Our purpose is to study the geometry of linear Weingarten spacelike hypersurfaces immersed in a locally symmetric Einstein spacetime, whose sectional curvature is supposed to obey some standard restrictions. In this setting, by using as main analytical tool a generalized maximum principle for complete non-compact Riemannian manifolds, we establish sufficient conditions to guarantee that such a hypersurface must be either totally umbilical or an isoparametric hypersurface with two distinct principal curvatures, one of which is simple. Applications to the de Sitter space are given.

We prove, using purely combinatorial methods, that there is a pairing

\begin{linenomath} $$\Sym^a\Q^2\times\Sym^a\Q^2\lra\Q$$ \end{linenomath}

This paper builds on work of Hochster and Yao that provides nice embeddings for finitely generated modules of finite G-dimension, finite projective dimension or locally finite injective dimension. We extend these results by providing similar embeddings in the relative setting, that is, for certain modules of finite GC-dimension, finite C-projective dimension, locally finite C-injective dimension or locally finite C-injective dimension where C is a semidualizing module. Along the way, we extend some results for modules of finite homological dimension to modules of locally finite homological dimension in the relative setting.

A brief survey of some aspects of noetherian Hopf algebras is given, concentrating on structure, homology, and classification, and accompanied by a panoply of open problems.

We obtain an improved bound for the 2k-th moment of a degree k Weyl sum, restricted to a set of minor arcs, when k is small. We then present some applications of this bound to some Diophantine problems, including a case of the Waring–Goldbach problem, and a particular family of Diophantine equations defined as the sum of a norm form and a diagonal form.

We will compute the subgroup permutability degree of the projective special linear groups PSL(2,pn).

We study the automorphism group of the algebra $\co_q(M_n)$ of n × n generic quantum matrices. We provide evidence for our conjecture that this group is generated by the transposition and the subgroup of those automorphisms acting on the canonical generators of $\co_q(M_n)$ by multiplication by scalars. Moreover, we prove this conjecture in the case when n = 3.

We give an easy diagrammatical description of the parabolic Kazhdan–Lusztig polynomials for the Weyl group Wn of type Dn with parabolic subgroup of type An and consequently an explicit counting formula for the dimension of morphism spaces between indecomposable projective objects in the corresponding category . As a by-product we categorify irreducible Wn-modules corresponding to the pairs of one-line partitions. Finally, we indicate the motivation for introducing the combinatorics by connections to the Springer theory, the category of perverse sheaves on isotropic Grassmannians, and to the Brauer algebras, which will be treated in two subsequent papers of the second author.

In this paper we investigate the following functional inequality

$ \begin{eqnarray*} \| f(x-y-z) - f(x-y+z) + f(y) +f(z)\| \leq \|f(x+y-z) - f(x)\| \end{eqnarray*}$

Let R be an affine PI-algebra over an algebraically closed field $\mathbb{k}$ and let G be an affine algebraic $\mathbb{k}$-group that acts rationally by algebra automorphisms on R. For R prime and G a torus, we show that R has only finitely many G-prime ideals if and only if the action of G on the centre of R is multiplicity free. This extends a standard result on affine algebraic G-varieties. Under suitable hypotheses on R and G, we also prove a PI-version of a well-known result on spherical varieties and a version of Schelter's catenarity theorem for G-primes.

A real algebraic curve of genus g is a pair (S,〈 τ 〉), where S is a closed Riemann surface of genus g and τ: S → S is a symmetry, that is, an anti-conformal involution. A Schottky uniformization of (S,〈 τ 〉) is a tuple (Ω,Γ,P:Ω → S), where Γ is a Schottky group with region of discontinuity Ω and P:Ω → S is a regular holomorphic cover map with Γ as its deck group, so that there exists an extended Möbius transformation $\widehat{\tau}$ keeping Ω invariant with P o $\widehat{\tau}$=τ o P. The extended Kleinian group K=〈 Γ, $\widehat{\tau}$〉 is called an extended Schottky groups of rank g. The interest on Schottky uniformizations rely on the fact that they provide the lowest uniformizations of closed Riemann surfaces. In this paper we obtain a structural picture of extended Schottky groups in terms of Klein–Maskit's combination theorems and some basic extended Schottky groups. We also provide some insight of the structural picture in terms of the group of automorphisms of S which are reflected by the Schottky uniformization. As a consequence of our structural description of extended Schottky groups, we get alternative proofs to results due to Kalliongis and McCullough (J. Kalliongis and D. McCullough, Orientation-reversing involutions on handlebodies, Trans. Math. Soc. 348(5) (1996), 1739–1755) on orientation-reversing involutions on handlebodies.

