The second kind Chebyshev–Newton–Cotes quadrature rule (open type) and its numerical improvement

The weighted Newton–Cotes quadrature rules of open type are denoted by

where w(x) is a positive function and is the step size. Various cases can be selected for the weight function of the above formula. In this paper, we consider as the main weight function and study the general formula:

The precision degree of the above formula is n + 1 for even n’s and is n for odd n’s but if one considers its upper and lower bounds as two additional variables, a nonlinear system will be derived whose solution improves the precision degree of above formula up to degree n + 2 numerically. In this way, some examples are given to show the numerical superiority of our idea.     

Existence of slip and stick periodic motions in a non-smooth dynamical system

The paper studies the existence of slip and stick periodic solutions to a second order equation with a discontinuous right-hand side,

     

A new sharpened version of the Neuberg–Pedoe inequality

In this work, we present a new sharpened version of the classical Neuberg–Pedoe inequality. As an application, the following improved Neuberg–Pedoe inequality is derived:

     

Limit behavior of global attractors for the complex Ginzburg–Landau equation on infinite lattices

In this work, the authors first show the existence of global attractors for the following lattice complex Ginzburg–Landau equation:

and for the following lattice Schrödinger equation:

Then they prove that the solutions of the lattice complex Ginzburg–Landau equation converge to that of the lattice Schrödinger equation as ε→0+. Also they prove the upper semicontinuity of as ε→0+ in the sense that .     

Asymptotic analysis of a perturbation problem

An asymptotic expansion is constructed for the solution of the initial-value problem

when t is restricted to the interval [0,T/ε], where T is any given number. Our analysis is mathematically rigorous; that is, we show that the difference between the true solution u(t,x;ε) and the Nth partial sum of the asymptotic series is bounded by ε^N+1 multiplied by a constant depending on T but not on x and t.     

Positive solutions of nonlinear fractional boundary value problems using Adomian decomposition method

We use Adomian decomposition method for solving the fractional nonlinear two-point boundary value problem

where D is Caputo fractional derivative, c is a constant, μ > 0, and F:[0,1]×[0,∞)→[0,∞) a continuous function. The fractional Bratu problem is solved as an illustrative example.     

A Chebyshev type inequality for fuzzy integrals

In this paper, we prove a Chebyshev type inequality for fuzzy integrals. More precisely, we show that:

where μ is the Lebesgue measure on and f,g:[0,1]→[0,∞) are two continuous and strictly monotone functions, both increasing or both decreasing. Also, some examples and applications are presented.     

On a class of fractal functions with graph Hausdorff dimension 2

We prove that the graph of the continuous function

has Hausdorff dimension 2, where λ > 1, β >  > 1, (x) = 2x, 0  x  1/2, (−x) = (x) and (x + 1) = (x).     

The oscillation of higher-ordernonlinear difference equations

The authors discuss the relation of the oscillation of the following two difference equations,

where m ≥ 2, τ : N → N, N isthe set of integers, |n − τ(n)| ≤ Mfor n N₀, M is a positive integer, is nondecreasing in x, xf(n, x)> 0, as x ≠ 0. Wewill show some relations of the oscillation of the above two equations. Especially, for m to be even, we establish the equivalenceof the oscillation of the above two difference equations.     

Existence and global attractivity of positive periodic solutions of functional differential equations with feedback control

Sufficient conditions are obtained for the existence and global attractivity of positive periodic solutions of the delay differential system with feedback control

The method involves the application of Krasnoselskii's fixed point theorem and estimates of uniform upper and lower bounds of solutions. When these results are applied to some special delay population models with multiple delays, some new results are obtained and some known results are generalized.     

Coding for a multiple access OR channel: A survey

The signature coding for M active users out of T total users over a multiple access OR channel is considered. The mathematical problem is equivalent to the M-cover-free problem of extremal set theory. We survey the upper and lower bounds on the minimal code word length n(T,M), and present some code constructions. According to the current state of the theory, for 1MT

so there is a huge gap between the upper and lower bounds. Moreover, there is no known construction approaching the upper bound.     

On the relation between interior critical points and parameters for a class of nonlinear problems with Neumann–Robin boundary conditions

We consider boundary value problem

where   0, λ > 0 are parameters and f  C²[0, ∞) such that f(0) < 0. In this paper we study for the cases p  (0, β) and p  (β, θ) (p is the value of the solution at x = 0 and β, θ are such that f(β) = 0, , the relation between λ and the number of interior critical points of the positive solutions of the above system.     

