In this work we consider a complete spacelike submanifold $M^{n}$ immersed in the De Sitter space $S_{p}^{n+p}(1)$ with parallel mean curvature vector. We use a Simons type inequality to obtain some rigidity results characterizing umbilical submanifolds and hyperbolic cylinders in $S_p^{n+p}(1)$.

Let $\Delta$ denote the (2,3,7)-group. We establish an upper bound for the number of congruence subgroups of index $n$ and a lower bound for the total number of subgroups of index $n$. Since the latter grows more quickly, there exist non-congruence subgroups of index $n$ for all $n$ greater than some $n_0$.

Let $\alpha$ be an irrational number and $\varphi$: $\mathbb{N} \to \mathbb{R^+}$ be a decreasing sequence tending to zero. Consider the set \[E_{\varphi}(\alpha)=\{\beta \in \mathbb{R}: \ \|n \alpha- \beta\|<\varphi(n)\ {\rm {holds\ for\ infinitely\ many}} \ n \in \mathbb{N}\}\], where $\|{\cdot}\|$ denotes the distance to the nearest integer. We show that for general error function $\varphi$, the Hausdorff dimension of $E_{\varphi}(\alpha)$ depends not only on $\varphi$, but also heavily on $\alpha$. However, recall that the Hausdorff dimension of $E_{\varphi}(\alpha)$ is independent of $\alpha$ when $\varphi(n) = n^{-\gamma}$ with $\gamma >1$.

We construct a compactly generated, totally disconnected, locally compact group whose Hecke algebra with respect to any compact open subgroup does not have a $C^*$-enveloping algebra.

It has been proved by D. E. Cohen [1] that the lattice of all varieties of metabelian groups is countable. In this paper, we show that the lattice of all varieties of completely simple semigroups with metabelian subgroups has the cardinality of the continuum. M. Petrich and N. R. Reilly have introduced in [6] the notion of near varieties of idempotent generated completely simple semigroups. The mapping assigning to every variety $\mathcal{V}$ of completely simple semigroups the class of all idempotent generated members of $\mathcal{V}$ is a complete lattice homomorphism of the lattice of all varieties of completely simple semigroups onto the lattice of all near varieties of idempotent generated completely simple semigroups. In this paper we show that, in fact, the lattice of all near varieties of idempotent generated completely simple semigroups with metabelian subgroups has itself the cardinality of the continuum.

It is proved that every abelian VNL-ring is an SVNL-ring, which gives a positive answer to a question of Osba et al. [7]. Some characterizations of duo VNL-rings are given and some main results of Osba et al. [7] on commutative VNL-rings are extended to right duo VNL-rings and even abelian GVNL-rings.

We consider expansions with respect to the multi-dimensional Hermite functions and to the multi-dimensional Hermite polynomials. They are respectively eigenfunctions of the Harmonic oscillator $\cal{L}= -\Delta +|x|^2$ and of the Ornstein-Uhlenbeck operator ${\bf L} = -\Delta +2x \cdot \nabla.$ The corresponding heat semigroups and Riesz transforms are considered and results on both aspects (polynomials and functions) are obtained.

We study a partial differential equation on a bounded domain $\Omega\subset\mathbb{R}^N$ with a $p(x)$-growth condition in the divergence operator and we establish the existence of at least two nontrivial weak solutions in the generalized Sobolev space $W_0^{1,p(x)}(\Omega)$. Such equations have been derived as models of several physical phenomena. Our proofs rely essentially on critical point theory combined with corresponding variational techniques.

We combine the theory of sectorial sesquilinear forms with the theory of unbounded subnormal operators in Hilbert spaces to characterize the Friedrichs extensions of multiplication operators (with analytic symbols) in certain functional Hilbert spaces. Such characterizations lead to abstract Galerkin approximations and generalized wave equations.

We introduce the concept of left APP-rings which is a generalization of left p.q.-Baer rings and right PP-rings, and investigate its properties. It is shown that the APP property is inherited by polynomial extensions and is a Morita invariant property.

We consider a one-dimensional weakly damped wave equation, with a damping coefficient depending on the displacement. We prove the existence of a regular connected global attractor of finite fractal dimension for the associated dynamical system, as well as the existence of an exponential attractor.

In this paper we present some results about $wV$ (weak property $V$ of Peł czyński) or property $wV^*$ (weak property $V^*$ of Peł czyński) in Banach spaces. We show that $E$ has property $wV$ if for any reflexive subspace $F$ of $E^*$, $^{\perp} {F}$ has property $wV$. It is shown that $G$ has property $wV$ if under some condition $K_{w^*}(E^*, F^*)$ contains the dual of $G$. Moreover, it is proved that $E^*$ contains a copy of $c_0$ if and only if $E$ contains a copy of $\ell_1$ where $E$ has property $wV^*$. Finally, the identity between $L(C(\Omega, E), F)$ and $WP(C(\Omega, E), F)$ is investigated.

