We introduce and study CLESS-modules, which subsume two generalizations of extending modules due to P.F. Smith and A. Tercan. A module $M$ will be called a CLESS-module if every closed submodule $N$ of $M$ (in the sense that $M/N$ is non-singular) with essential socle is a direct summand of $M$. Various properties concerning direct sums of CLESS-modules are established. We show that, over a Dedekind domain, a module is CLESS if and only if its torsion submodule is a direct summand. We also study the behaviour of CLESS-modules under excellent extensions of rings.

This paper is concerned with asymptotic behavior of a class of fourth-order neutral delay dynamic equations with a noncanonical operator on an arbitrary time scale. A new asymptotic criterion and an illustrative example are included.

Ruled submanifolds of Minkowski space with finite-type Gauss map are studied. Not having a parallel in Euclidean space, ruled submanifolds with degenerate rulings in Minkowski space drew our attention. We show that if non-cylindrical ruled submanifolds with non-degenerate rulings or ruled submanifolds with degenerate rulings have finite-type Gauss map, the Gauss map is one of the following: (1) harmonic; (2) of the so-called finite rank; (3) of null 2-type. For ruled submanifolds with degenerate rulings, we set up a relationship between finite-type immersions and immersions with finite-type Gauss map and introduce new examples of ruled submanifolds with degenerate rulings. We also characterize minimal ruled submanifolds with degenerate rulings in terms of finite-type Gauss map.

Algorithms approach to equilibrium problems and fixed points problems have been extensively studied in the literature. The purpose of this paper is devoted to consider the minimization problem of finding a point $x^†$ with the property \begin{equation*} x^†\in \Omega\quad {\rm and}\quad \|x^†\|^2=\min_{x\in \Omega}\|x\|^2, \end{equation*} where $\Omega$ is the intersection of the solution set of equilibrium problem and the fixed points set of nonexpansive mapping. For this purpose, we suggest two algorithms: \begin{eqnarray*} F(z_t,y)+\frac{1}{\lambda}\Big{\langle} y-z_t,z_t-\Big{(}(1-t)I-\lambda A\Big{)}Sz_t\Big{\rangle}\ge 0,\;\forall y\in C. \end{eqnarray*} and \begin{eqnarray*} \begin{cases} F(z_n,y)+\langle Ax_n,y-z_n\rangle+\frac{1}{\lambda_n}\Big{\langle}y-z_n,z_n-(1-\alpha_n)x_n\Big{\rangle}\ge 0,\;\forall y\in C,\\ x_{n+1}=\beta_nx_n+(1-\beta_n)Sz_n,\; n\ge 0. \end{cases} \end{eqnarray*} It is shown that under some mild conditions, the net $\{z_t\}$ and the sequences $\{z_n\}$ and $\{x_n\}$ converge strongly to $\tilde{x}$ which is the unique solution of the above minimization problem. It should be point out that our suggested algorithms solve the above minimization problem without involving projection.

In this paper, we prove that the transcendental entire solution of complex linear differential equation $f^{(k)}-e^{P(z)}f=Q(z)$, where $P(z)$ is a transcendental entire function and $Q(z)$ is a polynomial, is of infinite hyper-order under the hypothesis that the Fatou set of $P(z)$ has a multiply connected component.

By standard barrier solution method associated with Schauder fixed point theorem, we establish an existence theory for nonlinear second order nonlinear multi-point boundary value problem $(1.3)$, $(1.4)$. Through the pervious existence theorem, we mainly work out some asymptotic behaviors of solutions for semilinear singularly perturbed three-point boundary value problem $(1.1)$, $(1.2)$. Barrier solutions will be constructed explicitly when the boundary or interior layers occur, respectively.

In this paper we investigate the existence and multiplicity of classical homoclinic solutions for the following second order Hamiltonian systems $$ (\mbox{HS}) \ddot u-L(t)u+\nabla W(t,u)=0, $$ where $L\in C(\mathbb R,\mathbb R^{n^2})$ is a symmetric and positive definite matrix for all $t\in \mathbb R$, $W\in C^1(\mathbb R\times\mathbb R^n,\mathbb R)$ and $\nabla W(t,u)$ is the gradient of $W$ at $u$. The novelty of this paper is that, assuming that $L$ is bounded in the sense that there are constants $0\lt \tau_1\lt \tau_2$ such that $\tau_1 |u|^2\leq (L(t)u,u)\leq \tau_2 |u|^2$ for all $(t,u)\in \mathbb R\times \mathbb R^n$ and $W(t,u)$ is of subquadratic growth at infinity, we are able to establish two new criteria to guarantee the existence and multiplicity of classical homoclinic solutions for (HS), respectively. Recent results in the literature are extended and significantly improved.

