Golubev series for solutions of elliptic equations

Let be an elliptic system with real analytic coefficients on an open set and let be a fundamental solution of Given a locally connected closed set we fix some massive measure on . Here, a non-negative measure is called massive, if the conditions and imply that We prove that, if is a solution of the equation in then for each relatively compact open subset of and every there exist a solution of the equation in and a sequence () in satisfying such that for This complements an earlier result of the second author on representation of solutions outside a compact subset of

     

Homogeneous projective varieties with degenerate secants

The secant variety of a projective variety in , denoted by , is defined to be the closure of the union of lines in passing through at least two points of , and the secant deficiency of is defined by . We list the homogeneous projective varieties with under the assumption that arise from irreducible representations of complex simple algebraic groups. It turns out that there is no homogeneous, non-degenerate, projective variety with and , and the -variety is the only homogeneous projective variety with largest secant deficiency . This gives a negative answer to a problem posed by R. Lazarsfeld and A. Van de Ven if we restrict ourselves to homogeneous projective varieties.

     

Decomposition of

For any locally compact group , let and be the Fourier and the Fourier-Stieltjes algebras of , respectively. is decomposed as a direct sum of and , where is a subspace of consisting of all elements that satisfy the property: for any and any compact subset , there is an with and such that is characterized by the following: an element is in if and only if, for any there is a compact subset such that for all with and . Note that we do not assume the amenability of . Consequently, we have for all if is noncompact. We will apply this characterization of to investigate the general properties of and we will see that is not a subalgebra of even for abelian locally compact groups. If is an amenable locally compact group, then is the subspace of consisting of all elements with the property that for any compact subset , .

     

Distribution semigroups and abstract Cauchy problems

We present a new definition of distribution semigroups, covering in particular non-densely defined generators. We show that for a closed operator in a Banach space the following assertions are equivalent: (a) generates a distribution semigroup; (b) the convolution operator has a fundamental solution in where denotes the domain of supplied with the graph norm and denotes the inclusion ; (c) generates a local integrated semigroup. We also show that every generator of a distribution semigroup generates a regularized semigroup.

     

The stability of the equilibrium of reversible systems

In this paper, we consider the system

where , , and are continuous, even and 1-periodic in the time variable ; and are real analytic in a neighbourhood of the origin of -plane and continuous 1-periodic in . We also assume that the above system is reversible with respect to the involution . A sufficient and necessary condition for the stability in the Liapunov sense of the equilibrium is given.

     

Galois coverings of selfinjective algebras by repetitive algebras

In the representation theory of selfinjective artin algebras an important role is played by selfinjective algebras of the form where is the repetitive algebra of an artin algebra and is an admissible group of automorphisms of . If is of finite global dimension, then the stable module category of finitely generated -modules is equivalent to the derived category of bounded complexes of finitely generated -modules. For a selfinjective artin algebra , an ideal and , we establish a criterion for to admit a Galois covering with an infinite cyclic Galois group . As an application we prove that all selfinjective artin algebras whose Auslander-Reiten quiver has a non-periodic generalized standard translation subquiver closed under successors in are socle equivalent to the algebras , where is a representation-infinite tilted algebra and is an infinite cyclic group of automorphisms of .

     

On minimal parabolic functions and time-homogeneous parabolic -transforms

Does a minimal harmonic function remain minimal when it is viewed as a parabolic function? The question is answered for a class of long thin semi-infinite tubes of variable width and minimal harmonic functions corresponding to the boundary point of ``at infinity.' Suppose is the width of the tube units away from its endpoint and is a Lipschitz function. The answer to the question is affirmative if and only if . If the test fails, there exist parabolic -transforms of space-time Brownian motion in with infinite lifetime which are not time-homogenous.

     

On the number of radially symmetric solutions to Dirichlet problems with jumping nonlinearities of superlinear order

This paper is concerned with the multiplicity of radially symmetric solutions to the Dirichlet problem

on the unit ball with boundary condition on . Here is a positive function and is a function that is superlinear (but of subcritical growth) for large positive , while for large negative we have that , where is the smallest positive eigenvalue for in with on . It is shown that, given any integer , the value may be chosen so large that there are solutions with or less interior nodes. Existence of positive solutions is excluded for large enough values of .

     

Scrambled sets of continuous maps of 1-dimensional polyhedra

Let be a 1-dimensional simplicial complex in without isolated vertexes, be the polyhedron of with the metric induced by , and be a continuous map. In this paper we prove that if is finite, then the interior of every scrambled set of in is empty. We also show that if is an infinite complex, then there exist continuous maps from to itself having scrambled sets with nonempty interiors, and if or , then there exist maps of with the whole space being a scrambled set.

     

An algorithm for calculating the Nielsen number on surfaces with boundary

Let be a self-map of a hyperbolic surface with boundary. The Nielsen number, , depends only on the induced map of the fundamental group, which can be viewed as a free group on generators, . We determine conditions for fixed points to be in the same fixed point class and if these conditions are enough to determine the fixed point classes, we say that is -characteristic. We define an algebraic condition on the and show that ``most' maps satisfy this condition and that all maps which satisfy this condition are -characteristic. If is -characteristic, we present an algorithm for calculating and prove that the inequality holds, where denotes the Lefschetz number of and the Euler characteristic of , thus answering in part a question of Jiang and Guo.