Let $R$ be a commutative Noetherian ring and $M$ a finite $R$-module. In this paper, we consider Zariski-openness of the FID-locus of $M$, namely, the subset of $\mathrm{spec}\,R$ consisting of all prime ideals ${\mathfrak p}$ such that $M_{\mathfrak p}$ has finite injective dimension as an $R_{\mathfrak p}$-module. We prove that the FID-locus of $M$ is an open subset of $\mathrm{spec}\,R$ whenever $R$ is excellent.

The $\ell^{1}$-convolution algebra of a semilattice is known to have trivial cohomology in degrees 1, 2 and 3 whenever the coefficient bimodule is symmetric. We extend this result to all cohomology groups of degree $\geq 1$ with symmetric coefficients. Our techniques prove a stronger splitting result, namely that the splitting can be made natural with respect to the underlying semilattice.

In this paper, we prove that, for any integer $n\ge 2,$ and any $\delta > 0$ there exists an $\epsilon(n,\delta) \ge 0$ such that if $M$ is an $n$-dimensional complete manifold with sectional curvature $K_M \ge 1$ and if $M$ has conjugate radius $\rho \ge\frac{\pi}{2}+\delta $ and contains a geodesic loop of length $2(\pi-\epsilon(n,\delta))$ then $M$ is diffeomorphic to the Euclidian unit sphere $\mathbb{S}^{n}.$

We study the boundary value problem $\begin{array}{rcl} {\rm div}(|\n u|^{m-2}\n u) + u^av^b &=& 0\quad \mbox{ in } {\Omega}, \vspace{\jot}\\ {\rm div}(|\n v|^{m-2}\n v) + u^cv^d &=& 0 \quad \mbox{ in } {\Omega}, \vspace{\jot}\\ u =v &= & 0 \quad \mbox{ on } {\partial}{\Omega},\vspace{\jot}\\ \end{array}$ where ${\Omega}\subset\mathbb{R}^n$ ($n\ge2$) is a bounded connected smooth domain, and the exponents $m>1$ and $a,b,c,d\ge0$ are non-negative numbers. Under appropriate conditions on the exponents $m$, $a$, $b$, $c$ and $d$, a variety of results on a priori estimates and existence of positive solutions has been established.

We prove some theorems on locally graded groups and nilpotent-by-Chernikov groups.

We consider the standard action of the dihedral group $\bf{D}_n$ of order $2n$ on $\bf{C}$. This representation is absolutely irreducible and so the corresponding Hopf bifurcation occurs on $\bf{C} \oplus \bf{C}$. Golubitsky and Stewart (Hopf bifurcation with dihedral group symmetry: Coupled nonlinear oscillators. In: Multiparameter Bifurcation Series, M. Golubitsky and J. Guckenheimer, eds., Contemporary Mathematics 46, Am. Math. Soc., Providence, R.I. 1986, 131–173) and van Gils and Valkering (Hopf bifurcation and symmetry: standing and travelling waves in a circular chain. Japan J. Appl. Math.3, 207–222, 1986) prove the generic existence of three branches of periodic solutions, up to conjugacy, in systems of ordinary differential equations with $\bf{D}_n$-symmetry, depending on one real parameter, that present Hopf bifurcation. These solutions are found by using the Equivariant Hopf Theorem. We prove that generically, when $n\neq 4$ and assuming Birkhoff normal form, these are the only branches of periodic solutions that bifurcate from the trivial solution.

It is easy to imagine that a subvariety of a vector bundle, whose intersection with every fibre is a vector subspace of constant dimension, must necessarily be a sub-bundle. We give two examples to show that this is not true, and several situations in which the implication does hold. For example it is true if the base is normal and the field has characteristic zero. A convenient test is whether or not the intersections with the fibres are reduced as schemes.

Let $G$ be a compact $p$-adic analytic group and let $\Lambda_G$ be its completed group algebra with coefficient ring the $p$-adic integers $\mathbb{Z}_p$. We show that the augmentation ideal in $\Lambda_G$ of a closed normal subgroup $H$ of $G$ is localisable if and only if $H$ is finite-by-nilpotent, answering a question of Sujatha. The localisations are shown to be Auslander-regular rings with Krull and global dimensions equal to dim $H$. It is also shown that the minimal prime ideals and the prime radical of the $\mathbb{F}_p$-version $\Omega_G$ of $\Lambda_G$ are controlled by $\Omega_{\Delta^+}$, where $\Delta^+$ is the largest finite normal subgroup of $G$. Finally, we prove a conjecture of Ardakov and Brown [1].