The order and symbol of a distribution
Alan
Weinstein
1-54
Abstract: A definition is given, for an arbitrary distribution g on a manifold X, of the order and symbol of g at a point $({\chi ,\xi })$ of the cotangent bundle $ T^{\ast}X$. If $X = \textbf{R}^n$, the order of g at $ ({0,\xi})$ is the growth order as $\tau \to \infty $ of the distributions $ {g^\tau }(x) = {e^{ - i\sqrt \tau \langle x,\xi \rangle }}g\left( {x /\sqrt \tau } \right)$ ; if the order is less than or equal to N, the N-symbol of g is the family $ {g^\tau }$ modulo $O({{\tau ^{N - 1/2}}})$. It is shown that the order and symbol behave in a simple way when g is acted upon by a pseudo-differential operator. If g is a Fourier integral distribution, suitable identifications can be made so that the symbol defined here agrees with the bundle-valued symbol defined by Hörmander.
Groups of PL $\Lambda $-homology spheres
Gerald A.
Anderson
55-67
Abstract: Let $ \wedge \, = \,{{\textbf{Z}}_k}\,$ be a subring of Q. The group of $ {H_K}$-cobordism classes of closed PL n-manifolds with the $ \wedge$-homology of $ {S^n}$ is computed for $ n \ge 4$ (modulo K-torsion). The simply connected version is also computed.
Solution of the nonlinear problem $Au=N(u)$ in a Banach space
Martin
Schechter;
Jack
Shapiro;
Morris
Snow
69-78
Abstract: We solve a nonlinear problem $Au\, = \,N(u)$ where A is semi-Fredholm and N is a nonlinear compact operator.
Ultrafilter invariants in topological spaces
Victor
Saks
79-97
Abstract: Let $ \mathfrak{m}\, \geqslant \,{\aleph _0}$ and $X\, = \,\prod\nolimits_{i\, \in \,I} {{X_i}}$. Then X is $[{\aleph _0},\,\mathfrak{m}]$-compact if and only if $\prod\nolimits_{i\, \in \,J} {{X_i}}$ is $ [{\aleph _0},\,\mathfrak{m}]$-compact for all $ J\, \subset \,I$ with $\vert J\vert\, \leqslant \,{2^{{2^\mathfrak{m}}}}$. Let $\mathfrak{m}\, \geq \,{\aleph _0}$, $ ({x_\xi }:\,\xi \, < \,\mathfrak{m})$ a net in X, $p\, \in \,X$, and $\mathcal{D}\, \in \,\beta (\mathfrak{m})$. Then $p\, = \,\mathcal{D}\, - \,{\lim _{\xi < \mathfrak{m}}}\,{x_\xi }$ if $ \{ \xi \, < \,\mathfrak{m}:\,{x_\xi }\, \in \,U \} \, \in \, \mathcal{D}$ for every neighborhood U of p. Every topological space is characterized by its $\mathcal{D}$-limits. Several topological properties are described using ultrafilter invariants, including compactness and perfect maps. If X is a Hausdorff space and D is a discrete space equipotent with a dense subset of X, then X is a continuous perfect image of a subspace of $\beta D$ which contains D if and only if X is regular.
The BP homology of $H$-spaces
Richard
Kane
99-119
Abstract: The BP homology of 1-connected H-spaces whose loop space is torsion free is studied. It is shown that localizing in a suitable manner kills all torsion in $ B{P_{\ast}}(X)$. Similar results are then obtained for the bordism of X. Finally the K-theory of X is shown to have no p torsion.
Undecidable existential problems for addition and divisibility in algebraic number rings
L.
Lipshitz
121-128
Abstract: Existential formulas involving addition and divisibility are shown to be undecidable in the ring of integers of a real quadratic extension of the rationals. A weaker result is proved for extensions of higher degree.
Distances between two-state Markov processes attainable by Markov joinings
Martin H.
Ellis
129-153
Abstract: The function which assigns to each pair of two-state Markov processes the set of partition distances between them attainable by a Markov process on their joint atoms is computed. It is found that the infimum of these distances, the ``Markov distance'' between the pair, fails to satisfy the Triangle Inequality, hence fails to be a metric; thus in some cases the $\overline d$-distance between two two-state Markov processes cannot be attained by a Markov process on their joint atoms.
Singularly perturbed boundary value problems with angular limiting solutions
F. A.
Howes
155-182
Abstract: A basic result of Haber and Levinson which describes the behavior of solutions of $y(a,\varepsilon )$, $ y(b,\varepsilon )$, prescribed, in the presence of a reduced solution with corners is modified to treat related classes of problems. Under various stability assumptions, solutions are shown to remain, for small $\varepsilon \, > \,0$, in a o(l)-neighborhood of an angular reduced solution with the possible exception of narrow layers near the boundaries in some cases. Each aspect of the theory developed here is illustrated by several examples.
