Extremal problems of Bloch function spaces

Letf be analytic in a hyperbolic region . The Bloch constant _f off is defined by , where (z)|dz| is the Poincaré metric in . Suppose is hyperbolic and where . Then for allf withf() , we have _f 1/(). In this paper we study the extremal functions defined by _f =1/() and the existence of those functions.Supported by the National Natural Science Foundation of China.     

Algebraic systems of quadratic forms of number fields and function fields

In this paper we examine for which Witt classes ,..., _n over a number field or a function fieldF there exist a finite extensionL/F and ₂,..., _n L^* such thatT _L/F ()=1 andTr _L/F (_i)=_i fori=2,...n.     

Lattice Polytopes Associated to Certain Demazure Modules of sl n + 1

Let w be an element of the Weyl group of sl _{n + 1}. We prove that for a certain class of elements w (which includes the longest element w₀ of the Weyl group), there exist a lattice polytope R ^l(w) , for each fundamental weight _i of sl _{n + 1}, such that for any dominant weight = _{i = 1} ⁿ a _{i i} , the number of lattice points in the Minkowski sum _w = _{i = 1} ⁿ a _{i i} ^w is equal to the dimension of the Demazure module E _w (). We also define a linear map A ^w : R ^l(w) P _Z R where P denotes the weight lattice, such that char E _w () = e e^–A(x) where the sum runs through the lattice points x of ^w .     

Invariant equations defining coincident root loci

Let F(x) = xⁿ+₁ x^n-1+₂ x^n-2+ ··· +_n be a polynomial with complex coefficients, and suppose we are given a partition (₁,...,_r) of n. It is a classical problem to determine explicit algebraic conditions on the _i so that F may have roots with multiplicities ₁,...,_r. We give an invariant theoretic solution to this problem, to wit, we exhibit a set of covariants of F whose vanishing is a necessary and sufficient condition. The construction of such covariants is combinatorial, and involves associating a set of graphs on n vertices (called decisive graphs) to each .Received: 28 September 2003     

All Lambda-Designs With λ = 2 p are Type-1

A-design is a family B ₁,B ₂,...,B _v of subsets of X={1, 2,..., v} such that B _i B _j = for all i jand not all B _i are of the same size. Ryser's andWoodall's -design conjecture states thateach -design can be obtained from a symmetricblock design by a certain complementation procedure. Our mainresult is that the conjecture is true when is twice a prime number.     

On graphical partitions

An integer partition {₁,₂,..., _v } is said to be graphical if there exists a graph with degree sequence _i . We give some results corcerning the problem of deciding whether or not almost all partitions of even integer are non-graphical. We also give asymptotic estimates for the number of partitions with given rank.     

Quasi-conformal mapping theorem and bifurcations

LetH be a germ of holomorphic diffeomorphism at 0 . Using the existence theorem for quasi-conformal mappings, it is possible to prove that there exists a multivalued germS at 0, such thatS(ze ²ⁱ )=HS(z) (1). IfH is an unfolding of diffeomorphisms depending on (,0), withH ₀=Id, one introduces its ideal . It is the ideal generated by the germs of coefficients (a _i (), 0) at 0 ^k , whereH (z)–z=a _i ()z ⁱ . Then one can find a parameter solutionS (z) of (1) which has at each pointz ₀ belonging to the domain of definition ofS ₀, an expansion in seriesS (z)=z+b _i ()(z–z ₀) ⁱ with , for alli.This result may be applied to the bifurcation theory of vector fields of the plane. LetX be an unfolding of analytic vector fields at 0 ² such that this point is a hyperbolic saddle point for each . LetH (z) be the holonomy map ofX at the saddle point and its associated ideal of coefficients. A consequence of the above result is that one can find analytic intervals , , transversal to the separatrices of the saddle point, such that the difference between the transition mapD (z) and the identity is divisible in the ideal . Finally, suppose thatX is an unfolding of a saddle connection for a vector fieldX ₀, with a return map equal to identity. It follows from the above result that the Bautin ideal of the unfolding, defined as the ideal of coefficients of the difference between the return map and the identity at any regular pointz, can also be computed at the singular pointz=0. From this last observation it follows easily that the cyclicity of the unfoldingX , is finite and can be computed explicity in terms of the Bautin ideal.Dedicated to the memory of R. Mañé     

