Generation of finite groups of Lie type
Gary M.
Seitz
351-407
Abstract: Let $p$ be an odd prime and $G$ a finite group of Lie type in characteristic other than $p$. Fix an elementary abelian $p$-subgroup of $\operatorname{Aut} (G)$. It is shown that in most cases $G$ is generated by the centralizers of the maximal subgroups of $E$. Results are established concerning the notions of layer generation and balance, and the strongly $p$-embedded subgroups of $\operatorname{Aut} (G)$ are determined.
Symmetric skew balanced starters and complete balanced Howell rotations
Ding Zhu
Du;
F. K.
Hwang
409-413
Abstract: Symmetric skew balanced starters on $n$ elements have been previously constructed for $n = 4k + 3$ a prime power and $8k + 5$ a prime power. In this paper we give an approach for the general case $n = {2^m}k + 1$ a prime power with $k$ odd. In particular we show how this approach works for $m = 2$ and $3$. Furthermore, we prove that for $n$ of the general form and $k > 9 \cdot {2^{3m}}$, then a symmetric skew balanced starter always exists. It is known that a symmetric skew balanced starter on $ n$ elements, $ n$ odd, can be used to construct complete balanced Howell rotations (balanced Room squares) for $n$ players and $2(n + 1)$ players, and in the case that $ n$ is congruent to $ 3$ modulo $4$, also for $n + 1$ players.
Balanced Howell rotations of the twin prime power type
Ding Zhu
Du;
F. K.
Hwang
415-421
Abstract: We prove by construction that a balanced Howell rotation for $n$ players always exists if $n = {p^r}{q^s}$ where $p$ and $q \ne 3$ are primes and ${q^s} = {p^r} + 2$. This generalizes a much weaker previous result. The construction uses properties of a Galois domain which is a direct sum of two Galois fields.
Free products of $C\sp{\ast} $-algebras
Daniel
Avitzour
423-435
Abstract: Small ("spatial") free products of $ {C^{\ast}}$-algebras are constructed. Under certain conditions they have properties similar to those proved by Paschke and Salinas for the algebras $ C_r^{\ast}({G_1}{\ast}{G_2})$ where ${G_1}$, ${G_2}$ are discrete groups. The free-product analogs of noncommutative Bernoulli shifts are discussed.
Nilpotent inverse semigroups with central idempotents
G.
Kowol;
H.
Mitsch
437-449
Abstract: An inverse semigroup $S$ with central idempotents, i.e. a strong semilattice of groups, will be called nilpotent, if it is finite and if for each prime divisor ${p_i}$ of the orders of the structure groups of $S$ the sets ${P_i} = \{ s \in S\vert o(s) = p_i^{{k_s}},\,{k_s} \geqslant 0\}$ are subsemigroups of $ S$. If $S$ is a group, then ${P_i}$ are exactly the Sylow $ {p_i}$-subgroups of the group. A theory similar to that given by W. Burnside for finite nilpotent groups is developed introducing the concepts of ascending resp. descending central series in an inverse semigroup, and it is shown that almost all of the well-known properties of finite nilpotent groups do hold also for the class of finite inverse semigroups with central idempotents.
The Budan-Fourier theorem and Hermite-Birkhoff spline interpolation
T. N. T.
Goodman;
S. L.
Lee
451-467
Abstract: We extend the classical Budan-Fourier theorem to Hermite-Birkhoff splines, that is splines whose knots are determined by a finite incidence matrix. This is then applied to problems of interpolation by Hermite-Birkhoff splines, where the nodes of interpolation are also determined by a finite incidence matrix. For specified knots and nodes in a finite interval, conditions are examined under which there is a unique interpolating spline for any interpolation data. For knots and nodes spaced periodically on the real line, conditions are examined under which there is a unique interpolating spline of power growth for data of power growth.
