An extension of the Fuglede commutativity theorem modulo the Hilbert-Schmidt class to operators of the form $\sum M\sb{n}XN\sb{n}$
Gary
Weiss
1-20
Abstract: We study the operators $\Delta (X) = \sum\nolimits_1^n {{M_n}X{N_n}}$ and ${\Delta^{\ast}}(X) = \sum\nolimits_1^n {M_n^{\ast}XN_n^{\ast}}$ which map the algebra of all bounded linear operators on a separable Hubert space to itself, where $\langle {M_n}\rangle_1^m$ and $\langle {N_n}\rangle_1^m$ are separately commuting sequences of normal operators. We prove that (1) when $m \leqslant 2$, the Hilbert-Schmidt norms of $\Delta (X)$ and $ {\Delta^{\ast}}(X)$ are equal (finite or infinite); (2) for $m \geqslant 3$, if $ \Delta (X)$ and ${\Delta^{\ast}}(X)$ are Hilbert-Schmidt operators, then their Hilbert-Schmidt norms are equal; (3) if $ \Delta ,{\Delta^{\ast}}$ have the property that for each $X,\Delta (X) = 0$ implies ${\Delta^{\ast}}(X) = 0$, then for each $X$, if $ \Delta (X)$ is a Hilbert-Schmidt operator then ${\Delta^{\ast}}^2(X)$ is also and the latter has the same Hilbert-Schmidt norm as ${\Delta ^2}(X)$. Note that Fuglede's Theorem is immediate from $(1)$ in the case $m = 2,{M_1} = {N_2}$ and ${N_1} = I = - {M_2}$. The proofs employ the duality between the trace class and the class of all bounded linear operators and, unlike the early proofs of Fuglede's Theorem, they are free of complex function theory.
Semilinear parabolic problems define semiflows on $C\sp{k}$ spaces
Xavier
Mora
21-55
Abstract: Linear parabolic problems of a general class are proved to determine analytic semigroups on certain closed subspaces of ${C^k}(\overline \Omega )$ ($k$ integer); ${C^k}(\overline \Omega )$ denotes the space of functions whose derivatives or order $\leqslant k$ are bounded and uniformly continuous, with the usual supremum norm; the closed subspaces where the semigroups are obtained, denoted by ${\hat C^k}(\overline \Omega )$, are determined by the boundary conditions and a possible condition at infinity. One also obtains certain embedding relations concerning the fractional power spaces associated to these semigroups. Usually, results of this type are based upon the theory of solution of elliptic problems, while this work uses the corresponding theory for parabolic problems. The preceding results are applied to show that certain semilinear parabolic problems define semiflows on spaces of the type ${\hat C^k}(\overline \Omega )$.
Uniqueness of $\Gamma \sb{p}$ in the Gross-Koblitz formula for Gauss sums
Alan
Adolphson
57-63
Abstract: It is determined what continuous functions besides the $p$-adic gamma function make the Gross-Koblitz formula valid.
A Boolean algebra with few subalgebras, interval Boolean algebras and retractiveness
Matatyahu
Rubin
65-89
Abstract: Using $ {\diamondsuit_{{\aleph_1}}}$ we construct a Boolean algebra $B$ of power $ {\aleph_1}$, with the following properties: (a) $B$ has just $ {\aleph_1}$ subalgebras. (b) Every uncountable subset of $B$ contains a countable independent set, a chain of order type $\eta$, and three distinct elements $ a,b$ and $c$, such that $ a \cap b = c$. (a) refutes a conjecture of J. D. Monk, (b) answers a question of R. McKenzie. $B$ is embeddable in $ P(\omega )$. A variant of the construction yields an almost Jónson Boolean algebra. We prove that every subalgebra of an interval algebra is retractive. This answers affirmatively a conjecture of B. Rotman. Assuming MA or the existence of a Suslin tree we find a retractive BA not embeddable in an interval algebra. This refutes a conjecture of B. Rotman. We prove that an uncountable subalgebra of an interval algebra contains an uncountable chain or an uncountable antichain. Assuming CH we prove that the theory of Boolean algebras in Magidor's and Malitz's language is undecidable. This answers a question of M. Weese.
