Analysis on analytic spaces and non-self-dual Yang-Mills fields
N. P.
Buchdahl
431-469
Abstract: This paper gives a cohomological description of the Witten-Isenberg-Yasskin-Green generalization to the non-self-dual case of Ward's twistor construction for self-dual Yang-Mills fields. The groundwork for this description is presented in Part I: with a brief introduction to analytic spaces and differential forms thereon, it contains an investigation of the exactness of the holomorphic relative de Rham complex on formal neighbourhoods of submanifolds, results giving sufficient conditions for the invertibility of pull-back functors on categories of analytic objects, and a discussion of the extension problem for analytic objects in the context of the formalism earlier introduced. Part II deals with non-self-dual Yang-Mills fields: the Yang-Mills field and current are identified in terms of the Griffiths obstructions to extension, including a proof of Manin's result that "current = obstruction to third order". All higher order obstructions are identified, there being at most ${N^2}$ for a bundle of rank $N$. An ansatz for producing explicit examples of non-self-dual fields is obtained by using the correspondence. This ansatz generates $ {\text{SL}}(2,\mathbb{C})$ solutions with topological charge $1$ on ${S^4}$.
No $L\sb 1$-contractive metrics for systems of conservation laws
Blake
Temple
471-480
Abstract: Let $(\ast)$ $\displaystyle \quad {u_t} + F{(u)_x} = 0$ be any $2 \times 2$ system of conservation laws satisfying certain generic assumptions on $F$ in a neighborhood $ \mathcal{N}$ of $ u$-space. We prove that for every nondegenerate metric $D$ on $u$-space there exists states ${u_1}$ and ${u_2}$ in $ \mathcal{N}$ such that $\int_{ - \infty }^\infty {D(u(x,t),{u_1})\;dx}$ is a strictly increasing function of $t$ in a neighborhood of $ t = 0$, where $ u$ is the admissible solution of $( \ast )$ with initial data $\displaystyle u(x,0) = \left\{ {\begin{array}{*{20}{c}} {{u_1},} & {x \l... ...,} {{u_1},} & {x \geqslant 1.} \end{array} } \right.$ This contrasts with the case of a scalar equation in which $ \int_{ - \infty }^\infty {D(u(x,t),v(x,t))\;dx}$ is a decreasing function of $ t$ for all admissible solution pairs $u$ and $v$ when $D$ is taken to be the absolute value norm.
Semigroups in Lie groups, semialgebras in Lie algebras
Joachim
Hilgert;
Karl H.
Hofmann
481-504
Abstract: Consider a subsemigroup of a Lie group containing the identity and being ruled by one-parameter semigroups near the identity. We associate with it the set $W$ of its tangent vectors at the identity and obtain a subset of the Lie algebra $ L$ of the group. The set $ W$ has the following properties: (i) $W + W = W$, (ii) ${{\mathbf{R}}^ + }\,\cdot\;W \subset W$, (iii) ${W^ - } = W$, and, the crucial property, (iv) for all sufficiently small elements $ x$ and $y$ in $W$ one has $x \ast y = x + y + \frac{1} {2}[x,y] + \cdots$ (Campbell-Hausdorff!) $\in W$. We call a subset $W$ of a finite-dimensional real Lie algebra $L$ a Lie semialgebra if it satisfies these conditions, and develop a theory of Lie semialgebras. In particular, we show that a subset $W$ satisfying (i)-(iii) is a Lie semialgebra if and only if, for each point $x$ of $W$ and the (appropriately defined) tangent space ${T_x}$ to $W$ in $x$, one has $[x,{T_x}] \subset {T_x}$. (The Lie semialgebra $W$ of a subgroup is always a vector space, and for vector spaces $W$ we have ${T_x} = W$ for all $x$ in $W$, and thus the condition reduces to the old property that $W$ is a Lie algebra.) In the introduction we fully discuss all Lie semialgebras of dimension not exceeding three. Our methods include a full duality theory for closed convex wedges, basic Lie group theory, and certain aspects of ordinary differential equations.
