Coterminal families and the strong Markov property
A. O.
Pittenger;
C. T.
Shih
1-42
Abstract: Let ${E_\Delta }$ be a compact metric space and assume that a strong Markov process X is defined on ${E_\Delta }$. Under the assumption that X has right continuous paths with left limits, it is shown that a version of the strong Markov property extends to coterminal families, a class of random times which can be visualized as last exit times before t from a fixed subset of $ {E_\Delta }$. Since the random times are not Markov times, the conditioning $ \sigma$-field and the new conditional probabilities must be defined. If X is also assumed to be nearly quasileft continuous, i.e. branching points are permitted, two different conditionings are possible--one on the ``past'' of the random time and one on the ``past plus present"--and two different conditional probabilities must be defined.
On the congruence lattice characterization theorem
William A.
Lampe
43-60
Abstract: A simplified proof is given for the theorem characterizing the congruence lattice of a universal algebra.
Decomposition theories for abelian categories
Joe W.
Fisher;
Harvey
Wolff
61-69
Abstract: Both the classical approach to decomposition theories and Fisher's technique of constructing decomposition theories from radical functions are extended to and exploited in the context of abelian categories. These two different approaches to decomposition theories for abelian categories intertwine in one theorem from which flows necessary and sufficient conditions for the existence of the tertiary, primary, and Bourbaki's $\mathcal{P}$-primary decomposition theories.
A fixed point theorem, a perturbed differential equation, and a multivariable Volterra integral equation
David Lowell
Lovelady
71-83
Abstract: A fixed point theorem is obtained for an equation of the form $u = T[p,f + G[u]]$. This theorem is then applied to a functionally perturbed ordinary differential equation of the form \begin{displaymath}\begin{array}{*{20}{c}} {{u_1}(s,t) = {g_1}(s,t) + \int_0^s {... ...),{u_2}(x,t),{u_3}(x,t)} \right]} \;dx,} \end{array} \end{displaymath} and, as a corollary to this, a differential equation of the form \begin{displaymath}\begin{array}{*{20}{c}} {\frac{{{\partial ^2}}}{{\part... ... {u(s,0) = \phi (s),u(0,t) = \psi (t).} \end{array} \end{displaymath} These last two equations are set in a Banach space so as to allow applications to integrodifferential equations such as \begin{displaymath}\begin{array}{*{20}{c}} {\frac{{{\partial ^2}}}{{\partial s\p... ...u(s,0,z) = \sigma (s,z),u(0,t,z) = \tau (t,z).} \end{array} \end{displaymath}
Inverse semigroups which are separated over a subsemigroup
D. B.
McAlister
85-117
Abstract: An inverse semigroup T is separated over a subsemigroup S if T is generated, as an inverse semigroup, by S and for each $ a,b,\epsilon S$ there exists $ x\;\epsilon \;Sa \cap Sb$ such that ${a^{ - 1}}a{b^{ - 1}}b = {x^{ - 1}}x$ and dually for right ideals. For example, if T is generated as an inverse semigroup by a semigroup S whose principal left and right ideals form chains under inclusion, then T is separated over S. In this paper we investigate the structure of inverse semigroups T which are separated over subsemigroups S.
