On the automorphism group of a reduced primary Abelian group
Kai
Faltings
1-25
Abstract: The classical theorems concerning isomorphisms and automorphisms of full linear groups are generalized to reduced primary Abelian groups and their automorphism groups. Also, a duality theory for (not necessarily finite) reduced Abelian p-groups is presented.
Strong convergence of functions on K\"othe spaces
Gerald
Silverman
27-35
Abstract: Let $\Lambda$ be a rearrangement invariant Köthe space over a nondiscrete group G with Haar measure $\mu$. For a function $f \in \Lambda$ and relatively compact 0-neighborhood U in G the function $\displaystyle {T_U}f(x) = \frac{1}{{\mu (U)}} \cdot \int_{U + x} {f\,d\mu }$ is continuous and also belongs to $\Lambda$. The convergence ${T_U}f \to f$ (as $U \to 0$) for the strong Köthe topology on $ \Lambda$ is involved in establishing compactness criteria for subsets of a Köthe space. The main result of this paper is a necessary and sufficient condition for convergence ${T_U}f \to f$ in the strong topology on $ \Lambda$.
A constructive approach to the theory of stochastic processes
Yuen Kwok
Chan
37-44
Abstract: Some basic problems in probability theory will be considered with the constructive point of view. Among them are the construction of measurable stochastic processes from finite joint probabilities, and the construction of interesting random variables related to a given process. These random variables include (1) the first instant when a process has spent a definite length of time in a definite set, and (2) the value of another process at such an instant.
A method for shrinking decompositions of certain manifolds
Robert D.
Edwards;
Leslie C.
Glaser
45-56
Abstract: A general problem in the theory of decompositions of topological manifolds is to find sufficient conditions for the associated decomposition space to be a manifold. In this paper we examine a certain class of decompositions and show that the nondegenerate elements in any one of these decompositions can be shrunk to points via a pseudo-isotopy. It follows then that the decomposition space is a manifold homeomorphic to the original one. As corollaries we obtain some results about suspensions of homotopy cells and spheres, including a new proof that the double suspension of a Poincaré 3-sphere is a real topological 5-sphere.
Successive approximations in ordered vector spaces and global solutions of nonlinear Volterra integral equations
Terrence S.
McDermott
57-64
Abstract: Conditions are found under which a nonlinear operator in an ordered topological vector space will have a fixed point. This result is applied to study a nonlinear Volterra integral operator in the space of continuous, real valued functions on $ [0,\infty )$ equipped with the topology of uniform convergence on compact subsets. Two theorems on the global existence of solutions to the related Volterra integral equation as limits of successive approximations are proved in this manner.
The oscillation of an operator
Robert
Whitley
65-73
Abstract: Foiaş and Singer introduced the oscillation of a bounded linear operator mapping $C(S)$ into a Banach space. Using this concept we define a generalization of the Fredholm operators T with $\mathcal{K}(T) < \infty $ and a corresponding perturbation class which contains the weakly compact operators. We show that a bounded linear operator on c is a conservative summability matrix which sums every bounded sequence if and only if it has zero oscillation at infinity.
A Fredholm theory for a class of first-order elliptic partial differential operators in ${\bf R}\sp{n}$
Homer F.
Walker
75-86
Abstract: The objects of interest are linear first-order elliptic partial differential operators with domain ${H_1}({R^n};{C^k})$ in ${L_2}({R^n};{C^k})$, the first-order coefficients of which become constant and the zero-order coefficient of which vanishes outside a compact set in $ {R^n}$. It is shown that operators of this type are ``practically'' Fredholm in the following way: Such an operator has a finite index which is invariant under small perturbations, and its range can be characterized in terms of the range of an operator with constant coefficients and a finite index-related number of orthogonality conditions.
Characterization of precompact maps, Schwartz spaces and nuclear spaces
Dan
Randtke
87-101
Abstract: A general representation theorem for ``precompact'' seminorms on a locally convex space is proven. Using this representation theorem the author derives a representation theorem for precompact maps from one locally convex space into another, that is analogous to the spectral representation theorem for compact maps from one Hilbert space into another and that is applicable to a very extensive class of locally convex spaces. The author uses his representation theorem to derive new characterizations of Schwartz spaces and proves analogous results for nuclear and strongly nuclear spaces.
