Stationary configurations of point vortices
Kevin Anthony
O’Neil
383-425
Abstract: The motion of point vortices in a plane of fluid is an old problem of fluid mechanics, which was given a Hamiltonian formulation by Kirchhoff. Stationary configurations are those which remain self-similar throughout the motion. Results of two types are presented. Configurations which are in equilibrium or which translate uniformly are counted using methods of algebraic geometry, which establish necessary and sufficient conditions for existence. Relative equilibria (rigidly rotating configurations) which lie on a line are studied using a topological construction applicable to other power-law systems. Upper and lower bounds for such configurations are found for vortices with mixed circulations. Arrangements of three vortices which collide in finite time are well known. One-dimensional families of such configurations are shown to exist for more than three vortices. Stationary configurations of four vortices are examined in detail.
The Fraser-Horn and Apple properties
Joel
Berman;
W. J.
Blok
427-465
Abstract: We consider varieties $\mathcal{V}$ in which finite direct products are skew-free and in which the congruence lattices of finite directly indecomposables have a unique coatom. We associate with $ \mathcal{V}$ a family of derived varieties, $ d(\mathcal{V})$: a variety in $ d(\mathcal{V})$ is generated by algebras $ {\mathbf{A}}$ where the universe of $ {\mathbf{A}}$ consists of a congruence class of the coatomic congruence of a finite directly indecomposable algebra ${\mathbf{B}} \in \mathcal{V}$ and the operations of $ {\mathbf{A}}$ are those of ${\mathbf{B}}$ that preserve this congruence class. We also consider the prime variety of $\mathcal{V}$, denoted ${\mathcal{V}_0}$, generated by all finite simple algebras in $ \mathcal{V}$. We show how the structure of finite algebras in $\mathcal{V}$ is determined to a considerable extent by $ {\mathcal{V}_0}$ and $d(\mathcal{V})$. In particular, the free $\mathcal{V}$-algebra on $n$ generators, $ {{\mathbf{F}}_\mathcal{V}}(n)$, has as many directly indecomposable factors as $ {{\mathbf{F}}_{{\mathcal{V}_0}}}(n)$ and the structure of these factors is determined by the varieties $ d(\mathcal{V})$. This allows us to produce in many cases explicit formulas for the cardinality of $ {{\mathbf{F}}_\mathcal{V}}(n)$. Our work generalizes the structure theory of discriminator varieties and, more generally, that of arithmetical semisimple varieties. The paper contains many examples of algebraic systems that have been investigated in different contexts; we show how these all fit into a general scheme.
Graded Lie algebras of the second kind
Jih Hsin
Chêng
467-488
Abstract: The associated Lie algebra of the Cartan connection for an abstract CR-hypersurface admits a gradation of the second kind. In this article, we give two ways to characterize this kind of graded Lie algebras, namely, geometric characterization in terms of symmetric spaces and algebraic characterization in terms of root systems. A complete list of this class of Lie algebras is given.
The multiplicity of isolated two-dimensional hypersurface singularities
Henry B.
Laufer
489-496
Abstract: Consider an isolated two-dimensional complex analytic hypersurface singularity $(V,p)$. A relation is given between the abstract topology of $(V,p)$ and the multiplicity of $ (V,p)$, yielding an upper bound for the multiplicity. This relation is a necessary condition for a Gorenstein singularity to be a hypersurface.
The topology of resolution towers
Selman
Akbulut;
Henry
King
497-521
Abstract: An obstruction theory is given to determine when a space has a resolution tower. This can be used to decide whether or not the space is homeomorphic to a real algebraic set.
Produced representations of Lie algebras and Harish-Chandra modules
Michael J.
Heumos
523-534
Abstract: The comultiplication of the universal enveloping algebra of a Lie algebra is used to give modules produced from a subalgebra, an additional compatible structure of a module over an algebra of formal power series. When only the $ \mathfrak{k}$-finite elements of this algebra act on a module, conditions are given that insure that it is the Harish-Chandra module of a produced module. The results are then applied to Zuckerman derived functor modules for reductive Lie algebras. The main application describes a setting where the Zuckerman functors and production from a subalgebra commute.
