The Elastic Continuum Limit of the Tight Binding Model

The authors consider the simplest quantum mechanics model of solids,the tight binding model,and prove that in the continuum limit,the energy of tight binding model converges to that of the continuum elasticity model obtained using Cauchy-Born rule.The technique in this paper is based mainly on spectral perturbation theory for large matrices.     

Decorrelation Estimates for a 1D Tight Binding Model in the Localized Regime

In this article, we prove decorrelation estimates for the eigenvalues of a 1D discrete tight-binding model near two distinct energies in the localized regime. Consequently, for any integer n ≥ 2, the asymptotic independence for local level statistics near n distinct energies is obtained.     

Locality of connective constants

The connective constant $\mu \left(G\right)$ of a quasi-transitive graph $G$ is the exponential growth rate of the number of self-avoiding walks from a given origin. We prove a locality theorem for connective constants, namely, that the connective constants of two graphs are close in value whenever the graphs agree on a large ball around the origin (and a further condition is satisfied). The proof is based on a generalized bridge decomposition of self-avoiding walks, which is valid subject to the assumption that the underlying graph is quasi-transitive and possesses a so-called unimodular graph height function.     

Locality problem in quantum theory

We discuss the locality problem in relativistic and nonrelativistic quantum theory. We show that there exists a formulation of quantum theory that, on one hand, preserves the mathematical apparatus of the standard quantum mechanics and, on the other hand, ensures the satisfaction of the locality condition for each individual event including the measurement procedure. As an example, we consider the scattering from two slits. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 2, pp. 327–343, May, 2008.     

Testing for Constant Nonparametric Effects in General Semiparametric Regression Models with Interactions

We consider the problem of testing for a constant nonparametric effect in a general semiparametric regression model when there is a potential for interaction between the parametrically and nonparametrically modeled variables. The work was originally motivated by a unique testing problem in genetic epidemiology (Chatterjee et al., 2006) that involved a typical generalized linear model but with an additional term reminiscent of the Tukey 1-degree-of-freedom formulation, and their interest was in testing for main effects of the genetic variables, while gaining statistical power by allowing for a possible interaction between genes and the environment. Later work (Maity et al., 2009) involved the possibility of modeling the environmental variable nonparametrically, but they focused on whether there was a parametric main effect for the genetic variables. In this paper, we consider the complementary problem, where the interest is in testing for the main effect of the nonparametrically modeled environmental variable. We derive a generalized likelihood ratio test for this hypothesis, show how to implement it, and provide evidence that our method can improve statistical power when compared to standard partially linear models with main effects only. We use the method for the primary purpose of analyzing data from a case-control study of colorectal adenoma.     

蒙特卡罗方法在原子间多体势拟合中的应用

蒙特卡罗方法是求解非线性方程组的一种有效的方法.我们采用F inn is-S incla ir型的嵌入原子势,并运用蒙特卡罗方法拟合了金属T a的平衡晶格常数、结合能、弹性常数、空位形成能,给出了此元素的多体势函数的参数.     

Locality of optimal binary codes

The locality of locally repairable codes (LRCs) for a distributed storage system is the number of nodes that participate in the repair of failed nodes, which characterizes the repair cost. In this paper, we first determine the locality of MacDonald codes, then propose three constructions of LRCs with $r=1,2\text{ and }3$. Based on these results, for $2\le k\le 7$ and $n\ge k+2$, we give an optimal linear $[n,k,d]$ code with small locality. The distance optimality of these linear codes can be judged by the codetable of M. Grassl for $n<2(2^k-1)$ and by the Griesmer bound for $n\ge 2(2^k-1)$. Almost all the $[n,k,d]$ codes ($2\le k\le 7$) have locality $r\le 3$ except for the three codes, and most of the $[n,k,d]$ code with $n<2(2^k-1)$ achieves the Cadambe–Mazumdar bound for LRCs.     

Lower Bound for the Interatomic Distance in Lennard-Jones Clusters

We prove in this article lower bounds and upper bounds for the interatomic distance in cluster of atoms minimizing the Lennard-Jones energy. Our main result is in dimension three, but we also prove it in the two-dimensional case, since it seems interesting from a theoretical point of view.     

