Modular constructions for combinatorial geometries
Tom
Brylawski
1-44
Abstract: R. Stanley, in an investigation of modular flats in geometries (Algebra Universalis 1-2 (1971), 214-217), proved that the characteristic polynomial $\chi (x)$ of a modular flat $x$ divides the characteristic polynomial $ \chi (G)$ of a geometry $ G$. In this paper we identify the quotient: THEOREM. If $x$ is a modular flat of $G,\chi (G)/\chi (x) = \chi (\overline {{T_x}} (G))/(\lambda - 1)$, where $\overline {{T_x}} (G)$ is the complete Brown truncation of $G$ by $x$. (The lattice of $\overline {{T_x}} (G)$ consists of all flats containing $x$ and all flats disjoint from $x$, with the induced order from $ G$.) We give many characterizations of modular flats in terms of their lattice properties as well as by means of a short-circuit axiom and a modular version of the MacLane-Steinitz exchange axiom. Modular flats are shown to have many of the useful properties of points and distributive flats (separators) in addition to being much more prevalent. The theorem relating the chromatic polynomials of two graphs and the polynomial of their vertex join across a common clique generalizes to geometries: THEOREM. Given geometries $G$ and $H$, if $x$ is a modular flat of $G$ as well as a subgeometry of $ H$, then there exists a geometry $ P = {P_x}(G,H)$ which is a pushout in the category of injective strong maps and such that $\chi (P) = \chi (G)\chi (H)/\chi (x)$. The closed set structure, rank function, independent sets, and lattice properties of $P$ are characterized. After proving a modular extension theorem we give applications of our results to Crapo's single element extension theorem, Crapo's join operation, chain groups, unimodular geometries, transversal geometries, and graphs.
The automorphism group of a compact group action
W. D.
Curtis
45-54
Abstract: This paper contains results on the structure of the group, $\operatorname{Diff} _G^r(M)$, of equivariant $ {C^r}$-diffeomorphisms of a free action of the compact Lie group $G$ on $M$. $\operatorname{Diff} _G^r(M)$ is shown to be a locally trivial principal bundle over a submanifold of ${\operatorname{Diff} ^r}(X),X$ the orbit manifold. The structural group of this bundle is ${E^r}(G,M)$, the set of equivariant $ {C^r}$-diffeomorphisms which induce the identity on $X$. $ {E^r}(G,M)$ is shown to be a submanifold of ${\operatorname{Diff} ^r}(M)$ and in fact a Banach Lie group $(r < \infty )$.
Function algebras and flows. IV
Paul S.
Muhly
55-66
Abstract: The automorphisms of the algebra $ \mathfrak{A}$ of analytic functions associated with a flow (without periodic orbits) are completely determined. This result extends earlier work of Arens who determined the automorphisms of $ \mathfrak{A}$ when the flow is almost periodic. The Choquet boundary of the maximal ideal space of $ \mathfrak{A}$ is also determined under the hypothesis that the flow has no fixed points.
Duality theories for metabelian Lie algebras. II
Michael A.
Gauger
67-75
Abstract: In this paper I have replaced one of the axioms given in my Duality theory for metabelian Lie algebras (Trans. Amer. Math. Soc. 187 (1974), 89-102) concerning duality theories by a considerably more natural assumption which yields identical results--a uniqueness theorem.
Automorphisms of commutative rings
H. F.
Kreimer
77-85
Abstract: Let $B$ be a commutative ring with 1, let $ G$ be a finite group of automorphisms of $B$, and let $A$ be the subring of $G$-invariant elements of $B$. For any separable $A$-subalgebra $A'$ of $B$, the following assertions are proved: (1) $ A'$ is a finitely generated, protective $A$-module; (2) for each prime ideal $p$ of $A$, the rank of ${A'_p}$ over ${A_p}$ does not exceed the order of $G$; (3) there is a finite group $ H$ of automorphisms of $ B$ such that $ A'$ is the subring of $ H$-invariant elements of $ B$. If, in addition, $ A'$ is $G$-stable, then every automorphism of $ A'$ over $A$ is the restriction of an automorphism of $B$, and ${\operatorname{Hom} _A}(A',A')$ is generated as a left $A'$-module by those automorphisms of $ A'$ which are the restrictions of elements of $G$.
Under the degree of some finite linear groups. II
Harvey I.
