On the convexity of the sum of the first eigenvalues of operators depending on a real parameter

Let ₁ (k) ₂ (k) be the eigenvalues of an operator of a certain type depending on a real parameterk. The paper shows that under certain requirements on the operator and on the nature of its dependence onk, the sum ₁ (k)++ _N (k) is a concave function ofk, for any positive integerN.

---

Zusammenfassung Seien ₁ (k) ₂ (k) die Eigenwerte eines von einem reellen Parameterk abhängigen Operators. Man zeigt, daß unter gewissen Voraussetzungen über den Operator und seine Abhängigkeit vonk die Summe ₁ (k)++ _N (k) für jedesN eine konkave Funktion vonk ist.

     

Ultrafilter translations

We develop a method for extending results about ultrafilters into a more general setting. In this paper we shall be mainly concerned with applications to cardinality logics. For example, assumingV=L, Gödel's Axiom of Constructibility, we prove that if > then the logic with the quantifier there exist many is (,)-compact if and only if either is weakly compact or is singular of cofinality<. As a corollary, for every infinite cardinals and , there exists a (,)-compact non-(,)-compact logic if and only if either < orcf<cf or < is weakly compact.Counterexamples are given showing that the above statements may fail, ifV=L is not assumed.However, without special assumptions, analogous results are obtained for the stronger notion of [,]-compactness.     

Remarks on the perturbation of analytic matrix functions III

As in [N], [LN] the Newton diagram is used in order to get information about the first terms of the Puiseux expansions of the eigenvalues () of the perturbed matrix pencilT(, )=A()+B(, ) in the neighbourhood of an unperturbed eigenvalue () ofA(). In fact sufficient conditions are given which assure that the orders of these first terms correspond to the partial multiplicities of the eigenvalue ₀ ofA().     

Sur les égalités du miroir et certaines formes faibles de la conjecture de Greenberg

Let p be an odd prime. For any CM number field K containing a primitive p^th-root of unity, class field theory and Kummer theory put together yield the well known reflection inequality ^+– between the plus and minus parts of the -invariant of K. Greenbergs conjecture asserts that ⁺ is always trivial. We study here a weak form of this conjecture, namely ⁺=^– if and only if ⁺=^–=0.     

On a problem of A. Pleijel

In 1955, Arne Pleijel proposed the following problem which remains unsolved to this day: Given a closed plane convex curve C and a point x() at a fixed distance above the plane, as the point x() varies, characterize the point for which the conical surface with vertex x() and base C attains its minimum, and determine the limits as 0 and of this minimum point. The purpose of this paper is to solve the cases where approach its extremities and in the course of the solution, we obtain an interesting characterization of the limit points, which we shall call the Pleijel points of C. A consequence is that the inner Pleijel point provides an upper bound for the isoperimetric defect of C. We also generalize the problem to higher dimensional spaces, and obtain the corresponding characterizations of the limiting points for convex surfaces.     

Spectral parameter asymptotics of the Weyl solutions of Sturm-Liouville equations

In the paper one investigates the dependence of Weyl's solution ,)=c(,)+n()s(,) of the Sturm-Liouville equation y+q()y=²y on the spectral parameter . Under the condition that the potential q is bounded from below and q()exp(c₀+c[in1 ¦¦), it is proved for {ie217-01} for any positive values and A. If q()>1 and {ie217-02} for all >0, then in the semiplane >0 the Weyl solution (, ) is obtained from the Weyl solution (,x) is obtained from the Weyl solution e^ix with zero potential, with the aid of a generalization of B. Ya Levin's transformation operators.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 170, pp. 184–206, 1989.I express my sincere gratitude to L. A. Pastur and I. V. Ostrovskii for valuable advice and discussions.     

Prediction Problems for Square-Transformed Stationary Processes

This paper discusses the prediction problems for square-transformed process, Y _t = X _t ², where X _t is a stationary process with spectral density g(). The square-transformation is important in prediction of the volatility of ARCH models. First, we evaluate the mean square prediction error for square-transformed process when the predictor is constructed from the true spectral density g(). However, it is often that the true structure g() is not completely specified. Hence, we consider the problem of misspecified prediction when a conjectured spectral density f (), , is fitted to g(). Then, constructing the best linear predictor based on f (), we can evaluate the prediction error for square-transformed process. Also, we consider a bias adjusted prediction problem for the above two cases. Furthermore, we may suppose that X _t is a non-Gaussian process. Then, we evaluate the mean square prediction errors when the best linear predictor is constructed by the true spectral density g() and the conjectured spectral density f (), respectively. Since is usually unknown we estimate it by a quasi-MLE . The second-order asymptotic approximations of the mean square errors of the predictors based on g() and f () are given. Finally, we provide some numerical examples, which show some unexpected features.     