Set-valued measures
Zvi
Artstein
103-125
Abstract: A set-valued measure is a $\sigma$-additive set-function which takes on values in the nonempty subsets of a euclidean space. It is shown that a bounded and non-atomic set-valued measure has convex values. Also the existence of selectors (vector-valued measures) is investigated. The Radon-Nikodym derivative of a set-valued measure is a set-valued function. A general theorem on the existence of R.-N. derivatives is established. The techniques require investigations of measurable set-valued functions and their support functions.
Sets of uniqueness on the $2$-torus
Victor L.
Shapiro
127-147
Abstract: $ {H^{(J)}}$-sets are defined on the 2-torus and the following results are established: (1) ${H^{(J)}}$-sets are sets of uniqueness both for Abel summability and circular convergence of double trigonometric series; (2) a countable union of closed sets of uniqueness of type (A) (i.e., Abel summability) is also a set of uniqueness of type (A).
Principal local ideals in weighted spaces of entire functions
James J.
Metzger
149-158
Abstract: This paper deals with principal local ideals in a class of weighted spaces of entire functions of one variable. Let $\rho > 1$ and $q > 1$, and define ${E_I}[\rho ,q]$ (respectively, ${E_P}[\rho ,q]$) to be the space of all entire functions f of one variable which satisfy $\vert f(x + iy)\vert = O(\exp (A\vert x{\vert^\rho } + A\vert y{\vert^q}))$ for some (respectively, all) $A > 0$. It is shown that in each of the spaces $ {E_I}[\rho ,q]$ and ${E_P}[\rho ,q]$, the local ideal generated by any one function coincides with the closed ideal generated by the function. This result yields consequences for convolution on these spaces. It is also proved that when $\rho \ne q$ a division theorem fails to hold for either space $ {E_I}[\rho ,q]$ or ${E_P}[\rho ,q]$.
A representation theorem for functions holomorphic off the real axis
Albert
Baernstein
159-165
Abstract: Let f be holomorphic in the union of the upper and lower half planes, and let $p \in [1,\infty )$. We prove that there exists an entire function $\varphi$ and a sequence $\{ {f_n}\}$ in ${L^p}(R)$ satisfying $\left\Vert {{f_n}} \right\Vert _p^{1/n} \to 0$ such that $\displaystyle f(z) = \varphi (z) + \sum\limits_{n = 0}^\infty {\int_{ - \infty }^\infty {{{(t - z)}^{ - n - 1}}{f_n}(t)dt.} }$ This complements an earlier result of the author's on representation of function holomorphic outside a compact subset of the Riemann sphere. A principal tool in both proofs is the Köthe duality between the spaces of functions holomorphic on and off a subset of the sphere. A corollary of the present result is that each hyperfunction of one variable can be represented by a sum of Cauchy integrals over the real axis.
An energy inequality for higher order linear parabolic operators and its applications
David
Ellis
167-206
Abstract: A generalization of the classical energy inequality is obtained for evolution operators $(\partial /\partial t)I - H(t){\Lambda ^{2k}} - J(t)$, associated with higher order linear parabolic operators with variable coefficients. Here $H(t)$ and $J(t)$ are matrices of singular integral operators. The key to the result is an algebraic inequality involving matrices similar to the symbol of $H(t)$ having their eigenvalues contained in a fixed compact subset of the open left-half complex plane. Then a sharp estimate on the norms of certain imbedding maps is obtained. These estimates along with the energy inequality is applied to the Cauchy problem for higher order linear parabolic operators restricted to slabs in $ {R^{n + 1}}$.
