3N+1猜想周期数的平衡方程

提出奇数a的成分指数序列P(a)及成分指数序列的和∑P(a)的概念,得出成分指数序列的和∑P(a)的简单计算公式,并在此基础上证明3N+1猜想中的周期数必须满足的一个平衡方程,这个平衡方程对猜想的证明将起到很大作用.

短波通滤光片膜系设计

详细介绍用等效折射率概念设计短波通滤光片的原理和计算方法。根据原理和方法，选择二氧化钛（TiO2）作为高折射率材料、二氧化硅（SiO2）作为低折射率材料。首先从理论上计算出用这2种材料设计的波长A-950～1150nm的短波通滤光片所需要的周期数，然后给出短波通滤光片的主膜系和光谱曲线。由于据此周期数设计出的膜系光谱曲线在750～810nm处的透过率不符合要求，因此对该膜系进行了改进。依照改进的设计进行多次制备，最终制备出了符合要求的短波通滤光片，找到了最佳制备工艺和方法。最后，对制备出来的短波通滤光片薄膜进行了各种环境实验。实验结果表明，膜层的各项指标符合设计要求。

New Solutions to Generalized mKdV Equation

Trial function method is applied to solve generalized mKdV (GmKdV for short) equations. It is shownthat GmKdV equations with a real number parameter can be solved directly by this method without a transformation,and more new kinds of solitary wave solutions are obtained.

A Generalized Variable-Coefficient Algebraic Method Exactly Solving （3＋1）-Dimensional Kadomtsev-Petviashvilli Equation

A generalized variable-coefficient algebraic method is appfied to construct several new families of exact solutions of physical interest for （3＋1）-dimensional Kadomtsev-Petviashvilli （KP） equation. Among them, the Jacobi elliptic periodic solutions exactly degenerate to the soliton solutions at a certain limit condition. Compared with the existing tanh method, the extended tanh method, the Jacobi elliptic function method, and the algebraic method, the proposed method gives new and more general solutions.

Global exponential periodicity of a class of impulsive neural networks

By the Lyapunov function method, combined with the inequality techniques, some criteria are established to ensure the existence, uniqueness and global exponential stability of the periodic solution for a class of impulsive neural networks. The results obtained only require the activation functions to be globally Lipschitz continuous without assuming their boundedness, monotonicity or differentiability. The conditions are easy to check in practice and they can be applied to design globally exponentially periodic impulsive neural networks.

PERIODIC DYADIC GREEN’S FUNCTION FOR FIELD ANALYSIS IN EMC CHAMBER

Herein a novel Dyadic Green’s Function (DGF) is presented to calculate the field in ElectroMag-netic Compatibility (EMC) chamber. Due to the difficulty of simulating the whole chamber environment, the analysis combines the DGF formulation and the FEM method, with the latter deals with the reflection from absorbers. With DGF formulation for infinite periodic array structures, this paper investigates electromagnetic field in chamber with truncated arrays. The reflection from the absorber serves as the virtual source contribut-ing to the total field. Hence the whole chamber field calculation can be separated from the work of absorber model set-up. Practically the field homogeneity test and Normal Site Attenuation (NSA) test are carried out to evaluate the chamber performance. Based on the method in this paper, the simulation results agree well with the test, and predict successfully the victim frequency points of the chamber.

A generalized sinusoidal model and its applications

A physical model of sinusoidal function was established. It is generalized that the force is directly proportional to a power function of the distance in a classical spring-oscillator system. The differential equation of the generalized model was given. Simulations were conducted with different power values. The results show that the solution of the generalized equation is a periodic function. The expressions of the amplitude and the period(frequency) of the generalized equation were derived by the physical method. All the simulation results coincide with the calculation results of the derived expressions. A special function also was deduced and proven to be convergent in the theoretical analysis. The limit value of the special function also was derived. The generalized model can be used in solving a type of differential equation and to generate periodic waveforms.

A Data Assured Deletion Scheme in Cloud Storage

In order to provide a practicable solution to data confidentiality in cloud storage service,a data assured deletion scheme,which achieves the fine grained access control,hopping and sniffing attacks resistance,data dynamics and deduplication,is proposed.In our scheme,data blocks are encrypted by a two-level encryption approach,in which the control keys are generated from a key derivation tree,encrypted by an All-OrNothing algorithm and then distributed into DHT network after being partitioned by secret sharing.This guarantees that only authorized users can recover the control keys and then decrypt the outsourced data in an ownerspecified data lifetime.Besides confidentiality,data dynamics and deduplication are also achieved separately by adjustment of key derivation tree and convergent encryption.The analysis and experimental results show that our scheme can satisfy its security goal and perform the assured deletion with low cost.

ODD-PERIODIC AND PERIODIC COMPLEMENTARY BINARY SEQUENCES

Two or more sequences are called an Odd-Periodic Complementary binary sequences Set (OPCS) if the sum of their respective odd-periodic autocorrelation function is a delta function. In this paper, the definition of OPCS is given and the construction method of OPCS is discussed. The relation of the OPCS with the Periodic Complementary binary sequences Set (PCS) is pointed out, and some new PCSs are obtained based on such relation.

