In this work, we study spacelike surfaces in Minkowski space ${\bf E}_1^3$ foliated by pieces of circles that satisfy a linear Weingarten condition of type $aH + bK = c$, where $a,b$ and $c$ are constants and $H$ and $K$ denote the mean curvature and the Gauss curvature respectively. We show that such surfaces must be surfaces of revolution or surfaces with constant mean curvature $H=0$ or surfaces with constant Gauss curvature $K=0$.

In this paper we prove that slant surfaces in a non-flat Lorentzian complex space form must be Lagrangian. By applying this result, we completely classify pseudo-umbilical slant surfaces in Lorentzian complex space forms. Our classification results state that there exist two families of pseudo-umbilical slant surfaces in Lorentzian complex plane $\mathbb{C}^2_1$, three families in complex projective plane $CP^2_1$, and three families in complex hyperbolic plane $CH^2_1$.

We investigate shift preserving operators on locally compact abelian groups. We show that there is a one-to-one correspondence between shift preserving operators and range operators on $L^2(G)$ where $G$ is a locally compact abelian group. We conclude that a shift preserving operator has several properties in common with its associated range operator, especially compactness of one implies compactness of the other. Moreover, we obtain a necessary condition for a shift preserving operator to be Hilbert Schmidt or of finite trace in terms of its range function.

In this paper we study a problem for photon transport in a host medium (e.g. an interstellar cloud where a localized source is present), that occupies a compact convex region $V$ in $R^{3}$. We find the generalized solution of the photon transport problem by means of the theory of equicontinuous semigroup of bounded linear operators on a sequentially complete locally convex topological vector space.

We show in this paper that every bijective linear isometry between the continuous section spaces of two non-square Banach bundles gives rise to a Banach bundle isomorphism. This is to support our expectation that the geometric structure of the continuous section space of a Banach bundle determines completely its bundle structures. We also describe the structure of an into isometry from a continuous section space into an other. However, we demonstrate by an example that a non-surjective linear isometry can be far away from a subbundle embedding.

In this paper, we construct two algorithms for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for an $\alpha$-inverse-strongly monotone mapping in a Hilbert space. We show that the sequence converges strongly to a common element of two sets under the some mild conditions on parameters. As special cases of the above two algorithms, we obtain two schemes which both converge strongly to the minimum norm element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for an $\alpha$-inverse-strongly monotone mapping.

In this paper, as natural extensions of the generalized exponents of primitive digraphs introduced by R.A. Brualdi and B. Liu, we introduce the generalized $r$-exponents of primitive digraphs, where $1 \leq r \leq n$. Moreover, some properties and sharp upper bounds of the generalized $r$-exponents of primitive digraphs are obtained, respectively.

We deal with a predator-prey interaction model with Holling-type monotonic functional response and diffusion and which is endowed with a second homogeneous boundary condition. Via spectrum analysis and bifurcation theory, we investigate the local and global bifurcation solutions of the model which emanate from a positive constant solution by taking the growth rate as a bifurcation parameter. Basing on the fixed point index theory, we prove the existence of positive steady-state solutions of the model.

In this note, the author obtains heat kernels for the generalised Hermite operators $L = -\Delta + \langle Bx,x \rangle$ where $B$ is a (not necessarily symmetry) semi-positive definite matrix.

In this paper, we establish separate necessary and sufficient John-Nirenberg (JN) type inequalities for functions in $Q_{\alpha}^{\beta}(\mathbb{R}^{n})$ which imply Gagliardo-Nirenberg (GN) type inequalities in $Q_{\alpha}(\mathbb{R}^{n})$. Consequently, we obtain Trudinger-Moser type inequalities and Brezis-Gallouet-Wainger type inequalities in $Q_{\alpha}(\mathbb{R}^{n})$.

We consider the following generalized quasi-variational inequality problem introduced in [7]: given a real normed space $X$ with topological dual $X^*$, two sets $C,D \subseteq X$ and two multifunctions $S: C \to 2^D$ and $T: C \to 2^{X^*}$, find $(\hat x,\hat\varphi) \in C \times X^*$ such that $$\hat x \in S(\hat x), \quad\hat\varphi \in T(\hat x) \quad{\rm and} \quad \langle\hat\varphi,\hat x-y\rangle \le 0 \quad{\rm for\ all}\quad y\in S(\hat x).$$ We prove an existence theorem where $T$ is not assumed to have any continuity or monotonicity property, improving some aspects of the main result of [7]. In particular, the coercivity assumption is meaningfully weakened. As an application, we prove a theorem of the alternative for the fixed points of a Hausdorff lower semicontinuous multifunction. In particular, we obtain sufficient conditions for the existence of a fixed point which belongs to the relative boundary of the corresponding value.

