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.

     

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 .

     

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.

     

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.

     

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.

     

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 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.

     

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

     

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.

     

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.

     

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.

     

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 , .

     

Multi-separation, centrifugality and centripetality imply chaos

Let be an interval. need not be compact or bounded. Let be a continuous map, and be a trajectory of with or . Then there is a point such that . A point is called a centripetal point of relative to if or , and is centrifugal if or . In this paper we prove that if there exist centripetal points of in , then has periodic points of some odd () period . In addition, we also prove that if ) is multi-separated by Fix(), or there exists a centrifugal point of in , then is turbulent and hence has periodic points of all periods.

     

On sectional genus of quasi-polarized 3-folds

Let be a smooth projective variety over and a nef-big (resp. ample) divisor on . Then is called a quasi-polarized (resp. polarized) manifold. Then we conjecture that , where is the sectional genus of and is the irregularity of . In general it is unknown whether this conjecture is true or not, even in the case of . For example, this conjecture is true if and . But it is unknown if and . In this paper, we prove if and . Furthermore we classify polarized manifolds with , , and .

     

On locally linearly dependent operators and derivations

The first section of the paper deals with linear operators , , where and are vector spaces over an infinite field, such that for every , the vectors are linearly dependent modulo a fixed finite dimensional subspace of . In the second section, outer derivations of dense algebras of linear operators are discussed. The results of the first two sections of the paper are applied in the last section, where commuting pairs of continuous derivations of a Banach algebra such that is quasi-nilpotent for every are characterized.

     

Haar Measure and the Artin Conductor

Let be a connected reductive group, defined over a local, non-archimedean field . The group is locally compact and unimodular. In On the motive of a reductive group, Invent. Math. 130 (1997), by B. H. Gross, a Haar measure was defined on , using the theory of Bruhat and Tits. In this note, we give another construction of the measure , using the Artin conductor of the motive of over . The equivalence of the two constructions is deduced from a result of G. Prasad.

     

Class numbers of cyclotomic function fields

Let be a prime power and let be the finite field with elements. For each polynomial in , one could use the Carlitz module to construct an abelian extension of , called a Carlitz cyclotomic extension. Carlitz cyclotomic extensions play a fundamental role in the study of abelian extensions of , similar to the role played by cyclotomic number fields for abelian extensions of . We are interested in the tower of Carlitz cyclotomic extensions corresponding to the powers of a fixed irreducible polynomial in . Two types of properties are obtained for the -parts of the class numbers of the fields in this tower, for a fixed prime number . One gives congruence relations between the -parts of these class numbers. The other gives lower bound for the -parts of these class numbers.

     

Forcing minimal extensions of Boolean algebras

We employ a forcing approach to extending Boolean algebras. A link between some forcings and some cardinal functions on Boolean algebras is found and exploited. We find the following applications:

1) We make Fedorchuk's method more flexible, obtaining, for every cardinal of uncountable cofinality, a consistent example of a Boolean algebra whose every infinite homomorphic image is of cardinality and has a countable dense subalgebra (i.e., its Stone space is a compact S-space whose every infinite closed subspace has weight ). In particular this construction shows that it is consistent that the minimal character of a nonprincipal ultrafilter in a homomorphic image of an algebra can be strictly less than the minimal size of a homomorphic image of , answering a question of J. D. Monk.

2) We prove that for every cardinal of uncountable cofinality it is consistent that and both and exist.

3) By combining these algebras we obtain many examples that answer questions of J.D. Monk.

4) We prove the consistency of MA + CH + there is a countably tight compact space without a point of countable character, complementing results of A. Dow, V. Malykhin, and I. Juhasz. Although the algebra of clopen sets of the above space has no ultrafilter which is countably generated, it is a subalgebra of an algebra all of whose ultrafilters are countably generated. This proves, answering a question of Arhangelskii, that it is consistent that there is a first countable compact space which has a continuous image without a point of countable character.

5) We prove that for any cardinal of uncountable cofinality it is consistent that there is a countably tight Boolean algebra with a distinguished ultrafilter such that for every the algebra is countable and has hereditary character .

     

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.

     

Classes of singular integrals along curves and surfaces

This paper is concerned with singular convolution operators in , , with convolution kernels supported on radial surfaces . We show that if , then boundedness holds if and only if . This statement can be reduced to a similar statement about the multiplier in . We also construct smooth for which the corresponding operators are bounded for but unbounded for , for given . Finally we discuss some examples of singular integrals along convex curves in the plane, with odd extensions.