Characterizations of amenable groups
William R.
Emerson
183-194
Abstract: Generalizing a construction of Banach from 1923 we obtain new criteria for the amenability of a locally compact group G. The relationship of these new criteria to known characterizations is then investigated, and in particular a formally strenghthened version of the von Neumann/Dixmier condition for amenability is established.
Topologies on spaces of vector-valued continuous functions
Surjit Singh
Khurana
195-211
Abstract: Topologies ${\beta _0},{\beta _1},\beta ,{\beta _\infty },{\beta _{\infty c}}$ are defined on $ {C_b}(X,E)$, the space of all bounded, continuous functions from a completely regular Hausdorff space X, into E, a normed space, and their duals are determined. Also many properties of these topologies are proved.
$3$-pseudomanifolds with preassigned links
Amos
Altshuler
213-237
Abstract: A 3-pseudomanifold is a finite connected simplicial 3-complex $\mathcal{K}$ such that every triangle in $\mathcal{K}$ belongs to precisely two 3-simplices of $\mathcal{K}$, the link of every edge in $\mathcal{K}$ is a circuit, and the link of every vertex in $ \mathcal{K}$ is a closed 2-manifold. It is proved that for every finite set $ \sum$ of closed 2-manifolds, there exists a 3-pseudomanifold $\mathcal{K}$ such that the link of every vertex in $\mathcal{K}$ is homeomorphic to some $S\, \in \,\sum$, and every $S\, \in \,\sum$ is homeomorphic to the link of some vertex in $ \mathcal{K}$.
On the growth of the integral means of subharmonic functions of order less than one
Faruk F.
Abi-Khuzam
239-252
Abstract: Let u be a subharmonic function of order $\lambda (0 < \lambda < 1)$, and let ${m_s}(r,u)\, = \,{\left\{ {(1/2\pi )\int_{ - \pi }^\pi {{{\left\vert {u(r{e^{i\theta }})} \right\vert}^s}} d\theta } \right\}^{1/s}}$. We compare the growth of $ {m_s}(r,u)$ with that of the Riesz mass of u as measured by $N\,(r,u)\, = (1/2\pi )\int_{ - \pi }^\pi {u(r{e^{i\theta }})d\theta }$. A typical result of this paper states that the following inequality is sharp: $\displaystyle \underset{x\to \infty }{\mathop{\lim \,\inf }}\,\,\frac{{{m}_{s}}... ...N\left( r,\,u \right)}\,\leqslant\,{{m}_{s}}\left( {{\psi }_{\lambda }} \right)$ ($ \ast$) where $ \psi_\lambda (\theta )\, = \,(\pi \lambda /\sin \,\lambda )\cos \,\lambda \theta$. The case $s\, = \,1$ is due to Edrei and Fuchs, the case $ s\, = \,2$ is due to Miles and Shea and the case $ s\, = \,\infty$ is due to Valiron.
Standard and normal reductions
R.
Hindley
253-271
Abstract: Curry and Feys' original standardization proof for $\lambda \beta$-reduction is analyzed and generalized to $\lambda \beta \eta$-reductions with extra operators. There seem to be two slightly different definitions of 'standard reduction' in current use, without any awareness that they are different; it is proved that although these definitions turn out to be equivalent for $ \lambda \beta$-reduction, they become different for $\lambda \beta \eta$ and for reductions involving extra operators, for example the recursion operator. Normal reductions are also studied, and it is shown that the basic normal-reduction theorem stays true when fairly simple operators like Church's $\delta$ and Curry's iterator Z are added, but fails for more complicated ones like the recursion operator R. Finally, a table is given summarizing the results, and showing how far the main theorems on $ \lambda \beta$-reductions extend to reductions with various extra operators.
Weak uniqueness sets for discrete groups
Marek
Bożejko;
Tadeusz
Pytlik
273-282
Abstract: For discrete groups we introduce a new class of sets, called weak uniqueness sets, which for abelian groups contains the class of sets of uniqueness. Considered is the problem of determining groups for which every finite set is a weak uniqueness set. Some examples are given.
Partition theorems and ultrafilters
James E.
Baumgartner;
Alan D.
Taylor
283-309
Abstract: We introduce a class of ultrafilters on $\omega$ called k-arrow ultrafilters and characterized by the partition relation $U \to {(U,k)^2}$. These are studied in conjunction with P-points, Q-points, weakly Ramsey and Ramsey ultrafilters.