Weighted norm inequalities for the Hardy maximal function
Benjamin
Muckenhoupt
207-226
Abstract: The principal problem considered is the determination of all nonnegative functions, $U(x)$, for which there is a constant, C, such that $\displaystyle \int_J {{{[{f^ \ast }(x)]}^p}U(x)dx \leqq C\int_J {\vert f(x){\vert^p}U(x)dx,} }$ where $1 < p < \infty$, J is a fixed interval, C is independent of f, and ${f^ \ast }$ is the Hardy maximal function, $\displaystyle {f^ \ast }(x) = \mathop {\sup }\limits_{y \ne x;y \in J} \frac{1}{{y - x}}\int_x^y {\vert f(t)\vert dt.}$ The main result is that $ U(x)$ is such a function if and only if $\displaystyle \left[ {\int_I {U(x)dx} } \right]{\left[ {\int_I {{{[U(x)]}^{ - 1/(p - 1)}}dx} } \right]^{p - 1}} \leqq K\vert I{\vert^p}$ where I is any subinterval of J, $\vert I\vert$ denotes the length of I and K is a constant independent of I. Various related problems are also considered. These include weak type results, the problem when there are different weight functions on the two sides of the inequality, the case when $p = 1$ or $ p = \infty$, a weighted definition of the maximal function, and the result in higher dimensions. Applications of the results to mean summability of Fourier and Gegenbauer series are also given.
Sequences having an effective fixed-point property
T. H.
Payne
227-237
Abstract: Let $\alpha$ be any function whose domain is the set N of all natural numbers. A subset B of N precompletes the sequence $\alpha$ if and only if for every partial recursive function (p.r.f.) $\psi$ there is a recursive function f such that $\alpha f$ extends $ \alpha \psi$ and $ f[N - \operatorname{Dom} \psi ] \subset B$. An object e in the range of $ \alpha$ completes $ \alpha$ if and only if ${\alpha ^{ - 1}}[\{ e\} ]$ precompletes $ \alpha$. The theory of completed sequences was introduced by A. I. Mal'cev as an abstraction of the theory of standard enumerations. In this paper several results are obtained by refining and extending his methods. It is shown that a sequence is precompleted (by some B) if and only if it has a certain effective fixed-point property. The completed sequences are characterized, up to a recursive permutation, as the composition $F\varphi$ of an arbitrary function F defined on the p.r.f.'s with a fixed standard enumeration $\varphi$ of the p.r.f.'s. A similar characterization is given for the precompleted sequences. The standard sequences are characterized as the precompleted indexings which satisfy a simple uniformity condition. Several further properties of completed and precompleted sequences are presented, for example, if B precompletes $\alpha$ and S and T are r.e. sets such that ${\alpha ^{ - 1}}[\alpha [S]] \ne N$ and $ {\alpha ^{ - 1}}[\alpha [T]] \ne N$, then $ B - (S \cup T)$ precompletes $\alpha$.
Upon a convergence result in the theory of the Pad\'e table
P.
Wynn
239-249
Abstract: The main theorem of this paper is the following: Let $ {M_\nu },{b_\nu }(\nu = 1,2, \ldots ,n)$ be two sets of finite positive real numbers, with $ {b_1} > {b_2} > \cdots > {b_n}$, and let $\sigma (\varsigma )$ be a bounded nondecreasing function for $a \leqq \varsigma \leqq b$ where $0 \leqq a \leqq b < {b_n}$; denote the Padé quotients derived from the series expansion of the function $\displaystyle f(z) = \sum\limits_{\nu = 1}^n {\frac{{{M_\nu }}}{{(1 + {b_\nu }z)}} + \int_a^b {\frac{{d\sigma (\varsigma )}}{{1 + z\varsigma }}} }$ in ascending powers of z by $\{ {R_{i,j}}(z)\} ;$ let $ \mathfrak{D}$ be the open disc $\vert z\vert < {b^{ - 1}}$ cut along the real segment $( - {b^{ - 1}}, - b_1^{ - 1}];$ define a progressive sequence of Padé quotients to be one in which the successor $ {R_{i'',j''}}(z)$ to ${R_{i',j'}}(z)$ is such that either $i'' > i'$ and ${R_{n + r,n + n' + r}}(z)$, where $ n'$ is a finite nonnegative integer and $ r = 0,1, \ldots ,$ converge uniformly for $ z \in \mathfrak{D}$ to $ f(z)$. From a theorem of de Montessus de Ballore the row sequence ${R_{n,n + r}}(z)(r = 0,1, \ldots )$ converges uniformly for $z \in \mathfrak{D}$ to $f(z)$. From a result of the author the backward diagonal sequences ${R_{n + r,2m - n - r}}(z)(r = 0,1, \ldots ,m - n)$ and ${R_{n + r,2m - n - r + 1}}(z)(r = 0,1, \ldots ,m - n + 1)$, where m is a finite positive integer, are, when z is real and positive, respectively monotonically decreasing and monotonically increasing. Hence the result of the theorem is true for the restricted progressive sequences in question when z is real and positive. Using the result of de Montessus de Ballore, and extending a result of Nevanlinna to the theory of the Padé table in question, it is shown that there exists a finite positive integer $ r'$ such that all quotients ${R_{n + r,n + r'' + r}}(r = 0,1, \ldots ;r'' = r',r' + 1, \ldots )$ are uniformly bounded for $ \mathfrak{D}$ from which points lying in the neighborhood of the negative real axis have been excluded. Thus, using the Stieltjes-Vitali theorem, all progressive sequences of Padé quotients taken from the latter double array converge uniformly for $z \in \mathfrak{D}';$ that this result also holds for values of $ z \in \mathfrak{D}$ lying in the neighborhood of the negative real axis (and not, therefore, belonging to ${R_{n + r,n}}(z)(r = 0,1, \ldots )$. The two partial results are then combined.
