Compactifications of the generalized Jacobian variety
Tadao
Oda;
C. S.
Seshadri
1-90
Abstract: The generalized Jacobian variety of an algebraic curve with at most ordinary double points is an extension of an abelian variety by an algebraic torus. Using the geometric invariant theory, we systematically compactify it in finitely many different ways and describe their structure in terms of torus embeddings. Our compactifications include all known good ones.
Multiplications on cohomology theories with coefficients
Alvin Frank
Martin
91-120
Abstract: Araki and Toda have considered the existence and classification of multiplications on generalized cohomology theories with coefficients in the category of finite CW-complexes. We consider the same matters for representable cohomology theories in a category of stable CW-spectra, such as that constructed by Adams. We obtain similar, and in certain instances stronger, results than Araki and Toda, with methods of proof that are often simpler and more straightforward.
Natural endomorphisms of Burnside rings
Andreas
Blass
121-137
Abstract: The Burnside ring $ \mathcal{B}(G)$ of a finite group G consists of formal differences of finite G-sets. $ \mathcal{B}$ is a contravariant functor from finite groups to commutative rings. We study the natural endomorphisms of this functor, of its extension $\textbf{Q} \otimes \mathcal{B}$ to rational scalars, and of its restriction $ \mathcal{B} \upharpoonright {\text{Ab}}$ to abelian groups. Such endomorphisms are canonically associated to certain operators that assign to each group one of its conjugacy classes of subgroups. Using these operators along with a carefully constructed system of linear congruences defining the image of $ \mathcal{B}(G)$ under its canonical embedding in a power of Z, we exhibit a multitude of natural endomorphisms of $\mathcal{B}$, we show that only two of them map G-sets to G-sets, and we completely describe all natural endomorphisms of $ \mathcal{B} \upharpoonright {\text{Ab}}$.
On sufficient conditions for harmonicity
P. C.
Fenton
139-147
Abstract: Suppose that u is continuous in the plane and that given any complex number z there is a number $\rho = \rho (z) > 0$ such that $\displaystyle u(z) = \frac{1} {{2\pi }}\int_0^{2\pi } {u(z + \rho {e^{i\theta }})} d\theta$ (1) The main result is: if u possesses a harmonic majorant and $\rho (z)$ is continuous and satisfies a further condition (which may not be omitted) then u is harmonic. Another result in the same vein is proved.
An algebraic determination of closed orientable $3$-manifolds
William
Jaco;
Robert
Myers
149-170
Abstract: Associated with each polyhedral simple closed curve j in a closed, orientable 3-manifold M is the fundamental group of the complement of j in M, ${\pi _1}(M - j)$. The set, $\mathcal{K}(M)$, of knot groups of M is the set of groups $ {\pi _1}(M - j)$ as j ranges over all polyhedral simple closed curves in M. We prove that two closed, orientable 3-manifolds M and N are homeomorphic if and only if $\mathcal{K}(M) = \mathcal{K}(N)$. We refine the set of knot groups to a subset $\mathcal{F}(M)$ of fibered knot groups of M and modify the above proof to show that two closed, orientable 3-manifolds M and N are homeomorphic if and only if $\mathcal{F}(M) = \mathcal{F}(N)$.
The spaces of functions of finite upper $p$-variation
Robert R.
Nelson
171-190
Abstract: Let y be a Banach space, $1\, \leqslant \,p\, < \,\infty$, and $ {U_p}$ be the semi-normed space of Y-valued Bochner measurable functions of a real variable which have finite upper p-variation. Let $ {\tilde U_p}$ be the space of ${U_p}$-equivalence classes. An averaging operator is defined with the aid of the theory of helixes in Banach spaces, which enables us to show that the spaces ${\tilde U_p}$ are Banach spaces, to characterize their members, and to show that they are isometrically isomorphic to Banach spaces of Y-valued measures with bounded p-variation.
Differentiability of measures associated with parabolic equations on infinite-dimensional spaces
M. Ann
Piech
191-209
Abstract: The transition measures of the Brownian motion on manifolds modelled on abstract Wiener spaces locally correspond to fundamental solutions of certain infinite dimensional parabolic equations. We establish the existence of such fundamental solutions under a broad new set of hypotheses on the differential coefficients. The fundamental solutions can be approximated in total variation by fundamental solutions of ``almost'' finite dimensional parabolic equations. By the finite dimensional theory, the approximations are seen to be differentiable. We prove that the property of differentiability is closed under a particular type of sequential convergence, and conclude the differentiability of the fundamental solutions of the infinite dimensional parabolic equations. This result provides strong evidence in support of the conjecture that the transition measures of the Brownian motion are differentiable, and hence is of importance in the construction of infinite dimensional Laplace-Beltrami operators.