The John-Nirenberg inequalities on Lebesgue spaces with variable exponents are obtained.

A classic theorem of Approximation Theory states the uniform convergence of the best approximation polynomials, concerning the Banach space $C$ of all real-valued continuous functions defined on the interval $[-1,1]$ of $\mathbb{R}$, in supremum norm. By contrast, the main result of this paper highlights the phenomenon of double condensation of singularities (meaning unbounded divergence on large subsets of $C$ and $[-1,1]$, in topological sense) for the discrete best approximation on Chebyshev nodes.

We study multilinear operators $T(f_{1},f_{2},...,f_{m})$ that commutes with simultaneous translations and prove that if T is bounded from $L^{p_{1}} \times L^{p_{2}} \times ... \times L^{p_{m}}$ to $L^{p}$, then for any $r \geqslant p$, $0 \lt p,q \leqslant \infty$ and \[s \gt \left\{ \begin{array}{lll} n(1-1\wedge\frac{1}{q}), &(\frac{1}{p},\frac{1}{q})\in D_{1};\\ n(1\vee\frac{1}{p}\vee\frac{1}{q}-\frac{1}{q}), &(\frac{1}{p},\frac{1}{q})\in \mathbb{R}_{+}^{2}-D_{1}, \end{array} \right.\] ($D_{1}=\{(\frac{1}{p},\frac{1}{q})\in\mathbb{R}_{+}^{2}:\frac{1}{q}\geqslant\frac{2}{p},\frac{1}{p}\leqslant\frac{1}{2}\}$)T is bounded from $M_{p_{1},q}^{s}\times M_{p_{2},q}^{s}\times...\times M_{p_{m},q}^{s}$ to $M_{r,q}^{s}$ (which improves the results obtained by [5], [6].), where $M_{p,q}^{s}$ is the modulation spaces. Besides, we alsoobtain the similar results for Triebel-type spaces $N_{p,q}^{s}$ introduced by [21] (T is bounded from $N_{p,q}^{s} \times N_{p,q}^{s} \times ... \times N_{p,q}^{s}$ to $N_{p,q}^{s}$). As applications, we obtain the boundedness on the modulation spaces for the bilinear Hilbert transform, bilinear fractional integral, the pointwise product of functions, and the bilinear oscillatory integral along parabolas. Also, in modulation spaces and $N_{p,q}^{s}$, we study the well-posedness of the Cauchy problem for the fractional heat and Schrödinger equations with some new nonlinear terms. Such nonlinear well-posedness problems are not studied in other function spaces.

In this paper, we provide sufficient conditions to construct sense-preserving and univalent biharmonic mappings that arises from analytic functions which are not necessarily univalent in the unit disk $|z|\lt 1$. Also we state and prove several theorems under different weaker hypothesis in each case, leading to an affirmative answer to the radius problem posed by Y. Abu Muhanna in 2008.

In this paper, we consider two new subclasses, $S^*(a,b,B^n)$ and ${\mathcal A}S^*(a,b,B^n)$, of the class of starlike mappings on $B^n$ ($a,b\in \mathbb{R}$, $|a-1|\lt b\le a$, and $B^n$ is the Euclidean unit ball in $\mathbb{C}^n$). The class $S^*(a,b,B)$ is the $n$-dimensional version of Janowski class of one variable starlike functions. We obtain sharp growth results and upper distortion estimates for these two classes of starlike mappings. We also derive sufficient conditions for normalized holomorphic mappings (expressed in terms of their coefficient bounds) to belong to one of the classes $S^*(a,b,B^n)$, respectively ${\mathcal A}S^*(a,b,B^n)$. Finally, similar notions on the unit ball in a complex Hilbert space are analogously presented.

This paper is concerned with the following Schrödinger equation: $$ \left\{ \begin{array}{ll} -\triangle u+V(x)u=f(x, u), \ \ \ x\in\mathbb R^N,\\ u(x)\rightarrow0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ as \ \ \ \ |x| \rightarrow\infty, \end{array} \right. $$ where the potential $V$ and $f$ are periodic with respect to $x$ and $0$ is a boundary point of the spectrum $\sigma(-\triangle+V)$. By a generalized variant fountain theorem and an approximation technique, for old $f$, we are able to obtain the existence of infinitely many large energy solutions.

