Hermitian Lie algebras and metaplectic representations. I
Shlomo
Sternberg;
Joseph A.
Wolf
1-43
Abstract: A notion of ``hermitian Lie algebra'' is introduced which relates ordinary and graded Lie algebra structures. In the case of real-symplectic and arbitrary-signature-unitary Lie algebras, it leads to an analysis of the minimal dimensional coadjoint orbits, and then to the metaplectic representations and their restrictions to unitary groups of arbitrary signature and parabolic subgroups of these unitary groups.
Examples of nonintegrable analytic Hamiltonian vector fields with no small divisors
R.
Cushman
45-55
Abstract: Any analytic symplectic diffeomorphism $\Phi$ of a symplectic manifold M is the Poincaré map of a real analytic Hamiltonian vector field ${X_H}$. If $\Phi$ does not have an analytic integral, then $ {X_H}$ has no analytic integral which is not a power series in H. Let $M = {{\mathbf{R}}^2}$. If $ \Phi$ has a finite contact homoclinic point, then $\Phi$ is nonintegrable. Also Moser's polynomial mapping is nonintegrable.
On the number of real zeros of a random trigonometric polynomial
M.
Sambandham
57-70
Abstract: For the random trigonometric polynomial $\displaystyle \sum\limits_{n = 1}^N {{g_n}(t)\cos n\theta ,}$ where ${g_n}(t),0 \leqslant t \leqslant 1$, are dependent normal random variables with mean zero, variance one and joint density function ${M^{ - 1}}$ is the moment matrix with ${\rho _{ij}} = \rho ,0 < \rho < 1,i \ne j,i,j = 1,2, \ldots ,N$ and $\bar a$ is the column vector, we estimate the probable number of zeros.
Bounded point evaluations and smoothness properties of functions in $R\sp{p}(X)$
Edwin
Wolf
71-88
Abstract: Let X be a compact subset of the complex plane C. We denote by $ {R_0}(X)$ the algebra consisting of the (restrictions to X of) rational functions with poles off X. Let m denote 2-dimensional Lebesgue measure. For $p \geqslant 1$, let ${L^p}(X) = {L^p}(X,dm)$. The closure of $ {R_0}(X)$ in $ {L^p}(X)$ will be denoted by ${R^p}(X)$. Whenever p and q both appear, we assume that $ 1/p + 1/q = 1$. If x is a point in X which admits a bounded point evaluation on ${R^p}(X)$, then the map which sends f to $ f(x)$ for all $f \in {R_0}(X)$ extends to a continuous linear functional on ${R^p}(X)$. The value of this linear functional at any $f \in {R^p}(X)$ is denoted by $ f(x)$. We examine the smoothness properties of functions in ${R^p}(X)$ at those points which admit bounded point evaluations. For $p > 2$ we prove in Part I a theorem that generalizes the ``approximate Taylor theorem'' that James Wang proved for $R(X)$. In Part II we generalize a theorem of Hedberg about the convergence of a certain capacity series at a point which admits a bounded point evaluation. Using this result, we study the density of the set X at such a point.
Representation theory of algebras stably equivalent to an hereditary Artin algebra
María Inés
Platzeck
89-128
Abstract: Two artin algebras are stably equivalent if their categories of finitely generated modules modulo projectives are equivalent. The author studies the representation theory of algebras stably equivalent to hereditary algebras, using the notions of almost split sequences and irreducible morphisms. This gives a new unified approach to the theories developed for hereditary and radical square zero algebras by Gabriel, Gelfand, Bernstein, Ponomarev, Dlab, Ringel and Müller, as well as other algebras not covered previously. The techniques are purely module theoretical and do not depend on representations of diagrams. They are similar to those used by M. Auslander and the author to study hereditary algebras.
