Even-order linear dynamic equations with mixed derivatives

We consider the time scale $k$th-order differential operators $D_k^{\Delta }y≔\{\begin{array}{cc}{y}^{\Delta \nabla \dots \Delta \nabla }& k\text{ even ,}\\ y^{\Delta \nabla \dots \nabla \Delta }& k\text{ odd ,}\end{array}{\tilde{D}}_k^{\Delta }y≔\{\begin{array}{cc}{y}^{\nabla \Delta \dots \nabla \Delta }& k\text{ even ,}\\ y^{\Delta \nabla \dots \nabla \Delta }& k\text{ odd ,}\end{array}$$D_k^{\nabla }y≔\{\begin{array}{cc}{y}^{\nabla \Delta \dots \nabla \Delta }& k\text{ even ,}\\ y^{\nabla \Delta \dots \Delta \nabla }& k\text{ odd ,}\end{array}{\tilde{D}}_k^{\nabla }y≔\{\begin{array}{cc}{y}^{\Delta \nabla \dots \Delta \nabla }& k\text{ even ,}\\ y^{\nabla \Delta \dots \Delta \nabla }& k\text{ odd ,}\end{array}$ and the higher-order dynamic equations $L\left(y\right)≔\sum _{\nu =0}^n{(-1)}^{\nu }{\tilde{D}}_{\nu }^{\nabla }\left(r_{\nu }\left(t\right)D_{\nu }^{\Delta }y\right)=0\text{,}$$M\left(y\right)≔\sum _{\nu =0}^n{(-1)}^{\nu }{\tilde{D}}_{\nu }^{\Delta }\left(r_{\nu }\left(t\right)D_{\nu }^{\nabla }y\right)=0\text{.}$ We will show that these equations can be investigated as special cases of the so-called (delta or nabla) symplectic dynamic systems $z^{\Delta }=S\left(t\right)z\text{,}{z}^{\nabla }=S\left(t\right)z\text{,}$ whose qualitative theory is well developed. We also suggest further perspectives of the investigation of the qualitative properties of higher-order equations with mixed derivatives.     

Periodic conjugate direction algorithm for symmetric periodic solutions of general coupled periodic matrix equations

Analysis and design of linear periodic control systems are closely related to the periodic matrix equations. The conjugate direction (CD) method is a famous iterative algorithm to find the solution to nonsymmetric linear systems $Ax=b$. In this work, a new method based on the CD method is proposed for computing the symmetric periodic solutions $(X_1,X_2,\dots ,X_{\lambda })$ and $(Y_1,Y_2,\dots ,Y_{\lambda })$ of general coupled periodic matrix equations

for $i=1,2,\dots ,\lambda$. The key idea of the scheme is to extend the CD method by means of Kronecker product and vectorization operator. In order to assess the convergence properties of the method, some theoretical results are given. Finally two numerical examples are included to illustrate the efficiency and effectiveness of the method.     

Suslin’s algorithms for reduction of unimodular rows

A well-known lemma of Suslin says that for a commutative ring $A$ if $(v_1\left(X\right),\dots ,v_n\left(X\right))\in {\left(A\left[X\right]\right)}^n$ is unimodular where $v_1$ is monic and $n\ge 3$, then there exist ${\gamma }_1,\dots ,{\gamma }_{\ell }\in E_{n-1}\left(A\left[X\right]\right)$ such that the ideal generated by $\mathrm{Res}(v_1,e_1.{\gamma }_1{}^t(v_2,\dots ,v_n)),\dots ,\mathrm{Res}(v_1,e_1.{\gamma }_{\ell }{}^t(v_2,\dots ,v_n))$ equals $A$. This lemma played a central role in the resolution of Serre’s Conjecture. In the case where $A$ contains a set $E$ of cardinality greater than $deg{v}_1+1$ such that $y-y^{\prime }$ is invertible for each $y\ne y^{\prime }$ in $E$, we prove that the ${\gamma }_i$ can simply correspond to the elementary operations $L_1\to L_1+y_i{\sum }_{j=2}^{n-1}{u}_{j+1}{L}_j$, $1\le i\le \ell =deg{v}_1+1$, where $u_1{v}_1+\cdots +u_n{v}_n=1$. These efficient elementary operations enable us to give new and simple algorithms for reducing unimodular rows with entries in $K[X_1,\dots ,X_k]$ to ${}^t(1,0,\dots ,0)$ using elementary operations in the case where $K$ is an infinite field. Another feature of this paper is that it shows that the concrete local–global principles can produce competitive complexity bounds.     

