On sub-harmonic bifurcations

Under fairly general hypotheses, we investigate by elementary methods the structure of the p-periodic orbits of a family $h_u$ of transformations near $(u_0,x_0)$ when $h_{u_0}(x_0)=x_0$ and $\mathrm{d}{h}_{u_0}(x_0)$ has a simple eigenvalue which is a primitive p-th root of unity. To cite this article: M. Chaperon et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Some asymptotic properties for solutions of one-dimensional advection–diffusion equations with Cauchy data in Lp(R)

We state and discuss a number of fundamental asymptotic properties of solutions $u(?,t)$ to one-dimensional advection–diffusion equations of the form $u_t+f{(u)}_x=(a{(u)u_x)}_x$, $x\in \mathbb{R}$, $t>0$, assuming initial values $u(?,0)=u_0\in L^p(\mathbb{R})$ for some $1?p<\infty$. To cite this article: P. Braz e Silva, P.R. Zingano, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Life-span of classical solutions to one dimensional initial–boundary value problems for general quasilinear wave equations

This paper is devoted to studying the initial–boundary value problem for one dimensional general quasilinear wave equations $u_{tt}?u_{xx}=b(u,Du)u_{xx}+2a_0(u,Du)u_{tx}+F(u,Du)$ on exterior domain. We obtain the sharp lower bound of the life-span of classical solutions to the initial–boundary value problem with small initial data and zero boundary data for one dimensional general quasilinear wave equations.     

Positive solution for system of nonlinear first-order PBVPs on time scales

In this paper we are concerned with the following system of nonlinear first-order periodic boundary value problems on time scale $\mathbb{T}$where $f_i:[0,T]\times [0,+\infty {)}^n\to \mathbb{R}$ is continuous and there exists a constant $M_i>0$ such that$M_i{x}_i-f_i(t,x_1,x_2,\dots ,x_n)⩾0\text{for}(x_1,x_2,\dots ,x_n)\in [0,+\infty {)}^n,t\in [0,T]\text{.}$Some existence criteria of positive solution are established by using a fixed point theorem for operators on cone.     

Dynamics of epidemics in homogeneous/heterogeneous populations and the spreading of multiple inter-related infectious diseases: Constant-sign periodic solutions for the discrete model

We consider the following model that describes the dynamics of epidemics in homogeneous/heterogeneous populations as well as the spreading of multiple inter-related infectious diseases:$u_i(k)=\sum _{\ell =k-{\tau }_i}^{k-1}{g}_i(k,\ell )f_i(\ell ,u_1(\ell ),u_2(\ell ),\dots ,u_n(\ell )),k\in \mathbf{Z},1⩽i⩽n\text{.}$Our aim is to establish criteria such that the above system has one or multiple constant-sign periodic solutions $(u_1,u_2,\dots ,u_n)$, i.e., for each $1⩽i⩽n$, $u_i$ is periodic and ${\theta }_i{u}_i⩾0$ where ${\theta }_i\in \{1,-1\}$ is fixed. Examples are also included to illustrate the results obtained.     

On low-dimensional cancellation problems

A well-known cancellation problem of Zariski asks when, for two given domains (fields) $K_1$ and $K_2$ over a field k, a k-isomorphism of $K_1[t]$ ($K_1(t)$) and $K_2[t]$ ($K_2(t)$) implies a k-isomorphism of $K_1$ and $K_2$. The main results of this article give affirmative answer to the two low-dimensional cases of this problem:1. Let K be an affine field over an algebraically closed field k of any characteristic. Suppose $K(t)?k(t_1,t_2,t_3)$, then $K?k(t_1,t_2)$.2. Let M be a 3-dimensional affine algebraic variety over an algebraically closed field k of any characteristic. Let $A=K[x,y,z,w]/M$ be the coordinate ring of M. Suppose $A[t]?k[x_1,x_2,x_3,x_4]$, then $\mathrm{frac}(A)?k(x_1,x_2,x_3)$, where $\mathrm{frac}(A)$ is the field of fractions of A.In the case of zero characteristic these results were obtained by Kang in [Ming-chang Kang, A note on the birational cancellation problem, J. Pure Appl. Algebra 77 (1992) 141–154; Ming-chang Kang, The cancellation problem, J. Pure Appl. Algebra 47 (1987) 165–171]. However, the case of finite characteristic is first settled in this article, that answered the questions proposed by Kang in [Ming-chang Kang, A note on the birational cancellation problem, J. Pure Appl. Algebra 77 (1992) 141–154; Ming-chang Kang, The cancellation problem, J. Pure Appl. Algebra 47 (1987) 165–171].