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

Let ${\cal A}$ be a unital maximal full algebra of operator fields with base space $[0, 1]^k$ and fibre algebras $\{{\cal A}_t\}_{t\in[0, 1]}^{k}$. It is shown in this paper that the stable rank of ${\cal A}$ is bounded above by the quantity sup$_{t\in[0, 1]^k}\,{\rm sr}(C([0, 1]^k)\,{\otimes}\,{\cal A}_t)$, where ‘sr’ means stable rank. Using the above estimate, the stable ranks of the C$^*$-algebras of the (possibly higher rank) discrete Heisenberg groups are computed.

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.

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.

Let [Ascr ] be a unital von Neumann algebra of operators on a complex separable Hilbert space ([Hscr ]0, and let {Tt, t [ges ] 0} be a uniformly continuous quantum dynamical semigroup of completely positive unital maps on [Ascr ]. The infinitesimal generator [Lscr ] of {Tt} is a bounded linear operator on the Banach space [Ascr ]. For any Hilbert space [Kscr ], denote by [Bscr ]([Kscr ]) the von Neumann algebra of all bounded operators on [Kscr ]. Christensen and Evans [3] have shown that [Lscr ] has the form

formula here

where π is a representation of [Ascr ] in [Bscr ]([Kscr ]) for some Hilbert space [Kscr ], R: [Hscr ]0 → [Kscr ] is a bounded operator satisfying the ‘minimality’ condition that the set {(RX−π(X)R)u, u∈[Hscr ]0, X∈[Ascr ]} is total in [Kscr ], and K0 is a fixed element of [Ascr ]. The unitality of {Tt} implies that [Lscr ](1) = 0, and consequently K0 = iH−½R*R, where H is a hermitian element of [Ascr ]. Thus (1.1) can be expressed as

formula here

We say that the quadruple ([Kscr ], π, R, H) constitutes the set of Christensen–Evans (CE) parameters which determine the CE generator [Lscr ] of the semigroup {Tt}. It is quite possible that another set ([Kscr ]′, π′, R′, H′) of CE parameters may determine the same generator [Lscr ]. In such a case, we say that these two sets of CE parameters are equivalent. In Section 2 we study this equivalence relation in some detail.

A Radon measure $\mu$ on ${mathbb R}^n$ is said to be $k$-monotone if $r\mapsto{\mu(B(x,r))}/{r^k}$ is a non-decreasing function on $(0,\infty)$ for every $x\in {\mathbb R}^n$. (If $\mu$ is the $k$-dimensional Hausdorff measure restricted to a $k$-dimensional minimal surface then this important property is expressed by the monotonicity formula.) We give an example of a 1 -monotone measure $\mu$ in ${\mathbb R}^2$ with non-unique and non-conical tangent measures at a point. Furthermore, we show that $\mu$ can be the one-dimensional Hausdorff measure restricted to a closed set $A\subset {\mathbb R}^2$.

Let G and A be finite groups with coprime orders, and suppose that A acts on G by automorphisms. Let π(G, A)[ratio ]IrrA(G)→Irr (CG(A)) be the Glauberman–Isaacs correspondence. Let B[les ]A and χ∈IrrA(G). We exhibit a counterexample to the conjecture that χπ(G, A) is an irreducible constituent of the restriction of χπ(G, B) to CG(A).

Extending a theorem of S. Mori [6], we characterize two-dimensional factorial algebras which are graded by a finitely generated abelian group of rank one and which are affine over an algebraically closed field. The result is related to the classification of weighted projective lines [2, 5] and to the topic of canonical algebras [7].

Let P be an n-dimensional polytope admitting a finite reflection group G as its symmetry group. Consider the set [Hscr ]P(k) of all continuous functions on Rn satisfying the mean value property with respect to the k-skeleton P(k) of P, as well as the set [Hscr ]G of all G-harmonic functions. Then a necessary and sufficient condition for the equality [Hscr ]P(k)=[Hscr ]G is given in terms of a distinguished invariant basis, called the canonical invariant basis, of G.

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 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 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.

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.

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 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.

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.

I exhibit a pair of non-symmetric operads that, while not themselves isomorphic, induce isomorphic monads. The existence of such a pair implies that if ‘algebraic theory’ is understood as meaning ‘monad’ then operads cannot be regarded as algebraic theories of a special kind.

The topological disc (De Paepe's) \[ P=\{(z^2,\bar{z}^2+\bar{z}^3):|z|\le 1\}\subset {\bb C}^2 \]

is shown here to have non-trivial polynomially convex hull. In fact, the authors show that this holds for all discs of the form $X=\{(z^2,f(\bar{z})):|z|\le r\}$ , where $f$ is holomorphic on $|z|\le r$ , and $f(z)=z^2+a_3z^3+\ldots$ , with all coefficients $a_n$ real, and at least one $a_{2n+1}\ne 0$ .

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.

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.