Extremal problems of distance geometry related to energy integrals
Ralph
Alexander;
Kenneth B.
Stolarsky
1-31
Abstract: Let K be a compact set, $ \mathcal{M}$ a prescribed family of (possibly signed) Borel measures of total mass one supported by K, and f a continuous real-valued function on $K \times K$. We study the problem of determining for which $\mu \in \mathcal{M}$ (if any) the energy integral $I(K,\mu ) = \smallint_K {\smallint_K {f(x,y)d\mu (x)d\mu (y)} }$ is maximal, and what this maximum is. The more symmetry K has, the more we can say; our results are best when K is a sphere. In particular, when $ \mathcal{M}$ is atomic we obtain good upper bounds for the sums of powers of all $ (_2^n)$ distances determined by n points on the surface of a sphere. We make use of results from Schoenberg's theory of metric embedding, and of techniques devised by Pólya and Szegö for the calculation of transfinite diameters.
Invariant measures and growth conditions
Joseph Max
Rosenblatt
33-53
Abstract: Let G be a finitely-generated group acting on a set X and let A be a nonempty subset of X. If G has polynomial growth then there exists a finitely-additive G-invariant positive extended real-valued measure $\mu$ defined on all subsets of X such that $\mu (A) = 1$. When G is solvable, it has polynomial growth if and only if it does not contain a free subsemigroup on two generators. If G contains a free subsemigroup S on two generators, then G has exponential growth and there does not exist a measure $\mu$ as above with G acting on itself by multiplication and $A = S$.
Some thin sets in discrete abelian groups
Ron C.
Blei
55-65
Abstract: Let $\Gamma$ be a discrete abelian group, and $E \subset \Gamma $. For $F \subset E$, we say that $F \in \mathcal{P}(E)$, if for all $\Lambda$, finite subsets of $ \Gamma ,0 \notin \Lambda ,\Lambda + F \cap F$ is finite. Having defined the Banach algebra, $\tilde A(E) = c(E) \cap B(E)$, we prove the following: (i) $ E \subset \Gamma$ is a Sidon set if and only if every $F \in \mathcal{P}(E)$ is a Sidon set; (ii) $E \in \mathcal{P}(\Gamma )$ is a Sidon set if and only if $\tilde A(E) = A(E)$.
Inequalities for polynomials with a prescribed zero
A.
Giroux;
Q. I.
Rahman
67-98
Abstract: Inequalities for the derivative and for the maximum modulus on a larger circle of a polynomial with a given zero on the unit circle are obtained in terms of its degree and maximum modulus on the unit circle; examples are given to show that these are sharp with respect to the degree (best constants are not known). Inequalities for $ {L^p}$ norms, in particular ${L^2}$ norms, are also derived. Also certain functions of exponential type are considered and similar inequalities are obtained for them. Finally, the problem of estimating ${P_n}(r)$ (with $0 < r < 1$) given $ {P_n}(1) = 0$ is taken up.
Multiplier transformations on compact Lie groups and algebras
Robert S.
Strichartz
99-110
Abstract: Let G be a semisimple compact Lie group and $Tf = \sum \phi (m){d_{m \chi m}} \ast f$ a bi-invariant operator on ${L^2}(G)$, where ${\chi _m}$ and ${d_m}$ are the characters and dimensions of the irreducible representations of G, which are indexed by a lattice of points m in the Lie algebra $\mathfrak{G}$ in a natural way. If $ \Phi$ is a bounded ad-invariant function on $ \mathfrak{G}$ and $\displaystyle \phi {\text{(}}m{\text{) = }}\Phi {\text{(}}m{\text{ + }}\beta {\text{)}}\quad{\text{or}}$ ($\ast$) $\displaystyle \phi {\text{(}}m{\text{) = }}\int_G {\Phi (m + \beta - {\text{ad}}\;g\beta )dg}$ ($ \ast \ast$) $ \beta$ being half the sum of the positive roots, then various properties of T are related to properties of the Fourier multiplier transformation on $ \mathfrak{G}$ with multiplier $\Phi$. These properties include boundedness on $ {L^1}$, uniform boundedness on ${L^p}$ of a family of operators, and, in the special case $ G = {\text{SO}}(3)$, boundedness in ${L^p}$ for ad-invariant functions with $1 \leq p < 3/2$.