J-pseudo-spectral andJ-inner-pseudo-outer factorizations for matrix polynomials

For a comonic polynomialL() and a selfadjoint invertible matrixJ the following two factorization problems are considered: firstly, we parametrize all comonic polynomialsR() such that . Secondly, if it exists, we give theJ-innerpseudo-outer factorizationL()=()R(), where () isJ-inner andR() is a comonic pseudo-outer polynomial. We shall also consider these problems with additional restrictions on the pole structure and/or zero structure ofR(). The analysis of these problems is based on the solution of a general inverse spectral problem for rational matrix functions, which consists of finding the set of rational matrix functions for which two given pairs are extensions of their pole and zero pair, respectively.The work of this author was supported by the USA-Israel Binational Science Foundation (BSF) Grant no. 9400271.     

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.     

Nonnegative Hall Polynomials

The number of subgroups of type and cotype in a finite abelian p-group of type is a polynomialg with integral coefficients. We prove g has nonnegative coefficients for all partitions and if and only if no two parts of differ by more than one. Necessity follows from a few simple facts about Hall-Littlewood symmetric functions; sufficiency relies on properties of certain order-preserving surjections that associate to each subgroup a vector dominated componentwise by . The nonzero components of (H) are the parts of , the type of H; if no two parts of differ by more than one, the nonzero components of – (H) are the parts of , the cotype of H. In fact, we provide an order-theoretic characterization of those isomorphism types of finite abelian p-groups all of whose Hall polynomials have nonnegative coefficients.     

The dynamics of perturbations of the contracting Lorenz attractor

We show here that by modifying the eigenvalues ₂ < ₃ < 0 < ₁ of the geometric Lorenz attractor, replacing the usualexpanding condition ₃+₁ > 0 by acontracting condition ₃+₁ < 0, we can obtain vector fields exhibiting transitive non-hyperbolic attractors which are persistent in the following measure theoretical sense: They correspond to a positive Lebesgue measure set in a twoparameter space. Actually, there is a codimension-two submanifold in the space of all vector fields, whose elements are full density points for the set of vector fields that exhibit a contracting Lorenz-like attractor in generic two parameter families through them. On the other hand, for an open and dense set of perturbations, the attractor breaks into one or at most two attracting periodic orbits, the singularity, a hyperbolic set and a set of wandering orbits linking these objects.     

Conjugation for polynomial mappings

We consider Keller's functions, namely polynomial functionsf:C ⁿ C ⁿ with detf(x)=1 at allx C ⁿ. Keller conjectured that they are all bijective and have polynomial inverses. The problem is still open.Without loss of generality assumef(0)=0 andf'(0)=I. We study the existence of certain mappingsh , > 1, defined by power series in a ball with center at the origin, such thath(0)=I andh (f(x))=h (x). So eachh conjugates f to its linear part I in a ball where it is injective.We conjecture that for Keller's functionsf of the homogeneous formf(x)=x +g(x),g(sx)=s ^dg(x),g(x)ⁿ=0,xC ⁿ,sC the conjugationh for f is anentire function.     

Solving a nonlinear spectral problem for a matrix

This paper examines the solving of the eigenvalue problem for a matrix M () with a nonlinear occurrence of the spectral parameter. Two methods are suggested for replacing the equation dat M()=0 by a scalar equationf()=0. Here the functionf() is not written formally, but a rule for computingf() at a fixed point of the domain in which the desired roots lie is indicated. Müller's method is used to solve the equationf()=0. The eigenvalue found is refined by Newton's method based on the normalized expansion of matrix M(), and the linearly independent vectors corresponding to it are computed. An ALGOL program and test examples are presented.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 58, pp. 54–66, 1976.     