Blau
87-96
Abstract: Let $G$ be a finite group with a cyclic Sylow $ p$-subgroup for some prime $p \geq 13$. Assume that $ G$ is not of type $ {L_2}(p)$, and that $ G$ has a faithful indecomposable modular representation of degree $d \leq p$. Some known lower bounds for $d$ are improved, in case the center of the group is trivial, as a consequence of results on the degrees $ \pmod p$ of irreducible Brauer characters in the principal $p$-block.
${\rm PL}$ involutions of $S\sp{1}\times S\sp{1}\times S\sp{1}$
Kyung Whan
Kwun;
Jeffrey L.
Tollefson
97-106
Abstract: We prove that the $ 3$-dimensional torus $ {S^1} \times {S^1} \times {S^1}$ admits exactly nine nonequivalent PL involutions. With the exception of the four fixed point free ones, the involutions may be distinguished by their fixed point sets: (1) eight points, (2) two simple closed curves, (3) four simple closed curves, (4) one torus, (5) two tori.
Asymptotic values of modulus $1$ of Blaschke products
K. K.
Leung;
C. N.
Linden
107-118
Abstract: A sufficient condition is found for each subproduct of a Blaschke product to have an asymptotic value of modulus 1 along a prescribed arc of a specified type in the unit disc. The condition obtained is found to be necessary in the case of further restrictions of the arc, and the two results give rise to a necessary and sufficient condition for the existence of $ {T_\gamma }$-limits of modulus 1 for Blaschke products.
Asymptotic values of modulus $1$ of functions in the unit ball of $H\sp{\infty }$
Kar Koi
Leung
119-128
Abstract: The main purpose of this paper is to prove a theorem concerning a necessary and sufficient condition for an inner function to have a limiting value of modulus 1 along an arc inside the unit disc, terminating at a point of the unit circle.
Quasi-equivalence classes of normal representations for a separable $C\sp{\ast} $-algebra
Herbert
Halpern
129-140
Abstract: It is shown that the set of quasi-equivalence classes of normal representations of a separable ${C^\ast }$-algebra is a Borel subset of the quasi-dual with the Mackey Borel structure and forms a standard Borel space in the induced Borel structure. It is also shown that the set of factor states which induce normal representations forms a Borel set of the space of factor states with the ${w^\ast }$-topology and that this set has a Borel transversal.
On the dimension of varieties of special divisors
R. F.
Lax
141-159
Abstract: Let ${T_g}$ denote the Teichmüller space and let $V$ denote the universal family of Teichmüller surfaces of genus $g$ Let $ V_{{T_g}}^{(n)}$ denote the $n$th symmetric product of $V$ over ${T_g}$ and let $J$ denote the family of Jacobians over $ {T_g}$. Let $ f:V_{{T_g}}^{(n)} \to$ J$$ be the natural relativization over $ {T_g}$ of the classical map defined by integrating holomorphic differentials. Let $\displaystyle u:{f^\ast }\Omega _{\text{J} /{T_g}}^1 \to \Omega _{V_{{T_g}/{T_g}}^{(n)}}^1$ be the map induced by $f$. We define $G_n^r$ to be the analytic subspace of $V_{{T_g}}^{(n)}$ defined by the vanishing of ${ \wedge ^{n - r + 1}}u$. Put $\tau = (r + 1)(n - r) - rg$. We show that $G_n^1 - G_n^2$, if nonempty, is smooth of pure dimension $3g - 3 + \tau + 1$. From this result, we may conclude that, for a generic curve $X$, the fiber of $ G_n^1 - G_n^2$ over the module point of $X$, if nonempty, is smooth of pure dimension $ \tau + 1$, a classical assertion. Variational formulas due to Schiffer and Spencer and Rauch are employed in the study of $ G_n^r$.
Stable positive definite functions
K. R.
Parthasarathy;
K.
Schmidt
161-174
Abstract: This paper investigates the stability of positive definite functions on locally compact groups under one parameter groups of automorphisms. As an application of this it is shown that the only probability distributions on $ {R^n}$ which are stable under the full automorphism group GL$(n,R)$ of ${R^n}$ are the nondegenerate Gaussian distributions. It is furthermore shown that there are no nondegenerate probability distributions of $ {R^n}$ which are stable under $ {\text{SL}}(n,R)$.
