We discuss the representation of primes, almost-primes, and related arithmetic sequences as sums of kth powers of natural numbers. In particular, we show that on GRH, there are infinitely many primes represented as the sum of 2\lceil 4k/3\rceil positive integral kth powers, and we prove unconditionally that infinitely many P_2-numbers are the sum of 2k+1 positive integral kth powers. The sieve methods required to establish the latter conclusion demand that we investigate the distribution of sums of kth powers in arithmetic progressions, and our conclusions here may be of independent interest.

This paper discusses the fractional chromatic number of the direct product of graphs. It is proved that if H is a circulant graph G^k_d, or a Kneser graph, or a direct sum of such graphs, then for any graph G, \chi_f{\hskip1}(G\times H{\hskip1}) = {\text min}\{\chi_f{\hskip1}(G), \chi_f{\hskip1}(H{\hskip1})\}.

Given a number field K, we find two simple separate necessary and sufficient conditions on a given algebraic number for it to be expressible as a quotient (respectively as a difference) of two algebraic numbers conjugate over K.

The Kadets path distance between Banach spaces X and Y is defined to be the infimum of the lengths with respect to the Kadets distance of all curves joining X and Y. If there is no curve joining X and Y, the Kadets path distance between X and Y is defined to be infty .

Some approaches to estimates of the Kadets path distance from above and from below are developed. In particular, the Kadets path distances between the spaces l_p^n,\ p\in [1,+\rm (inf)ty ], n\in {\b (N)} are estimated.

In this paper, we obtain an exact formula for the Hausdorff and box dimensions of a class of self-affine sets in two dimensions, namely those with disjoint projections. We prove, in particular, that fractals in this class have a Hausdorff and box dimension that is equal to the maximum Hausdorff and box dimension of one of their projections.

Let B(H) denote the C*-algebra of all bounded linear operators on a separable Hilbert space H. For A,B∈B(H), the chordal transform f_(A,B), as an operator on B(H), is defined by {f_(A,B)(X)=(\vert A^*\vert ^2+I)^(-1/2){\delta_(A,B)(X)(\vert B\vert ^2+I)^(-1/2)}, where {\delta_(A,B)} is the generalized derivation defined on B(H) by {\delta _(A,B)(X)=AX-XB}. Orthogonality of the range and the kernel of f_(A,B), with respect to the unitarily invariant norms \vert \vert \vert .\vert \vert \vert , are discussed. It is shown that if A, B are self-adjoint, then {\vert \vert \vert f_(A,B)(X)\vert \vert \vert \le \vert \vert \vert X\vert \vert \vert for all X. Related norm inequalities comparing f_(A,B) and {\delta _(A,B) are also given.

This paper investigates finite p-groups, p \geq 5, in which every cyclic subgroup has defect at most two. This class of groups is often denoted by {\cal U}_{2,1}. The main result is a theorem which characterises these groups by identifying a family of groups in {\cal U}_{2,1}, and showing that any finite p-group in {\cal U}_{2,1}, with p \geq 5, must be a homomorphic image of one of these groups.

We prove the following result of existence of graphs with constant mean curvature in Euclidean space: given a convex bounded planar domain \Omega of area a(\Omega) and a real number H such that a(\Omega)H^2<\pi/2, there exists a graph on \Omega with constant mean curvature H and whose boundary is \partial\Omega.

We calculate the height of quantum determinantal ideals in the algebra of quantum matrices, and also the Gelfand-Kirillov, Krull and Classical Krull dimensions of the corresponding quantum determinantal factors.

Let \bf{R}_+ be the set of all non-negative real numbers, \bf(I}\in \{ \bf(R}, \bf(R}_+\} and {\cal U}_\bf(I}=\{ U(t, s): t\ge s\in I\} be a strongly measurable and exponentially bounded evolution family of bounded linear operators acting on a Banach space X. Let \phi :\bf(R}_+\to \bf(R}_+ be a strictly increasing function and E be a normed function space over \bf(I} satisfying some properties; see Section 2. We prove that if

\phi \circ (\chi _{[s, \infty )}(\cdot )||U(\cdot , s)x||)

defines an element of the space E for every s\in \bf(I} and all x\in X and if there exists M>0 such that

