Minimality of root vectors of operator functions analytic in an angle

We study the minimality of elementsx _h,j,k of canonical systems of root vectors. These systems correspond to the characteristic numbers _k of operator functionsL() analytic in an angle; we assume that operators act in a Hilbert space . In particular, we consider the case whereL()=I+T()c, >0,I is an identity operator,C is a completely continuous operator, (I- C)^–1c for ¦arg¦, 0<<, the operator functionT() is analytic, and T()c for ¦arg¦<. It is proved that, in this case, there exists >0 such that the system of vectorsC ^v x _h,j,k is minimal in for arbitrary positive <1+, provided that ¦_k¦>.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 5, pp. 545–566, May, 1994.This research was partially supported by the Ukrainian State Committee of Science and Technology.     

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.     

Reverse Functional Inequalities and Their Applications to Nonlinear Elliptic Boundary Value Problems

We establish some reverse inequalities. We give applications to nonlinear elliptic boundary value problems containing a parameter which have two branches of solutions u (0) and U (>0) of which the first is continuous at the origin and the second increases indefinitely as 0.     

Minimal coverings and maximal packings of (k−1)-subsets by k-subsets

This paper studies the asymptotic behavior of functions M(n, k, k–1, ) and m(n, k, k–1, ), equal to the respective cardinalities of the minimal -covering and maximal -packing of all (k–1)-subsets of the n-element set of its k-subsets. It is shown that, if sequence k=k(n) is such that k(n)/n 0 as then m(n, k, k–1, ).( _k–1 ⁿ ).k¹, and if as n , thenM(n,k,k–1,).( _k–1 ⁿ ).k^–1. A consequence of these results is the validity of the Erdös-Hanani conjecture concerning the asymptotic behavior of functions M(n, k, k–1, 1) and m(n, k, k–1, 1).Translated from Matematicheskie Zametki, Vol. 21, No. 4, pp. 565–571, April, 1977.In conclusion, the author wishes to thank A. A. Sapozhenko under whose direction the present work was achieved.     

Integrals involving Gegenbauer and Hermite polynomials on the imaginary axis

Summary The integral _- [C _2n (it)]^–2(1+t ²)^–-1/2 dt is evaluated for > –1/2 whereC _2n is the Gegenbauer polynomial of degree 2n. Letting gives the value _- [H _2n (it)]^–2 e ^{1-1/2t 2} dt involving the Hermite polynomialH _2n of degree 2n. The result is obtained using Gegenbauer functions of the second kind.     

Principal part of resolvent and factorization of an increasing nonanalytic operator-function

Let a selfadjoint operator-valued functionL() be given on the interval [a,b] such thatL(a)0,L(b)0,L()0 (ab), andL() has a certain smoothness (for instance, it satisfies Hölder's condition). It turns out that the spectral theory of the operator-valued functionL() can be reduced to the spectral theory of one operatorZ, the spectrum of which lies on (a, b) and which is similar to a selfadjoint operator. In particular, the factorization takes place:L()=M()(I–Z), where the operator-valued functionM() is invertible on [a, b]. Earlier similar results were known only for analytic operator-valued functions. The authors had to use new methods for the proof of the described theorem. The key moment is the decomposition ofL ^–1() into the sume of its principal and regular parts.     

On the Multiplicity of Eigenvalues of the Laplacian on Surfaces

We show that the multiplicity of the eigenvalues of the Laplace Beltrami operator on compact Riemannian surfaces with genus zero is bounded by m(k) 2k – 3 for k 3. Here we label the eigenvalues in the following way: 0 = 1 < 2 3 . . ..     

Thermodynamics,real and complex periods of the Volterra model

The Lotka-Volterra model is converted into a Hamiltonian system. For this system the Gibbs state and the thermodynamic functions as free energy, entropy, expectation and variances of the Hamiltonian and its summands are calculated together with expansions for low and high temperature ^–1.Since the canonical partition functionZ (, , ) is the Laplace transform of the energy-period functionT(E, , ), we obtain expansions ofT (E, , ) for small and large energyE=H (p, q) and arbitrary parameters, by inverse Laplace transformation of the partition functionZ (, , ). Expansions of the partial orbit times T_±±, i.e. the part of the periodT with predator and prey above or below equilibrium values, are available, too. By expressingT(E, , ) as a convolution integral, we derive global inequalities, e. g. the function (log (E/), log (E/)) log (ET) (E, , ) is convex on the whole domainR ² and (E/T) (T/E) (0, 1) globally. Finally the analytic continuation of the functionT=T (E, , ) to the left- as well as right halfplane ReE<0 and ReE>0 can be expressed by Laplace transforms.

