Normal structure of the one-point stabilizer of a doubly-transitive permutation group. I
Michael E.
O’Nan
1-42
Abstract: Let G be a doubly-transitive permutation group on a finite set X and x a point of X. Let ${N^x}$ be a normal subgroup of $ {G_x}$, the subgroup fixing x, such that ${N^x}$ is a T.I. set and not semiregular on $X - x$. Then, $ PSL(n,q) \subseteq G \subseteq P\Gamma L(n,q)$. Geometrical consequences of this result are also obtained.
Normal structure of the one-point stabilizer of a doubly-transitive permutation group. II
Michael E.
O’Nan
43-74
Abstract: The main result is that the socle of the point stabilizer of a doubly-transitive permutation group is abelian or the direct product of an abelian group and a simple group. Under certain circumstances, it is proved that the lengths of the orbits of a normal subgroup of the one point stabilizer bound the degree of the group. As a corollary, a fixed nonabelian simple group occurs as a factor of the socle of the one point stabilizer of at most finitely many doubly-transitive groups.
Inclusion maps of $3$-manifolds which induce monomorphisms of fundamental groups
Jože
Vrabec
75-93
Abstract: The main result is the following ``duality'' theorem. Let M be a 3-manifold, P a compact and connected polyhedral 3-submanifold of $\int M$, and X a compact and connected polyhedron in $\int P$. If ${\pi _1}(X) \to {\pi _1}(P)$ is onto, then $ {\pi _1}(M - P) \to {\pi _1}(M - X)$ is one-to-one. Some related results are proved, for instance: we can allow P to be noncompact if also X satisfies a certain noncompactness condition: if M lies in a 3-manifold W with $ {H_1}(W) = 0$, then the condition that $ {\pi _1}(X) \to {\pi _1}(P)$ is onto can be replaced by the weaker one that ${H_1}(X) \to {H_1}(P)$ is onto.
Partition theorems related to some identities of Rogers and Watson
Willard G.
Connor
95-111
Abstract: This paper proves two general partition theorems and several special cases of each with both of the general theorems based on four q-series identities originally due to L. J. Rogers and G. N. Watson. One of the most interesting special cases proves that the number of partitions of an integer n into parts where even parts may not be repeated, and where odd parts occur only if an adjacent even part occurs is equal to the number of partitions of n into parts $ \equiv \pm 2, \pm 3, \pm 4, \pm 5, \pm 6, \pm 7 \pmod 20$. The companion theorem proves that the number of partitions of an integer n into parts where even parts may not be repeated, where odd parts $> 1$ occur only if an adjacent even part occurs, and where 1's occur arbitrarily is equal to the number of partitions of n into parts $\equiv \pm 1, \pm 2, \pm 5, \pm 6, \pm 8, \pm 9 \pmod 20$.
Equisingular deformations of Puiseux expansions
Augusto
Nobile
113-135
Abstract: Parametrizations of a deformation (over a complete local ring) of an irreducible algebroid curve are studied. With these parametrizations another definition of equisingularity (equivalent to the known ones) is given, by using their characteristic numbers. These methods are also used in the complex analytic case. The following applications of these techniques are given: a proof of the existence of a versal equisingular deformation in the complex analytic case; a proof that an equisingular formal family of branches is determined by its $\nu $-truncation ($ \nu$ large enough, depending only on the characteristic of the special fiber).
${\rm BV}$-functions, positive-definite functions and moment problems
P. H.
Maserick
137-152
Abstract: Let S be a commutative semigroup with identity 1 and involution. A complex valued function f on S is defined to be positive definite if ${\Pi _j}{\Delta _j}f(1) \geqslant 0$ where the ${\Delta _j}$'s belong to a certain class of linear sums of shift operators. For discrete groups the positive definite functions defined herein are shown to be the classically defined positive definite functions. An integral representation theorem is proved and necessary and sufficient conditions for a function to be the difference of two positive-definite functions, i.e. a BV-function, are given. Moreover the BV-function defined herein agrees with those previously defined for semilattices, with respect to the identity involution. Connections between the positive-definite functions and completely monotonic functions are discussed along with applications to moment problems.
Weakly smooth continua
Lewis
Lum
153-167
Abstract: We define and investigate a class of continua called weakly smooth. Smooth dendroids, weakly smooth dendroids, generalized trees, and smooth continua are all examples of weakly smooth continua. We generalize characterizations of the above mentioned examples to weakly smooth continua. In particular, we characterize them as compact Hausdorff spaces which admit a quasi order satisfying certain properties.