Generating Uniform Random Vectors

In this paper, we study the rate of convergence of the Markov chain X _n+1=AX _n +b _n mod p, where A is an integer matrix with nonzero integer eigenvalues and {b _n } _n is a sequence of independent and identically distributed integer vectors. If _i±1 for all eigenvalues _i of A, then n=O((log p)²) steps are sufficient and n=O(log p) steps are necessary to have X _n sampling from a nearly uniform distribution. Conversely, if A has the eigenvalue ₁=±1, and _i±1 for all i1, n=O(p²) steps are necessary and sufficient.     

Irregular but nonfractal drums andn-dimensional weyl conjecture

In this paper, we study the asymptotics of the spectrum of the Dirichlet (or Neumann) Laplacian in a bounded open set R ⁿ (n 1) with irregular but nonfractal boundary. We give a partial resolution of the Weyl conjecture, i.e. for the counting functionN _i ()(i=0 : Dirichlet;i=1 : Neumann), we have got a precise estimate of the remainder term÷ _i ()=() –N _i () for large, where() is the Weyl term. This implies that for the irregular but nonfractal drum , not only the volume || _n is spectral invariant but also the area of boundary || _n–1 might be spectral invariant as well.Partially supported by the National Natural Science Foundation of China and the Grant of Chinese State Education Committee.     

Separation of two (possibly unbounded) components of the spectrum of a linear operator

LetX be a complex Banach space andA: D(A)X a densely defined closed linear operator whose resolvent set contains the real line and for which (–A)^–1 is bounded onR. We give a necessary and sufficient condition, in terms of the complex powers ofA and –A, for the existence of a decompositionX=X ₊X _–, whereX _± are closed subspaces, invariant forA, the spectra of the reduced operatorsA _± are {(A);Im>0} and {(A);Im<0} respectively, and (–A _±)^–1 is bounded forIm0.Finally we give an example of an operator in anL ^p-type space for which the decomposition exists if 1<p<+ and does not exist ifp=1.     

Regularity of ∫Ω|∇u|2 + λ∫Ω|u−f{2 and some gap phenomenon

We study the regularity of the minimizer u for the functional F (u,f)=|u|² + |u–f{² over all maps uH ¹(, S ²). We prove that for some suitable functions f every minimizer u is smooth in if ₀ and for the same functions f, u has singularities when is large enough.

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Résumé On étudie la régularité des minimiseurs u du problème de minimisation min_ueH ¹(,S²)(|u|² + |u–f{². On montre que pour certaines fonctions f, u est régulière lorsque ₀ et pour les mêmes f, si est assez grand, alors u possède des singularités.

     

Excesses of systems of exponential functions

Conditions on the closeness of real sequences {_n} and {_n} are studied which imply the equality of the excesses of the systems {exp(i_nx)} and {exp(i_nx)} in the space L²(–a, a). A theorem is formulated in terms of the difference of the sequences {_n} and {_n} enumerating the functions. In the corollaries of the theorem, conditions are given in terms of the behavior of the difference _n–_n0. An example is constructed showing that the condition _n–_n0 alone is not sufficient for equality of the excesses.Translated from Matematicheskie Zametki, Vol. 22, No. 6, pp. 803–814, December, 1977.     

The Müntz-Szasz problem

A condition is obtained on the placement of point _n (in some sense, the final point) with which completeness of the system of functionsexp (– _n x), Re_n>0, in spaces L^p, 1p<2. is equivalent to divergence of the series re_n(1+¦_n¦²)^–1.Translated from Matematicheskie Zametki, Vol. 23, No. 1, pp. 91–103, January, 1978.Deceased.     