New Rational Form Solutions to Coupled Nonlinear Wave Equations

The new rational form solutions to the elliptic equation are shown, and then these solutions to the elliptic equation are taken as a transformation and applied to solve nonlinear coupled wave equations. It is shown that more novel kinds of solutions are derived, such as periodic solutions of rational form, solitary wave solutions of rational form,and so on.

New Doubly Periodic Solutions for the Coupled Nonlinear Klein-Gordon Equations

By using the general solutions of a new coupled Riccati equations, a direct algebraic method is described to construct doubly periodic solutions (Jacobi elliptic function solution) for the coupled nonlinear Klein-Gordon equations.It is shown that more doubly periodic solutions and the corresponding solitary wave solutions and trigonometric function solutions can be obtained in a unified way by this method.

On the Jacobi Elliptic Function Expansion Method

The main idea of this method is to take full advantage of the elliptic equation that Jacobi elliptic functions satisfy and use its solutions to replace Jacobi elliptic functions in Jacobi elliptic function method. Some illustrative equations are investigated by this means.

PSEUDO ALMOST PERIODIC SOLUTIONS OF DIFFERENTIAL EQUATIONS WITHPIECEWISE CONSTANT ARGUMENT

The authors discuss the existence of pseudo almost periodic solutions of differential equations with piecewise constant argument by means of introducing new concept, pseudo almost periodic sequence.

MELNIKOV FUNCTIONS AND PERTURBATION OF A PLANAR HAMILTONIAN SYSTEM

In this paper, Melnikov functions which appear in the study of limit cycles of a perturbedplanar Hamiltonian system are studied. There are two main contributions here. The first oneis related to the explicit formulae for these functions: a new method is developed to achievethe goal for the second one (Theorem A). the authors also discover a close relation betweenMelnikov functions and focal quantities (Theorem B). This relation is useful in both judgingwhen a Melnikov function is identically zero and simplifying the computation of a Melnikovfunction (see 5). Despite these results, this paper also includes other related results, e.g. someestimations of the upper bound for the number of limit cycles in a perturbed Hamiltoniansystem.

Acyclic edge coloring of planar graphs without adjacent cycles

A proper edge coloring of a graph G is said to be acyclic if there is no bicolored cycle in G.The acyclic edge chromatic number of G,denoted byχ′a(G),is the smallest number of colors in an acyclic edge coloring of G.Let G be a planar graph with maximum degree.In this paper,we show thatχ′a(G)+2,if G has no adjacent i-and j-cycles for any i,j∈{3,4,5},which implies a result of Hou,Liu and Wu(2012);andχ′a(G)+3,if G has no adjacent i-and j-cycles for any i,j∈{3,4,6}.

Disjoint long cycles in a graph

We prove that ifG is a graph of order at least 2k with k≥9 and the minimum degree of G is at least k + 1,then G contains two vertex-disjoint cycles of order at least k.Moreover,the condition on the minimum degree is sharp.

On the primitive divisors of the recurrent sequence un＋1=(4cos^2(2π/7)-1)un-un-1 with applications to group theory

Consider the sequence of algebraic integers un given by the starting values u0=0,u1=1 and the recurrence u_(n+1)=(4cos^2(2π/7)-1)u_n-u_(n-1).We prove that for any n ■{1,2,3,5,8,12,18,28,30}the n-th term of the sequence has a primitive divisor in Z[2 cos(2π/7)].As a consequence we deduce that for any sufficiently large n there exists a prime power q such that the groupcan be generated by a pair x,y with χ~2=y^3=(xy)~7=1 and the order of the commutator[x,y]is exactly n.The latter result answers in affirmative a question of Holt and Plesken.

ALMOST PERIODIC SOLUTIONS OF NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS WITH NONAUTONOMOUS OPERATOR

This paper presents some existence theorems on almost periodic solutions for almost periodicneutral functional differential equation and its perturbed systems by means of Liapunov functional, and extends the corresponding results in 12,4,5,10,11].

Periodic Wave, Solitary Wave and Compacton Solutions of a Nonlinear Wave Equation with Degenerate Dispersion

In this letter, we investigate traveling wave solutions of a nonlinear wave equation with degenerate dispersion. The phase portraits of corresponding traveling wave system are given under different parametric conditions. Some periodic wave and smooth solitary wave solutions of the equation are obtained. Moreover, we find some new hyperbolic function compactons instead of well-known trigonometric function compactons by analyzing nilpotent points.

L(d_1,d_2,...,d_t)-Number λ(C_n;d_1,d_2,...,d_t) of Cycles

An L（d0,d2,...,dt）-labeling of a graph G is a function f from its vertex set V（G） to the set {0,1,..., k} for some positive integer k such that If（x） - f（y）l ≥di, if the distance between vertices x and y in G is equal to i for i = 1,2,...,t. The L（d1,d2,...,dt）-number λ（G;d1,d2,... ,dt） of G is the smallest integer number k such that G has an L（d1,d2,...,dr）- labeling with max{f （x）｜x ∈ V（G）} = k. In this paper, we obtain the exact values for λ（Cn; 2, 2, 1） and λ（Cn; 3, 2, 1）, and present lower and upper bounds for λ（Cn; 2,..., 2, 1,..., 1）