Exact traveling wave solutions and dynamical behavior for the (n+1)-dimensional multiple sine-Gordon equation

Using the methods of dynamical systems for the (n 1)-dimensional multiple sine-Gordon equation, the existences of uncountably infinite many periodic wave solutions and breaking bounded wave solutions axe obtained. For the double sine-Gordon equation, the exact explicit parametric representations of the bounded traveling solutions are given. To guarantee the existence of the above solutions, all parameter conditions are determined.     

Bifurcation analysis and exact traveling wave solutions for a generic two-dimensional sine-Gordon equation in nonlinear optics

We focus on investigating a generic two-dimensional sine-Gordon equation in nonlinear optics. Based on a viable transformation, the bifurcation analysis of the equation is carried out in this paper. The phase portraits are given and different kinds of traveling wave solutions are obtained. The analytical results are also numerically simulated.     

Composite Integers n for Which φ（n）｜n- 1

In this note, we show that the number of composite integers n ≤ x such that φ（n）｜n - 1 is at most O（x^1/2（loglog x）^1/2）, thus improving earlier results by Pomerance and by Shan.     

Dynamics and bifurcations of travelling wave solutions of R(<Emphasis Type="Italic">m,n</Emphasis>) equations

By using the bifurcation theory and methods of planar dynamical systems to R(m, n) equations, the dynamical behavior of different physical structures like smooth and non-smooth solitary wave, kink wave, smooth and non-smooth periodic wave, and breaking wave is obtained. The qualitative change in the physical structures of these waves is shown to depend on the systemic parameters. Under different regions of parametric spaces, various sufficient conditions to guarantee the existence of the above waves are given. Moreover, some explicit exact parametric representations of travelling wave solutions are listed.     

On the multiply connectedness of level lines <Emphasis Type="Italic">mπ</Emphasis> of <Emphasis Type="Italic">n</Emphasis>-soliton solutions of the sine-Gordon equation

The behavior of level lines z = mπ of multi-soliton solutions of the sine-Gordon equation is considered. __________ Translated from Fundamental’naya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 11, No. 1, Geometry, 2005.     

Construction of the Riemann function for the Bianchi equation in an <Emphasis Type="Italic">n</Emphasis>-dimensional space

In this paper we obtain a more general form of conditions which enable one to construct explicitly the Riemann function for the Bianchi equation.     

MATRIX EQUATION AXB = E WITH PX = sXP CONSTRAINT

The matrix equation AXB = E with the constraint PX=sXP is considered,where P is a given Hermitian matrix satisfying p~2=I and s=±1.By an eigenvalue decomposition of P,the constrained problem can be equivalently transformed to a well-known unconstrained problem of matrix equation whose coefficient matrices contain the corresponding eigenvector, and hence the constrained problem can be solved in terms of the eigenvectors of P.A simple and eigenvector-free formula of the general solutions to the constrained problem by generalized inverses of the coefficient matrices A and B is presented.Moreover,a similar problem of the matrix equation with generalized constraint is discussed.     

Conditions for periodic vibrations in a symmetric <Emphasis Type="Italic">n</Emphasis>-string

A symmetric N-string is a network of N ≥ 2 sections of string tied together at one common mobile extremity. In their equilibrium position, the sections of string form N angles of 2π/N at their junction point. Considering the initial and boundary value problem for small-amplitude oscillations perpendicular to the plane of the N-string at rest, we obtain conditions under which the solution will be periodic with an integral period.      

Directional <Emphasis Type="Italic">∇</Emphasis>-derivative and Curves on <Emphasis Type="Italic">n</Emphasis>-dimensional Time Scales

The general ideal in this paper is to study a differential calculus for multivariable functions, directional ∇-derivative and curves of parametric equations on n-dimensional time scales.      

On the value distribution of <Emphasis Type="Italic">ƒ</Emphasis> + <Emphasis Type="Italic">a</Emphasis>(<Emphasis Type="Italic">ƒ</Emphasis>′)<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Let ƒ be a transcendental meromorphic function, a a nonzero finite complex number, and n ⩾ 2 a positive integer. Then ƒ + a(ƒ′) ⁿ assumes every complex value infinitely often. This answers a question of Ye for n = 2. A related normality criterion is also given. This work was supported by the National Natural Science Foundation of China (Grant No. 10771076), the Natural Science Foundation of Guangdong Province, China (Grant No. 07006700) and by the German-Israeli Foundation for Scientific Research and Development (Grant No. G-809-234.6/2003)     

