Kernels for the tangential Cauchy-Riemann equations
Al
Boggess
1-49
Abstract: On certain codimension one and codimension two submanifolds in ${{\textbf{C}}^n}$, we can solve the tangential Cauchy-Riemann equations ${\bar \partial _b}u\, = \,f$ with an explicit integral formula for the solution. Let $M\, = \,\partial D$, where D is a strictly pseudoconvex domain in $ {{\textbf{C}}^n}$. Let $ \omega \, \subset \, \subset \,M$ be defined by $ \omega \, = \,\{ z\, \in \,M;\,\operatorname{Re} \,h(z)\, < \,0\}$, where h is holomorphic near D. Points on the boundary of $\omega$, $ \partial \omega$, where the tangent space of $ \partial \omega$ becomes complex linear, are called characteristic points. Theorem 1. Suppose $\partial \omega$ is admissible (in particular if $\partial \omega $ has two characteristic points). Suppose $f\, \in \,{\mathcal{E}}_M^{p,q}(\bar \omega )$, $1\, \leqslant \,q\, \leqslant \,n\, - \,3$, is smooth on $\omega$ and satisfies ${\bar \partial _M}f\, = \,0$ on $ \omega$; then there exists $u\, \in \,{\mathcal{E}}_M^{p,q - 1}(\omega )$ which is smooth on $\omega$ except possibly at the characteristic points on $\partial \omega$ and which solves the equation $ {\bar \partial _M}u\, = \,f$ on $\omega$. Theorem 2. Suppose $f\, \in \,{\mathcal{E}}_M^{p,q}(\omega )$, $2\, \leqslant \,q\, \leqslant \,n\, - \,3$, is smooth on $\omega$; vanishes near each characteristic point; and $ {\bar \partial _M}f\, = \,0$ on $\omega$. Then there exists $ u\, \in \,{\mathcal{E}}_M^{p,q - 1}(\omega )$ satisfying ${\bar \partial _M}u\, = \,f$ on $ \omega$. Theorem 3. Suppose $f\, \in \,{\mathcal{D}}_M^{p,q}(\omega )$, $2\, \leqslant \,q\, \leqslant \,n - \,3$, is smooth with compact support in $ \omega$, and ${\bar \partial _M}f\, = \,0$. Then there exists $u\, \in \,{\mathcal{D}}_M^{p,q - 1}(\omega )$ with compact support in $ \omega$ and which solves $ {\bar \partial _M}u\, = \,f$. In all three theorems we have an explicit integral formula for the solution. Now suppose $S\, = \,\partial \omega$. Let $ {C_s}$ be the set of characteristic points on S. We construct an explicit operator $ E:\,{\mathcal{D}}_S^{p,q}(S\, - \,{C_S})\, \to \,{\mathcal{E}}_S^{p,q - 1}(S\, - \,{C_S})$ with the following properties. Theorem 4. The operator E maps $ L_{p,\operatorname{comp} }^{\ast}(S\, - \,{C_S})\, \to \,L_{p,\operatorname{loc} }^{\ast}(S\, - {C_S})$ and if $f\, \in \,{\mathcal{D}}_S^{p,q}(S\, - \,{C_S})$, $1\, \leqslant \,q\, \leqslant \,n\, - \,3$, then $f\, = \,{\bar \partial _S}\{ E(f)\} \, + \,E({\bar \partial _S}f)$.
Countable ultrahomogeneous undirected graphs
A. H.
Lachlan;
Robert E.
Woodrow
51-94
Abstract: Let $G\, = \,\left\langle {{V_G},\,{E_G}} \right\rangle$ be an undirected graph. The complementary graph $\tilde G$ is $\left\langle {{V_G},\,{E_{\tilde G}}} \right\rangle$ where $({V_1},\,{V_2})\, \in \,{E_{\tilde G}}$ iff ${V_1}\, \ne \,{V_2}$ and $ ({V_1},\,{V_2})\, \notin \,{E_G}$. Let $K(n)$ be the complete undirected graph on n vertices and let E be the graph [ill] i.e. $\left\langle {\{ a,\,b,\,c\} ,\,\{ (b,\,c),\,(c,\,b)\} } \right\rangle$. G is ultrahomogeneous just in case every isomorphism of subgraph of smaller cardinality can be lifted to an automorphism of G. Let $\mathcal{D}\, = \,\{ K(n):\,n\, \in \,\omega \} \, \cup \,\{ E,\,\tilde E\} \, \cup \,\{ \tilde K(n):\,n\, \in \,\omega \}$. Theorem: Let ${G_1}$, ${G_2}$ be two countable (infinite) ultrahomogeneous graphs such that for each $H\, \in \,\mathcal{D}\,H$ can be embedded in ${G_1}$, just in case it can be embedded in $ {G_2}$. Then ${G_1}\, \cong \,{G_2}$. Corollary: There are a countable number of countable ultrahomogeneous (undirected) graphs.