On the structure of abelian $p$-groups
Paul
Hill
505-525
Abstract: A new kind of abelian $p$-group, called an $A$-group, is introduced. This class contains the totally projective groups and Warfield's $S$-groups as special cases. It also contains the $N$-groups recently classified by the author. These more general groups are classified by cardinal (numerical) invariants which include, but are not limited to, the Ulm-Kaplansky invariants. Thus the existing theory, as well as the classification, of certain abelian $p$-groups is once again generalized. Having classified $A$-groups (by means of a uniqueness and corresponding existence theorem) we can successfully study their structure and special properties. Such a study is initiated in the last section of the paper.
On isometric embeddings of graphs
R. L.
Graham;
P. M.
Winkler
527-536
Abstract: If $G$ is a finite connected graph with vertex set $V$ and edge set $E$, a standard way of defining a distance $ {d_G}$ on $G$ is to define ${d_G}(x,y)$ to be the number of edges in a shortest path joining $x$ and $y$ in $V$. If $(M,{d_M})$ is an arbitrary metric space, then an embedding $ \lambda :V \to M$ is said to be isometric if ${d_G}(x,y) = {d_M}(\lambda (x),\lambda (y))$ for all $x,y \in V$. In this paper we will lay the foundation for a theory of isometric embeddings of graphs into cartesian products of metric spaces.
Condensed Julia sets, with an application to a fractal lattice model Hamiltonian
M. F.
Barnsley;
J. S.
Geronimo;
A. N.
Harrington
537-561
Abstract: The Julia set for the complex rational map $z \to {z^2} - \lambda + \varepsilon /z$, where $ \lambda$ and $\varepsilon$ are complex parameters, is considered in the limit as $ \varepsilon \to 0$. The result is called the condensed Julia set for $z \to ({z^3} - \lambda z)/z$. The limit of balanced measures, associated functional equations and orthogonal polynomials are considered; it is shown, for example, that for $\lambda \geqslant 2$ the moments, orthogonal polynomials, and associated Jacobi matrix $\mathcal{J}$ can be calculated explicitly and are not those belonging to ${z^2} - \lambda$. The spectrum of $\mathcal{J}$ consists of a point spectrum $ P$ together with its derived set. The latter is the Julia set for ${z^2} - \lambda$, and carries none of the spectral mass when $ \lambda > 2$. When $\lambda = 2$, $P$ is dense in $[-2,2]$. A similar condensation in the case $\lambda = 15/4$ leads to a system which corresponds precisely to the spectrum and density of states of a two-dimensional Sierpinski gasket model Schrödinger equation. The basic ideas about condensation of Julia sets in general are described. If $R(z)$ is a rational transformation of degree greater than one, then condensation can be attached to $\displaystyle z \to R(z) + \varepsilon \sum\limits_{i = 1}^k {{{(z - {a_i})}^{ - {\gamma _i}}},}$ where the ${\gamma _i}$'s and $k$ are finite positive integers and the $ {a_i}$'s are complex numbers. If $\infty$ is an indifferent or attractive fixed point of $R(z)$, then all of the moments of the associated condensed balanced measure can be calculated explicitly, as can the orthogonal polynomials when the condensed Julia set is real. Sufficient conditions for the condensed measure $\sigma$ to be a weak limit of the balanced measures $ {\mu _\varepsilon }$ are given. Functional equations connected to the condensed measure are derived, and it is noted that their form typifies those encountered in statistical physics, in connection with partition functions for Ising hierarchical models.
Grade schemes and grade functions
Stephen
McAdam
563-590
Abstract: In recent years, two concepts similar to $R$-sequences have appeared, essential sequences and asymptotic sequences. This work explores the general nature of such sequences.
Index theory on curves
Peter
Haskell
591-604
Abstract: This paper constructs from the $\bar \partial $-operator on the smooth part of a complex projective algebraic curve a cycle in the analytically defined $K$ homology of the curve. The paper identifies the corresponding cycle in the topologically defined $ K$ homology.
Primitive group rings and Noetherian rings of quotients
Christopher J. B.
Brookes;
Kenneth A.
Brown
605-623
Abstract: Let $k$ be a field, and let $G$ be a countable nilpotent group with centre $Z$. We show that the group algebra $kG$ is primitive if and only if $ k$ is countable, $ G$ is torsion free, and there exists an abelian subgroup $A$ of $G$, of infinite rank, with $A \cap Z = 1$. Suppose now that $G$ is torsion free. Then $kG$ has a partial quotient ring $Q = kG{(kZ)^{ - 1}}$. The above characterisation of the primitivity of $kG$ is intimately connected with the question: When is $Q$ a Noetherian ring? We determine this for those groups $G$, as above, all of whose finite rank subgroups are finitely generated. In this case, $Q$ is Noetherian if and only if $ G$ has no abelian subgroup $ A$ of infinite rank with $A \cap Z = 1$.
