A combinatorial model for series-parallel networks
Thomas H.
Brylawski
1-22
Abstract: The category of pregeometries with basepoint is defined and explored. In this category two important operations are extensively characterized: the series connection $S(G,H)$, and the parallel connection $ P(G,H) = \tilde S(\tilde G,\tilde H)$; and the latter is shown to be the categorical direct sum. For graphical pregeometries, these notions coincide with the classical definitions. A pregeometry F is a nontrivial series (or parallel) connection relative to a basepoint p iff the deletion $F\backslash p$ (contraction $F/p$) is separable. Thus both connections are n-ary symmetric operators with identities and generate a free algebra. Elements of the subalgebra $A[{C_2}]$ generated by the two point circuit are defined as series-parallel networks, and this subalgebra is shown to be closed under arbitrary minors. Nonpointed series-parallel networks are characterized by a number of equivalent conditions: 1. They are in $A[{C_2}]$ relative to some point. 2. They are in $A[{C_2}]$ relative to any point. For any connected minor K of three or more points: 3. K is not the four point line or the lattice of partitions of a four element set. 4. K or $ \tilde K$ is not a geometry. 5. For any point e in K, $K\backslash e$ or $K/e$ is separable. Series-parallel networks can also be characterized in a universally constructed ring of pregeometries generalized from previous work of W. Tutte and A. Grothendieck. In this Tutte-Grothendieck ring they are the pregeometries for which the Crapo invariant equals one. Several geometric invariants are directly calculated in this ring including the complexity and the chromatic polynomial. The latter gives algebraic proofs of the two and three color theorems.
A Boolean algebra of regular closed subsets of $\beta X-X$
R. Grant
Woods
23-36
Abstract: Let X be a locally compact, $\sigma$-compact, noncompact Hausdorff space. Let $\beta X$ denote the Stone-Čech compactification of X. Let $R(X)$ denote the Boolean algebra of all regular closed subsets of the topological space X. We show that the map $A \to ({\text{cl}_{\beta X}}A) - X$ is a Boolean algebra homomorphism from $R(X)$ into $ R(\beta X - X)$. Assuming the continuum hypothesis, we show that if X has no more than $ {2^{{\aleph _0}}}$ zero-sets, then the image of a certain dense subalgebra of $ R(X)$ under this homomorphism is isomorphic to the Boolean algebra of all open-and-closed subsets of $ \beta N - N$ (N denotes the countable discrete space). As a corollary, we show that there is a continuous irreducible mapping from $\beta N - N$ onto $\beta X - X$. Some theorems on higher-cardinality analogues of Baire spaces are proved, and these theorems are combined with the previous result to show that if S is a locally compact, $\sigma $-compact noncompact metric space without isolated points, then the set of remote points of $\beta S$ (i.e. those points of $\beta S$ that are not in the $ \beta S$-closure of any discrete subspace of S) can be embedded densely in $\beta N - N$.
Ternary rings
W. G.
Lister
37-55
Abstract: We characterize those additive subgroups of rings which are closed under the triple ring product, then discuss their imbeddings in rings, their representation in terms of two types of modules, a radical theory, the structure of those which satisfy a minimum condition for certain ideals, and finally the classification of those which are simple ternary algebras over an algebraically closed or real closed field.
The method of least squares for boundary value problems
John
Locker
57-68
Abstract: The method of least squares is used to construct approximate solutions to the boundary value problem $\tau f = {g_0},{B_i}(f) = 0$ for $i = 1, \ldots ,k$, on the interval [a, b], where $\tau$ is an nth order formal differential operator, ${g_0}(t)$ is a given function in ${L^2}[a,b]$, and ${B_1}, \ldots ,{B_k}$ are linearly independent boundary values. Letting $ {H^n}[a,b]$ denote the space of all functions $f(t)$ in $ {C^{n - 1}}[a,b]$ with ${f^{(n - 1)}}$ absolutely continuous on [a, b] and ${f^{(n)}}$ in $ {L^2}[a,b]$, a sequence of functions ${\xi _i}(t)\;(i = 1,2, \ldots )$ in ${H^n}[a,b]$ is constructed satisfying the boundary conditions and a completeness condition. Assuming the boundary value problem has a solution, the approximate solutions $ {f_i}(t) = \Sigma _{j = 1}^ia_j^i{\xi _j}(t)\;(i = 1,2, \ldots )$ are constructed; the coefficients $a_j^i$ are determined uniquely from the system of equations $\displaystyle \sum\limits_{j = 1}^i {(\tau {\xi _j},\tau {\xi _l})a_j^i = ({g_0},\tau {\xi _l}),\quad l = 1, \ldots ,i,}$ where (f, g) denotes the inner product in ${L^2}[a,b]$. The approximate solutions are shown to converge to a solution of the boundary value problem, and error estimates are established.