Weak Chebyshev subspaces and continuous selections for the metric projection
Günther
Nürnberger;
Manfred
Sommer
129-138
Abstract: Let G be an n-dimensional subspace of $C[a,b]$. It is shown that there exists a continuous selection for the metric projection if for each f in $C[a,b]$ there exists exactly one alternation element ${g_f}$, i.e., a best approximation for f such that for some $a \leqslant {x_0} < \cdots < {x_n} \leqslant b$, $\displaystyle \varepsilon {( - 1)^i}(f - {g_f})({x_i}) = \left\Vert {f - {g_f}} \right\Vert,\quad i = 0, \ldots ,n,\varepsilon = \pm 1.$ Further it is shown that this condition is fulfilled if and only if G is a weak Chebyshev subspace with the property that each g in G, $g \ne 0$, has at most n distinct zeros. These results generalize in a certain sense results of Lazar, Morris and Wulbert for $n = 1$ and Brown for $n = 5$.
A Hopf global bifurcation theorem for retarded functional differential equations
Roger D.
Nussbaum
139-164
Abstract: We prove a result concerning the global nature of the set of periodic solutions of certain retarded functional differential equations. Our main theorem is an analogue, for retarded F.D.E.'s, of a result by J. Alexander and J. Yorke for ordinary differential equations.
Analytic left algebraic groups. II
Andy R.
Magid
165-177
Abstract: An analytic left algebraic group is a complex analytic group carrying a structure of affine algebraic variety such that left translations by fixed elements are morphisms. The core of such a group is the (algebraic) subgroup of all elements such that right translation by them is a morphism. It is shown that the core determines the left algebraic structure, and this is used to determine when left algebraic structures are conjugate by inner automorphisms.
Bender groups as standard subgroups
Robert L.
Griess;
David R.
Mason;
Gary M.
Seitz
179-211
Abstract: A subgroup X of a finite group G is called $^ \ast $-standard if $\tilde X = X/O(X)$ is quasisimple, $Y = {C_G}(X)$ is tightly embedded in G and ${N_G}(X) = {N_G}(Y)$. This generalizes the notion of standard subgroups. Theorem. Let G be a finite group with $O(G) = 1$. Suppose X is $^ \ast$-standard in G and $\tilde X/Z(\tilde X) \cong {L_2}({2^n}),{U_3}({2^n})$ or $ {\text{Sz}}({2^n})$. Assume $ X \ntriangleleft G$. Then $O(X) = 1$ and one of the following holds: $({\text{i}})\;E(G) \cong X \times X$. $ ({\text{ii}})\;X \cong {L_2}({2^n})$ and $E(G) \cong {L_2}({2^{2n}}),{U_3}({2^n})\;or\;{L_3}({2^n})$. $({\text{iii}})\;X \cong {U_3}({2^n})$ and $ E(G) \cong {L_3}({2^{2n}})$. $({\text{iv}})\;X \cong {\text{Sz}}({2^n})$ and $ E(G) \cong {\text{Sp}}(4,{2^n})$. $({\text{v}})\;X \cong {L_2}(4)$ and $E(G) \cong {M_{12}},{A_9},{J_1},{J_2},{A_7},{L_2}(25),{L_3}(5)\;or\;{U_3}(5)$. $ ({\text{vi}})\;X \cong {\text{Sz}}(8)$ and $E(G) \cong {\text{Ru}}$ (the Rudvalis group). $({\text{vii}})\;X \cong {L_2}(8)$ and $E(G) \cong {G_2}(3)$. $ ({\text{viii}})\;X \cong {\text{SL}}(2,5)$ and G has sectional 2-rank at most 4. In particular, if G is simple, $ G \cong {M_{12}},{A_9},{J_1},{J_2},{\text{Ru}},{U_3}(5),{L_3}(5),{G_2}(5), or\;{^3}{D_4}(5)$.
Maximum principles, gradient estimates, and weak solutions for second-order partial differential equations
William
Bertiger
213-227
Abstract: Weak solutions to second order elliptic equations and the first derivatives of these solutions are shown to satisfy $ {L^p}$ bounds. Classical second order equations with nonnegative characteristic form are also considered. It is proved that auxiliary functions of the gradient of a solution must satisfy a maximum principle. This result is extended to higher order derivatives and systems.
