In this paper we generalize the formula of Frobenius-Stickelberger (see (0.1) below) and the formula of Kiepert (see (0.2) below) to the genus-two case.

An element r in a ring R is clean if r is a sum of a unit and an idempotent. Camillo and Yu showed that unit regular rings are clean and in a very surprising development Nicholson and Varadarajan showed that linear transformations on countable dimension vector spaces over division rings are clean. These rings are very far from being unit regular.

Here we note that an idempotent is just a root of g(x)=x^{2}-x. For any g(x) we say R is g(x)-clean if every r in R is a sum of a root of g(x) and a unit. We show that if V is a countable dimensional vector space and over a division ring D and g(x) is any polynomial with coefficients in <formtex>K={\text Center}D and two distinct roots in K, then {\text End}V_D is g(x)-clean.

We give a method for embedding a large family of partially ordered simple groups of rank one into simple Riesz groups of rank one. In particular, we answer in the affirmative a question of Wehrung, by constructing a torsion-free, simple Riesz group G of rank one containing an interval D\varsubsetneqq G^+ such that 2D=G^+. We sketch some potential applications of this result in the context of monoids of intervals and K-Theory of rings.

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 sequence of solutions to the Galerkin approximation of a nonstationary magnetohydrodynamic system is proved to converge to a measure-valued solution, in the sense of R. J. DiPerna–A. J. Majda, to the three-dimensional stationary Euler equations.

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.

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.

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.

The paper deals with local existence, blow-up and global existence for the solutions of a wave equation with an internal nonlinear source and a nonlinear boundary damping. The typical problem studied is\cases{u_{tt}-\Delta u=|u|^{p-2}u \hfill& \rm{~}OPEN~7~in <hsp sp=0.25>[0,\rm{inf}ty )\times \Omega ,}\hfill \cr u=0 \hfill & \rm{~}OPEN~6~on <hsp sp=0.25>[0,\rm{inf}ty )\times \Gamma _0,}\hfill \cr \frac {\partial u}{\partial \nu }=-\alpha (x)|u_t|^{m-2}u_t \hfill& \rm{~}OPEN~2~on <hsp sp=0.25>[0,\rm{inf}ty )\times \Gamma _1,}\cr u(0,x)=u_0(x),u_t(0,x)=u_1(x) & \rm{~}OPEN~1~on<hsp sp=0.25>\Omega ,}\hfill }

where \Omega \subset ℝ^n (n\ge 1) is a regular and bounded domain, \partial \Omega =\Gamma _0\cup \Gamma _1, \lambda _{n-1}(\Gamma _0)>>;0, 2<>;p\le 2(n-1)/(n-2) (when n\ge 3), m>>;1, \alpha \in L^\rm{inf}ty (\Gamma _1), \alpha \ge 0, and the initial data are in the energy space. The results proved extend the potential well theory, which is well known when the nonlinear damping acts in the interior of \Omega, to this problem.

We characterize the pairs of operator spaces that occur as pairs of Morita equivalence bimodules between non-selfadjoint operator algebras in terms of the mutual relation between the spaces. We obtain a characterization of the operator spaces which are completely isometrically isomorphic to imprimitivity bimodules between some strongly Morita equivalent (in the sense of Rieffel) C*-algebras. As corollaries, we give representation results for such operator spaces.

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.

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.

If P is a Sylow-p-subgroup of a finite p-solvable group G, we prove that G^\prime \cap \bf{N}_G(P) \subseteq {P} if and only if p divides the degree of every irreducible non-linear p-Brauer character of G. More generally if π is a set of primes containing p and G is π-separable, we give necessary and sufficient group theoretic conditions for the degree of every irreducible non-linear p-Brauer character to be divisible by some prime in π. This can also be applied to degrees of ordinary characters.

In this paper we consider the integral Volterra operator on the space L^2(0,1). We say that a complex number \lambda is an extended eigenvalue ofV if there exists a nonzero operator X satisfying the equation XV=\lambda VX. We show that the set of extended eigenvalues of V is precisely the interval (0,\infty ) and the corresponding eigenvectors may be chosen to be integral operators as well.

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.

Let T be p-hyponormal or \rm{log}-hyponormal on a Hilbert space H. Then we have XT=T^*X whenever XT^*=TX for some X \in \scriptstyle{B}(\scriptstyle{H}). This is an extension of Patel's result. Also for p-hyponormal or \rm{log}-hyponormal T^*, dominant S and any X \in \scriptstyle{B}(\scriptstyle{H}) such that XT=SX, we have XT^*=S^*T.

In this paper we show that there are no real hypersurfaces in a non-flat complex space form with recurrent Ricci tensor.