Prime ideals in enveloping rings
D. S.
Passman
535-560
Abstract: Let $L$ be a Lie algebra over the field $ K$ of characteristic 0 and let $U(L)$ denote its universal enveloping algebra. If $R$ is a $K$-algebra and $L$ acts on $R$ as derivations, then there is a natural ring generated by $R$ and $U(L)$ which is denoted by $R\char93 U(L)$ and called the smash product of $ R$ by $U(L)$. The aim of this paper is to describe the prime ideals of this algebra when it is Noetherian. Specifically we show that there exists a twisted enveloping algebra $U(X)$ on which $L$ acts and a precisely defined one-to-one correspondence between the primes $P$ of $R\char93 U(L)$ with $ P \cap R = 0$ and the $ L$-stable primes of $ U(X)$. Here $X$ is a Lie algebra over some field $C \supseteq K$.
Strong multiplicity theorems for ${\rm GL}(n)$
George T.
Gilbert
561-576
Abstract: Let $\pi = \otimes {\pi _\upsilon }$ be a cuspidal automorphic representation of $GL(n,{F_A})$, where ${F_A}$ denotes the adeles of a number field $ F$. Let $E$ be a Galois extension of $ F$ and let $\{ g\}$ denote a conjugacy class of the Galois group. The author considers those cuspidal automorphic representations which have local components ${\pi _\upsilon }$ whenever the Frobenius of the prime $\upsilon$ is $\{ g\}$, showing that such representations are often easily described and finite in number. This generalizes a result of Moreno [Bull. Amer. Math. Soc. 11 (1984), pp. 180-182].
Proof of a conjecture of Kostant
Dragomir Ž.
Đoković
577-585
Abstract: Let ${\mathfrak{g}_0} = {\mathfrak{k}_0} + {\mathfrak{p}_0}$ be a Cartan decomposition of a semisimple real Lie algebra and $\mathfrak{g} = \mathfrak{k} + \mathfrak{p}$ its complexification. Denote by $G$ the adjoint group of $ \mathfrak{g}$ and by ${G_0},K,{K_0}$ the connected subgroups of $ G$ with respective Lie algebras $ {\mathfrak{g}_0},\mathfrak{k},{\mathfrak{k}_0}$. A conjecture of Kostant asserts that there is a bijection between the $ {G_0}$-conjugacy classes of nilpotent elements in $ {\mathfrak{g}_0}$ and the $ K$-orbits of nilpotent elements in $ \mathfrak{p}$ which is given explicitly by the so-called Cayley transformation. This conjecture is proved in the paper.
Asymptotic behavior and traveling wave solutions for parabolic functional-differential equations
Klaus W.
Schaaf
587-615
Abstract: This paper is a generalization of the theory of the KPP and bistable nonlinear diffusion equations. It is shown that traveling wave solutions exist for nonlinear parabolic functional differential equations (FDEs) which behave very much like the well-known solutions of the classical KPP and bistable equations. Among the techniques used are maximum principles, sub- and supersolutions, phase plane techniques for FDEs and perturbation of linear operators.
VMO, ESV, and Toeplitz operators on the Bergman space
Ke He
Zhu
617-646
Abstract: This paper studies the largest ${C^*}$-subalgebra $Q$ of ${L^\infty }({\mathbf{D}})$ such that the Toeplitz operators ${T_f}$ on the Bergman space $L_a^2({\mathbf{D}})$ with symbols $f$ in $Q$ have a symbol calculus modulo the compact operators. $Q$ is characterized by a condition of vanishing mean oscillation near the boundary. I also give several other necessary and sufficient conditions for a bounded function to be in $Q$. After decomposing $Q$ in a "nice" way, I study the Fredholm theory of Toeplitz operators with symbols in $Q$. The essential spectrum of ${T_f}(f \in Q)$ is shown to be connected and computable in terms of the Stone-Cěch compactification of $ {\mathbf{D}}$. The results in this article partially answer a question posed in [3] and give several new necessary and sufficient conditions for a bounded analytic function on the open unit disc to be in the little Bloch space ${\mathcal{B}_0}$.
The connectedness of the group of automorphisms of $L\sp 1(0,1)$
F.
