Let A be a quasi-accretive operator defined in a uniformly smooth Banach space. We present a necessary and sufficient condition for the strong convergence of the semigroups generated by – A and of the steepest descent methods to a zero of A.

Let A(z) be a transcendental entire function and f1, f2 be linearly independent solutions of

We prove that if A(z) has Nevanlinna deficiency δ(0, A) = 1, then the exponent of convergence of E: = flf2 is infinite. The theorems that we prove here are similar to those in Bank, Laine and Langley [3].

An asymptotic tract of a real function u harmonic and non-constant in ℂ is a component of the set {z:u(z)≠c}, for some real number c; a quasi-tractT(≠ℂ) is an unbounded simply-connected domain in ℂ such that there exists a function u that is positive, unbounded and harmonic in T such that, for each point ζ∈∂T∩ℂ,

and a ℱ-tract is an unbounded simply-connected domain T in ℂ whose every prime end that contains ∞ in its impression is of the first kind.

The authors study the growth of a harmonic function in one of its asymptotic tracts, and the question of whether a quasi-tract is an asymptotic tract. The branching of either type of tract is also taken into consideration.

The paper is devoted to studying one generalization of Steiner systems S(n, k, l) closely related to packings and coverings of l-tuples by k-tuples of an n-set. One necessary and one sufficient condition for the existence of such designs are obtained.

Let V be a left vector space over the arbitrary division ring D and G a locally nilpotent group of finitary automorphisms of V (automorphisms g of V such that dimDV(g-1)<∞) such that V is irreducible as D-G bimodule. If V is infinite dimensional we show that such groups are very rare, much rarer than in the finite-dimensional case. For example we show that if dimDV is infinite then dimDV = |G| = ℵ0 and G is a locally finite q-group for some prime q ≠ char D. Moreover G is isomorphic to a finitary linear group over a field. Examples show that infinite-dimensional such groups G do exist. Note also that there exist examples of finite-dimensional such groups G that are not isomorphic to any finitary linear group over a field. Generally the finite-dimensional examples are more varied.

We discuss the existence and non-existence of front, domain wall and pulse type traveling wave solutions of a Ginzburg-Landau equation with cubic terms containing spatial derivatives and a fifth order term, in both subcritical and supercritical cases. Our results appear to be the first rigorous existence and non-existence proofs for the full equation with all possible terms derived from second order perturbation theory present.

We consider rational functions of the form fm(z) = zm/(z – p) which are analytic in |z|<p, p>1, and establish that the asymptotic distribution of the zeros of their Taylor sections and Lagrange interpolants at uniformly distributed nodes is similar. This notion is also illustrated computationally. We conjecture that a similar result can be expected for any function analytic in |z| < p.

The result of Ballantine [1] to the effect that a singular matrix A is a product of k idempotent matrices if and only if the rank of I – A does not exceed k times the nullity of A is generalized to endomorphisms of a class of independence algebras.

A ℤG-module A is said to have an f-decomposition if in which A∫ is a ℤG-submodule of A such that each irreducible ℤG-factor of A∫ as an abelian group is finite and the ℤG-submodule A∫ has no finite irreducible ℤG-factors. In this paper, we prove that: if G is a hyperfinite group then any artinian ℤG-module A has an f-decomposition, which gives a positive answer to the question raised by D.I. Zaitzev in 1986.

For a multiplicative set S of a commutative ring R we define the co-localization functor HomR(Rs,⋅). It is a functor on the category of R-modules to the category of Rs-modules. It is shown to be exact on the category of Artinian R-modules. While the co-localization of an Artinian module is almost never an Artinian Rs-module it inherits many good properties of A, e.g. it has a secondary representation. The construction is applied to the dual of a result of Bourbaki, a description of asymptotic prime divisors and the co-support of an Artinian module.

Let Fq be the finite field of q elements. Let f(x) be a polynomial of degree d over Fq and let r be the least non-negative residue of q-1 modulo d. Under a mild assumption, we show that there are at most r values of c∈Fq, such that f(x) + cx is a permutation polynomial over Fq. This indicates that the number of permutation polynomials of the form f(x) +cx depends on the residue class of q–1 modulo d.

As an application we apply our results to the construction of various maximal sets of mutually orthogonal latin squares. In particular for odd q = pn if τ(n) denotes the number of positive divisors of n, we show how to construct τ(n) nonisomorphic complete sets of orthogonal squares of order q, and hence τ(n) nonisomorphic projective planes of order q. We also provide a construction for translation planes of order q without the use of a right quasifield.

It is known, for each 1<p<∞, p≠2, that there exist differential operators in LP(ℝN) which are not (unbounded) decomposable operators in the sense of C. Foiaş. In this note we exhibit large classes of differential (and unbounded multiplier operators which are decomposable in LP(ℝN) and hence have good spectral mapping properties; the arguments are based on the existence of a sufficiently rich functional calculus. The basic idea is to take advantage of existing classical results on p-multipliers and use them to generate appropriate functional calculi.

Let M be a hypersurface in Euclidean space and let the Ricci curvature of M be bounded below by some nonnegative constant. In this paper, we estimate the sectional curvature of M in terms of the lower bound of Ricci curvature and the upper bound of mean curvature.

It is proved that every space L2 (I1, ∪ I2), where I1 and I2 are finite intervals, has a Riesz basis of complex exponentials , {λk} a sequence of real numbers. A partial result for the corresponding problem for n≧3 finite intervals is also obtained.

The aim of this paper is to prove the Theorem: Let M be a complete non compact surface without boundary in the euclidean space 3. We suppose that all geodesies of M are congruent. Then M is an affine plane in 3.

Let Γ be a finite graph together with a group Gv at each vertex v. The graph productG(Γ) is obtained from the free product of all Gv by factoring out by the normal subgroup generated by for all adjacent v, w.

In this note we construct a projective resolution for G(Γ) given projective resolutions for each Gv, and obtain some applications.