Differentiable representations. I. Induced representations and Frobenius reciprocity
Johan F.
Aarnes
1-35
Abstract: In this paper we give the construction of the adjoint and the co-adjoint of the restriction functor in the category of differentiable G-modules, where G is a Lie group. Stated in terms of representation theory this means that two types of induced representations are introduced, both differing from the classical definition of differentiably induced representation given by Bruhat. The Frobenius reciprocity theorem is shown to hold. The main part of the paper is devoted to obtaining suitable realizations of the spaces of the induced representations. It turns out that they may be given as $ {E_K}(G,E)$ and
Norming $C(U)$ and related algebras
B. E.
Johnson
37-58
Abstract: The first result of the paper is that the question of defining a submultiplicative seminorm on the commutative unital $ {C^\ast}$ algebra $C(\Omega )$ is equivalent to that of putting a nontrivial submultiplicative seminorm on the algebra of infinitesimals in some nonstandard model of C. The extent to which the existence of such a norm on one $C(\Omega )$ implies the existence for others is investigated. Using the continuum hypothesis it is shown that the algebras of infinitesimals are isomorphic and that if such an algebra has a submultiplicative norm (or, equivalently, seminorm) then, for any totally ordered field $ \mathfrak{k}$ containing R, the R-algebra of infinitesimals in $\mathfrak{k}$ has a norm. A result of Allan is extended to show that in the particular case when $\mathfrak{k}$ is a certain field of Laurent series in several (possibly infinitely many) unknowns then the infinitesimals have a submultiplicative seminorm.
$T\sp{3}$-actions on simply connected $6$-manifolds. I
Dennis
McGavran
59-85
Abstract: We are concerned with ${T^3}$-actions on simply connected 6-manifolds $ {M^6}$. As in the codimension two case, there exists, under certain restrictions, a cross-section. Unlike the codimension two case, the orbit space need not be a disk and there can be finite stability groups. C. T. C. Wall has determined (Invent. Math. 1 (1966), 355-374) a complete set of invariants for simply connected 6-manifolds with ${H_\ast}({M^6})$ torsion-free and ${\omega _2}({M^6}) = 0$. We establish sufficient conditions for these two requirements to be met when M is a ${T^3}$-manifold. Using surgery and connected sums, we compute the invariants for manifolds satisfying these conditions. We then construct a $ {T^3}$-manifold $ {M^6}$ with invariants different than any well-known manifold. This involves comparing the trilinear forms (defined by Wall) for two different manifolds.
Some topics on equilibria
Ezio
Marchi
87-102
Abstract: In the present paper we introduce a proof for the existence of equilibrium points of a certain nonbilinear problem by using the Knaster-Kuratowski-Mazurkiewicz theorem, which turns out to be somewhat efficient for studies related to n-person games. As an application of this result, by embedding an n-person game in the ``cooperative'' set of action the existence of an equilibrium point in the strict noncooperative case and more general cases is obtained.
Differential equations on closed subsets of a Banach space
V.
Lakshmikantham;
A. Richard
Mitchell;
Roger W.
Mitchell
103-113
Abstract: The problem of existence of solutions to the initial value problem $f \in C[[{t_0},{t_0} + a] \times F,E]$, F is a locally closed subset of a Banach space E is considered. Nonlinear comparison functions and dissipative type conditions in terms of Lyapunov-like functions are employed. A new comparison theorem is established which helps in surmounting the difficulties that arise in this general setup.
Solution of Hallam's problem on the terminal comparison principle for ordinary differential inequalities
Giovanni
Vidossich
115-132
Abstract: We solve affirmatively the open problem raised by Hallam [3] and we apply this result to classical differential inequalities as well as to get existence and uniqueness theorems for the terminal value problem for ordinary differential equations.
Nonlinear differential inequalities and functions of compact support
Ray
Redheffer
133-157
Abstract: This paper is concerned with strongly nonlinear (and possibly degenerate) elliptic partial differential equations in unbounded regions. To broaden the class of problems for which solutions exist, the equation and boundary conditions are expressed by use of set-valued functions; this involves no technical complications. The concept of ``solution'' is so formulated that existence is needed only in bounded regions. Uniform boundedness is first established, and compactness of support is then deduced by a comparison argument, similar to that in recent work of Brezis, but simpler in detail. The central problems here are not associated with the comparison argument, but with the nonlinearities. Our hypotheses are given only when $ \vert{\text{grad}}\;u\vert$ is small, so that the minimal surface operator (for example) is just as tractable as the Laplacian. Further nonlinearity is allowed by the use of the Bernstein-Serrin condition on the quadratic form, and by a suitably generalized version of the Meyers-Serrin concept of essential dimension. Although the boundary can have corners, we allow nonlinear boundary conditions of mixed type. Counterexamples show that certain seemingly ad hoc distinctions are in fact necessary to the truth of the theorems.