On the positive spectrum of Schr\"odinger operators with long range potentials
G. B.
Khosrovshahi;
H. A.
Levine;
L. E.
Payne
211-228
Abstract: In this paper we are concerned with solutions of the equation $\Delta u\, + \,p(x)u\, = \,0$ in an unbounded domain $\Omega$ in ${R^n}$, $ \Omega \, \supset \,\{ x\vert\,\,\left\Vert x \right\Vert\, \geqslant \,{R_0}\}$. The main result is a determination of conditions on the asymptotic behavior of $p(x)$ sufficient to guarantee that no nontrivial ${L_2}$ solution exists. Our results contain those of previous authors as special cases. The principal application is to the determination of upper bounds for positive eigenvalues of Schrödinger operators.
Dixmier's representation theorem of central double centralizers on Banach algebras
Sin-ei
Takahasi
229-236
Abstract: The present paper is devoted to a representation theorem of central double centralizers on a complex Banach algebra with a bounded approximate identity. In particular, our result implies the representation theorem of the ideal center of an arbitrary $ {C^\ast}$-algebra established by J. Dixmier.
Dirichlet forms associated with hypercontractive semigroups
James G.
Hooton
237-256
Abstract: We exhibit a class of probability measures on ${\textbf{R}^n}$ such that the associated Dirichlet form is represented by a selfadjoint operator A and such that $ {e^{ - tA}}$ is a hypercontractive semigroup of operators. The measures are of the form $ d\mu \, = \,{\Omega ^2}\,dx$ where $\Omega$ has classical first derivatives and $ {L^p}$ second derivatives, p determined by n.
Dispersion points for linear sets and approximate moduli for some stochastic processes
Donald
Geman
257-272
Abstract: Let $\Gamma \, \in \,[0,\,1]$ be Lebesgue measurable; then $ \Gamma$ has Lebesgue density 0 at the origin if and only if $\displaystyle \int_\Gamma {{t^{ - 1}}\Psi ({t^{ - 1}}\,{\text{meas}}} \{ \Gamma \, \cap \,(0,\,t)\} )\,dt\, < \,\infty$ for some continuous, strictly increasing function $\Psi (t)\,(0\, \leqslant \,t\, \leqslant \,1)$ with $\Psi (0)\, = \,0$. This result is applied to the local growth of certain Gaussian (and other) proceses $\{ {X_t},\,t\, \geqslant \,0\}$ as follows: we find continuous, increasing functions $\phi (t)$ and $\eta (t)\,(t\, \geqslant \,0)$ such that, with probability one, the set $\{ t:\eta (t)\, \leqslant \,\left\vert {{X_t}\, - \,{X_0}} \right\vert\, \leqslant \,\phi (t)\}$ has density 1 at the origin.
Difference equations over locally compact abelian groups
G. A.
Edgar;
J. M.
Rosenblatt
273-289
Abstract: A homogeneous linear difference equation with constant coefficients over a locally compact abelian group G is an equation of the form $\Sigma_{j\, = \,1}^n {{c_j}f({t_j}x)\, = \,0}$ which holds for all $ x\, \in \,G$ where ${c_1}, \ldots ,\,{c_n}$ are nonzero complex scalars, ${t_1},\, \ldots \,,\,{t_n}$ are distinct elements of G, and f is a complex-valued function on G. A function f has linearly independent translates precisely when it does not satisfy any nontrivial linear difference equation. The locally compact abelian groups without nontrivial compact subgroups are exactly the locally compact abelian groups such that all nonzero $f\, \in \,{L_p}(G)$ with $1\, \leqslant \,p\, \leqslant \,2$ have linearly independent translates. Moreover, if G is the real line or, more generally, if G is $ {R^n}$ and the difference equation has a characteristic trigonometric polynomial with a locally linear zero set, then the difference equation has no nonzero solutions in $ {C_0}(G)$ and no nonzero solutions in ${L_p}(G)$ for $1\, \leqslant \,p\, < \,\infty$. But if G is some ${R^n}$ for $ n\, \geqslant \,2$ and the difference equation has a characteristic trigonometric polynomial with a curvilinear portion of its zero set, then there will be nonzero ${C_0}({R^n})$ solutions and even nonzero ${L_p}({R^n})$ solutions for $p\, > \,2n/(n - 1)$. These examples are the best possible because if $1\, \leqslant p\, < \,2n/(n - 1)$, then any nonzero function in $ {L_p}({R^n})$ has linearly independent translates. Also, the solutions to linear difference equations over the circle group can be simply described in a fashion which an example shows cannot be extended to all compact abelian groups.
Smooth orbit equivalence of ergodic ${\bf R}\sp{d}$ actions, $d\geq 2$
Daniel
Rudolph
291-302
Abstract: We show here that any two free ergodic finite measure preserving actions of $ {\textbf{R}^d}$, $d\, \geqslant \,2$, are orbit equivalent by a measure preserving map which on orbits is ${C^\infty }$.
$\sigma $-connectedness in hereditarily locally connected spaces
J.
