Let $G$ be an additively written abelian group, and let $S$ be a sequence in $G \setminus \{0\}$ with length $|S| \ge 4$. Suppose that $S$ is a product of two subsequences, say $S = B C$, such that the element $g+h$ occurs in the sequence $S$ whenever $g \cdot h$ is a subsequence of $B$ or of $C$. Then $S$ contains a proper zero-sum subsequence, apart from some well-characterized exceptional cases. This result is closely connected with restricted set addition in abelian groups. Moreover, it solves a problem on the structure of minimal zero-sum sequences, which recently occurred in the theory of non-unique factorizations.

In this paper, half-dimensional Einstein–Kähler submanifolds or locally reducible Kähler submanifolds are classified in a quaternion projective space.

Thanks to Szemerédi's theorem on sets with no long arithmetic progressions, an elementary trick is used here to show that for a given positive integer $h$ and a given set $U$ of residue classes modulo $n$ with positive density, there exists a dense subset $V$ of $U$; that is, $U\smallsetminus V$ is very small, such that the sumset $hV$ is included in the restricted sumset $h\times U$. The next step is to obtain information on the structure of $V$ from Kneser's theorem on the sum of sets in an abelian group, and to use this for studying the structure of $U$ itself. Finally, this idea is used in the paper to derive some new values of a function related to the Erdős–Ginzburg–Ziv theorem.

We prove that the mod [pfr ]-reduction of a Shimura subvariety of the moduli space of abelian varieties has a Zariski dense ordinary locus for almost all height 1 primes [pfr ] of the reflex field. We show further that the ordinary locus is empty at primes which are not of height 1.

For any holomorphic map in the unit disk, the set of radial limits at a Borel set on the unit circle is a Suslin-analytic set. Here it is proved that, for a conformal map, this set is, in fact, Borel. As a consequence, the sets of accessible boundary points, of cut points and of transition points are Borel. In addition, it is shown that the set of end points is a $G_{\delta}$-set.

By means of a simplification in a significant step of Zelmanov's solution of the restricted Burnside problem, the known bounds for the order of finite $m$ -generator groups of exponent $n$ are improved.

We study deformations of functions on isolated singularities. A unified proof of the equality of Milnor and Tjurina numbers for functions on isolated complete intersections singularities and space curves is given. As a consequence, the base space of their miniversal deformations is endowed with the structure of an $F$-manifold, and we can prove a conjecture of V. Goryunov, stating that the critical values of the miniversal unfolding of a function on a space curve are generically local coordinates on the base space of the deformation.

We present a Borel reduction from a subset shift equivalence relation of a countable group to a subgroup conjugacy relation of a free product. The technique gives a much shorter proof of an earlier result of Thomas and Velickovic.

The authors present a technique of deriving basic hypergeometric identities from specializations using fewer parameters, by using the classical Cauchy identity on the expansion of the power of $x$ in terms of the $q$-binomial coefficients. This method is referred to as ‘Cauchy augmentation’. Despite its simple appearance, the Cauchy identity plays a key role in parameter augmentation. For example, one can reach the $q$-Gauss summation formula from the Euler identity by using the Cauchy augmentation twice. This idea also applies to Jackson's $_2\phi_1$ to $_3\phi_1$ transformation formula. Moreover, a transformation formula analogous to Jackson's formula is obtained.

Let $H$ be a not necessarily separable Hilbert space, and let $\mathcal{B}(H)$ denote the space of all bounded linear operators on $H$. It is proved that a commutative lattice $\mathcal{D}$ of self-adjoint projections in $H$ that contains $0$ and $I$ is spatially complete if and only if it is a closed subset of $\mathcal{B}(H)$ in the strong operator topology. Some related results are obtained concerning commutative lattice-ordered cones of self-adjoint operators that contain $\mathcal{D}$.

A relatively long-standing problem concerning the best constants of approximation by entire functions of exponential type on the class $W^rH_{\omega}$ is solved. The central tool is a limit relation between the best constants of approximation by trigonometric polynomials and entire functions of exponential type on $W^rH_{\omega}$ and its periodic analogue.

In this paper we characterise the kernel of a little Hankel operator on the Bergman space L2a(D) in terms of the inner divisors, and obtain a characterisation for finite rank little Hankel operators using the invariant subspace theory technique.