Ghahramani
647-659
Abstract: For each of the radical Banach algebras $ {L^1}(0,1)$ and $ {L^1}(w)$ an integral representation for the automorphisms is given. This is used to show that the groups of the automorphisms of ${L^1}(0,1)$ and ${L^1}(w)$ endowed with bounded strong operator topology (BSO) are arcwise connected. Also it is shown that if $ \vert\vert\vert \cdot \vert\vert{\vert _p}$ denotes the norm of $B({L^p}(0,1)$, $ {L^1}(0,1))$, $1 < p \leq \infty$, then the group of automorphisms of $ {L^1}(0,1)$ topologized by $\vert\vert\vert \cdot \vert\vert{\vert _p}$ is arcwise connected. It is shown that every automorphism $\theta$ of $ {L^1}(0,1)$ is of the form $\theta = {e^{\lambda d}}{\operatorname{lim}}{e^{qn}}({\text{BSO}})$, where each ${q_n}$ is a quasinilpotent derivation. It is shown that the group of principal automorphisms of $ {l^1}(w)$ under operator norm topology is arcwise connected, and every automorphism has the form $ {e^{i\alpha d}}{({e^{\lambda d}}{e^D}{e^{ - \lambda d}})^ - }$, where $\alpha \in {\mathbf{R}}$, $\lambda > 0$, and $D$ is a derivation, and where $ {({e^{\lambda d}}{e^D}{e^{ - \lambda d}})^ - }$ denotes the extension by continuity of ${e^{\lambda d}}{e^D}{e^{ - \lambda d}}$ from a dense subalgebra of ${l^1}(w)$ to ${l^1}(w)$.
Convex subcones of the contingent cone in nonsmooth calculus and optimization
Doug
Ward
661-682
Abstract: The tangential approximants most useful in nonsmooth analysis and optimization are those which lie "between" the Clarke tangent cone and the Bouligand contigent cone. A study of this class of tangent cones is undertaken here. It is shown that although no convex subcone of the contingent cone has the isotonicity property of the contingent cone, there are such convex subcones which are more "accurate" approximants than the Clarke tangent cone and possess an associated subdifferential calculus that is equally strong. In addition, a large class of convex subcones of the contingent cone can replace the Clarke tangent cone in necessary optimality conditions for a nonsmooth mathematical program. However, the Clarke tangent cone plays an essential role in the hypotheses under which these calculus rules and optimality conditions are proven. Overall, the results obtained here suggest that the most complete theory of nonsmooth analysis combines a number of different tangent cones.
Degrees of splittings and bases of recursively enumerable subspaces
R. G.
Downey;
J. B.
Remmel;
L. V.
Welch
683-714
Abstract: This paper analyzes the interrelationships between the (Turing) of r.e. bases and of r.e. splittings of r.e. vector spaces together with the relationship of the degrees of bases and the degrees of the vector spaces they generate. For an r.e. subspace $V$ of $ {V_\infty }$, we show that $ \alpha$ is the degree of an r.e. basis of $V$ iff $\alpha$ is the degree of an r.e. summand of $ V$ iff $\alpha$ is the degree and dependence degree of an r.e. summand of $V$. This result naturally leads to explore several questions regarding the degree theoretic properties of pairs of summands and the ways in which bases may arise.
On the central limit theorem for dynamical systems
Robert
Burton;
Manfred
Denker
715-726
Abstract: Given an aperiodic dynamical system $ (X,T,\mu )$ then there is an $f \in {L^2}(\mu )$ with $\smallint fd\mu = 0$ satisfying the Central Limit Theorem, i.e. if $ {S_m}f = f + f \circ T + \cdots + f \circ {T^{m - 1}}$ and $ {\sigma _m} = {\left\Vert {{S_m}f} \right\Vert _2}$ then $\displaystyle \mu \left\{ {x\vert\frac{{{S_m}f(x)}}{{{\sigma _m}}} < u} \right\... ...fty }^u {{\text{exp}}} \left[ {\frac{{ - {\upsilon ^2}}}{2}} \right]d\upsilon .$ The analogous result also holds for flows.
