In this paper, we study the geometry associated with Schrödinger operator via Hamiltonian and Lagrangian formalism. Making use of a multiplier technique, we construct the heat kernel with the coefficient matrices of the operator both diagonal and non-diagonal. For applications, we compute the heat kernel of a Schrödinger operator with terms of lower order, and obtain a globally closed solution to a matrix Riccati equations as a by-product. Besides, we finally recover and generalise several classical results on some celebrated operators.

We give some simple criteria for uniform asymptotic stability and exponential asymptotic stability of linear Volterra-Stieltjes differential equations. These criteria are given in terms of the matrix measure or the spectral abscissa of certain matrices derived from the coefficient matrices. An application of obtained results to linear integro-differential equations with delay is presented.

We prove elementary inequalities for the Gram matrices and their equality conditions. As an application we show that inequalities for the Gram determinants hold for general reproducing kernel Hilbert spaces.

We give a new approach of constructing the generalized entropy solutions to the Riemann problem of scalar nonlinear balance laws with singular source terms. The source term is singular in the sense that it is a product of delta function and a discontinuous function, which is undefined in distribution. By re-formulating the source term, we study the corresponding perturbed Riemann problem. The existence and stability of perturbed Riemann solutions is established under some entropy condition so that the generalized entropy solutions of Riemann problem can be interpreted as the limit of corresponding perturbed Riemann solutions. The self-similarity of generalized entropy solutions is also obtained, which means that Lax's method in [13] can be extended to scalar nonlinear balance laws with singular source terms.

The main theme of this paper is the construction of complex powers of $C$-sectorial operators in the setting of sequentially complete locally convex spaces. We consider the constructed powers as the integral generators of equicontinuous analytic $C$-regularized resolvent families, and incorporate the obtained results in the study of incomplete higher order Cauchy problems.

In this paper, we shall adopt topological degree theory and critical point theory to study the existence of weak solutions for the $p$-Laplacian Dirichlet boundary value problem \[\begin{cases} -(|u'|^{p-2} u')' = f(t,u), \; \textrm{in } \Omega, \\ u(0) = u(1) = 0, \end{cases}\] with impulsive conditions $u(t_j^+) - u(t_j^-) = 0$, $\Delta |u'(t_j)|^{p-2} u'(t_j) = I_j(u(t_j))$, $j=1,2,\ldots,n$, where $p \in (1,+\infty)$, $\Omega = (0,1) \backslash \{t_1,\ldots,t_n\}$, $f \in C([0,1] \times \mathbb{R}, \mathbb{R})$ and $I_j \in C(\mathbb{R}, \mathbb{R})$ ($j=1,2,\ldots,n$).

About a decade ago Lutwak, Yang, and Zhang introduced the notion of $L_p$-projection body. More recently, Wang and Leng established an $L_p$-version of Petty's affine projection inequality. At the same time Ludwig discovered a family of general $L_p$-projection bodies and Haberl and Schuster established Petty's projection inequality for general $L_p$-projection bodies. In this paper we establish a general $L_p$-version of Petty's affine projection inequality for general $L_p$-projection bodies. Moreover, we obtain an analogous inequality for $L_p$-geominimal surface area.

As a generalization of covariant completely positive maps, we consider (projective) covariant $\alpha$-completely positive maps between locally $C^*$-algebras. We first study (projective) covariant $J$-representations of locally $C^*$-algebras on Krein modules over locally $C^*$-algebras. Secondly, we construct a covariant KSGNS type representation associated with a covariant $\alpha$-completely positive map on a locally $C^*$-algebra, and then study extensions to a locally $C^*$-crossed product of $\alpha$-completely positive maps on a locally $C^*$-algebra. The results provide (projective) covariant representations of a locally $C^*$-crossed product on a Krein module over a locally $C^*$-algebra.

In this paper, we study the oscillatory behavior of a class of third-order nonlinear delay differential equations $$ (a(t) (b(t) y'(t))')' + q(t) y^\gamma(\tau(t)) = 0. $$ Some new oscillation criteria are presented by transforming this equation to the first-order delayed and advanced differential equations. Employing suitable comparison theorems we establish new results on oscillation of the studied equation. Assumptions in our theorems are less restrictive, these criteria improve those in the recent paper [Appl. Math. Comput., 202 (2008), 102-112] and related contributions to the subject. Examples are provided to illustrate new results.

A new characterization of the exact minimax penalty function method is presented. The exactness of the penalization for the exact minimax penalty function method is analyzed in the context of saddle point criteria of the Lagrange function in the nonconvex differentiable optimization problem with both inequality and equality constraints. Thus, new conditions for the exactness of the exact minimax penalty function method are established under assumption that the functions constituting considered constrained optimization problem are invex with respect to the same function $\eta $ (exception with those equality constraints for which the associated Lagrange multipliers are negative - these functions should be assumed to be incave with respect to the same function $\eta $). The threshold of the penalty parameter is given such that, for all penalty parameters exceeding this treshold, the equivalence holds between a saddle point of the Lagrange function in the considered constrained extremum problem and a minimizer in its associated penalized optimization problem with the exact minimax penalty function.

