Finite Words,Infinite Wisdom

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Infinitely Many Solutions and a Ground-State Solution for Klein-Gordon Equation Coupled with Born-Infeld Theory

In this paper, we intend to consider a kind of nonlinear Klein-Gordon equation coupled with Born-Infeld theory. By using critical point theory and the method of Nehari manifold, we obtain two existing results of infinitely many high-energy radial solutions and a ground-state solution for this kind of system, which improve and generalize some related results in the literature.

Darboux transformation,infinite conservation laws,and exact solutions for the nonlocal Hirota equation with variable coefficients

This article presents the construction of a nonlocal Hirota equation with variable coefficients and its Darboux transformation.Using zero-seed solutions,1-soliton and 2-soliton solutions of the equation are constructed through the Darboux transformation,along with the expression for N-soliton solutions.Influence of coefficients that are taken as a function of time instead of a constant,i.e.,coefficient functionδ(t),on the solutions is investigated by choosing the coefficient functionδ(t),and the dynamics of the solutions are analyzed.This article utilizes the Lax pair to construct infinite conservation laws and extends it to nonlocal equations.The study of infinite conservation laws for nonlocal equations holds significant implications for the integrability of nonlocal equations.

TWO DISJOINT AND INFINITE SETS OF SOLUTIONS FOR AN ELLIPTIC EQUATION INVOLVING CRITICAL HARDY-SOBOLEV EXPONENTS

In this paper,by an approximating argument,we obtain two disjoint and infinite sets of solutions for the following elliptic equation with critical Hardy-Sobolev exponents■whereΩis a smooth bounded domain in RN with 0∈?Ωand all the principle curvatures of?Ωat 0 are negative,a∈C1(Ω,R*+),μ>0,0<s<2,1<q<2 and N>2(q+1)/(q-1).By2*:=2N/(N-2)and 2*(s):(2(N-s))/(N-2)we denote the critical Sobolev exponent and Hardy-Sobolev exponent,respectively.

Empirical Likelihood Statistical Inference for Compound Poisson Vector Processes under Infinite Covariance Matrix

The paper discusses the statistical inference problem of the compound Poisson vector process(CPVP)in the domain of attraction of normal law but with infinite covariance matrix.The empirical likelihood(EL)method to construct confidence regions for the mean vector has been proposed.It is a generalization from the finite second-order moments to the infinite second-order moments in the domain of attraction of normal law.The log-empirical likelihood ratio statistic for the average number of the CPVP converges to F distribution in distribution when the population is in the domain of attraction of normal law but has infinite covariance matrix.Some simulation results are proposed to illustrate the method of the paper.

Convergence of a Sinusoidal Series with an Infinite Integral

In this paper, we study the relationship between the convergence of the sinusoidal series and the infinity integrals (any real number α ∈[0,1], parameter p > 0). First of all, we study the convergence of the series (any real number α ∈[0,1], parameter p > 0), mainly using the estimation property of the order to obtain that the series diverges when 0 p ≤1-α, the series converges conditionally when 1-α p ≤1, and the series converges absolutely when p >1. In the next part, we study the convergence state of the infinite integral (any real number α ∈[0,1], parameter p > 0), and get that when 0 p ≤1-α, the infinite integral diverges;when 1-α p ≤1, the infinite integral conditionally converges;when p >1, the infinite integral absolutely converges. Comparison of the conclusions of the above theorem, it is not difficult to derive the theorem: the level of and the infinity integral with the convergence of the state (any real number α ∈[0,1], the parameter p >0), thus promoting the textbook of the two with the convergence of the state requires the function of the general term or the product of the function must be monotonically decreasing conditions.

Well-Posedness and Infinite Propagation Speed for the Fifth-Order Camassa-Holm Equation with Weakly Dissipative Term

In this paper, we study the fifth-order Camassa-Holm equation with weakly dissipative term. We first give the local well-posedness result and the blow up criterion. Then, we establish sufficient conditions to guarantee that the solution exists globally in time. Finally, the infinite propagation speed of this equation is also investigated.

A NOVEL ELLIPSOIDAL ACOUSTIC INFINITE ELEMENT

A novel ellipsoidal acoustic infinite element is proposed. It is based a new pressure representation, which can describe and solve the ellipsoidal acoustic field more exactly. The shape functions of this novel acoustic infinite element are similar to the (Burnett's) method, while the weight functions are defined as the product of the complex conjugates of the shaped functions and an additional weighting factor. The code of this method is cheap to generate as for 1-D element because only 1-D integral needs to be numerical. Coupling with the standard finite element, this method provides a capability for very efficiently modeling acoustic fields surrounding structures of virtually any practical shape. This novel method was deduced in brief and the conclusion was kept in detail. To test the feasibility of this novel method efficiently,in the examples the infinite elements were considered,excluding the finite elements relative. This novel ellipsoidal acoustic infinite element can deduce the analytic solution of an oscillating sphere. The example of a prolate spheroid shows that the novel infinite element is superior to the boundary element and other acoustic infinite elements. Analytical and numerical results of these examples show that this novel method is feasible.