The theory of $p$-spaces with an application to convolution operators.
Carl
Herz
69-82
Abstract: The class of p-spaces is defined to consist of those Banach spaces B such that linear transformations between spaces of numerical ${L_p}$-functions naturally extend with the same bound to B-valued ${L_p}$-functions. Some properties of p-spaces are derived including norm inequalities which show that 2-spaces and Hilbert spaces are the same and that p-spaces are uniformly convex for $1 < p < \infty$. An ${L_q}$-space is a p-space iff $p \leqq q \leqq 2$ or $p \geqq q \geqq 2$; this leads to the theorem that, for an amenable group, a convolution operator on $ {L_p}$ gives a convolution operator on ${L_q}$ with the same or smaller bound. Group representations in p-spaces are examined. Logical elementarity of notions related to p-spaces are discussed.
On some starlike and convex functions
G. M.
Shah
83-91
Abstract: In this paper we study functions of the form $\smallint _0^z(g(t)/\Pi _{k = 1}^n{(1 - t{z_k})^{{\alpha _k}}})$ for $\vert z\vert < 1$ and show under what conditions such a function is convex, convex in one direction and hence univalent in $\vert z\vert < 1$. We also study the functions $ g(z)$ where $g(0) = 1,g(z) \ne 0$ and
Analytic-function bases for multiply-connected regions
Victor
Manjarrez
93-103
Abstract: Let E be a nonempty (not necessarily bounded) region of finite connectivity, whose boundary consists of a finite number of nonintersecting analytic Jordan curves. Work of J. L. Walsh is utilized to construct an absolute basis $ ({Q_n},n = 0, \pm 1, \pm 2, \ldots )$ of rational functions for the space $ H(E)$ of functions analytic on E, with the topology of compact convergence; or the space $ H({\text{Cl}}\;(E))$ of functions analytic on $ {\text{Cl}}\;(E)$ = the closure of E, with an inductive limit topology. It is shown that $\Sigma _{n = 0}^\infty {Q_n}(z){Q_{ - n - 1}}(w) = 1/(w - z)$, the convergence being uniform for z and w on suitable subsets of the plane. A sequence $({P_n},n = 0, \pm 1, \pm 2, \ldots )$ of elements of $ H(E)$ (resp. $H({\text{Cl}}\;(E))$) is said to be absolutely effective on E(resp. $ {\text{Cl}}\;(E)$) if it is an absolute basis for $H(E)$ (resp. $ H({\text{Cl}}\;(E))$) and the coefficients arise by matrix multiplication from the expansion of $({Q_n})$. Conditions for absolute effectivity are derived from W. F. Newns' generalization of work of J. M. Whittaker and B. Cannon. Moreover, if $({P_n},n = 0,1,2, \ldots )$ is absolutely effective on a certain simply-connected set associated with E, the sequence is extended to an absolutely effective basis $({P_n},n = 0, \pm 1, \pm 2, \ldots )$ for $ H(E)$ (or $H({\text{Cl}}\;(E))$) such that $\Sigma _{n = 0}^\infty {P_n}(z){P_{ - n - 1}}(w) = 1/(w - z)$. This last construction applies to a large class of orthogonal polynomials.
