution locally in time will be obtained by the use of a fixed point theorem. Also the existence of the classical solution globally in time and a convergence result with respect to a parameter $\lambda$ will be proved under some reasonable assumptions.

We characterise the ternary rings of operators possessing a completely isometric representation whose range consists of normalisers or semi-normalisers between the ranges of some $^{*}$-representations of two fixed C$^{*}$-algebras. We give some corollaries of these results.

Let $p^m$ be a power of a prime number $p$, $\mathbb{Dacute;_{p^m}$ be the dihedral group of order $2p^m$ and $k$ be a field where $p$ is invertible and containing a primitive $2p^m$-th root of unity. The aim of this paper is computing the Brauer group $BM(k,\mathbb{D}_{p^m},R_z)$ of the group Hopf algebra of $\mathbb{D}_{p^m}$ with respect to the quasi-triangular structure $R_z$ arising from the group Hopf algebra of the cyclic group $\mathbb{Z}_{p^m}$ of order $p^m,$ for $z$ coprime with $p$. The main result states that $BM(k,\mathbb{D}_{p^m},R_z)\cong \mathbb{Z}_2 \times k^{\cdot}/k^{\cdot 2} \times Br(k)$ when $p$ is odd and when $p=2,$$BM(k,\mathbb{D}_{2^m},R_z) \cong \mathbb{Z}_2\times \mathbb{Z}_2 \times k^{\cdot}/k^{\cdot 2} \times k^{\cdot}/k^{\cdot 2} \times Br(k).$

We obtain estimates for the degree of bilipschitz determinacy of quasihomogeneous function-germs.

Using compact simple Lie groups and Heisenberg groups, we combine and generalize the constructions of complex structures on Kodaira surfaces and Hopf surfaces. We identify locally complete parameter spaces of deformations of these spaces and analyze the deformation of Kodaira manifolds in details.

Let $m, n\in\Bbb N$, $V$ be a $2m$-dimensional complex vector space. The irreducible representations of the Brauer's centralizer algebra $B_n(-2m)$ appearing in $V^{\otimes{n}}$ are in 1–1 correspondence to the set of pairs $(\,f,\lamda)$, where $f\in\Z$ with $0\leq f\leq [n/2]$, $and$ $\lam\vdash n-2f$ satisfying $\lam_1\leq m$. In this paper, we first show that each of these representations has a basis consists of eigenvectors for the subalgebra of $B_n(-2m)$ generated by all the Jucys-Murphy operators, and we determine the corresponding eigenvalues. Then we identify these representations with the irreducible representations constructed from a cellular basis of $B_n(-2m)$. Finally, an explicit description of the action of each generator of $B_n(-2m)$ on such a basis is also given, which generalizes earlier work of [15] for Brauer's centralizer algebra $B_n(m)$.

Let $G$ be a locally soluble-by-finite group in which every non-subnormal subgroup has finite rank. It is proved that either $G$ has finite rank or $G$ is soluble and locally nilpotent (and even a Baer group). On the other hand, a group $G$ is constructed that has infinite rank and satisfies the given hypothesis, but does not have every subgroup subnormal.

In this paper, we prove some Hardy-type inequalities for the degenerate operators, $L_{p,\alpha}u\,{=}\,{\rm div}_L(|\nabla_Lu|^{p-2}\nabla_Lu)$, where $\nabla_Lu\,{=}\,(\frac{\partial u}{\partial z_1},\ldots,\frac{\partial u}{\partial z_n},|z|^\alpha \frac{\partial u}{\partial t_1},\ldots,|z|^\alpha\frac{\partial u}{\partial t_m})$. These inequalities are established for the whole space, the pseudo-ball and the external domain of the pseudo-ball. We also give a generalization of a result in [8]. Finally, a sharp inequality for $L_{\alpha}\,{=}\,L_{2,\alpha}$ is obtained.