In this paper, the maximal dissipativeone dimensional singular Hamiltonian operators (in limit-circle case atsingular end point $b$) are considered in the Hilbert space $\mathcal{L}_{W}^{2}(\left[ a,b\right) ;\mathbb{C}^{2}) (-\infty\lt a\lt b\leq \infty ).$ The maximal dissipative operators with general boundary conditions are investigated. A selfadjoint dilation of the dissipative operator and its incoming and outgoing spectral representations are constructed. These representations allows us to determine the scattering matrix of the dilation. Further a functional model of the dissipative operator is constructed and its characteristic function in terms of the scattering matrix of dilation is considered. Finally, the theorem on completeness of the system of root vectors of the dissipative operators is proved.

When generalized harmonic functions belong to the weighted Lebesgue classes, we give the asymptotic behaviors of them at infinity in an $n$-dimensional cone. Meanwhile, the integral representations of them are also considered, which imply the known representations of classical harmonic functions in the upper half space.

Let $p$ be an odd prime. It is well known that $F_{p-(\frac {p}{5})} ≡0\; ({\rm mod}\ p)$, where $\{F_n\}_{n \geq 0}$ is the Fibonacci sequence and $(-)$ is the Jacobi symbol. In this paper we show that if $p \not= 5$ then we may determine $F_{p-(p/5)}$ mod $p^3$ in the following way: $$\sum_{k=0}^{(p-1)/2} \frac{\binom{2k}{k}}{(-16)^k} ≡ \left( \frac{p}{5} \right) \left( 1+\frac{F_{p-(\frac{p}{5})}}{2} \right) \pmod{p^3}.$$ We also use Lucas quotients to determine $\sum_{k=0}^{(p-1)/2} \binom{2k}{k}/m^k$ modulo $p^2$ for any integer $m \not≡ 0 \ (\mod\ p)$; in particular, we obtain $$\sum_{k=0}^{(p-1)/2} \frac{\binom{2k}{k}}{16^k} ≡ \left( \frac{3}{p} \right) \pmod{p^2}.$$ In addition, we pose three conjectures for further research.

In this paper, we define translation surfaces in the Riemannian product space $\mathbb{H}^2 \times \mathbb{R}$ and completely classify minimal translation surfaces in $\mathbb{H}^2 \times \mathbb{R}$.

For any prime $p$ and each admissible value $n$, a complete answer to the existence problem for cyclic $3p$-cycle systems of the complete graph $K_n$ is given.

In this paper, we present a recent approach via variational methodsand critical point theory to obtain the existence of solutions forthe nonautonomous second-order system on time scales with impulsive effects \[ \begin{cases} u^{\Delta^2}(t) = \nabla F(\sigma(t),u(\sigma(t))), & \Delta\textrm{-a.e.} \; t \in [0,T]_{\mathbb{T}}^{\kappa}; \\ u(0) - u(T) = \dot{u}(0) - \dot{u}(T) = 0, \\ (u^i)^{\Delta}(t_{j}^{+}) - (u^i)^{\Delta}(t_{j}^{-}) = I_{ij}(u^{i}(t_{j})), & i=1,2,\ldots,N, j=1,2,\ldots,p, \end{cases}\] where $t_{0} = 0 \lt t_{1} \lt t_{2} \lt \ldots \lt t_{p} \lt t_{p+1} = T$, $t_j \in [0,T]_\mathbb{T}$ ($j = 0,1,2,\ldots,p+1$), $u(t) = \big(u^{1}(t), u^{2}(t), \ldots, u^{N}(t)\big) \in \mathbb{R}^N$, $I_{ij} : \mathbb{R} \rightarrow \mathbb{R}$ ($i=1,2,\ldots,N$, $j=1,2,\ldots,p$) are continuous and $F: [0,T]_\mathbb{T} \times \mathbb{R}^{N} \rightarrow \mathbb{R}$. Finally, two examples are presented to illustrate the feasibility andeffectiveness of our results.