The integral closure of a Noetherian ring
James A.
Huckaba
159-166
Abstract: Let R be a commutative ring with identity and let $ R'$ denote the integral closure of R in its total quotient ring. The basic question that this paper is concerned with is: What finiteness conditions does the integral closure of a Noetherian ring R possess? Unlike the integral domain case, it is possible to construct a Noetherian ring R of any positive Krull dimension such that $ R'$ is non-Noetherian. It is shown that if $ \dim R \leqslant 2$, then every regular ideal of $R'$ is finitely generated. This generalizes the situation that occurs in the integral domain case. In particular, it generalizes Nagata's Theorem for two-dimensional Noetherian domains.
Topological measure theory for double centralizer algebras
Robert A.
Fontenot
167-184
Abstract: The classes of tight, $\tau$-additive, and $\sigma$-additive linear functionals on the double centralizer algebra of a ${C^\ast}$-algebra A are defined. The algebra A is called measure compact if all three classes coincide. Several theorems relating the existence of certain types of approximate identities in A to measure compactness of A are proved. Next, permanence properties of measure compactness are studied. For example, the $ {C^\ast}$-algebra tensor product of two measure compact $ {C^\ast}$-algebras is measure compact. Next, the question of weak-star metrizability of the positive cone in the space of tight measures is considered. In the last part of the paper, another topology is defined and is used to study the relationship of measure compactness of A and the property that the strict topology is the Mackey topology in the pairing of $M(A)$ with the tight functionals on $ M(A)$. Also, in the last section of the paper is an extension of a result of Glickberg about finitely additive measures on pseudocompact topological spaces.
On the integrability of Jacobi fields on minimal submanifolds
D. S. P.
Leung
185-194
Abstract: Let M be a minimal submanifold of a Riemannian manifold. It is proved that every Jacobi field on M is locally the deformation vector field along M of some one-parameter families of minimal submanifolds. This fact follows from a theorem on nonlinear elliptic systems which is also proved in this paper. The related global problems are also discussed briefly.
The law of infinite cardinal addition is weaker than the axiom of choice
J. D.
Halpern;
Paul E.
Howard
195-204
Abstract: We construct a permutation model of set theory with urelements in which $ {C_2}$ (the choice principle restricted to families whose elements are unordered pairs) is false but the principle, ``For every infinite cardinal m, $2m = m$'' is true. This answers in the negative a question of Tarski posed in 1924. We note in passing that the choice principle restricted to well-ordered families of finite sets is also true in the model.
Hypersurfaces of order two
Tibor
Bisztriczky
205-233
Abstract: A hypersurface ${S^{n - 1}}$ of order two in the real projective n-space is met by every straight line in maximally two points; cf. [1, p. 391]. We develop a synthetic theory of these hypersurfaces inductively, basing it upon a concept of differentiability. We define the index and the degree of degeneracy of an ${S^{n - 1}}$ and classify the ${S^{n - 1}}$ in terms of these two quantities. Our main results are (i) the reduction of the theory of the ${S^{n - 1}}$ to the nondegenerate case; (ii) the Theorem (A.5.11) that a nondegenerate ${S^{n - 1}}$ of positive index must be a quadric and (iii) a comparison of our theory with Marchaud's discussion of ``linearly connected'' sets; cf. [3].
Generalized super-solutions of parabolic equations
Neil A.
Eklund
235-242
Abstract: Let L be a linear, second order parabolic operator in divergence form and let Q be a bounded cylindrical domain in ${E^{n + 1}}$. Super-solutions of $ Lu = 0$ are defined and generalized to three equivalent forms. Generalized super-solutions are shown to satisfy a minimum principle and form a lattice.
Relativization of the theory of computational complexity
Nancy Ann
Lynch;
Albert R.
Meyer;
Michael J.