Lie-admissible, nodal, noncommutative Jordan algebras
D. R.
Scribner
105-111
Abstract: The main theorem is that if A is a central simple flexible algebra, with an identity, of arbitrary dimension over a field F of characteristic not 2, and if A is Lie-admissible and ${A^ + }$ is associative, then
Boundary conditions in the infinite interval and some related results.
Rao V.
Govindaraju
113-128
Abstract: The number of square-integrable solutions of a real, selfadjoint differential equation are determined using exclusively the elementary theory of matrices. Boundary conditions in the infinite interval are given a simple format and a relation between any two selfadjoint boundary conditions is deduced. Finally a lemma due to Titchmarsh, which forms the basis of eigenfunction expansions, is generalized.
Semiprimary hereditary algebras
Abraham
Zaks
129-135
Abstract: Let $\Sigma$ be a semiprimary k-algebra, with radical M. If $\Sigma$ admits a splitting then $ {\dim _k}\Sigma /M \leqq {\dim _k}\Sigma$. The residue algebra $\Sigma /{M^2}$ is finite (cohomological) dimensional if and only if all residue algebras are finite dimensional. If $ {\dim _k}\Sigma = 1$ then all residue algebras are finite dimensional.
On Spencer's cohomology theory for linear partial differential operators
Joseph
Johnson
137-149
Abstract: Let D be a linear partial differential operator between vector bundles on a differentiable manifold X of dimension n. Let $ \mathcal{D}$ be the sheaf of germs of differentiable functions on X. For every $h \in Z$ a spectral sequence ${(^h}{E^{pq}})$ is associated to D. When D satisfies appropriate regularity conditions these spectral sequences degenerate for all sufficiently large h and $^hE_2^{p0}$ is the pth Spencer cohomology for D. One can compute $^hE_2^{pq}$ as the cohomology at ${\Lambda ^p}{T^\ast}{ \otimes _\mathcal{O}}{R_{h - p,q}}$ of a complex $\displaystyle 0 \to {R_{hq}} \to {\Lambda ^1}{T^\ast}{ \otimes _\mathcal{O}}{R_... ... \to \cdots \to {\Lambda ^n}{T^\ast}{ \otimes _\mathcal{O}}{R_{h - n,q}} \to 0.$ When q = 0 this complex coincides with the usual (first) Spencer complex for D. These results give a generalization of Spencer's theory. The principal importance of this generalization is that it greatly clarifies the role played by homological algebra in the theory of overdetermined systems of linear partial differential equations.
Functorial characterizations of Pontryagin duality
David W.
Roeder
151-175
Abstract: Let $\mathcal{L}$ be the category of locally compact abelian groups, with continuous homomorphisms as morphisms. Let $\chi :\mathcal{L} \to \mathcal{L}$ denote the contravariant functor which assigns to each object in $\mathcal{L}$ its character group and to each morphism its adjoint morphism. The Pontryagin duality theorem is then the statement that $\chi \circ \chi$ is naturally equivalent to the identity functor in $ \mathcal{L}$. We characterize $\chi$ by giving necessary and sufficient conditions for an arbitrary contravariant functor $ \varphi :\mathcal{L} \to \mathcal{L}$ to be naturally equivalent to $\chi$. A sequence of morphisms is called proper exact if it is exact in the algebraic sense and is composed of morphisms each of which is open considered as a function onto its image. A pseudo-natural transformation between two functors in $\mathcal{L}$ differs from a natural transformation in that the connecting maps are not required to be morphisms in $\mathcal{L}$. We study and classify pseudo-natural transformations in $ \mathcal{L}$ and use this to prove that (R denotes the real numbers) $ \varphi$ is naturally equivalent to $\chi$ if and only if the following three statements are all true: (1) $ \varphi (R)$ is isomorphic to R, (2) $\varphi$ takes short proper exact sequences to short proper exact sequences, and (3) $\varphi$ takes inductive limits of discrete groups to projective limits and takes projective limits of compact groups to inductive limits. From this we prove that $\varphi$ is naturally equivalent to $\chi$ if and only if $\varphi$ is a category equivalence.
Torus invariance for the Clifford algebra. I
Michael C.