PLANE INFINITE ANALYTICAL ELEMENT AND HAMILTONIAN SYSTEM

It is not convenient to solve those engineering problems defined in an infinite field by using FEM. An infinite area can be divided into a regular infinite external area and a finite internal area. The finite internal area was dealt with by the FEM and the regular infinite external area was settled in a polar coordinate. All governing equations were transformed into the Hamiltonian system. The methods of variable separation and eigenfunction expansion were used to derive the stiffness matrix of a new infinite analytical element.This new element, like a super finite element, can be combined with commonly used finite elements. The proposed method was verified by numerical case studies. The results show that the preparation work is very simple, the infinite analytical element has a high precision, and it can be used conveniently. The method can also be easily extended to a three-dimensional problem.

DISCRETE TIME STOCHASTIC EQUILIBRIUM WITH INFINITE HORIZON INCOMPLETE ASSET MARKETS

This paper examines the existence of general equilibrium in a discrete time economy with the infinite horizon incomplete markets.There is a single good at each node in the event tree.The existence of general equilibrium for the infinite horizon economy is proved by taking limit of equilibria in truncated economies in which trade stops at a sequence of dates.

Integral Sequences of Infinite Length Whose Terms Are Relatively Prime

It is given in Weil and Rosenlicht ([1], p. 15) that (resp. 2) for all non-negative integers m and n with m≠n if c is any even (resp. odd) integer. In the present paper we generalize this. Our purpose is to give other integral sequences such that G.C.D.(ym,yn)=1 for all positive integers m and n with m≠n. Roughly speaking we show the following 1) and 2). 1) There are infinitely many polynomial sequences such that G.C.D.(fm(a),fn(a))=1 for all positive integers m and n with with m≠n and infinitely many rational?integers a. 2) There are polynomial sequences such that G.C.D.(gm(a,b),gn(a,b))=1 for all positive integers m and n with m≠n and arbitrary (rational or odd) integers a and b with G.C.D.(a,b)=1. Main results of the present paper are Theorems 1 and 2, and Corollaries 3, 4 and 5.

ON THE EXISTENCE OF INFINITELY MANY CRITICAL POINTS OF THE EVEN FUNCTIONAL integral from n=Q to (F(x,u,Du))+integral from n=■Q to (G(x,u))

In this paper we deal with the existence of infinitely many critical points of the even functional I(u)=integral from n=Q to (F(x,u,Du))+integral from n=(?)Q to (G(x,u)), u∈W^(1,p)(Ω),where G(x, u)=integral from n=o to u (g(x,t)dt), under the weak structure conditions on F(x, u, q) by the Mountain Pass Lemma.

Completeness of Eigenfunction Systems for Off-Diagonal Infinite-Dimensional Hamiltonian Operators

For the off-diagonal infinite dimensional Hamiltonian operators, which have at most countable eigenvalues, a necessary and sufficient condition of the eigenfunction systems to be complete in the sense of Cauchy principal value is presented by using the spectral symmetry and new orthogonal relationship of the operators. Moreover, the above result is extended to a more general case. At last, the completeness of eigenfunction systems for the operators arising from the isotropic plane magnetoelectroelastic solids is described to illustrate the effectiveness of the criterion. The whole results offer theoretical guarantee for separation of variables in Hamiltonian system for some mechanics equations.

Analysis of ground vibrations due to underground trains by 2.5D finite/infinite element approach

The 2.5D finite/infinite element approach is adopted to study wave propagation problems caused by underground moving trains. The irregularities of the near field, including the tunnel structure and parts of the soil, are modeled by the finite elements, and the wave propagation properties of the far field extending to infinity are modeled by the infinite elements. One particular feature of the 2.5D approach is that it enables the computation of the three-dimensional response of the half-space, taking into account the load-moving effect, using only a two-dimensional profile. Although the 2.5D finite/infinite element approach shows a great advantage in studying the wave propagation caused by moving trains, attention should be given to the calculation aspects, such as the rules for mesh establishment, in order to avoid producing inaccurate or erroneous results. In this paper, some essential points for consideration in analysis are highlighted, along with techniques to enhance the speed of the calculations. All these observations should prove useful in making the 2.5D finite/infinite element approach an effective one.