On constructing least squares solutions to two-point boundary value problems
John
Locker
175-183
Abstract: For an $ n$th order linear boundary value problem $Lf = {g_0}$ in the Hilbert space ${L^2}[a,b]$, a sequence of approximate solutions is constructed which converges to the unique least squares solution of minimal norm. The method is practical from a computational viewpoint, and it does not require knowing the null spaces of the differential operator $ L$ or its adjoint ${L^ \ast }$.
The factorization and representation of lattices
George
Markowsky
185-200
Abstract: For a complete lattice $L$, in which every element is a join of completely join-irreducibles and a meet of completely meet-irreducibles (we say $L$ is a jm-lattice) we define the poset of irreducibles $P(L)$ to be the poset (of height one) $ J(L) \cup M(L)(J(L)$ is the set of completely join-irreducibles and $ M(L)$ is the set of completely meet-irreducibles) ordered as follows: $a{ < _{P(L)}}b$ if and only if $a \in J(L),b \in M(L)$, and $a \nleqslant { _L}b$. For a jm-lattice $L$, the automorphism groups of $ L$ and $P(L)$ are isomorphic, $L$ can be reconstructed from $ P(L)$, and the irreducible factorization of $L$ can be gotten from the components of $ P(L)$. In fact, we can give a simple characterization of the center of a jm-lattice in terms of its separators (or unions of connected components of $P(L)$). Thus $P(L)$ extends many of the properties of the poset of join-irreducibles of a finite distributive lattice to the class of all jm-lattices. We characterize those posets of height 1 which are $P(L)$ for some jm-lattice $ L$. We also characterize those posets of height 1 which are $P(L)$ for a completely distributive jm-lattice, as well as those posets which are $ P(L)$ for some geometric lattice $L$. More generally, if $L$ is a complete lattice, many of the above arguments apply if we use ``join-spanning'' and ``meet-spanning'' subsets of $L$, instead of $J(L)$ and $M(L)$. If $L$ is an arbitrary lattice, the same arguments apply to ``join-generating'' and ``meet-generating'' subsets of $L$.
The regular ring and the maximal ring of quotients of a finite Baer $\sp{\ast} $-ring
Ernest S.
Pyle
201-213
Abstract: Necessary and sufficient conditions are obtained for extending the involution of a Baer $\ast$-ring to its maximal ring of quotients. Berberian's construction of the regular ring of a Baer $ \ast$-ring is generalized and this ring is identified (under suitable hypotheses) with the maximal ring of quotients.
Analytic structure in some analytic function algebras
William R.
Zame
215-226
Abstract: A complete description is given of the analytic structure of maximal dimension in the spectra of a wide class of concrete function algebras generated by analytic functions. A connection is also given with point derivations on such algebras.
An extremal length problem on a bordered Riemann surface
Jeffrey Clayton
Wiener
227-245
Abstract: Partition the contours of a compact bordered Riemann surface $ R'$ into four disjoint closed sets ${\alpha _0},{\alpha _1},{\alpha _2}$ and $ \gamma$ with ${\alpha _0}$ and $ {\alpha _1}$ nonempty. Let $ F$ denote the family of all locally rectifiable $1$-chains in ${\alpha _0}$ to $ {\alpha _1}$. The extremal length problem on $R'$ considers the existence of a real-valued harmonic function $u$ on $R'$ which is 0 on $ {\alpha _0},1$ on ${\alpha _1}$, a constant on each component $ {\nu _k}$ of ${\alpha _2}$ with ${\smallint _{{\nu _k}}}^ \ast du = 0$ and $^ \ast du = 0$ along $\gamma$ such that the extremal length of $ F$ is equal to the reciprocal of the Dirichlet integral of $u$, that is, $ \partial \bar R \subset S$. We consider the extremal length problem on $ \bar R$ (as a subset of $ S$) when ${\alpha _0},{\alpha _1}$, and ${\alpha _2}$ are relatively closed subarcs of $\partial \bar R$ and when ${\alpha _0},{\alpha _1}$ and ${\alpha _2}$ are closed subsets of $ \partial S = (S - \bar R) \cup \partial \bar R$.
Fuchsian manifolds
Su Shing
Chen
247-256
Abstract: Recently Eberlein and O'Neill have investigated Riemannian manifolds of negative sectional curvature. For visibility manifolds, they have obtained a classification into three types: parabolic, axial and fuchsian. Fundamental groups of fuchsian manifolds of finite type will be investigated. The main theorem is that isometry groups of certain (not necessarily compact) fuchsian manifolds are finite. Fundamental groups of fuchsian manifolds of finite type are not amenable. The spectral radius of the random matrix of the fundamental group of a compact Riemannian manifold of negative sectional curvature is less than one.
