Difference equations: disconjugacy, principal solutions, Green's functions, complete monotonicity
Philip
Hartman
1-30
Abstract: We find analogues of known results on nth order linear differential equations for nth order linear difference equations. These include the concept of disconjugacy, Pólya's criterion for disconjugacy, Frobenius factorizations, generalized Sturm theorems, existence and properties of principal solutions, signs of Green's functions, and completely monotone families of solutions of equations depending on a parameter.
Multidimensional quality control problems and quasivariational inequalities
Robert F.
Anderson;
Avner
Friedman
31-76
Abstract: A machine can manufacture any one of n m-dimensional Brownian motions with drift $ {\lambda _j}$, $P_x^{{\lambda _j}}$, defined on the space of all paths $x\left( t \right)\, \in \,C\left( {\left[ {0,\,\infty } \right);\,{R^m}} \right)$. It is given that the product is a random evolution dictated by a Markov process $ \theta \left( t \right)$ with n states, and that the product is $P_x^{{\lambda _j}}$ when $\theta \left( t \right)\, = \,j,\,1\, \leqslant \,j\, \leqslant \,n$. One observes the $\sigma $-fields of $x\left( t \right)$, but not of $\theta \left( t \right)$. With each product $P_x^{{\lambda _j}}$ there is associated a cost $ {c_j}$. One inspects $ \theta$ at a sequence of times (each inspection entails a certain cost) and stops production when the state $\theta \, = \,n$ is reached. The problem is to find an optimal sequence of inspections. This problem is reduced to solving a certain elliptic quasi variational inequality. The latter problem is actually solved in a rather general case.
Quality control for Markov chains and free boundary problems
Robert F.
Anderson;
Avner
Friedman
77-94
Abstract: A machine can manufacture any one of n Markov chains $P_x^{{\lambda _j}}\,\left( {1\, \leq \,j\, \leq \,n} \right)$; the $ P_x^{{\lambda _j}}$ are defined on the space of all sequences $x\, = \,\left\{ {x\left( m \right)} \right\}\,\left( {1\, \leq \,m\, \leq \,\infty } \right)$ and are absolutely continuous (in finite times) with respect to one another. It is assumed that chains $P_x^{{\lambda _j}}$ evolve in a random way, dictated by a Markov chain $\theta \left( m \right)$ with n states, so that when $\theta \left( m \right)\, = \,j$ the machine is producing $P_x^{{\lambda _j}}$. One observes the $ \sigma$-fields of $x\left( m \right)$ in order to determine when to inspect $ \theta \left( m \right)$. With each product $ P_x^{{\lambda _j}}$ there is associated a cost ${c_j}$. One inspects $\theta$ at a sequence of times (each inspection entails a certain cost) and stops production when the state $\theta \, = \,n$ is reached. The problem is to find an optimal sequence of inspections. This problem is reduced, in this paper, to solving a certain free boundary problem. In case $n\, = \,2$ the latter problem is solved.
On the free boundary of a quasivariational inequality arising in a problem of quality control
Avner
Friedman
95-110
Abstract: In some recent work in stochastic optimization with partial observation occurring in quality control problems, Anderson and Friedman [1], [2] have shown that the optimal cost can be determined as a solution of the quasi variational inequality \begin{displaymath}\begin{gathered}Mw\left( p \right)\, + \,f\left( p \right)\, ... ... \,\psi \left( {p;\,w} \right)} \right)\, = \,0 \end{gathered} \end{displaymath} in the simplex $ {p_i}\, > \,0$, $ \sum\nolimits_{i\, = \,1}^n {{p_i}\, = \,1}$. Here f, $\psi$ are given functions of p, $ \psi$ is a functional of w, and M is a given elliptic operator degenerating on the boundary. This system has a unique solution when M does not degenerate in the interior of the simplex. The aim of this paper is to study the free boundary, that is, the boundary of the set where $w\left( p \right)\, < \,\psi \left( {p;\,w} \right)$.
On regular semigroups and their multiplication
Pierre Antoine
Grillet
111-138
Abstract: A method is given for the construction of regular semigroups in terms of groups and partially ordered sets. This describes any regular semigroup S and its multiplication by means of triples $\left( {i,\,g,\,\lambda } \right)$ with $i\, \in \,S/{\mathcal{R}}$, $ \lambda \, \in \,S/{\mathcal{L}}$ and g in the Schützenberger group of the corresponding $ {\mathcal{D}}$-class. It is shown that the multiplication on S is determined by certain simple products. Furthermore the associativity of these simple products implies associativity of the entire multiplication.
