For any point x, we use the symbol x to denote {(x, x)}. By a triform, we mean a triple (X, Fx, S[X]) where X is a nonempty set, Fx is a family of subsets of X, [X] is a family of subsets of X x X and the following conditions are satisfied: .

A short and easy proof that the minimum number of generators of the nth direct power of a non-trival finite group of order s having automorphism group of order a is more than logsn + logsa, n > 1. On the other hand, for non-abelian simple G and large n, d(Gn) is within 1 + e of logsn + logsa.

Although varieties of groups can in theory be determined as well by the identical relations which the groups all satisfy as by some structural property inherited by subgroups, factor groups and cartesian products which the groups have in common, it seems in practice just as hard to answer questions about properties of a group from knowledge of identical relations as it is from, say, a presentation. Many of the important questions connected with Burnside's problems exemplify this difficulty: we still do not know if there is a bound on the derived length of finite groups of exponent 4, nor whether there is a bound on the nilpotency class of finite groups of exponent p (p ≧ 5, a fixed prime).

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}M$ be a commutative cancellative monoid. For $m$ a nonunit in $M$, the catenary degree of $m$, denoted $c(m)$, and the tame degree of $m$, denoted $t(m)$, are combinatorial constants that describe the relationships between differing irreducible factorizations of $m$. These constants have been studied carefully in the literature for various kinds of monoids, including Krull monoids and numerical monoids. In this paper, we show for a given numerical monoid $S$ that the sequences $\{c(s)\}_{s\in S}$ and $\{t(s)\}_{s\in S}$ are both eventually periodic. We show similar behavior for several functions related to the catenary degree which have recently appeared in the literature. These results nicely complement the known result that the sequence $\{\Delta (s)\}_{s\in S}$ of delta sets of $S$ also satisfies a similar periodicity condition.

Ein Teil von Philip Halls bekannter Theorie der auflösbaren Gruppen läßt sich, wie im folgenden gezeigt wird, auf beliebige endiliche Gruppen G ausdehnen. Wir fragen: Was kann man über die Normalstruktur (Hauptreihen, Kompositionsfaktoren) von G aussagen, wenn man mehrere Zer legungen von G in Produkte von Untergruppen mit gegebener Normalstruktur kennt? Eine Antwort gibt der folgende Satz, das Hauptergebnis dieser Note:

A duality theorem of Wolfe for nonlinear differentiable programs is extended to nondifferentiable programs with strong and weak convex functions, by replacing gradients by local subgradient. A converse duality theorem is also proved.

We prove here that if F is a finitely generated free associative algebra over the field and R is an ideal of F, then F/R2 is finitely presented if and only if F/R has finite dimension. Amitsur, [1, p. 136] asked whether a finitely generated algebra which is embeddable in matrices over a commutative f algebra is necessarily finitely presented. Let R = F′, the commutator ideal of F, then [4, theorem 6], F/F′2 is embeddable and thus provides a negative answer to his question. Another such example can be found in Small [6]. We also show that there are uncountably many two generator I algebras which satisfy a polynomial identity yet are not embeddable in any algebra of n xn matrices over a commutative algebra.

Let R be a commutative Noetherian ring with nonzero identity and let M be a finitely generated R-module. In this paper, we prove that if an ideal I of R is generated by a u.s.d-sequence on M then the local cohomology module (M) is I-cofinite. Furthermore, for any system of ideals Φ of R, we study the cofiniteness problem in the context of general local cohomology modules.

In this paper, we give a sufficient condition (Theorem) in order that one domain D1 bounded by a C2-smooth boundary can be enclosed in, or enclose, another domain D0 bounded by the same kind of boundary. A same kind of sufficient condition for convex bodies (Corollary) is also obtained.

A 2-handle addition on the boundary of a hyperbolic 3-manifold M is called degenerating if the resulting manifold is not hyperbolic. There are examples that some manifolds admit infinitely many degenerating handle additions. But most of them are not ‘basic’. (See Section 1 for definitions). Our first main theorem shows that there are only finitely many basic degenerating handle additions. We also study the case that one of the handle additions produces a reducible manifold, and another produces a ∂-reducible manifold, showing that in this case either the two attaching curves are disjoint, or they can be isotoped into a once-punctured torus. A byproduct is a combinatorial proof of a similar known result about degenerating hyperbolic structures by Dehn filling.

We consider solutions to the algebraic differential equation $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}f^nf'+Q_d(z,f)=u(z)e^{v(z)}$, where $Q_d(z,f)$ is a differential polynomial in $f$ of degree $d$ with rational function coefficients, $u$ is a nonzero rational function and $v$ is a nonconstant polynomial. In this paper, we prove that if $n\ge d+1$ and if it admits a meromorphic solution $f$ with finitely many poles, then

$$\begin{equation*} f(z)=s(z)e^{v(z)/(n+1)} \quad \mbox {and}\quad Q_d(z,f)\equiv 0. \end{equation*}$$

$f$

$f'p_k(f)+q_m(f)$

$\alpha $

$p_k(f), q_m(f)$

$f$

$k$

$m$

$k\ge m+1$

This note is concerned with a question arising from some work of D. R. Smart (2). In the introduction we indicate the nature of the problem. Notations will be explained only in as far as they differ from those of (2).