Computing of the Anchor by the Method of Three-Dimension Point-Radiate Infinite Elements

On the basis of the one-dimension infinite element theory, the coordinate translation and shape function of 3D point-radiate 8-node and 4-node infinite elements are derived. They are coupled with 20-node and 8-node finite elements to compute the compression distortion of the prestressed anchorage segment. The results indicate that when the prestressed force acts on the anchorage head and segment, the stresses and the displacements in the rock around the anchorage head and segment concentrate on the zone center with the anchor axis, and they decrease with exponential forms. Therefore,the stresses and the displacement spindles are formed. The calculating results of the infinite element are close to the theoretical results. This indicates the method is right. This article introduces a new way to study the mechanism of prestressed anchors. The obtained results have an important role in the research of the anchor mechanism and engineering application.

Infinitely many periodic solutions for second-order Hamiltonian systems

The existence of high energy periodic solutions for the second-order Hamiltonian system -ü（t）＋A（t）u（t）=▽F（t,u（t）） with convex and concave nonlinearities is studied, where F（t, u） = F1（t,u）＋F2（t,u）. Under the condition that F is an even functional, infinitely many solutions for it are obtained by the variant fountain theorem. The result is a complement for some known ones in the critical point theory.

基于Infinite integral的四象限探测器光斑位置检测算法

为了进一步提高基于四象限探测器的半主动式激光导引头的目标定位精度,本文以高斯分布的圆形光斑作为入射光斑模型,提出了一种基于Infinite integral拟合方法的改进算法(GII)。在Infinite integral拟合算法的基础上,考虑了外部光能和死区带来的误差影响,采用最小二乘法获得最佳拟合的解析表达式。为了简化算法的计算量,对部分解析表达式进行分段式多项式拟合,优化拟合区间,并得到GII算法的解析表达式。GII算法相较于传统Infinite integral拟合算法和8次多项式拟合法,精度大幅度提升。仿真和实验表明,该算法在确保良好实时性的同时,能增加四象限探测器的有效探测范围,提高不同光斑半径下四象限探测器的定位精度,为战场环境下精准激光制导武器的发展提供了理论参考。

Measurement of Infinite Diluted Activity Coefficient of Solvents in Polymer by Inverse Gas Chromatography Method

The infinite diluted activity coefficients of solvents in polyisopropyl methylacrylate was measured using inverse gas chromatography. The solvents used were benzene, toluene, ethyl benzene, methyl acetate, ethyl acetate, propyl acetate, butyl acetate, methanol, ethanol isopropyl alcohol, butyl alcohol, 1,2-dichloroethane, and chloroform. It was observed that the infinite diluted activity coefficient of alcohols are well above those of the other solvents investigated.

Mixed-solvothermal Synthesis, Crystal Structure and Luminescence of a New Dinuclear Yttrium(Ⅲ) Coordination Polymer with 1-D Wave-like Infinite Chains

A new dinuclear Y（3＋） coordination polymer{[Y2（H2O）2（C（14）H8O4）3（C（12）H8N2）2]·3 H2O}n （1, C（14）H8 O4 = 2,2＇-biphenyldicarboxylate, phen = 1,10-phenanthroline）, has been obtained by means of a mixed-solvothermal method using ethylene glycol and water as solvent. The compound was characterized by elemental analysis, energy-dispersive X-ray spectroscopy（EDS）, IR spectrum and single-crystal X-ray diffraction. The results reveal that 1 belongs to monoclinic system, space group C2/c with a = 24.249（3）, b = 12.069（48）, c = 22.7304（08） A, β = 113.480（7）°, Z = 4, V = 6102（2） A3, Dc = 1.462 g·cm^-3, F（000） = 2728, μ = 1.968 mm（-1）, the final R = 0.0673, w R = 0.1508 and S = 1.085. Its structure can be regarded as a 1-D coordination polymer constructed by Y^3＋ cations, 2,2A-biphenyldicarboxylate, 1,10-phenanthroline and water molecules. The compound not only contains two kinds of organic ligands, but also exhibits interesting wave-like infinite chains and 18-MR windows with the diameter of 4.070（7）A × 5.326（9）A. The structure is further stabilized by means of O–H…O hydrogen bonds and π-π stacking interactions. Furthermore, the luminescent properties（including emission spectrum, CIE chromaticity coordinate and decay curve） of 1 were also investigated in the solid-state at room temperature.

Preparation of Superconducting Thin Films of Infinite-Layer Nickelate Nd_(0.8)Sr_(0.2)NiO_(2)

The recent observation of superconductivity in thin films of infinite-layer nickelate Nd_(0.8)Sr_(0.2)NiO_(2) has received considerable attention.Despite the many efforts to understand the superconductivity in infinite-layer nickelates,a consensus on the underlying mechanism for the superconductivity has yet to be reached,partly owing to the challenges with the material synthesis.Here,we report the successful growth of superconducting infinite-layer Nd_(0.8)Sr_(0.2)NiO_(2) films by pulsed laser deposition and soft chemical reduction.The details on the growth process are discussed.