On Sums of a Prime, and a Square of Prime, and a κ-power of Prime

In 1923, Hardy and Littlewood[1] conjectured that each integer n can be written asp+m12+ m22 = n,and Linnik[2,3] proved that this conjecture is true. But if these mi with i = 1,2 are restricted to primes Pi, the corresponding result is out of reach at present. We consider the following Diophantine equation     

q-Hermite Polynomials and a q-Beta Integral

The main object of the present paper is to derive a formula for the q-Hahn.polynomials by using the q-Mehler formula about the q-Hermite polynomials.This result, together with other properties of the q-Hermite polynomials, provides an evalution of a q-beta integral.Both the celebrated Askey-Wilson integral and the Ismail-Stanton-Viennot integral are special cases of this integral.     

Sperner’s lemma and zero point theorems on a discrete simplex and a discrete simplotope

We show two discrete zero point theorems that are derived from Sperner’s lemma and a Sperner-like theorem (van der Laan and Talman [Math. Oper. Res. (1982)] [5]; Freund [Math. Oper. Res. (1986)]) [3]. Applications to economic and game models are also presented.     

On a Result of Pandey and Bhargava

Let S denote the class of functions f(z) which are analytic in the unit disc D={z:|z|<1} and normalized by the conditions f(0)=0=f′(0)-1. Let K(μ)and S (m,M ) be the subclasses of S and satisfying the conditions     

Modules with a“nice”endomorphism ring and a new characterization of semisimple modules and rings

In this work we introduce some classes of modules whose endomorphism rings have some of the properties of the endomorphism rings of vector spaces. Then we apply these notions to obtain new characterizations of semisimple rings and modules.     

Modelling ‘contracts’ between a terrorist group and a government in a sequential game

In this paper, we apply a sequential game to study the possibility of ‘contracts’ (or at least mutually beneficial arrangements) between a government and a terrorist group. We find equilibrium solutions for complete and incomplete information models, where the government defends and/or provides positive rent, and the terrorist group attacks. We also study the sensitivities of equilibria as a function of both players’ target valuations and preferences for rent. The contract option, if successful, may achieve (partial) attack deterrence, and significantly increase the payoffs not only for the government, but also for some types of terrorist groups. Our work thus provides some novel insights in combating terrorism.     

The Intersection of a Triangular Bézier Patch and a Plane

In this paper, the problem of finding the intersection of a triangular Bézier patch and a plane is studied. For the degree that one frequently encounters in practice, i.e. $n=2,3$, an efficient and reliable algorithm is obtained, and computational steps are presented.     

Hopf bifurcations in a predator–prey system with a discrete delay and a distributed delay

A delayed Lotka–Volterra two-species predator–prey system with discrete hunting delay and distributed maturation delay for the predator population described by an integral with a strong delay kernel is considered. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the asymptotic stability of the positive equilibrium is investigated and Hopf bifurcations are demonstrated. It is found that under suitable conditions on the parameters the positive equilibrium is asymptotically stable when the hunting delay is less than a certain critical value and unstable when the hunting delay is greater than this critical value. Meanwhile, according to the Hopf bifurcation theorem for functional differential equations (FDEs), we find that the system can also undergo a Hopf bifurcation of nonconstant periodic solution at the positive equilibrium when the hunting delay crosses through a sequence of critical values. In particular, by applying the normal form theory and the center manifold reduction for FDEs, an explicit algorithm determining the direction of Hopf bifurcations and the stability of bifurcating periodic solutions occurring through Hopf bifurcations is given. Finally, to verify our theoretical predictions, some numerical simulations are also included at the end of this paper.     

Stokes and Navier–Stokes problems in a half-space: the existence and uniqueness of solutions a priori nonconvergent to a limit at infinity

The Cauchy problem and the initial boundary value problem in the half-space of the Stokes and Navier–Stokes equations are studied. The existence and uniqueness of classical solutions (u, π) (considered at least C ² × C ¹ smooth with respect to the space variable and C ¹ × C ⁰ smooth with respect to the time variable) without requiring convergence at infinity are proved. A priori the fields u and π are nondecreasing at infinity. In the case of the Stokes problem, the existence, for any t > 0, and the uniqueness of solutions with kinetic field and pressure field are established for some β ∈ (0, 1) and γ ∈ (0, 1 − β). In the case of Navier–Stokes equations, the existence (local in time) and the uniqueness of classical solutions to the Navier–Stokes equations are shown under the assumption that the initial data are only continuous and bounded, by proving that, for any t ∈ (0, T), the kinetic field u(x, t) is bounded and, for any γ ∈ (0, 1), the pressure field π(x, t) is O(1 + |x| ^γ ). Bibliography: 20 titles. To V. A. Solonnikov on his 75th birthday Published in Zapiski Nauchnykh Seminarov POMI, Vol. 362, 2008, pp. 176–240.     