Somewhere locally flat codimension one manifolds with $1-{\rm ULC}$ complements are locally flat
T. M.
Price;
C. L.
Seebeck
111-122
Abstract: The purpose of this paper is to prove a taming theorem for a codimension one manifold that is locally flat at some point and has 1-ULC complement. We also prove that any two sufficiently close locally flat embeddings of a codimension one manifold are ambient isotopic. Since this paper was first submitted, R. Daverman has shown that, given any point on a codimension one manifold with 1-ULC complement, some neighborhood of that point lies on a codimension one sphere that is locally flat at some points and has 1-ULC complement. Hence the two papers combined prove that a codimension one manifold is locally flat if and only if its complement is 1-ULC.
Almost complex structures on complex projective spaces
Alan
Thomas
123-132
Abstract: In this paper we classify the almost complex structures on a complex projective space as roots of a certain polynomial equation.
A relation between $K$-theory and cohomology
Alan
Thomas
133-142
Abstract: It is well known that for X a CW-complex, $K(X)$ and ${H^{{\text{ev}}}}(X)$ are isomorphic modulo finite groups, although the ``isomorphism'' is not natural. The purpose of this paper is to improve this result for X a finite CW-complex.
Equisingular deformations of plane algebroid curves
Jonathan M.
Wahl
143-170
Abstract: We construct a formal versal equisingular deformation of a plane algebroid curve (in characteristic zero), and show it is smoothly embedded in the whole deformation space of the singularity. Closer analysis relates equisingular deformations of the curve to locally trivial deformations of a certain (nonreduced) projective curve. Finally, we prove that algebraic ${\pi _1}$ of the complement of a plane algebroid curve remains constant during formal equisingular deformation.
The undecidability of the word problems for projective geometries and modular lattices
L.
Lipshitz
171-180
Abstract: We show that the restricted word problems for finite-dimensional projective geometries and finite modular lattices and the word problem for modular lattices are undecidable.
A generalization of the ${\rm cos} \pi \,\rho $ theorem
Albert
Baernstein
181-197
Abstract: Let f be an entire function, and let $\beta$ and $\lambda$ be positive numbers with $\beta \leq \pi$ and $\beta \lambda < \pi$. Let $E(r) = \{ \theta :\log \vert f(r{e^{i\theta }})\vert > \cos \beta \lambda \log M(r)\}$. It is proved that either there exist arbitrarily large values of r for which $E(r)$ contains an interval of length at least $ 2\beta$, or else ${\lim _{r \to \infty }}{r^{ - \lambda }}\log M(r,f)$ exists and is positive or infinite. For $\beta = \pi$ this is Kjellberg's refinement of the cos $\pi \rho$ theorem.
Bounded mean oscillation and regulated martingales
Carl
Herz
199-215
Abstract: In the martingale context, the dual Banach space to ${H_1}$ is BMO in analogy with the result of Charles Fefferman [4] for the classical case. This theorem is an easy consequence of decomposition theorems for ${H_1}$-martingales which involve the notion of $ {L_p}$-regulated $ {L_1}$-martingales where $1 < p \leq \infty$. The strongest decomposition theorem is for $p = \infty$, and this provides full information about BMO. The weaker $p = 2$ decomposition is fundamental in the theory of martingale transforms.
On a Wedderburn principal theorem for the flexible algebras
Robert A.
Chaffer
217-229
Abstract: A strictly power-associative algebra A over a field K is said to have a Wedderburn decomposition if there is a subalgebra S of A such that $A = S + N$, where N is the nil radical of A, and $S = A - N$. A Wedderburn principal theorem for a class of algebras is a theorem which asserts that the algebras A, in the class, with $A - N$ separable have Wedderburn decompositions. It is known that there is no such theorem for the class of noncommutative Jordan algebras. A partial result in this direction is the following theorem. Theorem. Let A be a strictly power-associative, flexible algebra over a field F with characteristic not 2 or 3, with $A - N$ separable and such that $A = {A_1} \oplus {A_2} \oplus \cdots \oplus {A_n}$ where each ${A_i}$. has $ {A_i} - {N_i}$ simple and has more than two pairwise orthogonal idempotents. Then $A = S + N$ where S is a subalgebra of A.
