Let $\beta = \frac{q}{p}$ be a fixed rational number, where $p$ and $q$ are positive integers with $2 \leq p \lt q$ and $\gcd(p,q) = 1$. Consider two real-valued functions $\sigma(x) = \beta^x \mod 1$ and $\tau(x) = \beta x \mod 1$. For each positive integer $n$, let $s(n) = \sigma(n) = \frac{s(n)_1}{p} + \dots + \frac{s(n)_n}{p^n}$ and $t(n) = \tau^n(1) = \frac{t(n)_1}{p} + \dots + \frac{t(n)_n}{p^n}$ be the $p$-ary representation. In this paper, we study the periods of both sequences $S_k = \{s(n + k)_n\}_{n=1}^{\infty}$ and $T_k = \{t(n + k)_n\}_{n=1}^{\infty}$ for any non-negative integer $k$.

Let $q \gt 1$ and $E$ be a real $q$-uniformly smooth Banach space, $K$ be a nonempty closed convex subset of $E$ and $T: K \rightarrow K$ be a strictly pseudocontractive mapping in the sense of F. E. Browder and W.V. Pstryshyn [1]. Let $\{u_n\}$ be a bounded sequence in $K$ and $\{\alpha_n\}$ and $\{\beta_n\}$ be real sequences in $[0,1]$ satisfying some restrictions. Let $\{x_n\}$ be the sequence generated from an arbitrary $x_1 \in K$ by the Ishikawa iteration process with errors: $y_n = (1-\beta_n) x_n + \beta_n Tx_n$, $x_{n+1} = (1-\alpha_n) x_n + \alpha_n Ty_n + u_n$, $n \geq 1$. Sufficient and necessary conditions for the strong convergence $\{x_n\}$ to a fixed point of $T$ is established.

It is proved that the system $\left\{ \omega_{\ell,n,k}: \ell = j-m; n = 2^{m}, 2^{m}+1, \ldots, 2^{m+1}-1; \quad j,k \in \mathbb{Z} \right\}$ of wavelet packets is an unconditional basis for $ L^{p}(\mathbb{R})$, $1 \lt p \lt \infty$ and $\mathcal{H}^{1}(\mathbb{R})$, where $m=0$ if $j \leq 0$ and $m = 0,1,2,\ldots,j$ if $j \gt 0$, provided the orthonormal wavelet packets $\omega_{n}$ and its derivative $ \omega'_{n}$ have a common radial decreasing $L^{1}$-majorant satisfying the condition $\int_{0}^{\infty} sH(s) \, ds \lt \infty$.

The idea of difference sequence spaces was introduced by Kizmaz [12] and was generalized by Et and Colak [6]. In this paper we introduce and examine some properties of the sequence spaces $\left[ V,\lambda,f,p \right]_{0} \left( \Delta_{v}^{r},q \right)$, $\left[ V,\lambda,f,p \right]_{1} \left( \Delta_{v}^{r},q \right)$, $\left[ V,\lambda,f,p \right]_{\infty} \left( \Delta_{v}^{r},q \right)$, $S_{\lambda}(\Delta_{v}^{r},q)$ and give some inclusion relations on these spaces. We also show that the space $S_{\lambda}(\Delta_{v}^{r},q)$ may be represented as a $\left[ V,\lambda,f,p \right]_{1} \left( \Delta_{v}^{r},q \right)$ space. Furthermore, we compute Köthe-Toeplitz duals of the spaces of generalized Cesàro difference sequences spaces.

Let $M$ be a right $R$-module, and let $a \in End_R M$ be unit-regular. If $End_R(Ima)$ is an exchange ring and $End_R(Kera)$ has stable rank one, it is shown that there exist an idempotent $e \in End_R M$ and a left cancellable $u \in End_R M$ such that $a = e+u$ and $aM \cap eM = 0$.

We obtain upper bounds for the upper generalized exponents of digraphs in the class of ministrong digraphs and in the class of non-primitive ministrong digraphs, characterize the corresponding extremal digraphs, and discuss the numbers attainable as upper generalized exponents of ministrong digraphs.