Minimal D-splitting and SD-splitting for a Compression Body

In this paper, we fix the genus of SD-splittings of a compression body. We also discuss the uniqueness for minimal D-splittings and SD-splittings for a han-dlebody.     

On the Norm of a Self-Adjoint Operator and a New Bilinear Integral Inequality

In this paper, the expression of the norm of a self-adjoint integral operator T ： L^2（0, ∞） → L^2 （0, ∞） is obtained. As applications, a new bilinear integral inequality with a best constant factor is established and some particular cases are considered.     

Bifurcation and control of a bioeconomic model of a prey–predator system with a time delay

In this paper, we analyze the dynamical behaviour of a bioeconomic model system using differential algebraic equations. The system describes a prey–predator fishery with prey dispersal in a two-patch environment, one of which is a free fishing zone and other is a protected zone. It is observed that a singularity-induced bifurcation phenomenon appears when a variation of the economic interest of harvesting is taken into account. We have incorporated a state feedback controller to stabilize the model system in the case of positive economic interest. A discrete-type gestational delay of predators is incorporated, and its effect on the dynamical behaviour of the model is analyzed. The occurrence of Hopf bifurcation of the proposed model with positive economic profit is shown in the neighbourhood of the coexisting equilibrium point through considering the delay as a bifurcation parameter. Finally, some numerical simulations are given to verify the analytical results, and the system is analyzed through graphical illustrations.     

Turán Number of the Family Consisting of a Blow-up of a Cycle and a Blow-up of a Star

Let F={H₁,...,H_k}(k> 1) be a family of graphs.The Turán number of the family F is the maximum number of edges in an n-vertex {H₁,...,H_k)-free graph,denoted by ex(n,F) or ex(n,{H₁,H₂,...,H_k}).The blow-up of a graph H is the graph obtained from H by replacing each edge in H by a clique of the same size where the new vertices of the cliques are all different.In this paper we determine the Turán number of the family cons...     

On Sums of Two Squares of Primes and a Cube of a Prime

In this paper we are concerned with the exceptional set of the sum of two squares of primesand a cube of a prime P;+p;+p;.Noting that竹三1 or 3(mod 6)is a necessary conditionfor the solvability of the equation n=P}+P;+P;(see【1]),we define E(N)=Card{n:n≤N,礼∈三and n≠P;+Pi+P;for any Pi,1≤i s 3), (1)where三={n:n三1 or 3(mod 6)). This and similar problems have been studied by a number of authors.In 1937,Davenportand Heilbronn[2J proved that if后2 3 is an odd integer then almost all posi…     

Jacobi–Hecke algebras and a rationality theorem for a formal Hecke series

It is well known that the L-function associated to a Siegel eigenform f is equal to a Rankin-Selberg type zeta-integral involving f and a restricted Eisenstein series ([3], [14]). At some point in the proof one has to show the equality of a certain Dirichlet series and the L-function, which follows from a rationality theorem for a certain formal power series over the Hecke algebra. The main purpose of this paper is to develop a Hecke theory for the Jacobi group and to prove such a rationality theorem. Received: 17 August 1998 / Revised version: 17 February 1999     

On a Problem of Dobrowolski and Williams and the Pólya-Vinogradov Inequality

In [2] E. Dobrowolski and K.S. Williams considered a problem of obtaining estimates for the sum _n=a+1 ^a+N f(n),for a certain class of functions f. One specific application of their result is a new method for estimating character sums. In particular, they obtain a form of the Pólya-Vinogradov inequality with the constant 1/(2 log 2). In this note we improve their estimates and obtain, in particular, a form of the Pólya-Vinogradov inequality with the constant 1/(3 log 3). A nice feature of our estimate is that it is obtained by a very simple argument.     

Periodic solutions of Liénard equation with a singularity and a deviating argument

In this paper, we study the existence of periodic solutions of the Liénard equation with a singularity and a deviating argument $x^{″}+f\left(x\right)x^{\prime }+g(t,x(t-\sigma ))=0\text{.}$ When $g$ has a strong singularity at $x=0$ and satisfies a new small force condition at $x=\infty$, we prove that the given equation has at least one positive $T$-periodic solution.