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.     

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.     

Innovation based on Gaussian elimination to compute generalized inverse AT,S(2)

In this paper, we execute elementary row and column operations on the partitioned matrix $\left(\begin{array}{cc}\hfill GAG\hfill & \hfill G\hfill \\ \hfill G\hfill & \hfill 0\hfill \end{array}\right)$ into $\left(\begin{array}{cc}\hfill \left(\begin{array}{cc}\hfill I_s\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill ?A_{T,S}^{\left(2\right)}\hfill \end{array}\right)$to compute generalized inverse $A_{T,S}^{\left(2\right)}$ of a given complex matrix $A$, where $G$ is a matrix such that $R\left(G\right)=T$ and $N\left(G\right)=S$. The total number of multiplications and divisions operations is $T(m,n,s)=2m{n}^2+\frac{4m?s?1}{2}ns+(m?s)ns+mns$ and the upper bound of $T(m,n,s)$ is less than $6m{n}^2?\frac{3}{2}{n}^3?\frac{1}{2}{n}^2$ when $n\le m$. A numerical example is shown to illustrate that this method is correct.     

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.     

Global bifurcation of positive radial solutions for an elliptic system in reactor dynamics

In this paper, we are concerned with the existence of positive radial solutions of the elliptic system $\{\begin{array}{c}?\Delta u=uv?\lambda u+f(\left|x\right|,u)\text{,}{R}_1<\left|x\right|<R_2\text{,}x\in {\mathbb{R}}^N,N\ge 1\text{,}\hfill \\ ?\Delta v=\mu u\text{,}{R}_1<\left|x\right|<R_2\text{,}x\in {\mathbb{R}}^N,N\ge 1\text{,}\hfill \\ u=v=0\text{,}\text{on }\left|x\right|=R_1\text{ and }\left|x\right|=R_2\text{,}\hfill \end{array}$ where $\left|x\right|={\left({\sum }_{i=1}^N{x}_i^2\right)}^{\frac{1}{2}}$, $\lambda >0$ is a constant, $\mu >0$ is a parameter and $0<R_1<R_2<\infty$, $f:[R_1,R_2]\times [0,\infty )\to [0,\infty )$ is continuous and $f(t,s)>0$ for all $(t,s)\in [R_1,R_2]\times (0,\infty )$. Under some appropriate conditions on the nonlinearity $f$, we show that the above system possesses at least one positive radial solution for any $\mu \in (0,\infty )$. The proof of our main results is based upon bifurcation techniques.     

State complexity of power

The number of states in a deterministic finite automaton (DFA) recognizing the language $L^k$, where $L$ is regular language recognized by an $n$-state DFA, and $k?2$ is a constant, is shown to be at most $n{2}^{(k?1)n}$ and at least $(n?k)2^{(k?1)(n?k)}$ in the worst case, for every $n>k$ and for every alphabet of at least six letters. Thus, the state complexity of $L^k$ is $\Theta \left(n{2}^{(k?1)n}\right)$. In the case $k=3$ the corresponding state complexity function for $L^3$ is determined as $\frac{6n?3}{8}{4}^n?(n?1)2^n?n$ with the lower bound witnessed by automata over a four-letter alphabet. The nondeterministic state complexity of $L^k$ is demonstrated to be $nk$. This bound is shown to be tight over a two-letter alphabet.     

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.