It is well known (see Latyšev [1] for finite dimensional case) that the universal envelope of Lie Alaebra g over a commutative field ť of characteristic 0 is a PI-alaebra (i.e. possesses a nontrivial identity) if and only if this Lie algebra is abelian. On the other hand the recent results due to Passman [2] describe the conditions under which the group algebra of a group over an arbitrary commutative field is a PI-algebra. A. L. Šmel'kin suggested that I should find necessary and sufficient conditions for a Lie algebra g over a field of nonzero characteristic under which its universal envelope Ug should be a PI-algebra. These conditions are given in the following theorem.

Let $H$ be a Krull monoid with finite class group $G$ such that every class contains a prime divisor (for example, a ring of integers in an algebraic number field or a holomorphy ring in an algebraic function field). The catenary degree $\mathsf{c}(H)$ of $H$ is the smallest integer $N$ with the following property: for each $a\in H$ and each pair of factorizations $z,z^{\prime }$ of $a$, there exist factorizations $z=z_{0},\dots ,z_{k}=z^{\prime }$ of $a$ such that, for each $i\in [1,k]$, $z_{i}$ arises from $z_{i-1}$ by replacing at most $N$ atoms from $z_{i-1}$ by at most $N$ new atoms. To exclude trivial cases, suppose that $|G|\geq 3$. Then the catenary degree depends only on the class group $G$ and we have $\mathsf{c}(H)\in [3,\mathsf{D}(G)]$, where $\mathsf{D}(G)$ denotes the Davenport constant of $G$. The cases when $\mathsf{c}(H)\in \{3,4,\mathsf{D}(G)\}$ have been previously characterized (see Theorem A). Based on a characterization of the catenary degree determined in the paper by Geroldinger et al. [‘The catenary degree of Krull monoids I’, J. Théor. Nombres Bordeaux23 (2011), 137–169], we determine the class groups satisfying $\mathsf{c}(H)=\mathsf{D}(G)-1$. Apart from the extremal cases mentioned, the precise value of $\mathsf{c}(H)$ is known for no further class groups.

Suppose a preliminary set of m independent observations are drawn from a population in which a random variable x has a continuous but unknown cumulative distribution function F(x). Let y be the largest observation in this preliminary sample. Now suppose further observations are drawn one at a time from this population until an observation exceeding y is obtained. Let n be the number of further drawings required to achieve this objective. The problem is to determine the distribution function of the random variable n. More generally, suppose y is the r-th from the largest observation in the preliminary sample and let n denote the number of further trials required in order to obtain k observations which exceed y. What is the distribution function of n?

From an integer-valued function f we obtain, in a natural way, a matroid Mf on the domain of f. We show that the class of matroids so obtained is closed under restriction, contraction, duality, truncation and elongation, but not under direct sum. We give an excluded-minor characterization of and show that consists precisely of those transversal matroids with a presentation in which the sets in the presentation are nested. Finally, we show that on an n-set there are exactly 2n members of .

We show that a graph G is semi-stable at vertex v if and only if the set of vertices of G adjacent to v is fixed by the automorphism group of Gv, the subgraph of G obtained by deleting v and its incident edges. This result leads to a neat proof that regular graphs are semi-stable at each vertex. We then investigate stable regular graphs, concentrating mainly on stable vertex-transitive graphs. We conjecture that if G is a non-trivial vertex-transitive graph, then G is stable if and only if γ(G) contains a transposition, offering some evidence for its truth.

We study classes of Banach spaces where every set-valued mapping from a complete metric space into subsets of the Banach space which satisfies certain minimal properties, is single-valued and norm upper semi-continuous at the points of a dense Gδ subset of its domain. Characterisations of these classes are developed and permanence properties are established. Sufficiency conditions for membership of these classes are defined in terms of fragmentability and σ-fragmentability of the weak topology. A characterisation of non membership is used to show that l∞ (N) is not a member of our classe of generic continuity spaces.

Several morphisms of this lattice V(CR) are found, leading to decompostions of it, and various sublattices, into subdirect products of interval sublattices. For example the map V → V ∪ G (where G is the variety of groups) is shown to be a retraction of V(CR); from modularity of the lattice V(BG) of varieties of bands of groups it follows that the map V → (V ∪ V V G) is an isomorphism of V(BG).

In the terminology of J. R. Isbell [5], an element d of a semigroup S is dominated by a subsemigroup U of S if, for an arbitrary semigroup X and arbitrary homomorphisms α β, from S into X, α(u) = β(u) for every u in U implies α(d) = β(d). The set of elements of S dominated by U is a subsemigroup of S containing U and is called the dominion of U. It was shown by Isbell that if one takes two disjoint isomorphic copies S+, S− of S and forms their amalgamated free product S+ * US− that is to say, the quotient of the free product S+ * S− by the congruence p generated by , (u+ and u− being the images of u in S+, S− respectively) then the homomorphisms: μ+: S→ S+ * uS−, μ−: S→ S+ * uS− defined by are one-one. Moreover, μ+(s) = μ−(t) only if s = t, and μ+(s)= μ−(s) if and only if s is in the dominion of U. In other words, the two natural copies of S in S+ * uS− intersect precisely in the dominion of U. Thus, in particular, and in the terminology of [3], the amalgam [S+, S−; U] is embeddable if and only if U is self-dominating (that is to say, its own dominion) in S.