Discrete Spectrum of Differential Operators in Spectral Gaps under Nonnegative Perturbations of High Order

Let A be a self-adjoint elliptic second-order differential operator, let (, ) be an inner gap in the spectrum of A, and let B(t) = A + tW ^* W, where W is a differential operator of higher order. Conditions are obtained under which the spectrum of the operator B(t) in the gap (, ) is either discrete, or does not accumulate to the right-hand boundary of the spectral gap, or is finite. The quantity N(, A, W, ), (, ), > 0 (the number of eigenvalues of the operator B(t) passing the point (, ) as t increases from 0 to ) is considered. Estimates of N(, A, W, ) are obtained. For the perturbation W ^* W of a special form, the asymptotics of N(, A, W, ) as + is given. Bibliography: 5 titles.     

Universality of critical behaviour in a class of recurrent random walks

Let X₀=0, X₁, X₂,.. be an aperiodic random walk generated by a sequence ₁, ₂,... of i.i.d. integer-valued random variables with common distribution p(·) having zero mean and finite variance. For anN-step trajectory and a monotone convex functionV: withV(0)=0, define Further, let be the set of all non-negative paths compatible with the boundary conditionsX₀=a, X_N=b. We discuss asymptotic properties of under the probability distribution N and 0, Z^a,b_N,+, being the corresponding normalization. If V(·) grows not faster than polynomially at infinity, define H() to be the unique solution to the equation Our main result reads that as 0, the typical height of X_[, ^N] scales as H() and the correlations along decay exponentially on the scale H()². Using a suitable blocking argument, we show that the distribution tails of the rescaled height decay exponentially with critical exponent 3/2. In the particular case of linear potential V(·), the characteristic length H() is proportional to ^-1/3 as 0.Mathematics Subject Classification (2000):60G50, 60K35; 82B27, 82B41     

Reflecting stationary sets and successors of singular cardinals

REF is the statement that every stationary subset of a cardinal reflects, unless it fails to do so for a trivial reason. The main theorem, presented in Sect. 0, is that under suitable assumptions it is consistent that REF and there is a which is ⁺ⁿ -supercompact. The main concepts defined in Sect. 1 are PT, which is a certain statement about the existence of transversals, and the bad stationary set. It is shown that supercompactness (and even the failure of PT) implies the existence of non-reflecting stationary sets. E.g., if REF then for many PT(, ₁). In Sect. 2 it is shown that Easton-support iteration of suitable Levy collapses yield a universe with REF if for every singular which is a limit of supercompacts the bad stationary set concentrates on the right cofinalities. In Sect. 3 the use of oracle c.c. (and oracle proper—see [Sh-b, Chap. IV] and [Sh 100, Sect. 4]) is adapted to replacing the diamond by the Laver diamond. Using this, a universe as needed in Sect. 2 is forced, where one starts, and ends, with a universe with a proper class of supercompacts. In Sect. 4 bad sets are handled in ZFC. For a regular {<⁺ : cf<} is good. It is proved in ZFC that if=cf>₁ then {<⁺ : cf<} is the union of sets on which there are squares.     

Sur l'adjonction d'une masse de Dirac á une forme régulière et semi-classique

Résumé On montre que si L est une forme linéaire régulière et semi-classique, alors la forme u=L + _c où c C est quelconque, est encore régulière et semi-classique pour tout C en dehors d'un ensemble dénombrable de valeurs singulières. On donne l'équation différentielle linéaire du second ordre vérifiée par chaque polynôme de la suite orthogonale associée à u.

---

We show that, if L is a regular, semi-classical functional, then u=L + _c where c C, is also regular and semi-classical for every complex , except for a discret set. We give the second order linear differential equation satisfied by each polynomial of the orthogonal sequence associated with u.

     

A sandwich theorem and Hyers—Ulam stability of affine functions

It is shown that two real functionsf andg, defined on a real intervalI, satisfy the inequalitiesf(x + (1 – )y) g(x) + (1 – )g(y) andg(x + (1 – )y) f(x) + (1 – )f(y) for allx, y I and [0, 1], iff there exists an affine functionh: I such thatf h g. As a consequence we obtain a stability result of Hyers—Ulam type for affine functions.     

Cardinal Invariants of λ-Topologies

We study the basic cardinal-valued invariants of C (X) such as weight, density, network weight, i-weight, and tightness, where C (X) is the space of all continuous real functions on X in the -topology.     

A remark on a boundary value problem for force-free fields

Summary For the equation rotv–v=0 with not necessarily constant a boundary value problem of Neumann's type is solved by reducing it to an equivalent integral equation.

---

Zusammenfassung Für die Gleichung rotv–v=0 mit nicht notwendig konstantem wird ein Neumannsches Randwertproblem durch Zuruckführung auf eine äquivalente Integralgleichung gelöst.