Resolvable 4-cycle group divisible designs with two associate classes: Part size even

Let ${\lambda }_1{K}_a$ denote the graph on a vertices with ${\lambda }_1$ edges between every pair of vertices. Take p copies of this graph ${\lambda }_1{K}_a$, and join each pair of vertices in different copies with ${\lambda }_2$ edges. The resulting graph is denoted by $K(a,p;{\lambda }_1,{\lambda }_2)$, a graph that was of particular interest to Bose and Shimamoto in their study of group divisible designs with two associate classes. The existence of z-cycle decompositions of this graph have been found when $z\in \{3,4\}$. In this paper we consider resolvable decompositions, finding necessary and sufficient conditions for a 4-cycle factorization of $K(a,p;{\lambda }_1,{\lambda }_2)$ (when ${\lambda }_1$ is even) or of $K(a,p;{\lambda }_1,{\lambda }_2)$ minus a 1-factor (when ${\lambda }_1$ is odd) whenever a is even.     

Infinite Fujita exponents for nonlinear diffusion equations with localized sources

This paper deals with Cauchy problem to nonlinear diffusion $u_t=\Delta u^m+{\lambda }_1{u}^{p_1}(x,t)+{\lambda }_2{u}^{p_2}(x^1\left(t\right),t)$ with $m\ge 1$, $p_i,{\lambda }_i\ge 0$ ($i=1,2$) and $x^1\left(t\right)$ Hölder continuous. A new phenomenon is observed that the critical Fujita exponent $p_c=+\infty$ whenever ${\lambda }_2>0$. More precisely, the solution blows up under any nontrivial and nonnegative initial data for all $p=max\{p_1,p_2\}\in (1,+\infty )$. This result is then extended to a coupled system with localized sources as well as the cases with other nonlinearities.     

On uniqueness of stationary solutions to the Navier–Stokes equations in exterior domains

The aim of this paper is to prove a uniqueness criterion for solutions to the stationary Navier–Stokes equation in 3-dimensional exterior domains within the class $u\in L_{3,\infty }$ with $?u\in L_{3/2,\infty }$, where $L_{3,\infty }$ and $L_{3/2,\infty }$ are the Lorentz spaces. Our criterion asserts that if $u$ and $v$ are the solutions, $u$ is small in $L_{3,\infty }$ and $u,v\in L_p$ for some $p>3$, then $u=v$. The proof is based on analysis of the dual equation with the aid of the bootstrap argument.     

The zero-one law for a complete orthonormal system

A complete orthonormal system of functions $\Theta ={\left\{{\theta }_n\right\}}_{n=1}^{\infty }\text{,}{\theta }_n\in L_{[0\text{,}1]}^{\infty }$ is constructed such that ${\sum }_{n=1}^{\infty }{a}_n{\theta }_n$ converges almost everywhere on $[0\text{,}1]$ if ${\left\{a_n\right\}}_{n=1}^{\infty }\in l^2$ and ${\sum }_{n=1}^{\infty }{a}_n{\theta }_n$ diverges a.e. for any ${\left\{a_n\right\}}_{n=1}^{\infty }?l^2$. We also show that for any complete ONS ${\left\{f_n\right\}}_{n=1}^{\infty }$ of functions defined on $[0\text{,}1]$ there exists a fixed non decreasing subsequence ${\left\{n_k\right\}}_{k=1}^{\infty }$ of natural numbers such that for any $f\in L_{[0\text{,}1]}^0$ and some sequence of coefficients ${\left\{b_n\right\}}_{n=1}^{\infty }$,

$\sum _{n=1}^{n_k}{b}_n{f}_n\to f\text{a.e. when}k\to \infty \text{.}$

To cite this article: K. Kazarian, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Ramsey numbers of odd cycles versus larger even wheels

The generalized Ramsey number $R(G_1,G_2)$ is the smallest positive integer $N$ such that any red–blue coloring of the edges of the complete graph $K_N$ either contains a red copy of $G_1$ or a blue copy of $G_2$. Let $C_m$ denote a cycle of length $m$ and $W_n$ denote a wheel with $n+1$ vertices. In 2014, Zhang, Zhang and Chen determined many of the Ramsey numbers $R(C_{2k+1},W_n)$ of odd cycles versus larger wheels, leaving open the particular case where $n=2j$ is even and $k<j<3k∕2$. They conjectured that for these values of $j$ and $k$, $R(C_{2k+1},W_{2j})=4j+1$. In 2015, Sanhueza-Matamala confirmed this conjecture asymptotically, showing that $R(C_{2k+1},W_{2j})\le 4j+334$. In this paper, we prove the conjecture of Zhang, Zhang and Chen for almost all of the remaining cases. In particular, we prove that $R(C_{2k+1},W_{2j})=4j+1$ if $j?k\ge 251$, $k<j<3k∕2$, and $j\ge 212299$.     

A ratio-dependent predator–prey model with diffusion

This paper deals with the existence and nonexistence of nonconstant positive steady-state solutions to a ratio-dependent predator–prey model with diffusion and with the homogeneous Neumann boundary condition. We demonstrate that there exists $a_0(b)$ satisfying $0<a_0(b)<m_1$ for $0<b<m_1$, such that if $0<b<m_1$ and $a_0(b)<a<m_1$, then the diffusion can create nonconstant positive steady-state solutions; whereas the diffusion cannot do provided $a>m_1$.