In this work, we show that under certain conditions the values of some generalized lacunary power series with algebraic coefficients for Mahler's $U_{m}-$number arguments belong to either a certain algebraic number field or $\bigcup_{i=1}^{t} U_{i}$ in Mahler's classification of the complex numbers, where $t$ denotes a positive rational integer dependent on the coefficients of the given series and on the argument. Moreover, the obtained results are adapted to the field $\mathbb{Q}_{p}$ of $p-$adic numbers.

In this paper, we investigate the solutions of difference equation $f(z+1) = R \circ f(z)$ by utilizing Nevanlinna theory, where $R(z)$ is a rational function. And we also research the quantity of zeroes, poles, fixed points, and Borel exceptional values of the solutions.

By using functions of bounded variation we generalize the class of strongly starlike functions and related classes. The main object is to obtain characterizations and inclusion properties of these classes of functions.

Ruled submanifolds of Minkowski space with harmonic Gauss map are studied. Apart from ruled submanifolds in Euclidean space, ruled submanifolds with degenerate rulings in Minkowski space draw our attention. In particular, we completely classify ruled submanifolds with harmonic Gauss map and we also characterize minimal ruled submanifolds with degenerate rulings by means of harmonic Gauss map.

This paper deals with a class of stochastic partial functional differential equations with finite delay. We give some sufficient conditions to guarantee the existence of a unique random attractor which attracts any tempered random set in the phase space.

Efficient algorithms for computing mixed volumes, via the computation of mixed cells, have been implemented in DEMiCs [18] and MixedVol-2.0 [13]. While the approaches in those two packages are somewhat different, they follow the same theme and are both highly serial. To fit the need for the parallel computing, a reformulation of the algorithms is inevitable. This article proposes a reformulation of the algorithm for the mixed volume computation rooted from algorithms in graph theory, making it much more fine-grained and scalable. The resulting parallel algorithm can be readily adapted to both distributed and shared memory computing systems. Illustrated by the numerical results on several different architectures, the speedups of our parallel algorithms for the mixed volume computation are remarkable.

In this paper we continue our investigation of the recent summability notion of $I-$statistical convergence introduced in [8, 26] (where a new extension of the notion of natural density and statistical convergence was proposed using the notion of ideals of the set of positive integers $\mathbb{N}$) and introduce the notion of $I$-statistically pre-Cauchy sequences in line of [3]. We mainly show that $I-$statistical convergence implies $I-$statistical pre-Cauchy condition and give certain sufficient conditions for the converse to be true.

This paper is devoted to studying the singular integral operators associated to polynomial mappings as well as the corresponding compound submanifolds. By imposing a restrictive condition on the kernels of the operators in the radial direction, the boundedness for such operators on Triebel-Lizorkin spaces and Besov spaces are established, provided that the kernels satisfy a rather weak size condition on the unit sphere, which is distinct from the Hardy space functions. Some previous results are essentially improved and generalized.

In this paper, we consider weighted inequalities for linear and multilinear fractional integrals with homogeneous kernels on Morrey spaces. Recently, weighted inequalities without homogeneous kernels were proved by the authors. In this paper, we generalize ones with homogeneous kernels.

The aim of this paper is to continue the study of properties of an $\ell$-stable at a point vector function. We show that any $\ell$-stable at a point function from arbitrary normed linear space is strictly differentiable at the considered point.

The $CR$ $\delta$-invariant for $CR$-submanifolds was introduced by B.-Y. Chen in a recent article [13]. In this paper, we prove two new optimal inequalities for anti-holomorphic submanifolds in complex space forms involving the $CR$ $\delta$-invariant. Moreover, we obtain some classification results for certain anti-holomorphic submanifolds in complex space forms which satisfy the equality case of either inequality.

