Monotone reducibility over the Cantor space
Randall
Dougherty
433-484
Abstract: Define the partial ordering $\leqslant$ on the Cantor space ${}^\omega 2$ by $x \leqslant y$ iff $\forall n\,x(n) \leqslant y(n)$ (this corresponds to the subset relation on the power set of $ \omega$). A set $A \subseteq {}^\omega 2$ is monotone reducible to a set $B \subseteq {}^\omega 2$ iff there is a monotone (i.e., $x \leqslant y \Rightarrow f(x) \leqslant f(y)$) continuous function $f:{}^\omega 2 \to {}^\omega 2$ such that $ x \in A$ iff $f(x) \in B$. In this paper, we study the relation of monotone reducibility, with emphasis on two topics: (1) the similarities and differences between monotone reducibility on monotone sets (i.e., sets closed upward under $\leqslant$) and Wadge reducibility on arbitrary sets; and (2) the distinction (or lack thereof) between `monotone' and `positive,' where `positive' means roughly `a priori monotone' but is only defined in certain specific cases. (For example, a $\Sigma _2^0$-positive set is a countable union of countable intersections of monotone clopen sets.) Among the main results are the following: Each of the six lowest Wadge degrees contains one or two monotone degrees (of monotone sets), while each of the remaining Wadge degrees contains uncountably many monotone degrees (including uncountable antichains and descending chains); and, although `monotone' and `positive' coincide in a number of cases, there are classes of monotone sets which do not match any notion of `positive.'
Groups acting on affine algebras
Daniel R.
Farkas
485-497
Abstract: General actions of groups on commutative affine domains are studied. We prove a finiteness theorem for orbits of ideals and an ergodic theorem inspired by results from the theories of group algebras and universal enveloping algebras.
Finite basis theorems for relatively congruence-distributive quasivarieties
Don
Pigozzi
499-533
Abstract: $\mathcal{Q}$ is any quasivariety. A congruence relation $\Theta$ on a member $ {\mathbf{A}}$ of $\mathcal{Q}$ is a $ \mathcal{Q}$-congruence if ${\mathbf{A}}/\Theta \in \mathcal{Q}$. The set $ Co{n_\mathcal{Q}}{\mathbf{A}}$ of all $ \mathcal{Q}$-congruences is closed under arbitrary intersection and hence forms a complete lattice $ {\mathbf{Co}}{{\mathbf{n}}_\mathcal{Q}}{\mathbf{A}}$. $ \mathcal{Q}$ is relatively congruence-distributive if $ {\mathbf{Co}}{{\mathbf{n}}_\mathcal{Q}}{\mathbf{A}}$ is distributive for every $ {\mathbf{A}} \in \mathcal{Q}$. Relatively congruence-distributive quasivarieties occur naturally in the theory of abstract data types. $ \mathcal{Q}$ is finitely generated if it is generated by a finite set of finite algebras. The following generalization of Baker's finite basis theorem is proved. Theorem I. Every finitely generated and relatively congruence-distributive quasivariety is finitely based. A subquasivariety $ \mathcal{R}$ of an arbitrary quasivariety $ \mathcal{Q}$ is called a relative subvariety of $ \mathcal{Q}$ if it is of the form $\mathcal{V} \cap \mathcal{Q}$ for some variety $\mathcal{V}$, i.e., a base for $\mathcal{R}$ can be obtained by adjoining only identities to a base for $ \mathcal{Q}$. Theorem II. Every finitely generated relative subvariety of a relatively congruence-distributive quasivariety is finitely based. The quasivariety of generalized equality-test algebras is defined and the structure of its members studied. This gives rise to a finite algebra whose quasi-identities are finitely based while its identities are not. Connections with logic and the algebraic theory of data types are discussed.
The $q$-Selberg polynomials for $n=2$
Kevin W. J.
Kadell
535-553
Abstract: We have conjectured that Selberg's integral has a plethora of extensions involving the Selberg polynomials and proved that these are the Schur functions for $k = 1$. We prove this conjecture for $ n = 2$ and show that the polynomials are, in a formal sense, Jacobi polynomials. We conjecture an orthogonality relation for the Selberg polynomials which combines orthogonality relations for the Schur functions and Jacobi polynomials. We extend a basic Schur function identity. We give a $ q$-analogue of the Selberg polynomials for $n = 2$ using the little $q$-Jacobi polynomials.
Differential delay equations that have periodic solutions of long period
Steven A.
Chapin
555-566
Abstract: If $f:{\mathbf{R}} \to {\mathbf{R}}$ is a continuous odd function satisfying $xf(x) > 0$, $x \ne 0$, and $ f(x) = o({x^{ - 2}})$ as $x \to \infty$, then so-called periodic solutions of long period seem to play a prominent role in the dynamics of $({\ast})$ $xf(x) \geqslant 0$. These solutions have quite different qualitative features than in the odd case.
