We first prove a strong convergence theorem of Mann’s type for a commutative family of continuous linear operators in a Banach space by using strongly regular sequences of means on commutative semigroups. Using this, we obtain various strong convergence theorems for continuous linear operators in a Banach space.

We consider spike solutions of a second order differential equation with a forcing modeling a nonlinear circuit used in converting analog signals to digital ones. It is shown that the number of spikes which correspond to bits in digital signals can be provided by asymptotic expansions. Numerical results are also presented.

The boundedness, the oscillatory behavior and the global stability of the nonnegative solutions of the difference equation $$ x_{n+1} = \frac{\alpha + \beta x_{n-k}}{f(x_n,...,x_{n-k+1})}$$ is investigated, where $k \in \mathbf{N},$ the parameters $\alpha$ and $\beta$ are nonnegative real numbers and $f: \mathbf{R}^k_+ \to \mathbf{R}_+$ is a continuous function nondecreasing in each variable such that $f(0,...,0) \gt 0$.

Herstein’s theorems on Jordan homomorphisms and Jordan derivations on prime associative algebras are extended to graded prime associative algebras.

In this paper, we introduce some new sequence spaces combining lacunary sequence, invariant means and a sequence of Orlicz functions. We discuss some topological properties and establish some inclusion relations between these spaces. Also we studied connections between lacunary $\sigma-$statistical convergence with these spaces.

A nonincreasing sequence of positive integers $\langle m_1,m_2,\cdots,m_k \rangle$ is said to be $n$-realizable if the set $I_n = \{1,2,\cdots,n\}$ can be partitioned into $k$ mutually disjoint subsets $S_1,S_2,\cdots,S_k$ such that $\sum_{x \in S_i} x = m_i$ for each $1 \le i \le k$. In this paper, we will prove that a nonincreasing sequence of positive integers $\langle m_1,m_2,\cdots,m_k \rangle$ is $n$-realizable under the conditions that $\sum_{i=1}^k m_i = \binom{n+1}{2}$ and $m_{k-1} \ge n$.

An analysis of the problem of unsteady laminar boundary layer near a stagnation point is presented. The transformation group theoretic approach is applied to this problem. The application of two-parameter groups reduces the number of independent variables in the governing system consisting of partial differential equations and a set of auxiliary conditions from three to only one independent variable, and consequently the system of governing partial differential equations with boundary conditions reduces to a system of ordinary differential equations with appropriate boundary conditions. The possible forms of the main stream velocity and the surface temperature variations with position and time are derived. It is shown how not only we recover many of the previously known exact solutions but also find some completely new forms.

The restricted arc-connectivity $\lambda'$ of a strongly connected digraph $G$ is the minimum cardinality of an arc cut $F$ in $G$ such that every strongly connected component of $G-F$ contains at least two vertices. This paper shows that for a $d$-regular strongly connected digraph with order $n$ and diameter $k \geq 4$, if $\lambda'$ exists, then $$ \lambda'(G) \geq \min \left\{ \frac{(n-d^{k-1})(d-1)}{d^{k-1}+d-2},\ 2d-2 \right\}. $$ As consequences, the restricted arc-connectivity of the de Bruijn and Kautz digraph and the generalized de Bruijn and Kautz digraph are determined.

The $\lambda$-number of a graph $G$ is the minimum value $\lambda$ such that $G$ admits a labeling with labels from $\{0,1,\ldots,\lambda\}$ where vertices at distance two get different labels and adjacent vertices get labels that are at least two apart. Sierpiński graphs $S(n,k)$ generalize the Tower of Hanoi graphs---the graph $S(n,3)$ is isomorphic to the graph of the Tower of Hanoi with $n$ disks. It is proved that for any $n\geq 2$ and any $k \geq 3$, $\lambda(S(n,k)) = 2k$. To obtain the result (perfect) codes in Sierpiński graphs are studied in detail. In particular a new proof of their (essential) uniqueness is obtained.

Let $(X,\rho,\mu)_{d,\theta}$ be a space of homogeneous type which includes metric measure spaces and some fractals, namely, $X$ is a set, $\rho$ is a quasi-metric on $X$ satisfying that there exist constants $C_0 \gt 0$ and $\theta \in (0,1]$ such that for all $x,\ x',\ y \in X$, $$ |\rho(x,y)-\rho(x',y)| \le C_0 \rho(x,x')^\theta [\rho(x,y) + \rho(x',y)]^{1-\theta}, $$ and $\mu$ is a nonnegative Borel regular measure on $X$ satisfying that for some $d \gt 0$, all $x \in X$ and all $0 \lt r \lt \operatorname{diam} X$, \[ \mu(\{ y \in X: \rho(x,y) \lt r \}) \sim r^{d}. \] In this paper, we first obtain the boundedness of Calderón-Zygmund operators on spaces of test functions; and using this, we then establish the continuous Calderón reproducing formulae associated with a given para-accretive function, which is a key tool for developing the theory of Besov and Triebel-Lizorkin spaces associated with para-accretive functions. By the Calderón reproducing formulae, we finally obtain a Littlewood-Paley theorem on the inhomogeneous $g$-function which gives a new characterization of Lebesgue spaces $L^{p}(X)$ for $p \in (1,\infty)$ and generalizers a corresponding result of David, Journé and Semmes.

Let $X_1, X_2, \cdots, X_n, \cdots$ be independent, identically distributed Weibull random variables with an unknown scale parameter $\alpha$. If we define the reward sequence $Y_n = \max \{ X_1,X_2,\cdots,X_n \} - cn$ for $c \gt 0$, the optimal stopping rule for $Y_n$ depends on the unknown scale parameter $\alpha$. In this paper we propose an adaptive stopping rule that does not depend on the unknown scale parameter $\alpha$ and show that the difference between the optimal expected reward and the expected reward using the proposed adaptive stopping rule vanishes as $c$ goes to zero.