Persistently finite theories with hyperarithmetic models
Terrence
Millar
91-99
Abstract: Nerode asked if there could be a complete decidable theory with only finitely many countable models up to isomorphism, such that not all of the countable models were decidable. Morley, Lachlan, and Peretyatkin produced examples of such theories. However, all the countable models of those theories were decidable in $0^{\prime}$. The question then arose whether all countable models of such theories had to be, for example, arithmetic. In this paper we provide a negative answer to that question by showing that there are such examples with countable models of arbitrarily high hyperarithmetic degree. It is not difficult to show that any countable model of a hyperarithmetic theory which has only finitely many countable models must be decidable in some hyperarithmetic degree.
On quasiprojective covers
Theodore G.
Faticoni
101-113
Abstract: The main results determine the Goldie dimension of superfluous submodules of semiprime left Goldie rings and apply this to the study of quasiprojective covers of torsion free modules. Conditions are given to guarantee that a quasiprojective cover of a torsion free module is an isomorphism. A class of nonperfect rings is given such that finitely generated singular modules possess quasiprojective covers.
Analytic functionals with unbounded carriers and mean periodic functions
Alex
Meril
115-136
Abstract: We study certain ideals in some spaces of analytic functionals with unbounded carriers introduced by T. Kawaï, M. Morimoto and J. W. de Roever. Using Banach algebra methods, we show an example of space without spectral synthesis. Using Hörmander's ${L^2}$ estimates, we prove a spectral synthesis theorem for mean periodic functions.
A new bound for the Steiner ratio
D. Z.
Du;
F. K.
Hwang
137-148
Abstract: Let $V$ denote a given set of $n$ points in the euclidean plane. A Steiner minimal tree for $V$ is the shortest network (clearly, it has to be a tree) interconnecting $V$. Junctions of the network which are not in $ V$ are called Steiner points (those in $V$ will be called regular points). A shortest tree interconnecting $V$ without using any Steiner points is called a minimal tree. Let $ \sigma (V)$ and $ \mu (V)$ denote the lengths of a Steiner minimal tree and a minimal tree, respectively. Define $\rho$ to be the greatest lower bound for the ratio $ \sigma (V)/\mu (V)$ over all $V$. We prove $\rho > .8$.
Steiner minimal trees on zig-zag lines
D. Z.
Du;
F. K.
Hwang;
J. F.
Weng
149-156
Abstract: A Steiner minimal tree for a given set $P$ of points in the Euclidean plane is a shortest network interconnecting $P$ whose vertex set may include some additional points. The construction of Steiner minimal trees has been proved to be an $NP$-complete problem for general $P$. However, the $ NP$-completeness does not exclude the possibility that Steiner trees for sets of points with special structures can be efficiently determined. In this paper we determine the Steiner mimmal trees for zig-zag lines with certain regularity properties. We also give an explicit formula for the length of such a tree.
Earthquakes and tessellations of Teichm\"uller space
Peter
Waterman;
Scott
Wolpert
157-167
Abstract: A brief exposition of the Teichmüller and earthquake theory is given for the once punctured torus. Using these ideas, computer plots, of the tessellation of Teichmüller space by the mapping class group and of the earthquake paths leaving a point, were generated. The algorithms for generating the plots are discussed.
On the rate of convergence of moments in the central limit theorem for lattice distributions
Peter
Hall
169-181
Abstract: We derive sharp asymptotic expressions for moments of the type $ E\{ b(\vert{S_n}/{n^{1/2}}\vert)\}$, where ${S_n}$ is a sum of independent lattice-valued random variables with finite variance, and $b$ is a concave function. It is shown that the behaviour of $b$ at the origin has a profound effect on the behaviour of such moments, and that this influence accounts for the major difference between the properties of moments of lattice and nonlattice sums. Asymptotic expansions for moments of sums of lattice-valued variables are also derived.