On the double suspension homomorphism
Mark
Mahowald
169-178
Abstract: This paper studies the family of unstable Adams spectral sequences, $E_2^{s,t}({S^{2n + 1}})$. The main results deal with the range of filtrations for which these groups stabilize and for which the groups $ E_2^{s,t}({\Omega ^2}{S^{2n + 1}},{S^{2n - 1}})$ stabilize.
A codimension theorem for pseudo-Noetherian rings
Kenneth
McDowell
179-185
Abstract: M. Auslander and M. Bridger have shown that the depth of a Noetherian local ring is the sum of the Gorenstein dimension and the depth of any given nonzero finitely generated module of finite Gorenstein dimension. In this paper it is demonstrated that this result remains true when suitably interpreted for the class of coherent rings herein entitled pseudo-Noetherian rings. This class contains, among others, all Noetherian rings and valuation domains as well as non-Noetherian local rings of infinite depth.
The Gaussian law and the law of the iterated logarithm for lacunary sets of characters
E.
Dudley
187-214
Abstract: Salem and Zygmund showed that the Gaussian law holds for Hadamard sequences of real numbers while Mary Weiss proved a similar result for the law of the iterated logarithm. In the present paper, the author obtains corresponding results for lacunary sets of characters of an arbitrary infinite compact abelian group. It is shown that the laws are best satisfied for a certain class of lacunary sets but that modified results apply to more general classes.
Infinite convolutions on locally compact Abelian groups and additive functions
Philip
Hartman
215-231
Abstract: Let ${\mu _1},{\mu _2}, \ldots $ be regular probability measures on a locally compact Abelian group G such that $ \mu = {\mu _1} \ast {\mu _2} \ast \cdots = \lim {\mu _1} \ast \cdots \ast {\mu _n}$ exists (and is a probability measure). For arbitrary G, we derive analogues of the Lévy theorem on the existence of an atom for $\mu$ and of the ``pure theorems'' of Jessen, Wintner and van Kampen (dealing with discrete ${\mu _1},{\mu _2}, \ldots$) in the case $G = {R^d}$. These results are applied to the asymptotic distribution $\mu$ of an additive function $f:{Z_ + } \to G$ after generalizing the Erdös-Wintner result $(G = {R^1})$ which implies that $ \mu$ is an infinite convolution of discrete probability measures.
The bracket ring of a combinatorial geometry. II. Unimodular geometries
Neil L.
White
233-248
Abstract: The bracket ring of a combinatorial geometry G is a ring of generalized determinants which acts as a universal coordinatization object for G. Our main result is the characterization of a unimodular geometry as a binary geometry such that the radical of the bracket ring is a prime ideal. This implies that a unimodular geometry has a universal coordinatization over an integral domain, which domain we construct explicitly using multisets. An ideal closely related to the radical, the coordinatizing radical, is also defined and proved to be a prime ideal for every binary geometry. To prove these results, we use two major preliminary theorems, which are of interest in their own right. The first is a bracket-theoretic version of Tutte's Homotopy Theorem for Matroids. We then prove that any two coordinatizations of a binary geometry over a given field are projectively equivalent.
Adjoint groups, regular unipotent elements and discrete series characters
G. I.
Lehrer
249-260
Abstract: It is shown that if G is a finite Chevalley group or twisted type over a field of characteristic p and U is a maximal p-subgroup of G then any nonlinear irreducible character of U vanishes on regular elements. For groups of adjoint type the linear content of the restriction to U of a discrete series character J of G is calculated and it is deduced that J takes the value 0 or ${( - 1)^s}$ on regular elements of U $ (s = {\text{rank}}\;G)$.
The stability problem in shape, and a Whitehead theorem in pro-homotopy
David A.
Edwards;
Ross
Geoghegan
261-277
Abstract: Theorem 3.1 is a Whitehead theorem in pro-homotopy for finite-dimensional pro-complexes. This is used to obtain necessary and sufficient algebraic conditions for a finite-dimensional tower of complexes to be pro-homotopy equivalent to a complex (§4) and for a finite-dimensional compact metric space to be pointed shape equivalent to an absolute neighborhood retract (§5).
Spectral analysis of finite convolution operators
Richard
Frankfurt
279-301
Abstract: In this paper the similarity problem for operators of the form $( \ast )\;T:f(x) \to \smallint _0^xk(x - t)f(t)dt$ on $ {L^2}(0,1)$ is studied. Let $K(z) = \smallint _0^1\;k(t){e^{itz}}dt$. A function $C(z)$ is called a symbol for T if $ C(z)$ can be written in the form $ C(z) = K(z) + {e^{iz}}G(z)$, where $G(z)$ is a function bounded and analytic in a half plane $y > \delta $, for some real number $ \delta$. Under suitable restrictions, it is shown that two operators of the form $( \ast )$ will be similar if they possess symbols which are asymptotically close together as $z \to \infty$ in some half plane $y > \delta$.
