In this paper, we obtain some properties of signless Laplacian eigenvalues of general hypergraphs. We give the upper and the lower bound of edge connectivity of general hypergraphs in terms of average degree, minimum degree, the rank and the number of vertices, or analytic connectivity $\alpha(G)$, respectively. We also give the upper bound of analytic connectivity $\alpha(G)$ of general hypergraphs in terms of the degrees of vertices. Finally, we obtain the bounds of the smallest $H^{+}$-eigenvalue of the normalized Laplacian sub-tensors of general hypergraphs.

This paper was retracted by the publisher as it has numerous unfixable errors. In particular, the main result of the paper, Corollary 3.10, is unfortunately false.

Sharp multi-dimensional Hardy's inequality for the Laguerre functions of Hermite type is proved for the type parameter $\alpha \in [-1/2,\infty)^d$. As a consequence we obtain the corresponding result for the generalized Hermite expansions. In particular, it validates that the known version of Hardy's inequality for the Hermite functions is sharp.

In this paper we obtain almost sharp decay estimates for $L^2$ operator norm of strongly singular oscillatory integral operators in $\mathbb{R}^{n+1}$ for $n \geq 2$; we prove some necessary condition for $L^2$ estimates. Also, we prove that the operators are bounded on $L^p$ for some $p \neq 2$ and the range of $p$ depends on the hypersingularity of the operators.

The purpose of this paper is to give a well-posedness result for a boundary value problem of transmission type for the Stokes and generalized Brinkman systems in two complementary Lipschitz domains in $\mathbb{R}^3$. In the first part of the paper, we have introduced the classical and weighted $L^2$-based Sobolev spaces on Lipschitz domains in $\mathbb{R}^3$. Afterwards, the trace and conormal derivative operators are defined in the case of both Stokes and generalized Brinkman systems. Also, a summary of the main properties of the layer potential operators for the Stokes system, is provided. In the second part of the work, we exploit the well-posedness of another transmission problem concerning the Stokes system on two complementary Lipschitz domains in $\mathbb{R}^3$ which is based on the Potential Theory for the Stokes system. Then, certain properties of Fredholm operators will allow us to show our main well-posedness result in $L^2$-based Sobolev spaces.

Let $H(\mathbb{D})$ denote the space of holomorphic functions on the unit disk $\mathbb{D}$ of $\mathbb{C}$, $\psi,\varphi \in H(\mathbb{D})$, $\varphi(\mathbb{D}) \subset \mathbb{D}$ and $n \in \mathbb{N} \cup \{0\}$. Let $C_{\varphi}$, $M_{\psi}$ and $D^n$ denote the composition, multiplication and iterated differentiation operators, respectively. To treat the operators induced by products of these operators in a unified manner, we introduce a sum operator $\sum_{j=0}^n M_{\psi_j} C_{\varphi} D^j$. We characterize the boundedness and compactness of this sum operator mapping from a large class of Banach spaces of holomorphic functions into the $k$th weighted-type space $\mathcal{W}_{\mu}^{(k)}$ (or $\mathcal{W}_{\mu,0}^{(k)}$), $k \in \mathbb{N} \cup \{0\}$, and give its estimates of norm and essential norm. Our results show that the boundedness and compactness of the sum operator depend only on the symbols and the norm of the point-evaluation functionals on the domain space. Our results cover many known results in the literature. Moreover, we introduce the order boundedness of the sum operator and turn its study into that of the boundedness and compactness.

In the present paper, it is shown that the linear wave equation subject to Dirichlet boundary condition \[ u_{tt} - u_{xx} + \varepsilon V(\omega t, x) u = 0, \quad u(t,-\pi) = u(t,\pi) = 0 \] can be changed by a symplectic transformation into \[ v_{tt} - v_{xx} + \varepsilon M_{\xi} v = 0, \quad v(t,-\pi) = v(t,\pi) = 0, \] where $V$ is finitely smooth and time-quasi-periodic potential with frequency $\omega \in \mathbb{R}^n$ in some Cantor set of positive Lebeague measure and where $M_{\xi}$ is a Fourier multiplier. Moreover, it is proved that the corresponding wave operator $\partial_t^2 - \partial_x^2 + \varepsilon V(\omega t, x)$ possesses the property of pure point spectra and zero Lyapunov exponent.

In this paper, we continue the investigations done in the literature about the so called Bohr-Neugebauer property for almost periodic differential equations. More specifically, for a class of functional heat equations, we prove that each $L^2$-bounded solution is almost periodic. This extends a result in [5] to the delay case.

In this paper, we study an integral system \[ \begin{cases} u_{i}(x) = K_{i}(x) (|x|^{\alpha-n} \ast u^{p_{i+1}}_{i+1})(x), &i = 1,2,\ldots,m-1, \\ u_{m}(x) = K_{m}(x) (|x|^{\alpha-n} \ast u^{p_{1}}_{1})(x). \end{cases} \] When $\alpha \in (0,n)$, $p_{i} \gt 0$ ($i = 1,2,\ldots,m$), the Serrin-type condition is critical for existence of positive solutions for some double bounded functions $K_{i}(x)$ ($i = 1,2,\ldots,m$). When $\alpha \in (0,n)$, $p_{i} \lt 0$ ($i = 1,2,\ldots,m$), the system has no positive solution for any double bounded $K_{i}(x)$ ($i = 1,2,\ldots,m$). When $\alpha \gt n$, $p_{i} \lt 0$ ($i = 1,2,\ldots,m$), and $\max_{i} \{-p_{i}\} \gt \alpha/(\alpha-n)$, then the system exists positive solutions increasing with the rate $\alpha-n$.

Our main interest in this work is to detect a rigid inclusion immersed in an isotropic elastic body $\Omega$ from a single pair of Cauchy data on $\partial \Omega$ in two dimensions. We want to completely characterize the unknown rigid inclusion, namely, the shape and the location of inclusion. The idea is to rewrite the inverse problem as an optimization problem, where an energy like functional is minimized with respect to the presence of a small inclusion. A topological sensitivity analysis is derived for an energy like functional. We proposed a non-iterative reconstruction algorithm based on the topological gradient concept. The unknown rigid inclusion is defined by a level curve of a scalar function. The proposed numerical approach is very robust with respect to noisy data. Finally, in order to show the efficiency and accuracy of the proposed algorithm, we present some numerical results.

We investigate the boundary behavior of variational solutions of Dirichlet problems for prescribed mean curvature equations at smooth boundary points where certain boundary curvature conditions are satisfied (which preclude the existence of local barrier functions). We prove that if the Dirichlet boundary data $\phi$ is continuous at such a point (and possibly nowhere else), then the solution of the variational problem is continuous at this point.

This paper concerns the problem of modifying edge lengths of a network at minimum total costs so as to make a prespecified vertex become an optimal location in the modified environment. Here, we focus on the ordered median objective function with respect to the vector of multipliers $\lambda = (1,\ldots,1,0,\ldots,0)$ with $k$ 1's. This problem is called the inverse anti-$k$-centrum problem. We first show that the inverse anti-$k$-centrum problem is NP-hard even on tree networks. However, for the inverse anti-$k$-centrum problem on cycles, we formulate it as one or two linear programs, depending on odd or even integer $k$. Concerning the special cases with $k = 2,3,M$, we develop combinatorial algorithms that efficiently solve the problem, where $M$ is the number of vertices of the cycle.