Some $p$-groups of maximal class
R. J.
Miech
1-47
Abstract: This paper deals with the construction of some p-groups of maximal class.
Counting commutators
R. J.
Miech
49-61
Abstract: Let G be a group generated by x and y, ${G_2}$ be the commutator subgroup of G, and ${G_1}$ be the group generated by y and $ {G_2}$. This paper contains explicit expansions of $ {y^{{x^m}}}$ modulo [ ${G_2},{G_2},{G_2}$] and ${(xy)^m}$ modulo [ $ {G_1},{G_1},{G_1}$]. The motivation for these results stem from the p-groups of maximal class, for a large number of these groups have $[{G_1},{G_1},{G_1}] = 1$.
Fixed point theorems for certain classes of semigroups of mappings
Mo Tak
Kiang
63-76
Abstract: Fixed point theorems for commuting semigroups of self-mappings are considered in this paper. A generalization of the classical Markov-Kakutani theorem is first given. This is followed by a fixed point theorem for commutative semigroups of continuous asymptotically-nonexpansive self-mappings on a weakly compact, convex subset of a strictly convex Banach space.
Explicit class field theory for rational function fields
D. R.
Hayes
77-91
Abstract: Developing an idea of Carlitz, I show how one can describe explicitly the maximal abelian extension of the rational function field over $ {{\mathbf{F}}_q}$ (the finite field of q elements) and the action of the idèle class group via the reciprocity law homomorphism. The theory is closely analogous to the classical theory of cyclotomic extensions of the rational numbers.
Minimal sequences in semigroups
Mohan S.
Putcha
93-106
Abstract: In this paper we generalize a result of Tamura on $\delta$-indecomposable semigroups. Based on this, the concept of a minimal sequence between two points, and from a point to another, is introduced. The relationship between two minimal sequences between the same points is studied. The rank of a semigroup S is defined to be the supremum of the lengths of the minimal sequences between points in S. The semirank of a semigroup S is defined to be the supremum of the lengths of the minimal sequences from a point to another in S. Rank and semirank are further studied.
On the structure of the set of solutions of equations involving $A$-proper mappings
P. M.
Fitzpatrick
107-131
Abstract: Let X and Y be Banach spaces having complete projection schemes (say, for example, they have Schauder bases). We consider various properties of mappings $T:D \subset X \to Y$ which are either Approximation-proper (A-proper) or the uniform limit of such mappings. In §1 general properties, including those of the generalized topological degree, of such mappings are discussed. In §2 we give sufficient conditions in order that the solutions of an equation involving a nonlinear mapping be a continuum. The conditions amount to requiring that the generalized topological degree not vanish, and that the mapping involved be the uniform limit of well structured mappings. We devote §3 to proving a result connecting the topological degree of an A-proper Fréchet differentiable mapping to the degree of its derivative. Finally, in §4, various Lipschitz-like conditions are discussed in an A-proper framework, and constructive fixed point and surjectivity results are obtained.
Subgroups of groups of central type
Kathleen M.
Timmer
133-161
Abstract: Let $\lambda$ be a linear character on the center Z of a finite group Z of a finite group H, such that (1) ${\lambda ^H} = \sum\nolimits_{i = 1}^p {{\phi _i}(1){\phi _i}}$ where the ${\phi _i}$'s are inequivalent irreducible characters on H of the same degree, and (2) if $\sum\nolimits_{i = 1}^p {{m_i}{\phi _i}(x) = 0}$ for some $x \in H$ and nonnegative integers ${m_i}$, then either ${\phi _i}(x) = 0$ for all i or ${m_i} = {m_j}$ for all i, j. The object of the paper is to describe finite groups which satisfy conditions (1) and (2) in terms of the multiplication of the group. If S is a p Sylow subgroup of the group H, and $R = S \cdot Z$, then H satisfies conditions (1) and (2) if and only if (a) $\{ x \in H:{x^{ - 1}}{h^{ - 1}}xh \in Z \Rightarrow \lambda ({x^{ - 1}}{h^{ - 1}}xh) = 1,h \in H\} /Z$ consists of elements of order a power of p in $H/Z$, and these elements form p conjugacy classes of $H/Z$, and (b) the elements of $ \{ x \in R:{x^{ - 1}}{r^{ - 1}}xr \in Z \Rightarrow \lambda ({x^{ - 1}}{r^{ - 1}}xr) = 1,r \in R\} /Z$ form p conjugacy classes of $R/Z$.
