Singularities in the nilpotent scheme of a classical group
Wim
Hesselink
1-32
Abstract: If $(X,x)$ is a pointed scheme over a ring k, we introduce a (generalized) partition $ {\text{ord}}(x,X/k)$. If G is a reductive group scheme over k, the existence of a nilpotent subscheme $N(G)$ of $ {\text{Lie}}(G)$ is discussed. We prove that ${\text{ord}}(x,N(G)/k)$ characterizes the orbits in $ N(G)$, their codimension and their adjacency structure, provided that G is $G{l_n}$, or $S{p_n}$ and $1/2 \in k$. For $S{O_n}$ only partial results are obtained. We give presentations of some singularities of $ N(G)$. Tables for its orbit structure are added.
Classification theory of abelian groups. I. Balanced projectives
R. B.
Warfield
33-63
Abstract: We introduce in this paper a class of Abelian groups which includes the torsion totally projective groups and those torsion-free groups which are direct sums of groups of rank one. Characterizations of the groups in this class are given, and a complete classification theorem, in terms of additive numerical invariants, is proved.
Interposition and lattice cones of functions
Jörg
Blatter;
G. L.
Seever
65-96
Abstract: A lattice cone of functions on a set X is a convex cone of bounded real-valued functions on X which contains the constants and which is closed under the lattice operations. Our principal results concern the relation between closed lattice cones on a set X and certain binary relations, called inclusions, on the power set of X. These results are applied to interposition problems, Császár compactifications of quasi-proximity spaces, the compactification of Nachbin's completely regular ordered topological spaces, and a problem in best approximation.
Centralisers of $C\sp{\infty }$ diffeomorphisms
Boyd
Anderson
97-106
Abstract: It is shown that if F is a hyperbolic contraction of $ {R^n}$, coordinates may be chosen so that not only is F a polynomial mapping, but so is any diffeomorphism which commutes with F. This implies an identity principle for diffeomorphisms ${G_1}$ and ${G_2}$ commuting with an arbitrary Morse-Smale diffeomorphism F of a compact manifold M: if $ {G_1},{G_2} \in Z(F)$, then ${G_1} = {G_2}$ on an open subset of $M \Rightarrow {G_1} \equiv {G_2}$ on M. Finally it is shown that under a certain linearisability condition at the saddles of F, $Z(F)$ is in fact a Lie group in its induced topology.
Holomorphic convexity of compact sets in analytic spaces and the structure of algebras of holomorphic germs
William R.
Zame
107-127
Abstract: Let $(X,{\mathcal{O}_X})$ be a reduced analytic space and let K be a compact, holomorphically convex subset of X. It is shown that analogs of Cartan's Theorems A and B are valid for coherent analytic sheaves on K. This result is applied to the study of the algebra of germs on K of functions holomorphic near K. In particular, characterizations of finitely generated ideals, prime ideals and homomorphisms are obtained.
Principal co-fiber bundles
Elyahu
Katz
129-141
Abstract: Principal co-fiber bundles are defined in the category of topological groups. They are Eckmann-Hilton duals of principal fiber bundles. A classification theorem is provided as well as an example which almost represents the most general case.
On the blocks of $GL(n,q)$. I
Jørn B.
Olsson
143-156
Abstract: A study is made of the distribution of the ordinary irreducible characters of $ {\text{GL}}(n,q)$ into p-blocks for primes different from the characteristic. The paper gives a description of all possible defect groups for $p \ne 2$ and their normalizers. Various other results are obtained, including a classification of the blocks of defect 0.
Algebras of functions on semitopological left-groups
John F.
Berglund;
Paul
Milnes
157-178
Abstract: We consider various algebras of functions on a semitopological left-group $S = X \times G$, the direct product of a left-zero semigroup X and a group G. In §1 we examine various analogues to the theorem of Eberlein that a weakly almost periodic function on a locally compact abelian group is uniformly continuous. Several appealing conjectures are shown by example to be false. In the second section we look at compactifications of products $S \times T$ of semitopological semigroups with right identity and left identity, respectively. We show that the almost periodic compactification of the product is the product of the almost periodic compactifications, thus generalizing a result of deLeeuw and Glicksberg. The weakly almost periodic compactification of the product is not the product of the weakly almost periodic compactifications except in restrictive circumstances; for instance, when T is a compact group. Finally, as an application, we define and study analytic weakly almost periodic functions and derive the theorem, analogous to a classical theorem about almost periodic functions, that an analytic function which is weakly almost periodic on a single line is analytic weakly almost periodic on a whole strip.