Some remarks on the intrinsic measures of Eisenman
Ian
Graham;
H.
Wu
625-660
Abstract: This paper studies the intrinsic measures on complex manifolds first introduced by Eisenman in analogy with the intrinsic distances of Kobayashi. Some standard conjectures, together with several new ones, are considered and partial or complete answers are provided. Most of the counterexamples come from a closer examination of unbounded domains in complex euclidean space. In particular, a large class of unbounded hyperbolic domains are exhibited. Those unbounded domains of finite euclidean volume are also singled out for discussion.
Lipschitzian mappings and total mean curvature of polyhedral surfaces. I
Ralph
Alexander
661-678
Abstract: For a smooth closed surface $C$ in ${E^3}$ the classical total mean curvature is defined by $M(C) = \frac{1} {2}\int ({\kappa _1} + {\kappa _2})\;d\sigma (p)$, where ${\kappa _1},{\kappa _2}$ are the principal curvatures at $p$ on $C$. If $C$ is a polyhedral surface, there is a well known discrete version given by $M(C) = \frac{1} {2}\Sigma {l_i}(\pi - {\alpha _i})$, where ${l_i}$ represents edge length and ${\alpha _i}$ the corresponding dihedral angle along the edge. In this article formulas involving differentials of total mean curvature (closely related to the differential formula of L. Schláfli) are applied to several questions concerning Lipschitizian mappings of polyhedral surfaces. For example, the simplest formula $\Sigma {l_i}\,d{\alpha _i} = 0$ may be used to show that the remarkable flexible polyhedral spheres of R. Connelly must flex with constant total mean curvature. Related differential formulas are instrumental in showing that if $ f: {E^2} \to {E^2}$ is a distance-increasing function and $K \subset {E^2}$, then $ \operatorname{Per}(\operatorname{conv}\;K) \leqslant \operatorname{Per}(\operatorname{conv}\;f[K])$. This article (part I) is mainly concerned with problems in ${E^n}$. In the sequel (part II) related questions in ${S^n}$ and ${H^n}$, as well as ${E^n}$, will be considered.
Packing measure, and its evaluation for a Brownian path
S. James
Taylor;
Claude
Tricot
679-699
Abstract: A new measure on the subsets $E \subset {{\mathbf{R}}^d}$ is constructed by packing as many disjoint small balls as possible with centres in $E$. The basic properties of $\phi$-packing measure are obtained: many of these mirror those of $\phi$-Hausdorff measure. For $\phi (s) = {s^2}/(\log \,\log (1/s))$, it is shown that a Brownian trajectory in ${{\mathbf{R}}^d}(d \geqslant 3)$ has finite positive $\phi$-packing measure.
Simplexwise linear near-embeddings of a $2$-disk into ${\bf R}\sp 2$
Ethan D.
Bloch
701-722
Abstract: Let $K \subset {{\mathbf{R}}^2}$ be a finitely triangulated $2$-disk; a map $f:K \to {{\mathbf{R}}^2}$ is called simplexwise linear $(SL)$ if $f\vert\sigma$ is affine linear for each (closed) simplex $\sigma$ of $K$. Interest in $ {\text{SL}}$ maps originated with work of S. S. Cairns and subsequent work of R. Thom and N. H. Kuiper. Let $E(K) = \{ {\text{orientation preserving SL embeddings}}\;K \to {{\mathbf{R}}^2}\}$, $L(K) = \{ {\text{SL homeomorphism}}\;K \to K\;{\text{fixing}}\;\partial K\;{\text{pointwise}}\}$, and $\overline {E(K)} ,\overline {L(K)}$ denote their respective closures in the space of all ${\text{SL}}$ maps $K \to {{\mathbf{R}}^2}$ and the space of all ${\text{SL}}$ maps $K \to K$ fixing $ \partial K$. The main result of this paper is useful characterizations of maps in $\overline {L(K)}$ and some maps in $\overline {E(K)}$, including the relation of such maps to $ {\text{SL}}$ embeddings into the nonstandard plane.
