[8] is a deep study in the sharp lower bound estimate of the gap for the first two eigenvalues in Schrödinger operator on a smooth bounded convex domain in $\mathbb{R}^n$. In this paper we give another simple proof of the main result in [8]. Although the methods used in here due to [8] on the whole, to some extent we deal with the singularity of some function and also simplify greatly calculation in [8].

We study hypersurfaces either in the pseudo-Riemannian De Sitter space $\mathbb{S}_t^{n+1} \subset \mathbb{R}_t^{n+2}$ orin the pseudo-Riemannian anti De Sitter space $\mathbb{H}_t^{n+1} \subset \mathbb{R}_{t+1}^{n+2}$ whose position vector $\psi$ satisfies the condition $L_k \psi = A \psi + b$, where $L_k$ is the linearized operator of the $(k+1)$-th mean curvature of the hypersurface, for a fixed $k=0,\dots,n-1$, $A$ is an $(n+2) \times (n+2)$ constant matrix and $b$ is a constant vector in the corresponding pseudo-Euclidean space. For every $k$, we prove that when $H_k$ is constant, the only hypersurfaces satisfying that condition are hypersurfaces with zero $(k+1)$-th mean curvature and constant $k$-th mean curvature, open pieces of a totally umbilical hypersurface in $\mathbb{S}^{n+1}_t$ ($\mathbb{S}^n_{t-1}(r)$, $r \gt 1$; $\mathbb{S}^n_t(r)$, $0 \lt r \lt 1$; $\mathbb{H}^n_{t-1}(-r)$, $r \gt 0$; $\mathbb{R}^n_{t-1}$), open pieces of a totally umbilical hypersurface in $\mathbb{H}^{n+1}_t$ ($\mathbb{H}^n_t(-r)$, $r \gt 1$; $\mathbb{H}^n_{t-1}(-r)$, $0 \lt r \lt 1$; $\mathbb{S}^n_t(r)$, $r \gt 0$; $\mathbb{R}^n_t$), open pieces of a standard pseudo-Riemannian product in $\mathbb{S}_t^{n+1}$ ($\mathbb{S}_u^m(r) \times \mathbb{S}^{n-m}_v(\sqrt{1-r^2})$, $\mathbb{H}^m_{u-1}(-r) \times \mathbb{S}^{n-m}_v(\sqrt{1+r^2})$, $\mathbb{S}^m_u(r) \times \mathbb{H}^{n-m}_{v-1}(-\sqrt{r^2-1})$), open pieces of a standard pseudo-Riemannian product in $\mathbb{H}_t^{n+1}$ ($\mathbb{H}_u^m(-r) \times \mathbb{S}^{n-m}_v(\sqrt{r^2-1})$, $\mathbb{S}_u^m(r) \times \mathbb{H}^{n-m}_v(-\sqrt{1+r^2})$, $\mathbb{H}^m_u(-r) \times \mathbb{H}^{n-m}_{v-1}(-\sqrt{1-r^2})$) and open pieces of a quadratic hypersurface $\{x \in \mathbb{M}^{n+1}_t(c) \mid \langle Rx,x \rangle = d\}$, where $R$ is a self-adjoint constant matrix whose minimal polynomial is $\mu_R(z) = z^2 + az + b$, $a^2 - 4b \leq 0$, and $\mathbb{M}^{n+1}_t(c)$ stands for $\mathbb{S}_t^{n+1} \subset \mathbb{R}_t^{n+2}$ or $\mathbb{H}_t^{n+1} \subset \mathbb{R}_{t+1}^{n+2}$.

Variational relation problems were introduced by Luc in [1] as a general model fora large class of problems in nonlinear analysis and applied mathematics. Since this manner ofapproach provides unied results for several mathematical problems it has been used in manyrecent papers (see [2-9]). In this paper we investigate the existence of solutions for three typesof variational relation problems which encompass several generalized equilibrium problems, variational inequalities and variational inclusions studied in a long list of papers in the eld.

We prove dimension free $L^p$ estimates for Riesz transforms associated with multi-dimensional Laguerre function expansions of Hermite type. The range of the admissible Laguerre type multi-index $\alpha$ in these estimates depends on $p \in (1,\infty)$; for $1 \lt p \le 2$ this range is almost optimal. The proof is based on suitably defined square functions with Poisson and modified Poisson semigroups involved.