A remainder formula and limits of cardinal spline interpolants
T. N. T.
Goodman;
S. L.
Lee
469-483
Abstract: A Peano-type remainder formula $\displaystyle f(x) - {S_n}(f;\,x) = \int_{ - \infty }^\infty {{K_n}(x,\,t){f^{(n + 1)}}(t)\,dt}$ for a class of symmetric cardinal interpolation problems C.I.P. $ (E,\,F,\,{\mathbf{x}})$ is obtained, from which we deduce the estimate $ \vert\vert f - {S_{n,r}}(f;\,)\vert{\vert _\infty } \leqslant K\vert\vert{f^{(n + 1)}}\vert{\vert _\infty }$. It is found that the best constant $K$ is obtained when $ {\mathbf{x}}$ comprises the zeros of the Euler-Chebyshev spline function. The remainder formula is also used to study the convergence of spline interpolants for a class of entire functions of exponential type and a class of almost periodic functions.
Varieties of combinatorial geometries
J.
Kahn;
J. P. S.
Kung
485-499
Abstract: A hereditary class of (finite combinatorial) geometries is a collection of geometries which is closed under taking minors and direct sums. A sequence of universal models for a hereditary class $\mathcal{J}$ of geometries is a sequence $ ({T_n})$ of geometries in $\mathcal{J}$ with rank ${T_n} = n$, and satisfying the universal property: if $G$ is a geometry in $ \mathcal{J}$ of rank $ n$, then $G$ is a subgeometry of $ {T_n}$. A variety of geometries is a hereditary class with a sequence of universal models. We prove that, apart from two degenerate cases, the only varieties of combinatorial geometries are (1) the variety of free geometries, (2) the variety of geometries coordinatizable over a fixed finite field, and (3) the variety of voltage-graphic geometries with voltages in a fixed finite group.
On the construction of relative genus fields
Gary
Cornell
501-511
Abstract: We show how to construct the relative genus field in many cases. This is then applied to constructing fields with interesting class groups.
Existence of Chebyshev centers, best $n$-nets and best compact approximants
Dan
Amir;
Jaroslav
Mach;
Klaus
Saatkamp
513-524
Abstract: In this paper we investigate the existence and continuity of Chebyshev centers, best $n$-nets and best compact sets. Some of our positive results were obtained using the concept of quasi-uniform convexity. Furthermore, several examples of nonexistence are given, e.g., a sublattice $M$ of $C[0,\,1]$, and a bounded subset $B \subset M$ is constructed which has no Chebyshev center, no best $n$-net and not best compact set approximant.
Transitivity of families of invariant vector fields on the semidirect products of Lie groups
B.
Bonnard;
V.
Jurdjevic;
I.
Kupka;
G.
Sallet
525-535
Abstract: In this paper we give necessary and sufficient conditions for a family of right (or left) invariant vector fields on a Lie group $ G$ to be transitive. The concept of transitivity is essentially that of controllability in the literature on control systems. We consider families of right (resp. left) invariant vector fields on a Lie group $G$ which is a semidirect product of a compact group $ K$ and a vector space $ V$ on which $K$ acts linearly. If $\mathcal{F}$ is a family of right-invariant vector fields, then the values of the elements of $\mathcal{F}$ at the identity define a subset $ \Gamma$ of $L(G)$ the Lie algebra of $G$. We say that $\mathcal{F}$ is transitive on $G$ if the semigroup generated by ${ \cup _{X \in \Gamma }}\{ \exp (tX):t \geqslant 0\}$ is equal to $G$. Our main result is that $\mathcal{F}$ is transitive if and only if $ \operatorname{Lie} (\Gamma )$, the Lie algebra generated by $\Gamma$, is equal to $L(G)$.
