Irregularities of distribution. VIII
Wolfgang M.
Schmidt
1-22
Abstract: If ${x_1},{x_2} \ldots$ is a sequence in the unit interval $0 \leqslant x \leqslant 1$ and if $S$ is a subinterval, write $C(n,S)$ for the number of elements among ${x_1}, \ldots ,{x_n}$ which lie in $ S$, minus $n$ times the length of $S$. For a well distributed sequence, $ C(n,S)$ as a function of $ n$ will be small. It is shown that the lengths of the intervals $S$ for which $C(n,S)(n = 1,2, \ldots )$ is bounded form at most a countable set.
Consistency theorems for almost convergence
G.
Bennett;
N. J.
Kalton
23-43
Abstract: The concept of almost convergence of a sequence of real or complex numbers was introduced by Lorentz, who developed a very elegant theory. The purpose of the present paper is to continue Lorentz's investigations and obtain consistency theorems for almost convergence; this is achieved by studying certain locally convex topological vector spaces.
Cauchy problems for certain Isaacs-Bellman equations and games of survival
Robert J.
Elliott;
Nigel J.
Kalton
45-72
Abstract: Two person zero sum differential games of survival are considered; these terminate as soon as the trajectory enters a given closed set $F$, at which time a cost or payoff is computed. One controller, or player, chooses his control values to make the payoff as large as possible, the other player chooses his controls to make the payoff as small as possible. A strategy is a function telling a player how to choose his control variable and values of the game are introduced in connection with there being a delay before a player adopts a strategy. It is shown that various values of the differential game satisfy dynamic programming identities or inequalities and these results enable one to show that if the value functions are continuous on the boundary of $ F$ then they are continuous everywhere. To discuss continuity of the values on the boundary of $F$ certain comparison theorems for the values of the game are established. In particular if there are sub- and super-solutions of a related Isaacs-Bellman equation then these provide upper and lower bounds for the appropriate value function. Thus in discussing value functions of a game of survival one is studying solutions of a Cauchy problem for the Isaacs-Bellman equation and there are interesting analogies with certain techniques of classical potential theory.
Ordinary differential operators under Stieltjes boundary conditions
Richard C.
Brown;
Allan M.
Krall
73-92
Abstract: The operator $\int_0^1 {d\nu (t)y(t) = 0}$, is studied in $\mathcal{L}_n^p(0,1),1 \leqslant p < \infty$. It is shown that ${L_p}$ has a dense domain; hence there exists a dual operator $L_q^ +$ operating on $\mathcal{L}_n^q(0,1)$. After finding $L_q^ +$ we show that both $ {L_p}$ and $L_q^ +$ are Fredholm operators. This implies some elementary results concerning the spectrum and states of ${L_p}$. Finally two eigenfunction expansions are derived.
Linear transformations on matrices
D. Ž.
Djoković
93-106
Abstract: The real orthogonal group $O(n)$, the unitary group $U(n)$ and the symplectic group ${\text{sp(}}n{\text{)}}$ are embedded in a standard way in the real vector space of $n \times n$ real, complex and quaternionic matrices, respectively. Let $F$ be a nonsingular real linear transformation of the ambient space of matrices such that $F(G) \subset G$ where $G$ is one of the groups mentioned above. Then we show that either $ F(x) = a\sigma (x)b$ or $ F(x) = a\sigma ({x^\ast })b$ where $a,b \in G$ are fixed, ${x^\ast }$ is the transpose conjugate of the matrix $x$ and $\sigma$ is an automorphism of reals, complexes and quaternions, respectively.
Entire vectors and holomorphic extension of representations
Richard
Penney
107-121
Abstract: Let $G$ be a connected, simply connected real Lie group and let $U$ be a representation of $G$ in a complete, locally convex, topological vector space $\mathcal{J}$. If $G$ is solvable, it can be canonically embedded in its complexification ${G_c}$. A vector $v \in \mathcal{J}$ is said to be entire for $U$ if the map $ g \to {U_g}v$ of $ G$ into $\mathcal{J}$ is holomorphically extendible to ${G_c}$. The space of entire vectors is an invariant subspace of the space of analytic vectors. $ U$ is said to be holomorphically extendible iff the space of entire vectors is dense. In this paper we consider the question of existence of holomorphic extensions We prove Theorem. A unitary representation $ U$ is holomorphically extendible to ${G_C}$ iff $G$ modulo the kernel of $U$ is type $R$ in the sense of Auslander-Moore [1]. In the process of proving the above results, we develop several interesting characterizations of entire vectors which generalize work of Goodman for solvable Lie groups and we prove a conjecture of Nelson concerning the relationship between infinitesimal representations of Lie algebras and representations of the corresponding Lie groups.
On the structure of ideals of the dual algebra of a coalgebra
David E.
