It is shown that in a certain extensive class of algebras one can associate with each Lie ideal a corresponding associative ideal which facilitates the study of Lie ideals, especially for simple algebras. We apply this construction to obtain new, simpler proofs of some known results of Herstein [10] and others on the Lie structure of associative rings.

A finite group is said to have deficiency zero if it can be presented with an equal number of generators and relations. Finite metacyclic groups of deficiency zero have been classified, see [1] or [6]. Finite non-metacyclic groups of deficiency zero, which we denote by FD0-groups, are relatively scarce. In [3] I. D. Macdonald introduced a class of nilpotent FD0-groups all having nilpotent class≤8. The largest nilpotent class known for a Macdonald group is 7 [4]. Only a finite number of nilpotent FD0-groups, other than the Macdonald groups, seem to be known [5], [7]. In this note we exhibit a class of FD0-groups which contains nilpotent groups of arbitrarily large nilpotent class.

We give an elementary proof of the theorem of Saks that states that most functions in C[0, 1] have infinite unilateral derivatives at a continuum many points.

In this paper we prove: for any positive integers $a$ and $q$ with $\left( a,\,q \right)\,=\,1$, we have uniformly

$$\sum\limits_{\begin{matrix} n\le N \\ (n,q)=1,n\bar{n}\equiv 1(\,\bmod \,q) \\ \end{matrix}}{\mu (n)e(\frac{a\bar{n}}{q})\ll Nd(q)\left\{ \frac{{{\log }^{\frac{5}{2}}}N}{{{q}^{\frac{1}{2}}}}+\frac{{{q}^{\frac{1}{5}}}{{\log }^{\frac{13}{5}}}N}{{{N}^{\frac{1}{5}}}} \right\}.}$$

This improves the previous bound obtained by D. Hajela, A. Pollington and B. Smith [5].

We show that for compact orientable hyperbolic orbisurfaces, the Laplace spectrum determines the length spectrum as well as the number of singular points of a given order. The converse also holds, giving a full generalization of Huber's theorem to the setting of compact orientable hyperbolic orbisurfaces.

For a simple Lie algebra L over the complex numbers ℂ all irreducible representations admitting a highest weight have been constructed and characterized for example in [3, 6]. In [1] Bouwer considered the family of all irreducible representations of L admitting at least one one-dimensional weight space (this includes, of course, all those having a highest weight space) and showed, by construction, that this is a strictly larger class of representations.

Let μ∗ be an outer measure on (X, S) with σ- algebra S and let μ* be the inner measure induced by μ∗. A set M is a measurable cover of a set A ⊆ X if A ⊆ M, M is measurable, and μ∗ (M-A) = 0. We assume that every subset of X has a measurable cover; this holds, for example, if μ∗ is the outer measure induced by a measure which is σ- finite on X [2, theorem C, p. 50].

In this paper, another relationship between the quasi-ideal adequate transversals of an abundant semigroup is given. We introduce the concept of a weakly multiplicative adequate transversal and the classic result that an adequate transversal is multiplicative if and only if it is weakly multiplicative and a quasi-ideal is obtained. Also, we give two equivalent conditions for an adequate transversal to be weakly multiplicative. We then consider the case when $I$ and $\Lambda $ (defined below) are bands. This is analogous to the inverse transversal if the regularity condition is adjoined.

In [8] and subsequent papers, Jônsson (et al) developed a lattice identity which reflects precisely Desargues Law in projective geometry in that a projective geometry satisfies Desargues Law if and only if the geometry, qua lattice, satisfies this identity. This identity, appropriately called the Arguesian law, has become exceedingly important in recent investigations in the variety of modular lattices (see for example [2], [3], [9], and [12]). In this note, we supply two possible lattice identities for the Pappus' Law of projective geometry.