Let $\Cal F$ be a holomorphic foliation (possibly with singularities) on a non-singular manifold $M$, and let $V$ be a complex analytic subset of $M$. Usual residue theorems along $V$ in the theory of complex foliations require that $V$ be tangent to the foliation (that is, a union of leaves and singular points of $V$ and $\Cal F$); this is the case for instance for the blow-up of a non-dicritical isolated singularity. In this paper, residue theorems are introduced along subvarieties that are not necessarily tangent to the foliation, including the blow-up of the dicritical situation.

The following result is established.

THEOREM. Let G be a periodic, residually finite group with all subgroups sub-normal. Then G is nilpotent.

The well-known groups of Heineken and Mohamed [1] show that the hypothesis of residual finiteness cannot be omitted here, while examples in [5] show that a residually finite group with all subgroups subnormal need not be nilpotent. The proof of the Theorem will use the results of Möhres that a group with all subgroups subnormal is soluble [3] and that a periodic hypercentral group with all subgroups subnormal is nilpotent [4]. Borrowing an idea from [2], the plan is to construct certain subgroups H and K that intersect trivially, and to show that the subnormality of both leads to a contradiction.

We give a theoretical lower bound for the slope of a Siegel modular cusp form that is as least as good as Eichler's lower bound. In degrees $n=5,6$ and 7 we show that our new bound is strictly better. In the process we find the forms of smallest dyadic trace on the perfect core for ranks $n \le 8$. In degrees $n=5,6$ and 7 we settle the value of the generalized Hermite constant $\gamma_n'$ introduced by Bergé and Martinet and find all dual-critical pairs.

It is well known that no non-trivial Killing vector field exists on a compact Riemannian manifold of negative Ricci curvature; analogously, no non-trivial harmonic one-form exists on a compact manifold of positive Ricci curvature. One can consider the following, more general, problem. By reducing the assumption on the Ricci curvature to one on the scalar curvature, such vanishing theorems cannot hold in general. This raises the question: “What information can we obtain from the existence of non-trivial Killing vector fields (or, respectively, harmonic one-forms)?” This paper gives answers to this problem; the results obtained are optimal.

A complete characterization of the categorical quotients of $({\mathbb P}^1)^n$ by the diagonal action of $\SL(2, {\mathbb C})$ with respect to any polarization is given by M. Polito, in ‘$\SL(2, {\mathbb C})$-quotients de $({\mathbb P}^1)^n$’, C. R. Acad. Sci. Paris Sér. I 321 (1995) 1577–1582. In this paper, these categorical quotients are obtained by certain linear systems on ${\mathbb P}^{n-3}$?>, depending on the given polarization.

Let $J$ be a Jacobi real symmetric matrix on $l_{2}$ with zero diagonal and non-diagonal entries of the form $\{1+p_{n}\}$. If $p_{n-1}\pm p_{n}=O(n^{-\alpha})$ with some $\alpha>2/3$, then the existence of bounded solutions of $Ju=\lambda u$ is proved for almost every $\lambda\in(-2,2)$ with the WKB-type asymptotic behavior.

Regular homotopy classes of immersions of homotopy $n$-spheres into $(n+q)$-space form an Abelian group under connected summation. The subgroup of immersions of those homotopy spheres which bound parallelizable manifolds is computed for $n=4k-1$ and arbitrary codimension $q$. A consequence of this computation is that in codimensions $q<2k+1$, the group of immersed homotopy $(4k-1)$-spheres is not the direct sum of the group of immersed standard spheres and the group of homotopy spheres. Also, our results shed light on Brieskorn's famous equations: for any exotic sphere (bounding a parallelizable manifold), a countable number of distinct equations give rise to codimension-two embeddings of this exotic sphere. It follows from our results that these embeddings lie in pairwise distinct regular homotopy classes, and that any regular homotopy class containing embeddings is representable by a map arising from such an equation.

It is known that if a is an algebraic element of a Banach algebra A, then its spectrum σ(a) is finite, and there exists γ > 0 such that the Hausdorff distance to spectra of nearby elements satisfies

(formula here)

We prove that the converse is also true, provided that A is semisimple.