The space of framed functions
Kiyoshi
Igusa
431-477
Abstract: We define the notion of a ``framed function'' on a compact smooth manifold $ N$ and we show that the space of all framed functions on $N$ is $(\operatorname{dim} \,N - 1)$-connected. A framed function on $N$ is essentially a smooth function $N \to \mathbf{R}$ with only Morse and birth-death singularities together with certain additional structure.
Bordism of semifree circle actions on Spin manifolds
Lucília Daruiz
Borsari
479-487
Abstract: Using traditional methods in bordism theory, an almost complete description of the rational bordism groups of semifree circle actions on Spin manifolds is given. The single remaining problem, to describe the ideal of $\Omega _ \ast ^{{\operatorname{Spin}}}\, \otimes \,\mathbf{Q}$, generated by bordism classes of Spin manifolds admitting a semifree action of odd type, has been recently solved by S. Ochanine $[\mathbf{O}]$.
Induced group actions, representations and fibered skew product extensions
R. C.
Fabec
489-513
Abstract: Let $G$ be a locally compact group acting ergodically on $Y$. We introduce the notion of an action of this group action and study the notions of induced group actions, ergodicity, and fibered product extensions in this context. We also characterize fibered skew product actions built over a cocycle.
Lattice embeddings in the recursively enumerable truth table degrees
Christine Ann
Haught
515-535
Abstract: It is shown that every finite lattice, and in fact every recursively presentable lattice, can be embedded in the r.e. tt-degrees by a map preserving least and greatest elements. The decidability of the $1$-quantifier theory of the r.e. tt-degrees in the language with $\leqslant ,\, \vee ,\, \wedge ,\,0$, and 1 is obtained as a corollary.
Infinitely many traveling wave solutions of a gradient system
David
Terman
537-556
Abstract: We consider a system of equations of the form ${u_t} = {u_{xx}} + \nabla F(u)$. A traveling wave solution of this system is one of the form $u(x,\,t) = U(z),\,z = x + \theta t$. Sufficient conditions on $F(u)$ are given to guarantee the existence of infinitely many traveling wave solutions.
Stability of harmonic maps and eigenvalues of the Laplacian
Hajime
Urakawa
557-589
Abstract: The index and nullity of the Hessian of the energy for every harmonic map are estimated above by a geometric quantity. The stability theory of harmonic maps is developed and as an application, the Kähler version of the Lichnerowicz-Obata theorem about the first eigenvalue of the Laplacian is proved.
Koszul homology and the structure of low codimension Cohen-Macaulay ideals
Wolmer V.
Vasconcelos
591-613
Abstract: The relationship between the properties of the Koszul homology modules of two ideals connected by linkage is studied. If the ideal $I$ is either (i) a Cohen-Macaulay ideal of codimension 3, or (ii) a Gorenstein ideal of codimension 4, the one-dimensional Koszul module carries considerable information on the structural nature of the linkage class of $I$ in case (i), or on the conormal module of $ I$ in case (ii). Emphasis is given to the verification of the properties by computation.
Supersymmetry, twistors, and the Yang-Mills equations
Michael
Eastwood
615-635
Abstract: This article investigates a supersymmetric proof due to Witten of the twistor description of general Yang-Mills fields due to Green, Isenberg, and Yasskin. In particular, some rigor is added and the rather complicated calculations are given in detail.
Balanced subgroups of finite rank completely decomposable abelian groups
Loyiso G.
Nongxa
637-648
Abstract: It is proved that, if a finite rank completely decomposable group has extractable typeset of cardinality at most 5, all its balanced subgroups are also completely decomposable. Balanced Butler groups with extractable typeset of size at most 3 are almost completely decomposable and decompose into rank 1 and/or rank 3 indecomposable summands. We also construct an indecomposable balanced Butler group whose extractable typeset is of size 4 which fails to be almost completely decomposable.
On the M\"obius function
Helmut
Maier
649-664
Abstract: We investigate incomplete convolutions of the Möbius function of the form $\sum\nolimits_{d\vert n;d \leq z} {\mu (d)}$. It is shown that for almost all integers $n$ one can find $z$ for which this sum is large.
Pseudo-Chern classes of an almost pseudo-Hermitian manifold
Yasuo
Matsushita
665-677
Abstract: For an almost pseudo-Hermitian manifold, pseudo-Chern classes are defined on its complexified tangent bundle with the pseudo-Hermitian structure as represented by certain $ {\text{ad}}(U(p,\,q))$-invariant forms on the manifold. It is shown that such a manifold always admits an almost Hermitian structure, and hence that Chern classes are also defined on the complexified tangent bundle with such an almost Hermitian structure. A relation between the pseudo-Chern classes and the Chern classes is established. From the relation, the pseudo-Chern classes are considered as the characteristic classes which measure how the almost pseudo-Hermitian structure deviates from an almost Hermitian structure.
Quadratic geometry of numbers
Hans Peter
Schlickewei;
Wolfgang M.
Schmidt
679-690
Abstract: We give upper bounds for zeros of quadratic forms. For example we prove that for any nondegenerate quadratic form $ \mathfrak{F}({x_1}, \ldots ,\,{x_n})$ with rational integer coefficients which vanishes on a $d$-dimensional rational subspace $(d > 0)$ there exist sublattices ${\Gamma _0},\,{\Gamma _1},\, \ldots \,,{\Gamma _{n - d}}$ of $ {\mathbf{Z}^n}$ of rank $ d$, on which $\mathfrak{F}$ vanishes, with the following properties: $\displaystyle {\text{rank}}({\Gamma _0} \cap {\Gamma _i}) = d - 1,\quad {\text{rank}}({\Gamma _0} \cup {\Gamma _1} \cup \cdots \cup {\Gamma _{n - d}}) = n$ and $\displaystyle {(\det \,{\Gamma _0})^{n - d}}\det \,{\Gamma _1} \cdots \det \,{\Gamma _{n - d}} \ll {F^{{{(n - d)}^2}}}$ , where $F$ is the maximum modulus of the coefficients of $ \mathfrak{F}$.