A well known theorem by S. Banach states that a contractive function f: X → X on a complete metric space X has a fixed point, and that this fixed point is unique. This result has a partial extension to multi-functions: every contractive compact-valued multi-function on a complete metric space has a fixed point (see Definition 1 and Theorem 1 below). But simple examples show that this fixed point is no longer unique. We investigate some questions concerned with the properties of the fixed point set Φ of a contractive multi-function φ. Is, e.g., Φ connected if φ is connected-valued? Is Φ convex if φ is convex-valued? The answer is yes if X is the real line (§2), but examples in §3 and §4 show that in general the answer is no.

Given a finite group $G$, we attach to the character degrees of $G$ a graph whose vertex set is the set of primes dividing the degrees of irreducible characters of $G$, and with an edge between $p$ and $q$ if $pq$ divides the degree of some irreducible character of $G$. In this paper, we describe which graphs occur when $G$ is a solvable group of Fitting height 2.

A projective geometry of dimension (n - 1) can be defined as modular lattice with a spanning n-diamond of atoms (i.e.: n + 1 atoms in general position whose join is the unit of the lattice). The lemma we show is that one could equivalently define a projective geometry as a modular lattice with a spanning n-diamond that is (a) is generated (qua lattice) by this n-diamond and a coordinating diagonal and (b) every non-zero member of this coordinatizing diagonal is invertible. The lemma is applied to describe certain freely generated modular and Arguesian lattices.

In an earlier paper [3] we considered an inversion formula for the Gauss transformation G defined by

1.1

We noted there that formally G is inverted by,

1.2

and we showed that if e-D2 is interpreted via the power series for the exponential function, that is if

1.3

then under certain conditions on φ,

1.4

An orientation of a graph G is an assignment of a unique direction to each line of G. The result is called an oriented graph. Two orientations of a graph are regarded as equivalent if the resulting oriented graphs are isomorphic as directed graphs. For example, the graph C3 consisting of a cycle of length 3 (a triangle) shown in Figure 1(a), has exactly two orientations D1 and D2; see Figure 1(b) and (c).

We construct new examples of non-nil algebras with any number of generators, which are direct sums of two locally nilpotent subalgebras. Like all previously known examples, our examples are contracted semigroup algebras and the underlying semigroups are unions of locally nilpotent subsemigroups. In our constructions we make more transparent than in the past the close relationship between the considered problem and combinatorics of words.

Let D(n) denote a function of an integral variable n ≥ 0 such that

1. (1) D(1) = D(0) = 0

2. (2) D(p) = 1 for every prime p

3. (3) D(n1n2) = n1D(n2) + n2D(n1) for every pair of non-negative integers n1, n2.

J. Dixmier has observed [3, p. 1070] that a non-reflexive Banach space has non-rotund fourth conjugate space. It is the aim of this paper to improve Dixmier’s result by showing that a non-reflexive Banach space already has non-smooth third conjugate space in that the images under natural embedding of the continuous linear functionals which do not attain their norm on the unit sphere are non-smooth points of the third conjugate space.

In measure theory and probability it is often useful to be able to extend a set function g to a measure μ. One situation in which such an extension arises is that of obtaining limit measures for inverse (or projective) systems of measure spaces ([1], [5]).

We consider the class of polynomial differential systems of the form $\dot{x} =\,\lambda x\,-\,y\,+\,{{P}_{n}}\left( x,\,y \right)$ , $\dot{y} =\,x\,+\,\lambda y\,+\,{{Q}_{n}}\left( x,\,y \right)$ where ${{P}_{n}}$ and ${{Q}_{n}}$ are homogeneous polynomials of degree $n$ . For this class of differential systems we summarize the known results for the existence of limit cycles, and we provide new results for their nonexistence and existence.

In this brief note, we will show how in principle to find all units in the integral group ring ZG, whenever G is a finite group such that and Z(G) each have exponent 2, 3, 4 or 6. Special cases include the dihedral group of order 8, whose units were previously computed by Polcino Milies [5], and the group discussed by Ritter and Sehgal [6]. Other examples of noncommutative integral group rings whose units have been computed include , but in general very little progress has been made in this direction. For basic information on units in group rings, the reader is referred to Sehgal [7].