A degree theory, fixed point theorems, and mapping theorems for multivalued noncompact mappings
W. V.
Petryshyn;
P. M.
Fitzpatrick
1-25
Abstract: We define and study the properties of a topological degree for ultimately compact, multivalued vector fields defined on the closures of open subsets of certain locally convex topological vector spaces. In addition to compact mappings, the class of ultimately compact mappings includes condensing mappings, generalized condensing mappings, perturbations of compact mappings by certain Lipschitz-type mappings, and others. Using this degree we obtain fixed point theorems and mapping theorems.
Local decay of solutions of conservative first order hyperbolic systems in odd dimensional space
James V.
Ralston
27-51
Abstract: This paper deals with symmetric hyperbolic systems, $\partial u/\partial t = Lu$, where L is equal to the homogeneous, constant coefficient operator ${L_0}$ for $\vert x\vert > R$. Under the hypothesis that L has simple null bicharacteristics and these propagate to infinity, local decay of solutions and completeness of the wave operators relating solutions of $ \partial u/\partial t = Lu$ and solutions of $\partial u/\partial t = {L_0}u$ are established. Results of this type for elliptic L are due to Lax and Phillips. The proof here is based, in part, on a new estimate of the regularity of the $ {L^2}$-solutions of the equation $Lu + (i\lambda + \varepsilon )u = g$ for smooth g with support in $\vert x\vert \leq R$.
Equivariant endomorphisms of the space of convex bodies
Rolf
Schneider
53-78
Abstract: We consider maps of the set of convex bodies in d-dimensional Euclidean space into itself which are linear with respect to Minkowski addition, continuous with respect to Hausdorff metric, and which commute with rigid motions. Examples constructed by means of different methods show that there are various nontrivial maps of this type. The main object of the paper is to find some reasonable additional assumptions which suffice to single out certain special maps, namely suitable combinations of dilatations and reflections, and of rotations if $d = 2$. For instance, we determine all maps which, besides having the properties mentioned above, commute with affine maps, or are surjective, or preserve the volume. The method of proof consists in an application of spherical harmonics, together with some convexity arguments.
Transversally parallelizable foliations of codimension two
Lawrence
Conlon
79-102
Abstract: We study framed foliations such that the framing of the normal bundle can be chosen to be invariant under the linear holonomy of each leaf. In codimension one there is a strong structure theory for such foliations due, e.g., to Novikov, Sacksteder, Rosenberg, Moussu. An analogous theory is developed here for the case of codimension two.
Generators for $A(\Omega )$
N.
Sibony;
J.
Wermer
103-114
Abstract: We consider a bounded domain $\Omega$ in $ {{\mathbf{C}}^n}$ and the Banach algebra $ A(\Omega )$ of all continuous functions on $ \bar \Omega$ which are analytic in $\Omega$. Fix $ {f_1}, \ldots ,{f_k}$ in $A(\Omega )$. We say they are a set of generators if $A(\Omega )$ is the smallest closed subalgebra containing the ${f_i}$. We restrict attention to the case when $ \Omega$ is strictly pseudoconvex and smoothly bounded and the ${f_i}$ are smooth on $\bar \Omega$. In this case, Theorem 1 below gives conditions assuring that a given set ${f_i}$ is a set of generators.
Cohen-Macaulay rings and ideal theory in rings of invariants of algebraic groups
Ronald E.
Kutz
115-129
Abstract: Theorem. Let R be a commutative Noetherian ring with identity. Let M = $M = ({c_{ij}})$ be an s by s symmetric matrix with entries in R. Let I the be ideal of $t + 1$ by $t + 1$ minors of M. Suppose that the grade of I is as large as possible, namely, gr $I = g = s(s + 1)/2 - st + t(t - 1)/2$. Then I is a perfect ideal, so that $R/I$ is Cohen Macaulay if R is. Let G be a linear algebraic group acting rationally on $R = K[{x_1}, \ldots ,{x_n}]$. Hochster has conjectured that if G is reductive, then $ {R^G}$ is Cohen-Macaulay, where ${R^G}$ denotes the ring of invariants of the action of G. The above theorem provides a special case of this conjecture. For $G = O(t,K)$, the orthogonal group, and K a field of characteristic zero, the above yields: Corollary. For R and G as above, ${R^G}$ is Cohen-Macaulay for an appropriate action of G. In order to obtain these results it was necessary to prove a more general form of the theorem stated above, which in turn yields a more general form of the corollary.