In this paper, by using variational methods and critical point theory, we study the existence of semiclassical solutions for the following nonlinear Schrödinger-Maxwell equations \[\left\{\begin{array}{lll} -\varepsilon^{2}\triangle u+V(x)u+\phi u=K(x)|u|^4u+f(x,u), &in \, \mathbb R^{3}, \\ -\triangle\phi=4\pi u^{2}, & in \, \mathbb R^{3}, \end{array} \right.\] where $\varepsilon\gt 0$, $V(x)\geq0$ and $K(x)\gt 0$ for all $x\in\mathbb R^{3}$, under some more assumptions on $V$, $K$ and $f$, we prove that the system has at least one nontrivial solution for sufficient small $\varepsilon\gt 0$. Our approach is much more straightforward.

By using the Carathéodory-Pesin structure (C-P structure), the topological entropy on the whole space introduced for a proper map, is generalized to the cases of arbitrary subset, i.e., we introduce three notions of topological entropy. Some of the properties of these notions are provided. As some applications, for the proper map of locally compact separable metric space, we prove the following variational principles: (1) The upper capacity topological entropy on any subset and the minimum of the Bowen-Dinaburg entropies always coincide; (2) For any invariant probability measure, the measure-theoretic entropy and the infimum of the topological entropies on all sets which are of full measures always coincide; (3) The relationship between the topological entropies of level sets of the ergodic average of some continuous functions and the measure-theoretic entropies are given. These are the extensions of results of Patrão and Pesin, etc.

As a continuation of the study of cover-incomparability graphs of posets (C-I graphs), the notion of 2-colored diagrams is introduced and used in characterizations of posets whose C-I graphs belong to certain natural classes of graphs. As a particular instance, posets whose C-I graphs are chordal are characterized using a single 2-colored diagram. Some other instances are characterized in a similar way.

Let $M$ be a complete Riemannian manifold with infinite volume and $\Omega$ be a compact subdomain in $M.$ In this paper we obtain two upper bound estimates for the first eigenvalue of the Laplacian on the punctured manifold $M \setminus \Omega$ subject to volume growth and lower bound of Ricci curvature, respectively. The proof hinges on asymptotic behavior of solutions of second order differential equations, the max-min principle and Bishop volume comparison theorem.

In this paper, we first derive the CR Bochner formula and the CR Kato's inequality for pseudoharmonic maps. Secondly, by applying the CR Bochner formula and the CR Kato's inequality we are able to prove the Liouville property for pseudoharmonic maps with finite Dirichlet energy in a complete $(2n+1)$-pseudohermitian manifold. This is served as CR analogue to the Liouville theorem for harmonic maps in Riemannian Geometry.

In this paper, we study the so-called mean Minkowski measures, proposed and studied by Toth in a series of papers, for convex bodies of constant width. We show that, with respect to the mean Minkowski measure, the completions of regular simplices are, as well as for many other measures, the most asymmetric ones among all convex bodies of constant width.

Let $L:=-\mathrm{div}(A\nabla)+V$ be a Schrödinger type operator with the nonnegative potential $V$ belonging to the reverse Hölder class $RH_{q_0}(\mathbb{R}^n)$ for some $q_0\in[n,\infty)$ with $n\ge3$, where $A$ satisfies the uniformly elliptic condition. Assume that $\varphi:\,\mathbb{R}^n\times[0,\infty)\to[0,\infty)$ is a function such that $\varphi(x,\cdot)$ is an Orlicz function, $\varphi(\cdot,t)\in {\mathbb A}_{\infty}(\mathbb{R}^n)$ (the class of uniformly Muckenhoupt weights) and its uniformly critical lower type index $i(\varphi)\in(\frac{n}{n+\alpha_0},1]$, where $\alpha_0\in(0,1]$ measures the regularity of kernels of the semigroup generalized by $L_0:=-\mathrm{div}(A\nabla)$. In this article, we first prove that operators $VL^{-1}$, $V^{1/2}\nabla L^{-1}$ and $\nabla^2L^{-1}$ are bounded from the Musielak-Orlicz-Hardy space associated with $L$, $H_{\varphi,\,L}(\mathbb{R}^n)$, to the Musielak-Orlicz space $L^{\varphi}(\mathbb{R}^n)$. Moreover, we also obtain the boundedness of $VL^{-1}$ and $\nabla^2L^{-1}$ on $H_{\varphi,\,L}(\mathbb{R}^n)$. All these results are new even when $\varphi(x,t):=t^p$, with $p\in(\frac{n}{n+\alpha_0},1]$, for all $x\in\mathbb{R}^n$ and $t\in[0,\infty)$.