Grispolakis;
E. D.
Tymchatyn
303-315
Abstract: B. Knaster, A. Lelek and J. Mycielski [Colloq. Math. 6 (1958), 227-246] had asked whether there exists a hereditarily locally connected planar set, which is the union of countably many disjoint arcs. They gave an example of a locally connected, connected planar set, which is the union of a countable sequence of disjoint arcs. Lelek proved in a paper in Fund. Math. in 1959, that connected subsets of planar hereditarily locally connected continua are weakly $\sigma$-connected (i.e., they cannot be written as unions of countably many disjoint, closed connected subsets). In this paper we generalize the notion of finitely Suslinian to noncompact spaces. We prove that there is a class of spaces, which includes the class of planar hereditarily locally connected spaces and the finitely Suslinian spaces, and which are weakly $ \sigma$-connected, thus, answering the above question in the negative. We also prove that arcwise connected, hereditarily locally connected, planar spaces are locally arcwise connected. This answers in the affirmative a question of Lelek [Colloq. Math. 36 (1976), 87-96].
On the global asymptotic behavior of Brownian local time on the circle
E.
Bolthausen
317-328
Abstract: The asymptotic behavior of the local time of Brownian motion on the circle is investigated. For fixed time point t this is a (random) continuous function on ${S^1}$. It is shown that after appropriate norming the distribution of this random element in $ C({S^1})$ converges weakly as $t\, \to \,\infty $. The limit is identified as $ 2(B(x)\, - \,\int {B(y)\,dy)}$ where B is the Brownian bridge. The result is applied to obtain the asymptotic distribution of a Cramer-von Mises type statistic for the global deviation of the local time from the constant t on ${S^1}$.
The structure of supermanifolds
Marjorie
Batchelor
329-338
Abstract: The increasing recognition of Lie superalgebras and their importance in physics inspired a search to find an object, a ``supermanifold", which would realize the geometry implicit in Lie superalgebras. This paper analyzes the structure of supermanifolds as defined by B. Kostant. The result is the following structure theorem. The Main Theorem. If E is a real vector bundle over the smooth manifold X, let $\Lambda E$ be the associated exterior bundle and let $ \Gamma (\Lambda E)$ be the sheaf of sections of $\Lambda E$. Then every supermanifold over X is isomorphic to $ \Gamma (\Lambda E)$ for some vector bundle E over X. Although the vector bundle E is not unique but is determined only up to isomorphism, and the isomorphism guaranteed is not canonical, the existence of the isomorphism provides a base for a better understanding of geometry in the graded setting.
Pseudo-integral operators
A. R.
Sourour
339-363
Abstract: Let $(X,\,\mathcal{a},\,m)$ be a standard finite measure space. A bounded operator T on ${L^2}(X)$ is called a pseudo-integral operator if $(Tf)(x)\, = \,\int {f(y)\,\mu (x,\,dy)}$, where, for every x, $\mu (x,\, \cdot \,)$ is a bounded Borel measure on X. Main results: 1. A bounded operator T on $ {L^2}$ is a pseudo-integral operator with a positive kernel if and only if T maps positive functions to positive functions. 2. On nonatomic measure spaces every operator unitarily equivalent to T is a pseudo-integral operator if and only if T is the sum of a scalar and a Hilbert-Schmidt operator. 3. The class of pseudo-integral operators with absolutely bounded kernels form a selfadjoint (nonclosed) algebra, and the class of integral operators with absolutely bounded kernels is a two-sided ideal. 4. An operator T satisfies $ (Tf)(x)\, = \,\int {f(y)\,\mu (x,\,dy)}$ for $f\, \in \,{L^\infty }$ if and only if there exists a positive measurable (almost-everywhere finite) function $\Omega$ such that $\left\vert {(Tf)(x)} \right\vert\, \leqslant \,{\left\Vert f \right\Vert _\infty }\Omega (x)$.
Optimal stochastic switching and the Dirichlet problem for the Bellman equation
Lawrence C.
Evans;
Avner
Friedman
365-389
Abstract: Let ${L^i}$ be a sequence of second order elliptic operators in a bounded n-dimensional domain $ \Omega$, and let $ {f^i}$ be given functions. Consider the problem of finding a solution u to the Bellman equation ${\sup _i}({L^i}u\, - \,{f^i})\, = \,0$ a.e. in $ \Omega$, subject to the Dirichlet boundary condition $u\, = \,0$ on $ \partial \Omega$. It is proved that, provided the leading coefficients of the $ {L^i}$ are constants, there exists a unique solution u of this problem, belonging to ${W^{1,\infty }}(\Omega )\, \cap \,W_{{\text{loc}}}^{2,\infty }(\Omega )$. The solution is obtained as a limit of solutions of certain weakly coupled systems of nonlinear elliptic equations; each component of the vector solution converges to u. Although the proof is entirely analytic, it is partially motivated by models of stochastic control. We solve also certain systems of variational inequalities corresponding to switching with cost.
The atomic decomposition for parabolic $H\sp{p}$ spaces
Robert H.
Latter;
Akihito
Uchiyama
391-398
Abstract: The theorem of A. P. Calderón giving the atomic decomposition for certain parabolic ${H^p}$ spaces is extended to all such spaces. The proof given also applies to Hardy spaces defined on the Heisenberg group.