Structural stability for flows on the torus with a cross-cap
Carlos
Gutiérrez
311-320
Abstract: Let $ {{\mathcal{X}}^r}({\tilde M}), r \geq 1$, denote the space of ${C^r}$-vector fields on the torus with a cross-cap $\tilde M$. We show that the Morse-Smale vector fields of ${{\mathcal{X}}^r}({\tilde M})$ are dense on it. We also give a simple proof that a ${C^0}$-flow on the Klein bottle cannot support a nontrivial $\omega$-recurrent trajectory.
Regular Hjelmslev planes. II
Dieter
Jungnickel
321-330
Abstract: In this paper we introduce the notion of an x-partition for a Hjelmslev-matrix (H-matrix). This allows us to prove a new composition theorem for H-matrices. We obtain the existence of $ ({t,\,r})$-H-matrices and hence of regular $ ({t,\,r})$-H-planes for infinitely many series of invariants which were not yet known. In fact, many of these invariants were not even known to occur as the invariants of any H-plane at all (whether regular or not).
The semigroup of varieties of Brouwerian semilattices
Peter
Köhler
331-342
Abstract: It is shown that the semigroup of varieties of Brouwerian semilattices is free.
Desarguesian Klingenberg planes
P. Y.
Bacon
343-355
Abstract: Klingenberg planes are generalizations of Hjelmslev planes. If R is a local ring, one can construct a projective Klingenberg plane $ {\textbf{V}}(R)$ and a derived affine Klingenberg plane ${\textbf{A}}(R)$ from R. If V is a projective Klingenberg plane, if ${R_1},\,{R_2}$ and ${R_3}$ are local rings, if ${s_1},\,{s_2}$ and ${s_3}$ are the sides of a nondegenerate triangle in V, and if each of the derived affine Klingenberg planes $\mathcal{a}\left( {V,\,{s_i}} \right)$ is isomorphic to $ {\textbf{A}}({R_i}),\,$, $i\, = \,1,\,2,\,3$, then the rings $ {R_1},\,{R_2}$ and $ {R_3}$ are isomorphic, and V is isomorphic to ${\textbf{V}}({R_1});$; also, if g is a line of V, then the derived affine Klingenberg plane $ \mathcal{a}({V,\,g})$ is isomorphic to $ \textbf{A}({R_1})$. Examples are given of projective Klingenberg planes V, each of which has the following two properties: (1) V is not isomorphic to ${\textbf{V}}(R)$ for any local ring R; and (2) there is a flag $(B,\,b)$ of V, and a local ring S such that each derived affine Klingenberg plane $\mathcal{a}({V,\,m})$ is isomorphic to ${\textbf{A}}(S)$ whenever $m\, = \,b$, or m is a line through B which is not neighbor to b.
Isosingular loci and the Cartesian product structure of complex analytic singularities
Robert
Ephraim
357-371
Abstract: Let X be a (not necessarily reduced) complex analytic space, and let V be a germ of an analytic space. The locus of points q in X at which the germ $ {X_q}$ is complex analytically isomorphic to V is studied. If it is nonempty it is shown to be a locally closed submanifold of X, and X is locally a Cartesian product along this submanifold. This is used to define what amounts to a coarse partial ordering of singularities. This partial ordering is used to show that there is an essentially unique way to completely decompose an arbitrary reduced singularity as a cartesian product of lower dimensional singularities. This generalizes a result previously known only for irreducible singularities.
Uniqueness in the Cauchy problem for parabolic equations
Roger M.
Hayne
373-399
Abstract: In a classical paper S. Täcklind (Nova Acta Soc. Sci. Upsaliensis (4) 10 (1936), 1-57) closed the uniqueness question for the Cauchy problem for the heat equation with a general growth hypothesis which was both necessary and sufficient. Täcklind's proof of the sufficiency involved an ingenious bootstrapping comparison technique employing the maximum principle and a comparison function constructed from the Green's function for a half cylinder. G. N. Zolotarev (Izv. Vysš. Učebn. Zaved. Matematika 2 (1958), 118-135) has extended this result using essentially the same technique to show that Täcklind's uniqueness condition remains sufficient for a general second order parabolic equation provided the coefficients are regular enough to permit the existence and estimation of a Green's function. We have now shown, using a new approach which replaces the construction based upon a Green's function by an appropriate comparison solution of the maximizing equation (C. Pucci, Ann. Mat. Pura Appl. 72 (1966), 141-170), that Täcklind's condition is sufficient without any regularity conditions on the coefficients.
A characterization of upper-embeddable graphs
Mark
Jungerman
401-406
Abstract: It is proved that a pseudograph G is upper-embeddable if and only if it has a spanning tree T such that G - T has at most one component with an odd number of edges. This result is then used to show that all 4-edge connected graphs are upper-embeddable.