Bounds for integral solutions of diagonal cubic equations
Ka Hin
Leung
183-195
Abstract: It was proved by Davenport [3] that for the nonzero integral ${\lambda_i}$ the equation ${\lambda_1}x_1^3 + \cdots + {\lambda_8}x_8^3 = 0$ always has a nontrivial integral solution. In this paper, we investigate the bounds of nontrivial integral solutions in terms of ${\lambda_1}, \ldots ,{\lambda_8}$.
Stop rule inequalities for uniformly bounded sequences of random variables
Theodore P.
Hill;
Robert P.
Kertz
197-207
Abstract: If ${X_{0}},{X_1},\ldots$ is an arbitrarily-dependent sequence of random variables taking values in $ [0,1]$ and if $V({X_0},{X_1},\ldots)$ is the supremum, over stop rules $t$, of $E{X_t}$, then the set of ordered pairs $\{ (x,y):x = V({X_0},{X_1},\ldots,{X_n})$ and $y = E({\max_{j\, \leqslant \,n}}{X_j})$ for some $ {X_0},\ldots,{X_n}\}$ is precisely the set $\displaystyle {C_n} = \{ (x,y):x \leqslant y \leqslant x\,( {1 + n\,(1 - {x^{1/n}})} );0 \leqslant x \leqslant 1\} ;$ and the set of ordered pairs $\{ (x,y):x = V({X_{0}},{X_1},\ldots)$ and $ y = E({\sup_n}\;{X_n})$ for some $ {X_0},{X_1},\ldots\}$ is precisely the set $\displaystyle C = \bigcup\limits_{n = 1}^\infty {{C_n}} .$ As a special case, if ${X_0},{X_1},\ldots$ is a martingale with $E{X_0} = x$, then $ E({\max_{j \leqslant n}} X) \leqslant x + nx(1 - {x^{1\,/\,n}})$ and $ E({\sup_n}\;{X_n}) \leqslant x - x\ln \;x$, and both inequalities are sharp.
Spaces of complex null geodesics in complex-Riemannian geometry
Claude
LeBrun
209-231
Abstract: The notion of a complex - Riemannian $n$-manifold, meaning a complex $ n$-manifold with a nondegenerate complex quadratic form on each tangent space which varies holomorphically from point to point, is briefly developed. It is shown that, provided $n \geqslant 4$, every such manifold locally arises canonically as the moduli space of all quadrics of a fixed normal-bundle type in an associated space of complex null geodesies. This relationship between local geometry and global complex analysis is stable under deformations.
Toral actions on $5$-manifolds
Hae Soo
Oh
233-252
Abstract: We are mainly concerned with closed orientable manifolds of dimension $ 5$ supporting effective three-dimensional torus actions. We obtain a complete classification of simply-connected manifolds of this type and a partial classification for the nonsimply-connected case.
Bi-interpretable groups and lattices
M.
Jambu-Giraudet
253-269
Abstract: A large class of 0-$2$ transitive lattice-ordered groups is finitely axiomatizable as a class of groups and as a class of lattices. In each model, the group structure and the lattice structure plus one parameter are bi-interpretable, sometimes up to duality only. A characterization of lattice-automorphisms of the structures is also given.
Homology cobordisms, link concordances, and hyperbolic $3$-manifolds
Robert
Myers
271-288
Abstract: Let $M_0^3$ and $M_1^3$ be compact, oriented $3$-manifolds. They are homology cobordant (respectively relative homology cobordant) if $ \partial M_1^3 = \emptyset \;({\text{resp.}}\;\partial M_1^3 \ne \emptyset )$ and there is a smooth, compact oriented $4$-manifold ${W^4}$ such that $\partial {W^4} = M_0^3 - M_1^3$ (resp. $\partial {W^4} = M_0^3 - M_1^3) \cup (M_i^3 \times [0,1])$ and $ {H_{\ast}}(M_i^3;{\mathbf{Z}}) \to {H_{\ast}}({W^4};{\mathbf{Z}})$ are isomorphisms, $i = 0,1$. Theorem. Every closed, oriented $3$-manifold is homology cobordant to a hyperbolic $3$-manifold. Theorem. Every compact, oriented $3$-manifold whose boundary is nonempty and contains no $2$-spheres is relative homology cobordant to a hyperbolic $3$-manifold. Two oriented links $ {L_0}$ and ${L_1}$ in a $3$-manifold ${M^3}$ are concordant if there is a set ${A^2}$ of smooth, disjoint, oriented annuli in $M \times [0,1]$ such that $\partial {A^2} = {L_0} - {L_1}$, where ${L_{i}} \subseteq \;{M^3} \times \{ i\} ,i = 0,1$. Theorem. Every link in a compact, oriented $ 3$-manifold $ {M^3}$ whose boundary contains no $2$-spheres is concordant to a link whose exterior is hyperbolic. Corollary. Every knot in ${S^3}$ is concordant to a knot whose exterior is hyperbolic.