In this paper, the weighted anisotropic Hardy spaces associated with the general discrete group dilations, the weighted anisotropic atoms and molecules are introduced. Then the molecular decomposition of the weighted anisotropic Hardy spaces is obtained. As an application, the boundedness of Calderón-Zygmund operators on the weighted anisotropic Hardy spaces is discussed.

In this paper we investigate path properties for strictly stationary and linearly positive quadrant dependent (LPQD) or linearly negative quadrant dependent (LNQD) random fields with multidimensional indices taking values in $l^\infty$-space.

Let $p$ be the characteristic of $\mathbb{F}_q$ and let $q$ be a primitive root modulo a prime $r = 2n + 1$. Let $\beta \in \mathbb{F}_{q^{2n}}$ be a primitive $r$th root of unity. We prove that the multiplicative order of the Gauss period $\beta + \beta^{-1}$ is at least $(\log p)^{c\log n}$ for some $c \gt 0$. This improves the bound obtained by Ahmadi, Shparlinski and Voloch when $p$ is very large compared with $n$. We also obtain bounds for "most" $p$.

We survey main results of holomorphic submersions, anti-invariant submersions, slant submersions, semi-invariant submersions and semi-slant submersions defined on almost Hermitian manifolds. We also give an application of Riemannian submersions on redundant robotic chains obtained by Altafini and propose some open problems related to topics discussed in the paper.

Let $T$ be the multilinear Calderón-Zygmund operator and $T_q(\vec{f})$ be the vector-valued version of $T$ given by $T_q(\vec{f})(x) = \bigg( \sum_{k=1}^\infty |T(f_{1k}, \cdots, f_{mk})(x)|^q\bigg)^{{1}/{q}}$. In this paper, the weighted strong type and weighted end-point weak type estimates for the commutators of $T_q(\vec{f})$ were established respectively.

We obtain the weighted anisotropic Hardy space estimate for anisotropic singular integrals of convolution type, and apply it to derive Littlewood-Paley characterization of weighted anisotropic Hardy spaces.

In this paper, we investigate the problem of finding a common element of the solution set of a general system of variational inequalities, the solution set of a convex feasibility problem and the fixed point set of a strict pseudocontraction in a real Hilbert space. Based on the well-known extragradient method, viscosity approximation method and Mann iterative method, we propose and analyze a new relaxed extragradient method for computing a common element. Under very mild assumptions, we obtain a strong convergence theorem for three sequences generated by the proposed method. Our proposed method is quite general and flexible and includes the iterative methods considered in the earlier and recent literature as special cases. Our results represent the modification, supplement, extension and improvement of some corresponding results in the references.

In this paper, a new class of bilevel mixed equilibrium problems (for short, (BMEP)) is introduced and investigated in reflexive Banach space and some topological properties of solution sets for the lower level mixed equilibrium problem and the problem (BMEP) are established without coercivity. Subsequently, we construct a new iterative algorithm which can directly compute some solutions of the problem (BMEP). Some strong convergence theorems of the sequence generated by the proposed algorithm are also presented. Finally, the well-posedness and generalized well-posedness for the problem (BMEP) are introduced by an $\epsilon$-bilevel mixed equilibrium problem. Also, we explore the sufficient and necessary conditions for (generalized) well-posedness of the problem (BMEP) and show that, under some suitable conditions, the well-posedness and generalized well-posedness of (BMEP) are equivalent to the uniqueness and existence of its solutions, respectively. These results are new and improve some recent results in this field.

In this paper, via variational methods and critical point theory, we study the existence of heteroclinic orbits for the following second order nonautonomous Hamiltonian system $$ \ddot{u} - \triangledown{F(t,u)} = 0, $$ where $u \in R^{n}$ and $F \in C^{1}(R \times R^{n}, R)$, $F \geq 0$. $\mathcal{M} \subset R^{n}$ be set of isolated points and $\sharp{\mathcal{M}} \geq 2$. For each $\xi \in \mathcal{M}$, there exists a positive number $\rho_{0}$ such that if $y \in B_{\rho_{0}}(\xi)$, then $F(t,y) \geq F(t,\xi)$ for all $t \in R$, where $B_{\rho_{0}}(\xi) = \{y \in R^{n} \mid \mid y - \xi \mid \lt \rho_{0}\}$. Under some more assumptions on $F(t,x)$ and $\mathcal{M}$, we prove that each point in $\mathcal{M}$ is joined to another point in $\mathcal{M}$ by a solution of our system.