On the degrees and rationality of certain characters of finite Chevalley groups
C. T.
Benson;
C. W.
Curtis
251-273
Abstract: Let $\mathcal{S}$ be a system of finite groups with (B, N)-pairs, with Coxeter system (W, R) and set of characteristic powers $\{ q\}$ (see [4]). Let A be the generic algebra of the system, over the polynomial ring $\mathfrak{o} = Q[u]$. Let K be $Q(u)$, K an algebraic closure of K, and $ {\mathfrak{o}^ \ast }$ the integral closure of $ \mathfrak{o}$ in K. For the specialization $f:u \to q$ mapping $\mathfrak{o} \to Q$, let ${f^ \ast }:{\mathfrak{o}^ \ast } \to \bar Q$ be a fixed extension of f. For each irreducible character $\chi$ of the algebra $ {A^{\bar K}}$, there exists an irreducible character ${\zeta _{\chi ,{f^ \ast }}}$ of the group $ G(q)$ in the system corresponding to q, such that $({\zeta _{\chi ,{f^ \ast }}},1_{B(q)}^{G(q)}) > 0$, and $\chi \to {\zeta _{\chi ,{f^ \ast }}}$ is a bijective correspondence between the irreducible characters of ${A^{\bar K}}$ and the irreducible constituents of $ 1_{B(q)}^{G(q)}$. Assume almost all primes occur among the characteristic powers $ \{ q\}$. The first main result is that, for each $\chi$, there exists a polynomial ${d_\chi }(t) \in Q[t]$ such that, for each specialization $f:u \to q$, the degree ${\zeta _{\chi ,{f^ \ast }}}(1)$ is given by ${d_\chi }(q)$. The second result is that, with two possible exceptions in type ${E_7}$, the characters ${\zeta _{\chi ,{f^ \ast }}}$ are afforded by rational representations of $G(q)$.
On the asymptotic behaviour of nonnegative solutions of a certain integral inequality
Gunnar A.
Brosamler
275-289
Abstract: The asymptotic behaviour of nonnegative solutions of a certain integral inequality is discussed, in the framework of a probabilistic-potential theoretic boundary theory.
A generalized Weyl equidistribution theorem for operators, with applications
J. R.
Blum;
V. J.
Mizel
291-307
Abstract: The present paper is motivated by the observation that Weyl's equidistribution theorem for real sequences on a bounded interval can be formulated in a way which is also meaningful for sequences of selfadjoint operators on a Hilbert space. We shall provide general results on weak convergence of operator measures which yield this version of Weyl's theorem as a corollary. Further, by combining the above results with the von Neumann ergodic theorem, we will obtain a Cesàro convergence property, equivalently, an ``ergodic theorem", which is valid for all (projection-valued) spectral measures whose support is in a bounded interval, as well as for the more general class of positive operator-valued measures. Within the same circle of ideas we deduce a convergence property which completely characterizes those spectral measures associated with ``strongly mixing'' unitary transformations. The final sections are devoted to applications of the preceding results in the study of complex-valued Borel measures as well as to an extension of our results to summability methods other than Cesàro convergence. In particular, we obtain a complete characterization, in purely measure theoretic terms, of those complex measures on a bounded interval whose Fourier-Stieltjes coefficients converge to zero.
