On the torus theorem and its applications
C. D.
Feustel
1-43
Abstract: In this paper, we prove the torus theorem and that manifolds in a certain class of 3-manifolds with toral boundary are determined by their fundamental groups alone. Both of these results were reported by F. Waldhausen. We also give an extension of Waldhausen's generalization of the loop theorem.
On the torus theorem for closed $3$-manifolds
C. D.
Feustel
45-57
Abstract: In this paper we give the appropriate generalization of the torus theorem to closed, sufficiently large, irreducible, orientable 3-manifolds.
The multiplicative behavior of $\mathcal{H}$
Pierre Antoine
Grillet
59-86
Abstract: Various results are given describing the product of two $\mathcal{H}$-classes in an arbitrary semigroup in terms of groups and homomorphisms.
Adjoint abelian operators on $L\sp{p}$ and $C(K)$
Richard J.
Fleming;
James E.
Jamison
87-98
Abstract: An operator A on a Banach space X is said to be adjoint abelian if there is a semi-inner product $[ \cdot , \cdot ]$ consistent with the norm on X such that $ [Ax,y] = [x,Ay]$ for all $ x,y \in X$. In this paper we show that every adjoint abelian operator on $ C(K)$ or $ {L^p}(\Omega ,\Sigma ,\mu )\;(1 < p < \infty ,p \ne 2)$ is a multiple of an isometry whose square is the identity and hence is of the form $Ax( \cdot ) = \lambda \alpha ( \cdot )(x \circ \phi )( \cdot )$ where $\alpha$ is a scalar valued function with $\alpha ( \cdot )\alpha \circ \phi ( \cdot ) = 1$ and $\phi$ is a homeomorphism of K (or a set isomorphism in case of ${L^p}(\Omega ,\Sigma ,\mu ))$ with $\phi \circ \phi =$ identity (essentially).
Bernoulli convolutions and differentiable functions
R.
Kaufman
99-104
Abstract: Bernoulli convolutions, similar in structure to convolutions with a constant ratio, are considered in relation to differentiable transformations. A space of functions on the Cantor set leads to highly singular measures that nevertheless resemble absolutely continuous measures sufficiently to control their Fourier-Stieltjes transforms.
Convolution equations for vector-valued entire functions of nuclear bounded type
Thomas A. W.
Dwyer
105-119
Abstract: Given two complex Banach spaces E and F, convolution operators ``with scalar coefficients'' are characterized among all convolution operators on the space $ {H_{Nb}}(E';F)$. THEOREM 2. Solutions of homogeneous convolution equations of scalar type can be approximated in
Simplicial geometry and transportation polytopes
Ethan D.
Bolker
121-142
Abstract: The classical transportation problem is the study of the set of nonnegative matrices with prescribed nonnegative row and column sums. It is aesthetically satisfying and perhaps potentially useful to study more general higher dimensional rectangular arrays whose sums on some subarrays are specified. We show how such problems can be rewritten as problems in homology theory. That translation explains the appearance of bipartite graphs in the study of the classical transportation problem. In our generalization, higher dimensional cell complexes occur. That is why the general problem requires a substantial independent investigation of simplicial geometry, the name given to the class of theorems on the geometry of a cell complex which depend on a particular cellular decomposition. The topological invariants of the complex are means, not ends. Thus simplicial geometry attempts to do for complexes what graph theory does for graphs. The dual title of this paper indicates that we shall spend as much time studying simplicial geometry for its own sake as applying the results to transportation problems. Our results include formulas for inverting the boundary operator of an acyclic cell complex, and some information on the number of such subcomplexes of a given complex.
Coincidence index and multiplicity
B.
Laloux;
J.
Mawhin
143-162
Abstract: This paper is devoted to the extension, in the frame of coincidence degree theory in normed spaces, of the concept of Leray-Schauder index of an isolated fixed point. The generalization includes basic properties of the coincidence index, Krasnosel'skiĭ type theorems for the case of noninvertible linear part and a Leray-Schauder's type formula relating the index and spectral theory in the linear case. This last problem needs the introduction of the concept of characteristic value for some couples of linear mappings and of its multiplicity.
A special integral and a Gronwall inequality
Burrell W.
