On smoothing compact measure spaces by multiplication
Dorothy
Maharam
1-39
Abstract: Let $\mu$ be a regular Borel probability measure on a compact Hausdorff space $S$, and let $m$ be Lebesgue measure on the unit interval $ I$. It is proved that the measure-theoretic product $(S,\mu ) \times ({I^w},{m^w})$, where $ w$ is a large enough cardinal and ${m^w}$ denotes product Lebesgue measure, is ``pseudo-isometric'' to $ ({I^w},{m^w})$. Here a pseudo-isometry $\phi$ is a point-isometry except that, instead of $\phi (A)$ being measurable for every measurable $A$, it is required only that $A$ differ by a null set from a set with measurable image. If instead $\mu$ is a Baire probability measure and $ S$ is a Baire subset of $ {I^w}$, then $(S,\mu ) \times ({I^w},{m^w})$ is point-isometric to $ ({I^w},{m^w})$. Finally it is shown that (roughly speaking) continuous maps can be ``smoothed'' into projection maps (to within pseudo-isometries) by multiplication by suitable projection maps.
On the Alder polynomials and a new generalization of the Rogers-Ramanujan identities
George E.
Andrews
40-64
Abstract: In 1954, H. L. Alder obtained certain analytic generalizations of the Rogers-Ramanujan identities that involved a sequence of polynomials subsequently named the Alder polynomials. The object here is to obtain a new representation of the Alder polynomials from which a new generalization of the Rogers-Ramanujan identities is easily deduced.
Splitting an $\alpha $-recursively enumerable set
Richard A.
Shore
65-77
Abstract: We extend the priority method in $\alpha$-recursion theory to certain arguments with no a priori bound on the required preservations by proving the splitting theorem for all admissible $\alpha$. THEOREM: Let $C$ be a regular $\alpha$-r.e. set and $D$ be a nonrecursive $\alpha$-r.e. set. Then there are regular $ \alpha$-r.e. sets $ A$ and $B$ such that $ A \cup B = C,A \cap B = \phi ,A,B{ \leq _\alpha }C$ and such that $D$ is not $\alpha$-recursive in $A$ or $B$. The result is also strengthened to apply to $ { \leq _{c\alpha }}$, and various corollaries about the structure of the $ \alpha$ and $ c\alpha$ recursively enumerable degrees are proved.
On $J$-convexity and some ergodic super-properties of Banach spaces
Antoine
Brunel;
Louis
Sucheston
79-90
Abstract: Given two Banach spaces $F\vert\vert$ and $X\vert\vert\,\vert\vert$, write $F{\text{ fr }}X{\text{ iff}}$ for each finite-dimensional subspace $F'$ of $F$ and each number $ \varepsilon > 0$, there is an isomorphism $V$ of $F'$ into $X$ such that $\vert\vert x\vert - \vert\vert Vx\vert\vert\vert \leq \varepsilon$ for each $x$ in the unit ball of $F'$. Given a property ${\mathbf{P}}$ of Banach spaces, $X$ is called super- $ {\mathbf{P}}{\text{ iff }}F{\text{ fr }}X$ implies $F$ is $ {\mathbf{P}}$. Ergodicity and stability were defined in our articles On $ B$-convex Banach spaces, Math. Systems Theory 7 (1974), 294-299, and C. R. Acad. Sci. Paris Ser. A 275 (1972), 993, where it is shown that super-ergodicity and super-stability are equivalent to super-reflexivity introduced by R. C. James [Canad. J. Math. 24 (1972), 896-904]. $Q$-ergodicity is defined, and it is proved that super-$Q$-ergodicity is another property equivalent with super-reflexivity. A new proof is given of the theorem that $ J$-spaces are reflexive [Schaffer-Sundaresan, Math. Ann. 184 (1970), 163-168]. It is shown that if a Banach space $X$ is $B$-convex, then each bounded sequence in $ X$ contains a subsequence $ ({y_n})$ such that the Cesàro averages of $ {( - 1)^i}{y_i}$ converge to zero.
Rings of continuous functions with values in a topological field
George
Bachman;
Edward
Beckenstein;
Lawrence
Narici;
Seth
Warner
91-112
Abstract: Let $F$ be a complete topological field. We undertake a study of the ring $C(X,F)$ of all continuous $ F$-valued functions on a topological space $X$ whose topology is determined by $C(X,F)$, in that it is the weakest making each function in $C(X,F)$ continuous, and of the ring ${C^\ast }(X,F)$ of all continuous $F$-valued functions with relatively compact range, where the topology of $X$ is similarly determined by ${C^\ast }(X,F)$. The theory of uniform structures permits a rapid construction of the appropriate generalizations of the Hewitt realcompactification of $X$ in the former case and of the Stone-Čech compactification of $X$ in the latter. Most attention is given to the case where $F$ and $X$ are ultraregular; in this case we determine conditions on $F$ that permit a development parallel to the classical theory where $F$ is the real number field. One example of such conditions is that the cardinality of $ F$ be nonmeasurable and that the topology of $F$ be given by an ultrametric or a valuation. Measure-theoretic interpretations are given, and a nonarchimedean analogue of Nachbin and Shirota's theorem concerning the bornologicity of $ C(X)$ is obtained.