In this paper, we introduce a broad class of nonlinear mappings in a Hilbert space which covers nonexpansive mappings, nonspreading mappings, hybrid mappings and contractive mappings. Then we prove fixed point theorems for the class of such mappings. Using these results, we prove well-known and new fixed point theorems in a Hilbert space.

Let $N$ be a near-ring. In this paper, we associate a graph corresponding to the $3$-prime radical $\mathcal{I}$ of $N$, denoted by $\Gamma_\mathcal{I}(N)$. Further we obtain certain topological properties of $Spec(N)$, the spectrum of $3$-prime ideals of $N$ and graph theoretic properties of $\Gamma_\mathcal{I}(N)$. Using these properties, we discuss dominating sets and connected dominating sets of $\Gamma_\mathcal{I}(N)$.

In this paper, we study nonhomogeneous Schrödinger-Poisson systems \[ \begin{cases} -\Delta u + u + K(x) \phi(x) u = a(x) f(u) + h(x), & x \in \mathbb{R}^3, \\ -\Delta \phi = K(x) u^2, & x \in \mathbb{R}^3, \end{cases}\] where $f(t)$ is either asymptotically linear or asymptotically 3-linear with respect to $t$ at infinity. Under appropriate assumptions on $K, a, f$ and $ h$, the existence of two positive solutions of the above system is obtained by using the Ekeland's variational principle and the MountainPass Theorem in critical point theory.

Let $c(H)$ denote the number of components of graph $H$. The scattering number of a graph $G$ is the maximum of $c(G-S)-|S|$ taken over all cut-sets $S$ of $G$. In this note we explore the minimum and maximum scattering number for several families. For example, we show that the minimum scattering number of a triangle-free graph on $n$ vertices is approximately $-n/3$. We also consider the scattering number of some graph products.

In this paper, we introduce and study the concept of $\Delta^{m}-$summable sequence of fuzzy numbers by using a modulus function and $\Delta^{m}-$statistical convergence of sequences fuzzy numbers. Also we have defined $\Delta^{m}-$statistical pre-Cauchy sequences of fuzzy numbers.

In this paper, we introduce the vector generalizations of the well-known Vandermonde's convolution such as \begin{align*} & \sum_{\boldsymbol{j} = \boldsymbol{\alpha_{1}} + \cdots + \boldsymbol{\alpha_{k}}} \sum_{\boldsymbol{\ell} = \boldsymbol{\beta_{1}} + \cdots + \boldsymbol{\beta_{k}}} \prod_{i=1}^{k} \binom{|\boldsymbol{\alpha_{i}}| + |\boldsymbol{\beta_{i}}|}{|\boldsymbol{\alpha_{i}}|} M(\boldsymbol{\alpha_{i}}) M(\boldsymbol{\beta_{i}}) \\ &= M(\boldsymbol{j}) M(\boldsymbol{\ell}) \binom{|\boldsymbol{j}| + |\boldsymbol{\ell}| + k -1}{|\boldsymbol{j}|, |\boldsymbol{\ell}|, k - 1}, \end{align*} where $\boldsymbol{j}$, $\boldsymbol{\ell}$, $\boldsymbol{\alpha_{i}}$ and $\boldsymbol{\beta_{i}}$ are the vectors with nonnegative integer components, and $M(\boldsymbol{j})$ is the multinomial coefficient defined by $\binom {j_1+j_2+ \cdots +j_m}{j_1,\ldots, j_m}$ with the value $\frac {|j|!}{j_1! \cdots j_m!}$. The main interest in such generalization comes from the number of multiple zeta values in relations produced from the shuffle product of two sets of multiple zeta values in their iterated integral representations over simplices. Several generalizations of Vandermonde's convolution type are given as well.

