On parabolic Kazhdan–Lusztig R-polynomials for the symmetric group

Parabolic R-polynomials were introduced by Deodhar as parabolic analogues of ordinary R-polynomials defined by Kazhdan and Lusztig. In this paper, we are concerned with the computation of parabolic R-polynomials for the symmetric group. Let $S_n$ be the symmetric group on $\{1,2,\dots ,n\}$, and let $S=\{s_i|1\le i\le n?1\}$ be the generating set of $S_n$, where for $1\le i\le n?1$, $s_i$ is the adjacent transposition. For a subset $J?S$, let ${(S_n)}_J$ be the parabolic subgroup generated by J, and let ${(S_n)}^J$ be the set of minimal coset representatives for $S_n/{(S_n)}_J$. For $u\le v\in {(S_n)}^J$ in the Bruhat order and $x\in \{q,?1\}$, let $R_{u,v}^{J,x}(q)$ denote the parabolic R-polynomial indexed by u and v. Brenti found a formula for $R_{u,v}^{J,x}(q)$ when $J=S?\{s_i\}$, and obtained an expression for $R_{u,v}^{J,x}(q)$ when $J=S?\{s_{i?1},s_i\}$. In this paper, we provide a formula for $R_{u,v}^{J,x}(q)$, where $J=S?\{s_{i?2},s_{i?1},s_i\}$ and i appears after $i?1$ in v. It should be noted that the condition that i appears after $i?1$ in v is equivalent to that v is a permutation in ${(S_n)}^{S?\{s_{i?2},s_i\}}$. We also pose a conjecture for $R_{u,v}^{J,x}(q)$, where $J=S?\{s_k,s_{k+1},\dots ,s_i\}$ with $1\le k\le i\le n?1$ and v is a permutation in ${(S_n)}^{S?\{s_k,s_i\}}$.     

On a Liouville-type comparison principle for solutions of semilinear elliptic partial differential inequalities

This Note is devoted to the study of a Liouville-type comparison principle for entire weak solutions of semilinear elliptic partial differential inequalities of the form $Lu+{|u|}^{q?1}u?Lv+{|v|}^{q?1}v$, where $q>0$ is a given number and L is a linear (possibly non-uniformly) elliptic partial differential operator of second order in divergent form given formally by the relation

$L=\sum _{i,j=1}^n\frac{?}{?x_i}[a_{ij}(x)\frac{?}{?x_j}].$

We assume that $n?2$, that the coefficients $a_{ij}(x)$, $i,j=1,\dots ,n$, are measurable bounded functions on ${\mathbb{R}}^n$ such that $a_{ij}(x)=a_{ji}(x)$, and that the corresponding quadratic form is non-negative. The results obtained in this work complete similar results on solutions of quasilinear elliptic partial differential inequalities announced in Kurta [C. R. Acad. Sci. Paris, Ser. I 336 (11) (2003) 897–900]. To cite this article: V.V. Kurta, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

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.     

Periodicity in a nonlinear discrete predator–prey system with state dependent delays

With the help of a continuation theorem based on Gaines and Mawhin's coincidence degree, easily verifiable criteria are established for the global existence of positive periodic solutions of the following nonlinear discrete state dependent delays predator–prey systemwhere $a_i,c_j,d_i:Z\to R^{+}$ are positive $\omega$-periodic, $\omega$ is a fixed positive integer. $b_1,b_2:Z\to R^{+}$ are $\omega$-periodic and ${\sum }_{k=0}^{\omega -1}{b}_i(k)>0$. ${\tau }_i,{\sigma }_j,{\rho }_i:Z\times R\times R\to R(i=1,2,\dots ,n,j=1,2,\dots ,m)$ are $\omega$-periodic with respect to their first arguments, respectively. ${\alpha }_i,{\beta }_j,{\gamma }_i(i=1,2,\dots ,n,j=1,2,\dots ,m)$ are positive constants.     

ASYMPTOTIC STABILITY OF RAREFACTION WAVE FOR GENERALIZED BURGERS EQUATION

This paper is concerned with the stability of the rarefaction wave for the Burgers equationwhere 0 ≤ a < 1/4p (q is determined by (2.2)). Roughly speaking, under the assumption that u_ < u , the authors prove the existence of the global smooth solution to the Cauchy problem (I), also find the solution u(x, t) to the Cauchy problem (I) satisfying sup |u(x, t) -uR(x/t)| → 0 as t → ∞, where uR(x/t) is the rarefaction wave of the non-viscous Burgersequation ut f(u)x = 0 with Riemann initial data u(x, 0) =     

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.