Reed
177-183
Abstract: A problem in Quantum Field Theory leads to the study of a representation of the torus, ${T^3}$, as automorphisms of the infinite dimensional Clifford algebra. It is shown that the irreducible product representations of the Clifford algebra fall into two categories: the discrete representations where the automorphisms are unitarily implementable, and all the others in which the automorphisms are not implementable and which cannot even appear as subrepresentations of larger representations in which the automorphisms are implementable.
Conjugates in prime rings
Charles
Lanaki
185-192
Abstract: Let R be a prime ring with identity, center $Z \ne GF(2)$, and a nonidentity idempotent. If R is not finite and if $x \in R - Z$, then x has infinitely many distinct conjugates in R. If R has infinitely many Z-independent elements then $x \in R - Z$ has infinitely many Z-independent conjugates.
A Galois theory for inseparable field extensions
Nickolas
Heerema
193-200
Abstract: A Galois theory is obtained for fields k of characteristic $p \ne 0$ in which the Galois subfields h are those for which k/h is normal, modular, and for some nonnegative integer r, $h({k^{{p^{r + 1}}}})/h$ is separable. The related automorphism groups G are subgroups of the group A of automorphisms $\alpha$ on $k[\bar X] = k[X]/{X^{{p^{r + 1}}}}k[X]$, X an indeterminate, such that $\alpha (\bar X) = \bar X$. A subgroup G of A is Galois if and only if G is a semidirect product of subgroups ${G_k}$ and ${G_0}$, where ${G_k}$ is a Galois group of automorphisms on k (classical separable theory) and ${G_0}$ is a Galois group of rank $ {p^r}$ higher derivations on k (Jacobson-Davis purely inseparable theory). Implications of certain invariance conditions on a Galois subgroup of a Galois group are also investigated.
On $N$-parameter families and interpolation problems for nonlinear ordinary differential equations
Philip
Hartman
201-226
Abstract: Let $y = ({y_0}, \ldots ,{y_{N - 1}})$. This paper is concerned with the existence of solutions of a system of ordinary differential equations $(^\ast )\,{y_0}({t_j}) = {c_j}$ for $j = 1, \ldots$, N and ${t_1} < \cdots < {t_N}$. It is shown that, under suitable conditions, the assumption of uniqueness for all such problems and of ``local'' solvability (i.e., for $ {t_1}, \ldots ,{t_N}$ on small intervals) implies the existence for arbitrary ${t_1}, \ldots ,{t_N}$ and ${c_1}, \ldots ,{c_N}$. A result of Lasota and Opial shows that, in the case of a second order equation for $ {y_0}$, the assumption of uniqueness suffices, but it will remain undecided if the assumption of ``local'' solvability can be omitted in general. More general interpolation conditions involving N points, allowing coincidences, are also considered. Part I contains the statement of the principal results for interpolation problems and those proofs depending on the theory of differential equations. Actually, the main theorems are consequences of results in Part II dealing with ``N-parameter families'' and ``N-parameter families with pseudoderivatives.'' A useful lemma states that if F is a family of continuous functions $\{ {y^0}(t)\}$ on an open interval (a, b), then F is an N-parameter family (i.e., contains a unique solution of the interpolation conditions $(^\ast)$ for arbitrary ${t_1} < \cdots < {t_N}$ on (a, b) and $ {c_1}, \ldots ,{c_N}$) if and only if (i) $ {y^0},{z^0} \in F$ implies $ {y^0} - {z^0} \equiv 0$ or ${y^0} - {z^0}$ has at most N zeros; (ii) the set $ \Omega \equiv \{ ({t_1}, \ldots ,{t_N},{y^0}({t_1}), \ldots ,{y^0}({t_N})):a < {t_1} < \cdots < {t_N} < b$ and ${y^0} \in F\}$ is open in ${R^{2N}}$; (iii) ${y^1},{y^2}, \ldots , \in F$ and the inequalities $ {y^n}(t) \leqq {y^{n + 1}}(t)$ for $n = 1,2, \ldots $ or ${y^n}(t) \geqq {y^{n + 1}}(t)$ for $n = 1,2, \ldots$ on an interval $[\alpha ,\beta ] \subset (a,b)$ imply that either $ {y^0}(t) = \lim {y^n}(t)$ exists on (a, b) and ${y^0} \in F$ or $\lim \vert{y^n}(t)\vert = \infty$ on a dense set of (a, b); and finally, (iv) the set $ S(t) = \{ {y^0}(t):{y^0} \in F\}$ is not bounded from above or below for $ a < t < b$. The notion of pseudoderivatives permits generalizations to interpolation problems involving some coincident points.