Invariant differential operators on a real semisimple Lie algebra and their radial components
Mohsen
Pazirandeh
119-131
Abstract: Let $S({\mathfrak{g}_C})$ be the symmetric algebra over the complexification $ {\mathfrak{g}_C}$ of the real semisimple Lie algebra $ \mathfrak{g}$. For $u\;\epsilon \;S({\mathfrak{g}_C}),\partial (u)$ is the corresponding differential operator on $ \mathfrak{g}$. $ \mathcal{D}(\mathfrak{g})$ denotes the algebra generated by $\partial (S({\mathfrak{g}_C}))$ and multiplication by polynomials on ${\mathfrak{g}_C}$. For any open set $U \subset \mathfrak{g},{\text{Diff}}(U)$ is the algebra of differential operators with ${C^\infty }$-coefficients on U. Let $ \mathfrak{h}$ be a Cartan subalgebra of $ \pi = {\Pi _{\alpha \epsilon P}}\alpha$, P some positive system of roots. Let $\mathfrak{g}$. Harish-Chandra showed that, for each $D\;\epsilon \;{\text{Diff}}(W)$, there is a unique differential operator $(Df){\left\vert {_\mathfrak{h}' = \delta {'_\mathfrak{h}}(D)(f} \right\vert _\mathfrak{h}})$ for all G-invariant $f\epsilon \;{C^\infty }(W)$, and that if $D\;\epsilon \mathcal{D}(\mathfrak{h})$, then $\bar D\epsilon \mathcal{D}(\mathfrak{g})$. In particular $\overline {\partial (u)} = \partial (u{\vert _\mathfrak{h}}),u\;\epsilon \;S({\mathfrak{g}_C})$ and invariant. We prove these results by different, yet simpler methods. We reduce evaluation of ${\pi ^{ - 1}} \circ \delta '(D) \circ \pi \;\epsilon \;\mathcal{D}(\mathfrak{h})$ using properties of derivations $E \to \left[ {\partial (u),E} \right]$ of $ \mathcal{D}(\mathfrak{g})$ induced by $\partial (u)\;(u\;\epsilon \;S({\mathfrak{g}_C}))$ and of the algebra of polynomials on ${\mathfrak{h}_C}$ invariant under the Weyl group.
Completely seminormal operators with boundary eigenvalues
Kevin
Clancey
133-143
Abstract: For $f \in {L^2}(E)$ we consider the singular integral operator $ {T_E}f(s) = sf(s) + {\pi ^{ - 1}}{\smallint _E}f(t){(t - s)^{ - 1}}dt$. These singular integral operators are a special case of operators acting on a Hilbert space with one dimensional self-commutator. We discover generalized eigenfunctions of the equation ${T_E}f = 0$ and, for $p < 2$, we will give an ${L^p}(E)$ solution of the equation ${T_E}f = {\chi _E}$. The main result of the paper is an example of a nonzero ${L^2}(E)$ solution of ${T_E}f = 0$, with $ \lambda = 0$ a boundary point of the spectrum of ${T_E}$.
A Wedderburn theorem for alternative algebras with identity over commutative rings
W. C.
Brown
145-158
Abstract: In this paper, we study alternative algebras ${\mathbf{\Lambda }}$ over a commutative, associative ring R with identity. When ${\mathbf{\Lambda }}$ is finitely generated as an R-module, we define the radical J of ${\mathbf{\Lambda }}$. We show that matrix units and split Cayley algebras can be lifted from ${\mathbf{\Lambda }}/J$ to ${\mathbf{\Lambda }}$ when R is a Hensel ring. We also prove the following Wedderburn theorem: Let $ {\mathbf{\Lambda }}$ be an alternative algebra over a complete local ring R of equal characteristic. Suppose ${\mathbf{\Lambda }}$ is finitely generated as an R-module, and ${\mathbf{\Lambda }}/J$ is separable over $ \bar R$ ($\bar R$ the residue class field of R). Then there exists an $\bar R$-subalgebra S of $ {\mathbf{\Lambda }}$ such that $ S + J = {\mathbf{\Lambda }}$ and $ S \cap J = 0$.
The automorphism group of an abelian $p$-group and its normal $p$-subgroups
Jutta
Hausen
159-164
Abstract: Let ${\mathbf{\Gamma }}$ be the automorphism group of a nonelementary reduced abelian p-group, $p \geq 5$. It is shown that every noncentral normal subgroup of ${\mathbf{\Gamma }}$ contains a noncentral normal subgroup $ {\mathbf{\Delta }}$ of ${\mathbf{\Gamma }}$ such that ${{\mathbf{\Delta }}^p} = 1$. Furthermore, every cyclic normal subgroup of ${\mathbf{\Gamma }}$ is contained in the center of ${\mathbf{\Gamma }}$.
Existence of sum and product integrals
Jon C.
