A Remark on Real Parameter Global Bifurcation

We study the nonlinear eigenvalue problem F(x,) = L()x +R(x,) = 0 where F : X × R X with X a Hilbert space. IfL() is a polynomial in , then it is shown that 0> 0 is a global bifurcation point of the eigenvalue problem provided astandard transversality condition is satisfied, the dimension of the nullspace of L(0) is an odd number and L() is composed of asequence of positive operators on the finite dimensional null space ofL(0).     

On the existence of strongly normal ideals overP κ λ

For every uncountable regular cardinal and any cardinal,P denotes the set . Furthermore, < denotes=" the=" binary=" operation=" defined=">P byx<> iffxy¦x<>.By anideal over P we mean a proper, non-principal,-complete ideal overP extending the ideal dual to the filter generated by . For any idealI overP ,I ⁺ denotes the setP –I, andI ^* the filter dual toI.     

Reconstruction of a rational nonsquare matrix function from local data

In many problems the local zero-pole structure (i.e. locations of zeros and poles together with their orders) of a scalar rational functionw is a key piece of structure. Knowledge of the order of the pole or zero of the rational functionw at the point is equivalent to knowledge of the -module (where is the space of rational functions analytic at ). For the more intricate case of a rationalp×m matrix functionW, we consider the structure of the module as the appropriate analogue of zero-pole structure (location of zeros and poles together with directional information), where is the set of column vectors of heightm with entries equal to rational functions which are analytic at . Modules of the form in turn can be explicitly parametrized in terms of a collection of matrices (C ,A ,B ,B , ) together with a certain row-reduced(p–m)×m matrix polynomialP(z) (which is independent of ) which satisfy certain normalization and consistency conditions. We therefore define the collection (C ,A ,Z ,B , ,P(z)) to be the local spectral data set of the rational matrix functionW at . We discuss the direct problem of how to compute the local spectral data explicitly from a realizationW(z)=D+C(z–A) ^–1 B forW and solve the inverse problem of classifying which collections (C ,A ,Z ,B , ,P(z)) satisfying the local consistency and normalization conditions arise as the local spectral data sets of some rational matrix functionW. Earlier work in the literature handles the case whereW is square with nonzero determinant.     

The existence of non-trivial hyperfactorizations ofK 2n

A -hyperfactorization ofK _2n is a collection of 1-factors ofK _2n for which each pair of disjoint edges appears in precisely of the 1-factors. We call a -hyperfactorizationtrivial if it contains each 1-factor ofK _2n with the same multiplicity (then =(2n–5)!!). A -hyperfactorization is calledsimple if each 1-factor ofK _2n appears at most once. Prior to this paper, the only known non-trivial -hyperfactorizations had one of the following parameters (or were multipliers of such an example)

A New Plethysm Formula for Symmetric Functions

This paper gives a new formula for the plethysm of power-sum symmetric functions and Schur symmetric functions with one part. The form of the main result is that for b,

The Spectral Function and Principal Eigenvalues for Schrödinger Operators

Let m , 0 m₊ in Kato's class. We investigate the spectral function s( + m) where s( + m) denotes the upper bound of the spectrum of the Schrödinger operator + m. In particular, we determine its derivative at 0. If m_- is sufficiently large, we show that there exists a unique ₁ > 0 such that s( + ₁m) = 0. Under suitable conditions on m⁺ it follows that 0 is an eigenvalue of + ₁m with positive eigenfunction.     

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.     

Norms in quadratic fields and their relations to non commuting 2×2 matrices I

LetA be a fixed 2×2 integral matrix with irrational characteristic roots. LetB be an arbitrary 2×2 integral matrix. It was previously shown that -det (AB-BA)=norm, where is in the field of the characteristic roots ofA. It is now shown that the 's corresponding to varyingB's can be chosen to form a fractional ideal in this field.     

Bifurcation of periodic solutions to variational inequalities in R κ based on Alexander-Yorke theorem

Variational inequalities are studied, where K is a closed convex cone in , 3, B is a × matrix, G is a small perturbation, a real parameter. The assumptions guaranteeing a Hopf bifurcation at some ₀ for the corresponding equation are considered and it is proved that then, in some situations, also a bifurcation of periodic solutions to our inequality occurs at some _{I 0}. Bifurcating solutions are obtained by the limiting process along branches of solutions to penalty problems starting at ₀ constructed on the basis of the Alexander-Yorke theorem as global bifurcation branches of a certain enlarged system.     