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Zusammenfassung Das Lotka-Volterra Modell wird in ein Hamiltonsches System umgeformt. Für dieses System kann man den Gibbszustand und die thermodynamischen Funktionen wie freie Energie, Entropie sowie Erwartungswerte und Varianzen der Hamiltonfunktion und ihrer Summanden berechnen und Entwicklungen für niedrige als auch hohe Temperatur ^–1 angeben.Da die kanonische ZustandssummeZ (, , ) die Laplacetransformierte der Energie-Perioden FunktionT=T(E, , ) ist, erhält man durch Umkehrung der Laplacetransformation Entwicklungen vonT (E, , ) für kleine sowie große EnergieE=H(p, q). Durch spezielle Manipulationen bekommt man sogar Entwicklungen der PeriodenteileT _±±, das sind die Intervalle während denen Räuber bzw. Beute ober- bzw. unterhalb der Gleichgewichte liegen. Mit Hilfe einer Darstellung vonT(E, , ) als Faltungsintegral beweist man globale Ungleichungen, z. B. ist die Funktion (log (E/), log (E/)) log (ET) (E, , ) auf dem ganzen DefinitionsbereichR ² konvex und (E/T) (T/E) (0,1).Schlußendlich läßt sich die analytische Fortsetzung der FunktionT(E, , ) in die linke sowie rechte Halbebene ReE< 0 bzw. ReE> 0 durch ein Laplaceintegral darstellen.

     

Continuous Singularity of the Weak Limit of Convolution Powers of a Discrete Probability Measure on 2 × 2 Stochastic Matrices

Let be a probability measure on n 2 × 2 stochastic matrices, n an arbitrary positive integer, and = (w) lim _n ⁿ , such that the support of consists of 2 × 2 stochastic matrices of rank one, and as such, can be regarded as a probability measure on [0, 1]. We present simple sufficient conditions for to be continuous singular w.r.t. the Lebesgue measure on [0, 1]. We also determine , given .     

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 .     

On the almost everywhere convergence of double Fourier series of integrable functions

, . . Q _k [0,2],k=1,2, — . F(x, y)L(T), T=[0, 2]², G(x, y)L(T) , G(x,y)=F(x,y) Q=Q ₁ ×Q ₂ - .     

Spectrum of an elliptic equation

It is shown that the spectrum for the first boundary-value problem for a second-order elliptic equation always lies in the half-plane ₀ Re z, where is the leading eigenvalue to which there corresponds a nonnegative eigenfunction. Apart from ₀, there are no other points of the spectrum on the straight line Re z=₀.Translated from Matematicheskie Zametki, Vol. 7, No. 4, pp. 495–502, April, 1970.The authors are grateful to V. S. Vladimirov for discussing the results of the present paper and for pointing out the proposition in [3], which made it possible to shorten the proof that the leading eigenvalue of the problem (1) is simple.     

A Residue Calculus for Root Systems

Let V be a finite-dimensional real vector space on which a root system is given. Consider a meromorphic function on V =V+iV, the singular locus of which is a locally finite union of hyperplanes of the form V , = s, , s . Assume is of suitable decay in the imaginary directions, so that integrals of the form _+iV , d make sense for generic V. A residue calculus is developed that allows shifting . This residue calculus can be used to obtain Plancherel and Paley–Wiener theorems on semisimple symmetric spaces.     

The spectral bound of Schrödinger operators

Let V: R ^N [0, ] be a measurable function, and >0 be a parameter. We consider the behaviour of the spectral bound of the operator 1/2–V as a function of . In particular, we give a formula for the limiting value as , in terms of the integrals of V over subsets of R ^N on which the Laplacian with Dirichlet boundary conditions has prescribed values. We also consider the question whether this limiting value is attained for finite .     

Subordination des résolvantes

Résumé Soit (V ₎₀ une résolvante définie sur un espace mesurable telle que le noyau initial est borné; on trouve une condition nécéssaire et suffisante pour qu'un noyau borné U possède une résolvante (U ₎₀ telle que U V pour tout 0. On donne plusieurs applications de ce résultat.     

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.     

The Selberg trace formula for SL(3,Z)

In this article the first step toward the generalization of the Selberg trace formula to the case of a rank 2 symmetric space S and a discrete group for which the fundamental region \S goes to infinity nontrivially appears. For S we use the space SL(3,)/SO(3) and for we use SL(3,). The fundamental results are Theorems 9 and 10, in which is calculated the contribution to the matrix trace of the operator K which appears in the right side of the trace formula of the expression h()d^c(), where ^c() is the continuous part of the spectral measure of the quasiregular representation on the space IL₂(\S).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 63, pp. 8–66, 1976.     

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.     

Preassigning the boundary of diametrically-complete sets

In a Minkowski space with unit ballS, a setD() is diametrically complete ifD()={. In this paper a method is given for generating such setsD() which contain an arbitrary setX. This technique is then employed to find necessary and sufficient conditions for a setA to lie on the boundary of some suchD() and bounds are given on the acceptable values of .     

Factorization of a selfadjoint nonanalytic operator function II. Single spectral zone

We consider a selfadjoint and smooth enough operator-valued functionL() on the segment [a, b]. LetL(a)0,L(b)0, and there exist two positive numbers and such that the inequality |(L()f, f)|< ([a, b] f=1) implies the inequality (L'()f, f)>. Then the functionL() admits a factorizationL()=M()(I-Z) whereM() is a continuous and invertible on [a, b] operator-valued function, and operatorZ is similar to a selfadjoint one. This result was obtained in the first part of the present paper [10] under a stronge conditionL()0 ( [a,b]). For analytic functionL() the result of this paper was obtained in [13].