Compactifications of ${\bf C}\sp{n}$
L.
Brenton;
J.
Morrow
139-153
Abstract: Let X be a compactification of $ {{\text{C}}^n}$. We assume that X is a compact complex manifold and that $ A\, = \,X\, - \,{{\text{C}}^n}$ is a proper subvariety of X. If we suppose that A is a Kähler manifold, then we prove that X is projective algebraic, ${H^{\ast}}\left( {A,\,{\textbf{Z}}} \right)\, \cong \,{H^{\ast}}\left( {{{\textbf{P}}^{n\, - \,1}},\,{\textbf{Z}}} \right)$, and ${H^{\ast}}\left( {X,\,{\textbf{Z}}} \right)\, \cong \,{H^{\ast}}\left( {{{\textbf{P}}^n},\,{\textbf{Z}}} \right)$. Various additional conditions are shown to imply that $X\, = \,{{\textbf{P}}^n}$. It is known that no additional conditions are needed to imply $ X\, = \,{{\textbf{P}}^n}$ in the cases $ n\, = \,1,\,2$. In this paper we prove that if $n\, = \,3$, $X\, = \,{{\textbf{P}}^3}$.
Automorphisms of ${\rm GL}\sb{n}(R)$
B. R.
McDonald
155-171
Abstract: Let R denote a commutative ring having 2 a unit. Let $ {\text{G}}{{\text{L}}_n}\left( R \right)$ denote the general linear group of all $n\, \times \,n$ invertible matrices over R. Let $\wedge$ be an automorphism of $ {\text{G}}{{\text{L}}_n}\left( R \right)$. An automorphism $\wedge$ is ``stable'' if it behaves properly relative to families of commuting involutions (see §IV). We show that if R is connected, i.e., 0 and 1 are only idempotents, then all automorphisms $\wedge$ are stable. Further, if $n\, \geqslant \,3$, R is an arbitrary commutative ring with 2 a unit, and $\wedge$ is a stable automorphism, then we obtain a description of $\wedge$ as a composition of standard automorphisms.
Quantization and projective representations of solvable Lie groups
Henri
Moscovici;
Andrei
Verona
173-192
Abstract: Kostant's quantization procedure is applied for constructing irreducible projective representations of a solvable Lie group from symplectic homogeneous spaces on which the group acts. When specialized to a certain class of such groups, including the exponential ones, the technique exposed in the present paper provides a complete parametrization of all irreducible projective representations.
The Fuglede commutativity theorem modulo the Hilbert-Schmidt class and generating functions for matrix operators. I
Gary
Weiss
193-209
Abstract: We prove the following statements about bounded linear operators on a separable, complex Hilbert space: (1) Every normal operator N that is similar to a Hilbert-Schmidt perturbation of a diagonal operator D is unitarily equivalent to a Hilbert-Schmidt perturbation of D; (2) For every normal operator N, diagonal operator D and bounded operator X, the Hilbert-Schmidt norms (finite or infinite) of $NX\, - \,XD$ and ${N^{\ast}}X\, - \,X{D^{\ast}}$ are equal; (3) If $NX\, - \,XN$ and ${N^{\ast}}X\, - \,X{N^{\ast}}$ are Hilbert-Schmidt operators, then their Hilbert-Schmidt norms are equal; (4) If X is a Hilbert-Schmidt operator and N is a normal operator so that $NX\, - \,XN$ is a trace class operator, then Trace $ \left( {NX\, - \,XN} \right)\, = \,0$; (5) For every normal operator N that is a Hilbert-Schmidt perturbation of a diagonal operator, and every bounded operator X, the Hilbert-Schmidt norms (finite or infinite) of $NX\, - \,XN$ and ${N^{\ast}}X\, - \,X{N^{\ast}}$ are equal. The main technique employs the use of a new concept which we call 'generating functions for matrices'.
Geometric convexity. III. Embedding
John
Cantwell;
David C.
Kay
211-230
Abstract: The straight line spaces of dimension three or higher which were considered by the first author in previous papers are shown to be isomorphic with a strongly open convex subset of a real vector space. To achieve this result we consider the classical descriptive geometry studied in various papers and textbooks by Pasch, Hilbert, Veblen, Whitehead, Coxeter, Robinson, and others, with the significant difference that the geometry considered here is not restricted to 3 dimensions. Our main theorem (which is well known in dimension 3) is that any such geometry is isomorphic to a strongly open convex subset of a real vector space whose ``chords'' play the role of lines.
