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.     

Periodic solutions of singularly perturbed delay equations

We consider a general class of singularly perturbed delay differential systems depending on a singular parameter and another parameter . For =0, the equation defines a mapT which undergoes a generic period doubling at =0. If the bifurcation is supercritical (subcritical), these period two points define a stable period two square wave (unstable period two pulse wave). We give conditions on the vector field such that there is a sectorS in the , plane such that there is a unique periodic orbit if the parameters are inS, the orbit is stable (unstable) if the period doubling bifurcation is supercritical (subcritical) and approaches the square (pulse) wave as 0.Partially supported by NSF and DARPA.     

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 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.     

A Proof of the Jungnickel-Tonchev Conjecture on Quasi-Multiple Quasi-Symmetric Designs

A proof of the following conjecture of Jungnickel and Tonchev on quasi-multiple quasi-symmetric designs is given: Let D be a design whose parameter set (v,b,r,k,) equals (v,sv,sk,k, s) for some positive integer s and for some integers v,k, that satisfy (v-1) = k(k-1) (that is, these integers satisfy the parametric feasibility conditions for a symmetric (v,k,)-design). Further assume that D is a quasi-symmetric design, that is D has at most two block intersection numbers. If (k, (s-1)) = 1, then the only way D can be constructed is by taking multiple copies of a symmetric (v,k, )-design.     

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 . . ..     

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].     

Inequalities Between the Radii of Some Spheres That Are Connected with a Convex Surface in Space of Constant Curvature

With a convex surface in space of constant curvature, we associate four numbers (,M,), where is the radius of a largerst sphere freely rolling over the interior side of , is the inradius of , M is the outradius of , and is the radius of a sphere over whose interior may roll freely. Exact inequalities connecting these four numbers are found.     

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 .     

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.     

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.     

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.     

A minimax test for hypotheses on a spectral density

Assume that on the set of the values t = 1, , k there is given a realization of the Gaussian stationary process X_t with spectral density f(), (0, 1). There arises the problem of the minimax testing of the hypothesis H₀: f() = p().Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 184, pp. 115–125, 1990.     

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.     

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.     

Linearization of operator functions on arbitrary open sets

In this paper it is shown that after a suitable extension an operator function A, which is holomorphic on an open set in , is equivalent on to a linear pencil S-V. If is bounded, then V turns out to be right invertible, and in that case a further extension is equivalent on to a linear pencil T-I.     

The Solution Set to BVP for Some Functional Differential Inclusions

We consider a multivalued BVP x'(t) A(t)x((t))+ F(t,x((t))), Lx= . Under appropriate assumptions on A, L and F, we prove that for sufficiently small the set of solutions to this problem is a nonempty infinite dimensional AR-space (Theorem 4).     

A recursive method of construction of resolvable BIB-designs

A theorem is proved that every resolvable BIB-design (v,k,) with =k–1 and the parameters v and k such that there exists a set of k–1 pairwise orthogonal Latin squares of order v can be embedded in a resolvable BIB-design ((k+1)v, k, k–1). An analogous theorem is established for the class of arbitrary BIB-designs. As a consequence is deduced the existence of resolvable BIB-designs (v,k,) with =k–1 and =(k–1)/2.Translated from Matematicheskie Zametki, Vol. 21, No. 5, pp. 707–715, May, 1977.     

A complete asymptotic expansion of the jacobi functions with error bounds

In this paper we give a complete asymptotic expansion of the Jacobi functions ^{(, )} (t) as + . The method we employed to get the complete expansion follows that of Olver in treating similar problems. By using a Gronwall-Bellman type inequality for an improper integral in which the integrand is an unbounded function and contains a parameter, we get an error bound of the asymptotic approximation which is different from that of Olver's.     

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.