In this paper, we study the nonlocal anisotropic $\overrightarrow{p}(x)$-Laplacian problem of the following form \begin{gather*} - \sum_{i=1}^N M_{i} \Big( \int_{\Omega} \frac{|\partial_{x_i} u|^{p_i(x)}}{p_i(x)} dx \Big) \partial_{x_i} \Big( |\partial_{x_i} u|^{p_i(x)-2} \partial_{x_i} u \Big) = f(x,u) \quad \text{in } \Omega, \\ u=0 \quad \text{on } \partial \Omega. \end{gather*} By means of a direct variational approach and the theory of the anisotropic variable exponent Sobolev space, we obtain the existence and multiplicity of weak energy solution. Moreover, we get much better results with $f$ in a special form.

In this paper, new concepts of monotonicity, namely $\left( \Phi, \rho \right)$-monotonicity, $\left( \Phi, \rho \right)$-pseudo-monotonicity and $\left( \Phi, \rho \right)$-quasi-monotonicity are introduced for functions defined in Banach spaces. Series of necessary conditions are also given that relate $\left( \Phi, \rho \right)$-invexity and generalized $\left( \Phi, \rho \right)$-invexity of the function with $\left( \Phi, \rho \right)$-monotonicity and generalized $\left( \Phi, \rho \right)$-monotonicity of its gradient.

Let $R$ be a prime ring with nontrivial idempotents. Assume $\ast$ is an involution of $R$. In this note we characterize the additive map $\delta \colon R \to R$ such that $\delta(x) y^\ast + x \delta(y)^\ast = 0$ whenever $xy^\ast = 0$ and $\phi \colon R \to R$ such that $\phi(x) \phi(y)^\ast = 0$ whenever $xy^\ast = 0$.

Let $m \geq 2$ and $r \geq 1$ be integers and let $c \in Z_m = \{0, 1,\dots, m - 1\}$. In this paper, we give an upper bound and a lower bound for the number of unordered solutions $x_1, \dots, x_n \in Z_m$ of the congruence $x_1 + x_2 + \cdots + x_r \equiv c \mod m$. Exact formulae are also given when $m$ or $r$ is prime. This solution number involves the Catalan number or generalized Catalan number in some special cases. Moreover, the enumeration problem has relationship with the restricted integer partition.

In this paper, we obtain a necessary and sufficient condition for boundedness of composition operator $ C_{\varphi}$, between two different weighted spaces of holomorphic functions on the upper half-plane, whenever one of the weights satisfy certain growth condition.

We study the topological structure of the space of all bounded composition operators from $F(p,q,s)$ to $\mathcal{B}_\mu$ on the unit disk $\mathbb{D}$ in the operator norm topology. At the same time, we characterizes the boundedness and compactness of the differences of two composition operators.

In this paper, we proposed some lower bounds for the minimum eigenvalue of the Fan product of $M$-matrices, and a upper bound for the spectral radius of the Hadamard product of nonnegative matrices. These improve two existing results. To illustrate our results, two simple examples are considered.

In this paper, we mainly parameterize $g$-frames in Hilbert space in terms of operators. Firstly, under the condition that there exists a $g$-orthonormal basis we establish the one to one correspondence between different kinds of $g$-frames with certain kinds of operators. Then we parameterize $g$-frames without the above restriction. Finally, we consider some special $g$-frames, and their operator parameterizations are established. We also obtain some interesting results on $g$-bases and $g$-dual frames of the transforms by applying the operator parameterizations of $g$-frames.

Let $\alpha$ be a nonzero algebraic integer of degree $d$, all of whose conjugates $\alpha_i$ are confined to a sector $\left|\arg(\alpha_i)\right|\leq\theta$ with $0\lt \theta\lt \pi /2$. Let $P=X^d+b_1X^{d-1}+\cdots +b_d$ be the minimal polynomial of $\alpha$. We give in this paper the greatest lower bounds $\rho_{\mathcal {L}}(\theta)$ of the absolute length $\mathcal{L}(P) = (1+\sum_{i=1}^d |b_i|)^{1/d}$ of all but finitely many such $\alpha$, for ten different values of $\theta$.