On Kolmogorov's inequalities $\tilde{f}_p \leq C_p$, $f_1$, $0<p<1$
Burgess
Davis
179-192
Abstract: Let $\mu$ be a signed measure on the unit circle A of the complex plane satisfying $\vert\mu \vert(A) < \infty$, where $\vert\mu \vert(A)$ is the total variation of $ \mu$, and let $\tilde \mu$ be the conjugate function of $ \mu$. A theorem of Kolmogorov states that for each real number p between 0 and 1 there is an absolute constant $ {C_p}$ such that ${({(2\pi )^{ - 1}}\smallint _0^{2\pi }\vert\tilde \mu ({e^{i\theta }}){\vert^p}d\theta )^{1/p}} \leqslant {C_p}\vert\mu \vert(A)$. Here it is shown that measures putting equal and opposite mass at points directly opposite from each other on the unit circle, and no mass any place else, are extremal for all of these inequalities, that is, if $\nu$ is one of these measures the number ${({(2\pi )^{ - 1}}\smallint _0^{2\pi }\vert\tilde \nu ({e^{i\theta }}){\vert^p}d\theta )^{1/p}}/\vert\nu \vert(A)$ is the smallest possible value for $ {C_p}$. These constants are also the best possible in the analogous Hilbert transform inequalities. The proof is based on probability theory.
A Paley-Wiener theorem for locally compact abelian groups
Gunar E.
Liepins
193-210
Abstract: Extending the Paley-Wiener theorem to locally compact Abelian groups involves both finding a suitable Laplace transform and a suitable analogue for analytic functions. The Laplace transform is defined in terms of complex characters, and differentiability is defined with use of one-parameter subgroups. The resulting theorem is much as conjectured by Mackey [7],($^{1}$) the major differences being that the theorem is very much an ${L^2}$ theorem and that the problem exhibits a surprising finite dimensional nature.
$T$ measure of Cartesian product sets. II
Lawrence R.
Ernst
211-220
Abstract: It is proven that there exists a subset A of Euclidean 2-space such that the 2-dimensional T measure of the Cartesian product of an interval of unit length and A is less than the 1-dimensional T measure of A. In a previous paper it was shown that there exists a subset of Euclidean 2-space such that the reverse inequality holds. T measure is the first measure of its type for which it has been shown that both of these relations are possible.
Universal properties of Prym varieties with an application to algebraic curves of genus five
Leon
Masiewicki
221-240
Abstract: It is proved that every morphism of a curve with an involution into an Abelian variety, anticommuting with the involution, factors through the associated Prym variety. This result is used to show that Jacobians of curves of genus five arise as Prym varieties associated to a certain class of curves.
Asymmetric maximal ideals in $M(G)$
Sadahiro
Saeki
241-254
Abstract: Let G be a nondiscrete LCA group, $M(G)$ the measure algebra of G, and $ {M_0}(G)$ the closed ideal of those measures in $M(G)$ whose Fourier transforms vanish at infinity. Let $ {\Delta _G},{\Sigma _G}$ and ${\Delta _0}$ be the spectrum of $ M(G)$, the set of all symmetric elements of $ {\Delta _G}$, and the spectrum of ${M_0}(G)$, respectively. In this paper this is shown: Let $\Phi$ be a separable subset of $M(G)$. Then there exist a probability measure $\tau$ in ${M_0}(G)$ and a compact subset X of $ {\Delta _0}\backslash {\Sigma _G}$ such that for each $\vert c\vert \leqslant 1$ and each $\displaystyle \nu \in \Phi \;{\text{Card}}\;\{ f \in X:\hat \tau (f) = c\;{\text{and}}\;\vert\hat \nu (f)\vert = r(\nu )\} \geqslant {2^{\text{c}}}.$ Here $ r(\nu ) = \sup \{ \vert\hat \nu (f)\vert:f \in {\Delta _G}\backslash \hat G\}$. As immediate consequences of this result, we have (a) every boundary for ${M_0}(G)$ is a boundary for $M(G)$ (a result due to Brown and Moran), (b) ${\Delta _G}\backslash {\Sigma _G}$ is dense in $ {\Delta _G}\backslash \hat G$, (c) the set of all peak points for $M(G)$ is $\hat G$ if G is $\sigma$-compact and is empty otherwise, and (d) for each $\mu \in M(G)$ the set $ \hat \mu ({\Delta _0}\backslash {\Sigma _G})$ contains the topological boundary of $ \hat \mu ({\Delta _G}\backslash \hat G)$ in the complex plane.
