Unfreezing Casimir invariants: singular perturbations giving rise to forbidden instabilities

The infinite-dimensional mechanics of fluids and plasmas can be formulated as "noncanonical" Hamiltonian systems on a phase space of Eulerian variables. Singularities of the Poisson bracket operator produce singular Casimir elements that foliate the phase space, imposing topological constraints on the dynamics. Here we proffer a physical interpretation of Casimir elements as \emph{adiabatic invariants} ---upon coarse graining microscopic angle variables, we obtain a macroscopic hierarchy on which the separated action variables become adiabatic invariants. On reflection, a Casimir element may be \emph{unfrozen} by recovering a corresponding angle variable; such an increase in the number of degrees of freedom is, then, formulated as a \emph{singular perturbation}. As an example, we propose a canonization of the resonant-singularity of the Poisson bracket operator of the linearized magnetohydrodynamics equations, by which the ideal obstacle (resonant Casimir element) constraining the dynamics is unfrozen, giving rise to a tearing-mode instability

General Rule and Materials Design of Negative Effective U System for High-T_c Superconductivity

Based on the microscopic mechanisms of (1) charge-excitation-induced negative effective U in s^1 or d^9 electronic configurations, and (2) exchange-correlation-induced negative effective U in d^4 or d^6 electronic configurations, we propose a general rule and materials design of negative effective U system in itinerant (ionic and metallic) system for the realization of high-T_c superconductors. We design a T_c-enhancing layer (or clusters) of charge-excitation-induced negative effective $U$ connecting the superconducting layers for the realistic systems.Comment: 11 pages, 1 figures, 2 tables, APEX in printin

Lower Bounds on Query Complexity for Testing Bounded-Degree CSPs

In this paper, we consider lower bounds on the query complexity for testing CSPs in the bounded-degree model. First, for any ``symmetric'' predicate $P:{0,1}^{k} \to {0,1}$ except \equ where $k\geq 3$, we show that every (randomized) algorithm that distinguishes satisfiable instances of CSP(P) from instances $(|P^{-1}(0)|/2^k-\epsilon)$-far from satisfiability requires $\Omega(n^{1/2+\delta})$ queries where $n$ is the number of variables and $\delta>0$ is a constant that depends on $P$ and $\epsilon$. This breaks a natural lower bound $\Omega(n^{1/2})$, which is obtained by the birthday paradox. We also show that every one-sided error tester requires $\Omega(n)$ queries for such $P$. These results are hereditary in the sense that the same results hold for any predicate $Q$ such that $P^{-1}(1) \subseteq Q^{-1}(1)$. For EQU, we give a one-sided error tester whose query complexity is $\tilde{O}(n^{1/2})$. Also, for 2-XOR (or, equivalently E2LIN2), we show an $\Omega(n^{1/2+\delta})$ lower bound for distinguishing instances between $\epsilon$-close to and $(1/2-\epsilon)$-far from satisfiability. Next, for the general k-CSP over the binary domain, we show that every algorithm that distinguishes satisfiable instances from instances $(1-2k/2^k-\epsilon)$-far from satisfiability requires $\Omega(n)$ queries. The matching NP-hardness is not known, even assuming the Unique Games Conjecture or the $d$-to-$1$ Conjecture. As a corollary, for Maximum Independent Set on graphs with $n$ vertices and a degree bound $d$, we show that every approximation algorithm within a factor d/\poly\log d and an additive error of $\epsilon n$ requires $\Omega(n)$ queries. Previously, only super-constant lower bounds were known

Localization for Linear Stochastic Evolutions

We consider a discrete-time stochastic growth model on the $d$-dimensional lattice with non-negative real numbers as possible values per site. The growth model describes various interesting examples such as oriented site/bond percolation, directed polymers in random environment, time discretizations of the binary contact path process. We show the equivalence between the slow population growth and a localization property in terms of "replica overlap". The main novelty of this paper is that we obtain this equivalence even for models with positive probability of extinction at finite time. In the course of the proof, we characterize, in a general setting, the event on which an exponential martingale vanishes in the limit

Stochastic shear thickening fluids: Strong convergence of the Galerkin approximation and the energy equality

We consider a stochastic partial differential equation (SPDE) which describes the velocity field of a viscous, incompressible non-Newtonian fluid subject to a random force. Here, the extra stress tensor of the fluid is given by a polynomial of degree p-1 of the rate of strain tensor, while the colored noise is considered as a random force. We focus on the shear thickening case, more precisely, on the case $p\in [1+{\frac{d}{2}},{\frac{2d}{d-2}})$, where d is the dimension of the space. We prove that the Galerkin scheme approximates the velocity field in a strong sense. As a consequence, we establish the energy equality for the velocity field.Comment: Published in at http://dx.doi.org/10.1214/11-AAP794 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org