Approximation of solutions of generalized equations of Hammerstein type

Let $H$ be a real Hilbert space. For each $i=1,2,\dots ,m$, let $F_i,K_i:H\to H$ be bounded and monotone mappings. Assume that the generalized Hammerstein equation $u+{\sum }_{i=1}^m{K}_i{F}_{i}u=0$ has a solution in $H$. We construct a new explicit iterative sequence and prove strong convergence of the sequence to a solution of the generalized Hammerstein equation.     

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.     

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.     

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.     

Trees with minimal Laplacian coefficients

Let $G$ be a simple undirected graph with the characteristic polynomial of its Laplacian matrix $L\left(G\right)$, $P(G,\mu )={\sum }_{k=0}^n{(?1)}^k{c}_k{\mu }^{n?k}$. It is well known that for trees the Laplacian coefficient $c_{n?2}$ is equal to the Wiener index of $G$, while $c_{n?3}$ is equal to the modified hyper-Wiener index of the graph. In this paper, we characterize $n$-vertex trees with given matching number $m$ which simultaneously minimize all Laplacian coefficients. The extremal tree $A(n,m)$ is a spur, obtained from the star graph $S_{n?m+1}$ with $n?m+1$ vertices by attaching a pendant edge to each of certain $m?1$ non-central vertices of $S_{n?m+1}$. In particular, $A(n,m)$ minimizes the Wiener index, the modified hyper-Wiener index and the recently introduced Incidence energy of trees, defined as $IE\left(G\right)={\sum }_{k=0}^n\sqrt{{\mu }_k}$, where ${\mu }_k$ are the eigenvalues of signless Laplacian matrix $Q\left(G\right)=D\left(G\right)+A\left(G\right)$. We introduced a general $\rho$ transformation which decreases all Laplacian coefficients simultaneously. In conclusion, we illustrate on examples of Wiener index and Incidence energy that the opposite problem of simultaneously maximizing all Laplacian coefficients has no solution.