On a $4$-manifold homology equivalent to a bouquet of surfaces
Akio
Kawauchi
95-112
Abstract: This paper gives some algebraic invariants for a piecewise linear imbedding of a surface into some 4-manifold inducing a Z or Q-homology isomorphism. Several examples are obtained by using these invariants.
Approximation of $G$-maps by maps in equivariant general positions and imbeddings of $G$-complexes
Sören
Illman
113-157
Abstract: Let G be a finite group. In this paper we consider maps $ f:P \to M$ from equivariant polyhedra into equivariant p.l. manifolds. We prove an equivariant general position result which shows how to approximate a given continuous proper equivariant (or isovariant) map $f:P \to M$ by a G-map which is in equivariant general position. We also apply this equivariant general position result to get a general G-imbedding theorem. Applied to the case of G-imbeddings of simplicial G-complexes into euclidean representation space this general G-imbedding theorem gives a result which provides a good hold on the required dimension of the euclidean representation space. For example in the case when $G = {Z_m}$ we prove that there exists a representation space $ {{\textbf{R}}^{r(k,m)}}(\rho)$ with the property that any k-dimensional simplicial ${Z_m}$-complex X admits a proper p.l. $ {Z_m}$-imbedding into $ {{\textbf{R}}^{r(k,m)}}(\rho)$ and we also show that the dimension $r(k,m)$ is best possible, i.e., one cannot find a euclidean representation space of lower dimension than $r(k,m)$ with the same property as $ {{\textbf{R}}^{r(k,m)}}(\rho)$. Simple explicit expressions for the dimension $r(k,m)$ are given. We also consider the case of semi-free actions with a given imbedding of the fixed point set into some euclidean space. Furthermore we show that the p.l. G-imbeddings of equivariant p.l. manifolds into euclidean representation space obtained by our G-imbedding results are in general equivariantly locally knotted although they are locally flat in the ordinary sense. This phenomenon can occur in arbitrarily high codimensions.
Invariance of solutions to invariant parametric variational problems
John E.
Brothers
159-179
Abstract: Let G be a compact Lie group of diffeomorphisms of a connected orientable manifold M of dimension $n + 1$. Assume the orbits of highest dimension to be connected. Let $ \Psi$ be a convex positive even parametric integrand of degree n on M which is invariant under the action of G. Let T be a homologically $\Psi$-minimizing rectifiable current of dimension n on M, and assume there exists a G-invariant rectifiable current $T'$ which is homologous to T. It is shown that T is G-invariant provided $ \Psi$ satisfies a symmetry condition which makes it no less efficient for the tangent planes of T to lie along the orbits. This condition is satisfied by the area integrand in case G is a group of isometries of a Riemannian metric on M. Consequently, one obtains the corollary that if a rectifiable current T is a solution to the n-dimensional Plateau problem in M with G-invariant boundary $\partial T$, and if $ \partial T$ bounds a G-invariant rectifiable current $T'$ such that $T - T'$ is a boundary, then T is G-invariant. An application to the Plateau problem in ${{\textbf{S}}^3}$ is given.
Hyperbolic Birkhoff centers
I. P.
Malta
181-193
Abstract: The purpose of this paper is to show that if f is a diffeomorphism of a compact manifold whose Birkhoff center, $ c(f)$, is hyperbolic and has no cycles, then f satisfies Axiom A and is $ \Omega$-stable. To obtain a filtration for $c(f)$, the concept of an isolated set for a homeomorphism of a compact metric space is introduced. As a partial converse it is proved that if $c(f)$ is hyperbolic and f is $ \Omega$-stable, then $ c(f)$ has the no cycle property. A characterization of $\Omega$-stability when $c(f)$ is finite is also given.