On some solutions to the Klein-Gordon equation related to an integral of Sonine
Stuart
Nelson
227-237
Abstract: An integral due to Sonine is used to obtain an expansion for special solutions $W(x,t)$ of the Klein-Gordon equation. This expansion is used to estimate the ${L_p}$ norms $\left\Vert W( \cdot ,t)\right\Vert _p$ as $t \to \infty$. These estimates yield results on the time decay of a fairly wide class of solutions to the Klein-Gordon equation.
Integrally closed subrings of an integral domain
Robert
Gilmer;
Joe
Mott
239-250
Abstract: Let D be an integral domain with identity having quotient field K. This paper gives necessary and sufficient conditions on D in order that each integrally closed subring of D should belong to some subclass of the class of integrally closed domains; some of the subclasses considered are the completely integrally closed domains, Prüfer domains, and Dedekind domains.
Representations for transformations continuous in the ${\rm BV}$ norm
J. R.
Edwards;
S. G.
Wayment
251-265
Abstract: Riemann and Lebesgue-type integrations can be employed to represent operators on normed function spaces whose norms are not stronger than sup-norm by $T(f) = \smallint f\,d\mu $ where $\mu$ is determined by the action of T on the simple functions. The real-valued absolutely continuous functions on [0, 1] are not in the closure of the simple functions in the BV norm, and hence such an integral representation of an operator is not obtainable. In this paper the authors develop a v-integral whose structure depends on fundamental functions different than simple functions. This integral is as computable as the Riemann integral. By using these fundamental functions, the authors are able to obtain a direct, analytic representation of the linear functionals on AC which are continuous in the BV norm in terms of the v-integral. Further, the v-integral gives a characterization of the dual of AC in terms of the space of fundamentally bounded set functions which are convex with respect to length. This space is isometrically isomorphically identified with the space of Lipschitz functions anchored at zero with the norm given by the Lipschitz constant, which in turn is isometrically isomorphic to ${L^\infty }$. Hence a natural identification exists between the classical representation and the one given in this paper. The results are extended to the vector setting.
Conditions on an operator implying ${\rm Re}\,\sigma (T)=\sigma ({\rm Re}\,T)$
S. K.
Berberian
267-272
Abstract: It is shown that the equation of the title is valid for certain classes of not necessarily normal operators (including Toeplitz operators, and operators whose spectrum is a spectral set), and a new proof is given of C. R. Putnam's theorem that it is valid for seminormal operators.
Generic stability properties of periodic points
K. R.
Meyer
273-277
Abstract: A classification of the periodic points of a generic area-perserving diffeomorphism which depends on a parameter is given. The stability properties of each periodic point in the classification is decided.
Strongly locally setwise homogeneous continua and their homeomorphism groups
Beverly L.
Brechner
279-288
The closed ideals in a function algebra
Charles M.
Stanton
289-300
Abstract: We give a new method of determining the closed ideals in the algebra of functions continuous on a finite Riemann surface and analytic in its interior. Our approach is based on Ahlfors' mapping of a finite Riemann surface onto the unit disc.
Compactness properties of topological groups
S. P.
Wang
301-314
Abstract: In a paper of R. Baer and later in a paper of B. H. Neumann, finiteness properties of groups have been studied. In the present paper, we develop their analogous notions in topological groups and even sharpen some of their results.