Extensions for AF $C\sp{\ast}$ algebras and dimension groups
David
Handelman
537-573
Abstract: Let $A$, $C$ be approximately finite dimensional $ ({\text{AF)}}\,{C^{\ast}}$ algebras, with $A$ nonunital and $C$ unital; suppose that either (i) $A$ is the algebra of compact operators, or (ii) both $A$, $C$ are simple. The classification of extensions of $A$ by $C$ is studied here, by means of Elliott's dimension groups. In case (i), the weak Ext group of $ C$ is shown to be ${\operatorname{Ext} _{\mathbf{Z}}}({K_0}(C),\,{\mathbf{Z}})$, and the strong Ext group is an extension of a cyclic group by the weak Ext group; conditions under which either Ext group is trivial are determined. In case (ii), there is an unnatural and complicated group structure on the classes of extensions when $ A$ has only finitely many pure finite traces (and somewhat more generally).
Generalized skew polynomial rings
John
Dauns
575-586
Abstract: For a totally ordered cancellative semigroup $\Gamma$, a skew field $K$, let $ K[\Gamma ;\phi ]$ be a skew semigroup ring. If $x \in \Gamma ,\,k \in K$, then $kx = x{k^x}$, where $k \to {k^x}$ is an endomorphism of $K$ depending on $x$. Ideals of $ K[\Gamma ;\phi ]$ are investigated for various semigroups or groups $ \Gamma$.
Saturation properties of ideals in generic extensions. II
James E.
Baumgartner;
Alan D.
Taylor
587-609
Abstract: The general type of problem considered here is the following. Suppose $ I$ is a countably complete ideal on ${\omega _1}$ satisfying some fairly strong saturation requirement (e.g. $I$ is precipitous or ${\omega _2}$-saturated), and suppose that $ P$ is a partial ordering satisfying some kind of chain condition requirement (e.g. $ P$ has the c.c.c. or forcing with $P$ preserves $ {\omega _1}$). Does it follow that forcing with $P$ preserves the saturation property of $ I$? In this context we consider not only precipitous and ${\omega _2}$-saturated ideals, but we also introduce and study a class of ideals that are characterized by a property lying strictly between these two notions. Some generalized versions of Chang's conjecture and Kurepa's hypothesis also arise naturally from these considerations.
Theta-characteristics on algebraic curves
Joe
Harris
611-638
Abstract: The theory of theta-characteristics is developed algebraically, so that it may be applied to possibly singular and/or reducible algebraic curves. The configuration of theta-characteristics on a curve is described in terms of its singularities, with applications to the geometry of plane quartic curves and the problem of Appolonius. Some results on Gorenstein local rings are appended.
Minimal immersions of closed Riemann surfaces
J.
Sacks;
K.
Uhlenbeck
639-652
Abstract: Let $M$ be a closed orientable surface of genus larger than zero and $N$ a compact Riemannian manifold. If $u:M \to N$ is a continuous map, such that the map induced by it between the fundamental groups of $M$ and $N$ contains no nontrivial element represented by a simple closed curve in its kernel, then there exists a conformal branched minimal immersion $ s:M \to N$ having least area among all branched immersions with the same action on ${\pi _1}(M)$ as $u$. Uniqueness within the homotopy class of $ u$ fails in general: It is shown that for certain $3$-manifolds which fiber over the circle there are at least two geometrically distinct conformal branched minimal immersions within the homotopy class of any inclusion map of the fiber. There is also a topological discussion of those $3$-manifolds for which uniqueness fails.
Singular elliptic operators of second order with purely discrete spectra
Roger T.
Lewis
653-666
Abstract: The Friedrichs extension of a second order singular elliptic operator is considered on a weighted $\mathcal{L}_w^2(\Omega )$ space. The region $ \Omega$ is not necessarily bounded. Necessary conditions and sufficient conditions on the coefficients that will insure a discrete spectrum are given with a certain degree of sharpness achieved. The boundary conditions include the Dirichlet, Neumann, and mixed Dirichlet-Neumann boundary value problems.
Star-finite representations of measure spaces
Robert M.
