Let p be a prime. This paper deals with solutions of functional equations in either formal Laurent series or in analytic functions. Examples connected to special modular functions are considered.

The extended diagram of a two-bridged knot is introduced, and it is shown how the coefficients of the Alexander polynomial of the knot may be read straight from this diagram. Using this result, it is shown by diagram manipulation that a conjecture of Fox about the coefficients of the Alexander polynomial of an alternating knot is true at least for two-bridged knots (which are all alternating).

1980 Mathematics subject classification (Amer. Math. Soc.): 57 M 25.

The aim of this paper is to give solutions of some of the problems raised in Wiegold (1974). The main content is the proof of Theorems A and B following, which are improvements of Theorems 4.5 and 3.4 respectively of Wiegold (1974). All unexplained notation will be as in that paper, which would perhaps best be viewed as a preliminary to this one.

The core of a graph Γ is the smallest graph Δ that is homomorphically equivalent to Γ (that is, there exist homomorphisms in both directions). The core of Γ is unique up to isomorphism and is an induced subgraph of Γ. We give a construction in some sense dual to the core. The hull of a graph Γ is a graph containing Γ as a spanning subgraph, admitting all the endomorphisms of Γ, and having as core a complete graph of the same order as the core of Γ. This construction is related to the notion of a synchronizing permutation group, which arises in semigroup theory; we provide some more insight by characterizing these permutation groups in terms of graphs. It is known that the core of a vertex-transitive graph is vertex-transitive. In some cases we can make stronger statements: for example, if Γ is a non-edge-transitive graph, we show that either the core of Γ is complete, or Γ is its own core. Rank-three graphs are non-edge-transitive. We examine some families of these to decide which of the two alternatives for the core actually holds. We will see that this question is very difficult, being equivalent in some cases to unsolved questions in finite geometry (for example, about spreads, ovoids and partitions into ovoids in polar spaces).

An R-module M is called coretractable if  HomR(M/K,M)≠0 for any proper submodule K of M. In this paper we study coretractable modules and their endomorphism rings. It turns out that if all right R-modules are coretractable, then R is a right Kasch and (two-sided) perfect ring. However, the converse holds for commutative rings. Also, for a semi-injective coretractable module MR with S=EndR(M), we show that u.dim(SS)=corank(MR).

Let ${\mathcal{S}}$ be the family of analytic and univalent functions $f$ in the unit disk $\mathbb{D}$ with the normalization $f(0)=f^{\prime }(0)-1=0$, and let $\unicode[STIX]{x1D6FE}_{n}(f)=\unicode[STIX]{x1D6FE}_{n}$ denote the logarithmic coefficients of $f\in {\mathcal{S}}$. In this paper we study bounds for the logarithmic coefficients for certain subfamilies of univalent functions. Also, we consider the families ${\mathcal{F}}(c)$ and ${\mathcal{G}}(c)$ of functions $f\in {\mathcal{S}}$ defined by

$$\begin{eqnarray}\text{Re}\biggl(1+{\displaystyle \frac{zf^{\prime \prime }(z)}{f^{\prime }(z)}}\biggr)>1-{\displaystyle \frac{c}{2}}\quad \text{and}\quad \text{Re}\biggl(1+{\displaystyle \frac{zf^{\prime \prime }(z)}{f^{\prime }(z)}}\biggr)<1+{\displaystyle \frac{c}{2}},\quad z\in \mathbb{D},\end{eqnarray}$$

$c\in (0,3]$

$c\in (0,1]$

$|\unicode[STIX]{x1D6FE}_{n}|$

$n=1,2,3$

$f$

${\mathcal{F}}(c)$

${\mathcal{G}}(c)$

• ∙ If $f\in {\mathcal{F}}(-1/2)$, then $|\unicode[STIX]{x1D6FE}_{n}|\leq 1/n(1-(1/2^{n+1}))$ for $n\geq 1$, and

  $$\begin{eqnarray}\mathop{\sum }_{n=1}^{\infty }|\unicode[STIX]{x1D6FE}_{n}|^{2}\leq {\displaystyle \frac{\unicode[STIX]{x1D70B}^{2}}{6}}+{\displaystyle \frac{1}{4}}~\text{Li}_{2}\biggl({\displaystyle \frac{1}{4}}\biggr)-\text{Li}_{2}\biggl({\displaystyle \frac{1}{2}}\biggr),\end{eqnarray}$$
  where $\text{Li}_{2}(x)$ denotes the dilogarithm function.

• ∙ If $f\in {\mathcal{G}}(c)$, then $|\unicode[STIX]{x1D6FE}_{n}|\leq c/2n(n+1)$ for $n\geq 1$.

Let G be a finite group of order g having exactly k conjugate classes. Let π(G) denote the set of prime divisors of g. K. A. Hirsch [4] has shown that By the same methods we prove g ≡ k modulo G.C.D. {(p–1)2 p ∈ π(G)} and that if G is a p-group, g = h modulo (p−1)(p2−1). It follows that k has the form (n+r(p−1)) (p2−1)+pe where r and n are integers ≧ 0, p is a prime, e is 0 or 1, and g = p2n+e. This has been established using representation theory by Philip Hall [3] (see also [5]). If then simple examples show (for 6 ∤ g obviously) that g ≡ k modulo σ or even σ/2 is not generally true.

Let G be a finite group with d(G) = α, d(G/G′) = β≥1. If G has non-abelian simple images, let s denote the order of a smallest such image. Then d(Gn) = βn provided that βn≥α + 1 + log8n. If all simple images of G are abelian, then d(Gn) = βn provided that βn≥α. If G is non-trivial and perfect, with s again denoting the order of a smallest non-abelian simple image, then d(Gsn)≼d(G) + n for all n≥0. These results improve on results in previous papers with similar titles.

