A Note on a Boundary Value Problem

In this note, we prove that, for Robins boundary value problem, a unique solution exists if f_x(t, x, x), f_x(t, x, x), (t), and (t) are continuous, and f_x -(t), f_x -(t), 4(t) ² + ²(t) ++ 2(t), and 4(t) ² + ²(t) + 2(t).AMS Subject Classification (2000) 34B15     

Survival of Threshold Contact Processes

We consider the d-dimensional threshold contact process. Suppose that a vacant site becomes occupied at rate one when there are at least occupied sites in its neighborhood, and the death rate at any site is >0. We will explicitly give two integers ab with the following properties: For a the process survives starting from finite configurations when is small, but for >a the process dies out starting from any finite configuration with any positive death rate. For b the process has a nontrivial invariant measure when is small, but for >b the only invariant measure is the all-zero configuration for any positive death rate.     

Derivatives of the Spectral Function and Sobolev Norms of Eigenfunctions on a Closed Riemannian Manifold

Let e(x, y, ) be the spectral function and the unit spectral projection operator, with respect to the Laplace–Beltrami operator on a closed Riemannian manifold M. We generalize the one-term asymptotic expansion of e(x, x, ) by Hörmander (Acta Math. 88 (1968), 341–370) to that of _{x y} e(x,y,)| _x=y for any multiindices , in a sufficiently small geodesic normal coordinate chart of M. Moreover, we extend the sharp (L ²,L ^p) (2 p) estimates of by Sogge (J. Funct. Anal. 77 (1988), 123–134; London Math. Soc. Lecture Note Ser. 137, Cambridge University Press, Cambridge, 1989; Vol. 1, pp. 416–422) to the sharp (L ², Sobolev L ^p) estimates of .     

Commutation Relations in an Indefinite-Metric Space

We describe the irreducible regular representations of the algebra of operators a and b defined by [a,b]=1 and ba=a ⁺ b ⁺ in an arbitrary nondegenerate closed indefinite-metric space. We find the relation of this algebra to the generalized Heisenberg algebra.     

Approximation of one class of transcendental numbers

The author studies the approximation of fixed-degree transcendental numbers e a by algebraic numbers where 0, a 0; and 1 and irrational are algebraic numbers.Translated from Matematicheskie Zametki, Vol. 5, No. 1, pp. 117–128, January, 1969.     

Conditions for non-oscillation of singular linear differential equations of second order

Conditions are found in the fulfillment of which each non-trivial solution of the equation u+ (t)u+(t)u=0, where(t) L(a, b) and (t–a)(t–b)(t) L(a, b) has not more than one zero on the interval atb.Translated from Matematicheskie Zametki, Vol. 6, No. 5, pp. 633–639, November, 1969.     

Local Stochastic Channeling Theory: Kinetic Functions in the Case of Interaction Between Fast Particles and Lattice Atoms

We investigate the motion of high-energy particles in a crystal with regard to their interaction with the thermal vibrations of the lattice atoms using analytic methods in the theory of Markov processes including the local Fokker–Planck equation. We construct a local matrix of random actions, which is used to introduce the main kinetic functions in the traverse-energy space, namely, the function a() of energy losses due to the dynamic friction and the diffusion function b(). We show that the singularities of the functions a() and b() are related to the distinction between the contributions to the kinetics from particles moving in three different regimes, namely, in the channeling, quasichanneling, and chaotic motion modes.     

About generalized ideals in a distributive lattice

Let L be a distributive lattice characterized by a ternary operation (, ,), where (a,b,c)=(ab)(bc)(ac)=(ab)(ac)(bc), a,b,cL. The note considers convex sublattices of L, called generalized ideals of L generated by the operation (, ,). Some remarks have been stated about the graph of a distributive lattice.     

Hausdorff Dimension of Spectrum of One-Dimensional Schrödinger Operator with Sturmian Potentials

Let (0,1) be an irrational, and [a ₁,a ₂,...] be the continued fraction expansion of . Let H be the one-dimensional Schrödinger operator with Sturmian potentials. We show that if the potential strength V>20, then the Hausdorff dimension of the spectrum (H ) is strictly great than zero for any irrational , and is strictly less than 1 if and only if liminf _k(a ₁ a ₂a _k ))^1/k <.     