Undecidability and definability for the theory of global fields
R. S.
Rumely
195-217
Abstract: We prove that the theory of global fields is essentially undecidable, using predicates based on Hasse's Norm Theorem to define valuations. Polynomial rings or the natural numbers are uniformly defined in all global fields, as well as Gödel functions encoding finite sequences of elements.
$\sp{\ast} $-valuations and ordered $\sp{\ast} $-fields
Samuel S.
Holland
219-243
Abstract: We generalize elementary valuation theory to *-fields (division rings with involution), apply the generalized theory to the task of ordering *-fields, and give some applications to Hermitian forms.
Noninvariance of an approximation property for closed subsets of Riemann surfaces
Stephen
Scheinberg
245-258
Abstract: A closed subset E of an open Riemann surface M is said to have the approximation property $\mathcal{a}$ if each continuous function on E which is analytic at all interior points of E can be approximaed uniformly on E by functions which are everywhere analytic on M. It is known that $\mathcal{a}$ is a topological invariant (i.e., preserved by homeomorphisms of the pair $ (M,E)$) when M is of finite genus but not in general, not even for ${C^\infty }$ quasi-conformal automorphisms of M. The principal result of this paper is that $\mathcal{a}$ is not invariant even under a real-analytic isotopy of quasi-conformal automorphisms (of a certain M). M is constructed as the two-sheeted unbranched cover of the plane minus a certain discrete subset of the real axis, and the isotopy is induced by $(x + \,iy,\,t) \mapsto x + \,ity$, for $t > 0$; E can be taken to be that portion of M which lies over a horizontal strip.
Applications of the Fourier-Wiener transform to differential equations on infinite-dimensional spaces. I
Yuh Jia
Lee
259-283
Abstract: Let $(H,i,B)$ be an abstract Wiener space and $ {p_t}$ be the Wiener measure on B with variance t. Let [B] be the complexification of B and ${\mathcal{E}_a}$ be the class of exponential type analytic functions defined on [B]. We define the Fourier-Wiener c-transform for any f in $ {\mathcal{E}_a}$ by $\displaystyle {F_c}f(y)\, = \,\int_\textbf{B} {f(x\, + \,iy){p_c}(dx)}$ and the inverse transform by $\mathcal{F}_c^{ - 1}f(y)\, = \,{\mathcal{F}_c}f( - y)$. Then the inversion formula holds and ${\mathcal{F}_2}$ extends to ${L^2}(B,{p_1})$ as a unitary operator. Next, we apply the above transform to investigate the existence, uniqueness and regularity of solutions for Cauchy problems associated with the following two equations: (1) $ {u_t}\, = \, - {\mathcal{N}^k}u$, (2) ${u_{tt}}\, = \, - {\mathcal{N}^k}u$; and the elliptic type equation (3) $- {N^k}u\, = \,f(k\, \geqslant \,1)$, where $ \Delta$ is the Laplacian and $ \mathcal{N}u(x)\, = \, - \Delta u(x)\, + \,(x,Du(x))$.
Stability of the null solution of parabolic functional inequalities
Ray
Redheffer;
Wolfgang
Walter
285-302
Abstract: Uniqueness and stability theorems are established for coupled systems of parabolic differential equations which may involve a Volterra-type dependence on the past history of the process. We allow retarded or deviating arguments, convolution-type memory terms, and strong coupling. (This means that all the space derivatives up to a given order can occur in all the equations.) Our results for strong coupling depend on the concept of ``admissible monomial'' which is here introduced for the first time and has no counterpart in the linear case. It is possible for uniqueness to fail in general, but to be restored (relative to a tolerably large class of functions of $ (x,t)$) if a single solution independent of x exists. Another curious feature of these theorems, depending again on the concept of admissible monomial, is that conditions for uniqueness can involve derivatives of order much higher than those occurring in the equation. Examples given elsewhere show that the results are, in various respects, sharp. Thus, the seemingly peculiar hypotheses do not arise from deficient technique, but from the actual behavior of strongly coupled systems. The paper concludes with a new method of dealing with unbounded regions for the difficult case in which the functional occurs in the boundary operator as well as in the differential equation.