The Mackey problem for the compact-open topology
Robert F.
Wheeler
255-265
Abstract: Let ${C_c}(T)$ denote the space of continuous real-valued functions on a completely regular Hausdorff space T, endowed with the compact-open topology. Well-known results of Nachbin, Shirota, and Warner characterize those T for which ${C_c}(T)$ is bornological, barrelled, and infrabarrelled. In this paper the question of when $ {C_c}(T)$ is a Mackey space is examined. A convex strong Mackey property (CSMP), intermediate between infrabarrelled and Mackey, is introduced, and for several classes of spaces (including first countable and scattered spaces), a necessary and sufficient condition on T for $ {C_c}(T)$ to have CSMP is obtained.
Finite groups with prime $p$ to the first power
Zon I
Chang
267-288
Abstract: Earlier D. G. Higman classified the finite groups of order n, such that n is divisible by 3 to the first power, with the assumption that the centralizer $ {C_G}(X)$ of X, where X is a subgroup of order 3, is a cyclic trivial intersection set of even order 3s. In this paper the theorem is generalized to include all prime numbers greater than 3. With an additional assumption: $\vert{N_G}(X):{C_G}(X)\vert = 2$, we have proved that one of the following holds for these groups, hereafter designated as G: (A) G is isomorphic to ${L_2}(q)$, where $q = 2ps \pm 1$; (B) there exists a normal subgroup ${G_0}$ of odd index in G, and a normal subgroup N of ${G_0}$ of index 2 such that $G = N\langle \sigma \rangle $ where ${C_G}(X) = X \times \langle \sigma \rangle$.
Effective lower bounds for some linear forms
T. W.
Cusick
289-301
Abstract: It is proved that if $1, \alpha ,\beta $ are numbers, linearly independent over the rationals, in a real cubic number field, then given any real number $d \geqslant 2$, for any integers ${x_0},{x_1},{x_2}$ such that $\vert{\text{norm}}({x_0} + \alpha {x_1} + \beta {x_2})\vert \leqslant d$, there exist effectively computable numbers $c > 0$ and $k > 0$ depending only on $\alpha$ and $\beta$ such that $\vert{x_1}{x_2}\vert{(\log \vert{x_1}{x_2}\vert)^{k\log d}}\vert{x_0} + \alpha {x_1} + \beta {x_2}\vert > c$ holds whenever $ {x_1}{x_2} \ne 0$. It would be of much interest to remove the dependence on d in the exponent of $\log \vert{x_1}{x_2}\vert$, for then, among other things, one could deduce, for cubic irrationals, a stronger and effective form of Roth's Theorem.
The module of indecomposables for finite $H$-spaces
Richard
Kane
303-318
Abstract: The module of indecomposables obtained from the $\bmod p$ cohomology of a finite H-space is studied for p odd. General structure theorems are obtained, first, regarding the possible even dimensions in which this module can be nonzero and, secondly, regarding how the Steenrod algebra acts on the module.
An approximation theory for generalized Fredholm quadratic forms and integral-differential equations
J.
Gregory;