Weak distributive laws and their role in lattices of congruences and equational theories

By a result of Pigozzi and Kogalovskii, every algebraic latticeL having a completely join —irreducible top element can be represented as the lattice L() of equational theories extending some fixed theory . Conversely, strengthening a recent result due to Lampe, we show that such a representationL=L() forcesL to satisfy the following condition: if the top element ofL is the join of a nonempty subsetB ofL then there are elementsb..., B such thata=(... (((b₁ a) b₂) a) ... b_n) a for alla L. In presence of modularity, this equation reduces to the identitya=(a b₁) ... (a b_n). Motivated by these facts, we study several weak forms of distributive laws in arbitrary lattices and related types of prime elements. The main tool for applications to universal algebra is a generalized version of Lampe's Zipper Lemma.Presented by Ralph Freese.     

On complexity of subset interconnection designs

Given a setX and subsetsX ₁,...,X _m, we consider the problem of finding a graphG with vertex setX and the minimum number of edges such that fori=1,...,m, the subgraphG _i; induced byX _i is connected. Suppose that for any pointsx ₁,...,x X, there are at mostX _i 's containing the set {x₁,...,x }. In the paper, we show that the problem is polynomial-time solvable for ( 2, 2) and is NP-hard for (3,=1), (=l,6), and (2,3).Support in part by the NSF under grant CCR-9208913 and CCR-8920505.Part work was done while this author was visiting at DIMACS and on leave from Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing.     

Cut-elimination and interpolation for Ω-logic

In 1978, Girard introduced-logic to generalize-logic. The basic category of-logic is the categoryON of ordinals. For geometric structure reasons, Girard changed the basic categoryON into the more general categoryWF of well-founded orders (1983). The logic he obtained was called-logic. Here, we extend (unpublished) results of-logic to-logic.     

On a Problem of D. H. Lehmer and Kloosterman Sums

Let q3 be an odd number, a be any fixed positive integer with (a, q)=1. For each integer b with 1b<q and (b, q)=1, it is clear that there exists one and only one c with 0<c<q such that bca (mod q). Let N(a, q) denote the number of all solutions of the congruent equation bca (mod q) for 1b, c<q in which b and c are of opposite parity, and let . The main purpose of this paper is to study the distribution properties of E(a, q), and to give a sharper hybrid mean value formula involving E(a, q) and Kloosterman sums.Received January 24, 2002; in revised form August 12, 2002 Published online February 28, 2003     

Über die asymptotische Darstellung der Aufspaltung von Paaren benachbarter Eigenwerte der Differentialgleichung der Sphäroidfunktionen

Summary Let be the difference of a pair of adjacent eigen-values of the differential equation (1) for the spheroidal functions. Until now only the two first terms of the asymptotic expansion (2) of for large ^*2=–² had been known. In this note the next term, i. e. the coefficient of ^*–2, is given, and the way of calculating it is described.

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The preparation of this note containing partial results, was sponsored by the European Office Air Research and Development Command, U. S. Air Force, Project No. AF 61 (514)-443.     

Concerning the solution set ofAx=b whereP≦A≦Q andp≦b≦q

Summary LetPQ ben×n real matrices so that ifPAQ for some matrixA, thenA is nonsingular. Letp andq ben-dimensional real column vectors. This paper determines the set of all solutionsx to the equationAx=b for allA andb so thatPAQ andpbq.     

Improving the rate of convergence of ‘high order finite elements’ on polygons and domains with cusps

Summary. Let u and u_V V be the solution and, respectively, the discrete solution of the non-homogeneous Dirichlet problem u=f on , u|=0. For any m and any bounded polygonal domain , we provide a construction of a new sequence of finite dimensional subspaces V_n such that where f H^m–1() is arbitrary and C is a constant that depends only on and not on n (we do not assume u H^m+1()). The existence of such a sequence of subspaces was first proved in a ground–breaking paper by Babuka [8]. Our method is different; it is based on the homogeneity properties of Sobolev spaces with weights and the well–posedness of non-homogeneous Dirichlet problem in suitable Sobolev spaces with weights, for which we provide a new proof, and which is a substitute of the usual shift theorems for boundary value problems in domains with smooth boundary. Our results extended right away to domains whose boundaries have conical points. We also indicate some of the changes necessary to deal with domains with cusps. Our numerical computation are in agreement with our theoretical results.The authors were supported in part by the NSF grant DMS 02-09497. Victor Nistor was also partially supported by NSF grant DMS 02-00808.     