On modular functions in characteristic $p$
Wen Ch’ing Winnie
Li
231-259
Abstract: Let $k\, = \,{{\textbf{F}}_q}\left( T \right)$ be a function field of one variable over a finite field $ {{\textbf{F}}_q}$. For a nonzero polynomial $A\, \in \,{{\textbf{F}}_q}\left[ T \right]$ one can define the modular group $\Gamma \left( A \right)$. In this paper, we continue a theme introduced by Weil, and study the $\lambda $-harmonic modular functions for $ \Gamma \left( A \right)$. The main purpose of this paper is to give a natural definition of $\lambda$-harmonic Eisenstein series for $\Gamma \left( A \right)$ so that we obtain a decomposition theory of $\lambda$-harmonic modular functions, analogous to the classical results of Hecke. That is, we prove $\displaystyle {\text{Modular}}\,{\text{Function}}\,{\text{ = }}\,{\text{Eisenstein}}\,{\text{series}}\, \oplus \,{\text{Cusp Functions}}{\text{.}}$ Moreover, the dimension of the space generated by $\lambda$-harmonic Eisenstein series for $\Gamma \left( A \right)$ is equal to the number of cusps of $\Gamma \left( A \right)$, and so is independent of $\lambda$. For the definition of $\lambda $-harmonic Eisenstein series and the proof of decomposition theory, we consider two cases: (i) $\lambda \, \ne \, \pm 2\sqrt q$ and (ii) $ \lambda \, = \, \pm 2\sqrt q$, separately. Case (i) is treated in the usual way. Case (ii), being a ``degenerate'' case, is more interesting and requires more complicated analysis.
Persistent manifolds are normally hyperbolic
Ricardo
Mañé
261-283
Abstract: Let M be a smooth manifold, $f:\,M\,\mid$ a$\,{{C}^{1}}$ diffeomorphism and $V \subset M\,{\text{a}}\,{{\text{C}}^1}$ compact submanifold without boundary invariant under f (i.e. $f\left( V \right)\, = \,V$). We say that V is a persistent manifold for f if there exists a compact neighborhood U of V such that ${ \cap _{n\, \in \,{\textbf{z}}}}\,{f^n}\left( U \right)\, = \,V$, and for all diffeomorphisms $g:\,M\,\mid$ near to f in the $ {C^1}$ topology the set ${V_g}\, = \,{ \cap _{n\, \in \,{\textbf{z}}}}{g^n}\left( U \right)$ is a ${C^1}$ submanifold without boundary $ {C^1}$ near to V. Several authors studied sufficient conditions for persistence of invariant manifolds. Hirsch, Pugh and Shub proved that normally hyperbolic manifolds are persistent, where normally hyperbolic means that there exist a Tf-invariant splitting $ TM/V\, = \,{N^s}V\, \oplus \,{N^u}V\, \oplus \,TV$ and constants $K\, > \,0$, $0\, < \,\lambda \, < \,1$ such that: \begin{displaymath}\begin{gathered}\left\Vert {{{\left( {Tf} \right)}^n}/N_x^sV}... ... \right)}}V} \right\Vert\, \leq \,K{\lambda ^n} \end{gathered} \end{displaymath} for all $n\, > \,0$, $ x\, \in \,V$. In this paper we prove the converse result, namely that persistent manifolds are normally hyperbolic.
The spectral theory of distributive continuous lattices
Karl H.
Hofmann;
Jimmie D.
Lawson
285-310
Abstract: In this paper various properties of the spectrum (i.e. the set of prime elements endowed with the hull-kernel topology) of a distributive continuous lattice are developed. It is shown that the spectrum is always a locally quasicompact sober space and conversely that the lattice of open sets of a locally quasicompact sober space is a continuous lattice. Algebraic lattices are a special subclass of continuous lattices and the special properties of their spectra are treated. The concept of the patch topology is extended from algebraic lattices to continuous lattices, and necessary and sufficient conditions for its compactness are given.
