Free energy asymptotics of the quantum Heisenberg spin chain

We consider the ferromagnetic quantum Heisenberg model in one dimension, for any spin $$S\ge 1/2$$ S ≥ 1 / 2 . We give upper and lower bounds on the free energy, proving that at low temperature it is asymptotically equal to the one of an ideal Bose gas of magnons, as predicted by the spin-wave approximation. The trial state used in the upper bound yields an analogous estimate also in the case of two spatial dimensions, which is believed to be sharp at low temperature.

An Optimal Bound for Nonlinear Eigenvalues and Torsional Rigidity on Domains with Holes

In this paper we prove an optimal upper bound for the first eigenvalue of a Robin-Neumann boundary value problem for the p-Laplacian operator in domains with convex holes. An analogous estimate is obtained for the corresponding torsional rigidity problem.

SMOOTH APPROXIMATION FOR THE SOLUTIONS OF ELLIPTIC SYSTEMS

The paper is devoted to the problem of piecewise-linear approximation of mapping for the elliptic system of solutions defined on triangular grids. A mapping is build to approximate the differential, and the estimate of error of approximation that does not depend on the degree of degeneracy of triangles of the grid. An analogous estimate is obtained for mappings which approximate the differential of solution of Beltrami equation.

ON POWER MOMENTS OF THE HECKE MULTIPLICATIVE FUNCTIONS

In a recent paper, Soundararajan has proved the quantum unique ergodicity conjecture by getting a suitable estimate for the second order moment of the so-called ‘Hecke multiplicative’ functions. In the process of proving this he has developed many beautiful ideas. In this paper we generalize his arguments to a general$k$th power and provide an analogous estimate for the$k$th power moment of the Hecke multiplicative functions. This may be of general interest.

Number of Benzenoid Hydrocarbons

The number of benzenoid hydrocarbons with h hexagons can be estimated by means of the formula Bh = 0.045-3/2 (5 .4 )h . The analogous estimate for the number of catacondensed benzenoids is Ch = 0.049 h-5/4 (4.27)h.