Sign changes in harmonic analysis on reductive groups
Robert E.
Kottwitz
289-297
Abstract: Let $G$ be a connected reductive group over a field $F$. In this note the author constructs an element $ e(G)$ of the Brauer group of $F$. The square of this element is trivial. For a local field, $e(G)$ may be regarded as an element of $\{ \pm 1\}$ and is needed for harmonic analysis on reductive groups over that field. For a global field there is a product formula.
Existence of infinitely many solutions for a forward backward heat equation
Klaus
Höllig
299-316
Abstract: Let $\phi$ be a piecewise linear function which satisfies the condition $ s\phi (s) \geqslant c{s^2},c > 0,s \in {\mathbf{R}}$, and which is monotone decreasing on an interval $(a,b) \subset {{\mathbf{R}}_ + }$. It is shown that for $f \in {C^2}[0,1]$, with $\max f^\prime > a$, there exists a $T > 0$ such that the initial boundary value problem $\displaystyle {u_t} = \phi \,{({u_x})_x},\qquad {u_x}(0,t) = {u_x}(1,t) = 0,\qquad u( \cdot ,0) = f,$ has infinitely many solutions $u$ satisfying $\parallel \;u\;{\parallel_{\alpha }},\parallel \;{u_x}{\parallel_{\infty }},\parallel \;{u_t}{\parallel_{2}} \leqslant c(f,\phi )$ on $[0,1] \times [0,T]$.
Inverting the half-jump
S.
Homer;
G. E.
Sacks
317-331
Abstract: Assume $ \beta$ is weakly admissible over 0 and $ {0^{1\,/\,2}}$. It follows that the $\beta$-recursively enumerable degrees are dense. In addition each $\beta$-recursively enumerable degree above ${0^{1\,/\,2}}$ is the half-jump of some tamely $ \beta$-recursively enumerable degree below $ {0^{1\,/\,2}}$.
Convergence of linear and nonlinear Pad\'e approximants from series of orthogonal polynomials
D. S.
Lubinsky;
A.
Sidi
333-345
Abstract: Analogues of the Nuttall-Pommerenke theorem and Wallin-type theorems for classical Padé approximants, are proved for linear and nonlinear Padé approximants formed from series of orthogonal polynomials, corresponding to a distribution $ d\alpha (x)$ with at most finitely many sign changes.
The structure of rings with faithful nonsingular modules
J. M.
Zelmanowitz
347-359
Abstract: It is shown that the existence of a faithful nonsingular uniform module characterizes rings which have a full linear maximal quotient ring. New information about the structure of these rings is obtained and their maximal quotient rings are constructed in an explicit manner. More generally, rings whose maximal quotient rings are finite direct sums of full linear rings are characterized by the existence of a faithful nonsingular finite dimensional module.
On the generators of the first homology with compact supports of the Weierstrass family in characteristic zero
Goro C.