Anderson
667-687
Abstract: In nonstandard analysis, $^{\ast}$-finite sets are infinite sets which nonetheless possess the formal properties of finite sets. They permit a synthesis of continuous and discrete theories in many areas of mathematics, including probability theory, functional analysis, and mathematical economics. $^{\ast}$-finite models are particularly useful in building new models of economic or probabilistic processes. It is natural to ask what standard models can be obtained from these $^{\ast}$-finite models. In this paper, we show that a rich class of measure spaces, including the Radon spaces, are measure-preserving images of $ ^{\ast}$-finite measure spaces, using a construction introduced by Peter A. Loeb [15]. Moreover, we show that a number of measure-theoretic constructs, including integrals and conditional expectations, are naturally expressed in these models. It follows that standard models which can be expressed in terms of these measure spaces and constructs can be obtained from $^{\ast}$-finite models.
Minimum simplicial complexes with given abelian automorphism group
Zevi
Miller
689-718
Abstract: Let $K$ be a pure $n$-dimensional simplicial complex. Let ${\Gamma _0}(K)$ be the automorphism group of $ K$, and let ${\Gamma _n}(K)$ be the group of permutations on $ n$-cells of $K$ induced by the elements of ${\Gamma _0}(K)$. Given an abelian group $A$ we consider the problem of finding the minimum number of points $ M_0^{(n)}(A)$ in $ K$ such that ${\Gamma _0}(K) \cong A$, and the minimum number of $ n$-cells $M_1^{(n)}(A)$ in $K$ such that ${\Gamma _n}(K) \cong A$. Write $A = {\prod _{{p^\alpha }}}{\mathbf{Z}}_{{p^\alpha }}^{e({p^\alpha })}$, where each factor ${{\mathbf{Z}}_{{p^\alpha }}}$ appears $e({p^\alpha })$ times in the canonical factorization of $A$. For $A$ containing no factors ${{\mathbf{Z}}_{{p^\alpha }}}$ satisfying ${p^\alpha } < 17$ we find that $M_1^{(n)}(A) = M_0^{(2)}(A) = {\sum _{{p^\alpha }}}{p^\alpha }e({p^\alpha })$ when $n \geqslant 4$, and we derive upper bounds for $M_1^{(n)}(A)$ and $ M_0^{(n)}(A)$ in the remaining possibilities for $A$ and $n$.
Brownian motion with partial information
Terry R.
McConnell
719-731
Abstract: We study the following problem concerning stopped $N$-dimensional Brownian motion: Compute the maximal function of the process, ignoring those times when it is in some fixed region $R$. Suppose this modified maximal function belongs to ${L^q}$. For what regions $R$ can we conclude that the unrestricted maximal function belongs to ${L^q}$? A sufficient condition on $R$ is that there exist $p > q$ and a function $u$, harmonic in $R$, such that $\displaystyle \vert x{\vert^p} \leqslant u(x) \leqslant C\vert x{\vert^p} + C,\qquad x \in R,$ for some constant $C$. We give applications to analytic and harmonic functions, and to weak inequalities for exit times.
Interior and boundary continuity of weak solutions of degenerate parabolic equations
William P.
Ziemer
733-748
Abstract: In this paper we consider degenerate parabolic equations of the form $ ({\ast})$ $\displaystyle \beta {(u)_t} - \operatorname{div} A(x,\,t,\,u,\,{u_x}) + B(x,\,t,\,u,\,{u_x}) \ni 0$ where $A$ and $B$ are, respectively, vector and scalar valued Baire functions defined on $U \times {R^1} \times {R^n}$, where $ U$ is an open subset of ${R^{n + 1}}(x,\,t)$. The functions $ A$ and $B$ are subject to natural structural inequalities. Sufficiently general conditions are allowed on the relation $\beta \subset {R^1} \times {R^1}$ so that the porus medium equation and the model for the two-phase Stefan problem can be considered. The main result of the paper is that weak solutions of $({\ast})$ are continuous throughout $U$. In the event that $U = \Omega \times (0,\,T)$ where $ \Omega$ is an open set of $ {R^n}$, it is also shown that a weak solution is continuous at $ ({x_0},{t_0}) \in \partial \Omega \times (0,\,T)$ provided ${x_0}$ is a regular point for the Laplacian on $\Omega$.