Let Mn be an n-dimensional closed hypersurface with constant mean curvature H satisfying |H| ≤ ϵ(n) in a unit sphere Sn+1(1), n ≤ 8 and S the square of the length of the second fundamental form of M. There exists a constant δ(n, H) > 0, which depends only on n and H such that if S0 ≤ S ≤ S0 + δ(n, H), then S ≡ S0 and M is isometric to a Clifford hypersurface, where ϵ(n) is a sufficiently small constant depending on n and .

Let L be a convex body in n and z an interior point of L. We associate with L and z a new, convex and centrally symmetric, body CI(L, z). This generalizes the classical intersection bodyI(L, z) (whose radial function at u ∈ Sn−1 is the volume of the hyperplane section of L through z, orthogonal to u). CI(L, z) coincides with I(L, z) if and only if L is centrally symmetric about z. We study the properties of CI(L, z).

We prove the CR version of the Obata's result for the first eigenvalue of the sub-Laplacian in the setting of a compact strictly pseudoconvex pseudohermitian three-dimensional manifold with non-negative CR-Paneitz operator which satisfies a Lichnerowicz-type condition. We show that if the first positive eigenvalue of the sub-Laplacian takes the smallest possible value, then, up to a homothety of the pseudohermitian structure, the manifold is the standard Sasakian three-dimensional unit sphere.

Let be a finite dimensional complex semi-simple Lie algebra with Weyl group W and simple reflections S. For I ⊆ S let I be the corresponding semi-simple subalgebra of . Denote by WI the Weyl group of I and let w○ and wI○ be the longest elements of W and WI, respectively. In this paper we show that the answer to Kostant's problem, i.e. whether the universal enveloping algebra surjects onto the space of all ad-finite linear transformations of a given module, is the same for the simple highest weight I-module LI(x) of highest weight x ⋅ 0, x ∈ WI, as the answer for the simple highest weight -module L(xwI○w○) of highest weight xwI○w○ ⋅ 0. We also give a new description of the unique quasi-simple quotient of the Verma module Δ(e) with the same annihilator as L(y), y ∈ W.

In this paper we study a non-linear elliptic equation involving p(x)-growth conditions and satisfying a Neumann boundary condition on a bounded domain. For that equation we establish the existence of two solutions using as a main tool an abstract linking argument due to Brézis and Nirenberg.

We prove uniqueness of positive solutions for the boundary value problems

\[ \{\begin{array}{ll} -\Delta u=\lambda f(u)\ \ &\text{in}\Omega, \ \ \ \ \ u=0 &\text{on \partial \Omega, \]

Let M be an n-dimensional closed hypersurface with constant mean curvature H satisfying |H| ≤ ϵ(n) in a unit sphere Sn+1, n ≤ 7, and S the square of the length of the second fundamental form of M. There exists a constant δ(n, H) > 0, which depends only on n and H, such that if S0 ≤ S ≤ S0 + δ(n, H), then S ≡ S0 and M is isometric to a Clifford hypersurface, where ϵ(n) is a sufficiently small constant depending on n and .

R is called a right WV-ring if each simple right R-module is injective relative to proper cyclics. If R is a right WV-ring, then R is right uniform or a right V-ring. It is shown that a right WV-ring R is right noetherian if and only if each right cyclic module is a direct sum of a projective module and a CS (complements are summands, a.k.a. ‘extending modules’) or noetherian module. For a finitely generated module M with projective socle over a V-ring R such that every subfactor of M is a direct sum of a projective module and a CS or noetherian module, we show M = X ⊕ T, where X is semisimple and T is noetherian with zero socle. In the case where M = R, we get R = S ⊕ T, where S is a semisimple artinian ring and T is a direct sum of right noetherian simple rings with zero socle. In addition, if R is a von Neumann regular ring, then it is semisimple artinian.

A number theoretical aspect of the fundamental cycle of a periodic box-ball system is investigated. Using the formulae for the fundamental cycle of a class of initial states, we point out that the asymptotic behaviour of the fundamental cycle is closely related to the celebrated Riemann hypothesis.

The extension E of degree n over the Galois field F={\text GF}(q)} is called regular over F, if {\text ord}_r(q) and n have greatest common divisor 1 for all prime divisors r of n which are different from the characteristic p of F (here, \order{r}{q} denotes the multiplicative order of q modulo r). Under the assumption that E is regular over F and that q-1 is divisible by 4 if q is odd and n is even, we prove the existence of a primitive element w \in E which is also completely normal over F (the latter means that w simultaneously generates a normal basis for E over every intermediate field K of E/F). Our result achieves, for the class of extensions under consideration, a common generalization of the theorem of Lenstra and Schoof on the existence of primitive normal bases [12] and the theorem of Blessenohl and Johnsen on the existence of complete normal bases [1].

