We study the set of all m-tuples (λ(1), . . ., λ(m)) of possible types of finite abelian p-groups Mλ(1), . . ., Mλ(m) for which there exists a long exact sequence Mλ(1) → ⋅⋅⋅ → Mλ(m). When m=3, we recover W. Fulton's (Eigenvalues of majorized Hermitian matrices and Littlewood-Richardson coefficients (Special Issue: Workshop on Geometric and combinatorial Methods in the Hermitian Sum Spectral Problem), Linear Algebra Appl. 319(1–3) (2000), 23–36) results on the possible eigenvalues of majorized Hermitian matrices.

We extend the concept of umbilicity to higher order umbilicity in Riemannian manifolds saying that an isometric immersion is k-umbilical when APk−1(A) is a multiple of the identity, where Pk(A) is the kth Newton polynomial in the second fundamental form A with P0(A) being the identity. Thus, for k=1, one-umbilical coincides with umbilical. We determine the principal curvatures of the two-umbilical isometric immersions in terms of the mean curvatures. We give a description of the two-umbilical isometric immersions in space forms which includes the product of spheres embedded in the Euclidean sphere S2k+1 of radius 1. We also introduce an operator φk which measures how an isometric immersion fails to be k-umbilical, giving in particular that φ1 ≡ 0 if and only if the immersion is totally umbilical. We characterize the two-umbilical hypersurfaces of a space form as images of isometric immersions of Einstein manifolds.

Let x0 < x1 < x2 < ⋅⋅⋅ be an increasing sequence of positive integers given by the formula xn=⌊βxn−1 + γ⌋ for n=1, 2, 3, . . ., where β > 1 and γ are real numbers and x0 is a positive integer. We describe the conditions on integers bd, . . ., b0, not all zero, and on a real number β > 1 under which the sequence of integers wn=bdxn+d + ⋅⋅⋅ + b0xn, n=0, 1, 2, . . ., is bounded by a constant independent of n. The conditions under which this sequence can be ultimately periodic are also described. Finally, we prove a lower bound on the complexity function of the sequence qxn+1 − pxn ∈ {0, 1, . . ., q−1}, n=0, 1, 2, . . ., where x0 is a positive integer, p > q > 1 are coprime integers and xn=⌈pxn−1/q⌉ for n=1, 2, 3, . . . A similar speculative result concerning the complexity of the sequence of alternatives (F:x↦x/2 or S:x↦(3x+1)/2) in the 3x+1 problem is also given.

We prove some results on algebras, satisfying many generic relations. As an application we show that there are Golod–Shafarevich algebras which cannot be homomorphically mapped onto infinite dimensional algebras with finite Gelfand–Kirillov dimension. This answers a question of Zelmanov (Some open problems in the theory of infinite dimensional algebras, J. Korean Math. Soc. 44(5) 2007, 1185–1195).

For any rotation-invariant positive regular Borel measure ν on the closed unit ball whose support contains the unit sphere , let L2a be the closure in L2 = L2(, dν) of all analytic polynomials. For a bounded Borel function f on , the Toeplitz operator Tf is defined by Tf(ϕ) = P(fϕ) for ϕ ∈ L2a, where P is the orthogonal projection from L2 onto L2a. We show that if f is continuous on , then Tf is compact if and only if f(z) = 0 for all z on the unit sphere. This is well known when L2a is replaced by the classical Bergman or Hardy space.

Let K be a perfect field of characteristic p > 0; A1 := K〈x, ∂|∂x−x∂=1〉 be the first Weyl algebra; and Z:=K[X:=xp, Y:=∂p] be its centre. It is proved that (i) the restriction map res : AutK(A1)→ AutK(Z), σ ↦ σ|Z is a monomorphism with im(res) = Γ := {τ ∈ AutK(Z)|(τ)=1}, where (τ) is the Jacobian of τ, (Note that AutK(Z)=K* ⋉ Γ, and if K is not perfect then im(res) ≠ Γ.); (ii) the bijection res : AutK(A1) → Γ is a monomorphism of infinite dimensional algebraic groups which is not an isomorphism (even if K is algebraically closed); (iii) an explicit formula for res−1 is found via differential operators (Z) on Z and negative powers of the Fronenius map F. Proofs are based on the following (non-obvious) equality proved in the paper:

In this paper, we introduce an integral operator on the unit ball . The boundedness and compactness of the operator from the Zygmund space to the Bloch-type space or the little Bloch-type space are investigated.