Schauder bases in the Banach spaces $C\sp{k}({\bf T}\sp{q})$
Steven
Schonefeld
309-318
Abstract: A Schauder basis is constructed for the space $ {C^k}({T^q})$ of k-times continuously differentiable functions on $ {T^q}$, the product of q copies of the one-dimensional torus. This basis has the property that is also a basis for the spaces $ {C^1}({T^q}),{C^2}({T^q}), \ldots ,{C^{k - 1}}({T^q})$, and an interpolating basis for $C({T^q})$.
A characterization of compact multipliers
Gregory F.
Bachelis;
Louis
Pigno
319-322
Abstract: Let G be a compact abelian group and $\varphi$ a complex-valued function defined on the dual $\Gamma$. The main result of this paper is that $ \varphi$ is a compact multiplier of type $(p,q),1 \leqq p < \infty $ and $1 \leqq q \leqq \infty$, if and only if it satisfies the following condition: Given $\varepsilon > 0$ there corresponds a finite set $K \subset \Gamma $ such that $\vert\sum {a_\gamma }{b_\gamma }\varphi (\gamma )\vert < \varepsilon$ whenever $P = \sum {a_\gamma }\gamma $ and $Q = \sum {b_\gamma }\gamma $ are trigonometric polynomials satisfying
The local behavior of principal and chordal principal cluster sets
John T.
Gresser
323-332
Abstract: Let K be the unit circle, and let f be a function whose domain is the open unit disk and whose range is a subset of the Riemann sphere. We define a set, called the boundary principal cluster set of f at ${\zeta _0} \in K$, which characterizes the behavior of the principal cluster sets of f at points $\zeta \in K$ which are near ${\zeta _0}$ and distinct from ${\zeta _0}$. It is shown that if f is continuous, then the principal and boundary principal cluster sets of f at $ {\zeta _0}$ are equal for nearly every point $ {\zeta _0} \in K$. A similar result holds for chordal principal cluster sets. Examples are provided that indicate directions in which the result cannot be improved. Some results concerning points that are accessible through sets which are unions of arcs are also presented.
Removable sets for pointwise solutions of elliptic partial differential equations
Jim
Diederich
333-352
Abstract: We prove that dense sets of zero newtonian capacity are removable for bounded generalized pointwise solutions of second order elliptic equations.
Relative imaginary quadratic fields of class number $1$ or $2$
Larry Joel
Goldstein
353-364
Abstract: Let K be a normal totally real algebraic number field. It is shown how to effectively classify all totally imaginary quadratic extensions of class number 1. Let K be a real quadratic field of class number 1, whose fundamental unit has norm $- 1$. Then it is shown how to effectively classify all totally imaginary quadratic extensions of class number 2.
The $(\phi\sp{2n})\sb{2}$ field Hamiltonian for complex coupling constant
Lon
Rosen;
Barry
Simon
365-379
Abstract: We consider hamiltonians ${H_\beta } = {H_0} + \beta {H_I}(g)$, where $ {H_0}$ is the hamiltonian of a free Bose field $\phi (x)$ of mass $m > 0$ in two-dimensional space-time, ${H_I}(g) = \smallint g(x):P(\phi (x)):dx$ where $ g \geqq 0$ is a spatial cutoff and P is an arbitrary polynomial which is bounded below, and the coupling constant $ \beta$ is in the cut plane, i.e. $\beta \ne$ negative real. We show that ${H_\beta }$ generates a semigroup with hypercontractive properties and satisfies higher order estimates of the form $\left\Vert {{H_0}{N^r}R_\beta ^s} \right\Vert < \infty$, where N is the number operator, ${R_\beta } = {({H_\beta } + b)^{ - 1}}$, r a positive integer, and $\beta$, s, and b are suitably chosen. For any $ 0 \leqq \Theta < \pi$, ${R_\beta }$ converges in norm to ${R_0}$ as $\vert\beta \vert \to 0$ with $\vert\arg \beta \vert \leqq \Theta$. Finally we discuss applications of these results and establish asymptotic series and Borel summability for various objects in the real $\beta$ theory.
$L\,\sb{p}$ derivatives and approximate Peano derivatives
Michael J.