Global existence and blow-up solutions and blow-up estimates for a non-local quasilinear degenerate parabolic system

This paper deals with p-Laplacian systems

with null Dirichlet boundary conditions in a smooth bounded domain ΩR^N, where p,q>1, , and a,b>0 are positive constants. We first get the non-existence result for a related elliptic systems of non-increasing positive solutions. Secondly by using this non-existence result, blow-up estimates for above p-Laplacian systems with the homogeneous Dirichlet boundary value conditions are obtained under Ω=B_R={xR^N:|x|<R}(R>0). Then under appropriate hypotheses, we establish local theory of the solutions and obtain that the solutions either exists globally or blow-up in finite time.     

Numerical solution of Hammerstein integral equations by using combination of spline-collocation method and Lagrange interpolation

The numerical solutions to the nonlinear integral equations of the Hammerstein-type:

with using B-splines functions are investigated. An interpolation method based on B-splines functions combined with a new collocation method is presented. Finally, for showing efficiency of the method we give some numerical examples.     

Stability properties of non-negative solutions to a non-autonomous p-Laplacian equation

We study the stability of non-negative stationary solutions of

where Δ_p denotes the p-Laplacian operator defined by Δ_pz = div(z^p−2z); p > 2, Ω is a bounded domain in R^N(N  1) with smooth boundary where [0,1],h:∂Ω→R⁺ with h = 1 when  = 1, λ > 0, and g:Ω×[0,∞)→R is a continuous function. If g(x, u)/u^p−1 be strictly increasing (decreasing), we provide a simple proof to establish that every non-trivial non-negative solution is unstable (stable).     

Existence and global attractivity of a periodic solution for a prey–predator model with sex-structure

In this paper, we investigate the following periodic sex-structure model:

The sufficient and realistic conditions are obtained for the existence of positive periodic solution of above system. Further, by constructing a V functional and using the result of the existence of positive periodic solutions, the global attractivity of a positive periodic solution for above system is obtained.     

Oscillation criteria for a class of nonlinear system of differential equations

Using the integral average method, we establish some oscillation criteria of Kamenev type and Yan type for the nonlinear system of differential equation

where the functions b_i(t) (i = 1, 2) are nonnegative and summable on each finite segment of the interval Z0, ∞), λ_i > 0 (i = 1,2) with λ₁ λ₂ = 1.     

Generalizing the Quinn–Wójs theorem on distinct multiplets of composite Fermions

The combinatorial tool of generating functions for restricted partitions is used to generalize a quantum physics theorem relating distinct multiplets of different angular momenta in the composite Fermion model of the fractional quantum Hall effect. Specifically, if g_ℓ(N,M) denotes the number of distinct multiplets of angular momentum ℓ and total angular momentum M, we prove that

where the sum is taken over all positive divisors of N and L(k)=kℓ-kN/2+3k/2-N+N/(2k)-1/2. The original Quinn–Wójs theorem results when k=1 and it appears that this generalization will be useful in further investigations of nuclear shells modeling elementary particle interactions when the particles are clustered together.     

On a singular perturbation problem with two second-order turning points

In this paper, we study the singular perturbation problem

where 0<ε1 is a small positive parameter, p(x) and q(x) are sufficiently smooth and strictly positive functions. The main feature of this equation is that there are two second-order turning points in the interval (0,1). Based on the rigorous results on singular perturbation problems with one second-order turning point in our previous work, we obtain a uniform asymptotic approximation for the general solution of the above equation by means of a matching technique.     

Almost periodic solutions of differential equations with piecewise constant argument of generalized type

We consider the existence and stability of an almost periodic solution of the following hybrid system:

(1)

where if θ_i≤t<θ_i+1,i=…−2,−1,0,1,2,…, is an identification function, θ_i is a strictly ordered sequence of real numbers, unbounded on the left and on the right, p_j,j=1,2,…,m, are fixed integers, and the linear homogeneous system associated with (1) satisfies exponential dichotomy. The deviations of the argument are not restricted by any sign assumption when existence is considered. A new technique of investigation of equations with piecewise argument, based on integral representation, is developed.