     

On the diophantine equation

In this paper we prove that the equation , , , , , has only the solutions and with is a prime power. The proof depends on some new results concerning the upper bounds for the number of solutions of the generalized Ramanujan-Nagell equations.

     

The nilpotence height of for odd primes

K. G. Monks has recently shown that the element has nilpotence height in the mod Steenrod algebra. Here the method and result are generalized to show that for an odd prime the element has nilpotence height in the mod Steenrod algebra.

     

The problem on domains with piecewise smooth boundaries with applications

Let be a bounded domain in such that has piecewise smooth boudnary. We discuss the solvability of the Cauchy-Riemann equation

where is a smooth -closed form with coefficients up to the bundary of , and . In particular, Equation (0.1) is solvable with smooth up to the boundary (for appropriate degree if satisfies one of the following conditions:

i)

is the transversal intersection of bounded smooth pseudoconvex domains.

ii)

where is the union of bounded smooth pseudoconvex domains and is a pseudoconvex convex domain with a piecewise smooth boundary.

iii)

where is the intersection of bounded smooth pseudoconvex domains and is a pseudoconvex domain with a piecewise smooth boundary.

The solvability of Equation (0.1) with solutions smooth up to the boundary can be used to obtain the local solvability for on domains with piecewise smooth boundaries in a pseudoconvex manifold.

     

Combinatorial families that are exponentially far from being listable in Gray code sequence

Let be a collection of subsets of . In this paper we study numerical obstructions to the existence of orderings of for which the cardinalities of successive subsets satisfy congruence conditions. Gray code orders provide an example of such orderings. We say that an ordering of is a Gray code order if successive subsets differ by the adjunction or deletion of a single element of . The cardinalities of successive subsets in a Gray code order must alternate in parity. It follows that if is the difference between the number of elements of having even (resp. odd) cardinality, then is a lower bound for the cardinality of the complement of any subset of which can be listed in Gray code order. For , the collection of -blockfree subsets of is defined to be the set of all subsets of such that if and . We will construct a Gray code order for . In contrast, for we find the precise (positive) exponential growth rate of with as . This implies is far from being listable in Gray code order if is large. Analogous results for other kinds of orderings of subsets of are proved using generalizations of . However, we will show that for all , one can order so that successive elements differ by the adjunction and/or deletion of an integer from . We show that, over an -letter alphabet, the words of length which contain no block of consecutive letters cannot, in general, be listed so that successive words differ by a single letter. However, if and or if and , such a listing is always possible.

     

Ultrafilters on -their ideals and their cardinal characteristics

For a free ultrafilter on we study several cardinal characteristics which describe part of the combinatorial structure of . We provide various consistency results; e.g. we show how to force simultaneously many characters and many -characters. We also investigate two ideals on the Baire space naturally related to and calculate cardinal coefficients of these ideals in terms of cardinal characteristics of the underlying ultrafilter.

     

Witten-Helffer-Sjöstrand theory for -equivariant cohomology

Given an -invariant Morse function and an -invariant Riemannian metric , a family of finite dimensional subcomplexes , , of the Witten deformation of the -equivariant de Rham complex is constructed, by studying the asymptotic behavior of the spectrum of the corresponding Laplacian as . In fact the spectrum of can be separated into the small eigenvalues, finite eigenvalues and the large eigenvalues. Then one obtains as the complex of eigenforms corresponding to the small eigenvalues of . This permits us to verify the -equivariant Morse inequalities. Moreover suppose is self-indexing and satisfies the Morse-Smale condition, then it is shown that this family of subcomplexes converges as to a geometric complex which is induced by and calculates the -equivariant cohomology of .

     

Vertex operators for twisted quantum affine algebras

We construct explicitly the -vertex operators (intertwining operators) for the level one modules of the classical quantum affine algebras of twisted types using interacting bosons, where for (), for , for (), and for (). A perfect crystal graph for is constructed as a by-product.

     

Compressions of resolvents and maximal radius of regularity

Suppose that is left invertible in for all , where is an open subset of the complex plane. Then an operator-valued function is a left resolvent of in if and only if has an extension , the resolvent of which is a dilation of of a particular form. Generalized resolvents exist on every open set , with included in the regular domain of . This implies a formula for the maximal radius of regularity of in terms of the spectral radius of its generalized inverses. A solution to an open problem raised by
J. Zemánek is obtained.

     

On the non-vanishing of cubic twists of automorphic -series

Let be a normalised new form of weight for over and , its base change lift to . A sufficient condition is given for the nonvanishing at the center of the critical strip of infinitely many cubic twists of the -function of . There is an algorithm to check the condition for any given form. The new form of level is used to illustrate our method.

     

Quasitriangular + small compact = strongly irreducible

Let be a bounded linear operator acting on a separable infinite dimensional Hilbert space. Let be a positive number. In this article, we prove that the perturbation of by a compact operator with can be strongly irreducible if is a quasitriangular operator with the spectrum connected. The Main Theorem of this article nearly answers the question below posed by D. A. Herrero.

Suppose that is a bounded linear operator acting on a separable infinite dimensional Hilbert space with connected. Let be given. Is there a compact operator with such that is strongly irreducible?