Lattices of topological extensions
John
Mack;
Marlon
Rayburn;
Grant
Woods
163-174
Abstract: For completely regular Hausdorff spaces, we consider topological properties P which are akin to compactness in the sense of Herrlich and van der Slot and satisfy the equivalent of Mrowka's condition (W). The algebraic structure of the family of tight extensions of X (which have P and contain no proper extension with that property) is studied. Where X has P locally but not globally, the relations between the complete lattice $ {P^ \ast }(X)$ of those tight extensions which are above the maximal one-point extension and the topology of the P-reflection are investigated and conditions found under which ${P^\ast}(X)$ characterizes $\gamma X - X$. The results include those of Magill on the lattice of compactifications of a locally compact space, and other applications are considered.
Equivariant method for periodic maps
Wu Hsiung
Huang
175-183
Abstract: The notion of coherency with submanifolds for a Morse function on a manifold is introduced and discussed in a general way. A Morse inequality for a given periodic transformation which compares the invariants called qth Euler numbers on fixed point set and the invariants called qth Lefschetz numbers of the transformations is thus obtained. This gives a fixed point theorem in terms of qth Lefschetz number for arbitrary q.
Asymptotic solutions of linear Volterra integral equations with singular kernels
J. S. W.
Wong;
R.
Wong
185-200
Abstract: Volterra integral equations of the form $a(t) \in C(0,\infty ) \cap {L_1}(0,1)$. Explicit asymptotic forms are obtained for the solutions, when the kernels $a(t)$ have a specific asymptotic representation.
Rank $r$ solutions to the matrix equation $XAX\sp{T}=C,\,A$ alternate, over ${\rm GF}(2\sp{y})$
Philip G.
Buckhiester
201-209
Abstract: Let ${\text{GF}}(q)$ denote a finite field of characteristic two. Let ${V_n}$ denote an n-dimensional vector space over $ {\text{GF}}(q)$. An $n \times n$ symmetric matrix A over ${\text{GF}}(q)$ is said to be an alternate matrix if A has zero diagonal. Let A be an $n \times n$ alternate matrix over ${\text{GF}}(q)$ and let C be an $s \times s$ symmetric matrix over ${\text{GF}}(q)$. By using Albert's canonical forms for symmetric matrices over fields of characteristic two, the number $ N(A,C,n,s,r)$ of $s \times n$ matrices X of rank r over $ {\text{GF}}(q)$ such that $XA{X^T} = C$ is determined. A symmetric bilinear form on $ {V_n} \times {V_n}$ is said to be alternating if $ f(x,x) = 0$, for each x in ${V_n}$. Let f be such a bilinear form. A basis $({x_1}, \ldots ,{x_\rho },{y_1}, \ldots ,{y_\rho }),n = 2\rho$, for ${V_n}$ is said to be a symplectic basis for $ {V_n}$ if $f({x_i},{x_j}) = f({y_i},{y_j}) = 0$ and $ f({x_i},{y_j}) = {\delta _{ij}}$, for each i, $j = 1,2, \ldots ,\rho$. In determining the number $N(A,C,n,s,r)$, it is shown that a symplectic basis for any subspace of ${V_n}$, can be extended to a symplectic basis for $ {V_n}$. Furthermore, the number of ways to make such an extension is determined.
The structure of completely regular semigroups
Mario
Petrich
211-236
Abstract: The principal result is a construction of completely regular semigroups in terms of semilattices of Rees matrix semigroups and their translational hulls. The main body of the paper is occupied by considerations of various special cases based on the behavior of either Green's relations or idempotents. The influence of these special cases on the construction in question is studied in considerable detail. The restrictions imposed on Green's relations consist of the requirement that some of them be congruences, whereas the restrictions on idempotents include various covering conditions or the requirement that they form a subsemigroup.
Volumes of images of varieties in projective space and in Grassmannians
H.
Alexander
237-249
Abstract: If V is a complex analytic subvariety of pure dimension k in the unit ball in $ {{\mathbf{C}}^n}$ which does not contain the origin, then the 2k-volume of V equals the measure computed with multiplicity of the set of $(n - k)$-complex subspaces through the origin which meet V. The measure of this set computed without multiplicity is a smaller quantity which is nevertheless bounded below by a number depending only on the distance from V to the origin. As an application we characterize normal families in the unit ball as those families of analytic functions whose restrictions to each complex line through the origin are normal. The complex analysis which we shall need will be developed in the context of uniform algebras.
Approximation of analytic functions on compact sets and Bernstein's inequality
M. S.
Baouendi;
C.