Conservative dynamical systems involving strong forces
William B.
Gordon
113-135
Abstract: We consider conservative dynamical systems associated with potentials $ V$ which have singularities at a set $ S:V(x) \to - \infty$ as $x \to S$. It is shown that various ``action'' integrals satisfy Condition $ C$ of Palais and Smale provided that the potential satisfy a certain strong force (SF) condition. Hence, e.g., we establish the existence in SF systems of periodic trajectories which wind around $S$ and have arbitrary given topological (homotopy) type and which have arbitrary given period, and also periodic trajectories which make arbitrarily tight loops around $S$. Similar results are also obtained concerning the existence of trajectories which wind around $ S$ and join two given points. The SF condition is shown to be closely related to the completeness (in the riemannian sense) of certain Jacobi metrics associated with the potential $ V$, and this fact permits the use of the standard results of riemannian geometry in the analysis of SF systems. The SF condition excludes the gravitational case, and the action integrals do not satisfy the Palais-Smale condition in the gravitational case. The Jacobi metrics associated with gravitational potentials are not complete. For SF systems there exist trajectories which join two given points and make arbitrarily tight loops around $S$, and this is not the case in the gravitational two body problem. On the other hand, for SF systems any smooth family of $\lambda$-periodic trajectories ($\lambda$ fixed) is bounded away from $ S$, and this also is not the case for gravitational systems. Thus the definition of the SF condition is ``well motivated", and leads to the disclosure of certain differences between the behavior of SF systems and gravitational (and other weak force) systems.
The Brauer group of graded Azumaya algebras. II. Graded Galois extensions
Lindsay N.
Childs
137-160
Abstract: This paper continues the study of the Brauer group ${B_\phi }(R,G)$ of $G$-graded Azumaya $R$-algebras begun in [5]. A group $ {\operatorname{Galz} _\phi }(R,G)$ of graded Galois extensions is constructed which always contains, and often equals, the cokernel of ${B_\phi }(R,G)$ modulo the usual Brauer group of $R$. Sufficient conditions for equality are found. The structure of ${\operatorname{Galz} _\phi }(R,G)$ is studied, and ${\operatorname{Galz} _\phi }(R,{(Z/{p^e}Z)^r})$ is computed. These results are applied to give computations of a Brauer group of dimodule algebras constructed by F. W. Long.
Symmetries of spherical harmonics
Roberto De Maria Nunes
Mendes
161-178
Abstract: Let $G$ be a group of linear transformations of $ {R^n}$ and ${H_k}(G)$ the vector space of spherical harmonics invariant under $G$. The Pálya function is the formal power series ${\Sigma _{k \geq 0}}{t^k}\dim {H_k}(G)$. In this paper, after classifying all closed subgroups of $O(4)$, we compute the Pólya functions for these groups. These functions have recently proved to be of interest in quantum mechanics and elementary particle physics.
The radicals of a semigroup
Rebecca
Slover
179-195
Abstract: This paper investigates various radicals and radical congruences of a semigroup. A strongly prime ideal is defined. It is shown that the nil radical of a semigroup is the intersection of all strongly prime ideals of the semigroup. Furthermore, a semigroup with zero element is nil if and only if it has no strongly prime ideals. We investigate the question of when the left and right radical congruence relations of various radicals are equal. Some theorems analogous to theorems concerning the radicals of rings are also proved.
Topological dynamics on $C\sp *$-algebras
Diane
Laison;
Gary
Laison
197-205
Abstract: Dynamical properties of a group of homeomorphisms of a compact Hausdorff space $X$ can be interpreted in terms of the commutative $ {C^\ast }$-algebra $ C(X)$. We investigate a noncommutative topological dynamics extending dynamical concepts to the context of a group of automorphisms on a general ${C^\ast }$-algebra with unit. Such concepts as minimality, almost periodicity, and point-wise almost periodicity are extended to this situation. Theorems are obtained extending commutative dynamical results and relating the noncommutative dynamics to the transformation groups induced on the state space and the weak* closure of the pure states. We show, for example, that the group acts almost periodically on the $ {C^\ast }$-algebra if and only if each of these induced transformation groups is almost periodic.
