Properties of Defect Modes in One-Dimensional Optically Induced Photonic Lattices

In this article, localized defect modes in one-dimensional optically induced photonic lattices are studied comprehensively. First, the origin of these defect modes is investigated analytically in the weak-defect limit by perturbation methods. It is shown that in an attractive defect where the lattice light intensity at the defect site is higher than that of nearby sites, a defect mode bifurcates from the left edge of every Bloch band; while in a repulsive defect, a defect mode bifurcates from the right edge of every Bloch band. When the defect is not weak, defect modes are examined by numerical methods. It is shown that in a repulsive defect, the strongest confinement of defect modes arises when the lattice light intensity at the defect site is nonzero rather than zero. In addition, as the potential strength increases, defect modes disappear from lower bandgaps and appear in higher bandgaps. In an attractive defect, however, defect modes persist in every bandgap as the potential strength increases. Using a piecewise-constant potential model, defect modes are calculated analytically for a general defect. The analytical results qualitatively explain the main features in numerical results.     

Analog of the Kolmogorov Equations for One-Dimensional Stochastic Differential Equations Controlled by Fractional Brownian Motion with Hurst Exponent $$Hin (0,1)$$

Differential Equations - Analogs of the Kolmogorov equations for the expectations and distribution densities of solutions of one-dimensional stochastic differential equations controlled by...     

Stabilization of a One-Dimensional Dam-River System: Nondissipative and Noncollocated Case

In this paper, we consider a one-dimensional dam-river system, described by a diffusive-wave equation and often used in hydraulic engineering to model the dynamic behavior of the unsteady flow in a river for shallow water when the flow variations are not important. We propose an integral boundary control which leads to a nondissipative closed-loop system with noncollocated actuators and sensors; hence, two main difficulties arise: first, how to show the C ₀-semigroup generation and second, how to achieve the stability of the system. To overcome this situation, the Riesz basis methodology is adopted to show that the closed-loop system generates an analytic semigroup. Concerning the stability, the shooting method is applied to assign the spectrum of the system in the open left-half plane and ensure its exponential stability as well as the output regulation. Numerical simulations are presented for a family of system parameters. The authors express their sincere thanks to Boumenir Amin for valuable comments and suggestions. The first author acknowledges the support of Sultan Qaboos University. The second author was supported by the National Natural Science Foundation of China.     

The Physics of Self-Adjoint Extensions: One-Dimensional Scattering Problem for the Coulomb Potential

We consider a one-dimensional single-center scattering problem on the entire axis with the original potential |x|^–1. This problem reduces to seeking admissible self-adjoint extensions. Using conservation laws at the singularity point as necessary conditions and taking the analytic structure of fundamental solutions into account allows obtaining exact expressions for the wave functions (i.e., for the boundary conditions), scattering coefficients, singular corrections to the potential, and also the corresponding spectrum of bound states. It then turns out that pointlike -corrections to the potential must necessarily be involved for any choice of the admissible self-adjoint extension. The form of these corrections corresponds to the form of the renormalization terms obtained in quantum electrodynamics. The proposed method therefore indicates a 1:1 relation between boundary conditions, scattering coefficients, and -like additions to the potential and demonstrates the general possibilities arising in the analysis of self-adjoint extensions of the corresponding Hamilton operator. In the part pertaining to the renormalization theory, it can be considered a generalization of the renormalization method of Bogoliubov, Parasyuk, and Hepp.     

我国通货膨胀持久性的测度:中位数回返方法

本文提出了一种新的通货膨胀持久性的非参数测度方法:中位数回返方法。相对于现有的持久性测度方法,该方法不需要模型假定和参数估计,具有较强的稳健性。我们利用2001年1月至2014年2月的月度CPI同比数据,对我国居民消费价格指数以及八大类分指数的持久性问题进行了实证分析,研究结果表明,我国的通货膨胀具有较强的持久性。具体到某个行业来看,烟酒及用品类的通胀持久性最强,其次是医疗保健和个人用品类以及居住类。相对而言,衣着类、家庭设备用品及维修服务类的通胀持久性最弱。     

Classification of Simple C-Algebras: Inductive Limits of Matrix Algebras Over One-Dimensional Spaces

In this article, we will give a complete classification of simple C^*-algebras which can be written as inductive limits of algebras of the form A_n=⊕_i=1^k_nM_[n,i](C(X_n,i)), where X_n,i are arbitrary variable one-dimensional compact metrizable spaces. The results unify and generalize the previous results for the case X_n,i=S¹ and for the case of X_n,i being trees. We obtain our classification results by reducing the case of general one-dimensional spaces to the case of circles. The techniques in this paper play important roles in the study of the case of higher-dimensional spaces.     

A. N. Kolmogorov: Historian and philosopher of mathematics on the occasion of his 80th birthday

The contributions to the philosophy and history of mathematics of one of the most eminent contemporary mathematicians, A. N. Kolmogorov, are analyzed. Although these contributions are not numerous, they contain discussions of the gradual generalization of the subject of mathematics, a partition of the development of mathematics—in particular, of the theory of probability—into periods, and, also, an evaluation of the works of a number of great savants, such as Newton, Lobatchevsky, etc.     

Reconstruction of Signals from Magnitudes of Redundant Representations: The Complex Case

This paper is concerned with the question of reconstructing a vector in a finite-dimensional complex Hilbert space when only the magnitudes of the coefficients of the vector under a redundant linear map are known. We present new invertibility results as well as an iterative algorithm that finds the least-square solution, which is robust in the presence of noise. We analyze its numerical performance by comparing it to the Cramer–Rao lower bound.     