On the derived quotient module
C. N.
Winton
315-321
Abstract: With every R-module M associate the direct limit of $ {\operatorname{Hom}_R}(D,M)$ over the dense right ideals of R, the derived quotient module $ \mathcal{D}(M)$ of M. $ \mathcal{D}(M)$ is a module over the complete ring of right quotients of R. Relationships between $ \mathcal{D}(M)$ and the torsion theory of Gentile-Jans are explored and functorial properties of $ \mathcal{D}$ are discussed. When M is torsion free, results are given concerning rational closure, rational completion, and injectivity.
Barycenters of measures on certain noncompact convex sets
Richard D.
Bourgin
323-340
Abstract: Each norm closed and bounded convex subset K of a separable dual Banach space is, according to a theorem of Bessaga and Pełczynski, the norm closed convex hull of its extreme points. It is natural to expect that this theorem may be reformulated as an integral representation theorem, and in this connection we have examined the extent to which the Choquet theory applies to such sets. Two integral representation theorems are proved and an example is included which shows that a sharp result obtains for certain noncompact sets. In addition, the set of extreme points of K is shown to be $ \mu$-measurable for each finite regular Borel measure $\mu$, hence eliminating certain possible measure-theoretic difficulties in proving a general integral representation theorem. The last section is devoted to the study of a class of extreme points (called pinnacle points) which share important geometric properties with extreme points of compact convex sets in locally convex spaces. A uniqueness result is included for certain simplexes all of whose extreme points are pinnacle points.
Extensions of locally compact abelian groups. I
Ronald O.
Fulp;
Phillip A.
Griffith
341-356
Abstract: This paper is concerned with the development of a (discrete) group-valued functor Ext defined on $\mathcal{L} \times \mathcal{L}$ where $\mathcal{L}$ is the category of locally compact abelian groups such that, for A and B groups in $ \mathcal{L}$, Ext (A, B) is the group of all extensions of B by A. Topological versions of homological lemmas are proven to facilitate the proof of the existence of such a functor. Various properties of Ext are obtained which include the usual long exact sequence which connects Hom to Ext. Along the way some applications are obtained one of which yields a slight improvement of one of the Noether isomorphism theorems. Also the injectives and projectives of the category of locally compact abelian totally disconnected groups are obtained. They are found to be necessarily discrete and hence are the same as the injectives and projectives of the category of discrete abelian groups. Finally we obtain the structure of those connected groups C of $\mathcal{L}$ which are direct summands of every G in $ \mathcal{L}$ which contains C as a component.
Extensions of locally compact abelian groups. II
Ronald O.
Fulp;
Phillip A.
Griffith
357-363
Abstract: It is shown that the extension functor defined on the category $\mathcal{L}$ of locally compact abelian groups is right-exact. Actually $ {\text{Ext}^n}$ is shown to be zero for all $n \geqq 2$. Various applications are obtained which deal with the general problem as to when a locally compact abelian group is the direct product of a connected group and a totally disconnected group. One such result is that a locally compact abelian group G has the property that every extension of G by a connected group in $ \mathcal{L}$ splits iff $G = {(R/Z)^\sigma } \oplus {R^n}$ for some cardinal $\sigma$ and positive integer n.
Groups of embedded manifolds
Max K.
Agoston
365-375
Abstract: This paper defines a group $ \theta ({M^n},{\nu _\varphi })$ which generalizes the group of framed homotopy n-spheres in $ {S^{n + k}}$. Let $ {M^n}$ be an arbitrary 1-connected manifold satisfying a weak condition on its homology in the middle dimension and let ${\nu _\varphi }$ be the normal bundle of some imbedding $ \varphi :{M^n} \to {S^{n + k}}$, where $ 2k \geqq n + 3$. Then $ \theta ({M^n},{\nu _\varphi })$ is the set of h-cobordism classes of triples $ (F,{V^n},f)$, where $ F:{S^{n + k}} \to T({\nu _\varphi })$ is a map which is transverse regular on M, $ {V^n} = {F^{ - 1}}({M^n})$, and $f = F\vert{V^n}$ is a homotopy equivalence. ( $ T({\nu _\varphi })$ is the Thom complex of $ {\nu _\varphi }$.) There is a natural group structure on $\theta ({M^n},{\nu _\varphi })$, and $ \theta ({M^n},{\nu _\varphi })$ fits into an exact sequence similar to that for the framed homotopy n-spheres.
