The $2$-transitive permutation representations of the finite Chevalley groups
Charles W.
Curtis;
William M.
Kantor;
Gary M.
Seitz
1-59
Abstract: The permutation representations in the title are all determined, and no surprises are found to occur.
Mean convergence of Fourier series on compact Lie groups
Robert J.
Stanton
61-87
Abstract: The main result is an ${L^p}$ mean convergence theorem for the partial sums of the Fourier series of a class function on a compact semi-simple Lie group. A central element in the proof is a Lie group-Lie algebra analog of the theorems in classical Fourier analysis that allow one to pass back and forth between multiplier operators for Fourier series in several variables and multiplier operators for the Fourier transform in Euclidean space. To obtain the ${L^p}$ mean convergence theorem, the theory of the Hilbert transform with weight function is needed.
Reversible diffeomorphisms and flows
Robert L.
Devaney
89-113
Abstract: We generalize the classical notion of reversibility of a mechanical system. The generic qualitative properties of symmetric orbits of such systems are studied using transversality theory. In particular, we prove analogues of the closed orbit, Liapounov, and homoclinic orbit theorems for R-reversible systems.
On polar relations of abstract homogeneous polynomials
Neyamat
Zaheer
115-131
Abstract: In this paper we generalize, to vector spaces over algebraically closed fields of characteristic zero, two well-known classical results due to Laguerre and Grace, concerning, respectively, the relative location of the zeros of a complex-valued polynomial and its polar-derivative and the relative location of the zeros of two apolar polynomials. Vector space analogues of their results were generalized, to a certain degree, by Hörmander, Marden, and Zervos. Our results in this paper further generalize their results and, in the complex plane, improve upon those of Laguerre and Grace. Besides, the present treatment unifies their completely independent approaches into an improved and more systematic and abstract theory. We have also shown that our results are best possible in the sense that they cannot be further generalized in certain directions.
On the structure of certain subalgebras of a universal enveloping algebra
Bertram
Kostant;
Juan
Tirao
133-154
Abstract: The representation theory of a semisimple group G, from an algebraic point of view, reduces to determining the finite dimensional representation of the centralizer ${U^\mathfrak{k}}$ of the maximal compact subgroup K of G in the universal enveloping algebra U of the Lie algebra $ \mathfrak{g}$ of G. The theory of spherical representations has been determined in this way since by a result of Harish-Chandra $ {U^\mathfrak{k}}$ modulo a suitable ideal I is isomorphic to the ring of Weyl group W invariants $U{(\mathfrak{a})^W}$ in a suitable polynomial ring $U(\mathfrak{a})$. To deal with the general case one must determine the image of ${U^\mathfrak{k}}$ in $U(\mathfrak{k}) \otimes U(\mathfrak{a})$, where $\mathfrak{k}$ is the Lie algebra of K. We prove that if W is replaced by the Kunze-Stein intertwining operators $\tilde W$ then $ {U^\mathfrak{k}}$ suitably localized and completed is indeed isomorphic to $U(\mathfrak{k}) \otimes U{(\mathfrak{a})^{\tilde W}}$ suitably localized and completed.
Hermite series as boundary values
G. G.
Walter
155-171
Abstract: The relations between Hermite series expansions of functions and tempered distributions on the real axis and holomorphic or harmonic functions or generalizations of them in the upper half plane are studied. The Hermite series expansions of $ {H^2}$ functions are characterized in terms of their coefficients. Series of analytic representations of Hermite functions, series of Hermite functions of the second kind, and combined series of Hermite functions of the first and second kind are investigated. The functions to which these series converge in the upper half plane are shown to approach (in various ways) the distributions or functions whose Hermite series have the same coefficients.
Analytic hypoellipticity of certain second-order evolution equations with double characteristics
Mario
Tosques
173-196
Abstract: The present article establishes the analytic hypoellipticity (Definition 1.2) of a class of abstract evolution equations of order two, with double characteristics, under the hypothesis that the coefficients are analytic (in a suitable sense; see §2). The noteworthy feature of the main result (Theorem 4.1) is that analytic hypoellipticity holds whenever hypoellipticity does, even when one of the asymptotic eigenvalues $ {c^j}(A)$ fails to be elliptic of order one.