In this article we introduce the notion of $\Delta$-statistical boundedness for fuzzy real numbers and examine its some properties. We also give some relations related to this concept and construct some interesting examples.

Let $\mathbb D = \{ z \in \mathbb C: |z| \lt 1\}$ be the open unit disk in the complex plane $\mathbb C$, $H(\mathbb D)$ be the space of all analytic functions on $\mathbb D$, $\varphi$ be an analytic self-map of $\mathbb D$ and $u \in H(\mathbb D)$. Define operators by $DW_{\varphi,u}f = (u \cdot f \circ \varphi)'$ and $W_{\varphi,u}Df = (u \cdot f' \circ \varphi)$ for $f \in H(\mathbb D)$. In this paper we characterize bounded operators $DW_{\varphi,u}$ and $W_{\varphi,u}D$ from weighted Bergman space to Zygmund-type space, Bloch-type space and Bers-type space on the open unit disk. We also give some sufficient and necessary conditions for these operators to be compact operators in terms of inducing maps $\varphi$ and $u$.

By using critical point theory, we establish some existence criteria to guarantee the second-order discrete $p$-Laplacian systems $\triangle (\varphi_p(\Delta u(n-1))) - a(n) |u(n)|^{p-2} u(n) + \nabla W(n, u(n)) = 0$ have at least one homoclinic orbit, where $p \gt 1$, $n \in \mathbb{Z}$, $u \in \mathbb{R}^{N}$, $a \in C(\mathbb{Z}, \mathbb{R})$ and $W \in C^{1}(\mathbb{Z} \times \mathbb{R}^{N}, \mathbb{R})$ are no periodic in $n$.

In this paper, we investigate some analogous results on the existence of entire solutions of a certain type of nonlinear differential and differential-difference equations of the following form $$ f^n(z) + P_d(f) = p_1(z) e^{\alpha_1 z} + p_2(z) e^{\alpha_2 z},$$ where $P_d(f)$ is a differential polynomial or differential-difference polynomial in $f(z)$. And we find out its entire solutions or prove that it has no entire solution for some special $P_d(f)$.

An edge-weighted digraph $(\vec{G},\ell)$ is a strict digraph $\vec{G}$ together with a function $\ell$ assigning a real weight $\ell_{uv}$ to each arc $uv$. $(\vec{G},\ell)$ is symmetric if $uv$ is an arc implies that so is $vu$. A circular $r$-coloring of $(\vec{G},\ell)$ is a function $\varphi$ assigning each vertex of $\vec{G}$ a point on a circle of perimeter $r$ such that, for each arc $uv$ of $\vec{G}$, the length of the arc from $\varphi(u)$ to $\varphi(v)$ in the clockwise direction is at least $\ell_{uv}$. The circular chromatic number $\chi_c(\vec{G},\ell)$ of $(\vec{G},\ell)$ is the infimum of real numbers $r$ such that $(\vec{G},\ell)$ has a circular $r$-coloring. Suppose that $(\vec{G},\ell)$ is an edge-weighted symmetric digraph with positive weights on the arcs. Let $T$ be a $\{0,1\}$-function on the arcs of $\vec{G}$ with the property that $T(uv)+T(vu)=1$ for each arc $uv$ in $\vec{G}$. In this note we show that if ${\sum_{uv\in E(\vec{C})} \ell_{uv}/\sum_{uv\in E(\vec{C})} T(uv)}\leq r$ for each dicycle $\vec{C}$ of $\vec{G}$ satisfying $0 \lt (\sum_{uv \in E(\vec{C})} \ell_{uv})\bmod r\lt \max \{\ell_{xy} + \ell_{yx}: xy \in E(\vec{G})\}$, then $(\vec{G},\ell)$ has a circular $r$-coloring. Our result generalizes the work of Zhu, J. Comb. Theory, Ser. B, 86 (2002), 109-113, and also strengthens the workof Mohar, J. Graph Theory, 43 (2003), 107-116.

Let $E$ be a smooth, strictly convex and reflexive Banach space, let $C_*$ be a closed linear subspace of the dual space $E^*$ of $E$ and let $\Pi_{C_*}$ be the generalized projection of $E^*$ onto $C_*$. In this paper, we study the mapping $R$ defined by $R = J^{-1} \Pi_{C_*}J$, where $J$ is the normalized duality mapping from $E$ into $E^*$. We obtain some results which are related to conditional expectations and martingales in the probability theory.

Using the weakly Picard operator technique, we present some abstract Ulam-Hyers stability results for operatorial equations and inclusions involving nonself single-valued and multivalued operators.

An integral representation for a class of superfunctions, associated with the Schrödinger operator, is investigated. Meanwhile, growth properties of them are also proved outside of some exceptional sets.