Product of ring varieties and attainability
Awad A.
Iskander
231-238
Abstract: The class of all rings that are Everett extensions of a ring in a variety $ \mathfrak{U}$ by a ring in a variety $ \mathfrak{B}$ is a variety $ \mathfrak{U} \cdot \mathfrak{B}$. With respect to this operation the set of all ring varieties is a partially ordered groupoid (under inclusion), that is not associative. A variety is idempotent iff it is the variety of all rings, or generated by a finite number of finite fields. No families of polynomial identities other than those equivalent to $x = x$ or $x = y$ are attainable on the class of all rings or on the class of all commutative rings.
The genera of edge amalgamations of complete bigraphs
Seth R.
Alpert
239-247
Abstract: If G and H are graphs, then $G \vee H$ is defined to be a graph obtained by identifying some edge of G with some edge of H. It is shown that for all m, n, p, and q the genus $g({K_{m,n}} \vee {K_{p,q}})$ is either $ g({K_{m,n}}) + g({K_{p,q}})$ or else $g({K_{m,n}}) + g({K_{p,q}}) - 1$. The latter value is attained if and only if both ${K_{m,n}}$ and ${K_{p,q}}$ are critical in the sense that the deletion of any edge results in a graph whose genus is one less than the genus of the original graph.
The group of $PL$-homeomorphisms of a compact $PL$-manifold is an $1\sp{f}\sb{2}$-manifold
James
Keesling;
David C.
Wilson
249-256
Abstract: In this paper it is shown that if M is a compact PL-manifold and ${H_{PL}}(M)$ is the group of PL-homeomorphisms of M onto itself, then $ {H_{PL}}(M)$ is an $ l_2^f$-manifold. Here $ {l_2}$ is the Hilbert space of all real-valued square-summable sequences and $l_2^f = \{ ({x_i}) \in {l_2}:{x_i} = 0$ for almost all i.
On totally real submanifolds
Bang-yen
Chen;
Koichi
Ogiue
257-266
Abstract: Complex analytic submanifolds and totally real submanifolds are two typical classes among all submanifolds of an almost Hermitian manifold. In this paper, some characterizations of totally real submanifolds are given. Moreover some classifications of totally real submanifolds in complex space forms are obtained.
A general representation theorem for analytic solutions of first-order algebraic differential equations in sectors
Steven B.
Bank
267-289
Abstract: In this paper, we obtain precise asymptotic representations for a broad class of solutions of first-order algebraic differential equations whose coefficients belong to a certain type of function field.
Irreducible congruences over ${\rm GF}(2)$
C. B.
Hanneken
291-301
Abstract: In characterizing and determining the number of conjugate sets of irreducible congruences of degree m belonging to $ GF(p)$ relative to the group $G(p)$ of linear fractional transformations with coefficients belonging to the same field, the case $p = 2$ has been consistently excluded from considerations. In this paper we consider the special case $p = 2$ and determine the number of conjugate sets of m-ic congruences belonging to $ GF(2)$ relative to $ G(2)$.
Riesz points of the spectrum of an element in a semisimple Banach algebra
Lynn D.
Pearlman
303-328
Abstract: Let A be a semisimple Banach algebra with unit element and let $ {S_A}$ denote the socle of A. For an element y in A, let ${L_y}[{R_y}]$ denote the operator of left [right] multiplication by y on A. The operational calculus and A. E. Taylor's theory of the ascent $\alpha (T)$ and descent $ \delta (T)$ of an operator T on A are used to show that the following conditions on a number $\lambda$ in the spectrum of an element x in A are all equivalent. (1) $\lambda$ is a pole of the resolvent mapping $z \to {(z - x)^{ - 1}}$ and the spectral idempotent f, for x at $ \lambda$ is in $ {S_A}$. (2) $\lambda - x - c$ is invertible in A for some c in the closure of ${S_A}$ such that $cx = xc$. (3) $ \lambda - x$ is invertible modulo the closure of ${S_A}$ and $ 0 < \alpha ({L_{(\lambda - x)}}) = \delta ({L_{(\lambda - x)}}) < \infty$. (4) $ \lambda - x$ is invertible modulo the closure of ${S_A}$ and $0 < \alpha ({R_{(\lambda - x)}}) = \delta ({R_{(\lambda - x)}}) = \alpha ({L_{(\lambda - x)}}) = \delta ({L_{(\lambda - x)}}) < \infty$. Such numbers $\lambda$ are called Riesz points. An element x is called a Riesz element of A if it is topologically nilpotent modulo the closure of ${S_A}$. It is shown that x is a Riesz element if and only if every nonzero number in the spectrum of x is a Riesz point.