Factoring operators satisfying $p$-estimates
Stan
Byrd
567-582
Abstract: Necessary and sufficient conditions for a positive operator to factor through a Banach lattice satisfying upper and lower estimates are presented. These conditions are then combined to give a necessary condition for a positive operator to factor through a super-reflexive Banach lattice. An example is given to show that, in spite of the name given by Beauzamy, uniformly convexifying operators need not factor through any uniformly convex lattice
The resolvent parametrix of the general elliptic linear differential operator: a closed form for the intrinsic symbol
S. A.
Fulling;
G.
Kennedy
583-617
Abstract: Nonrecursive, explicit expressions are obtained for the term of arbitrary order in the asymptotic expansion of the intrinsic symbol of a resolvent parametrix of an elliptic linear differential operator, of arbitrary order and algebraic structure, which acts on sections of a vector bundle over a manifold. Results for the conventional symbol are included as a special case.
Cobordism classes of manifolds with category four
Harpreet
Singh
619-628
Abstract: The Lusternik-Schnirelmann category of a manifold $M$ is the smallest integer $ k$ such that $ M$ can be covered by $ k$ open sets each of which is contractible in $M$. The classification up to cobordism of manifolds with category $3$ was completed by the author in 1985. The object of this paper is to attempt a similar classification of manifolds with category $4$.
The classifying topos of a continuous groupoid. I
Ieke
Moerdijk
629-668
Abstract: We investigate some properties of the functor $B$ which associates to any continuous groupoid $ G$ its classifying topos $ BG$ of equivariant $ G$-sheaves. In particular, it will be shown that the category of toposes can be obtained as a localization of a category of continuous groupoids.
Decidability and invariant classes for degree structures
Manuel
Lerman;
Richard A.
Shore
669-692
Abstract: We present a decision procedure for the $ \forall \exists$-theory of $\mathcal{D}$. The decision procedure follows easily from these results. As a corollary to the $\forall \exists $-decision procedure for $\mathcal{D}$, we show that no degree ${\mathbf{a}} > {\mathbf{0}}$ is definable by any $ \exists \forall$-formula of degree theory. As a start on restricting the formulas which could possibly define the various jump classes we classify the generalized jump classes which are invariant for any $\forall$ or $\exists$-formula. The analysis again uses the decision procedure for the $ \forall \exists$-theory of $\mathcal{D}$. A similar analysis is carried out for the high/low hierarchy using the decision procedure for the $ \forall \exists$-theory of $\mathcal{C}$ is $\sigma $-invariant if $ \sigma ({\mathbf{a}})$ holds for every $ {\mathbf{a}}$ in $\mathcal{C}$.)
Markov-Duffin-Schaeffer inequality for polynomials with a circular majorant
Q. I.
Rahman;
G.
Schmeisser
693-702
Abstract: If $p$ is a polynomial of degree at most $ n$ such that $\vert p(x)\vert \leqslant \sqrt {1 - {x^2}}$ for $- 1 \leqslant x \leqslant 1$, then for each $ k$, $\max \vert{p^{(k)}}(x)\vert$ on $[ - 1,\,1]$ is maximized by the polynomial $({x^2} - 1){U_{n - 2}}(x)$ where $ {U_m}$ is the $ m$th Chebyshev polynomial of the second kind. The purpose of this paper is to investigate if it is enough to assume $\vert p(x)\vert \leqslant \sqrt {1 - {x^2}}$ at some appropriately chosen set of $n + 1$ points in $[ - 1,\,1]$. The problem is inspired by a well-known extension of Markov's inequality due to Duffin and Schaeffer.
On inductive limits of certain $C\sp *$-algebras of the form $C(X)\otimes F$
Cornel
Pasnicu
703-714
Abstract: A certain class of $ {\ast}$-homomorphisms $ C(X) \otimes A \to C(Y) \otimes B$, called compatible with a map defined on $ Y$ with values in the set of all closed nonempty subsets of $X$, is studied. A local description of ${\ast}$-homomorphisms $C(X) \otimes A \to C(Y) \otimes B$ is given considering separately the cases $X = {\text{point}}$ and $A = {\mathbf{C}}$; this is done in terms of continuous "quasifields" of $ {C^{\ast}}$-algebras. Conditions under which an inductive limit $\underrightarrow {\lim }(C({X_k}) \otimes {A_k},\,{\Phi _k})$, where each ${\Phi _k}$ is of the above type, is $ {\ast}$-isomorphic with the tensor product of a commutative ${C^{\ast}}$-algebra with an AF algebra are given. For such inductive limits the isomorphism problem is considered.