Necessary and sufficient conditions for the incompleteness of exponential system in Cα are characterised, where Cα is the weighted Banach space of complex continuous functions f defined on ℝn with f(t)exp(−α(t)) vanishing at infinity in the uniform norm.

The mean value inequality is characteristic for upper semi-continuous functions to be subharmonic. Quasinearly subharmonic functions generalise subharmonic functions. We find the necessary and sufficient conditions under which subsets of balls are big enough for the characterisation of non-negative, quasinearly subharmonic functions by mean value inequalities. Similar result is obtained also for generalised mean value inequalities where, instead of balls, we consider arbitrary bounded sets, which have non-void interiors and instead of the volume of ball some functions depending on the radius of this ball.

We study on a contact metric manifold M2n+1(ϕ, ξ, η, g) such that g is a Ricci soliton with potential vector field V collinear with ξ at each point under different curvature conditions: (i) M is of pointwise constant ξ-sectional curvature, (ii) M is conformally flat.

We prove that special ample line bundles on toric varieties arising from root systems are projectively normal. Here the maximal cones of the fans correspond to the Weyl chambers, and special means that the bundle is torus-equivariant such that the character of the line bundle that corresponds to a maximal Weyl chamber is dominant with respect to that chamber. Moreover, we prove that the associated semi-group rings are quadratic.

We introduce the notion of upper and lower frequently universal sequences and see that ‘most’ of the universal approximations are obtained by sets of indices which have upper density 1 and lower density 0. We also show that a class of universal series related to lower density is of first category.

A module M over a commutative ring R has an almost trivial dual if there is no homomorphism from M onto a free R-module of countable infinite rank. Using a new combinatorial principle (the ℵn-Black Box), which is provable in ordinary set theory, we show that for every natural number n, there exist arbitrarily large ℵn-free R-modules with almost trivial duals, when R is a complete discrete valuation domain. A corresponding result for torsion modules is also obtained.

We show that over every countable algebraically closed field $\mathbb{K}$ there exists a finitely generated $\mathbb{K}$-algebra that is Jacobson radical, infinite-dimensional, generated by two elements, graded and has quadratic growth. We also propose a way of constructing examples of algebras with quadratic growth that satisfy special types of relations.

We show that PQε-projectivity of two Riemannian metrics introduced in [15] (P. J. Topalov, Geodesic compatibility and integrability of geodesic flows, J. Math. Phys.44(2) (2003), 913–929.) implies affine equivalence of the metrics unless ε ∈ {0,−1,−3,−5,−7,. . .}. Moreover, we show that for ε=0, PQε-projectivity implies projective equivalence.

Let H be a real Hilbert space and denote by S its unit sphere. Consider the nonlinear eigenvalue problem Ax + ε B(x) =δ x, where A: H → H is a bounded self-adjoint (linear) operator with nontrivial kernel Ker A, and B: H → H is a (possibly) nonlinear perturbation term. A unit eigenvector x0 ∈ S∩ Ker A of A (thus corresponding to the eigenvalue δ=0, which we assume to be isolated) is said to be persistent, or a bifurcation point (from the sphere S∩ Ker A), if it is close to solutions x ∈ S of the above equation for small values of the parameters δ ∈ ℝ and ε ≠ 0. In this paper, we prove that if B is a C1 gradient mapping and the eigenvalue δ=0 has finite multiplicity, then the sphere S∩ Ker A contains at least one bifurcation point, and at least two provided that a supplementary condition on the potential of B is satisfied. These results add to those already proved in the non-variational case, where the multiplicity of the eigenvalue is required to be odd.