This paper deals with the approximate controllability of fractional order neutral stochastic integro-differential system with nonlocal conditions and infinite delay in Hilbert spaces under the assumptions that the corresponding linear system is approximately controllable. The control function for this system is suitably constructed by using the infinite dimensional controllability operator. With this control function, the sufficient conditions for approximate controllability of the proposed probelm in Hilbert space is established. Further, the results are obtained by using fractional calculus, stochastic analysis techniques, Sadovskii fixed point theorem and similar to the classical linear growth condition and the Lipschitz condition. Finally an example is provided to illustrate the application of the obtained results.

Using the theory of measures of noncompactness and applying a new method, we prove the existence of nondecreasing solutions of a quadratic integral equation of Volterra type in $C(I)$.

In this paper, stability results for generalized $f$-projection operators with parametric perturbations are given in reflexive, smooth, and strictly convex Banach spaces. By using the stability of generalized $f$-projection operators and the degree theory for the generalized set-valued variational inequality introduced by Wang and Huang [30], the lower continuity of the solution mapping for the parametric generalized set-valued variational inequality is established under some suitable conditions.

In this paper, the authors characterize the boundedness and compactness of the following Riemann-Stieltjes operator $$ L_g(f)(z) = \int_0^1 {\cal R} f(tz) g(tz) \, \frac{dt}{t}, z \in B,$$ where ${\cal R} f(z)$ is the radial derivative of function $f$ at $z$, from mixed norm spaces $H(p,\, q,\, \phi)$ to Zygmund-type spaces on the unit ball.

Using multiple stochastic integrals, we study the chaotic expansion and the regularity in the Sobolev-Watanabe spaces of the local time of the solution to the stochastic heat equation with fractional-colored noise.

In this paper, we prove that two nonconstant meromorphic functions are identical provided that they share five distinct small functions IM in some angular domains.

We derive one unified formula for Ricci curvature tensor on arbitrary warped product manifold (WPM) by introducing a new notation for the lift vector and the Levi-Civita connection. Using well-established formula, we consider two questions on WPM related to Ricci flow (RF) and hyperbolic geometric flow (HGF). Firstly, we discuss the preserved flow-type problem which says that when the first factor $(M,g)$ and the second factor $(N,h)$ are solutions to the RF (or HGF), the singly WPM $M \times_\lambda N$ is still solution to the RF (or HGF). We obtain some characteristic PDEs satisfied by warping function and also construct some simple examples. Next, we discuss the evolution equations for warping function $\lambda$ and Ricci curvature tensor etc. under RF/HGF. We gain some interesting results, especially adding an assumption with Einstein metric to the second factor.

In this paper we are concerned with an extension operator $\Phi_{n,\alpha,\beta}$ that provides a way of extending a locally univalent function $f$ on the unit disc $U$ to a locally biholomorphic mapping $F\in H(B^{n})$. By using the method of Loewner chains, we prove that if $f$ can be embedded as the first element of a $g$-Loewner chain on the unit disc, where $g(\zeta)=\frac{1-\zeta}{1+(1-2\gamma)\zeta}$ for $|\zeta|\lt 1$ and $\gamma \in (0,1)$, then $F=\Phi_{n,\alpha,\beta}(f)$ can also be embedded as the first element of a $g$-Loewner chain on $B^n$, whenever $\alpha\in [0,1]$, $\beta \in [0,1/2]$, $\alpha +\beta \leq 1$. In particular, if $f$ is starlike of order $\gamma \in (0,1)$ on $U$, then $F=\Phi_{n,\alpha,\beta}(f)$ is also starlike of order $\gamma$ on $B^n$. Also, if $f$ is spirallike of type $\delta$ and order $\gamma$ on $U$, where $\delta\in (-\pi/2,\pi/2)$ and $\gamma \in (0,1)$, then $F=\Phi_{n,\alpha,\beta}(f)$ is spirallike of type $\delta$ and order $\gamma$ on $B^n$. We also obtain a subordination preserving result under the operator $\Phi_{n,\alpha,\beta}$ and we consider some radius problems associated with this operator.

The split feasibility problem and fixed point problem is considered. New algorithm is presented for solving this split problem. Some analytical techniques are demonstrated and strong convergence results are obtained.