Square-integrable representations and the Mackey theory
Terje
Sund
131-139
Abstract: It is the purpose of this paper to clarify the relationship between the square-integrable irreducible representations of a 2nd countable unimodular locally compact group G and a closed normal subgroup N using the Mackey theory relating the dual spaces $\hat G$ and $\hat N$.
Pseudo-boundaries and pseudo-interiors in Euclidean spaces and topological manifolds
Ross
Geoghegan;
R. Richard
Summerhill
141-165
Abstract: The negligibility theorems of infinite-dimensional topology have finite-dimensional analogues. The role of the Hilbert cube $ {I^\omega }$ is played by euclidean n-space ${E^n}$, and for any nonnegative integer $k < n$, k-dimensional dense $ {F_\sigma }$-subsets of $ {E^n}$ exist which play the role of the pseudo-boundary of ${I^\omega }$. Their complements are $(n - k - 1)$-dimensional dense ${G_\delta }$ pseudo-interiors of $ {E^n}$. Two kinds of k-dimensional pseudo-boundaries are constructed, one from universal compacta, the other from polyhedra. All the constructions extend to topological manifolds.
A nonlinear Boltzmann equation in transport theory
C. V.
Pao
167-175
Abstract: The method of successive approximations is used to show the existence of a unique solution to a model of a nonlinear Boltzmann equation under the homogeneous boundary and typical initial conditions. An explicit formula in terms of the prescribed functions for the calculation of an approximate solution and its error estimate are given. This formula reveals an interesting analogy between the initial-boundary value problem of the Boltzmann equation and the Cauchy problem for ordinary differential equations. Numerical results for approximate solutions of the problem can be computed by using a computer. The linear Boltzmann equation is considered as a special case and a similar formula for the calculation of approximate solutions is included.
Regularly varying functions and convolutions with real kernels
G. S.
Jordan
177-194
Abstract: Let $\phi$ be a positive, measurable function and k a real-valued function on $(0,\infty ),k \in {L^1}(dt/t)$. We give conditions on $\phi$ and k sufficient to deduce the regular variation of $\phi$ from the assumption that $\displaystyle \alpha = \mathop {\lim }\limits_{x \to \infty } \frac{1}{{\phi (x... ...\frac{x}{2}} \right)} \;\frac{{dt}}{t}\;{\text{exits}}\;(\alpha \ne 0,\infty ).$ The general theorems extend in certain ways results of other authors and yield a new theorem on the relation between the radial growth and zero-distribution of those entire functions which are canonical products of nonintegral order with negative zeros.
On Borel mappings and Baire functions
R. W.
Hansell
195-211
Abstract: This paper studies conditions under which classes of Borel mappings (i.e., mappings such that the inverse image of open sets are Borel sets) coincide with certain classes of Baire functions (i.e., functions which belong to the smallest family containing the continuous functions and closed with respect to pointwise limits). Generalizations of the classical Lebesgue-Hausdorff and Banach theorems are obtained for the class of mappings which we call ``$\sigma$-discrete". These results are then applied to the problem of extending Borel mappings over Borel sets, and generalizations of the theorems of Lavrentiev and Kuratowski are obtained.
Parabolic equations associated with the number operator
M. Ann
Piech
213-222
Abstract: We study existence and uniqueness of solutions of the Cauchy problem for $\dot u = Nu$ where N is the number operator on abstract Wiener space.
Algebras of analytic operator valued functions
Kenneth O.