Goulaouic
251-261
Abstract: The characterization of analytic functions defined on a compact set K in $ {{\mathbf{R}}_N}$ by their polynomial approximation is possible if and only if K satisfies some ``Bernstein type inequality", estimating any polynomial P in some neighborhood of K using the supremum of P on K. Some criterions and examples are given. Approximation by more general sets of analytic functions is also discussed.
Galois theory for fields $K/k$ finitely generated
Nickolas
Heerema;
James
Deveney
263-274
Abstract: Let K be a field of characteristic $p \ne 0$. A subgroup G of the group $ {H^t}(K)$ of rank t higher derivations $ (t \leq \infty )$ is Galois if G is the group of all d in $ {H^t}(K)$ having a given subfield h in its field of constants where K is finitely generated over h. We prove: G is Galois if and only if it is the closed group (in the higher derivation topology) generated over K by a finite, abelian, independent normal iterative set F of higher derivations or equivalently, if and only if it is a closed group generated by a normal subset possessing a dual basis. If $t < \infty$ the higher derivation topology is discrete. M. Sweedler has shown that, in this case, h is a Galois subfield if and only if $ K/h$ is finite modular and purely inseparable. Also, the characterization of Galois groups for $t < \infty $ is closely related to the Galois theory announced by Gerstenhaber and and Zaromp. In the case $t = \infty$, a subfield h is Galois if and only if $K/h$ is regular. Among the applications made are the following: (1) ${ \cap _n}h({K^{{p^n}}})$ is the separable algebraic closure of h in K, and (2) if $K/h$ is algebraically closed, $K/h$ is regular if and only if $K/h({K^{{p^n}}})$ is modular for $n > 0$.
Quasi-bounded and singular functions
Maynard
Arsove;
Heinz
Leutwiler
275-302
Abstract: A general formulation is given for the concepts of quasi-bounded and singular functions, thereby extending to a much broader class of functions the concepts initially formulated by Parreau in the harmonic case. Let $\Omega$ be a bounded Euclidean region. With the underlying space taken as the class $\mathcal{M}$ of all nonnegative functions u on $\Omega$ admitting superharmonic majorants, an operator S is introduced by setting Su equal to the regularization of the infimum over $\lambda \geq 0$ of the regularized reduced functions for $ {(u - \lambda )^ + }$. Quasi-bounded and singular functions are then defined as those u for which $Su = 0$ and $Su = u$, respectively. A development based on properties of the operator S leads to a unified theory of quasi-bounded and singular functions, correlating earlier work of Parreau (1951), Brelot (1967), Yamashita(1968), Heins (1969), and others. It is shown, for example, that a nonnegative function u on $ \Omega$ is quasi-bounded if and only if there exists a nonnegative, increasing, convex function $\varphi$ on $ [0,\infty ]$ such that $\varphi (x)/x \to + \infty$ as $x \to \infty$ and $ \varphi \circ u$ admits a superharmonic majorant. Extensions of the theory are made to the vector lattice generated by the positive cone of functions u in $ \mathcal{M}$ satisfying $ Su \leq u$.
Differentiability of solutions to hyperbolic initial-boundary value problems
Jeffrey B.
Rauch;
Frank J.
Massey
303-318
Abstract: This paper establishes conditions for the differentiability of solutions to mixed problems for first order hyperbolic systems of the form $ (\partial /\partial t - \sum {A_j}\partial /\partial {x_j} - B)u = F$ on $[0,T] \times \Omega ,Mu = g$ on $[0,T] \times \partial \Omega ,u(0,x) = f(x),x \in \Omega$. Assuming that ${\mathcal{L}_2}$ a priori inequalities are known for this equation, it is shown that if $F \in {H^s}([0,T] \times \Omega ),g \in {H^{s + 1/2}}([0,T] \times \partial \Omega ),f \in {H^s}(\Omega )$ satisfy the natural compatibility conditions associated with this equation, then the solution is of class ${C^p}$ from [0, T] to $ {H^{s - p}}(\Omega ),0 \leq p \leq s$. These results are applied to mixed problems with distribution initial data and to quasi-linear mixed problems.
Canonical forms and principal systems for general disconjugate equations
William F.