Homogenization and concentrated capacity for the heat equation with non-linear variational data in reticular almost disconnected structures and applications to visual transduction

Out of a right, circular cylinder of height H and cross-section a disc of radius R+ one removes a stack of nH/ parallel, equi-spaced cylinders C_j,j=1,2,...,n, each of radius R and height . Here , are fixed positive numbers and is a positive parameter to be allowed to go to zero. The union of the C_j almost fills in the sense that any two contiguous cylinders C_j are at a mutual distance of the order of and that the outer shell, i.e., the gap S=-_o has thickness of the order of (_o is obtained from by formally setting =0). The cylinder from which the C_j are removed, is an almost disconnected structure, it is denoted by , and it arises in the mathematical theory of phototransduction.For each >0 we consider the heat equation in the almost disconnected structure , for the unknown function u, with variational boundary data on the faces of the removed cylinders C_j. The limit of this family of problems as 0 is computed by concentrating heat capacity and diffusivity on the outer shell, and by homogenizing the u within the limiting cylinder _o.It is shown that the limiting problem consists of an interior diffusion in _o and a boundary diffusion on the lateral boundary S of _o. The interior diffusion is governed by the 2-dimensional heat equation in _o, for an interior limiting function u. The boundary diffusion is governed by the Laplace–Beltrami heat equation on S, for a boundary limiting function u_S. Moreover the exterior flux of the interior limit u provides the source term for the boundary diffusion on S. Finally the interior limit u, computed on S in the sense of the traces, coincides with the boundary limit u_S. As a consequence of the geometry of , local arguments do not suffice to prove convergence in _o, and also we have to take into account the behavior of the solution in S. A key, novel idea consists in extending equi-bounded and equi-Hölder continuous functions in -dependent domains, into equi-bounded and equi-Hölder continuous functions in the whole ^N, by means of the Kirzbraun–Pucci extension technique.The biological origin of this problem is traced, and its application to signal transduction in the retina rod cells of vertebrates is discussed. Mathematics Subject Classification (2000) 35B27, 35K50, 92C37     

Estimation of Local Smoothness Coefficients for Continuous Time Processes

We consider processes that satisfied a local Hölder condition with coefficient ₀. According to the sampling times of observations given by i _n with i=0,...,n–1, we study two general classes of estimators for ₀. Their almost sure rates of convergence depend on asymptotic independence of the observed processes, on _n and eventually on an extra parameter ₀. Since this last parameter is in general unknown, we construct a family of preliminary estimators for ₀ with their rates of almost sure convergence. Finally we present some numerical simulations in order to compare the behaviour of our various estimators.     

Mittelwert der Eulerschen φ-Funktion und des Quadrates der Dirichletschen Teilerfunktion in algebraischen Zahlkörpern

LetK be an algebraic number field, and for every integer K let () andd(), respectively, denote the number of relatively prime residue classes and the number of divisors of the principal ideal (). Asymptotic equalities are proved for the sums () and d ²(), where runs through certain finite sets of integers ofK.     

Über den algorithmischen Nachweis von Ungleichungen I

We shall develop a method to prove inequalities in a unified manner. The idea is as follows: It is quite often possible to find a continuous functional : ⁿ , such that the left- and the right-hand side of a given inequality can be written in the form (u)(v) for suitable points,v=v(u). If one now constructs a map ^{n n} , which is functional increasing (i.e. for each x ⁿ (which is not a fixed point of ) the inequality (x)<((x)) should hold) one specially gets the chain (u)( u))( ²(u))... ⁿ (u)). Under quite general conditions one finds that the sequence { ⁿ (u)} _n converges tov=v(u). As a consequence one obtains the inequality (u)(v).