On a degenerate principal series of representations of ${\rm U}(2, 2)$
Yang
Hua
229-252
Abstract: A degenerate principal series of representations $T(\rho ,m; \cdot ),(\rho ,m) \in {\mathbf{R}} \times {\mathbf{Z}}$, of $U(2,2)$, is realized on the Hilbert space of all square integrable functions on the space X of $2 \times 2$ Hermitian matrices. Using Fourier analysis, gamma functions, and Mellin analysis, we spectrally analyze the operator equation $AT(\rho ,m;g) = T(\rho ,m;g)A$ for all $g \in \mathfrak{G} = U(2,2)$ on an invariant subspace of ${L^2}(X)$, and obtain the first main result: For $\rho \ne 0$ or m odd, $T(\rho ,m; \cdot )$ is irreducible. Then we define certain integral transforms on ${L^2}(X)$ the analytic continuation of which leads to the second main result: $T(0,2n; \cdot )$ is reducible.
On the structure of principal ideals of operators
G. D.
Allen;
L. C.
Shen
253-270
Abstract: This paper considers various types of principal ideals generated by single compact operators on a separable Hilbert space. In particular, necessary and sufficient condtions that a principal ideal be normable are given. Relations between principal ideals and duals of Lorentz and Orlicz spaces are also given. All conditions are expressed using the singular numbers of the operator.
Growth hyperspaces of Peano continua
D. W.
Curtis
271-283
Abstract: For X a nondegenerate Peano continuum, let ${2^X}$ be the hyperspace of all nonempty closed subsets of X, topologized with the Hausdorff metric. It is known that ${2^X}$ is homeomorphic to the Hilbert cube. A nonempty closed subspace $ \mathcal{G}$ of $ {2^X}$ is called a growth hyperspace provided it satisfies the following condition: if $A \in \mathcal{G}$, and $B \in {2^X}$ such that $B \supset A$ and each component of B meets A, then also $B \in \mathcal{G}$. The class of growth hyperspaces includes many previously considered subspaces of $ {2^X}$. It is shown that if X contains no free arcs, and $\mathcal{G}$ is a nontrivial growth hyperspace, then $\mathcal{G}\backslash \{ X\}$ is a Hilbert cube manifold. A corollary characterizes those growth hyperspaces which are homeomorphic to the Hilbert cube. Analogous results are obtained for growth hyperspaces with respect to the hyperspace ${\text{cc}}(X)$ of closed convex subsets of a convex n-cell X.
Periodic solutions for a differential equation in Banach space
James H.
Lightbourne
285-299
Abstract: Suppose X is a Banach space, $ \Omega \subset X$ is closed and convex, and $A:[0,\infty ) \times \Omega \to X$ is continuous. Then if $\displaystyle \mathop {\lim }\limits_{h \to 0} \vert x + hA(t,x);\Omega \vert/h = 0\quad {\text{for}}\;{\text{all}}\;(t,x) \in [0,\infty ) \times \Omega ,$ there exist approximate solutions to the initial value problem $A(t,x) = B(t,x) + C(t,x)$, where B satisfies a dissipative condition and C is compact, we obtain a growth estimate on the measure of noncompactness of trajectories for a class of approximate solutions. This estimate is employed to obtain existence of periodic solutions to (IVP).
Translation planes of order $q\sp{2}$: asymptotic estimates
Gary L.
Ebert
301-308
Abstract: R. H. Bruck has pointed out the one-to-one correspondence between the isomorphism classes of certain translation planes, called subregular, and the equivalence classes of disjoint circles in a finite miquelian inversive plane $ IP(q)$. The problem of determining the number of isomorphism classes of translation planes is old and difficult. Let q be an odd prime-power. In this paper, a study of sets of disjoint circles in $IP(q)$ enables the author to find an asymptotic estimate of the number of isomorphism classes of translation planes of order ${q^2}$ which are subregular of index 3 or 4. It is conjectured (and proved for $n \leqslant 3$) that, given a set of n disjoint circles in $IP(q)$, the numbers of circles disjoint from each of the given n circles is asymptotic to ${q^3}/{2^n}$. This conjecture, if true, would allow one to estimate the number of subregular translation planes of order ${q^2}$ with any positive index.
Central twisted group algebras
Harvey A.