Helton
165-174
Abstract: Functions are from $R \times R$ to R, where R represents the set of real numbers. If c is a number and either (1) $ \smallint _a^b{G^2}$ exists and $\smallint _a^bG$ exists, (2) $\smallint _a^bG$ exists and $_a{{\mathbf{\Pi }}^b}(1 + G)$ exists and is not zero or (3) each of $_a{{\mathbf{\Pi }}^b}(1 + G)$ and $_a{\Pi ^b}(1 - G)$ exists and is not zero, then $\smallint _a^bcG$ exists, $\smallint _a^b\vert cG - \smallint cG\vert = 0{,_x}{{\mathbf{\Pi }}^y}(1 + cG)$ exists for $a \leq x < y \leq b$ and $ \smallint _a^b\vert 1 + cG - {\mathbf{\Pi }}(1 + cG)\vert = 0$. Furthermore, if H is a function such that ${\lim _{x \to {p^ - }}}H(x,p),{\lim _{x \to {p^ + }}}H(p,x),{\lim _{x,y \to {p^ - }}}H(x,y)$ and $ {\lim _{x,y \to {p^ + }}}H(x,y)$ exist for each $p \in [a,b],n \geq 2$ is an integer, and G satisfies either (1), (2) or (3) of the above, then $\smallint _a^bH{G^n}$ exists, $\smallint _a^b\vert H{G^n} - \smallint H{G^n}\vert = 0{,_x}{{\mathbf{\Pi }}^y}(1 + H{G^n})$ exists for $a \leq x < y \leq b$ and $ \smallint _a^b\vert 1 + H{G^n} - {\mathbf{\Pi }}(1 + H{G^n})\vert = 0$.
Overrings of commutative rings. III. Normal pairs
Edward D.
Davis
175-185
Abstract: A pair of integral domains (A, B) is a normal (resp., QR-) pair provided that A is a subring of B and all intermediate rings are normal in B (resp., rings of quotients of A). The special case of B the field of fractions of A (e.g., Prüfer domains and Dedekind domains with torsion class group) has been studied in detail. It is shown that any domain A possesses a unique overring B maximal with respect to forming a normal (resp., QR-) pair with A. An explicit description of this overring and all the intermediate rings in terms of localizations A is obtained, and further details are provided in the presence of a noetherian-like condition on A. In addition, the ``overring'' characterizations of Prüfer domains are extended to ``intermediate ring'' characterizations of normal pairs.
On the asymptotic behavior of nonlinear wave equations
Robert T.
Glassey
187-200
Abstract: Positive energy solutions of the Cauchy problem for the equation $\square u = {m^2}u + F(u)$ are considered. With $ G(u) = \smallint _0^uF(s)ds$, it is proven that $G(u)$ must be nonnegative in order for uniform decay and the existence of asymptotic ``free'' solutions to hold. When $G(u)$ is nonnegative and satisfies a growth restriction at infinity, the kinetic and potential energies (with m = 0) are shown to be asymptotically equal. In case $F(u)$ has the form $ \vert u{\vert^{p - 1}}u$, scattering theory is shown to be impossible if $1 < p \leq 1 + 2{n^{ - 1}}\;(n \geq 2)$.
New criteria for freeness in abelian groups
Paul
Hill
201-209
Abstract: A new criterion is established for an abelian group to be free. The criterion applies to an ascending chain of free subgroups. The result is used to construct groups that are almost free but not free. In particular, we construct examples that show that the class of free abelian groups is not definable in the logical language ${L_{\infty \kappa }}$ if $\kappa \leq {\aleph _2}$. In doing so, we take advantage of a recent theorem of P. Eklof.
The Cauchy problem for Douglis-Nirenberg elliptic systems of partial differential equations
Richard J.
Kramer
211-225
Abstract: Several partial answers are given to the question: Suppose U is a solution of the Douglis-Nirenberg elliptic system $LU = F$ where F is analytic and L has analytic coefficients. If $U = 0$ in some appropriate sense on a hyperplane (or any analytic hypersurface) must U vanish identically? One answer follows from introducing a so-called formal Cauchy problem for Douglis-Nirenberg elliptic systems and establishing existence and uniqueness theorems. A second Cauchy problem, in some sense a more natural one, is discussed for an important subclass of the Douglis-Nirenberg elliptic systems. The results in this case give a second partial answer to the original question. The methods of proof employed are largely algebraic. The systems are reduced to systems to which the Cauchy-Kowalewski theorem applies.