Freudenthal [5, 7] defined a compactification of a rim-compact space, that is, a space having a base of open sets with compact boundary. The additional points are called ends and Freudenthal showed that a connected locally compact non-compact group having a countable base has one or two ends. Later, Freudenthal [8], Zippin [16], and Iwasawa [11] showed that a connected locally compact group has two ends if and only if it is the direct product of a compact group and the reals.

An estimate for q2|α – p/;q| is obtained by considering the relation between the continued fractions for α and p / q. This leads to an extension of the standard result “q2|α – p/q| < 1 imples that for some n, p/q = (ipn + pn-1) / (iqn + qn-1) where i = 0, 1 or an+1 −1”.

The concept of critical group was introduced by D. C. Cross (as reported by G. Higman in [5]): a finite group is called critical if it is not contained in the variety generated by its proper factors. (The factors of a group G are the groups H/K where K H ≦ G, and H/K is a proper factor of G unless H = G and K =1). Some investigations concerning finite groups and varieties depend on the investigator's ability to decide whether a given group is critical or not. (For instance, one of the crucial points in the important paper [9] of Sheila Oates and M. B. Powell is a necessary condition of criticality: their Lemma 2.4.2.) An obvious necessary condition is that the group should have only one minimal normal subgroup: the group is then called monolithic, and the minimal normal subgroup its monolith. This is, however, far from being a sufficient condition, and it is the purpose of the present paper to give some sufficient conditions for the criticality of monolithic groups. (We consider the trivial group neither monolithic nor critical.) The basis of our results is an analysis of the following situation.

In this paper we shall discuss the boundary value problem consisting of the nonlinear ordinary differential equation of the second order, and the boundary conditions.

A graph is found which is 4-chromatic, has girth 5, and can be obtained by taking a set S of points in the plane and joining two of the points whenever their distance is 1. The points in S are not found explicitly, but their existence is demonstrated by use of a computer.

Subject classification (Amer. Math. Soc. (MOS) 1970): 05 C 15, 05 C 10.

In [1] Neumann and Dey prove that the free product of two finitely generated Hopf groups is Hopf and ask whether a similar result holds for direct products. It is the purpose of this paper to show that this is not the case. We prove

THEOREM A. There exists a finitely generated group G satisfying the following conditions:

(i) G is isomorphic to a proper direct factor of itself;

(ii) G is the direct product of two Hopf groups.

Mathématical and computational techniques are described for constructing and enumerating generalized Bhaskar Rao designs (GBRD's). In particular, these methods are applied to GBRD(k + 1, k, 1(k − 1); G)'s for 1 ≥ 1. Properties of the enumerated designs, such as automorphism groups, resolutions and contracted designs are tabulated. Also described are applications to group divisible designs, multi-dimensional Howell cubes, generalized Room squares, equidistant permutation arrays, and doubly resolvable two-fold triple systems.

In this paper, we give a generalization of Hardy's theorems for the Dunkl transform ℱD on ℝd. More precisely for all a > 0, b > 0 and p, q ∈ [1, + ∞], we determine the measurable functions f on ℝd such that where are the Lebesgue spaces associated with the Dunkl transform.

If CS(respectively, O) denotes the class of all completely simple semigroups (respectively, semigroups that are orthodox unions of groups) then CS(respectively, O) is a variety of algebras with respect to the operations of multiplication and inversion. The main result shows that the lattice of subvarieties of is a precisely determined subdirect product of the lattice of subvarieties of CSand the lattice of subvarieties of O. A basis of identities is obtained for any variety in terms of bases of identities for . Several operators on the lattice of subvarieties of are also introduced and studied.

Metric regularity theory lies in the very heart of variational analysis, a relatively new discipline whose appearance was, to a large extent, determined by the needs of modern optimization theory in which such phenomena as nondifferentiability and set-valued mappings naturally appear. The roots of the theory go back to such fundamental results of the classical analysis as the implicit function theorem, Sard theorem and some others. The paper offers a survey of the state of the art of some principal parts of the theory along with a variety of its applications in analysis and optimization.

We deal with questions about the possible embeddings of two given groups A and B in a group P such that the intersection of A and B is a given subgroup H. The data, consisting of the “constituents” A and B with the “amalgamated” subgroup H, form an amalgam.1 According to a classical theorem of Otto Schreier [5], every amalgam of two groups can be embedded in a group F, the “free product of A and B with amalgamated subgroup H” or the “generalized free product” of the amalgam. This has the property that every group P in which the amalgam is embedded and which is generated by the amalgam, is a homomorphic image of it. Hence theorems on the existence of certain embedding groups P can be interpreted also as theorems on the existence of certain normal subgroups of F.

This paper is concerned with a new notion of coherency for monoids. A monoid S is right coherent if the first order theory of right S-sets is coherent; this is equivalent to the property that every finitely generated S-subset of every finitely presented right S-set is finitely presented. If every finitely generated right ideal of S is finitely presented we say that S is weakly right coherent. As for the corresponding situation for modules over a ring, we show that our notion of coherency is related to products of flat left S-sets, although there are some marked differences in behaviour from the case for rings. Further, we relate our work to ultraproducts of flat left S-sets and so to the question of axiomatisability of certain classes of left S-sets.

We show that a monoid S is weakly right coherent if and only if the right annihilator congruence of every element is finitely generated and the intersection of any two finitely generated right ideals is finitely generated. A similar result describes right coherent monoids. We use these descriptions to recognise several classes of (weakly) right coherent monoids. In particular we show that any free monoid is weakly right (and left) coherent and any free commutative monoid is right (and left) coherent.