Smith
309-320
Abstract: A twisted group algebra $ {L^1}(A,G;T,\alpha )$ is central iff T is trivial and A commutative. (Group algebras of central extension of G are such.) We show that if ${H^2}(G)$ is discrete any central ${L^1}(A,G;\alpha )$ is a direct sum of closed ideals ${L^1}({A_i},G;{\alpha _i})$ having as duals fibre bundles over the duals of closed ideals ${A_i}$ in A, with fibres projective duals of G, and principal ${G^\wedge}$ bundles (where ${G^\wedge}$ denotes the group of characters of G) satisfying the conditions which define characteristic bundles for G abelian. (If G is compact ${H^2}(G)$ is always discrete, the direct sum is countable, and the bundles are locally trivial.) Applications are made to the duals of central extensions of groups and in particular to duals of ``central'' groups. For G commutative, ${H^2}(G)$ discrete, and A a $ {C^\ast}$-algebra with identity, all central twisted group algebras ${L^1}(A,G;\alpha )$ (and their duals) are classified in purely algebraic terms involving ${H^2}(G)$, the group G, and the first Čech cohomology group of the dual of A. This result allows us, in principle, to construct all the central $ {L^1}(A,G;\alpha )$ and their duals where A is a ${C^\ast}$-algebra with identity and G a compact commutative group.
Galerkin methods in the constructive solvability of nonlinear Hammerstein equations with applications to differential equations
P. M.
Fitzpatrick;
W. V.
Petryshyn
321-340
Abstract: We consider the solution of abstract Hammerstein equations by means of a Galerkin approximating scheme. The convergence of the scheme is proven by first establishing an equivalent scheme in a Hilbert space and then proving a convergence result for firmly monotone operators in a Hilbert space. The general results are applied to the case when the involved linear mapping is angle-bounded, and also to the treatment of certain differential equations.
Generic properties of eigenfunctions of elliptic partial differential operators
Jeffrey H.
Albert
341-354
Abstract: The problem considered here is that of describing generically the zeros, critical points and critical values of eigenfunctions of elliptic partial differential operators. We consider operators of the form $L + \rho$, where L is a fixed, second-order, selfadjoint, $ {C^\infty }$ linear elliptic partial differential operator on a compact manifold (without boundary) and $\rho$ is a $ {C^\infty }$ function. It is shown that, for almost all $\rho$, i.e. for a residual set, the eigenvalues of $L + \rho$ are simple and the eigenfunctions have the following properties: (1) they are Morse functions; (2) distinct critical points have distinct critical values; (3) 0 is not a critical value.
Semimodular functions and combinatorial geometries
Hien Quang
Nguyen
355-383
Abstract: A point-lattice $\mathfrak{L}$ being given, to any normalized, nondecreasing, integer-valued, semimodular function f defined on $ \mathfrak{L}$, we can associate a class of combinatorial geometries called expansions of f. The family of expansions of f is shown to have a largest element for the weak map order, $E(f)$, the free expansion of f. Expansions generalize and clarify the relationship between two known constructions, one defined by R. P. Dilworth, the other by J. Edmonds and G.-C. Rota. Further applications are developed for solving two extremal problems of semimodular functions: characterizing (1) extremal rays of the convex cone of real-valued, nondecreasing, semimodular functions defined on a finite set; (2) combinatorial geometries which are extremal for the decomposition into a sum.
Taming and the Poincar\'e conjecture
T. L.
Thickstun
385-396
Abstract: L. Glaser and L. Siebenmann have shown that the double suspension of a homotopy 3-sphere is homeomorphic to the 5-sphere. This result, together with a well-known characterization of $ {S^3}$ due to R. H. Bing, is used to establish a relationship between the Poincaré conjecture and two conjectures concerned with taming embeddings in higher dimensions. One of the two conjectures, each of which implies the Poincaré conjecture, states, in effect, that a codimension two sphere is tame if it is tame ``modulo'' a tame disk contained in it.
A Gross measure property
Lawrence R.
Ernst
397-406
Abstract: We prove that there exists a subset E of $[0,1] \times {{\mathbf{R}}^2}$ such that the 2-dimensional Gross measure of E is 0, while the 1-dimensional Gross measure of $\{ z:(y,z) \in E\}$ is positive for all $y \in [0,1]$. It is known that for Hausdorff measures no set exists satisfying these conditions.