Asymptotic Properties of Self-Consistent Estimators with Mixed Interval-Censored Data

Mixed interval-censored (MIC) data consist of n intervals with endpoints L _i and R _i , i = 1, ..., n. At least one of them is a singleton set and one is a finite non-singleton interval. The survival time X _i is only known to lie between L _i and R _i , i = 1, 2, ..., n. Peto (1973, Applied Statistics, 22, 86–91) and Turnbull (1976, J. Roy. Statist. Soc. Ser. B, 38, 290–295) obtained, respectively, the generalized MLE (GMLE) and the self-consistent estimator (SCE) of the distribution function of X with MIC data. In this paper, we introduce a model for MIC data and establish strong consistency, asymptotic normality and asymptotic efficiency of the SCE and GMLE with MIC data under this model with mild conditions.     

Locality and time irreversibility in quantum processes

We discuss problems arising in three very different physical processes: an electron scattering on a nucleus, an experiment with delayed choice, and the cosmological Big Bang. We describe the role of soft and supersoft photons in solutions of the arising problems.     

The Index Locality Principle in Elliptic Theory

We prove a general theorem on the behavior of the relative index under surgery for a wide class of Fredholm operators, including relative index theorems for elliptic operators due to Gromov--Lawson, Anghel, Teleman, Booß-Bavnbek–Wojciechowski, et al. as special cases. In conjunction with some additional conditions (like symmetry conditions), this theorem permits computing the analytical index of a given operator. In particular, we obtain new index formulas for elliptic pseudodifferential operators and quantized canonical transformations on manifolds with conical singularities.     

Positon Solutions of the KdV Equation with Self-Consistent Sources

We construct a binary Darboux transformation with an arbitrary time function for the KdV equation with self-consistent sources. With this transformation, we obtain positon solutions of the KdV equation with self-consistent sources. We also discuss the properties of these solutions.     

Hilbert空间中的紧K-框架

K-框架是框架的一种推广.本文在Hilbert空间将紧框架推广到K-框架上,引入紧K-框架的概念.通过紧K-框架的算子K和合成算子给出紧K-框架的算子刻画,并利用紧K-框架的算子K给出紧K-框架成为紧框架的一个充要条件.还讨论紧K-框架的构造以及两个紧K-框架集的包含与涉及的算子K的相互关系.     

Locality in Free String Field Theory - II

We study the covariant free bosonic string field theory and explore its locality (causality) properties. We find string fields which are strictly local and covariant, but act on an unconstrained Hilbert space with an indefinite inner product. From these we construct observable fields which act on the physical Hilbert space with a definite inner product. These are shown to be approximately local.     

Dynamical Locality of the Free Scalar Field

Dynamical locality is a condition on a locally covariant physical theory, asserting that kinematic and dynamical notions of local physics agree. This condition was introduced in arXiv:1106.4785, where it was shown to be closely related to the question of what it means for a theory to describe the same physics on different spacetimes. In this paper, we consider in detail the example of the free minimally coupled Klein–Gordon field, both as a classical and quantum theory (using both the Weyl algebra and a smeared field approach). It is shown that the massive theory obeys dynamical locality, both classically and in quantum field theory, in all spacetime dimensions n ≥ 2 and allowing for spacetimes with finitely many connected components. In contrast, the massless theory is shown to violate dynamical locality in any spacetime dimension, in both classical and quantum theory, owing to a rigid gauge symmetry. Taking this into account (equivalently, working with the massless current) dynamical locality is restored in all dimensions n ≥ 2 on connected spacetimes, and in all dimensions n ≥ 3 if disconnected spacetimes are permitted. The results on the quantized theories are obtained using general results giving conditions under which dynamically local classical symplectic theories have dynamically local quantizations.     

Dimensions of Tight Spans

Given a finite metric, one can construct its tight span, a geometric object representing the metric. The dimension of a tight span encodes, among other things, the size of the space of explanatory trees for that metric; for instance, if the metric is a tree metric, the dimension of the tight span is one. We show that the dimension of the tight span of a generic metric is between

Dynamical Locality of the Free Maxwell Field

The extent to which the non-interacting and source-free Maxwell field obeys the condition of dynamical locality is determined in various formulations. Starting from contractible globally hyperbolic spacetimes, we extend the classical field theory to globally hyperbolic spacetimes of arbitrary topology in two ways, obtaining a ‘universal’ theory and a ‘reduced’ theory of the classical free Maxwell field and their corresponding quantisations. We show that the classical and the quantised universal theory fail local covariance and dynamical locality owing to the possibility of having non-trivial radicals in the classical pre-symplectic spaces and non-trivial centres in the quantised *-algebras. The classical and the quantised reduced theory are both locally covariant and dynamically local, thus closing a gap in the discussion of dynamical locality and providing new examples relevant to the question of how theories should be formulated so as to describe the same physics in all spacetimes.