Radford
123-137
Abstract: The weak-* topology is seen to play an important role in the study of various finiteness conditions one may place on a coalgebra $C$ and its dual algebra $ {C^ \ast }$. Here we examine the interplay between the topology and the structure of ideals of ${C^ \ast }$. The basic theory has been worked out for the commutative and almost connected cases (see [2]). Our basic tool for reducing to the almost connected case is the classical technique of lifting idempotents. Any orthogonal set of idempotents modulo a closed ideal of $\operatorname{Rad} {C^\ast }$ can be lifted. This technique is particularly effective when $C = {C_1}$. The strongest results we obtain concern ideals of $C_1^ \ast$. Using the properties of idempotents we show that ${C_1} = \Sigma_{x,y} {{C_x}\Lambda {C_y}}$ where ${C_x}$ and ${C_y}$ run over the simple subcoalgebras of $ C$. Our first theorem states that a coalgebra $C$ is locally finite and ${C_0}$ is reflexive if and only if every cofinite ideal of ${C^ \ast }$ contains a finitely generated dense ideal. We show in general that a cofinite ideal $I$ which contains a finitely generated dense ideal is not closed. (In fact either equivalent condition of the theorem does not imply $C$ reflexive.) The preceding statement is true if $C = {C_1}$, or more importantly if $ I \supset \operatorname{Rad} {C^\ast }$ and $ {C^ \ast }/I$ is algebraic. The second theorem characterizes the closure of an ideal with cofinite radical which also contains a finitely generated dense ideal.
The Wedderburn principal theorem for a generalization of alternative algebras
Harry F.
Smith
139-154
Abstract: A generalized alternative ring I is a nonassociative ring $ R$ in which the identities $ (wx,y,z) + (w,x,[y,z]) - w(x,y,z) - (w,y,z)x;([w,x],y,z) + (w,x,yz) - y(w,x,z) - (w,x,y)z$; and $(x,x,x)$ are identically zero. It is here demonstrated that if $A$ is a finite-dimensional algebra of this type over a field $F$ of characteristic # 2, 3, then $A$ a nilalgebra implies $ A$ is nilpotent. A generalized alternative ring II is a nonassociative ring $ R$ in which the identities $(wx,y,z) + (w,x,[y,z]) - w(x,y,z) - (w,y,z)x$ and $ (x,y,x)$ are identically zero. Let $A$ be a finite-dimensional algebra of this type over a field $F$ of characteristic # 2. Then it is here established that (1) $A$ a nilalgebra implies $A$ is nilpotent; (2) $A$ simple with no nonzero idempotent other than 1 and $F$ algebraically closed imply $A$ itself is a field; and (3) the standard Wedderburn principal theorem is valid for $ A$.
Initial-boundary value problems for hyperbolic systems in regions with corners. II
Stanley
Osher
155-175
Abstract: In the previous paper in this series we obtained conditions equivalent to the validity of certain energy estimates for a general class of hyperbolic systems in regions with corners. In this paper we examine closely the phenomena which occur near the corners if these conditions are violated. These phenomena include: the development of strong singularities (lack of existence), travelling waves which pass unnoticed through the corner (lack of uniqueness), existence and uniqueness if and only if additional conditions are imposed at the corner, and weak solutions which are not strong solutions. We also systematically analyze the conditions for certain important problems. We discuss the physical and computational significance of these results.
Parabolic It\^o equations
Robert
Marcus
177-190
Abstract: A parabolic Itô equation is an equation of the form $\displaystyle (\partial u/\partial t)(t,\omega ) = Lu(t,\omega ) + f(u(t,\omega )) + \alpha (t,\omega ),\quad u(0) = {u_0},{u_0},u \in H.$ $H$ is a Hilbert space with scalar product $u \cdot \upsilon$ and norm $\vert \cdot \vert$. $L$ is a linear time-independent negative-definite operator from $H$ to $H$. $f$ is a Lipschitz continuous operator from $ H$ to $H$. $ \alpha (t,\omega )$ is a white noise process in $H$. Under suitable technical conditions the following results are obtained: I. A unique nonanticipating solution of (1) exists with ${\sup _t}E\{ \vert u{\vert^2}\} < \infty$. II. $u(t,\omega ) = R(t,\omega ) + V(t,\omega )$ where $R(t,\omega )$ is a stationary process and $\displaystyle \mathop {\lim }\limits_{t \to \infty } E\{ \vert V(t,\omega ){\vert^2}\} = 0.$ III. If $L$ is selfadjoint and $f$ is the gradient of a smooth functional then an explicit expression is found for the stationary density of $R(t,\omega )$ on $H$. IV. For the equation $ (\partial u/\partial t)(t,\omega ) = Lu(t,\omega ) + f(u(t,\omega )) + \varepsilon \alpha (t,\omega )$ an asymptotic expansion in $\varepsilon$ is proven which holds uniformly in $ t$.
Recurrent random walk of an infinite particle system
Frank
Spitzer
191-199
Abstract: Let $p(x,y)$ be the transition function for a symmetric irreducible recurrent Markov chain on a countable set $S$. Let ${\eta _t}$ be the infinite particle system on $ S$ moving according to simple exclusion interaction with the one particle motion determined by $p$. Assume that $p$ is such that any two particles moving independently on $S$ will sooner or later meet. Then it is shown that every invariant measure for ${\eta _t}$ is a convex combination of Bernoulli product measures $ {\mu _\alpha }$ on ${\{ 0,1\} ^s}$ with density $ 0 \leqslant \alpha = \mu [\eta (x) = 1] \leqslant 1$. Ergodic theorems are proved concerning the convergence of the system to one of the ${\mu _\alpha }$.
