Newton's Polyhedron and Weyl's Formula for the Spectrum of the Schrödinger Operator with Polynomial Potential

Introduce the notation: V is a nonnegative polynomial, Q(V) is the corresponding Newton polyhedron, L= -+V is the Schrodinger operator on , N(,L) is the number of eigenvalues of L that are less than . The upper estimate for N(,L) is derived in terms of Q(V), and the Weyl asymptotic formula is established under certain assumptions on the geometry of Q(V). The case where the behavior of V(x) is not regular as is also considered. Bibliography: 6 titles.     

On the length of validity of averaging and the uniform approximation of elbow orbits,with an application to delayed passage through resonance

Consider a system of differential equations of the form , wheref is 2-periodic int and the averaged system is Morse-Smale. We show by geometrical resoning that in general no asymptotic approximation to the initial value problem is valid for larger thant=O(1/). This comes about because the stable manifolds vary with. But in the neighborhood of the stable and unstable manifolds of a hyperbolic periodic solution, every orbit (a so-called elbow orbit) has an approximation which is valid for as long as it remains in the neighborhood, although it does not satisfy the same initial conditions. This result is applied to passage through resonance.     

Product-integration with the Clenshaw-Curtis and related points

Summary This paper is concerned with the theoretical properties of a productintegration method for the integral , wherek is absolutely integrable andf is continuous. The integral is approximated by , where the points are given byx _ni =cos(i/n, 0in, and where the weightsw _ni are chosen to make the rule exact iff is any polynomial of degree n. The principal result is that ifkL _p [–1, 1] for somep>1, then the rule converges to the exact result asn for all continuous (or indeed R-integrable) functionsf, and moreover that the sum of the absolute values of the weights converges to the least possible value, namely . A limiting expression for the individual weights is also obtained, under certain assumptions. The results are exteded to other point sets of a similar kind, including the classical Chebyshev points.     

Generalized Quasi-Variational Inequalities in Infinite-Dimensional Normed Spaces

In this paper, we deal with the following problem: given a real normed space E with topological dual E*, a closed convex set XE, two multifunctions :X2^X and , find such that We extend to the above problem a result established by Ricceri for the case (x)X, where in particular the multifunction is required only to satisfy the following very general assumption: each set (x) is nonempty, convex, and weakly-star compact, and for each yX–:X the set is compactly closed. Our result also gives a partial affirmative answer to a conjecture raised by Ricceri himself.     

Extremal problems of Bloch function spaces

Letf be analytic in a hyperbolic region . The Bloch constant _f off is defined by , where (z)|dz| is the Poincaré metric in . Suppose is hyperbolic and where . Then for allf withf() , we have _f 1/(). In this paper we study the extremal functions defined by _f =1/() and the existence of those functions.Supported by the National Natural Science Foundation of China.     

On a theorem of Seidel and Walsh

Given a sequence ( _n ) _n in with there are functions such that , is a dense subset of , and the set of functions with this property is residual in . We will show that in and some related Banach spaceX there are functionsf with is dense in , and we will give a sufficient condition when the set of such functions is residual inX.     

Some extremal results on circles containing points

We define (n) to be the largest number such that for every setP ofn points in the plane, there exist two pointsx, y P, where every circle containingx andy contains (n) points ofP. We establish lower and upper bounds for (n) and show that [n/27]+2(n)[n/4]+1. We define for the special case where then points are restricted to be the vertices of a convex polygon. We show that .     

A stability theorem for a class of linear evolution systems

Dynamical systems of the form ,u(0)=u ₀, where,B is a densely defined linear operator mapping its domainD (B ) into — the infinitesimal generator of a semigroup of operatorsT (t, B) of classC ₀ — are investigated, such that for each solutionu to , whereP is the spectral eigenprojection onto the null space ofB.It is shown that under some general hypotheses concerning spectral properties ofB the above stability condition is equivalent with the following situation: There exist (i) a normal generating coneK such thatT(t;B)KK fort0 and (ii) a strictly positive element in the dual coneK such that , whereB denotes the dual ofB. Condition (ii) implies the so called total concentration time preservation, i. e. .     

Interior points of the convex hull of few points in\mathbb{E}^d

We show that ifP , |P|=d+k,dk1 andO int convP, then there exists a simplexS of dimension with vertices inP, satisfyingO rel intS, the bound being sharp. We give an upper bound for the minimal number of vertices of facets of a (j-1)-neighbourly convex polytope in withv vertices.Research (partially) supported by Hung. Nat. Found. for Sci. Research, grant no. 1817Research (partially) supported by Hung. Nat. Found. for Sci. Research, grant no. 326-0213     

Tensor ideals in the category of tilting modules

We study the tensor category of tilting modules over a quantum groupU _q with divided powers. The setX ₊ of dominant weights is a union of closed alcoves numbered by the elementswW ^f of a certain subset of affine Weyl groupW. G. Lusztig and N. Xi defined a partition ofW ^f into canonical right cells and the right order _R on the set of cells. For a cellAW ^f we consider a full subcategory formed by direct sums of tilting modulesQ() with highest weights . We prove that is a tensor ideal in , generalizing H. Andersen's theorem about the ideal of negligible modules which in our notations is nothing else then . The proof is an application of a recent result by W. Soergel who has computed the characters of tilting modules.This material is based upon work supported by the U.S. Civilian Research and Development Foundation under Award No. RM1-265.     