In this paper we use the penalty approach in order to study two constrained nonconvex minimization problems with locally Lipschitzian objective and constraint functions in a Banach space. We show that a local minimizer of the constrained minimization problem which is not a critical point of the constraint function is also a local minimizer of a corresponding unconstrained penalized problem if a penalty coefficient is large enough.

Let $M$ be a compact manifold with boundary and $N$ be compact manifold without boundary. Let $u(x,t)$ be a smooth solution of the harmonic heat equation from $M$ to $N$ with Dirichlet or Neumann condition. Suppose that $M$ is strictly convex, we will prove a monotonicity formula for $u$. Moreover, if $T$ is the blow-up time for $u$, and $\sup_M |Du|^2(x,t) \leq C/(T-t)$, we prove that a subsequence of the rescaled solutions converges to a homothetically shrinking soliton.

In this paper we characterize and classify surfaces in $\mathbb{H}^2 \times \mathbb{R}$ which have a canonical principal direction. Here $\mathbb{H}^2$ denotes the hyperbolic plane. We study some geometric properties such as minimality and flatness. Several examples are given to complete the study.

We study the nonlinear degenerate problem $-\sum_{i=1}^N \partial_{x_i} a_i \left(x,\partial_{x_i}u \right) = f(x,u)$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega \subset {\mathbb R}^N$ ($N \geq 3$) is a bounded domain with smooth boundary, $\sum_{i=1}^N \partial_{x_i} a_i \left(x,\partial_{x_i}u \right)$ is a $\overset{\rightarrow} p(\cdot)$ - Laplace type operator and the nonlinearity $f$ is $(P_+^+-1)$ - superlinear at infinity (with $\overset{\rightarrow} p(x) = (p_1(x), p_2(x), ... p_N(x))$ and $P_+^+ = \max_{i \in \{1,...,N\}} \left\{\sup_{x \in \Omega} p_i(x) \right\}$). By means of the symmetric mountain-pass theorem of Ambrosetti and Rabinowitz, we establish the existence of a sequence of weak solutions in appropriate anisotropic variable exponent Sobolev spaces.

In this paper, four types of Levitin-Polyak well-posedness of generalized vector equilibrium problems with both abstract set constraints and functional constraints are investigated. Criteria and characterizations for these types of Levitin-Polyak well-posedness of generalized vector equilibrium problems are obtained.

There is a theory of continued fractions for formal power series in $x^{−1}$ with coefficients in a field $\mathbb{F}_q$. This theory bears a close analogy with classical continued fractions for real numbers with formal power series playing the role of real numbers and the sum of the terms of non-negative degree in $x$ playing the role of the integral part. We give a family of cubic power series over $\mathbb{F}_2$ with unbounded partial quotients. To be more precise, let $f \in \mathbb{F}_2((x^{−1}))$ such that $f$ is not polynomial but $f^3$ is polynomial with degreed $d \in 3\mathbb{N}$, we prove that the continued fraction expansion of $f$ is unbounded.

$M$ is said to be a min-coherent (resp. $PS$, $FS$) module if its every simple submodule is finitely presented (resp. projective, flat). In this article, we study the properties of min-coherent, $PS$ and $FS$ modules. Some known results are generalized.

A theorem is proved on oscillation of the components of the eigenvector-functions of a one boundary value problem for the canonical onedimentional Dirac system.

This paper is concerned with the problem of best simultaneous $\tau_C$-approximations in terms of the Minkowski functional. The notions of simultaneous sun and simultaneous regular set are extended to suit the case of simultaneous $\tau_C$-approximation problems. Characterization and uniqueness results are established for simultaneous $\tau_C$-suns.

In this paper, we introduce and investigate each of the following subclasses: $$\mathcal{S}_g(\lambda, \gamma)\,\,\,\, \text{and} \,\,\,\, \mathcal{K}_g(\lambda, \gamma, m; u) \,\,\,\, \Big(0\!\leqq\! \lambda\! \leqq\! 1; u\!\in\! \mathbb{R}\setminus (-\infty, -1]; \ m\in \mathbb{N}\setminus\{1\}\Big) $$ of analytic functions of complex order $\gamma \in \mathbb{C} \setminus \{0\}$, $g: \mathbb{U} \rightarrow \mathbb{C}$ being some suitably constrained convex function in the open unit disk $\mathbb{U}$. We obtain coefficient bounds and coefficient estimates involving the Taylor-Maclaurin coefficients of the function $f(z)$ when $f(z)$ is in the class $\mathcal{S}_g(\lambda,\gamma)$ or in the class $\mathcal{K}_g(\lambda,\gamma,m;u)$. The various results, which are presented in this paper, would generalize and improve those in related works of several earlier authors.