Conjugacy correspondences: a unified view
L.
McLinden
257-274
Abstract: As preparation for a duality theory for saddle programs, a partial conjugacy correspondence is developed among equivalence classes of saddle functions. Three known conjugacy correspondences, including Fenchel's correspondence among convex functions and Rockafellar's extension of it to equivalence classes of saddle functions, are shown to be degenerate special cases. Additionally, two new correspondences are brought to light as further special cases. Various questions are answered concerning the lower and upper closures and effective domain of the resulting equivalence class, as well as the effect the correspondence has on the related purely convex function and the subdifferential mapping.
On entire functions of fast growth
S. K.
Bajpai;
G. P.
Kapoor;
O. P.
Juneja
275-297
Abstract: Let $\displaystyle (\ast )\quad f(z) = \sum\limits_{n = 0}^\infty {{a_n}{z^{{\lambda _n}}}}$ be a transcendental entire function. Set $\displaystyle M(r) = \mathop {max}\limits_{\vert z\vert = r} \vert f(z)\vert,\q... ...(r) = \mathop {\max }\limits_{n \geq 0} \{ \vert{a_n}\vert{r^{{\lambda _n}}}\}$ and $\displaystyle N(r) = \mathop {\max }\limits_{n \geq 0} \{ {\lambda _n}\vert m(r) = \vert{a_n}\vert{r^{{\lambda _n}}}\} .$ Sato introduced the notion of growth constants, referred in the present paper as $ {S_q}$-order $ \lambda$ and $ {S_q}$-type $T$, which are generalizations of concepts of classical order and type by defining $\displaystyle (\ast \ast )\quad \lambda = \mathop {\lim }\limits_{r \to \infty } \sup ({\log ^{[q]}}M(r)\vert\log r)$ and if $0 < \lambda < \infty $, then $\displaystyle (\ast \ast \ast )\quad T = \mathop {\lim }\limits_{r \to \infty } \sup ({\log ^{[q - 1]}}M(r)\vert{r^\lambda })$ for $q = 2,3,4, \cdots$ where ${\log ^{[0]}}x = x$ and ${\log ^{[q]}}x = \log ({\log ^{[q - 1]}}x)$. Sato has also obtained the coefficient equivalents of $(\ast \ast )$ and $(\ast \ast \ast )$ for the entire function $ (\ast )$ when ${\lambda _n} = n$. It is noted that Sato's coefficient equivalents of $\lambda$, and $T$ also hold true for $(\ast )$ if $n$'s are replaced by $ {\lambda _n}$'s in his coefficient equivalents. Analogous to $(\ast \ast )$ and $ (\ast \ast \ast )$ lower $ {S_q}$-order $ v$ and lower $ {S_q}$-type $t$ for entire function $f(z)$ are introduced here by defining $\displaystyle v = \mathop {\lim }\limits_{r \to \infty } \inf ({\log ^{[q]}}M(r)\vert\log r)$ and if $0 < \lambda < \infty$ then $\displaystyle t = \mathop {\lim }\limits_{r \to \infty } \inf ({\log ^{[q - 1]}}M(r)\vert{r^\lambda }),\quad q = 2,3,4, \cdots .$ For the case $q = 2$, these notions are due to Whittakar and Shah respectively. For the constant $v$, two complete coefficient characterizations have been found which generalize the earlier known results. For $t$ coefficient characterization only for those entire functions for which the consecutive principal indices are asymptotic is obtained. Determination of a complete coefficient characterization of $ t$ remains an open problem. Further ${S_q}$-growth and lower ${S_q}$-growth numbers for entire function $ f(z)$ we defined \begin{displaymath}\begin{array}{*{20}{c}} \delta \mu \end{array} = \matho... ...\inf } \end{array} ({\log ^{[q - 1]}}N(r)\vert{r^\lambda }),\end{displaymath} for $ q = 2,3,4, \cdots$ and $0 < \lambda < \infty$. Earlier results of Juneja giving the coefficients characterization of $\delta$ and $\mu$ are extended and generalized. A new decomposition theorem for entire functions of $ {S_q}$-regular growth but not of perfectly ${S_q}$-regular growth has been found.