Leland
223-239
Abstract: This paper proves and generalizes the following characterization of the algebra $A(K,K)$ of complex analytic functions on open subsets of the complex plane K into K to the case of algebras of functions on a real Euclidean space E into a real Banach algebra B. Theorem. Let $F(K,K)$ be the algebra of all continuous functions on open subsets of K into K, and F a subalgebra of $ F(K,K)$ with nonconstant elements such that ${ \cup _{f \in F}}$ range $f = K,F$ is closed under uniform convergence on compact sets and domain transformations of the form $z \to {z_0} + z\sigma ,z,{z_0},\sigma \in K$. Then F is $F(K,K)$ or $A(K,K)$ or $\bar A(K,K) = \{ \bar f;f \in A(K,K)\}$. In the general case conditions on B are studied that insure that either F contains an embedment of $ F(R,R)$ and thus contains quite arbitrary continuous functions or that the elements of F are analytic and F can be expressed as a direct sum of algebras ${F_1}, \ldots ,{F_n}$ such that for $i = 1, \ldots ,n$, there exist complexifications $ {M_i}$ of E and $ {N_i}$ of ${\cup _{f \in {F_i}}}$ range f, such that with respect to ${M_i}$ and ${N_i}$ the elements of ${F_i}$ are complex differentiable.
Group algebras whose simple modules are injective
Daniel R.
Farkas;
Robert L.
Snider
241-248
Abstract: Let F be either a field of char 0 with all roots of unity or a field of char $p > 0$. Let G be a countable group. Then all simple $F[G]$-modules are injective if and only if G is locally finite with no elements of order char F and G has an abelian subgroup of finite index.
Rational points of commutator subgroups of solvable algebraic groups
Amassa
Fauntleroy
249-275
Abstract: Let G be a connected algebraic group defined over a field k. Denote by $G(k)$ the group of k-rational points of G. Suppose that A and B are closed subgroups of G defined over k. Then $ [A,B](k)$ is not equal to $[A(k),B(k)]$ in general. Here [A,B] denotes the group generated by commutators $ab{a^{ - 1}}{b^{ - 1}},a \in A,b \in B$. We say that a field of k of characteristic p is p-closed if given any additive polynomial $f(x)$ in $k[x]$ and any element c in k, there exists an element $\alpha$ in k such that $f(\alpha ) = c$. Theorem 1. Let G be a connected solvable algebraic group defined over the p-closed field k. Let A and B be closed connected subgroups of G, which are also defined over k, and suppose A normalizes B. $Then\;[A,B]\;(k) = [A(K),B(K)]$. 2. If G, A and B are as above and k is only assumed to be perfect then there exists a finite extension $ {k_0}$ of k such that if K is the maximal p-extension of $ {k_0}$, then $[A,B](K) = [A(K),B(K)]$.
Power residues and nonresidues in arithmetic progressions
Richard H.
Hudson
277-289
Abstract: Let k be an integer $\geq 2$ and p a prime such that ${v_k}(p) = (k,p - 1) > 1$. Let $bn + c(n = 0,1, \ldots ;b \geq 2,1 \leq c < b,(b,p) = (c,p) = 1)$ be an arithmetic progression. We denote the smallest kth power nonresidue in the progression $bn + c$ by $ g(p,k,b,c)$, the smallest quadratic residue in the progression $bn + c$ by $ {r_2}(p,b,c)$, and the nth smallest prime kth power nonresidue by $ {g_n}(p,k),n = 0,1,2, \ldots$. If $C(p)$ is the multiplicative group consisting of the residue classes $\bmod\;p$, then the kth powers $ \bmod\;p$ form a multiplicative subgroup, ${C_k}(p)$. Among the ${v_k}(p)$ cosets of ${C_k}(p)$ denote by T the coset to which c belongs (where c is the first term in the progression $bn + c)$, and let $ h(p,k,b,c)$ denote the smallest number in the progression $bn + c$ which does not belong to T so that $ h(p,k,b,c)$ is a natural generalization of $ g(p,k,b,c)$. We prove by purely elementary methods that $h(p,k,b,c)$ is bounded above by $ {2^{7/4}}{b^{5/2}}{p^{2/5}} + 3{b^3}{p^{1/5}} + {b^2}$ if p is a prime for which either b or $p - 1$ is a kth power nonresidue. The restriction on b and $p - 1$ may be lifted if $p > {({g_1}(p,k))^{7.5}}$. We further obtain a similar bound for ${r_2}(p,b,c)$ for every prime p, without exception, and we apply our results to obtain a bound of the order of ${p^{2/5}}$ for the nth smallest prime kth power nonresidue of primes which are large relative to $ \Pi _{j = 1}^{n - 1}{g_j}(p,k)$.
