In this paper, we give a classification on spacelike linear Weingarten hypersurfaces in de Sitter space $S^{n+1}_{1}(1)$ according to the sectional curvature or the length of the second fundamental form.

A structure theorem for spherically symmetric associated homogeneous distributions (SAHDs) based on $R^{n}$ is given. It is shown that any SAHD is the pullback, along the function $\left\vert \mathbf{x}\right\vert ^{\lambda }$,\ $\lambda \in \mathbf{C}$, of an associated homogeneous distribution (AHD) on $R$. The pullback operator is found not to be injective and its kernel is derived (for $\lambda =1$). Special attention is given to the basis SAHDs, $D_{z}^{m}\left\vert \mathbf{x}\right\vert ^{z}$, which become singular when their degree of homogeneity $z=-n-2p$, $\forall p\in \mathbb{N}$. It is shown that $\left( D_{z}^{m}\left\vert \mathbf{x} \right\vert ^{z}\right) _{z=-n-2p}$ are partial distributions which can be non-uniquely extended to distributions $\left( \left( D_{z}^{m}\left\vert \mathbf{x}\right\vert ^{z}\right) _{e}\right) _{z=-n-2p}$ and explicit expressions for their evaluation are derived. These results serve to rigorously justify distributional potential theory in $R^{n}$.

Chebyshev subspaces of $ \mathcal{K}(c_0, c_0)$ are studied. A k-dimensional non-interpolating Chebyshev subspace is constructed. The unicity of best approximation in non-Chebyshev subspaces is considered.

We describe an explicit model for the blow-up construction in the smooth (or real analytic) category. We use it to prove the following functoriality property of the blow-up: Let $M$ and $N$ be smooth (real analytic) manifolds, with submanifolds $A$ and $B$ respectively. Let $f\colon M\to N$ be a smooth (real analytic) function such that $f^{-1}(B)=A$, and such that $f$ induces a fiberwise injective map from the normal space of $A$ to the normal space of $B$. Then $f$ has a unique lift to a smooth (real analytic) map between the blow-ups. In this way, the blow-up construction defines a continuous functor. As an application, we show how an action of a Lie group on a manifold lifts, under minimal hypotheses, to an action on a blow-up.

We study positive solutions to the steady state reaction diffusion equation with Dirichlet boundary condition of the form: \begin{equation} \left\{ \begin{aligned} -\Delta u &= au-bu^2-c \dfrac{u^p}{1+u^p}-K, \quad x \in \Omega \\u &= 0, \quad x \in\partial\Omega. \end{aligned} \right. \end{equation} Here $\Delta u=div \big(\nabla u\big)$ is the Laplacian of u, $a, b, c, p, K$ are positive constants with $p\geq2$ and $\Omega$ is a smooth bounded region with $\partial\Omega$ in $C^2$. This model describes the steady states of a logistic growth model with grazing and constant yield harvesting. It also describes the dynamics of the fish population with natural predation and constant yield harvesting. We study the existence of positive solutions to this model. We prove our results by the method of sub-super solutions.

Some existence theorems are obtained by the least action principle for periodic solutions of nonautonomous second-order differential systems with $(q,p)$--Laplacian.

In this paper, we show that certain families with relative property (T) have trivial first $\ell^2$-Betti number. We apply this to the elementary matrix group $\textnormal{EL}_n(\mathcal{R})$ where $\mathcal{R}$ is any countable unital ring of characteristic 0.

In this paper we classify the centers, the cyclicity of its Hopf bifurcation and the isochronicity of the polynomial differential systems in $\mathbb{R}^2$ of degree $d\geq 7$ odd that in complex notation $z=x+ i y$ can be written as \[ \dot z = (\lambda+i) z + (z \overline z)^{\frac{d-7}2} (A z^6 \overline z + B z^4 \overline z^3 + C z^2 \overline z^5 +D \overline z^7), \] where $\lambda \in \mathbb{R}$, and $A,B,C,D \in \mathbb{C}$.

We study an anisotropic partial differential equation on a bounded domain $\Omega\subset\mathbb R^N$. We prove the existence of at least two nontrivial weak solutions using as main tools the mountain pass lemma and Ekeland's variational principle.

Let us deal with the positive solutions of \begin{equation*} \frac{\partial u(t)}{\partial t}=k(t)\Delta _{\alpha }u(t)+h(t)u^{1+\beta }(t),\text{ \ }u(0,x)=\varphi (x)\geq 0,\text{ }x\in \mathbb{R}^{d}, \end{equation*} where $\Delta _{\alpha }$ is the fractional Laplacian, $0<\alpha \leq 2$, and $\beta >0$ is a constant. We prove that under certain regularity condition on $\varphi $, $h$ and $k$ any non-trivial positive solution blows up in finite time. In this way we answer, in particular, the question raised in [4] for the critical case.

The theory of approach spaces has set the context in which numerical topological concepts exist. The successful interaction between frames and topology on the one hand and the search for a good notion of sobriety in the context of approach theory on the other hand was the motivation to develop a theory of approach frames. The original definition of approach frames was given in terms of an implicitly defined set of equations. In this work, we describe a subset of this by a finite axiom scheme (of only six types of equations) which implies all the equations originally involved and hence provides a substantial simplification of the definition of approach frames. Furthermore we show that the category of approach frames is the Eilenberg-Moore category for the monad determined by the functor which takes each approach frame to the set of its regular functions.

{\em Quasi-quadrics} were introduced by Penttila, De Clerck, O'Keefe and Hamilton in [2]. They are defined as point sets which have the same intersection numbers with respect to hyperplanes as non-singular quadrics. We extend this definition in two ways. The first extension is to {\em quasi-Hermitian varieties}, which are point sets which have the same intersection numbers with respect to hyperplanes as non-singular Hermitian varieties. The second one is to {\em singular quasi-quadrics}, i.e. point sets $\mathcal{K}$ which have the same intersection numbers with respect to hyperplanes as singular quadrics. Our starting point was to investigate whether every singular quasi-quadric is a cone over a non-singular quasi-quadric. This question is tackled in the case of a point set $\mathcal{K}$ with the same intersection numbers with respect to hyperplanes as a point over an ovoid.

In this paper, using the Kakutani-Fan-Glicksberg fixed point theorem, we obtain an existence theorem of a point which is simultaneously fixed point for a given mapping and equilibrium point for a very general vector equilibrium problem. Finally some particular cases are discussed and three applications are given.

The aim of this paper is to prove the complete monotonicity of a class of functions arising from Kazarinoff's inequality [Edinburgh Math. Notes 40 (1956) 19-21]. As applications, new sharp inequalities for the gamma and digamma functions are established.

In this paper, we investigate the uniqueness problem of meromorphic functions sharing four distinct values $IM$ and a finite set in an angular domain. One result which we obtained generalizes and extends the former results.

Some results and criteria of existence concerning bounded approximate identities in Banach algebras are extended to the topological algebras setting. We thereby prove that the bidual of a commutative locally C*-algebra with either of the two Arens products is a unital commutative algebra, and that a quasinormable Fréchet m-convex algebra has a left (resp. right) bounded approximate identity if and only if it can be represented as an inverse limit of Banach algebras each of which has a left (resp. right) bounded approximate identity.