An Arc as the Inverse Limit of a Single Bonding Map of Type N on [0,1]

Let I = [0, 1], c ₁, c ₂ ∈ (0, 1) with c ₁ < c ₂ and f : I⊸I be a continuous map satisfying: are both strictly increasing and is strictly decreasing. Let A = {x ∈ [0, c ₁]∣f(x) = x}, a=max A, a ₁ =max(A\{a}), and B = {x ∈ [c ₂, 1]∣f(x) = x}, b=minB, b ₁ =min(B\{b}). Then the inverse limit (I, f) is an arc if and only if one of the following three conditions holds: (1) If c ₁ < f (c ₁) ≤ c ₂ (resp. c ₁ ≤ f (c ₂) < c ₂), then f has a single fixed point, a period two orbit, but no points of period greater than two or f has more than one fixed point but no points of other periods, furthermore, if A≠φ and B≠φ, then f (c ₂) > a (resp. f (c ₁) < b). (2) If f (c ₁) ≤ c ₁ (resp. f (c ₂) ≥ c ₂), then f has more than one fixed point, furthermore, if B≠φ and A\ {a} ≠φ, f (c ₂) ≥ a or if a ₁ < f (c ₂) < a, f ² (c ₂) > f (c ₂), (resp. f has more than one fixed point, furthermore, if A≠φ and B\{b}≠φ, f (c ₁) ≤ b or if b < f (c ₂) < b ₁, f ² (c ₁) < f (c ₁)). (3) If f (c ₁) > c ₂ and f (c ₂) < c ₁, then f has a single fixed point, a single period two orbit lying in I\(u, v) but no points of period greater than two, where u, v ∈ [c ₁, c ₂] such that f (u) = c ₂ and f (v) = c ₁. Supported by the National Natural Science Foundation of China (No. 19961001, No. 60334020) and Outstanding Young Scientist Research Fund. (No. 60125310)     

跳跃非线性项依赖于导数的二阶微分方程周期解的存在性

本文研究二阶微分方程χ"+aχ+-bχ-+f(χ)g(χ'=p(t)周期解的存在性,这里χ+=max{χ,0},χ-=max{-χ,0},a,6是正常数并且点(a,b)位于某一条Fucik谱曲线上.当g(χ)的极限lim.g(χ)=g(+∞),lim g(χ)=g(-∞)和f(χ)的极限lim,g(χ)=f(+∞),lim f(χ)=f(-∞)都存在且有限时,给出了此方程存在周期解的充分条件.     

On periodic solutions of nonlinear integral equations modelling infectious disease on measure chain

The modelling of the spread of infectious disease is carried out for time t on a measure chain T. Our approach unifies the continuous case and the discrete case . The model is described by the integral equation

where x(t) represents the proportion of the population infected at time t, f(t,x(t)) denotes the proportion of the population newly infected per unit time, and τ is the length of time an individual remains infectious. Using the measure chain calculus, we shall develop criteria for the existence of a nontrivial and nonnegative periodic solution for the modelling equation. The criteria can be implemented numerically, for this we shall give an algorithm as well as illustrative examples.     

Computer verification of the Ankeny-Artin-Chowla Conjecture for all primes less than

Let be a prime congruent to 1 modulo 4, and let be rational integers such that is the fundamental unit of the real quadratic field . The Ankeny-Artin-Chowla conjecture (AAC conjecture) asserts that will not divide . This is equivalent to the assertion that will not divide , where denotes the th Bernoulli number. Although first published in 1952, this conjecture still remains unproved today. Indeed, it appears to be most difficult to prove. Even testing the conjecture can be quite challenging because of the size of the numbers ; for example, when , then both and exceed . In 1988 the AAC conjecture was verified by computer for all . In this paper we describe a new technique for testing the AAC conjecture and we provide some results of a computer run of the method for all primes up to .

     

Phonon frequency shift and effect of correlation on the electron-phonon interaction in heavy fermion systems

The electron-phonon interaction in the periodic Anderson model (PAM) is considered. The PAM incorporates the effect of onsite Coulomb interaction (U) between f-electrons. The influence of Coulomb correlation U on the phonon response of the system is studied by evaluating the phonon spectral function for various parameters of the model. The numerical evaluation of the spectral function is carried out in the long wavelength limit at finite temperatures keeping only linear terms in U. The observed behaviour is found to agree well with the general features obtained experimentally for some heavy fermion (HF) systems.     

Heat flow disturbed by periodic array of insulated interface cracks

The thermal effects of an interface crack between two dissimilar half-spaces is considered. The interface cracks are partially or fully insulated, and spaced in a periodic array. Using the complex variable technique, the temperature and fluxes are found in closed form, and the interactions between heat flows due to nearby cracks are determined.     

Periodic solution of a chemostat model with variable yield and impulsive state feedback control

In this paper, a chemostat model with variable yield and impulsive state feedback control is considered. We obtain sufficient conditions of the globally asymptotical stability of the system without impulsive state feedback control. We also obtain that the system with impulsive state feedback control has periodic solution of order one. Sufficient conditions for existence and stability of periodic solution of order one are given. In some cases, it is possible that the system exists periodic solution of order two. Our results show that the control measure is effective and reliable.     

Interpolation on Sparse Grids and Tensor Products of Nikol'skij–Besov Spaces

We investigate the order of convergence of periodic interpolation on sparse grids (blending interpolation) in the framework of tensor products of Nikol'skij–Besov spaces. To this end, we make use of the uniformity of the considered tensor norms and provide a unified approach to error estimates for the interpolation of univariate periodic functions from Nikol'skij–Besov spaces.     

Titchmarsh's asymptotic formula for periodic eigenvalues and an extension to the p-Laplacian

An asymptotic formula for periodic eigenvalues, due to Titchmarsh, is taken to a higher degree of accuracy using a Prüfer transformation method, not depending on Floquet theory. With the idea of the rotation number, this method is then extended to the p-Laplacian.