Weyl-type hybrid subconvexity bounds for twisted LL-functions and Heegner points on shrinking sets

We prove a Weyl-type subconvexity bound for the central value of the $L$-function of a Hecke-Maass form or a holomorphic Hecke eigenform twisted by a quadratic Dirichlet character, uniform in the archimedean parameter as well as the twisting parameter. A similar hybrid bound holds for quadratic Dirichlet $L$-functions, improving on a result of Heath-Brown. As a consequence of these new bounds, we obtain explicit estimates for the number of Heegner points of large odd discriminant in shrinking sets.Comment: 27 pages. v2: Numerous improvements to the exposition. v3: refereed version with further improvement

Underscreened Kondo Necklace

It has been suggested recently by Gan, Coleman, and Andrei that studying the underscreened Kondo problem may help to understand the nature of magnetism in heavy fermion systems. Motivated by Doniach's work on the S=1/2 Kondo necklace, we introduce the underscreened Kondo necklace models with S>1/2. The underscreened Kondo necklace is the simplest lattice model on which the competition between Kondo spin compensation, and magnetic ordering due to an RKKY-type interaction can be examined. We used the mean-field approximation to determine the phase diagram, and found that the low-temperature phase is always an x-y antiferromagnet. This contention is further supported by the derivation of the exact form of the effective hamiltonian in the limit of very large Kondo coupling: it is found to be an antiferromagnetic x-y model for the residual (S-1/2)-spins. In general, the degree of moment compensation depends on both the Kondo coupling, and on S.Comment: 15 pages (2 figures upon request from [email protected]), LATEX, to appear in Modern Physics Letters

Convergence of Monte Carlo Simulations to Equilibrium

We give two direct, elementary proofs that a Monte Carlo simulation converges to equilibrium provided that appropriate conditions are satisfied. The first proof requires detailed balance while the second is quite general.Comment: 4 pages. v2: published versio

Complexity of several constraint satisfaction problems using the heuristic, classical, algorithm, WalkSAT

We determine the complexity of several constraint satisfaction problems using the heuristic algorithm, WalkSAT. At large sizes N, the complexity increases exponentially with N in all cases. Perhaps surprisingly, out of all the models studied, the hardest for WalkSAT is the one for which there is a polynomial time algorithm.Comment: 5 pages, 4 figure