Fischer
243-287
Abstract: The axiomatic treatment of the computational complexity of partial recursive functions initiated by Blum is extended to relatively computable functions (as computed, for example, by Turing machines with oracles). Relativizations of several results of complexity theory are carried out. A recursive relatedness theorem is proved, showing that any two relative complexity measures are related by a fixed recursive function. This theorem allows proofs of results for all measures to be obtained from proofs for a particular measure. Complexity-determined reducibilities are studied. Truth-table and primitive recursive reducibilities are proved to be reducibilities of this type. The concept of a set ``helping'' the computation of a function (by causing a saving in resource when used as an oracle in the computation of the function) is formalized. Basic properties of the helping relation are given, including nontransitivity and bounds on the amount of help certain sets can provide. Several independence results (results about sets that do not help each other's computation) are proved; they are subrecursive analogs to degrees-of-unsolvability theorems with proofs using diagonalization and priority arguments. In particular, the existence of a ``universally-helped set'' is discussed; partial results are obtained in both directions. The deepest result in the paper is a finite-injury priority argument (without an apparent recursive bound on the number of injuries) which produces sets preserving an arbitrary lower bound on the complexity of any given set.
Exact colimits and fixed points
John
Isbell;
Barry
Mitchell
289-298
Abstract: In this paper we shall give details of some work sketched in [6] on the exactness of the functor colim: $ {\text{Ab}}^\mathcal{\text{C}} \to {\text{Ab}}$. We shall also investigate the connection between this work and a paper of J. Adámek and J. Reiterman [1] characterizing those categories $ \mathcal{\text{C}}$ with the property that every endomorphism of an indecomposable functor $ \mathcal{\text{C}} \to$ Sets has a fixed point. Exactness of colim implies the fixed point property, and in some cases (such as when $ \mathcal{\text{C}}$ has only finitely many objects) both conditions turn out to be equivalent to the components of $\mathcal{\text{C}}$ being filtered. We do not expect that the two conditions are equivalent in general, although we have no example. However the category of finite ordinals and order preserving injections is an example of a connected, nonfiltered category relative to which colim is exact. This was conjectured by Mitchell, and is proved by Isbell in [5].
Favard's solution is the limit of $W\sp{k,p}$-splines
C. K.
Chui;
P. W.
Smith;
J. D.
Ward
299-305
Abstract: The purpose of this paper is to affirm a conjecture of C. de Boor, namely: The ${W^{k,p}}$-splines converge in ${W^{k,r}}[a,b]$ for all $r,1 \leqslant r < \infty$, to the Favard solution as p tends to infinity.
The Lebesgue decomposition for group-valued set functions
Tim
Traynor
307-319
Abstract: A Lebesgue-type decomposition is obtained for finitely additive set functions on a ring with values in topological groups. Corresponding results for Fréchet-Nikodým topologies are included. This generalizes Darst's result for real-valued set functions and a result of Drewnowski.
The multiplicity function of a local ring
James
Hornell
321-341
Abstract: Let A be a local ring with maximal ideal m. Let $f \in A$, and define ${\mu _A}(f)$ to be the multiplicity of the A-module $A/Af$ with respect to m. Under suitable conditions ${\mu _A}(fg) = {\mu _A}(f) + {\mu _A}(g)$. The relationship of ${\mu _A}$ to reduction of A, normalization of A and a quadratic transform of A is studied. It is then shown that there are positive integers ${n_1}, \ldots ,{n_s}$ and rank one discrete valuations $ {v_1}, \ldots ,{v_s}$ of A centered at m such that $ {\mu _A}(f) = {n_1}{v_1}(f) + \cdots + {n_s}{v_s}(f)$ for all regular elements f of A.