Periodic points and measures for Axiom $A$ diffeomorphisms
Rufus
Bowen
377-397
Dense sigma-compact subsets of infinite-dimensional manifolds
T. A.
Chapman
399-426
Abstract: In this paper four classes of separable metric infinite-dimensional manifolds are studied; those which are locally the countable infinite product of lines, those which are locally open subsets of the Hubert cube, and those which are locally one of two dense sigma-compact subsets of the Hilbert cube. A number of homeomorphism, product, characterization, and embedding theorems are obtained concerning these manifolds.
Weakening a theorem on divided powers
Moss E.
Sweedler
427-428
Abstract: We show that if a Hopf algebra has finite dimensional primitives and a primitive lies in arbitrarily long finite sequences of divided powers then it lies in an infinite sequence of divided powers.
Differential-boundary operators
Allan M.
Krall
429-458
Abstract: Differential-boundary systems occur naturally as adjoints for ordinary differential systems involving integral boundary conditions. In this paper such systems are generalized so that the adjoint system has the same form as the original. Interior boundary points are introduced and removed, and the integrals, used in the boundary conditions, are also removed. Selfadjoint systems are classified, and an eigenfunction expansion is derived. Finally, nonselfadjoint systems are discussed and again, an eigenfunction expansion is derived.
The influence on a finite group of the cofactors and subcofactors of its subgroups
Larry R.
Nyhoff
459-491
Abstract: The effect on a finite group G of imposing a condition $ \theta$ on its proper subgroups has been studied by Schmidt, Iwasawa, Itô, Huppert, and others. In this paper, the effect on G of imposing $\theta$ on only the cofactor $H/{\text{cor}_G}\;H$ (or more generally, the subcofactor $ H/{\text{scor}_G}\;H$) of certain subgroups H of G is investigated, where $ {\text{cor}_G}\;H\;({\text{scor}_G}\;H)$ is the largest G-normal (G-subnormal) subgroup of H. It is shown, for example, that if (a) $H/{\text{scor}_G}\;H$ is p-nilpotent for all self-normalizing $H < G$, or if (b) $H/{\text{scor}_G}\;H$ is p-nilpotent for all abnormal $H < G$ and p is odd or the p-Sylows of G are abelian, then in either case, G has a normal p-subgroup P for which G/P is p-nilpotent. Results of this type are also derived for $\theta =$ nilpotent, nilpotent of class $\leqq n$, solvable of derived length $\leqq n,\sigma $-Sylow-towered, supersolvable. In some cases, additional structure in G is obtained by imposing $\theta$ not only on these ``worst'' parts of the ``bad'' subgroups of G (from the viewpoint of normality), but also on the ``good'' subgroups, those which are normal in G or are close to being normal in that their cofactors are small. Finally, this approach is in a sense dualized by an investigation of the influence on G of the outer cofactors of its subgroups. The consideration of nonnormal outer cofactors is reduced to that of the usual cofactors. The study of normal outer cofactors includes the notion of normal index of maximal subgroups, and it is proved, for example, that G is p-solvable iff the normal index of each abnormal maximal subgroup of G is a power of p or is prime to p.
Topological properties of analytically uniform spaces
C. A.
Berenstein;
M. A.
Dostál
493-513
Abstract: In the first part of the article we study certain topological properties of analytically uniform spaces (AU-spaces, cf. L. Ehrenpreis, Fourier transforms in several complex variables, Interscience, New York, 1970). In particular we prove that AU-spaces and their duals are always nuclear. From here one can easily obtain some important properties of these spaces, such as the Fourier type representation of elements of a given AU-space, etc. The second part is devoted to one important example of AU-space which was not investigated in the aforementioned monograph: the scale of Beurling spaces ${\mathcal{D}_\omega }$ and ${\hat{\mathcal{D}}_\omega }$. This shows that the spaces of Beurling distributions are AU-spaces. Moreover, it leads to some interesting consequences and new problems.