Intertwining differential operators for ${\rm Mp}(n,\,{\bf R})$ and ${\rm SU}(n,\,n)$
Hans Plesner
Jakobsen
311-337
Abstract: For each of the two series of groups, three series of representations $ {U_n}$, ${D_n}$, and ${H_n}(n \in Z)$ are considered. For each series of representations there is a differential operator with the property, that raised to the nth power $ (n > 0)$, it intertwines the representations indexed by $- n$ and n. The operators are generalizations of the d'Alembertian, the Diracoperator and a combination of the two. Unitarity of subquotients of representations indexed by negative integers is derived from the intertwining relations.
The heat equation on a compact Lie group
H. D.
Fegan
339-357
Abstract: Recently there has been much work related to Macdonald's $\eta $-function identities. In the present paper the aim is to give another proof of these identities using analytical methods. This is done by using the heat equation to obtain Kostant's form of the identities. The basic idea of the proof is to look at subgroups of the Lie group which are isomorphic to the group $SU(2)$. When this has been done the problem has essentially been reduced to that for the group $ SU(2)$, which is a classical result.
Hardy spaces of close-to-convex functions and their derivatives
Finbarr
Holland;
John B.
Twomey
359-372
Abstract: Let $ f(z) = \sum\nolimits_1^\infty {{a_n}} {z^n}$ be close-to-convex on the unit disc. It is shown that (a) if $\lambda > 0$, then f belongs to the Hardy space $ {H^\lambda }$ if and only if ${\sum {{n^{\lambda - 2}}\left\vert {{a_n}} \right\vert} ^\lambda }$ is finite and that (b) if $0 < \lambda < 1$, then $ \sum {{n^{2\lambda - 2}}} {\left\vert {{a_n}} \right\vert^\lambda }$ or, equivalently,
Invariance of the $L$-regularity of compact sets in ${\bf C}\sp{N}$ under holomorphic mappings
W.
Pleśniak
373-383
Abstract: The property for a polynomially convex compact set E in $ {C^N}$ that the Siciak extremal function ${\Phi _E}$ be continuous or, equivalently, that E satisfy some Bernstein type inequality, is proved to be invariant under a large class of holomorphic mappings with values in $ {C^M}(M \leqslant N)$ including all open holomorphic mappings. Local specifications of this result are also given.
On the first occurrence of values of a character
G.
Kolesnik;
E. G.
Straus
385-394
Abstract: Let $\chi$ be a character of order $k\,(\bmod\, n)$, and let $ {g_m}(\chi )$ be the smallest positive integer at which $\chi$ attains its $(m + 1)$st nonzero value. We consider fixed k and large n and combine elementary group-theoretic considerations with the known results on character sums and sets of integers without large prime factors to obtain estimates for ${g_m}(\chi )$.
Amalgamation and inverse and regular semigroups
T. E.
Hall
395-406
Abstract: A method for proving the embeddability of semigroup amalgams is introduced. After providing necessary and sufficient conditions in terms of representations for the weak embeddability of a semigroup amalgam, it successfully deals with the embedding of inverse semigroup amalgams into inverse semigroups and the embedding of an amalgam of regular semigroups whose core is full in each member.
A characterization and sum decomposition for operator ideals
Andreas
Blass;
Gary
Weiss
407-417
Abstract: Let $L(H)$ be the ring of bounded operators on a separable Hubert space. Assuming the continuum hypothesis, we prove that in $L(H)$ every two-sided ideal that contains an operator of infinite rank is the sum of two smaller two-sided ideals. The proof involves a new combinatorial description of ideals of $L(H)$. This description is also used to deduce some related results about decompositions of ideals. Finally, we discuss the possibility of proving our main theorem under weaker assumptions than the continuum hypothesis and the impossibility of proving it without the axiom of choice.
Dense subgroups of Lie groups. II
David
Zerling
419-428
Abstract: Let G be a dense analytic subgroup of an analytic group L. Then G contains a maximal (CA) closed normal analytic suhgroup M and a closed abelian subgroup $A = Z(G) \times E$, where E is a closed vector subgroup of G, such that $G = M \cdot A$, $M \cap A = Z(G)$, $\overline M = M \cdot \overline {Z(G)}$, and $L = M \cdot \overline A$. We also indicate the extent to which a (CA) analytic group is uniquely determined by its center and a dense analytic subgroup.
Liftings and the construction of stochastic processes
Donald L.