Characterization of privileged polydomains
Yum Tong
Siu
329-357
Abstract: This paper gives a number of equivalent conditions for a bounded polydomain to be privileged with respect to a coherent analytic sheaf in the sense of Douady. One of the equivalent conditions is in terms of the homological codimensions of the sheaf at the boundary of the polydomain. In the case of a polydisc, this condition about homological codimensions coincides with a conjecture of Douady. The other equivalent conditions concern some weaker concepts of privilegedness and the existence of privileged sets at the boundary.
A finitely additive generalization of the Fichtenholz-Lichtenstein theorem
George Edward
Sinclair
359-374
Abstract: Let $\mu$ and $\nu$ be bounded, finitely additive measures on algebras over sets X and Y, respectively. Conditions are determined for a bounded function $ f:X \times Y \to {\mathbf{R}}$, without assuming bimeasurability, so that the iterated integrals $\smallint_X {\smallint_Y {fd\mu d\mu } }$ and $\smallint_Y {\smallint_X {fd\mu d\nu } }$ exist and are equal. This result is then used to construct a product algebra and finitely additive product measure for $ \mu$ and $\nu$. Finally, a simple Fubini theorem with respect to this product algebra and product measure is established.
Primitive elements and one relation algebras
Catherine
Aust
375-387
Abstract: Let F be a free algebra in a variety V. An element p of F is called primitive if it is contained in some free generating set for F. In 1936, J. H. C. Whitehead proved that a group with generators $ {g_1}, \ldots ,{g_n}$ and one relation $r = 1$ is free if and only if the relator r is primitive in the free group on ${g_1}, \ldots ,{g_n}$. In tnis paper, tne question of whether there is an analogous theorem for other varieties is considered. A necessary and sufficient condition that a finitely generated, one relation algebra be free is proved for any Schreier variety of nonassociative linear algebras and for any variety defined by balanced identities. An identity $ u({x_1}, \ldots ,{x_n}) = v({x_1}, \ldots ,{x_n})$ is called balanced if each of u and v has the same length and number of occurrences of each ${x_i}$. General sufficiency conditions that a finitely generated, one relation algebra be free are given, and all of the known results analogous to the Whitehead theorem are shown to be equivalent to a general necessary condition. Also an algebraic proof of Whitehead's theorem is outlined to suggest the line of argument for other varieties.
Contracting spaces of maps on the countable direct limit of a space
Richard E.
Heisey
389-411
Abstract: We give conditions sufficient to imply the contractibility of the space of maps, with compact-open topology, on the countable direct limit of a space. Applying these conditions we obtain the following: Let F be the conjugate of a separable infinite-dimensional Banach space with bounded weak-$^\ast$ topology, or the countable direct limit of the real line. Then there is a contraction of the space of maps on F which simultaneously contracts the subspaces of open maps, embeddings, closed embeddings, and homeomorphisms. Corollaries of our work are that any homeomorphism on F, F as above, is invertibly isotopic to the identity, and the general linear group of the countable direct limit of lines is contractible.
Conditions for the absolute continuity of two diffusions
Steven
Orey
413-426
Abstract: Consider two diffusion processes on the line. For each starting point x and each finite time t, consider the measures these processes induce in the space of continuous functions on [0, t]. Necessary and sufficient conditions on the generators are found for the induced measures to be mutually absolutely continuous for each x and t. If the first process is Brownian motion, the second one must be Brownian motion with drift $b(x)$, where $b(x)$ is locally in ${L_2}$ and satisfies a certain growth condition at $\pm \infty$.
Waring's problem for twenty-two biquadrates
Henry E.
Thomas
427-430
Abstract: That every natural number is the sum of at most twenty-two biquadrates is proven by ascent from machine results on sums of six fourth powers.