The Lindenstrauss formula

which states a strong relationship between the (Clarkson) modulus of uniform convexity δx of a Banach space X and the modulus of uniform smoothness px* of the conjugate space X*, is well known. Following the idea of the definitions of nearly uniform smooth space by S. Prus and modulus of uniform smoothness we define a modulus of nearly uniform smoothness and prove some Lindenstrauss type formulae concerning this modulus and the modulus of nearly uniform convexity for some measures of noncompactness.

Let E be an arbitrary (non-empty) set and S the restricted symmetric group on E, that is the group of all permutations of E which keep all but a finite number of elements of E fixed. If Φ is any commutative ring with unit element, let Γ = Φ(S) be the group algebra of S over Φ,Γ ⊃ Φ and let M be the free Φ-module having E as Φ-base. The “natural” representation of S is obtained by turning M into a Γ-module in the obvious manner, namely by writing for α∈S, λ1∈Φ,

Let K be an arbitrary field with characteristic p > 0, G a finite group of order pag′ with (p, g′) = 1, P a p-Sylow subgroup of G and G′ the commutator subgroup of G. For a ring R denote by J(R) the Jacobson radical of R and by Z(R) the centre of R. We write KG for the group algebra of G over K.

Talagrand has shown [4, p. 76] that there exists a continuous linear operator from L1[0, 1] to c0 which is not a Dunford-Pettis operator. In contrast to this result, Gretsky and Ostroy [2] have recently proved that every positive operator from L[0, 1] to c0 is a Dunford-Pettis operator, hence that every regular operator between these spaces (i.e. a difference of positive operators) is Dunford-Pettis.

This paper deals with the following degenerate and singular equation with non-local source and absorption. The existence of a unique classical non-negative solution is established and the sufficient conditions for the solution that exists globally or blows up in finite time are obtained.

The covers of cyclic acts over monoids were investigated by Mahmoudi and Renshaw (M. Mahmoudi and J. Renshaw, On covers of cyclic acts over monoids, Semigroup Forum77 (2008), 325–338) and the authors posed some open problems. In the present paper, we give answers to their problems 1 and 5, and we also give a sufficient and necessary condition that a cyclic act has a weakly pullback flat cover.

Let $A$ (resp. $B$) be a bounded linear operator on a complex Hilbert space $ {\mathcal H}$ (resp. $ {\mathcal K}$). We show that the tensor product $ A \otimes B $ is log-hyponormal if and only if $A$ and $B$ are log-hyponormal, and that a similar result holds for class $A(s,t)$ operators.

We give a necessary and sufficient condition for $k$-step nilmanifolds associated with graphs $(k \geq 3)$ to admit Anosov automorphisms. We also prove the nonexistence of Anosov automorphisms on certain classes of 2-step and 3-step nilmanifolds.

We study properties of -free numbers, that is numbers that are not divisible by any member of a set . First we formulate the most-used procedure for finding them (in a given set of integers) as easy-to-apply propositions. Then we use the propositions to consider Diophantine properties of -free numbers and their distribution on almost all short intervals. Results on -free numbers have implications to non-vanishing Fourier coefficients of cusp forms, so this work also gives information about them.

The problem of determining whether or not two operators are unitarily equivalent has been around for many years and considerable work has been done in attempting to solve this problem (see for example [1], [3], [5], [6], [7], [10], [11], [12], [15], [16], [17], [18], [19], [20], [21] and [22]). In many cases, a complete set of unitary invariants is furnished for a certain class of operators. Here we just mention two of such results which are related to what we are going to discuss. The first one was due to Arveson, who showed that two irreducible compact operators are unitarily equivalent if and only if they have the same nth algebraic matricial ranges, for each n≧1 ([1] and Theorem 2.4.3 of [3]). The second one was due to Parrott, who showed that two compact operators with zero reducing null spaces are unitarily equivalent if and only if they have the same nth spatial matricial ranges, for each n≧1 ([5, p. 146]). In this paper, we investigate the closures of the spatial matricial ranges of compact operators and obtain a complete set of unitary invariants for compact operators, from which Parrott's result follows easily.