Piecewise monotone interpolation and approximation with Muntz polynomials
Eli
Passow;
Louis
Raymon;
Oved
Shisha
197-205
Abstract: The possibility (subject to certain restrictions) of solving the following approximation and interpolation problem with a given set of ``Muntz polynomials'' on a real interval is demonstrated: (i) approximation of a continuous function by a ``copositive'' Muntz polynomial; (ii) approximation of a continuous function by a ``comonotone'' Muntz polynomial; (iii) approximation of a continuous function with a monotone kth difference by a Muntz polynomial with a monotone kth derivative; (iv) interpolation by piecewise monotone Muntz polynomials--i. e., polynomials that are monotone on each of the intervals determined by the points of interpolation. The strong interrelationship of these problems is shown implicitly in the proofs.
Certain continua in $S\sp{n}$ of the same shape have homeomorphic complements
Vo Thanh
Liem
207-217
Abstract: As a consequence of Theorem 1 of this paper, we see that if X and Y are globally 1-alg continua in ${S^n}\;(n \geqslant 5)$ having the shape of the real projective space ${P^k}\;(k \ne 2,2k + 2 \leqslant n)$, then ${S^n} - X \approx {S^n} - Y$. (For ${P^1} = {S^1}$, this establishes the last case of such a result for spheres.) We also show that if X and Y are globally 1-alg continua in ${S^n},n \geqslant 6$, which have the shape of a codimension $ \geqslant 3$, closed, $0 < (2m - n + 1)$-connected, PL-manifold $ {M^m}$, then ${S^n} - X \approx {S^n} - Y$.
Positive definite measures with applications to a Volterra equation
Olof J.
Staffans
219-237
Abstract: We study the asymptotic behavior of the solutions of the nonlinear Volterra integrodifferential equation
Tauberian theorems for a positive definite form, with applications to a Volterra equation
Olof J.
Staffans
239-259
Abstract: We study the relation between the condition $\displaystyle \mathop {\sup }\limits_{T > 0} \int_{[0,T]} {\bar \varphi (t)} \int_{[t - T,t]} {\varphi (t - s)\;dv (s)\;dt < \infty }$ and the asymptotic behavior of the bounded function $\varphi$ when $\nu$ is a positive definite measure. Earlier we have proved that if $\nu$ is strictly positive definite and $\varphi$ satisfies a tauberian condition, then $ \varphi (t) \to 0$ as $t \to \infty$. Here we characterize the spectrum of the limit set of $\varphi$ in the case when $\nu$ is not strictly positive definite. Applying this theory to a nonlinear Volterra equation we get some new results on the asymptotic behavior of its bounded solutions.
Pointwise bounded approximation and analytic capacity of open sets
Steven
Jacobson
261-283
Abstract: We examine the semi-additivity question for analytic capacity by studying the relation between the capacities of bounded open sets and their closures.
Hyperfinite extensions of bounded operators on a separable Hilbert space
L. C.
Moore
285-295
Abstract: Let H be a separable Hilbert space and Ĥ the nonstandard hull of H with respect to an ${\aleph _1}$-saturated enlargement. Let S be a $^\ast$-finite dimensional subspace of $^\ast H$ such that the corresponding hyperfinite dimensional subspace Ŝ of Ĥ contains H. If T is a bounded operator on H, then an extension  of T to Ŝ where  is obtained from an internal $^\ast$-linear operator on S is called a hyperfinite extension of T. It is shown that T has a compact (selfadjoint) hyperfinite extension if and only if T is compact (selfadjoint). However T has a normal hyperfinite extension if and only if T is subnormal. The spectrum of a hyperfinite extension  equals the point spectrum of Â, and if T is quasitriangular, A can be chosen so that the spectrum of  equals the spectrum of T. A simple proof of the spectral theorem for bounded selfadjoint operators is given using a hyperfinite extension.
Principal quadratic functionals
E. C.
Tomastik
297-309
Abstract: The general theory of the existence of a minimum limit in the fixed and variable end point problems of singular quadratic functionals of n dependent variables is developed, generalizing the one dimensional results of Marston Morse, Walter Leighton and A. D. Martin and completing a phase of the n-dimensional theory initiated recently by the author.