Cohn
429-438
Abstract: It is shown that if the continuum hypothesis holds, then the use of liftings to construct modifications of stochastic processes can replace measurable processes with nonmeasurable ones. The use of liftings to choose the paths, rather than the random variables, of a stochastic process is investigated.
Rational subspaces of induced representations and nilmanifolds
R.
Penney
439-450
Abstract: Recently, Auslander and Brezin developed a technique of distinguishing between certain unitarily equivalent irreducible subspaces of ${L^2}$ of the Heisenberg nilmanifold. In this paper we extend the Auslander-Brezin technique to arbitrary induced representations of arbitrary locally compact groups. We then return to nilmanifolds, showing that the existence of a ``nice'' theory of distinguished subspaces is equivalent to the existence of square integrable representations for the group.
Parabolic function spaces with mixed norm
V. R.
Gopala Rao
451-461
Abstract: The spaces $ \mathcal{H}_\alpha ^p$ of parabolic Bessel potentials were introduced by B. F. Jones and R. J. Bagby. We prove a Sobolev-type imbedding theorem for $\mathcal{H}_\alpha ^{{p_1},{p_2}}$ (multinormed versions of $ \mathcal{H}_\alpha ^p$) when $\alpha$ is a positive integer k, $1 < {p_1}$, ${p_2} < \infty$. In particular this theorem holds for $W_{2l,l}^p$, since $ \mathcal{H}_{2l}^p \equiv W_{2l,l}^p$. We use the concepts of parabolic Riesz transforms and half-time derivatives introduced by us elsewhere.
On the existence of uniformly distributed sequences in compact topological spaces. I
V.
Losert
463-471
Abstract: We prove the existence of uniformly distributed sequences for an arbitrary probability measure on a separable dyadic space, e.g. on a separable compact topological group. Some counterexamples for the nonexistence of u.d. sequences in certain dense subsets are given.
The fixed-point construction in equivariant bordism
Russell J.
Rowlett
473-481
Abstract: Consider the bordism $ {\Omega _ {\ast}}(G)$ of smooth G-actions. If K is a subgroup of G, with normalizer NK, there is a standard $NK/K$-action on $ {\Omega _ {\ast}}(K)$(All, Proper). If M has a smooth G-action, a tubular neighborhood of the fixed set of K in M represents an element of $ {\Omega _ {\ast}}(K){({\text{All, Proper}})^{NK/K}}$. One thus obtains the ``fixed point homomorphism'' $\phi$ carrying $ {\Omega _ {\ast}}(G)$ to the sum of the ${\Omega _ {\ast}}(K){({\text{All, Proper}})^{NK/K}}$, summed over conjugacy classes of subgroups K. Let P be the collection of primes not dividing the order of G. We show that the P-localization of $ \phi$ is an isomorphism, and give several applications.
On a theorem of Steinitz and Levy
Gadi
Moran
483-491
Abstract: Let $ \sum\nolimits_{n\,\, \in \,\omega } {h(n)}$ be a conditionally convergent series in a real Banach space B. Let $S(h)$ denote the set of sums of the convergent rearrangements of this series. A well-known theorem of Riemann states that $ S(h)\, = \,B$ if $ B\, = \,R$, the reals. A generalization of Riemann's Theorem, due independently to Levy [L] and Steinitz [S], states that if B is finite dimensional, then $ S(h)$ is a linear manifold in B of dimension $> \,0$. Another generalization of Riemann's Theorem [M] can be stated as an instance of the Levy-Steinitz Theorem in the Banach space of regulated real functions on the unit interval I. This instance generalizes to the Banach space of regulated B-valued functions on I, where B is finite dimensional, implying a generalization of the Levy-Steinitz Theorem.
Some metric properties of piecewise monotonic mappings of the unit interval
Sherman
Wong
493-500
Abstract: In this note, the result of Lasota and Yorke on the existence of invariant measures for piecewise ${C^2}$ functions is extended to a larger class of piecewise continuous functions. Also the result of Li and Yorke on the existence of ergodic measures for piecewise ${C^2}$ functions is extended for the above class of functions.
Even triangulations of $S\sp{3}$ and the coloring of graphs
Jacob Eli
Goodman;
Hironori
Onishi
501-510
Abstract: A simple necessary and sufficient condition is given for the vertices of a graph, planar or not, to be properly four-colorable. This criterion involves the notion of an ``even'' triangulation of ${S^3}$ and generalizes, in a natural way, a corresponding criterion for the three-colorability of planar graphs.