\sup \limits _{s\in \bf(I}}|\phi \circ (\chi _{[s, \infty )} (\cdot )||U(\cdot , s)x||)|_{E}= M<\infty \quad \forall x\in X\ {\text with}\ ||x||\le 1,

then {\cal U}_\bf(I} is uniformly exponentially stable. In particular if \psi :\bf(R}_+\to \bf(R}_+ is a nondecreasing function such that \psi (t)>0, for all t>0, and if there exists K>0 such that

\sup \limits _{s\in \bf(I}}\int \limits _s^{\infty }\psi (||U(t, s)x||)dt=K <\infty ,\quad \forall x\in X\ \text(with)\ ||x||\le 1, then

{\cal U}_\bf(I} is uniformly exponentially stable. For \bf(I}=\bf(R}_+, \psi continuous and {\cal U}_{\bf(R}_+} strongly continuous this last result is due to S. Rolewicz. Some related results for periodic evolution families are also proved.

In this paper we prove that the ideal property and the projection property do not conincide in general even in the separable case (despite the fact that, as we proved before, they are the same for GAH algebras-and, in particular, for AH algebras-and for separable LB algebras). We also study the behaviour of the projection property with respect to several natural operations.

It is shown that the creation operator is the only (up to a multiplicative constant) injective weighted shift all of whose translations (or at least one) are still injective weighted shifts regardless of what the weight sequences and the bases are. A similar result is true for the annihilation operator as well as for the Heisenberg and Schrödinger couples.

We show that if R is a commutative ring and (S, \leq ) a strictly totally ordered monoid, then the ring [[R^{S, \leq }]] of generalized power series is Baer if and only if R is Baer.

We study the conditions which force a semiperfect ring to admit a Nakayama permutation of its basic idempotents. We also give a few necessary and sufficient conditions for a semiperfect ring R, which cogenerates every 2-generated right R-module, to be right pseudo-Frobenius.

Compact hyperbolic surfaces of given genus g containing discs of the maximum radius have been studied from various points of view. In this paper we connect these different approaches and observe some properties of the Fuchsian groups uniformizing both compact and punctured extremal surfaces. We also show that extremal surfaces of genera g=2,3 may contain one or several extremal discs, while an extremal disc is necessarily unique for g \ge 4. Along the way we also construct explicit families of extremal surfaces, one of which turns out to be free of automorphisms.

In this paper, we establish two results assuring that \lambda =0 is a bifurcation point in L^ \rm{inf} ty (\Omega ) for the Hammerstein integral equation

u(x)=\lambda \int _\Omega k(x,y)f({}y,u({}y))dy.

We also present an application to the two-point boundary value problem

\cases{ -u''=\lambda f(x,u)\hfill \hbox {a.e. in [0,1] } \cr u(0)=u(1)=0 } \right.

A Banach space is called (almost) transitive if the isometry group acts (almost) transitively on the unit sphere. The main problems around transitivity are the Banach-Mazur conjecture that the only separable and transitive Banach spaces are the Hilbert ones (1930) and the Wood conjecture that C_0(L) cannot be almost transitive in its natural supremum norm unless L is a singleton (1982). In this note we give necessary and sufficient conditions on the locally compact space L for the (almost) transitivity of C_0(L). This will clarify the topological content of Wood's problem.

We prove that if X is a real Banach space, with dim X\ge 3, which contains subspace of codimension 1 which is 1-complemented in X and whose group of isometries is almost transitive then X is isometric to a Hilbert space. This partially answers the Banach-Mazur rotation problem and generalizes some recent related results.

In this paper we construct upper bounds for the solutions u({\bf{x}},t) and its gradient |\nabla u| of a class of parabolic initial-boundary value problems in terms of the solution \psi ({\bf{x}}) of the {\text S}^t-Venant problem. These bounds are sharp in the sense that they coincide with the exact values of u and |\nabla u| for appropriate geometry and appropriate initial conditions.

The first non-abelian cohomology of groups introduced by Guin is extended to any dimensions and for a substantially wider class of coefficients called G-partially crossed P-modules. The first and the second non-abelian cohomologies of groups are described in terms of torsors and extensions of groups respectively. Higher non-abelian cohomology pointed sets are described in terms of cotriple right derived functors of the group of derivations with respect to the first contravariant variable. For any short exact coefficient sequence a long exact cohomology sequence is obtained extending the well-known exact cohomology sequences and higher cohomology of groups with coefficients in any G-group is introduced.