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 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 .     

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.     

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.     

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.     

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.     

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].     

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 . . ..     

The existence of some resolvable block designs with divisibility into groups

This paper proves the existence of resolvable block designs with divisibility into groups GD(v; k, m; ₁, ₂) without repeated blocks and with arbitrary parameters such that ₁ = k, (v–1)/(k–1) ₂ v^k–2 (and also ₁ k/2, (v–1)/(2(k–1)) ₂ v^k–2 in case k is even) k 4 andp=1 (mod k–1), k < p for each prime divisor p of number v. As a corollary, the existence of a resolvable BIB-design (v, k, ) without repeated blocks is deduced with X = k (and also with = k/2 in case of even k) k , where a is a natural number if k is a prime power and=1 if k is a composite number.Translated from Matematicheskie Zametki, Vol. 19, No. 4, pp. 623–634, April, 1976.     

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.     

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.     

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     

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.     

Approximation of differentiable functions by polynomial splines in domains with external peaks

For unbounded domains with external power-type peaks, we propose a method for the approximation of functionsf(x) w _p ^r () by polynomial splines in the metricw _p ^r (), 1pq, and present the corresponding estimates.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 9, pp. 1224–1233, September, 1994.     

Identities of an algebra of triangular matrices

This paper deals with the ideals of identities of certain associative algebras over a field F of characteristic zero. An algebra W of matrices of the form ,,,M, where and , are F-algebras with unity and M is a (,)-bimodule, is considered. Under certain natural restrictions on M one obtains the equality of ideals of identities T(W)=T()T(), if [[x₁,x₂], x₃[x₄,x₅]]T().Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 114, pp. 7–27, 1982.     

Estimates of Entire Functions of Exponential Type Less than π in Terms of Logarithmic Sums over Real Duffin and Schaeffer Sequences

We give uniform estimates of entire functions of exponential type less than having sufficiently small logarithmic sums over real sequences { _n } satisfying | _n –n|L and _n+1– _n for fixed positive constants L and . We thereby generalize results about logarithmic sums over the set of integers and so-called relatively h-dense sequences.     

A trace theorem and a linearization method for operator polynomials

Let p() = ⁿI - ^n-1 A₁ - A_n, where A_i are operators on a Banach space X with (1/i)^th power summable approximation numbers. Then the sum of the eigenvalues of the operator polynomial p() equals the trace of A₁. This is shown by a linearization method which is further used to generalize results on the completeness of the eigenvalues of p() under suitable conditions on the resolvent.     

Difference Sets with n = 2pm

Let D be a (v,k,) difference set over an abelian group G with even n = k - . Assume that t N satisfies the congruences t q _i ^fi (mod exp(G)) for each prime divisor q_i of n/2 and some integer f_i. In [4] it was shown that t is a multiplier of D provided that n > , (n/2, ) = 1 and (n/2, v) = 1. In this paper we show that the condition n > may be removed. As a corollary we obtain that in the case of n= 2p^a when p is a prime, p should be a multiplier of D. This answers an open question mentioned in [2].     

D λ -cycles in 3-cyclable graphs

Letk and be positive integers, andG a 2-connected graph of ordern with minimum degree and independence number. A cycleC ofG is called aD -cycle if every component ofG – V(C) has order smaller than. The graphG isk-cyclable if anyk vertices ofG lie on a common cycle. A previous result of the author is that if k 2, G isk-connected and every connected subgraphH ofG of order has at leastn +k ² + 1/k + 1 – vertices outsideH adjacent to at least one vertex ofH, thenG contains aD -cycle. Here it is conjectured that k-connected can be replaced by k-cyclable, and this is proved fork = 3. As a consequence it is shown that ifn 4 – 6, or ifG is triangle-free andn 8 – 10, thenG contains aD ₃-cycle orG , where denotes a well-known class of nonhamiltonian graphs of connectivity 2. As an analogue of a result of Nash-Williams it follows that ifn 4 – 6 and – 1, thenG is hamiltonian orG . The results are all best possible and compare favorably with recent results on hamiltonicity of graphs which are close to claw-free.     

Minimality of derivative chains,corresponding to a boundary value problem on a finite segment

One investigates the minimality of derivative chains, constructed from the root vectors of polynomial pencils of operators, acting in a Hilbert space. One investigates in detail the quadratic pencil of operators. In particular, for L()=L₀+L₁+²L₂ with bounded operators L₀0, L₂0 and Re L₁0, one shows the minimality in the space_173-02 of the system {x_k, _kekx_k}, where x_k are eigenvectors of L(), corresponding to the characteristic numbers _kin the deleted neighborhoods of which one has the representation L^–1()=(–_k)^–1R_K+W_K() with one-dimensional operators R_k and operator-valued functions W_K(), k=1, 2, ..., analytic for =_k.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 2, pp. 195–205, February, 1990.