Over a field of characteristic $p \gt 2$; the low-dimensional cohomology groups of the special linear Lie superalgebra $\mathbf{A}(1;0)$ with coefficients in Witt superalgebra are computed by means of a direct sum decomposition of submodules and the weight space decomposition of the Witt superalgebra viewed as $\mathbf{A}(1;0)$-module. As application, the lowdimensional cohomology groups of $\mathbf{A}(1;0)$ with coefficients in Special superalgebra are also determined.

Let $M^{2}$ be a compact $F$-Willmore surface in the $n$-dimensional space form $\mathbb{N}^{n}(c)$ of constant curvature $c$. Denote by $\phi_{ij}^{\alpha}$ the trace free part of the second fundamental form $h=(h_{ij}^{\alpha})$, and by $\mathbf{H}$ the mean curvature vector of $M^{2}$. Let $\Phi$ be the square of the length of $\phi_{ij}^{\alpha}$ and $H = |\mathbf{H}|$. If $F^{\prime}(\Phi)\geq 0$, then $\int_{M} \Big\{F^{\prime\prime}(\Phi)\Big[\frac{1}{2}|\nabla\Phi|^{2} - \sum_{\alpha,i,j} \phi_{ij}^{\alpha}\Phi_{j}H_{i}^{\alpha}\Big]+F(\Phi)H^{2} + F^{\prime}(\Phi) (2c-K(n)\Phi)\Phi\Big\} dv \leq 0$. The constant function $K(n) = 1$ when $n=3$ and $K(n) = \frac{3}{2}$ when $n\geq 4$. Similarly, $\int_{M} \Big\{F^{\prime\prime}(\Phi) \Big[\frac{1}{2}|\nabla\Phi|^{2} - \sum_{\alpha,i,j} \phi_{ij}^{\alpha} \Phi_{j}H_{i}^{\alpha}\Big] + F(\Phi)H^{2} + F^{\prime}(\Phi) (2c-K(n)\Phi)\Phi\Big\} dv \geq 0$, if $F^{\prime}(\Phi) \leq 0$. We also prove the following: If $M^{2}$ is a compact Willmore surface in the $n$-dimensional space form $\mathbb{N}^{n}(c)$. Then $\int_{M}\Phi (C(n)(c+\frac{H^{2}}{2})-\Phi)\leq 0$, where $C(n)=2$ when $n=3$ and $C(n)=\frac{4}{3}$ when $n\geq 4$. If $0\leq\Phi\leq C(n)(c+\frac{H^{2}}{2})$, then either $\Phi=0$ and $M$ is totally umbilical sphere, or $\Phi=C(n)(c+\frac{H^{2}}{2})$. In the latter case, either $M$ is the Clifford torus in $S^{3}$ of $\mathbb{N}^{n}(c)$, or $M$ is the Veronese surface in $S^{4}$ of $\mathbb{N}^{n}(c)$.

In this paper, we consider the free R-module R⊕R, where R is an arbitrary commutative ring with identity. We give a full characterization for prime submodules of R⊕R and a useful primeness test for a finitely generated submodule of R⊕R. We study the existence of primary decomposition of a submodule of R⊕R and characterize the minimal primary decomposition. As applications of our results, we give some examples of primary decompositions in R⊕R.

We study the $(W^{2,p},W^{1,p})$-mild well-posedness of the second order differential equation $(P_2): u'' = Au+f$ on the real line $\mathbb{R}$, where $A$ is a densely defined closed operator on a Banach space $X$. We completely characterize the $(W^{2,p}, W^{1,p})$-mild well-posedness of $(P_2)$ by $L^p$-Fourier multiplier defined by the resolvent of $A$.

In this paper, we obtain the complete characterizations of the boundedness and compactness of the products of the multiplication and the radial derivative operator ${\cal R} M_u$ from $F(p,q,s)$ to weighted-type spaces on the unit ball.

Using the discrete Calderón reproducing formula in [9] and the Plancherel-Pôlya characterization of the inhomogeneous Besov and Triebel-Lizorkin spaces developed in [5], we prove pointwise multipliers on inhomogeneous Besov and Triebel-Lizorkin spaces in the setting of spaces of homogeneous type.

In this paper, we determine the distance 3 labeling number for the direct product $G = K_n \times K_m \times K_2$ ($n \geq m \geq 3$) of 3 complete graphs under various conditions on $j$ and $k$. As a quence, we have the radio number $\textrm{rn}(G) = 2mn-1$.