The fixed-point property of $(2m-1)$-connected $4m$-manifolds
S. Y.
Husseini
343-359
Abstract: Suppose that M is a $(2m - 1)$-connected smooth and compact manifold of dimension 4m. Assume that its intersection pairing is positive definite, and denote its signature by $\sigma$. Two notions are introduced. The first is that of a $ (\xi ,\lambda )$-map $ f:M \to M$ where $\xi \in K(M)$ and $\lambda$ an integer. It describes the concept of f preserving $\xi$ up to multiplication by $ \lambda$ outside a point. The second notion is that of $\xi$ being sufficiently asymmetric. It describes in terms of the Chern class of $\xi$ the concept that the restrictions of $ \xi$ to the 2m-spheres realizing a basis for ${H_{2m}}(M;Z)$ are sufficiently different so that no map which preserves $\xi$ can move the spheres among themselves. One proves that $ (\xi ,\lambda )$-maps with $ \xi$ being sufficiently asymmetric have fixed points, except possibly when $\sigma = 2$. On taking $\xi$ to be the complexification of the tangent bundle of M, one sees that mainfolds with sufficiently asymmetric tangent structures have the fixed point property with respect to a family of maps which includes diffeomorphisms. The question of the existence of $(\xi ,\lambda )$-maps as well as the question of the preservation of the fixed-point property under products are also discussed.
Decomposability of homotopy lens spaces and free cyclic group actions on homotopy spheres
Kai
Wang
361-371
Abstract: Let $\rho$ be a linear ${Z_n}$ action on ${{\text{C}}^m}$ and let $\rho$ also denote the induced ${Z_n}$ action on $ {S^{2p - 1}} \times {D^{2q}},{D^{2p}} \times {S^{2q - 1}}$ and ${S^{2p - 1}} \times {S^{2q - 1}}$ where $ p = [m/2]$ and $ q = m - p$. A free differentiable ${Z_n}$ action $({\Sigma ^{2m - 1}},\mu )$ on a homotopy sphere is $ \rho$-decomposable if there is an equivariant diffeomorphism $ \Phi$ of $ ({S^{2p - 1}} \times {S^{2q - 1}},\rho )$ such that $({\Sigma ^{2m - 1}},\mu )$ is equivalent to $(\Sigma (\Phi ),A(\Phi ))$ where $ \Sigma (\Phi ) = {S^{2p - 1}} \times {D^{2q}}{ \cup _\Phi }{D^{2p}} \times {S^{2q - 1}}$ and $A(\Phi )$ is a uniquely determined action on $\Sigma (\Phi )$ such that $A(\Phi )\vert{S^{2p - 1}} \times {D^{2q}} = \rho$ and $A(\Phi )\vert{D^{2p}} \times {S^{2q - 1}} = \rho$. A homotopy lens space is $\rho$-decomposable if it is the orbit space of a $ \rho$-decomposable free $ {Z_n}$ action on a homotopy sphere. In this paper, we will study the decomposabilities of homotopy lens spaces. We will also prove that for each lens space $ {L^{2m - 1}}$, there exist infinitely many inequivalent free $ {Z_n}$ actions on ${S^{2m - 1}}$ such that the orbit spaces are simple homotopy equivalent to $ {L^{2m - 1}}$.
One-parameter groups of isometries on Hardy spaces of the torus
Earl
Berkson;
Horacio
Porta
373-391
Abstract: The strongly continuous one-parameter groups of isometries on ${H^p}$ of the torus $(1 \leqslant p < \infty ,p \ne 2)$, as well as their generators, are classified and concretely described.
Weakly normal filters and irregular ultrafilters
A.
Kanamori
393-399
Abstract: For a filter over a regular cardinal, least functions and the consequent notion of weak normality are described. The following two results, which make a basic connection between the existence of least functions and irregularity of ultrafilters, are then proved: Let U be a uniform ultrafilter over a regular cardinal $ \kappa$. (a) If $\kappa = {\lambda ^ + }$, then U is not $(\lambda ,{\lambda ^ + })$-regular iff U has a least function f such that $\{ \xi < {\lambda ^ + }\vert{\text{cf}}(f(\xi )) = \lambda \} \in U$. (b) If $\omega \leqslant \mu < \kappa$ and U is not $(\omega ,\mu )$-regular, then U has a least function.
Free topological groups and dimension
Charles
Joiner
401-418
Abstract: For a completely regular space X we denote by $F(X)$ and $A(X)$ the free topological group of X and the free Abelian topological group of X, respectively, in the sense of Markov and Graev. Let X and Y be locally compact metric spaces with either $A(X)$ topologically isomorphic to $A(Y)$ or $F(X)$ topologically isomorphic to $F(Y)$. We show that in either case X and Y have the same weak inductive dimension. To prove these results we use a Fundamental Lemma which deals with the structure of the topology of $ F(X)$ and $A(X)$. We give other results on the topology of $F(X)$ and $A(X)$ and on the position of X in $ F(X)$ and $A(X)$.