Homeomorphisms of Hilbert cube manifolds
T. A.
Chapman
227-239
Abstract: It is shown in this paper that the homeomorphism group of any compact Hubert cube manifold is locally contractible. The proof uses some standard infinite-dimensional techniques along with an infinite-dimensional version of the torus-homeomorphism idea which was used by Edwards and Kirby to establish a corresponding finite-dimensional result.
On the Green's function for the biharmonic equation in an infinite wedge
Joseph B.
Seif
241-260
Abstract: The Green's function for the biharmonic equation in an infinite angular wedge is considered. The main result is that if the angle a is less than ${a_1} \cong 0.812\pi$, then the Green's function does not remain positive; in fact it oscillates an infinite number of times near zero and near $\infty$. The method uses a number of transformations of the problem including the Fourier transform. The inversion of the Fourier transform is accomplished by means of the calculus of residues and depends on the zeros of a certain transcendental function. The distribution of these zeros in the complex plane gives rise to the determination of the angle $ {a_1}$. A general expression for the asymptotic behavior of the solution near zero and near infinity is obtained. This result has the physical interpretation that if a thin elastic plate is deflected downward at a point, the resulting shape taken by the plate will have ripples which protrude above the initial plane of the plate.
Existence of solutions of ordinary differential equations with generalized boundary conditions
Stephen R.
Bernfeld;
V.
Lakshmikantham
261-274
Abstract: An investigation of the existence of solutions of the nonlinear boundary value problem $x' = f(t,x,y),y' = g(t,x,y),AV(a,x(a),y(a)) + BW(a,x(a),y(a)) = {C_1},CV(b,x(b),y(b)) + DW(b,x(b),y(b)) = {C_2}$, is made. Here we assume $ g,f:[a,b] \times {R^p} \times {R^q} \to {R^p}$ are continuous, and $ V,W:[a,b] \times {R^p} \times {R^q} \to R$ are continuous and locally Lipschitz. The main techniques used are the theory of differential inequalities and Lyapunov functions.
Some highly homogeneous groups
Larry
Dornhoff
275-301
Abstract: We study finite rank 4 permutation groups which are solvable and imprimitive, with a regular normal subgroup. This means determining those groups N with a solvable automorphism group A which has only three orbits on the nonidentity elements of N.
Inequalities associated with regular and singular problems in the calculus of variations
J. S.
Bradley;
W. N.
Everitt
303-321
Abstract: An inequality of the form $- \infty < a < b \leq \infty$. The inequality is first established for functions in the domain of an operator in the Hilbert function space $ {L^2}(a,b)$ that is associated with the differential equation $- (py')' + qy = \lambda y$, and the number $ {\mu _0}$ in the inequality is the smallest number in the spectrum of this operator. An approximation theorem is given that allows the inequality to be established for the larger set of functions D. An extension of some classical results from the calculus of variations and some spectral theory is then used to give necessary and sufficient conditions for equality and to show that the constant $ {\mu _0}$ is best possible. Certain consequences of these conclusions are also discussed.
Fixed point theorems for various classes of $1$-set-contractive and $1$-ball-contractive mappings in Banach spaces
W. V.
Petryshyn
323-352
Abstract: Let X be a real Banach space, D a bounded open subset of X, and $\bar D$ the closure of D. In §1 of this paper we establish a general fixed point theorem (see Theorem 1 below) for 1-set-contractions and 1-ball-contractions $ T:\bar D \to X$ under very mild conditions on T. In addition to classical fixed point theorems of Schauder, Leray and Schauder, Rothe, Kransnoselsky, Altman, and others for T compact, Theorem 1 includes as special cases the earlier theorem of Darbo as well as the more recent theorems of Sadovsky, Nussbaum, Petryshyn, and others (see §1 for further contributions and details) for T k-set-contractive with $k < 1$, condensing, and 1-set-contractive. In §§2, 3, 4, and 5 of this paper Theorem 1 is used to deduce a number of known, as well as some new, fixed point theorems for various special classes of mappings (e.g. mappings of contractive type with compact or completely continuous perturbations, mappings of semicontractive type introduced by Browder, mappings of pseudo-contractive type, etc.) which have been recently extensively studied by a number of authors and, in particular, by Browder, Krasnoselsky, Kirk, and others (see §1 for details),
On rationally convex hulls
Richard F.