Helton
163-181
Abstract: This paper considers a special integral $(I)\smallint _a^b(fdg + H)$ which is a subdivision-refinement-type limit of the approximating sum $\displaystyle \sum\limits_1^n {\{ f({t_i})[g({x_i}) - g({x_{i - 1}})] + H({x_{i - 1}},{x_i})\} ,}$ where ${x_{i - 1}} < {t_i} < {x_i}$. The author shows, with appropriate restrictions, that $(I)\smallint _a^b(fdg + H)$ exists if and only if $\displaystyle (R)\smallint _x^y(fdg + H - {A^ - }) = (L)\smallint _x^y(fdg + H + {A^ + })$ for $a \leqslant x < y \leqslant b$, where $ A(p,q) = [f(q) - f(p)][g(q) - g(p)],{A^ - }(p,q) = A({q^ - },q)$ and ${A^ + }(p,q) = A(p,{p^ + })$. Furthermore, if either of the equivalent statements is true, then all the integrals are equal. These equivalent statements are used to prove an integration-by-parts theorem and to solve a Gronwall inequality involving this special integral. Product integrals are used in the solution of the Gronwall inequality.
The semilattice tensor product of distributive lattices
Grant A.
Fraser
183-194
Abstract: We define the tensor product $A \otimes B$ for arbitrary semilattices A and B. The construction is analogous to one used in ring theory (see [4], [7], [8]) and different from one studied by A. Waterman [12], D. Mowat [9], and Z. Shmuely [10]. We show that the semilattice $A \otimes B$ is a distributive lattice whenever A and B are distributive lattices, and we investigate the relationship between the Stone space of $A \otimes B$ and the Stone spaces of the factors A and B. We conclude with some results concerning tensor products that are projective in the category of distributive lattices.
Compactifications of spaces of functions and integration of functionals
L. Š.
Grinblat
195-223
Abstract: For a locally compact space there exists a compactification such that all its points are effectively describable, namely, Alexandroff's onepoint compactification. The effective construction of compactifications for numerous standard separable metric spaces is already a very nontrivial problem. We propose a method of compactification which enables us to effectively construct compactifications of some spaces of functions (for example, of a ball in ${L_p}( - \infty ,\infty )$). It will be shown that the study of compactifications of spaces of functions is of principal importance in the theory of integration of functionals and in limit theorems for random processes.
Existence of periodic solutions of nonlinear differential equations
R.
Kannan
225-236
Abstract: The nonlinear differential equation $ x'' = f(t,x(t))$, f being $2\pi$-periodic in t, is considered for the existence of $2\pi$-periodic solutions. The equation is reduced to an equivalent system of two Hammerstein equations. The case of nonlinear perturbation at resonance is also discussed.
Real prime flows
H. B.
Keynes;
D.
Newton
237-255
Abstract: In this paper, we construct examples of real-type prime flows and study these examples in detail. General properties of prime flows are studied, with emphasis on proximality conditions and properties of automorphisms. Examples of prime flows which are not POD are shown to exist, and results analogous to number-theoretic properties, such as a ``unique factorisation'' theorem, are shown to hold for prime flows.
On the connectedness of homomorphisms in topological dynamics
D.
McMahon;
T. S.
Wu
257-270
Abstract: Let (X, T) be a minimal transformation group with compact Hausdorff phase space. We show that if $\phi :(X,T) \to (Y,T)$ is a distal homomorphism and has a structure similar to the structure Furstenberg derived for distal minimal sets, then for T belonging to a class of topological groups T, the homomorphism $X \to X/S(\phi )$ has connected fibers, where $S(\phi )$ is the relativized equicontinuous structure relation. The class T is defined by Sacker and Sell as consisting of all groups T with the property that there is a compact set $K \subseteq T$ such that T is generated by each open neighborhood of K. They show that for such T, a distal minimal set which is a finite-to-one extension of an almost periodic minimal set is itself an almost periodic minimal set. We provide an example that shows that the restriction on T cannot be dropped. As one of the preliminaries to the above we show that given $\phi :(X,T) \to (Y,T)$, the relation $Rc(\phi )$ induced by the components in the fibers relative to $\phi$, i.e.,
Light open and open mappings on manifolds. II
John J.
Walsh
271-284
Abstract: Sufficient conditions are given for the existence of light open mappings between p.l. manifolds. In addition, it is shown that a mapping f from a p.l. manifold ${M^m},m \geqslant 3$, to a polyhedron Q is homotopic to an open mapping of M onto Q iff the index of $ {f_\ast}({\pi _1}(M))$ in ${\pi _1}(Q)$ is finite. Finally, it is shown that an open mapping of ${M^m}$ onto a p.l. manifold ${N^n},n \geqslant m \geqslant 3$, can be approximated by a light open mapping of M onto N.
A local spectral theory for operators. V. Spectral subspaces for hyponormal operators
Joseph G.
