Oscillatory Perturbations of Linear Problems at Resonance

This paper is concerned with a study of bounded perturbations of resonant linear problems. It follows from our results that for certain types of bounded domains Ω ? Rⁿ, n ≥ 2, the Dirichlet problem $\matrix{\Delta u+\lambda_{1}u+g(u)=h(x),\ \ \ x\in\Omega\cr \quad\quad\quad\quad\quad\quad u=0,\ \ \ x\in\partial\Omega,}$ has infinitely many positive solutions, in case λ₁ is the principal eigenvalue of ?Δ subject to trivial Dirichlet boundary conditions, g is a nontrivial periodic nonlinearity of zero mean and ∫_03A9h(x)?(x)dx = 0, where ? is an eigenfunction corresponding to λ₁.     

Automata accepting primitive words

LeA be an automaton whose set of inputs equalsX (|X|≧2) and whose cardinality of the set of states equalsn (n≧2), and letQ be the set of all primitive words overX. ByT(A) we denote the language accepted byA. In this paper, we give the following results:

On the contact dimensions of graphs

Every simple graphG=(V, E) can be represented by a family of equal nonoverlapping spheres {S _v :v V} in a Euclidean spaceR ⁿ in such a way thatuv E if and only ifS _u andS _v touch each other. The smallest dimensionn needed to representG in such a way is called the contact dimension ofG and it is denoted by cd(G). We prove that (1) cd(T)<(7.3) log |T| for every treeT, and (2)

A proof of Freud's conjecture for exponential weights

LetW (x) be a function nonnegative inR, positive on a set of positive measure, and such that all power moments ofW ²(x) are finite. Let {p _n (W ²;x)} ₀ denote the sequence of orthonormal polynomials with respect to the weightW ²(x), and let {A _n } ₁ and {B _n } ₁ denote the coefficients in the recurrence relation

On the width of an orientation of a tree

There are 2 ^n-1 ways in which a tree on n vertices can be oriented. Each of these can be regarded as the (Hasse) diagram of a partially ordered set. The maximal and minimal widths of these posets are determined. The maximal width depends on the bipartition of the tree as a bipartite graph and it can be determined in time O(n). The minimal width is one of [/2] or [/2]+1, where is the number of leaves of the tree. An algorithm of execution time O(n + ² log ) to construct the minimal width orientation is given.This research was partially funded by the National Science and Engineering Research Council of Canada under Grant Number A4219.     

A short proof that N 3 is not a circle containment order

A partially ordered set P is called a circle containment order provided one can assign to each xP a circle C _x so that . We show that the infinite three-dimensional poset N ³ is not a circle containment order and note that it is still unknown whether or not [n]³ is such an order for arbitrarily large n.     

Gelfand theory of certainC *-algebras on nonasymptotically flat manifolds

Let be a Riemannian manifold with finitely many conical ends. Under certain conditions which do not require to be asymptotically flat, we study aC ^*-algebra containing pseudodifferential operators on . Results on compact commutators and the Gelfand space are presented. Criteria for differential operators or systems within reach to be Fredholm are just simple consequences of Atkinson's theorem and our results.     

Valence bond ground states in isotropic quantum antiferromagnets

Haldane predicted that the isotropic quantum Heisenberg spin chain is in a massive phase if the spin is integral. The first rigorous example of an isotropic model in such a phase is presented. The Hamiltonian has an exactSO(3) symmetry and is translationally invariant, but we prove the model has a unique ground state, a gap in the spectrum of the Hamiltonian immediately above the ground state and exponential decay of the correlation functions in the ground state. Models in two and higher dimension which are expected to have the same properties are also presented. For these models we construct an exact ground state, and for some of them we prove that the two-point function decays exponentially in this ground state. In all these models exact ground states are constructed by using valence bonds.Supported in part by N.S.F. Grant PHY-80-19754. Fellow of the A.P. Sloan Foundation and the Canadian Institute for Advanced ResearchN.S.F. Post-doctoral FellowSupported in part by N.S.F. Grant PHY-85-15288-A01     

The stability and instability of relativistic matter

We consider the quantum mechanical many-body problem of electrons and fixed nuclei interacting via Coulomb forces, but with a relativistic form for the kinetic energy, namelyp ²/2m is replaced by (p ² c ²+m ² c ⁴)^1/2–mc ². The electrons are allowed to haveq spin states (q=2 in nature). For one electron and one nucleus instability occurs ifz>2/, wherez is the nuclear charge and is the fine structure constant. We prove that stability occurs in the many-body case ifz2/ and <1/(47q). For smallz, a better bound on is also given. In the other direction we show that there is a critical _c (no greater than 128/15) such that if > _c then instability always occurs forall positivez (not necessarily integral) when the number of nuclei is large enough. Several other results of a technical nature are also given such as localization estimates and bounds for the relativistic kinetic energy.Work partially supported by U.S. National Science Foundation grant PHY-85-15288-A02The author thanks the Institute for Advanced Study for its hospitality and the U.S. National Science Foundation for support under grant DMS-8601978     

Approximate neutrality of large-Z ions

LetN(Z) denote the number of electrons which a nucleus of chargeZ can bind in non-relativistic quantum mechanics (assuming that electrons are fermions). We prove thatN(Z)/Z1 asZ.Research partially supported by the NSERC under Grant NA7901 and by the USNSF under Grants DMS-8416049 and PHY 85-15288-A01