In this paper, we examine various reproducing kernel Hilbert spaces $\mathcal{H}_{K_1}$ and $\mathcal{H}_{K_2}$ such that the inequality \[\det \left[\langle F_i G_i, F_j G_j \rangle_{\mathcal{H}_{K_1 K_2}} \right]_{i,j=1}^m \le C \det \left[\langle F_i,F_j \rangle_{\mathcal{H}_{K_1}} \langle G_i,G_j \rangle_{\mathcal{H}_{K_2}} \right]_{i,j=1}^m\] holds for all $F_j \in \mathcal{H}_{K_1}$, $G_j \in \mathcal{H}_{K_2}$, where $m$ is a positive integer, $C$ is a constant which is independent on $F_j$ and $G_j$ for all $j=1,2,...,m,$ and $\mathcal{H}_{K_1 K_2}$ is the Hilbert space admitting the reproducing kernel $K_1 K_2$.

Let $\tau = (\cal T, \cal F)$ be a torsion theory and $M$ an $R$-module. $M$ is a $\tau$-lifting module if for any submodule $N$ of $M$ there exists a decomposition $M = A \oplus B$ such that $A \leq N$ and $N \cap B$ is $\tau$-small in $M$. This definition unifies several definitions on generalizations of lifting property of modules. The aim of the present paper is to investigate properties of $\tau$-lifting modules. Various results are developed, many extending known results.

Combining the Skew-Hamiltonian Isotropic implicitly Restarted Arnoldi algorithm (SHIRA) due to Mehrmann and Waktins and the refined projection principle proposed by the first author, we present a Skew-Hamiltonian Isotropic implicitly Restarted Refined Arnoldi algorithm (SHIRRA) for the skew-Hamiltonian/Hamiltonian (SHH) pencil eigenvalue problem. Within SHIRRA, we propose new shifts, called refined shifts, that are theoretically better and numerically more efficient than the exact shifts used within SHIRA. Numerical examples illustrate the efficiency and superiority of SHIRRA.

An independent set of a graph $G$ is a subset of the vertices of $G$ such that no two vertices in the subset are joined by an edge in $G$. The independence number of $G$ is the cardinality of a maximum independent set of $G$. In this paper we give the exact value of the independence number for a (claw, $K_4$)-free 4-regular graph G without cut vertices.

In this paper, by using a alternative theorem, we establish Lagrangian conditions and duality results for set-valued vector optimization problems when the objective and constant are nearly cone-subconvexlike multifunctions in the sense of $E$-weak minimizer.

Considering two different non-convexity measures, we obtain some new Hardy-type inequalities for non-convex domains $\Omega \subset \mathbb{R}^n$. We study the three-dimensional case and then generalise the approach to the $n-$dimensional case.

This paper is concerned with a class of problem of best simultaneous approximations from $RS$-sets in Banach spaces $X$. It is shown that the best simultaneous approximations from an $RS$-set is strongly unique in the case when $X$ is a real Banach space, and strongly unique of order $\alpha \geq 2$ in the case when $X$ is a complex Banach space.

In this paper we prove the variation of parameters formula for linear Volterra integro-differential equations driven by multifractional Brownian motion. To do this, an approximate result for the Stratonovich stochastic integral with respect to the multifractional Brownian motion is given. Based on our obtained results we study almost surely exponentially convergence of the solution. Also, the existence and uniqueness of the solution of a multifractional Volterra integro-differential equation with time delay are proved.

We characterize characteristic polynomials of elements in a central simple algebra. We also give an account for the theory of rational canonical forms for separable linear transformations over a central division algebra, and a description of separable conjugacy classes of the multiplicative group.

The goal of this paper is to show a generalization to Sturm--Picone theorem for a pair of second-order nonlinear differential equations \begin{gather*} (p_1(t) x'(t))' + q_1(t) f_{1}(x(t)) = 0. \\ (p_2(t) y'(t))' + q_2(t) f_{2}(y(t)) = 0, \; t_1 \lt t \lt t_2. \end{gather*} This work generalizes well-known comparison theorems [C. Sturm, J. Math. Pures. Appl. 1 (1836), 106--186; M. Picone, Ann. Scoula Norm. Sup. Pisa, 11 (1909),39; W. Leighton, Proc. Amer. Math. Soc.13 (1962), 603--610], which play a key role in the qualitative behavior of solutions. We establish the generalization to a pair of nonlinear singular differential equations and elliptic partial differential equations also. We show generalization via the quadratic functionals associated to the above pair of equations. The celebrated Sturm--Picone theorem for a pair of linear differential equations turns out to be a particular case of our result. We also use these comparison theorems to ensure the oscillatory as well as nonoscillatory behavior of solutions for a class of nonlinear equations.