Stampfli
285-296
Abstract: In the first part of the paper we show that the local resolvent of a hyponormal operator satisfies a rather stringent growth condition. This result enables one to show that under a mild restriction, hyponormal operators satisfy Dunford's C condition. This in turn leads to a number of corollaries concerning invariant subspaces. In the second part we consider the local spectrum of a subnormal operator. The third section is concerned with the study of quasi-triangular hyponormal operators.
$L$-functions of a quadratic form
T.
Callahan;
R. A.
Smith
297-309
Abstract: Let Q be a positive definite integral quadratic form in n variables, with the additional property that the adjoint form ${Q^\dag }$ is also integral. Using the functional equation of the Epstein zeta function, we obtain a symmetric functional equation of the L-function of Q with a primitive character $\omega \bmod q$ (additive or multiplicative) defined by $ \Sigma \omega (Q({\text{x}}))Q{({\text{x}})^{ - s}},\operatorname{Re} (s) > n/2$, where the summation extends over all $ {\text{x}} \in {Z^n},{\text{x}} \ne 0$; our result does not depend upon the usual restriction that q be relatively prime to the discriminant of Q, but rather on a much milder restriction.
Hypoellipticity of certain degenerate elliptic boundary value problems
Yakar
Kannai
311-328
Abstract: The concept of hypoellipticity for degenerate elliptic boundary value problems is defined, and its relation with the hypoellipticity of certain pseudo-differential operators on the boundary is discussed (for second order equations). A theorem covering smoothness of solutions of boundary value problems such as $ a(x)\partial u/\partial n + b(x)u = f(x)$ for the Laplace equation is proved. An almost complete characterization of hypoelliptic boundary value problems for elliptic second order equations in two dimensions is given via analysis of hypoelliptic pseudo-differential operators in one variable.
Generalized Kloosterman sums and the Fourier coefficients of cusp forms
L. Alayne
Parson
329-350
Abstract: Certain generalized Kloosterman sums connected with congruence subgroups of the modular group and suitably restricted multiplier systems of half-integral degree are studied. Then a Fourier coefficient estimate is obtained for cusp forms of half-integral degree on congruence subgroups of the modular group and the Hecke groups $G(\sqrt 2 )$ and $ G(\sqrt 3 )$.
On the jump of an $\alpha $-recursively enumerable set
Richard A.
Shore
351-363
Abstract: We discuss the proper definition of the jump operator in $\alpha $-recursion theory and prove a sample theorem: There is an incomplete $\alpha$-r.e. set with jump $0''$ unless there is precisely one nonhyperregular $\alpha$-r.e. degree. Thus we have a theorem in the first order language of Turing degrees with the jump which fails to generalize to all admissible $ \alpha$.
Continuous cohomology for compactly supported vectorfields on $R\sp{n}$
Steven
Shnider
364-377
Abstract: In this paper we study the Gelfand-Fuks cohomology of the Lie algebra of compactly supported vectorfields on ${{\mathbf{R}}^n}$ and establish the degeneracy of a certain spectral sequence at the ${E_1}$ level. We apply this result to the study of another spectral sequence introduced by Resetnikov for the cohomology of the algebra of vectorfields on ${S^n}$.
Characteristic classes for the deformation of flat connections
Huei Shyong
Lue
379-393
Abstract: In this paper, we study the secondary characteristic classes derived from flat connections. Let M be a differential manifold with flat connection ${\omega _0}$. If f is a diffeomorphism of M, then ${\omega _1} = {f^\ast}{\omega _0}$ is another flat connection. Denote by $\alpha$ the difference of these two connections. Then $\alpha$ and its exterior covariant derivative $ D\alpha$ are both tensorial forms on M. To each invariant polynomial $ \varphi$ of $ {\text{GL}}(n,{\text{R}})$, where $n = \dim M,\varphi (\alpha ;D\alpha )$ is a globally defined form on M. The class $ \{ \varphi (\alpha ;D\alpha )\} \in H(M;{\text{R}})$ for $\deg \varphi > 1$ gives rise to an obstruction of the deformability from $ {\omega _0}$ to ${\omega _1}$. In particular, we prove that $ ( + )$ and $( - )$ connections, in the sense of E. Cartan, cannot be deformed to each other.
Estimates for the $\bar \partial $-Neumann operator in weighted Hilbert spaces
Sidney L.
Hantler
395-406
Abstract: Estimates for the $\bar \partial$ operator are used to derive estimates for the Neumann operator in weighted Hilbert spaces. The technique is similar to that used to prove regularity of solutions of elliptic partial differential equations. A priori estimates are first obtained for smooth compactly supported forms and these estimates are then extended by suitable approximation results. These estimates are applied to give new bounds for the reproducing kernels in the subspaces of entire functions.