A characterization of the invariant measures for an infinite particle system with interactions. II
Thomas M.
Liggett
201-213
Abstract: Let $p(x,y)$ be the transition function for a symmetric, irreducible Markov chain on the countable set $ S$. Let $\eta (t)$ be the infinite particle system on $S$ with the simple exclusion interaction and one-particle motion determined by $p$. The present author and Spitzer have determined all of the invariant measures of $\eta (t)$, and have obtained ergodic theorems for $\eta (t)$, under two different sets of assumptions. In this paper, these problems are solved in the remaining case.
Oscillation criteria and growth of nonoscillatory solutions of even order ordinary and delay-differential equations
R.
Grimmer
215-228
Abstract: A number of results are presented on oscillation and growth of nonoscillatory solutions of the differential equation $ {x^{(n)}}(t) + f(t,x(t)) = 0$. It is shown that a nonoscillatory solution satisfies a first-order integral inequality while its $ (n - 1)$st derivative satisfies a first-order differential inequality. By applying the comparison principle, results are obtained by analyzing the two associated first-order scalar differential equations. In the last section it is shown that these results can be easily extended to delay-differential equations.
Geometry of Lebesgue-Bochner function spaces---smoothness
I. E.
Leonard;
K.
Sundaresan
229-251
Abstract: There exist real Banach spaces $E$ such that the norm in $E$ is of class $ {C^\infty }$ away from zero; however, for any $p,1 \leqslant p \leqslant \infty$, the norm in the Lebesgue-Bochner function space ${L_p}(E,\mu )$ is not even twice differentiable away from zero. The main objective of this paper is to give a complete determination of the order of differentiability of the norm function in this class of Banach spaces.
On vector measures
J.
Diestel;
B.
Faires
253-271
Abstract: The four sections of this paper treat four different but somewhat related topics in the theory of vector measures. In §1 necessary and sufficient conditions for a Banach space $X$ to have the property that bounded additive $ X$-valued maps on $ \sigma$-algebras be strongly bounded are presented, namely, $X$ can contain no copy of ${l_\infty }$. The next two sections treat the Jordan decomposition for measures with values in $ {L_1}$-spaces on ${c_0}(\Gamma )$ spaces and criteria for integrability of scalar functions with respect to vector measures. In particular, a different proof is presented for a result of D. R. Lewis to the effect that scalar integrability implies integrability is equivalent to noncontainment of ${c_0}$. The final section concerns the Radon-Nikodym theorem for vector measures. A generalization of a result due to E. Leonard and K. Sundaresan is given, namely, if a Banach space $X$ has an equivalent very smooth norm (in particular, a Fréchet differentiable norm) then its dual has the Radon-Nikodym property. Consequently, a $C(\Omega )$ space which is a Grothendieck space (weak-star and weak-sequential convergence in dual coincide) cannot be renormed smoothly. Several open questions are mentioned throughout the paper.
On sums and products of unbounded operators in Hilbert space
M. J. J.
Lennon
273-285
Abstract: The characteristic matrices (in the sense of Stone) of the sum and product of two closed linear operators in Hilbert space are found in terms of the characteristic matrix of each operator. From these, necessary and sufficient conditions for the domain of the sum or product to be dense are found, and a new simple condition for the density of the domain of the sum is proved. The ideas developed are applied to the direct integral decomposition of closed linear operators.
An intermediate theory for a purely inseparable Galois theory
James K.
Deveney
287-295
Abstract: Let $K$ be a finite dimensional purely inseparable modular extension of $F$, and let $L$ be an intermediate field. This paper is concerned with an intermediate theory for the Galois theory of purely inseparable extensions using higher derivations [4]. If $L$ is a Galois intermediate field and $ M$ is the field of constants of all higher derivations on $L$ over $F$, we prove that every higher derivation on $ L$ over $F$ extends to $K$ if and only if $K = L{ \otimes _M}J$ for some field $J$. Similar to classical Galois theory the distinguished intermediate fields are those which are left invariant under a standard generating set for the group of all rank $t$ higher derivations on $ K$ over $F$. We prove: $L$ is distinguished if and only if $L$ is $M$-homogeneous (4.9).
Majorization-subordination theorems for locally univalent functions. III
Douglas Michael
Campbell
297-306
Abstract: A quantitative majorization-subordination result of Goluzin and Tao Shah for univalent functions is generalized to ${\mathfrak{n}_\alpha }$, the linear invariant family of locally univalent functions of finite order $ \alpha$. If $ f(z)$ is subordinate to $ F(z)$ in the open unit disc, ${\mathfrak{n}_\alpha },1.65 \leqslant \alpha < \infty$, then $f'(z)$ is majorized by $F'(z)$ in $ \vert z\vert \leqslant (\alpha + 1) - {({\alpha ^2} + 2\alpha )^{1/2}}$. The result is sharp.