Blow-up analysis in quasilinear reaction–diffusion problems with weighted nonlocal source

In this paper, we consider the blow-up of solutions to a class of quasilinear reaction–diffusion problems

where $\Omega$ is a bounded convex domain in ${\mathbb{R}}^n(n\ge 2)$, weighted nonlocal source satisfies $a\left(x\right)f\left(u(x,t)\right)\le a_1+a_2{\left(u(x,t)\right)}^p{\left({\int }_{\Omega }{\left(u(x,t)\right)}^l\mathrm{d}x\right)}^m,$ and $a_1,a_2,p,l,$ and $m$ are positive constants. By utilizing a differential inequality technique and maximum principles, we establish conditions to guarantee that the solution remains global or blows up in a finite time. Moreover, an upper and a lower bound for blow-up time are derived. Furthermore, two examples are given to illustrate the applications of obtained results.     

Relations between matrix sets generated from linear matrix expressions and their applications

Let $A+BXC$ and $A+BX+YC$ be two linear matrix expressions, and denote by $\{A+BXC\}$ and $\{A+BX+YC\}$ the collections of the two matrix expressions when $X$ and $Y$ run over the corresponding matrix spaces. In this paper, we study relationships between the two matrix sets $\{A_1+B_1{X}_1{C}_1\}$ and $\{A_2+B_2{X}_2{C}_2\}$, as well as the two sets $\{A_1+B_1{X}_1+Y_1{C}_1\}$ and $\{A_2+B_2{X}_2+Y_2{C}_2\}$, by using some rank formulas for matrices. In particular, we give necessary and sufficient conditions for the two matrix set inclusions $\{A_1+B_1{X}_1{C}_1\}?\{A_2+B_2{X}_2{C}_2\}$ and $\{A_1+B_1{X}_1+Y_1{C}_1\}?\{A_2+B_2{X}_2+Y_2{C}_2\}$ to hold. We also use the results obtained to characterize relations of solutions of some linear matrix equations.     

Existence of positive ground state solutions for a critical Kirchhoff type problem with sign-changing potential

In this work, we are interested in studying the following Kirchhoff type problem

where $\Omega ?{\mathbb{R}}^N(N\ge 3)$ is a smooth bounded domain, $2^1=\frac{2N}{N?2}$ is the critical Sobolev exponent, $0<q<1,\lambda >0$, and $f\in L^{\infty }\left(\Omega \right)$ with the set $\{x\in \Omega :f\left(x\right)>0\}$ of positive measures, and $g\in L^{\infty }\left(\Omega \right)$ with $g\left(x\right)\ge 0,g?0.$ By the Nehari method and variational method, the existence of positive ground state solutions is obtained.     

A new approach toward stabilization in a two-species chemotaxis model with logistic source

This paper aims at providing an alternative approach to study global dynamic properties for a two-species chemotaxis model, with the main novelty being that both populations mutually compete with the other on account of the Lotka–Volterra dynamics. More precisely, we consider the following Neumann initial–boundary value problem

in a bounded domain $\Omega ?{\mathbb{R}}^n,n\ge 1$, with smooth boundary, where $d_1,d_2,d_3,{\chi }_1,{\chi }_2,{\mu }_1,{\mu }_2,a_1,a_2$ are positive constants.When $a_1\in (0,1)$ and $a_2\in (0,1)$, it is shown that under some explicit largeness assumptions on the logistic growth coefficients ${\mu }_1$ and ${\mu }_2$, the corresponding Neumann initial–boundary value problem possesses a unique global bounded solution which moreover approaches a unique positive homogeneous steady state $(u^1,v^1,w^1)$ of above system in the large time limit. The respective decay rate of this convergence is shown to be exponential.When $a_1\ge 1$ and $a_2\in (0,1)$, if ${\mu }_2$ is suitable large, for all sufficiently regular nonnegative initial data $u_0$ and $v_0$ with $u_0?0$ and $v_0?0$, the globally bounded solution of above system will stabilize toward $(0,1,1)$ as $t\to \infty$ in algebraic.