G. C.
Lopez
319-335
Abstract: An approximation theory is given for a very general class of elliptic quadratic forms which includes the study of 2nth order (usually in integrated form), selfadjoint, integral-differential equations. These ideas follows in a broad sense from the quadratic form theory of Hestenes, applied to integral-differential equations by Lopez, and extended with applications for approximation problems by Gregory. The application of this theory to a variety of approximation problem areas in this setting is given. These include focal point and focal interval problems in the calculus of variations/optimal control theory, oscillation problems for differential equations, eigenvalue problems for compact operators, numerical approximation problems, and finally the intersection of these problem areas. In the final part of our paper our ideas are specifically applied to the construction and counting of negative vectors in two important areas of current applied mathematics: In the first case we derive comparison theorems for generalized oscillation problems of differential equations. The reader may also observe the essential ideas for oscillation of many nonsymmetric (indeed odd order) ordinary differential equation problems which will not be pursued here. In the second case our methods are applied to obtain the ``Euler-Lagrange equations'' for symmetric tridiagonal matrices. In this significant new result (which will allow us to reexamine both the theory and applications of symmetric banded matrices) we can construct in a meaningful way, negative vectors, oscillation vectors, eigenvectors, and extremal solutions of classical problems as well as faster more efficient algorithms for the numerical solution of differential equations. In conclusion it appears that many physical problems which involve symmetric differential equations are more meaningful presented as integral differential equations (effects of friction on physical processes, etc.). It is hoped that this paper will provide the general theory and present examples and methods to study integral differential equations.
Weak bases and metrization
Harold W.
Martin
337-344
Abstract: Several weak base (in the sense of A. V. Arhangel'skiĭ) metrization theorems are established, including a weak base generalization of the Nagata-Smirnov Metrization Theorem.
Frobenius calculations of Picard groups and the Birch-Tate-Swinnerton-Dyer conjecture
Raymond T.
Hoobler
345-352
Abstract: Let $Y \subset {{\text{P}}^m}$ be a subvariety of codimension d defined by an ideal I in char $p > 0$ with $ {H^1}(Y,\mathcal{O}( - 1)) = 0$. If t is an integer greater than ${\log _p}(d)$ and ${H^i}(Y,{I^n}/{I^{n + 1}}) = 0$ for $n > > 0$ and $i = 1,2$, then ${\text{Pic}}(Y)$ is an extension of a finite p-primary group of exponent at most ${p^t}$ by $ Z[\mathcal{O}(1)]$ and $ \dim Y < p$ and $ p \ne 2$, then the B-T-SD conjecture holds for cycles of codimension 1. These results are proved by studying the etale cohomology of the Frobenius neighborhoods of Y in ${{\text{P}}^m}$.
$SK\sb{1}$ of $n$ lines in the plane
Leslie G.
Roberts
353-365
Abstract: We calculate $ S{K_1}(A)$ where A is the coordinate ring of the reduced affine variety consisting of n straight lines in the plane.
Sobolev inequalities for weight spaces and supercontractivity
Jay
Rosen
367-376
Abstract: For $\phi \in {C^2}({{\mathbf{R}}^n})$ with $ \phi (x) = a\vert x{\vert^{1 + s}}$ for $\vert x\vert \geqslant {x_0},a,s > 0$, define the measure $d\mu = \exp ( - 2\phi ){d^n}x$ on ${{\mathbf{R}}^n}$. We show that for any $k \in {{\mathbf{Z}}^ + }$ \begin{displaymath}\begin{array}{*{20}{c}} {\int {\vert f{\vert^2}\vert\lg(\vert... ...mu )}){\vert^{2sk/(s + 1)}}} } \right\}} \end{array} \end{displaymath} As a consequence we prove ${e^{ - t{\nabla ^\ast} \cdot \nabla }}:{L_q}({{\mathbf{R}}^n},d\mu ) \to {L_p}({{\mathbf{R}}^n},d\mu ),p,q \ne 1,\infty$, is bounded for all $ t > 0$.
Characterizations of continua in which connected subsets are arcwise connected
E. D.
Tymchatyn
377-388
Abstract: The purpose of this paper is to give several characterizations of the continua in which all connected subsets are arcwise connected. The methods used are those developed by B. Knaster and K. Kuratowski, G. T. Whyburn and the author. These methods depend on Bernstein's decomposition of a topologically complete metric space into totally imperfect sets and on Whyburn's theory of local cutpoints. Some properties of connected sets in finitely Suslinian spaces are obtained. Two questions raised by the author are answered. Several partial results of Whyburn are obtained as corollaries of the main result.
Stability theorems in shape and pro-homotopy
David A.
Edwards;
Ross
Geoghegan
389-403
Abstract: Conditions are given under which a topological space has the pointed shape of a CW complex. These are derived from analogous conditions in pro-homotopy.
Erratum to: ``The involutions on homotopy spheres and their gluing diffeomorphisms'' (Trans. Amer. Math. Soc. {\bf 215} (1976), 363--391)
Chao Chu
Liang
405