On the construction of split-face topologies
Alan
Gleit
291-299
Abstract: We give a general theorem to facilitate the construction of interesting examples of split-face topologies of compact, convex sets.
An extension of Weyl's lemma to infinite dimensions
Constance M.
Elson
301-324
Abstract: A theory of distributions analogous to Schwartz distribution theory is formulated for separable Banach spaces, using abstract Wiener space techniques. A distribution T is harmonic on an open set U if for any test function f on U, $T(\Delta f) = 0$, where $\Delta f$ fis the generalized Laplacian of f. We prove that a harmonic distribution on U can be represented as a unique measure on any subset of U which is a positive distance from $ {U^C}$. In the case where the space is finite dimensional, it follows from Weyl's lemma that the measure is in fact represented by a ${C^\infty }$ function. This functional representation cannot be expected in infinite dimensions, but it is shown that the measure has smoothness properties analogous to infinite differentiability of functions.
Analytic domination with quadratic form type estimates and nondegeneracy of ground states in quantum field theory
Alan D.
Sloan
325-336
Abstract: We present a theorem concerning the analytic domination by a semi-bounded selfadjoint operator H of another linear operator A which requires only the quadratic form type estimates $\displaystyle \left\Vert {{H^{ - 1/2}}({{({\text{ad}}\;A)}^n}H){H^{ - 1/2}}u} \right\Vert \leq {c_n}\left\Vert u \right\Vert$ instead of the norm estimates $\displaystyle \left\Vert {{{({\text{ad}}\;A)}^n}Hu} \right\Vert \leq {c_n}\left\Vert {Hu} \right\Vert$ usually required for this type of theorem. We call the new estimates ``quadratic form type", since they are sometimes equivalent to $\displaystyle \vert({({\text{ad}}\;A)^n}Hu,u)\vert \leq {c_n}\vert(Hu,u)\vert.$ The theorem is then applied with H the Hamiltonian for the spatially cutoff boson field model with real, bounded below, even ordered polynomial self-interaction in one space dimension and $A = \pi (g)$, the conjugate momentum to the free field. When the underlying Hilbert space of this model is represented as $ {L^2}(Q,dq)$ where dq is a probability measure on Q, the spectrum of the von Neumann algebra generated by bounded functions of certain field operators, then ${e^{ - tH}}$ maximizes support in the sense that ${e^{ - tH}}f$ is nonzero almost everywhere whenever f is not identically zero.
On the tensor product of $W\sp{\ast} $ algebras
Bruce B.
Renshaw
337-347
Abstract: We develop the algebra underlying the reduction theory of von Neumann in the language and spirit of Sakai's abstract ${W^ \ast }$ algebras, and using the maximum spectrum of an abelian von Neumann algebra rather than a measure-theoretic surrogate. We are thus enabled to obtain the basic fact of the von Neumann theory as a special case of a weaker general decomposition theorem, valid without separability or type restrictions, and adapted to comparison with Wright's theory in the finite case.
Shape theory and compact connected abelian topological groups
James
Keesling
349-358
Abstract: Let C denote the category of compact Hausdorff spaces and continuous maps. Let $S:C \to SC$ denote the functor of shape in the sense of Holsztyński from C to the shape category SC determined by the homotopy functor $ H:C \to HC$ from C to the homotopy category HC. Let A, B, and D denote compact connected abelian topological groups. In this paper it is shown that if G is a morphism in the shape category from A to B, then there is a unique continuous homomorphism $g:A \to B$ such that $S(g) = G$. This theorem is used in a study of shape properties of continua which support an abelian topological group structure. The following results are shown: (1) The spaces A and B are shape equivalent if and only if $A \simeq B$. (2) The space A is movable if and only if A is locally connected. (3) The space A shape dominates $B,S(A) \geq S(B)$, if and only if there is a D such that $A \simeq B \times D$. (4) The fundamental dimension of A is the same as the dimension of $A,{\text{Sd}}(A) = \dim A$. In an Appendix it is shown that the Holsztyński approach to shape and the approach of Mardešić and Segal using ANR-systems are equivalent. Thus, the results apply to either theory and to the Borsuk theory in the metrizable case.