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 relationship between the Mahler measure and the discriminant of algebraic numbers

In this note we show that in the well-known Dobrowolski estimate lnM() (ln lnd/ lnd)3,d , where is a nonzero algebraic number of degreed that is not a root of unity andM() is its Mahler measure, the parameterd can be replaced by the quantity=d/() ^1/d, where () is the modulus of the discriminant of. To this end, must satisfy the condition deg ^p=deg for any primep.Translated fromMatematicheskie Zametki, Vol. 59, No. 3, pp. 415–420, March, 1996.     

The likelihood ratio for densities with singularities

Let t_n be the Bayesian estimator of the parameter constructed from independent observations in the case of infinite information and probability distribution density with some singularities. It is shown that under certain conditions on the behavior of the densities near their singularities, the normalizing factor (n) ensuring that _n=(n) (t_n-) has a nontriviallimiting distribution for n is regularly varying in Karamata's sense. The limiting distribution of _n is determined.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 74, pp. 66–82, 1977.I would like to acknowledge the guidance of I. A. Ibragimov in the course of this research.     

Tangential Limits of Potentials on Homogeneous Trees

Let T be a homogeneous tree of homogeneity q+1. Let denote the boundary of T, consisting of all infinite geodesics b=[b ₀,b ₁,b ₂,] beginning at the root, 0. For each b, 1, and a0 we define the approach region _,a (b) to be the set of all vertices t such that, for some j, t is a descendant of b _j and the geodesic distance of t to b _j is at most (–1)j+a. If >1, we view these as tangential approach regions to b with degree of tangency . We consider potentials Gf on T for which the Riesz mass f satisfies the growth condition _T f ^p (t)q ^–|t|<, where p>1 and 0<<1, or p=1 and 0<1. For 11/, we show that Gf(s) has limit zero as s approaches a boundary point b within _,a (b) except for a subset E of of -dimensional Hausdorff measure 0, where H (E)=sup_>0inf _i q ^{–|t i|}:E a subset of the boundary points passing through t _i for some i,|t _i |>log _q (1/).     

Field Extensions and Extensions of Linear Differential Operators

Let K R P be a tower of fields, N be a P-module, and : R N be a K-linear differential operator. The aim of this paper is to investigate whether the operator has an extension to P, i.e. if these exists a differential operator : P N such that |_R = . The results of this paper were published in Russian in Mat. Zametki 30(2) (1981), 237–248.     

On a Problem of Karpilovsky

Let G be a finite elementary group. Let ⁿ (G) denote the nth power of the augmentation ideal (G) of the integral group ring G. In this paper, we give an explicit basis of the quotient group Q_n(G) = ⁿ(G)/ⁿ⁺¹ (G) and compute the order of Qⁿ (G).2000 Mathematics Subject Classification: 16S34, 20C05     

A limiting absorption principle for Schrödinger operators with Von Neumann-Wigner type potentials

In this paper we prove the main step in establishing a limiting absorption principle for von Neumann-Wigner type Schrödinger Hamiltonians of the form –+csinb|x|/|x|+V(x), whereV(x) is a short range potential. The first fundamental step is to obtain a limiting absorption principal for the free operator –+csinb.|x|/|x|. The free operator is unitarily equivalent to a direct sum of ordinary differential operators. We obtain uniform estimates for the resolvents of these ordinary differential operators. by obtaining uniform estimates for the Weyl-Green kernels of these resolvents. In turn, these latter estimates require uniform estimates on the Wronskians of certain generalized eigen-solutions of these differential operators.This paper is dedicated to the memory of the late Professor Charles C. Conley.     

Isoperimetric inequalities in a boundary value problem on a Riemannian manifold

In this paper the problem u+1=0 in ,u=0 on is considered. Here is a finite domain on a Riemannian manifold and the associated Laplace-Beltrami operator. By means of maximum principles isoperimetric bounds for the maximum ofu and the maximum of the absolute value of the gradient ofu, as well as some related bounds are derived.

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Zusammenfassung Diese Arbeit behandelt das Problem u+1=0 in ,u=0 auf , wobei ein Gebiet auf einer zweidimensionalen Riemann'schen Mannigfaltigkeit ist, und der zugehörige Laplace-Beltrami Operator. Es werden isoperimetrische Schranken für das Maximum vonu und |u| aus gewissen Maximumsprinzipien hergeleitet, sowie einige verwandte Resultate.

     

Eigenfunctions of the Beltrami-Laplace operator on a hyperboloid of one sheet

The Beltrami-Laplace operator on a one-sheeted hyperboloid is hyperbolic. There is taken the set of functions bounded outside some neighborhood of two isotropic lines, intersecting at infinity. A necessary and sufficient condition is derived that a function of this set be an eigenfunction of the operator .Translated from Matematicheskie Zametki, Vol. 7, No. 2, pp. 255–263, February, 1970.     

Inequalities in Products of Minors of Totally Nonnegative Matrices

Let _I,I be the minor of a matrix which corresponds to row set I and column set I. We give a characterization of the inequalities of the form _{I,I K,K J,J L,L} which hold for all totally nonnegative matrices. This generalizes a recent result of Fallat, Gekhtman, and Johnson.     