Strictly convex simplexwise linear embeddings of a $2$-disk
Ethan D.
Bloch
723-737
Abstract: Let $K \subset {{\mathbf{R}}^2}$ be a finitely triangulated $2$-disk; a map $f:K \to {{\mathbf{R}}^2}$ is called simplexwise linear $(SL)$ if $f\vert\sigma$ is affine linear for each (closed) $ 2$-simplex $\sigma$ of $K$. Let $ E(K) = \{ {\text{orientation preserving SL embeddings}}\;K \to {{\mathbf{R}}^2}\}$, $ {E_{{\text{SC}}}}(K) = \{ f \in E(K)\vert f(K)\;{\text{is strictly convex}}\}$, and let $ \overline {E(K)}$ and $ \overline {{E_{{\text{SC}}}}(K)}$ denote their closures in the space of all ${\text{SL}}$ maps $K \to {{\mathbf{R}}^2}$. A characterization of certain elements of $ \overline {E(K)}$ is used to prove that $ {E_{{\text{SC}}}}(K)$ has the homotopy type of ${S^1}$ and to characterize those elements of $\overline {E(K)}$ which are in $ \overline {{E_{{\text{SC}}}}(K)}$, as well as to relate such maps to ${\text{SL}}$ embeddings into the nonstandard plane.
General position properties satisfied by finite products of dendrites
Philip L.
Bowers
739-753
Abstract: Let $\bar A$ be a dendrite whose endpoints are dense and let $A$ be the complement in $\bar A$ of a dense $\sigma$-compact collection of endpoints of $ \bar A$. This paper investigates various general position properties that finite products of $\bar A$ and $A$ possess. In particular, it is shown that (i) if $X$ is an $L{C^n}$-space that satisfies the disjoint $ n$-cells property, then $X \times \bar A$ satisfies the disjoint $ (n + 1)$-cells property, (ii) $ {\bar A^n} \times [ - 1,1]$ is a compact $(n + 1)$-dimensional $ {\text{AR}}$ that satisfies the disjoint $n$-cells property, (iii) ${\bar A^{n + 1}}$ is a compact $(n + 1)$-dimensional ${\text{AR}}$ that satisfies the stronger general position property that maps of $ n$-dimensional compacta into ${\bar A^{n + 1}}$ are approximable by both $Z$-maps and ${Z_n}$-embeddings, and (iv) ${A^{n + 1}}$ is a topologically complete $ (n + 1)$-dimensional ${\text{AR}}$ that satisfies the discrete $ n$-cells property and as such, maps from topologically complete separable $ n$-dimensional spaces into ${A^{n + 1}}$ are strongly approximable by closed ${Z_n}$-embeddings.
Varieties of automorphism groups of orders
W. Charles
Holland
755-763
Abstract: The group $A(\Omega )$ of automorphisms of a totally ordered set $\Omega$ must generate either the variety of all groups or the solvable variety of class $n$. In the former case, $A(\Omega )$ contains a free group of rank ${2^{{\aleph _0}}}$; in the latter case, $A(\Omega )$ contains a free solvable group of class $n - 1$ and rank ${2^{{\aleph _0}}}$.
Helical minimal immersions of compact Riemannian manifolds into a unit sphere
Kunio
Sakamoto
765-790
Abstract: An isometric immersion of a Riemannian manifold $M$ into a Riemannian manifold $\overline M$ is called helical if the image of each geodesic has constant curvatures which are independent of the choice of the particular geodesic. Suppose $ M$ is a compact Riemannian manifold which admits a minimal helical immersion of order $4$ into the unit sphere. If the Weinstein integer of $ M$ equals that of one of the projective spaces, then $M$ is isometric to that projective space with its canonical metric.
Homomorphisms between generalized Verma modules
Brian D.
Boe
791-799
Abstract: Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra and $ \mathfrak{p}$ a parabolic subalgebra. The first result is a necessary and sufficient condition, in the spirit of the Bernstein-Gelfand-Gelfand theorem on Verma modules, for Lepowsky's "standard map" between two generalized Verma modules for $\mathfrak{g}$ to be zero. The main result gives a complete description of all homomorphisms between the generalized Verma modules induced from one-dimensional $ \mathfrak{p}$-modules, in the "hermitian symmetric" situation.