Kato
361-368
Abstract: Let ${{\mathbf{W}}_{\mathbf{Q}}} = \operatorname{Proj}({\mathbf{Q}}[{{\text{g}}_2},... ...eous ideal generated by }} - {Y^2}Z + 4\,{X^3} - {g_2}\,X{Z^2} - {g_3}\,{Z^3}))$. This is said to be the Weierstrass Family over the field $ {\mathbf{Q}}$. Then the first homology with compact supports of the Weierstrass Family is computed explicitly, i.e., it is generated by ${\{ {C^{ - k}}\,dX\, \wedge \;dY\}_{k \geqslant 1}}$ and ${\{ X{C^{ - k}}dX \wedge \,dY\}_{k\, \geqslant 1}}$ over the ring $ {\mathbf{Q}}[{g_{2}},{g_3}]$, where $C$ is a polynomial ${Y^2} - 4{X^3} + {g_2}X + {g_3}$. When one tensors the homology of the Weierstrass Family with ${\Delta ^{ - 1}}\,{\mathbf{Q}}[{g_2},{g_3}]$, being localized at the discriminant $\Delta = g_2^3 - 27g_3^2$, over $ {\mathbf{Q}}[{{\text{g}}_2},{g_3}]$, the first homology is generated by $ {C^{ - 1}}dX\; \wedge \;dY$ and $ X{C^{ - 1}}dX\, \wedge dY$. One also obtains the first homologies with compact supports of singular fibres over $\wp = ({g_2} = {g_{3}} = 0)$ and $\wp = \,({g_2} = 3,{g_{3}} = 1)$ as corollaries.
A note on Michael's problem concerning the Lindel\"of property in the Cartesian products
K.
Alster
369-375
Abstract: In this note we present a sketch of a negative solution of the Michael's conjecture which says that if the product $Y \times X$ is Lindelöf for every hereditarily Lindelöf space $Y$, then $ Y \times {X^\omega }$ is Lindelöf for every hereditarily Lindelöf space $Y$.
Equilibrium states of an elastic conductor in a magnetic field: a paradigm of bifurcation theory
Peter
Wolfe
377-387
Abstract: In this paper we study the equilibrium states of a nonlinearly elastic conducting wire in a magnetic field. The wire is perfectly flexible and is suspended between fixed supports. The wire carries an electric current and is subjected to a constant magnetic field whose direction is parallel to the line between the supports. We solve this problem exactly and show that the set of solutions gives rise to a paradigmatic bifurcation diagram. We then carry out a study of the equations obtained by linearization about the nontrivial solutions in order to gain some insight into the stability of the various solution branches.
The nest subgroups of Kleinian groups
Takehiko
Sasaki
389-399
Abstract: The residual limit points of finitely generated Kleinian groups are classified into two types: the first kind and the second kind. To each point of the second kind, Abikoff associated a web subgroup. We shall classify the points of the first kind into two types and associate to each point of one type a finitely generated subgroup, the nest subgroup. To the points of the other type we shall give a significance showing that they are important for the sets of generators.
Solvable groups acting on the line
J. F.
Plante
401-414
Abstract: Actions of discrete groups on the real line are considered. When the group of homeomorphisms is solvable several sufficient conditions are given which guarantee that the group is semiconjugate to a subgroup of the affine group of the line. In the process of obtaining these results sufficient conditions are also determined for the existence of invariant (quasi-invariant) measures for abelian (solvable) groups acting on the line. It turns out, for example, that any solvable group of real analytic diffeomorphisms or a polycyclic group of homeomorphisms has a quasi-invariant measure, and that any abelian group of $ {C^2}$ diffeomorphisms has an invariant measure. An example is given to show that some restrictions are necessary in order to obtain such conclusions. Some applications to the study of codimension one foliations are indicated.
The Gauss-Lucas theorem and Jensen polynomials
Thomas
Craven;
George
Csordas
415-429
Abstract: A characterization is given of the sequences $\{ {\gamma_k}\}_{k = 0}^\infty$ with the property that, for any complex polynomial $f(z) = \Sigma \,{a_k}{z^k}$ and convex region $K$ containing the origin and the zeros of $ f$, the zeros of $ \Sigma \,{\gamma_k}{a_k}{z^k}$ again lie in $K$. Many applications and related results are also given. This work leads to a study of the Taylor coefficients of entire functions of type I in the Laguerre-Pólya class. If the power series of such a function is given by $ \Sigma \,{\gamma_k}{z^k}/k!$ and the sequence $ \{ {\gamma_k}\}$ is positive and increasing, then the sequence satisfies an infinite collection of strong conditions on the differences, namely ${\Delta ^n}{\gamma_k} \geqslant 0$ for all $ n$, $k$.