In this paper, some dual Brunn-Minkowski inequalities are established for intersection bodies for the harmonic Blaschke additions and p-radial additions.

The purpose of this paper is to give analogous definitions of Apostol type (see T. M. Apostol [Pacific J. Math. 1 (1951), 161-167]) for the so-called Apostol-Euler numbers and polynomials of higher order. We establish their elementary properties, obtain several explicit formulas involving the Gaussian hypergeometric function and the Stirling numbers of the second kind, and deduce their special cases and applications that lead to the corresponding formulas of the classical Euler numbers and polynomials of higher order.

$d$-separable, $\overline{d}$-separable and $d$-disjunct matrices are the major tools in constructing pooling designs which has many applications to DNA experiments, for example, the clone library screening problem. While there exists a simple decoding for $d$-disjunct matrices, only brute-force methods are known for the other two. In this paper we identify structures in these two matrices which lead to significant improvements for decoding.

In this paper we show some generalizations of estimations of difference of two integral means, Milovanović-Pečarić-Fink inequality.

The purpose of this paper is to investigate the iterative methods for a class of general mixed quasivariational inequalities in a Hilbert space. Utilizing the alternative equivalent formulation between general mixed quasivariational inequalities and implicit fixed-point problems, we suggest and analyze a new modified self-adaptive resolvent method with errors for solving this class of general mixed quasivariational inequalities in conjunction with a technique updating the solution. Moreover, we give the convergence analysis of this method in a Hilbert space. Since this class of general mixed quasivariational inequalities includes a number of known classes of variational inequalities as special cases, our results are more general than some earlier and recent ones in the literature.

We apply the two-scale convergence method introduced by G. Nguetseng and G. Allaire to study the homogenization of a first order linear differential equation. We show that it generates memory effects and the memory kernel is described by a Volterra integral equation. The explicit form of the memory kernel is given in terms of a Radon measure.

We generalize and give a simple proof of a Theorem of Blasco.

The initial boundary value problem for an integro-differential equation with strong damping in a bounded domain is considered. The existence, asymptotic behavior and blow-up of solutions are discussed under some conditions. The decay estimates of the energy function and the estimates of the lifespan of blow-up solutions are given.

Let $G$ be an outerplanar graph with maximum degree $\Delta(G) \ge 3$. We prove that the chromatic number $\chi(G^2)$ of the square of $G$ is at most $\Delta(G)+2$. This confirms a conjecture of Wegner [8] for outerplanar graphs. The upper bound can be further reduced to the optimal value $\Delta(G)+1$ when $\Delta(G) \ge 7$.

In this paper the multi-valued versions of some well-known hybrid fixed point theorems of Dhage [6, 7] in Banach algebras are proved. As an application, an existence theorem for a certain integral inclusion in Banach algebras is proved.

An optimal stopping rule is a rule that stops the sampling process at a sample size n that maximizes the expected reward. In this paper we will study the approximation to optimal stopping rule for Gumbel random variables, because the Gumbel-type distribution is the most commonly referred to in discussions of extreme values. Let $X_1, X_2,\cdots X_n,\cdots$ be independent, identically distributed Gumbel random variables with unknown location and scale parameters,$\alpha$ and $\beta$. 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 location and scale parameters $\alpha$ and $\beta$. We propose an adaptive stopping rule that does not depend on the unknown location and scale parameters 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. Also, we use simulation in statistics to verify the results.

The aim of this paper is to study one special quotient in the class of nearring modules. Contrast to the well known noetherian quotient, this quotient is less noticed. Results and applications for this quotient are developed and an induced topology in the class of type-1 $N$-modules is introduced.

In this paper, the three estimators for three parameters of the generalized gamma distribution are proposed by using its characterization, and shown to be more convenient and more efficient than the maximum likelihood estimator for small samples. Furthermore, the estimators for the parameter of its special distribution such as gamma and Weibull distributions which are the most important in the reliability field and survival analysis are obtained.

In this paper, we discuss three constructions: attaching construction, adjoining construction and the join of two graphs to obtain larger sequential graphs from small sequential or strongly $r$-indexable ones.