On the Stickelberger ideal and the relative class number
Tatsuo
Kimura;
Kuniaki
Horie
727-739
Abstract: Let $k$ be any imaginary abelian field, $ R$ the integral group ring of $G = {\text{Gal}}(k/\mathbb{Q})$, and $S$ the Stickelberger ideal of $k$. Roughly speaking, the relative class number ${h^ - }$ of $k$ is expressed as the index of $S$ in a certain ideal $A$ of $R$ described by means of $G$ and the complex conjugation of $k;{c^ - }{h^ - } = [A:S]$, with a rational number ${c^ - }$ in $\frac{1} {2}\mathbb{N} = \{ n/2;n \in \mathbb{N}\}$, which can be described without ${h^ - }$ and is of lower than $ {h^ - }$ if the conductor of $k$ is sufficiently large (cf. [6, 9, 10]; see also [5]). We shall prove that $2{c^ - }$, a natural number, divides $ 2{([k:\mathbb{Q}]/2)^{[k:\mathbb{Q}]/2}}$. In particular, if $ k$ varies through a sequence of imaginary abelian fields of degrees bounded, then ${c^ - }$ takes only a finite number of values. On the other hand, it will be shown that ${c^ - }$ can take any value in $\frac{1} {2}\mathbb{N}$ when $k$ ranges over all imaginary abelian fields. In this connection, we shall also make a simple remark on the divisibility for the relative class number of cyclotomic fields.
Une minoration de la norme de l'op\'erateur de Cauchy sur les graphes lipschitziens
Guy
David
741-750
Abstract: It was shown by T. Murai that the norm of the operator defined by the Cauchy kernel on the graph of a Lipschitz function $ A$ is less than
$K$-theory and right ideal class groups for HNP rings
Timothy J.
Hodges
751-767
Abstract: Let $R$ be an hereditary Noetherian prime ring, let $S$ be a "Dedekind closure" of $R$ and let $ \mathcal{T}$ be the category of finitely generated $S$-torsion $R$-modules. It is shown that for all $i \geq 0$, there is an exact sequence $0 \to {K_i}(\mathcal{T}) \to {K_i}(R) \to {K_i}(S) \to 0$. If $i = 0$, or $R$ has finitely many idempotent ideals then this sequence splits. A notion of "right ideal class group" is then introduced for hereditary Noetherian prime rings which generalizes the standard definition of class group for hereditary orders over Dedekind domains. It is shown that there is a decomposition $ {K_0}(R) \cong {\text{Cl}}(R) \oplus F$ where $F$ is a free abelian group whose rank depends on the number of idempotent maximal ideals of $ R$. Moreover there is a natural isomorphism ${\text{Cl}}(R) \cong {\text{Cl}}(S)$ and this decomposition corresponds closely to the splitting of the above exact sequence for ${K_0}$.
The normal subgroup structure of the Picard group
Benjamin
Fine;
Morris
Newman
769-786
Abstract: The Picard group $ \Gamma$ is $PS{L_2}(Z[i])$, the group of linear fractional transformations with Gaussian integer coefficients. We examine the structure of the normal subgroups of $ \Gamma$. In particular we give a complete classification of the normal subgroups for indices less than $60$ and show that beyond this there are large gaps in the possible indices. This classification depends on the structure of the derived series. Finally we give examples of normal noncongruence subgroups.
An infinite-dimensional Hamiltonian system on projective Hilbert space
Anthony M.
Bloch
787-796
Abstract: We consider here the explicit integration of a Hamiltonian system on infinite-dimensional complex projective space. The Hamiltonian, which is the restriction of a linear functional to this projective space, arises in the problem of line fitting in complex Hilbert space (or, equivalently, the problem of functional approximation) or as the expectation value of a model quantum mechanical system. We formulate the system here as a Lax system with parameter, showing how this leads to an infinite set of conserved integrals associated with the problem and to an explicit formulation of the flow in action-angle form via an extension of some work of J. Moser. In addition, we find the algebraic curve naturally associated with the system.
Holomorphic mappings on $l\sb 1$
Raymond A.
Ryan
797-811
Abstract: We describe the holomorphic mappings of bounded type, and the arbitrary holomorphic mappings from the complex Banach space $ {l_1}$ into a complex Banach space $X$. It is shown that these mappings have monomial expansions and the growth of the norms of the coefficients is characterized in each case. This characterization is used to give new descriptions of the compact open topology and the Nachbin ported topology on the space $ \mathcal{H}({l_1};X)$ of holomorphic mappings, and to prove a lifting property for holomorphic mappings on ${l_1}$. We also show that the monomials form an equicontinuous unconditional Schauder basis for the space $(\mathcal{H}({l_1}),{\tau _0})$ of holomorphic functions on ${l_1}$ with the topology of uniform convergence on compact sets.