In this paper, we improve the result by Harper on the lower bound of the bandwidth of connected graphs. In addition, we prove that considerating the interior boundary and the exterior boundary when estimating the bandwidth of connected graphs gives the same results.
We use the general Riemann approach to define the Stratonovich integral with respect to Brownian motion. Our new definition of Stratonovich integral encompass the classical Stratonovich integral and more importantly, satisfies the ideal Itô formula without the ''tail'' term, that is, f(Wt) = f(W0) + ∫ t 0 f ′ (Ws) ◦ dWs. Further, the condition on the integrands in this paper is weaker than the classical one.
In this paper the definition of Hermite-Hermite matrix polynomials is introduced starting from the Hermite matrix polynomials. An explicit representation, a matrix recurrence relation for the Hermite-Hermite matrix polynomials are given and differential equations satisfied by them is presented. A new expansion of the matrix exponential for a wide class of matrices in terms of Hermite-Hermite matrix polynomials is proposed.
Positioned eco-grammar systems (PEG systems, for short) were introduced in our previous papers. In this paper we engage in a new field of research, the hierarchy of PEG systems, namely in the hierarchy of the PEG systems according to the number of agents presented in the environment and according to the number of types of agents in the system.
We discuss the interior Hölder everywhere regularity for minimizers of quasilinear functionals of the type A(u; Ω) = Z Ω A αβ ij (x,u)Dαu iDβu j dx whose gradients belong to the Morrey space L 2,n−2 (Ω, RnN ).
Let D ' ⊂ Cn−1 be a bounded domain of Lyapunov and f(z ' , zn) a holomorphic function in the cylinder D = D' × Un and continuous on D. If for each fixed a 0 in some set E ⊂ ∂D' , with positive Lebesgue measure mes E > 0, the function f(a ' , zn) of zn can be continued to a function holomorphic on the whole plane with the exception of some finite number (polar set) of singularities, then f(z ' , zn) can be holomorphically continued to (D ' × C) \ S, where S is some analytic (closed pluripolar) subset of D ' × C.
It is shown that every almost linear Pexider mappings $f$, $g$, $h$ from a unital $C^*$-algebra $\mathcal A$ into a unital $C^*$-algebra $\mathcal B$ are homomorphisms when $f(2^n uy)=f(2^n u)f(y)$, $g(2^n uy)=g(2^nu)g(y)$ and $h(2^n uy)=h(2^n u)h(y)$ hold for all unitaries $u \in \mathcal A$, all $y \in \mathcal A$, and all $n\in \mathbb{Z}$, and that every almost linear continuous Pexider mappings $f$, $g$, $h$ from a unital $C^*$-algebra $\mathcal A$ of real rank zero into a unital $C^*$-algebra $\mathcal B$ are homomorphisms when $f(2^n uy)=f(2^n u)f(y)$, $g(2^n uy)=g(2^n u)g(y)$ and $h(2^n uy)=h(2^n u)h(y)$ hold for all $u \in \lbrace v\in \mathcal A\mid v=v^*\hspace{5.0pt}\text{and}\hspace{5.0pt}v\hspace{5.0pt}\text{is} \text{invertible}\rbrace $, all $y\in \mathcal A$ and all $n\in \mathbb{Z}$. Furthermore, we prove the Cauchy-Rassias stability of $*$-homomorphisms between unital $C^*$-algebras, and $\mathbb{C}$-linear $*$-derivations on unital $C^*$-algebras.
A set $\mathcal{S}=\lbrace x_1,\ldots ,x_n\rbrace $ of $n$ distinct positive integers is said to be gcd-closed if $(x_{i},x_{j})\in \mathcal{S}$ for all $1\le i,j\le n $. Shaofang Hong conjectured in 2002 that for a given positive integer $t$ there is a positive integer $k(t)$ depending only on $t$, such that if $n\le k(t)$, then the power LCM matrix $([x_i,x_j]^t)$ defined on any gcd-closed set $\mathcal{S}=\lbrace x_1,\ldots ,x_n\rbrace $ is nonsingular, but for $n\ge k(t)+1$, there exists a gcd-closed set $\mathcal{S}=\lbrace x_1,\ldots ,x_n\rbrace $ such that the power LCM matrix $([x_i,x_j]^t)$ on $\mathcal{S}$ is singular. In 1996, Hong proved $k(1)=7$ and noted $k(t)\ge 7$ for all $t\ge 2$. This paper develops Hong’s method and provides a new idea to calculate the determinant of the LCM matrix on a gcd-closed set and proves that $k(t)\ge 8$ for all $t\ge 2$. We further prove that $k(t)\ge 9$ iff a special Diophantine equation, which we call the LCM equation, has no $t$-th power solution and conjecture that $k(t)=8$ for all $t\ge 2$, namely, the LCM equation has $t$-th power solution for all $t\ge 2$.
This paper is concerned with the distributed filtering problem for nonlinear time-varying systems over wireless sensor networks under random link failures. To achieve consensus estimation, each sensor node is allowed to communicate with its neighboring nodes according to a prescribed communication topology. Firstly, a new hybrid consensus-based filtering algorithm under random link failures, which affect the information exchange between sensors and are modeled by a set of independent Bernoulli processes, is designed via redefining the interaction weights. Second, a novel observability condition, called parameterized jointly uniform observability, is proposed to ensure the stochastic boundedness of the error covariances of the hybrid consensus-based filtering algorithm. Finally, an example is given to demonstrate the effectiveness of the derived theoretical results.