Approximations for the <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">GI</Emphasis>/<Emphasis Type="Italic">N</Emphasis>+<Emphasis Type="Italic">GI</Emphasis> type call center

In this paper, we propose approximations to compute the steady-state performance measures of the M/GI/N+GI queue receiving Poisson arrivals with N identical servers, and general service and abandonment-time distributions. The approximations are based on scaling a single server M/GI/1+GI queue. For problems involving deterministic and exponential abandon times distributions, we suggest a practical way to compute the waiting time distributions and their moments using the Laplace transform of the workload density function. Our first contribution is numerically computing the workload density function in the M/GI/1+GI queue when the abandon times follow general distributions different from the deterministic and exponential distributions. Then we compute the waiting time distributions and their moments. Next, we scale-up the M/GI/1+GI queue giving rise to our approximations to capture the behavior of the multi-server system. We conduct extensive numerical experiments to test the speed and performance of the approximations, which prove the accuracy of their predictions.      

Generalization of the Kneser theorem on zeros of solutions of the equation <Emphasis Type="Italic">y</Emphasis>″ + <Emphasis Type="Italic">p</Emphasis>(<Emphasis Type="Italic">t</Emphasis>)<Emphasis Type="Italic">y</Emphasis> = 0

We establish conditions for the oscillation of solutions of the equation y″ + p(t)Ay = 0 in a Banach space, where A is a bounded linear operator and p: ℝ+ → ℝ+ is a continuous function. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 4, pp. 571–576, April, 2007.     

Asymptotic observers for <Emphasis Type="Italic">n</Emphasis>-dimensional bilinear systems

We consider the construction of exponential observers for n-dimensional bilinear dynamical systems with arbitrary n. An algorithm is proposed to construct observers with any prespecified degree of stability for a system with a multidimensional output. Observers are constructed for systems with a scalar output and a singular bilinearity matrix. __________ Translated from Nelineinaya Dinamika i Upravlenie, No. 3, pp. 19-26, 2003.     

Consecutive customer losses in regular and oscillating <Emphasis Type="Italic">M</Emphasis><Superscript><Emphasis Type="Italic">X</Emphasis></Superscript>/<Emphasis Type="Italic">G</Emphasis>/1/<Emphasis Type="Italic">n</Emphasis> systems

We derive fast recursions to compute the probability that k or more consecutive customer losses take place during a busy period of a queue, the so called k-CCL probability, for regular and oscillating M ^X /G/1/n systems.     

DYNAMICS OF A NONLINEAR NON-AUTONOMOUS n-PATCHES PREDATOR-PREY DISPERSION-DELAY MODEL

In this paper,a nonlinear nonautonomous predator-prey dispersion model with continuous distributed delay is studied,where all parameters are time-dependent.In this system consisting of n-patches the prey species can disperse among n-patches,but the predator species is confined to one patch and cannot disperse.It is proved that the system is uniformly persistent under any dispersion rate effect.Furthermore,some sufficient conditions are established for the existence of a unique almost periodic solution of the system.The example shows that the criteria in the paper are new,general and easily verifiable.     

<Emphasis Type="Italic">m</Emphasis> Components-admissible solutions of systems of higher-order partial differential equations on <Emphasis Type="Italic">C</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Using value distribution theory and techniques in several complex variables,we investigate the problem of existence of m components-admissible solutions of a class of systems of higher-order partial differential equations in several complex variables and estimate the number of admissible components of solutions.Some related results will also be obtained.     

An efficient algorithm for the generalized centro-symmetric solution of matrix equation <Emphasis Type="Italic">A X B</Emphasis> = <Emphasis Type="Italic">C</Emphasis>

In this paper, an iterative algorithm is constructed for solving linear matrix equation AXB = C over generalized centro-symmetric matrix X. We show that, by this algorithm, a solution or the least-norm solution of the matrix equation AXB = C can be obtained within finite iteration steps in the absence of roundoff errors; we also obtain the optimal approximation solution to a given matrix X ₀ in the solution set of which. In addition, given numerical examples show that the iterative method is efficient.     

Algebraic convergence theorems of <Emphasis Type="Italic">n</Emphasis>-dimensional Kleinian groups

Let {G _r,i } be a sequence of r-generator n-dimensional Kleinian groups and G _r be its algebraic limit group. In this paper, we prove that G _r is a Kleinian group if {G _{r, i} } satisfies some conditions. Our results are generalizations of the corresponding known ones. The research was partly supported by Program for NCET (No. 04-0783) and NSFs of China (No. 10771059, No. 10571048) and Hunan (No. 05JJ10001).