Basener
353-381
Abstract: For a compact set $X \subseteq {{\mathbf{C}}^n}$, let ${h_r}(X)$ denote the rationally convex hull of X; let $ {\mathbf{\Delta }}$ denote the closed unit disk in C; and, following Wermer, for a compact set S such that $\partial {\mathbf{\Delta }} \subseteq S \subseteq {\mathbf{\Delta }}$ let ${X_S} = S \times S \cap \partial {{\mathbf{\Delta }}^2}$. It is shown that $\displaystyle {h_r}({X_S}) = \{ (z,w) \in S \times S\vert{u_S}{(z)^ + }{u_S}(w) \leq 1\}$ where ${u_S}$ is a function on S which, in the case when S is smoothly bounded, is specified by requiring $ {u_S}{\vert _{\partial {\mathbf{\Delta }}}} = 0,{u_S}{\vert _{\partial S\backslash \partial {\mathbf{\Delta }}}} = 1$ and ${u_S}{\vert _{\operatorname{int} S}}$ harmonic. In particular this provides a precise description of ${h_r}(X)$ for certain sets $X \subseteq {{\mathbf{C}}^2}$ with the property that $ {h_r}(X) \ne X$, but $ {h_r}(X)$ does not contain analytic structure (as Wermer demonstrated, there are S for which $X = {X_S}$ has these properties). Furthermore, it follows that whenever ${h_r}({X_S}) \ne {X_S}$ then there is a Gleason part of ${h_r}({X_S})$ for the algebra $ R({X_S})$ with positive four-dimensional measure. In fact, the Gleason part of any point $(z,w) \in {h_r}({X_S}) \cap \operatorname{int} {{\mathbf{\Delta }}^2}$ such that ${u_S}(z) + {u_S}(w) < 1$ has positive four-dimensional measure. A similar idea is then used to construct a compact rationally convex set $ Y \subseteq {{\mathbf{C}}^2}$ such that each point of Y is a peak point for $ R(Y)$ even though $R(Y) \ne C(Y)$; namely, $Y = {\tilde X_T} = \{ (z,w) \in {{\mathbf{C}}^2}\vert z \in T,\vert w\vert = \sqrt {1 - \vert z{\vert^2}} \}$ where T is any compact subset of $\operatorname{int} {\mathbf{\Delta }}$ having the property that $ R(T) \ne C(T)$ even though there are no nontrivial Jensen measures for $ R(T)$. This example is more concrete than the original example of such a uniform algebra which was discovered by Cole. It is possible to show, for instance, that $ R({\tilde X_T})$ is not even in general locally dense in $C({\tilde X_T})$, a possibility which had been suggested by Stuart Sidney. Finally, smooth examples (3-spheres in $ {{\mathbf{C}}^6}$) with the same pathological properties are obtained from $ {X_S}$ and ${\tilde X_T}$.
Limit properties of Poisson kernels of tube domains
Lawrence J.
Dickson
383-401
Abstract: If certain local boundary conditions hold near $P \in \partial \Gamma$, the Poisson kernel belonging to a proper cone $\Gamma \subset {{\mathbf{R}}^n}$ converges to a tight $C_0^\ast$ limit as its parameter converges admissibly to P in $\Gamma$. This limit can be identified with a lower-dimensional Poisson kernel. The result always works for polytopic and ``rounded'' cones; for these, a result on the decrease at infinity is obtained which in fact implies convergence almost everywhere in the appropriate sense of the Poisson integral to certain of its boundary values.
Visitations of ruled sums
Leonard E.