Some norm estimates for the solutions of a nonlinear diffusion problem

Summary By means of Rellich's identity bounds for the spectrum of the nonlinear problem v+e ^v=0 are derived and certain norms for the solutions are estimated.

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Zusammenfassung Mit Hilfe einer Rellichschen Identität werden Schranken für das Spektrum des nichtlinearen Problems v+e ^v=0 angegeben. Ferner werden gewisse Normen für die Lösungen abgeschätzt.

     

Colouring a graph frugally

We prove that any graph with maximum degree sufficiently large, has a proper vertex colouring using +1 colours such that each colour class appears at most log⁸ times in the neighbourhood of any vertex. We also show that for 1, the minimum number of colours required to colour any such graph so that each vertex appears at most times in the neighbourhood of any vertex is (+^1+1//), showing in particular that when =log/loglog, such a colouring cannot always be achieved with O() colours. We also provide a polynomial time algorithm to find such a colouring. This has applications to the total chromatic number of a graph.The second two authors were supported by NATO Collaborative Research Grant #CRG950235.     

A Strong Maximum Principle for some quasilinear elliptic equations

In its simplest form the Strong Maximum Principle says that a nonnegative superharmonic continuous function in a domain ⁿ ,n 1, is in fact positive everywhere. Here we prove that the same conclusion is true for the weak solutions of – u + (u) = f with a nondecreasing function ,(0)=0, andf0 a.e. in if and only if the integral((s)s) ^–1/2 ds diverges ats=0+. We extend the result to more general equations, in particular to – _p u + (u) =f where _p (u) = div(|Du| ^p-2 Du), 1 <p < . Our main result characterizes the nonexistence of a dead core in some reaction-diffusion systems.This work was partly done while the author was visiting the University of Minnesota as a Fulbright Scholar.     

A note on smoothed estimating functions

The kernel estimate of regression function in likelihood based models has been studied in Staniswalis (1989,J. Amer. Statist. Assoc.,84, 276–283). The notion of optimal estimation for the nonparametric kernel estimation of semimartingale intensity (t) is proposed. The goal is to arrive at a nonparametric estimate of ₀=(t ₀) for a fixed pointt ₀ [0, 1]. We consider the estimator that is a solution of the smoothed optimal estimating equation is the optimal estimating function as in Thavaneswaran and Thompson (1986,J. Appl. Probab.,23, 409–417).     

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.     

Partial Results in Δ 3 0 -Categoricity in Linear Orderings and Boolean Algebras

The notion of ₃ ⁰ -categoricity in linear orderings and Boolean algebras is examined. We provide a proof for the fact that there are uncountably many relatively ₃ ⁰ -categorical linear orderings, and furnish a proof of another fact which suggests that the (unrelatively) ₃ ⁰ -categorical linear orderings may be very difficult to classify. In stark contrast to these results for linear orderings, a complete classification of the relatively ₃ ⁰ -categorical Boolean algebras is given.     

Perturbation bounds for theLDL H andLU decompositions

LetA andA+A be Hermitian positive definite matrices. Suppose thatA=LDL ^H and (A+A)=(L+L)(D+D)(L+L)^H are theLDL ^H decompositons ofA andA+A, respectively. In this paper upper bounds on |D| _F and |L| _F are presented. Moreover, perturbation bounds are given for theLU decomposition of a complexn ×n matrix.     

Lower bounds for faithful,preinjective modules

Let be the path algebra for some representation-infinite quiver over some field k. There exists a bound such that ^mI is faithful for all indecomposable injective -modules I and all , and such that there exists an indecomposable injective -module J satisfying that J is not faithful, denotes the Auslander-Reiten-translation. Let m() be the maximum of the taken over all possible orientations of the underlying graph . In this article we determine the bounds m() for representation-infinite quivers for which is a tree.     

The local nature of Δ-coloring and its algorithmic applications

Given a connected graphG=(V, E) with |V|=n and maximum degree such thatG is neither a complete graph nor an odd cycle, Brooks' theorem states thatG can be colored with colors. We generalize this as follows: letG-v be -colored; then,v can be colored by considering the vertices in anO(log n) radius aroundv and by recoloring anO(log n) length augmenting path inside it. Using this, we show that -coloringG is reducible inO(log³ n/log) time to (+1)-vertex coloringG in a distributed model of computation. This leads to fast distributed algorithms and a linear-processorNC algorithm for -coloring.A preliminary version of this paper appeared as part of the paper Improved Distributed Algorithms for Coloring and Network Decomposition Problems, in theProceedings of the ACM Symposium on Theory of Computing pages 581–592, 1992. This research was done when the authors were at the Computer Science Department of Cornell University. The research was supported in part by NSF PYI award CCR-89-96272 with matching funds from UPS and Sun Microsystems.     

Conditions for eigenvectors of a selfadjoint operator-function to form a basis

Consider a functionL() defined on an interval of the real axis whose values are linear bounded selfadjoint operators in a Hilbert spaceH. A point ₀ and a vector ₀ H( ₀ 0) are called eigenvalue and eigenvector ofL() ifL() ifL(₀) ₀ = 0. Supposing that the functionL() can be represented as an absolutely convergent Fourier integral, the interval is sufficiently small and the derivative will be positive at some point, it has been proved that all the eigenvectors of the operator-functionL() corresponding to the eigenvalues from the interval form an unconditional basis in the subspace spanned by them.