The spectrum of an elliptic operator of second order

Under minimal requirements on the coefficients and the boundary of the domain it is proved that the spectrum of the first boundary-value problem for an elliptic operator of second order always lies in the half-plane Re , where is the leading eigenvalue to which there corresponds a nonnegative eigenfunction. On the line Re = , there are no other points of the spectrum.Translated from Matematicheskie Zametki, Vol. 20, No. 3, pp. 351–358, September, 1976.     

Stretching a surface in a rotating fluid

Summary A surface is stretched in a rotating fluid. The solution to the governing set of nonlinear differential equations depends on a parameter which is the ratio of the rotation rate to the stretching rate. Perturbation solutions for small and large compare well with exact numerical integration.     

Vector measure range duality and factorizations of (<Emphasis Type="Italic">D,p</Emphasis>)-summing operators from Banach function spaces

We characterize the relationship between the space L₁() and the dual L₁() of the space L₁(), where (, ) is a dual pair of vector measures with associated spaces of integrable functions L₁() and L₁() respectively. Since the result is rather restrictive, we introduce the notion of range duality in order to obtain factorizations of operators from Banach function spaces that are dominated by the integration map associated to the vector measure . We obtain in this way a generalization of the Grothendieck-Pietsch Theorem for p-summing operators.*The research was partially supported by MCYT DGI project BFM 2001-2670.**The research was partially supported by MCYT DGI project BFM 2000-1111.     

Holomorphic Solutions of Some Functional Equations II

We study a certain type of functional equation, which is of significance from the view point of systems of difference equations. Let the characteristic values of the system be and The case that either || > 1 or 0 < || < 1 has been treated in a former paper. The case that = 1, || = 1 with 1 will be given in another paper. The present note deals with the case = = 1, the most difficult case.AMS Subject Classification (1991): 39A10 39B05     

Asymptotic behavior of the spectrum of a pseudodifferential operator with periodic bicharacteristics

Let _j be the eigenvalues of a positive elliptic pseudodifferential operator of order m > 0 on a closed compact d-dimensional C-manifold and let N()=#{j:_j^m}. It is shown that for each > 0 we have

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.     

Extremal feedback perturbations for families of closed operators

Let T- S, be a family of not necessarily bounded semi-Fredholm operators, where T and S are operators acting between Banach spaces X and Y, and where S is bounded with D(S) D(T). For compact sets , as well as for certain open sets , we investigate existence and minimal rank of bounded feedback perturbations of the form F=BE such that min.ind (T-S+F)=0 for all . Here B is a given operator from a linear space Z to Y and E is some operator from X to Z.We give a simple characterization of that situation, when such regularizing feedback perturbations exist and show that for compact sets the minimal rank never exceeds max { min.ind (T-S) }+1. Moreover, an example shows that the minimal rank, in fact, may increase from max {...} to max {...}+1, if the given B enforces a certain structure of the feedbachk perturbation F.However, the minimal rank is equal to max { min.ind (T-S) }, if is an open set such that min.ind (T-S) already vanishes for all but finitely many points . We illustrate this result by applying it to the stabilization of certain infinite-dimensional dynamical systems in Hilbert space.     

On a fixed-point theorem on locally convex linear topological spaces

Iff is a self mapping on a closed convex subsetK of a separated quasicomplete locally convex linear topological spaceE such that (i)E is strictly convex, (ii)f (K) is contained in a compact subset ofK and (iii)f satisfies a contraction condition, then it is shown that for eachxK, the sequence of {U ⁿ (x)} _n =1 of iterates, whereU KK is defined byU (y)=f(y)+(1-) y, yK, converges to a fixed point off.     

On factorization representations for Avakumović--Karamata functions with nondegenerate groups of regular points

Avakumovi-Karamata functions f are generalized regularly varying functions (so--called ORV functions) such that f*()= limsup _x f(x)/f(x) is finite for all >0. In this paper, we investigate classes of ORV functions with "nondegenerate groups of regular points", that is, having points 1, for which f*() exists as a positive and finite limit (instead of limsup) on a nontrivial subgroup of the positive real axis. Certain factorization representations, characterizations and uniform convergence theorems are proved, describing both the structure of ORV functions f as well as that of their limit functions f*. Some well-known results from regular variation theory are covered by this general approach.     

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.