Origin and meaning of the Fermi contact interaction

The Fermi-contact interaction (FCI) can easily be derived from 1st order perturbation theory applied to the non-relativistic wave equation for a spin-(1/2) particle of Lévy-Leblond, with the nuclear spin described by the field of an external magnetic dipole, and it results from the fact that the turn-over-rule for the operator is only valid if the derivatives implicit in are taken in the distribution sense. If one avoids to apply the turn-over-rule, the FCI is obtained without the need to introduce a -function. It is also shown that the formulation of a magnetic point dipole as the limit of an extended nucleus directly leads to the FCI. Traditional methods of the derivation of the FCI are analyzed in the light of this new interpretation. It is then explained why the perturbation expansions in powers of the magnetic moment of the nucleus necessarily diverges, but that the expression for the 1st order energy on which the concept of the FCI is based, can nevertheless be justified by means of the Hellmann-Feynman theorem with a correction term if singular wave functions are involved. Finally some comments on a theory beyond first order are made.Dedicated to Professor J. Koutecký on the occasion of his 65th birthday     

Unifying hidden-variable problems from quantum mechanics by logics of dependence and independence

We study hidden-variable models from quantum mechanics and their abstractions in purely probabilistic and relational frameworks by means of logics of dependence and independence, which are based on team semantics. We show that common desirable properties of hidden-variable models can be defined in an elegant and concise way in dependence and independence logic. The relationship between different properties and their simultaneous realisability can thus be formulated and proven on a purely logical level, as problems of entailment and satisfiability of logical formulae. Connections between probabilistic and relational entailment in dependence and independence logic allow us to simplify proofs. In many cases, we can establish results on both probabilistic and relational hidden-variable models by a single proof, because one case implies the other, depending on purely syntactic criteria. We also discuss the ‘no-go’ theorems by Bell and Kochen-Specker and provide a purely logical variant of the latter, introducing non-contextual choice as a team-semantical property.     

Tensor products of coherent configurations

A Cartesian decomposition of a coherent configuration ✗ is defined as a special set of its parabolics that form a Cartesian decomposition of the underlying set. It turns out that every tensor decomposition of ✗ comes from a certain Cartesian decomposition. It is proved that if the coherent configuration ✗ is thick, then there is a unique maximal Cartesian decomposition of ✗; i.e., there is exactly one internal tensor decomposition of ✗ into indecomposable components. In particular, this implies an analog of the Krull–Schmidt theorem for the thick coherent configurations. A polynomial-time algorithm for finding the maximal Cartesian decomposition of a thick coherent configuration is constructed.     

A Review of Results on Axially Symmetric Navier-Stokes Equations,with Addendum by X. Pan and Q. Zhang

In this paper, we give a brief survey of recent results on axially symmetric Navier-Stokes equations (ASNS) in the following categories: regularity criterion, Liouville property for ancient solutions, decay and vanishing of stationary solutions. Some discussions also touch on the full 3 dimensional equations. Two results, closing of the scaling gap for ASNS and vanishing of homogeneous D solutions in 3 dimensional slabs will be described in more detail.In the addendum, two new results in the 3rd category will also be presented, which are generalizations of recently published results by the author and coauthors.     

Monge-Ampère Equation with Bounded Periodic Data

We consider the Monge-Ampère equation det $(D^2u) = f$ in $\mathbb{R}^n,$ where $f$ is a positive bounded periodic function. We prove that $u$ must be the sum of a quadratic polynomial and a periodic function. For $f ≡ 1,$ this is the classic result by Jörgens, Calabi and Pogorelov. For $f ∈ C^α,$ this was proved by Caffarelli and the first named author.     

The Regularity of Stochastic Convolution Driven by Tempered Fractional Brownian Motion and Its Application to Mean-field Stochastic Differential Equations

In this paper, some properties of a stochastic convolution driven by tempered fractional Brownian motion are obtained. Based on this result, we get the existence and uniqueness of stochastic mean-field equation driven by tempered fractional Brownian motion. Furthermore, combining with the Banach fixed point theorem and the properties of Mittag-Leffler functions, we study the existence and uniqueness of mild solution for a kind of time fractional mean-field stochastic differential equation driven by tempered fractional Brownian motion.     

Limit cycles of a class of Liénard systems derived from state-dependent impulses

Discontinuous phenomena, in which objects may behave continuously and sometimes discretely are not only found in nature and under laboratory conditions but also in simple, familiar contexts. For example, this phenomenon is skillfully incorporated into the internal structure of mechanical wristwatches. Unless an extremely small amount of state-dependent impulse is applied intermittently, the reciprocating rotational movement of the balance and hairspring, which is the heart of the mechanical wristwatch, cannot be maintained. The small amount of state-dependent impulse, which is often overlooked, can make a significant difference; however, very few studies have examined this subject. This study assumes the underlying cause of discontinuous behaviors as impulses generated when an object reaches a particular state, assuming that the continuous behavior follows the Liénard system, which is widely studied in the field of electrical circuits. The main theorem provides the conditions under which the effect of the impulses causes a stable limit cycle in the Liénard system, even if no limit cycle exists when there are no impulses. The Poincaré–Bendixson theorem for discontinuous dynamical systems and phase plane analysis are used to prove the main theorem. Several examples and their simulations are provided to illustrate the main theorem.     

Estimation of a quantile in some nonstandard cases

A necessary condition for the asymptotic normality of the sample quantile estimator isf(Q(p))=F(Q(p))>0, whereQ(p) is thep-th quantile of the distribution functionF(x). In this paper, we estimate a quantile by a kernel quantile estimator when this condition is violated. We have shown that the kernel quantile estimator is asymptotically normal in some nonstandard cases. The optimal convergence rate of the mean squared error for the kernel estimator is obtained with respect to the asymptotically optimal bandwidth. A law of the iterated logarithm is also established.This research was partially supported by the new faculty award from the University of Oregon.     

Large deviations for martingales with some applications

Let(X _i ) be a martingale difference sequence. LetY be a standard normal random variable. We investigate the rate of uniform convergence

Algebraic spatial correlations in lattice gas automata violating detailed balance

In this paper we discuss the existence of generic long-range correlations in spatially homogeneous and stable equilibrium states of closed lattice gas automata whose stochastic collision rules violate the symmetry conditions of detailed balance and in addition satisfy local conservation laws. Such correlations occur even though the collision rules are strictly local and invariant under all symmetries of the lattice. First a phenomenological (Langevin equation) approach is discussed. Next we present a theoretical analysis on the basis of an approximate microscopic (ring kinetic) theory. This theory is used to calculate the amplitude ofr ^– tails in the spatial correlations, and the result is compared with computer simulations.