Trench
319-327
Abstract: It is shown that the disconjugate equation (1) $Lx \equiv (1/{\beta _n})(d/dt) \cdot (1/{\beta _{n - 1}}) \cdots (d/dt)(1/{\beta _1})(d/dt)(x/{\beta _0}) = 0$ , a $< t < b$, where ${\beta _i} > 0$, and (2) ${\beta _i} \in C(a,b)$, can be written in essentially unique canonical forms so that ${\smallint ^b}{\beta _i}dt = \infty ({\smallint _a}{\beta _i}dt = \infty )$ for $1 \leq i \leq n - 1$. From this it follows easily that (1) has solutions ${x_1}, \ldots ,{x_n}$ which are positive in (a, b) near $b(a)$ and satisfy ${\lim _{t \to b}} - {x_i}(t)/{x_j}(t) = 0({\lim _{t \to a}} + {x_i}(t)/{x_j}(t) = \infty )$ for $ 1 \leq i < j \leq n$. Necessary and sufficient conditions are given for (1) to have solutions $ {y_1}, \ldots ,{y_n}$ such that $ {\lim _{t \to b}} - {y_i}(t)/{y_j}(t) = {\lim _{t \to a}} + {y_j}(t)/{y_i}(t) = 0$ for $ 1 \leq i < j \leq n$. Using different methods, P. Hartman, A. Yu. Levin and D. Willett have investigated the existence of fundamental systems for (1) with these properties under assumptions which imply the stronger condition
Pairs of compacta and trivial shape
Sibe
Mardešić
329-336
Abstract: Let (X, Y, A), ${\text{sh}}\;X = {\text{sh}}\;X' = 0$ and ${\text{sh}}(X,A) = {\text{sh}}(X',A')$, which in view of an example of Borsuk shows that for compact metric pairs the ANR-system approach to shapes differs from the Borsuk approach.
An equiconvergence theorem for a class of eigenfunction expansions
C. G. C.
Pitts
337-350
Abstract: A recent result of Muckenhoupt concerning the convergence of the expansion of an arbitrary function in terms of the Hermite series of orthogonal polynomials is generalised to a class of orthogonal expansions which arise from an eigenfunction problem associated with a second-order linear differential equation.
Fundamental constants for rational functions
S. J.
Poreda;
E. B.
Saff;
G. S.
Shapiro
351-358
Abstract: Suppose R is a rational function with n poles all of which lie inside $\Gamma$, a closed Jordan curve. Lower bounds for the uniform norm of the difference $R - p$ on $\Gamma$, where p is any polynomial, are obtained (in terms of the norm of R on $\Gamma$). In some cases these bounds are independent of $\Gamma$ as well as R and p. Some related results are also given.
A multiplier theorem for Fourier transforms
James D.
McCall
359-369
Abstract: A function f analytic in the upper half-plane ${\Pi ^ + }$ is said to be of class ${E_p}({\Pi ^ + })(0 < p < \infty )$ if there exists a constant C such that $\smallint _{ - \infty }^\infty \vert f(x + iy){\vert^p}dx \leq C < \infty $ for all $y > 0$. These classes are an extension of the ${H_p}$ spaces of the unit disc U. For f belonging to ${E_p}({\Pi ^ + })(0 < p \leq 2)$, there exists a Fourier transform f with the property that $f(z) = 2{(\pi )^{ - 1}}\smallint _0^\infty \hat f(t){e^{izt}}dt$. This makes it possible to give a definition for the multiplication of ${E_p}({\Pi ^ + })(0 < p \leq 2)$ into ${L_q}(0,\infty )$ that is analogous to the multiplication of ${H_p}(U)$ into ${l_q}$. In this paper, we consider the case $ 0 < p < 1$ and $ p \leq q$ and derive a necessary and sufficient condition for multiplying ${E_p}({\Pi ^ + })$ into ${L_q}(0,\infty )$.
Weakly almost periodic and uniformly continuous functionals on the Fourier algebra of any locally compact group
Edmond E.
Granirer
371-382
Abstract: We define for any locally compact group G, the space of bounded uniformly continuous functionals on Ĝ, $UCB(\hat G)$, in the context of P. Eymard [Bull. Soc. Math. France 92 (1964), 181-236. MR 37 #4208] (for notations see next section). For $u \in A(G)$ let $ {u^ \bot } = \{ \phi \in VN(G);\phi [A(G)u] = 0\}$. Theorem. If for some norm separable subspace $X \subset VN(G)$ and some positive definite $0 \ne u \in A(G),UCB(\hat G) \subset$ norm closure $ [W(\hat G) + X + {u^ \bot }]$ then G is discrete. If G is discrete then $UCB(\hat G) \subset AP(\hat G) \subset W(\hat G)$.