Evans
381-388
Abstract: It is known that approximate derivatives and kth Peano derivatives share several interesting properties with ordinary derivatives. In this paper the author points out that kth ${L_p}$ derivatives also share these properties. Furthermore, a definition for a kth approximate Peano derivative is given which generalizes the notions of a kth Peano derivative, a kth $ {L_p}$ derivative, and an approximate derivative. It is then shown that a kth approximate Peano derivative at least shares the property of belonging to Baire class one with these other derivatives.
Trace algebras
R. P.
Sheets
389-423
Abstract: We give an algebraic unification for those mathematical structures which possess the abstract properties of finite-dimensional vector spaces: scalars, duality theories, trace functions, etc. The unifying concept is the ``trace algebra,'' which is a set with a ternary operation which satisfies certain generalized associativity and identity laws. Every trace algebra induces naturally an object which (even though no additive structure may be available) possesses a summation operator and inner product which obey the Fourier expansion and other familiar properties. We construct the induced object in great detail. The ultimate results of the paper are: a theorem which shows that the induced object of a ``well-behaved'' trace algebra determines it uniquely; and a theorem which shows that well-behaved trace algebras look, formally, like the trace algebras associated with finite-dimensional vector spaces.
Linearization for the Boltzmann equation
F. Alberto
Grünbaum
425-449
Abstract: In this paper we compare the nonlinear Boltzmann equation appearing in the kinetic theory of gases, with its linearized version. We exhibit an intertwining operator for the two semigroups involved. We do not assume from the reader any familiarity with Boltzmann's equation but rather start from scratch.
$2$-groups of normal rank $2$ for which the Frattini subgroup has rank $3$
Marc W.
Konvisser
451-469
Abstract: All finite 2-groups G with the following property are classified: Property. The Frattini subgroup of G contains an abelian subgroup of rank 3, but G contains no normal abelian subgroup of rank 3. The method of classification involves showing that if G is such a group, then G contains a normal abelian subgroup W isomorphic to ${Z_4} \times {Z_4}$, and that the centralizer C of W in G has an uncomplicated structure. The groups with the above property are then constructed as extensions of C.
Existence theorems for infinite particle systems
Thomas M.
Liggett
471-481
Abstract: Sufficient conditions are given for a countable sum of bounded generators of semigroups of contractions on a Banach space to be a generator. This result is then applied to obtain existence theorems for two classes of models of infinite particle systems. The first is a model of a dynamic lattice gas, while the second describes a lattice spin system.
Integrability of expected increments of point processes and a related random change of scale
F.
Papangelou
483-506
Abstract: Given a stationary point process with finite intensity on the real line R, denote by $N(Q)$ (Q Borel set in R) the random number of points that the process throws in Q and by ${\mathcal{F}_t}(t \in R)$ the $\sigma$-field of events that happen in $( - \infty ,t)$. The main results are the following. If for each partition $ \Delta = \{ b = {\xi _0} < {\xi _1} < \cdots < {\xi _{n + 1}} = c\}$ of an interval [b, c] we set ${S_\Delta }(\omega ) = \sum\nolimits_{\nu = 0}^n {E(N[{\xi _\nu },{\xi _{\nu + 1}})\vert{\mathcal{F}_{{\xi _\nu }}})}$ then ${\lim _\Delta }{S_\Delta }(\omega ) = W(\omega ,[b,c))$ exists a.s. and in the mean when ${\max _{0 \leqq \nu \leqq n}}({\xi _{\nu + 1}} - {\xi _\nu }) \to 0$ (the a.s. convergence requires a judicious choice of versions). If the random transformation $ t \Rightarrow W(\omega ,[0,1))$ of $ [0,\infty )$ onto itself is a.s. continuous (i.e. without jumps), then it transforms the nonnegative points of the process into a Poisson process with rate 1 and independent of ${\mathcal{F}_0}$. The ratio $ {\varepsilon ^{ - 1}}E(N[0,\varepsilon )\vert{\mathcal{F}_0})$ converges a.s. as $\varepsilon \downarrow 0$. A necessary and sufficient condition for its convergence in the mean (as well as for the a.s. absolute continuity of the function $W[0,t)$ on $ (0,\infty ))$ is the absolute continuity of the Palm conditional probability $ {P_0}$ relative to the absolute probability P on the $\sigma $-field ${\mathcal{F}_0}$. Further results are described in §1.