On eigenvalues of perturbed quadratic matrix polynomials

Among other results, it is shown that ifC andK are arbitrary complexn×n matrices and if det( ₀ ² I₀ C+K)=0 for some ₀0 (resp. ₀=0), then the Newton diagram of the polynomialt(, ) = det(² I+(1+)C+K expanded in (–₀) and , has at least a point on or below the linex+y=b (resp. has no expanded in (–₀) and , has at least a point on or below the of 0 as an eigenvalue of ₀ ² I+₀ C+K. These are extensions of similar results deu to H. Langer, B. Najman, and K. Veseli proved for diagonable matricesC, and shed light on the eigenvalues of the perturbed quadratic matrix polynomials. Our proofs are independent and seem to be simpler     

Boundary Modified Contact Processes

We introduce a one dimensional contact process for which births to the right of the rightmost particle and to the left of the leftmost particle occur at rate _e (where e is for external). Other births occur at rate _i (where i is for internal). Deaths occur at rate 1. The case _e= _i is the well known basic contact process for which there is a critical value _c>1 such that if the birth rate is larger than _c the process has a positive probability of surviving. Our main motivation here is to understand the relative importance of the external birth rates. We show that if _e1 then the process always dies out while if _e>1 and if _i is large enough then the process may survive. We also show that if _i< _c the process dies out for all _e. To extend this notion to d>1 we introduce a second process that has an epidemiological interpretation. For this process each site can be in one of three states: infected, a susceptible that has never been infected, or a susceptible that has been infected previously. Furthermore, the rates at which the two types of susceptible become infected are different. We obtain some information about the phase diagram about this case as well.     

An Improvement of an Inequality of Fiedler Leading to a New Conjecture on Nonnegative Matrices

Suppose that A is an n × n nonnegative matrix whose eigenvalues are = (A), ₂, ..., _n. Fiedler and others have shown that \det( I -A) ⁿ - ⁿ, for all > with equality for any such if and only if A is the simple cycle matrix. Let a _i be the signed sum of the determinants of the principal submatrices of A of order i × i, i=1, ..., n - 1. We use similar techniques to Fiedler to show that Fiedler's inequality can be strengthened to: for all . We use this inequality to derive the inequality that: . In the spirit of a celebrated conjecture due to Boyle-Handelman, this inequality inspires us to conjecture the following inequality on the nonzero eigenvalues of A: If ₁ = (A), ₂,...,_k are (all) the nonzero eigenvalues of A, then . We prove this conjecture for the case when the spectrum of A is real.     

Resolution of the Symmetric Nonnegative Inverse Eigenvalue Problem for Matrices Subordinate to a Bipartite Graph

There is a symmetric nonnegative matrix A, subordinate to a given bipartite graph G on n vertices, with eigenvalues _{12 n} if and only if, ₁ + _n 0, ₂ + _n-10,..., _m + _{n - m + 1}0, _{m + 1}0,..., _{n - m} 0, in which m is the matching numberof G. Other observations are also made about the symmetric nonnegative inverse eigenvalue problem with respect to a graph     

The Real Period Function of A3 Singularity and Perturbations of the Spherical Pendulum

We prove that the Hessian matrix of the real period function () associated with the real versal deformation f (x)=±x ⁴+₂ x ²+₁ x+₀ of a singularity of type A ₃, is nondegenerate, provided that ³ does not belong to the discriminant set of the singularity. We explain the relation between this result and the perturbations of the spherical pendulum.     

On extremal perturbations of semi-Fredholm operators

Given a bounded linear operatorA in an infinite dimensional Banach space and a compact subset of a connected component of its semi-Fredholm domain, we construct a finite rank operatorF such that –A+F is bounded below (or surjective) for each ,F ²=0 and rankF=max min{dimN(–A), codimR(–A)}, if ind(–A)0 (or ind(–A)0, respectively) for each .     

Some generalizations of the classical moment problem

The article is devoted to two generalizations of the classical power moment problem, namely: 1) instead of representing the moment sequence by ⁿ , a representation by polynomialsP _n (), ¹, connected with a Jacobi matrix, appears; 2) in the representation, instead of ⁿ , the expression ⁿ figures, where is a real generalized function (i.e., we investigate some infinite-dimensional moment problem).The work is partially supported by the DFG, Project 436 UKR 113/39/0 and by the CRDF, Project UM1-2090.