Piecewise continuous almost periodic functions and mean motions
Jing Bo
Xia
801-811
Abstract: In this paper, we prove the existence of mean motion for certain noncontinuous almost periodic functions.
A relation between invariant means on Lie groups and invariant means on their discrete subgroups
John R.
Grosvenor
813-825
Abstract: Let $G$ be a Lie group, and let $ D$ be a discrete subgroup of $G$ such that the right coset space $D\backslash G$ has finite right-invariant volume. We will exhibit an injection of left-invariant means on ${l^\infty }(D)$ into left-invariant means on the left uniformly continuous bounded functions of $ G$. When $G$ is an abelian Lie group with finitely many connected components, we also show surjectivity, and when $G$ is the additive group ${{\mathbf{R}}^n}$ and $D$ is $ {{\mathbf{Z}}^n}$, the bijection will explicitly take the form of an integral over the unit cube ${[0,1]^n}$.
The bidual of the compact operators
Theodore W.
Palmer
827-839
Abstract: Let $X$ be a Banach space such that $ {X^\ast}$ has the Radon-Nikodým property. If ${X^\ast}$ also has the approximation property, then the Banach algebra $B({X^{ \ast \ast }})$ of all bounded linear operators on $ {X^{\ast\ast}}$ is isometrically isomorphic (as an algebra) to the double dual ${B_K}{(X)^{ \ast \ast }}$ of the Banach algebra of compact operators on $X$ when ${B_K}{(X)^{ \ast \ast }}$ is provided with the first Arens product. The chief result of this paper is a converse to the above statement. The converse is formulated in a strong fashion and a number of other results, including a formula for the second Arens product, are also given.
Cascade of sinks
Clark
Robinson
841-849
Abstract: In this paper it is proved that if a one-parameter family $\{ {F_t}\}$ of ${C^1}$ dissipative maps in dimension two creates a new homoclinic intersection for a fixed point $ {P_t}$ when the parameter $t = {t_0}$, then there is a cascade of quasi-sinks, i.e., there are parameter values ${t_n}$ converging to ${t_0}$ such that, for $t = {t_n}$, ${F_t}$ has a quasi-sink ${A_n}$ with each point $q$ in ${A_n}$ having period $n$. A quasi-sink ${A_n}$ for a map $F$ is a closed set such that each point $ q$ in ${A_n}$ is a periodic point and $ {A_n}$ is a quasi-attracting set (à la Conley), i.e., ${A_n}$ is the intersection of attracting sets $ A_n^j, {A_n} = { \cap _j}A_n^j$, where each $A_n^j$ has a neighborhood $U_n^j$ such that $\cap \{ {F^k}(U_n^j):k \geqslant 0\} = A_n^j$. Thus, the quasi-sinks ${A_n}$ are almost attracting sets made up entirely of points of period $n$. Gavrilov and Silnikov, and later Newhouse, proved this result when the new homoclinic intersection is created nondegenerately. In this case the sets ${A_n}$ are single, isolated (differential) sinks. In an earlier paper we proved the degenerate case when the homoclinic intersections are of finite order tangency (or the family is real analytic), again getting a cascade of sinks, not just quasi-sinks.
Restricted ramification for imaginary quadratic number fields and a multiplicator free group
Stephen B.
Watt
851-859
Abstract: Let $K$ be an imaginary quadratic number field with unit group ${E_K}$ and let $\ell$ be a rational prime such that $\ell \nmid \left\vert {{E_K}} \right\vert$. Let $S$ be any finite set of finite primes of $ K$ and let $K(\ell ,S)$ denote the maximal $\ell $-extension of $ K$ (inside a fixed algebraic closure of $K$) which is nonramified at the finite primes of $ K$ outside $S$. We show that the finitely generated pro-$\ell$-group $\Omega (\ell ,S) = \operatorname{Gal}(K(\ell ,S)/K)$ has the property that a complete set of defining relations for $\Omega (\ell ,S)$ as a pro-$\ell$-group can be obtained by lifting the nontrivial abelian or torsion relations in the maximal abelian quotient group $\Omega {(\ell ,S)^{{\text{ab}}}}$. In addition we use the key idea of the proof to derive some interesting results on towers of fields over $ K$ with restricted ramification.