Differential geometry on simplicial spaces
Michael A.
Penna
303-323
Abstract: A simplicial space M is a separable Hausdorff topological space equipped with an atlas of linearly related charts of varying dimension; for example every polyhedron is a simplicial space in a natural way. Every simplicial space possesses a natural structure complex of sheaves of piecewise smooth differential forms, and the homology of the corresponding de Rham complex of global sections is isomorphic to the real cohomology of M. A cosimplicial bundle is a continuous surjection $ \xi :E \to M$ from a topological space E to a simplicial space M which satisfies certain criteria. There is a category of cosimplicial bundles which contains a subcategory of vector bundles. To every simplicial space M a cosimplicial bundle $\tau (M)$ over M is associated; $ \tau (M)$ is the cotangent object of M since there is an isomorphism between the module of global piecewise smooth one-forms on M and sections of $\tau (M)$.
Two applications of twisted wreath products to finite soluble groups
Trevor O.
Hawkes
325-335
Abstract: The group construction sometimes known as the twisted wreath product is used here to answer two questions in the theory of finite, soluble groups: first to show that an arbitrary finite, soluble group may be embedded as a subgroup of a group whose upper nilpotent series is a chief series; second to construct an A-group whose Carter subgroup is ``small'' relative to its nilpotent length.
On a Galois theory for inseparable field extensions
John N.
Mordeson
337-347
Abstract: Heerema has developed a Galois theory for fields L of characteristic $p \ne 0$ in which the Galois subfields K are those for which $L/K$ is normal, modular and, for some nonnegative integer $e,K({L^{{p^{e + 1}}}})/K$ is separable. The related automorphism groups G are subgroups of a particular group A of automorphisms on $L[x]/{x^{{p^e} + 1}}L[x]$ where x is an indeterminate over L. For $H \subseteq G$ Galois subgroups of A, we give a necessary and sufficient condition for H to be G-invariant. An extension of a result of the classical Galois theory is also given as is a necessary and sufficient condition for every intermediate field of $L/K$ to be Galois where K is a Galois subfield of L.
On angular momentum Helmholtz theorems and cohomology of Lie algebras
Henrik
Stetkaer
349-374
Abstract: Helmholtz' 2nd theorem (that every vector field on ${{\mathbf{R}}^3}$ with vanishing curl is gradient of a function) can be viewed as a statement about the group of translations of $ {{\mathbf{R}}^3}$. We prove similar theorems for other Lie transformation groups, in particular for semidirect products of abelian and compact semisimple groups. Using Hodge theory we also obtain results analogous to the 1st Helmholtz theorem, but only for compact Lie transformation groups.
Ruelle's operator theorem and $g$-measures
Peter
Walters
375-387
Abstract: We use g-measures to give a proof of a convergence theorem of Ruelle. The method of proof is used to gain information about the ergodic properties of equilibrium states for subshifts of finite type.
$o$-weakly compact mappings of Riesz spaces
P. G.
Dodds
389-402
Abstract: A characterization is given of linear mappings from a Riesz space to a Banach space which map order intervals to relatively weakly compact sets. The characterization is based on recent results of Burkinshaw and Fremlin. A number of applications are made to known results concerning weakly compact mappings and to results in the theory of Banach space-valued measures.
Diffeomorphisms obtained from endomorphisms
Louis
Block
403-413
Abstract: It is shown that if f is a differentiable map of a compact manifold, and the singularities of f satisfy a certain condition, then there is a diffeomorphism (of a different manifold) whose orbit structure is closely related to that of f. This theorem is then used to extend several results on the orbit structure of diffeomorphisms to the noninvertible case.
On the structure of $S$ and $C(S)$ for $S$ dyadic
James
Hagler
415-428
Abstract: A dyadic space S is defined to be a continuous image of ${\{ 0,1\} ^\mathfrak{m}}$ for some infinite cardinal number $ \mathfrak{m}$. We deduce Banach space properties of $C(S)$ and topological properties of S. For example, under certain cardinality restrictions on $\mathfrak{m}$, we show: Every dyadic space of topological weight $ \mathfrak{m}$ contains a closed subset homeomorphic to ${\{ 0,1\} ^\mathfrak{m}}$. Every Banach space X isomorphic to an $\mathfrak{m}$ dimensional subspace of $ C(S)$ (for S dyadic) contains a subspace isomorphic to ${l^1}(\Gamma )$ where $\Gamma$ has cardinality $ \mathfrak{m}$.