Right orders in full linear rings
K. C.
O’Meara
299-318
Abstract: In this paper a right order $R$ in an infinite dimensional full linear ring is characterized as a ring satisfying: (1) $ R$ is meet-irreducible (with zero right singular ideal) and contains uniform right ideals; (2) the closed right ideals of $ R$ are right annihilator ideals, and each such right ideal is essentially finitely generated; (3) $R$ possesses a reducing pair (i.e. a pair $({\beta _1},{\beta _2})$ of elements for which ${\beta _1}R,{\beta _2}R$ and $\beta _1^r + \beta _2^r$ are large right ideals of $ R$); (4) for each $a \in R$ with ${a^l} = 0,aR$ contains a regular element of $ R$. A second characterization of $R$ is also given. This is in terms of the right annihilator ideals of $R$ which have the same (uniform) dimension as $ {R_R}$. The problem of characterizing right orders in (infinite dimensional) full linear rings was posed by Carl Faith. The Goldie theorems settled the finite dimensional case.
On the Frattini subgroups of generalized free products and the embedding of amalgams
R. B. J. T.
Allenby;
C. Y.
Tang
319-330
Abstract: In this paper we shall prove a basic relation between the Frattini subgroup of the generalized free product of an amalgam $\mathfrak{A} = (A,B;H)$ and the embedding of $\mathfrak{A}$ into nonisomorphic groups, namely, if $ \mathfrak{A}$ can be embedded into two non-isomorphic groups ${G_1} = \langle A,B\rangle $ and ${G_2} = \langle A,B\rangle $ then the Frattini subgroup of $G = {(A \ast B)_H}$ is contained in $H$. We apply this result to various cases. In particular, we show that if $A,B$ are locally solvable and $H$ is infinite cyclic then $ \Phi (G)$ is contained in $ H$.
Two model theoretic proofs of R\"uckert's Nullstellensatz
Volker
Weispfenning
331-342
Abstract: Rückert's Nullstellensatz for germs of analytic functions and its analogue for germs of real analytic functions are proved by a combination of nonstandard analysis with a model theoretic transfer principle. It is also shown that Rückert's Nullstellensatz is constructive essentially relative to the Weierstrass preparation theorem.
Torus actions on a cohomology product of three odd spheres
Christopher
Allday
343-358
Abstract: The main purpose of this paper is to describe how a torus group may act on a space, $X$, whose rational cohomology ring is isomorphic to that of a product of three odd-dimensional spheres, in such a way that the fixed point set is nonempty, and $X$ is not totally nonhomologous to zero in the associated $X$-bundle, $ {X_T} \to {B_T}$. In the first section of the paper some general results on the cohomology theory of torus actions are established. In the second section the cohomology theory of the above type of action is described; and in the third section the results of the first two sections are used to prove a Golber formula for such actions, which, under certain conditions, bears an interesting interpretation in terms of rational homotopy.
Isolated invariant sets for flows on vector bundles
James F.
Selgrade
359-390
Abstract: This paper studies isolated invariant sets for linear flows on the projective bundle associated to a vector bundle, e.g., the projective tangent flow to a smooth flow on a manifold. It is shown that such invariant sets meet each fiber, roughly in a disjoint union of linear subspaces. Isolated invariant sets which are intersections of attractors and repellers (Morse sets) are discussed. We show that, over a connected chain recurrent set in the base space, a Morse filtration gives a splitting of the projective bundle into a direct sum of invariant subbundles. To each factor in this splitting there corresponds an interval of real numbers (disjoint from those for other factors) which measures the exponential rate of growth of the orbits in that factor. We use these results to see that, over a connected chain recurrent set, the zero section of the vector bundle is isolated if and only if the flow is hyperbolic. From this, it follows that if no equation in the hull of a linear, almost periodic differential equation has a nontrivial bounded solution then the solution space of each equation has a hyperbolic splitting.
Sequential convergence in the order duals of certain classes of Riesz spaces
P. G.
Dodds
391-403
Abstract: Several results of Hahn-Vitali-Saks type are given for sequences in the order dual of an Archimedean Riesz space with separating order dual. The class of Riesz spaces considered contains those which are Dedekind $\sigma $-complete, or have the projection property or have an interpolation property introduced by G. L. Seever. The results depend on recent work of O. Burkinshaw and some results of uniform boundedness type.
Acknowledgement of priority: ``Asymptotic behavior of solutions of linear stochastic differential systems'' (Trans. Amer. Math. Soc. {\bf 181} (1973), 1--22)
Avner
Friedman;
Mark A.
Pinsky
405