Absolute values of the coefficients of the polynomials in Weierstrass's approximation theorem

The following problem, bound up with Weierstrass's classical approximation theorem, is solved definitively: to determine the sequence of positive numbersM _k such that, for anyf(z)c[0,1] and > 0 there exists the polynomial that f–P< and _k <M _k ,k=1, ...,n.Translated from Matematicheskii Zametki, Vol. 22, No. 2, pp. 269–276, August, 1977.     

Linear extension majority cycles on partial orders

Let P define a partial order on a set X of cardinalityn. A linear extensionL ofP is a linear order withP G L, and is the set of all linear extensions ofP. denotes that subset of withxLy forx, y X. A linear extension majority (LEM) relationM onX is defined byxMy if . Similarly,M is defined byxMy if . An LEM cycle exists if there arex, y, z X withxMyMzMx, and an LEM quasi-cycle exists ifxMyMzMx and the equality part of the definition ofM holds for exactly one pair in the triple. The study shows that no semiorders have LEM cycles or LEM quasi-cycles, and that every interval order has a maximal element under theM relation. LEM cycles and LEM quasi-cycles are also considered for partial orders with specific structures. Simulation is used to determine the relative likelihood with which LEM cycles and LEM quasi-cycles are observed when connected partial orders are generated at random by a specific procedure.Dr. Gehrlein's research was supported through a fellowship from the Center for Advanced Study of the University of Delaware.     

Product integration with the Clenshaw-Curtis points: Implementation and error estimates

Summary This paper is concerned with the practical implementation of a product-integration rule for approximating , wherek is integrable andf is continuous. The approximation is , where the weightsw _ni are such as to make the rule exact iff is any polynomial of degree n. A variety of numerical examples, fork(x) identically equal to 1 or of the form |–x| with >–1 and ||1, or of the form cosx or sinx, show that satisfactory rates of convergence are obtained for smooth functionsf, even ifk is very singular or highly oscillatory. Two error estimates are developed, and found to be generally safe yet quite accurate. In the special casek(x)1, for which the rule reduces to the Clenshaw-Curtis rule, the error estimates are found to compare very favourably with previous error estimates for the Clenshaw-Curtis rule.     

Saturation of Kantorovich type operators

It is proved that an integrable functionf can be approximated by the Kantorovich type modification of the Szász—Mirakjan and Baskakov operators inL ¹ metric in the optimal order {n ^–1} if and only if ² f is of bounded variation where and , respectively.     

On periodic groups of automorphisms of extremal groups

It is proved that if a periodic group has an extremal normal divisor , determining a complete abelian factor group , then the center of the group contains a complete abelian subgroup , satisfying the relation and intersecting on a finite subgroup. It is also established with the aid of this proposition that every periodic group of automorphisms of an extremal group is a finite extension of a contained in it subgroup of inner automorphisms of the group .Translated from Matematicheskie Zametki, Vol. 4, No. 1, pp. 91–96, July, 1968.     

Un théorème de Vitali-Hahn-Saks pour les semimartingales

Summary Let (, , P) be a complete probability space; let _t0 be an increasing right-continuous family of -complete sub--fields of ; let be a sequence of semimartingales. Assume that for all positive t and for all bounded predictable processes H, the r.v.'s converge in probability to a limit J(t, H) when n tends to infinity. Then there exists a semimartingale X such that, for all t and H, J(t, H)= .     

Class Numbers of Indefinite Binary Quadratic Forms

Let be the class number of properly equivalent primitive binary quadratic forms of discriminant . The case of indefinite forms is considered. Assuming that the extended Riemann hypothesis for some fields of algebraic numbers holds, the following results are proved. 1. Let be an arbitrarily slow monotonically increasing function such that . Then

Conditions of Local Asymptotic Normality for Gaussian Stationary Processes

Let {\bold x}[] be a stationary Gaussian process with zero mean and spectral density f, let be the -algebra induced by the random variables {\bold x}[], D(R¹), and let _t, t > 0, be the -algebra induced by the random variables x[],supp [-t,t]. Denote by (f) the Gaussian measure on generated by {\bold x}. Let _t(f) be the restriction of (f) to _t. Let f and g be nonnegative functions such that the measures _t(f) and _t(g) are absolutely continuous. Put

Polynomial invariants of graphs II

Negami and Kawagoe has already defined a polynomial associated with each graphG as what discriminates graphs more finely than the polynomialf(G) defined by Negami and the Tutte polynomial. In this paper, we shall show that the polynomial includes potentially the generating function counting the independent sets and the degree sequence of a graphG, which cannot be recognized fromf(G) in general, and discuss on of treesT with observations by computer experiments.     

Finitely generated relatively universal varieties of Heyting algebras

Given distinct varieties and of the same type, we say that is relatively -universal if there exists an embedding :K from a universal categoryK such that for every pairA, B ofK-objects, a homomorphismf:A B has the formf=_g for someK-morphismg:A B if and only if Im(f) . Finitely generated relatively -universal varieties of Heyting algebras are described for the variety of Boolean algebras, the variety generated by a three element chain, and for the variety generated by the four element Boolean algebra with an added greatest element.Dedicated to the memory of Alan DayPresented by J. Sichler.The support of the NSERC is gratefully acknowledged.