On linear Volterra equations of parabolic type in Banach spaces
Jan
Prüss
691-721
Abstract: Linear integrodifferential equations of Volterra type in a Banach space are studied in case the main part of the equation generates an analytic ${C_0}$-semigroup. Under very general assumptions it is shown that a resolvent operator exists and that many of the solution properties of parabolic evolution equations are inherited. The results are then applied to integro-partial differential equations of parabolic type.
Scalar curvature functions in a conformal class of metrics and conformal transformations
Jean-Pierre
Bourguignon;
Jean-Pierre
Ezin
723-736
Abstract: This article addresses the problem of prescribing the scalar curvature in a conformal class. (For the standard conformal class on the $2$-sphere, this is usually referred to as the Nirenberg problem.) Thanks to the action of the conformal group, integrability conditions due to J. L. Kazdan and F. W. Warner are extended, and shown to be universal. A counterexample to a conjecture by J. L. Kazdan on the role of first spherical harmonics in these integrability conditions on the standard sphere is given. Using the action of the conformal groups, some existence results are also given.
Prime ideals in polycyclic crossed products
D. S.
Passman
737-759
Abstract: In this paper, we describe the prime ideals $P$ in crossed products $R \ast G$ with $R$ a right Noetherian ring and with $G$ a polycyclic-by-finite group. This is achieved through a series of reductions. To start with, we may assume that $P \cap R = 0$ so that $ R$ is a $G$-prime ring. The first step uses a technique of M. Lorenz and the author to reduce to a prime ring and a subgroup of finite index in $ G$. Next if $R$ is prime, then we show that the prime ideals of $R \ast G$ disjoint from $R$ are explicitly determined by the primes of a certain twisted group algebra of a normal subgroup of $G$. Finally the prime ideals in twisted group algebras of polycyclic-by-finite groups are studied by lifting the situation to ordinary group algebras where the results of J. E. Roseblade can be applied.
Positive forms and dilations
Wacław
Szymański
761-780
Abstract: By using the quadratic form and unbounded operator theory a new approach to the general dilation theory is presented. The boundedness condition is explained in terms of the Friedrichs extension of symmetric operators. Unbounded dilations are introduced and discussed. Applications are given to various problems involving positive definite functions.
Special functions of matrix argument. I. Algebraic induction, zonal polynomials, and hypergeometric functions
Kenneth I.
Gross;
Donald St. P.
Richards
781-811
Abstract: Hypergeometric functions of matrix argument arise in a diverse range of applications in harmonic analysis, multivariate statistics, quantum physics, molecular chemistry, and number theory. This paper presents a general theory of such functions for real division algebras. These functions, which generalize the classical hypergeometric functions, are defined by infinite series on the space $S = S(n,\,\mathbf{F})$ of all $n \times n$ Hermitian matrices over the division algebra $ \mathbf{F}$. The theory depends intrinsically upon the representation theory of the general linear group $G = GL(n,\,\mathbf{F})$ of invertible $n \times n$ matrices over $ \mathbf{F}$, and the theme of this work is the full exploitation of the inherent group theory. The main technique is the use of the method of ``algebraic induction'' to realize explicitly the appropriate representations of $ G$, to decompose the space of polynomial functions on $S$, and to describe the ``zonal polynomials'' from which the hypergeometric functions are constructed. Detailed descriptions of the convergence properties of the series expansions are given, and integral representations are provided. Future papers in this series will develop the fine structure of these functions.
Toeplitz operators on the Segal-Bargmann space
C. A.
Berger;
L. A.
Coburn
813-829
Abstract: In this paper, we give a complete characterization of those functions on $2n$-dimensional Euclidean space for which the Berezin-Toeplitz quantizations admit a symbol calculus modulo the compact operators. The functions in question are characterized by a condition of ``small oscillation at infinity'' .
Hardy spaces of heat functions
H. S.
Bear
831-844
Abstract: We consider spaces of solutions of the one-dimensional heat equation on appropriate bounded domains in the $ (x,\,t)$-plane. The domains we consider have the property that they are parabolically star-shaped at some point; i.e., each downward half-parabola from some center point intersects the boundary exactly once. We introduce parabolic coordinates $(r,\,\theta)$ in such a way that the curves $ \theta =$constant are the half-parabolas, and dilation by multiplying by $ r$ preserves heat functions. An integral kernel is introduced by specializing to this situation the very general kernel developed by Gleason and the author for abstract harmonic functions. The combination of parabolic coordinates and kernel function provides a close analogy with the Poisson kernel and polar coordinates for harmonic functions on the disc, and many of the Hardy space theorems for harmonic functions generalize to this setting. Moreover, because of the generality of the Bear-Gleason kernel, much of this theory extends nearly verbatim to other situations where there are polar-type coordinates (such that the given space of functions is preserved by the ``radial'' expansion) and the maximum principle holds. For example, most of these theorems hold unchanged for harmonic functions on a radial star in $ {\mathbf{R}^n}$. As ancillary results we give a simple condition that a boundary point of a plane domain be regular, and give a new local Phragmén-Lindelöf theorem for heat functions.
Nonsingular quadratic differential equations in the plane
M. I. T.
Camacho;
C. F. B.
Palmeira
845-859
Abstract: We consider the problem of determining the number of inseparable leaves of nonsingular polynomial differential equations of degree two. As a corollary of a classification theorem for the foliation defined by these equations, we prove that this number is at most 2.