On James' type spaces
Abderrazzak
Sersouri
715-745
Abstract: We study the spaces $ E$ which are isometric to their biduals $ {E^{{\ast}{\ast}}}$, and satisfy $\dim ({E^{{\ast}{\ast}}}/E) < \infty$. We show that these spaces have several common points with the usual James' space. Our study leads to a kind of classification of these spaces and we show that there are essentially four different basic structures for such spaces in the complex case, and five in the real case.
Ordinal rankings on measures annihilating thin sets
Alexander S.
Kechris;
Russell
Lyons
747-758
Abstract: We assign a countable ordinal number to each probability measure which annihilates all $H$-sets. The descriptive-set theoretic structure of this assignment allows us to show that this class of measures is coanalytic non-Borel. In addition, it allows us to quantify the failure of Rajchman's conjecture. Similar results are obtained for measures annihilating Dirichlet sets.
Electrophoretic traveling waves
P. C.
Fife;
O. A.
Palusinski;
Y.
Su
759-780
Abstract: An existence-uniqueness-approximability theory is given for a prototypical mathematical model for the separation of ions in solution by an imposed electric field. The separation is accomplished during the formation of a traveling wave, and the mathematical problem consists in finding a traveling wave solution of a set of diffusion-advection equations coupled to a Poisson equation. A basic small parameter $ \varepsilon$ appears in an apparently singular manner, in that when $\varepsilon = 0$ (which amounts to assuming the solution is everywhere electrically neutral), the last (Poisson) equation loses its derivative, and becomes an algebraic relation among the concentrations. Since this relation does not involve the function whose derivative is lost, the type of "singular" perturbation represented here is nonstandard. Nevertheless, the traveling wave solution depends in a regular manner on $\varepsilon$, even at $\varepsilon = 0$; and one of the principal aims of the paper is to show this regular dependence.
The continuation theory for Morse decompositions and connection matrices
Robert D.
Franzosa
781-803
Abstract: The continuation theory for ($<$-ordered) Morse decompositions and the indices defined on them--the homology index braid and the connection matrices--is established. The equivalence between $<$-ordered Morse decompositions and $ <$-consistent attractor filtrations is displayed. The spaces of ($ <$-ordered) Morse decompositions for a product parametrization of a local flow are introduced, and the local continuation of ($ <$-ordered) Morse decompositions is obtained via the above-described equivalence and the local continuation of attractors. The homology index braid and the connection matrices of an admissible ordering of a Morse decomposition are shown to be invariant on path components of the corresponding space of $<$-ordered Morse decompositions. This invariance is used to prove that the collection of connection matrices of a Morse decomposition is upper semicontinuous over the space of Morse decompositions (and over the parameter space) under local continuation.
Generalizations of Cauchy's summation theorem for Schur functions
G. E.
Andrews;
I. P.
Goulden;
D. M.
Jackson
805-820
Abstract: Cauchy's summation theorem for Schur functions is generalized, and a number of related results are given. The result is applied to a combinatorial problem involving products of pairs of permuations, by appeal to properties of the group algebra of the symmetric group.
Multilinear convolutions defined by measures on spheres
Daniel M.
Oberlin
821-835
Abstract: Let $\sigma$ be Lebesgue measure on ${\Sigma _{n - 1}}$ and write $\sigma = ({\sigma _1}, \ldots ,{\sigma _n})$ for an element of ${\Sigma _{n - 1}}$. For functions ${f_1}, \ldots ,{f_n}$ on $ {\mathbf{R}}$, define $\displaystyle T({f_1}, \ldots ,{f_n})(x) = \int_{{\Sigma _{n - 1}}} {{f_1}(x - {\sigma _1}) \cdots {f_n}(x - {\sigma _n})\,d\sigma ,\qquad x \in {\mathbf{R}}.}$ This paper partially answers the question: for which values of $p$ and $q$ is there an inequality $\displaystyle \vert\vert T({f_1}, \ldots ,{f_n})\vert{\vert _q} \leqslant C\vert\vert{f_1}\vert{\vert _p} \cdots \vert\vert{f_n}\vert{\vert _p}?$
Approximation properties for orderings on $*$-fields
Thomas C.
Craven
837-850
Abstract: The goal of this paper is to extend the main theorems on approximation properties of the topological space of orderings from formally real fields to skew fields with an involution $ ^{\ast}$. To accomplish this, the concept of $^{\ast}$-semiordering is developed and new theorems are obtained for lifting $^{\ast}$orderings from the residue class field of a real valuation.