Mean convergence of generalized Walsh-Fourier series
Wo Sang
Young
311-320
Abstract: Paley proved that Walsh-Fourier series converges in ${L^p}(1 < p < \infty )$. We generalize Paley's result to Fourier series with respect to characters of countable direct products of finite cyclic groups of arbitrary orders.
Some results on orientation preserving involutions
David E.
Gibbs
321-332
Abstract: The bordism of orientation preserving differentiable involutions is studied by use of the signature-like invariant ${\text{ab}}: {\mathcal{O}_\ast}({Z_2}) \to {W_0}({Z_2};Z)$. The equivariant Witt ring ${W_0}({Z_2};Z)$ is calculated and is shown to be isomorphic under ab to the effective part of ${\mathcal{O}_4}({Z_2})$. Modulo 2 relations are established between the representation of the involution on $ {H^{2k}}({M^{4k}};Z)/{\operatorname{torsion}}$ and $ {\chi _0}(F)$ and ${\chi _2}(F)$, where $ {\chi _i}(F)$ is the Euler characteristic of those components of the fixed point set with dimensions congruent to i modulo 4. For manifolds of dimension $4k + 2$, it is shown that $ {\chi _0}(F) \equiv {\chi _2}(F) \equiv 0\;(\bmod 2)$. Finally the ideal ${E_0}({Z_2};Z)$ consisting of those elements of $ {W_0}({Z_2};Z)$ admitting a representative of type II is determined.
Total mean curvature of immersed surfaces in $E\sp{m}$
Bang-yen
Chen
333-341
Abstract: Total mean curvature and value-distribution of mean curvature for certain pseudo-umbilical surfaces are studied.
Slowly varying functions in the complex plane
Monique
Vuilleumier
343-348
Abstract: Let f be analytic and have no zeros in $\vert\arg z\vert < \alpha \leqslant \pi$; f is called slowly varying if, for every $ \lambda > 0,f(\lambda z)/f(z) \to 1$ uniformly in $\vert\arg z\vert \leqslant \beta < \alpha$, when $\vert z\vert \to \infty$. One shows that f is slowly varying if and only if $\vert\arg z\vert \leqslant \beta < \alpha$, when $\vert z\vert \to \infty$.
Cohomology of finite covers
Allan
Calder
349-352
Abstract: For a finite dimensional CW-complex, X, and $q > 1$, it is shown that the qth Čech cohomology group based on finite open covers of X, $H_f^q(X)$, is naturally isomorphic to ${H^q}(X)$, the qth Čech cohomology of X (i.e. based on locally finite covers), and for reasonable X, ${H^1}(X)$ can be obtained algebraically from $ H_f^1(X)$.
Isotropic transport process on a Riemannian manifold
Mark A.
Pinsky
353-360
Abstract: We construct a canonical Markov process on the tangent bundle of a complete Riemannian manifold, which generalizes the isotropic scattering transport process on Euclidean space. By inserting a small parameter it is proved that the transition semigroup converges to the Brownian motion semigroup provided that the latter preserves the class $ {C_0}$. The special case of a manifold of negative curvature is considered as an illustration.
Representations of the $l\sp{1}$-algebra of an inverse semigroup
Bruce A.
Barnes
361-396
Abstract: In this paper the star representations on Hilbert space of the $ {l^1}$-algebra of an inverse semigroup are studied. It is shown that the set of all irreducible star representations form a separating family for the ${l^1}$-algebra. Then specific examples of star representations are constructed, and some theory of star representations is developed for the $ {l^1}$-algebra of a number of the most important examples of inverse semigroups.
Boundary value problems for second order differential equations in convex subsets of a Banach space
Klaus
Schmitt;
Peter
Volkmann
397-405
Abstract: Let E be a real Banach space, C a closed, convex subset of E and $f:[0,1] \times E \times E \to E$ be continuous. Let ${u_0},{u_1} \in C$ and consider the boundary value problem $u:[0,1] \to C$.
Erratum to: ``Homology with multiple-valued functions applied to fixed points'' (Trans. Amer. Math. Soc. {\bf 213} (1975), 407--427)
Richard
Jerrard
406