A canonical form for symmetric and skew-symmetric extended symplectic modular matrices with applications to Riemann surface theory
Robert
Zarrow
207-227
Abstract: The (extended) symplectic modular group $({ \wedge _n}){\Gamma _n}$ is the set of all $2n \times 2n$ integer matrices $M$ such that $\displaystyle (M{J^t}M = \pm J),M{J^t}M = J,J = \left[ {\begin{array}{*{20}{c}} 0 & I { - I} & 0 \end{array} } \right],$ $I$ being the $n \times n$ identity matrix. Let ${S_n} = \{ M \in { \wedge _n} - {\Gamma _n}\vert M = - {}^tM\}$ and ${T_n} = \{ M \in { \wedge _n} - {\Gamma _n}\vert M = {}^tM\}$. We say $M \sim N$ if there exists $K \in {\Gamma _n}$ such that $M = KN{}^tK$. This defines an equivalence relation on each of these sets separately and we obtain a canonical form for this equivalence. We use this canonical form to study two types of Riemann surfaces which are conformally equivalent to their conjugates and obtain characterizations of their period matrices. We also obtain characterizations of the symplectic matrices which the conformal equivalence induces on the first homology group. One type of surface dealt with is the symmetric Riemann surfaces, i.e. those surfaces which have a conjugate holomorphic self-map of order 2. The other type of surface studied we we call pseudo-symmetric surfaces. These are the hyperelliptic surfaces with the property that the sheet interchange is the square of a conjugate holomorphic automorphism.
On the $2$-realizability of $2$-types
Micheal N.
Dyer
229-243
Abstract: A $2$-type is a triple $(\pi ,{\pi _2},k)$, where $\pi$ is a group, ${\pi _2}$ a $\pi$-module and $k \in {H^3}(\pi ,{\pi _2})$. The following question is studied: When is a $2$-type $ (\pi ,{\pi _2},k)$ realizable by $2$-dimensional CW-complex $X$ such that the $2$-type $({\pi _1}X,{\pi _2}X,k(X))$ is equivalent to $(\pi ,{\pi _2},k)$? A long list of necessary conditions is given (2.2). One necessary and sufficient condition (3.1) is proved, provided $ \pi$ has the property that stably free, finitely generated $\pi$-modules are free. ``Stable'' $ 2$-realizability is characterized (4.1) in terms of the Wall invariant of [15]. Finally, techniques of [5] are used to extend C. T. C. Wall's Theorem F of [15] to a space $X$ which is dominated by a finite CW-complex of dimension 2, provided ${\pi _1}X$ is finite cyclic. Under these conditions $X$ has the homotopy type of a finite $2$-complex if and only if the Wall invariant vanishes.
Integration of ordinary linear differential equations by Laplace-Stieltjes transforms
James
D’Archangelo;
Philip
Hartman
245-266
Abstract: Let $R$ be a constant $N \times N$ matrix and $g(t)$ an $N \times N$ matrix of functions representable as absolutely convergent Laplace-Stieltjes transforms for $t > 0$. The paper gives sufficient conditions for certain solutions of the system
Equations with constant coefficients invariant under a group of linear transformations
André
Cerezo
267-298
Abstract: If $P$ is a linear differential operator on $ {{\mathbf{R}}^n}$ with constant coefficients, which is invariant under a group $ G$ of linear transformations, it is not true in general that the equation $ Pu = f$ always has a $ G$-invariant solution $ u$ for a $G$-invariant $f$. We elucidate here the particular case of a ``big'' group $G$, and we count the invariant solutions when they exist (see Corollary 28 and Theorems 32, 33). The case, of special interest, of the wave equation and the Lorentz group is covered (Corollary 27). The theory of hyperfunctions provides the frame for the work.
On a class of new inequalities
Daniel T.
Shum
299-341
Abstract: Inequalities of considerable interest are associated with the names of Beesack, Benson, Boyd, Calvert, Das, Hardy, Hua, Opial, Wong and Yang. In this note an elementary method used in a recent paper by Benson will be further investigated. The resultant new class of inequalities will bring a great number of inequalities--such as inequalities of Hardy's and those of Opial's--under one roof, so to speak.
Generalized quantifiers and compact logic
Saharon
Shelah
342-364
Abstract: We solve a problem of Friedman by showing the existence of a logic stronger than first-order logic even for countable models, but still satisfying the general compactness theorem, assuming e.g. the existence of a weakly compact cardinal. We also discuss several kinds of generalized quantifiers.
Obstruction theory with coefficients in a spectrum
Mark
Mahowald;
Robert
Rigdon
365-384
Abstract: In this paper an obstruction theory with coefficients in a spectrum is developed. An idea of orientability of a fiber bundle with respect to a spectrum is introduced and for bundles orientable with respect to the spectrum a resolution is produced which corresponds to a modified Postnikov tower in the classical case.
Surgery on a curve in a solid torus
J. P.
Neuzil
385-406
Abstract: We consider the following surgery question: If a regular neighborhood of a polyhedral knot in a solid torus is removed and then sewn back differently, what manifold results? We consider two classes of knots, torus knots and what we call doubly twisted knots. We obtain some related results on surgery on knots in ${S^3}$.