Baum;
H. H.
Stratton
403-430
Abstract: Let $\{ {X_i}\}$ be a sequence of independent identically distributed random variables and for $D \subseteq {I^ + }$ let ${S_D} = {\Sigma _{i \in D}}{X_i}$. A rule $ (\;)$ is a mapping $ {I^ + } \to {2^{{I^ + }}}:\forall n\vert(n)\vert = n$ and ${S_{(\;)}} = \{ {S_{(n)}}\} $ is its associated ruled sum. Ruled sums generalize ordinary sums $ {S_n}$. Indeed, all a.e. results for ${S_n}$ can be investigated for $ {S_{(n)}}$ frequently requiring different methods and sometimes yielding different conclusions. In a previous paper we studied strong laws of large numbers and the law of the iterated logarithm. In this paper we study infinite visitation. Under suitable hypotheses on the basic distribution function F of the ${X_i}$ we show that, for all rules $(\;),{S_{(n)}}$ visits each integer infinitely often a.e. in the lattice case (or has all points of the real line as accumulation points in the nonlattice case). In fact we obtain a ``rate of visitation.'' There follows extensions of the Pólya theorem on encounters in the plane and 3-space from random walks to these ruled sums. Some equivalence relations and partial orderings on rules are defined. For normal variables this leads to an extension of the previously mentioned result for ruled sums of the type of the iterated logarithm law.
Perfect open and distinguishable multivalued maps
Eric John
Braude
431-441
Abstract: It is shown that perfect open multivalued maps preserve $\mathcal{Z}$-analytic sets (which include compact zero sets) as well as other objects of descriptive set theory. The concept of ``distinguishability", introduced by Frolík, is applied to multivalued maps, yielding a new class of such maps with similar preservation properties. That the projection of a compact zero set is a zero set is one corollary, and another is a generalized $ {\mathcal{G}_\delta }$ diagonal metrization theorem.
Inner product modules over $B\sp{\ast} $-algebras
William L.
Paschke
443-468
Abstract: This paper is an investigation of right modules over a $ {B^\ast}$-algebra B which posses a B-valued ``inner product'' respecting the module action. Elementary properties of these objects, including their normability and a characterization of the bounded module maps between two such, are established at the beginning of the exposition. The case in which B is a $ {W^\ast}$-algebra is of especial interest, since in this setting one finds an abundance of inner product modules which satisfy an analog of the self-duality property of Hilbert space. It is shown that such self-dual modules have important properties in common with both Hilbert spaces and $ {W^\ast}$-algebras. The extension of an inner product module over B by a $ {B^\ast}$-algebra A containing B as a $^\ast$-subalgebra is treated briefly. An application of some of the theory described above to the representation and analysis of completely positive maps is given.
On Green's function of an $n$-point boundary value problem
K. M.
Das;
A. S.
Vatsala
469-480
Abstract: The Green's function ${g_n}(x,s)$ for an n-point boundary value problem, $ {y^{(n)}}(x) = 0,y({a_1}) = y({a_2}) = \cdots = y({a_n}) = 0$ is explicitly given. As a tool for discussing $\operatorname{sgn} g_n(x,s)$ on the square $ [{a_1},{a_n}] \times [{a_1},{a_n}]$, some results about polynomials with coefficients as symmetric functions of a's are obtained. It is shown that $\displaystyle \int_{{a_1}}^{{a_n}} {\vert{g_n}(x,s)\vert ds}$ is a suitable polynomial in x. Applications to n-point boundary value problems and lower bounds for ${a_m}\;(m \geq n)$ are included.
On the integrals of Perron type
P. S.
Bullen;
C. M.
Lee
481-501
Abstract: An abstract derivate system is defined axiomatically, and then a naturally corresponding Perron integral theory is developed, unifying all the existing integral theories of Perron type of first order. A new scale of approximately mean-continuous integrals and a new scale of symmetric Cesàro-Perron integrals are obtained as examples of the general theory. Also, the MZ-integral and the SCP-integral are proved to be equivalent.
Bernstein approximation problem for differentiable functions and quasi-analytic weights
Guido
Zapata
503-509
Abstract: The Bernstein problem for differentiable functions is considered. Sufficient conditions in order that a decreasing family of weights be fundamental are given. Some of these conditions are also related to the concept of quasi-analytic weight.