Spatial Point Process Models of Defensive Strategies: Detecting Changes

The study of stochastic processes can take many forms. Theoretical properties are important to ensure consistent model definition. Statistical inference on unknown parameters is equally important but can be difficult. This is principally because many of the standard assumptions for proving consistency and asymptotic normality of estimators involve independence and homogeneity. In the case where inference is concerned with detecting change in a spatial process from one time point to another, a statistical-computing approach can be rewarding. Regardless of the complexity of the stochastic process, if simulating from it is relatively easy, then detecting change is possible using a Monte Carlo approach. The methodology is applied in a military scenario, where a country’s defensive posture changes as a function of its perceived threat. For tactical-decision purposes, it is extremely important to know whether the country’s perceived threat level has changed.     

Poset Entropy Versus Number of Linear Extensions: The Width-2 Case

Kahn and Kim (J. Comput. Sci. 51, 3, 390–399, 1995) have shown that for a finite poset P, the entropy of the incomparability graph of P (normalized by multiplying by the order of P) and the base-2 logarithm of the number of linear extensions of P are within constant factors from each other. The tight constant for the upper bound was recently shown to be 2 by Cardinal et al. (Combinatorica 33, 655–697, 2013). Here, we refine this last result in case P has width 2: we show that the constant can be replaced by 2?ε if one also takes into account the number of connected components of size 2 in the incomparability graph of P. Our result leads to a better upper bound for the number of comparisons in algorithms for the problem of sorting under partial information.     

Group Analysis of the One-Dimensional Boltzmann Equation: IV. Complete Group Classification in the General Case

Theoretical and Mathematical Physics - We consider the one-dimensional Boltzmann equation $$f_t+cf_x+(\mathcal{F}f)_c=0$$ with a function $$\mathcal{F}$$ depending on (t,x,c,f) and obtain the...     

Preservation of persistence and stability under intersections and operations,part 1: Persistence

New results about convergence of sets and functions in possibly infinite-dimensional spaces are derived in a simple and unified way from two results dealing with the continuity with respect to a parameter of the intersection of two convex sets depending on this parameter.     

Delone Sets with Finite Local Complexity: Linear Repetitivity Versus Positivity of Weights

We consider Delone sets with finite local complexity. We characterize the validity of a subadditive ergodic theorem by uniform positivity of certain weights. The latter can be considered to be an averaged version of linear repetitivity. In this context, we show that linear repetitivity is equivalent to positivity of weights combined with a certain balancedness of the shape of return patterns.     

Polar Decomposition of the Wiener Measure: Schwarzian Theory Versus Conformal Quantum Mechanics

Theoretical and Mathematical Physics - We find an explicit form of the polar decomposition of the Wiener measure and obtain an equation relating functional integrals in conformai quantum mechanics...     

Indices Versus Transcendental Functions in Seasonal Forecasting: Reaping the Benefits of Both

Two different statistical models have been frequently used for short-term forecasting. The first, involving seasonal indices, is easy to understand but can lead to problems of instability. The second, transcendental functions (mixtures of sine waves), tends to be much more stable but is difficult for the user to understand. In this paper, we explain an approach that combines the benefits of the two types of models.     

One-Dimensional Cutting Stock Problem with a Given Number of Setups: A Hybrid Approach of Metaheuristics and Linear Programming

One-dimensional cutting stock problem (1D-CSP) is one of the representative combinatorial optimization problems, which arises in many industrial applications. Since the setup costs for switching different cutting patterns become more dominant in recent cutting industry, we consider a variant of 1D-CSP, called the pattern restricted problem (PRP), to minimize the number of stock rolls while constraining the number of different cutting patterns within a bound given by users. For this problem, we propose a local search algorithm that alternately uses two types of local search processes with the 1-add neighborhood and the shift neighborhood, respectively. To improve the performance of local search, we incorporate it with linear programming (LP) techniques, to reduce the number of solutions in each neighborhood. A sensitivity analysis technique is introduced to solve a large number of associated LP problems quickly. Through computational experiments, we observe that the new algorithm obtains solutions of better quality than those obtained by other existing approaches.     

Asymptotic Solutions of Nonrelativistic Equations of Quantum Mechanics in Curved Nanotubes: I. Reduction to Spatially One-Dimensional Equations

We consider equations of nonrelativistic quantum mechanics in thin three-dimensional tubes (nanotubes). We suggest a version of the adiabatic approximation that permits reducing the original three-dimensional equations to one-dimensional equations for a wide range of energies of longitudinal motion. The suggested reduction method (the operator method for separating the variables) is based on the Maslov operator method. We classify the solutions of the reduced one-dimensional equation. In Part I of this paper, we deal with the reduction problem, consider the main ideas of the operator separation of variables (in the adiabatic approximation), and derive the reduced equations. In Part II, we will discuss various asymptotic solutions and several effects described by these solutions.     

Group Analysis of the One-Dimensional Boltzmann Equation: III. Condition for the Moment Quantities to Be Physically Meaningful

We present the group classification of the one-dimensional Boltzmann equation with respect to the function F = F(t, x, c) characterizing an external force field under the assumption that the physically meaningful constraints dx = c dt, dc = F dt, dt = 0, and dx =0 are imposed on the variables. We show that for all functions F, the algebra is finite-dimensional, and its maximum dimension is eight, which corresponds to the equation with a zero F.