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A Relation Between Pointwise Convergence of Functions and Convergence of Functionals
We show that if f n is a sequence of uniformly L p-bounded functions on a measure space, and if f n → f pointwise a.e., then lim for all 0 < p < ∞. This result is also generalized in Theorem 2 toExpand
Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities
A maximizing function, f, is shown to exist for the HLS inequality on R': 11 IXI - * fIq < Np f A , Iif IIwith Nbeing the sharp constant and i/p + X/n = 1 + 1/q, 1 <p, q, n/X < x. When p =q' or p = 2Expand
Two Soluble Models of an Antiferromagnetic Chain
Two genuinely quantum mechanical models for an antiferromagnetic linear chain with nearest neighbor interactions are constructed and solved exactly, in the sense that the ground state, all theExpand
Existence and Uniqueness of the Minimizing Solution of Choquard's Nonlinear Equation
The equation dealt with in this paper is in three dimensions. It comes from minimizing the functional which, in turn, comes from an approximation to the Hartree-Fock theory of a plasma. It describesExpand
On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation
We extend the Prekopa-Leindler theorem to other types of convex combinations of two positive functions and we strengthen the Prekopa—Leindler and Brunn-Minkowski theorems by introducing the notion ofExpand
Exact Analysis of an Interacting Bose Gas. II. The Excitation Spectrum
We continue the analysis of the one-dimensional gas of Bose particles interacting via a repulsive delta function potential by considering the excitation spectrum. Among other things we show that: (i)Expand
Harmonic maps with defects
Two problems concerning maps ϕ with point singularities from a domain Ω C ℝ3 toS2 are solved. The first is to determine the minimum energy of ϕ when the location and topological degree of theExpand
EXACT ANALYSIS OF AN INTERACTING BOSE GAS. I. THE GENERAL SOLUTION AND THE GROUND STATE
A gas of one-dimensional Bose particles interacting via a repulsive delta-function potential has been solved exactly. All the eigenfunctions can be found explicitly and the energies are given by theExpand
Theory of monomer-dimer systems
We investigate the general monomer-dimer partition function,P(x), which is a polynomial in the monomer activity,x, with coefficients depending on the dimer activities. Our main result is thatP(x) hasExpand
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