We consider families of quasiplatonic Riemann surfaces characterised by the fact that – as in the case of Fermat curves of exponent n – their underlying regular (Walsh) hypermap is an embedding of the complete bipartite graph Kn,n, where n is an odd prime power. We show that these surfaces, regarded as algebraic curves, are all defined over abelian number fields. We determine their orbits under the action of the absolute Galois group, their minimal fields of definition and in some easier cases their defining equations. The paper relies on group – and graph – theoretic results by G. A. Jones, R. Nedela and M. Škoviera about regular embeddings of the graphs Kn,n [7] and generalises the analogous results for maps obtained in [9], partly using different methods.

We consider Ptolemy's inequality in a metric space setting. It is not hard to see that CAT(0) spaces satisfy this inequality. Although the converse is not true in full generality, we show that if our Ptolemaic space is either a Riemannian or Finsler manifold, then it must also be CAT(0). Ptolemy's inequality is closely related to inversions of metric spaces. We exploit this link to establish a new characterization of Euclidean space amongst all Riemannian manifolds.

We obtain a new characterisation for weighted Bergman spaces Apα on the unit ball n of ℂn in terms of a double integral of the functions |f(z) − f(w)|/|z − w| and |f(z) − f(w)|/|1 − 〈 z, w〉|.

Let M be an n-dimensional totally real minimal submanifold in CPn. We prove that if M is semi-parallel and the scalar curvature τ, , then M is an open part of the Clifford torus Tn ⊂ CPn. If M is semi-parallel and the scalar curvature τ, , then M is an open part of the real projective space RPn.

It is proved that if the normal closure of every element of a group G has rank at most r, then the derived subgroup of G has r-bounded rank.

It is shown that every finitely generated projective module PR over a semiprime ring R has the smallest FI-extending essential module extension (called the absolute FI-extending hull of PR) in a fixed injective hull of PR. This module hull is explicitly described. It is proved that , where is the smallest right FI-extending right ring of quotients of End(PR) (in a fixed maximal right ring of quotients of End(PR). Moreover, we show that a finitely generated projective module PR over a semiprime ring R is FI-extending if and only if it is a quasi-Baer module and if and only if End(PR) is a quasi-Baer ring. An application of this result to C*-algebras is considered. Various examples which illustrate and delimit the results of this paper are provided.

Let G be a finite group. A minimal subgroup of G is a subgroup of prime order. A subgroup of G is called S-quasinormal in G if it permutes with each Sylow subgroup of G. A group G is called an MS-group if each minimal subgroup of G is S-quasinormal in G. In this paper, we investigate the structure of minimal non-MS-groups (non-MS-groups all of whose proper subgroups are MS-groups).

In this paper we show existence of positive solutions for a class of quasi-linear problems with Neumann boundary conditions defined in a half-space and involving the critical exponent.

In this paper we describe a special class of self-adjoint operators associated with the singular self-adjoint second-order differential expression ℓ. This class is defined by the requirement that the sesquilinear form q(u, v) obtained from ℓ by integration by parts once agrees with the inner product 〈ℓu, v〉. We call this class Type I operators. The Friedrichs Extension is a special case of these operators. A complete characterization of these operators is given, for the various values of the deficiency index, in terms of their domains and the boundary conditions they satisfy (separated or coupled).

Let B be a submodule of an R-module M. The intersection of all prime (resp. weakly prime) submodules of M containing B is denoted by rad(B) (resp. wrad(B)). A generalisation of 〈E(B)〉 denoted by UE(B) of M will be introduced. The inclusions 〈E(B)〉 ⊆ UE(B) ⊆ wrad(B) ⊆ rad(B) are motivations for studying the equalities UE(B) = wrad(B) and UE(B) = rad(B) in this paper. It is proved that if R is an arithmetical ring, then UE(B) = wrad(B). In Theorem 2.5, a generalisation of the main result of [11] is given.

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 .