A formula for semigroups, with an application to branching diffusion processes
Stanley A.
Sawyer
1-38
Abstract: A Markov process $P = \{ {x_t}\}$ proceeds until a random time $ \tau$, where the distribution of $\tau$ given $P$ is $ \exp ( - {\phi _t})$ for finite additive functional $\{ {\phi _t}\}$, at which time it jumps to a new position given by a substochastic kernel $K({x_\tau },A)$. A new time $\tau '$ is defined, the process again jumps at a time $(P,\{ {\phi _t}\} ,K)$.
Koszul resolutions
Stewart B.
Priddy
39-60
Abstract: Resolutions, which generalize the classical Koszul resolutions, are constructed for a large class of augmented algebras including the Steenrod algebra and the universal enveloping algebras. For each such algebra $A$, an explicit differential algebra $\bar K\ast (A)$ is described such that (1) $\bar K^ \ast(A)$ is a small quotient algebra of the cobar complex and (2) the homology of $\bar K^ \ast(A)$ is the cohomology algebra $H ^ \ast (A)$. The resolution of May for restricted Lie algebras in characteristic 2 is retrieved and a simple derivation of the resolution of Kan et al. of the Steenrod algebra is given.
On the ideal structure of $C(X)$
William E.
Dietrich
61-77
Abstract: The ideal structure of $C(X)$, the algebra of continuous functions from a completely regular Hausdorff space $X$ to the scalars is analyzed by examining for fixed $A \subset \beta X$ (the Stone-Čech compactification of $X$) the structure of the quotient ${I^A}/{F^A}$, where $ {I^A}[{F^A}]$ is the ideal of maps $f \in C(X)$ for which $\displaystyle A \subset {\text{cl} _{\beta X}}Z(f)\quad [A \subset {\operatorname{int} _{\beta X}}{\text{cl} _{\beta X}}Z(f)].$ Unless it vanishes, $ {I^A}/{F^A}$ has no minimal or maximal ideals, and its Krull dimension is infinite. If $J$ is an ideal of $C(X)$ strictly between ${F^A}$ and ${I^A}$, there are ideals $ \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{J} $ and $\bar J$ of $C(X)$ for which ${F^A} \subset \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{J} \subset J \subset \bar J \subset {I^A}$ with all inclusions proper. For $K \subset C(X)$, let $Z(K) = \cap \{ {\text{cl} _{\beta X}}Z(f):f \in K\}$. If $ J \subsetneqq I$ are ideals of $C(X)$ with $ Z(J) = Z(I)$ and if $ I$ is semiprime, there is an ideal $K$ strictly between $J$ and $I$. If $I$ and $J$ are $Z$-ideals, $K$ can be chosen to be of the form $P \cap I, P$ a prime ideal of $C(X)$. The maximal ideals of a semiprime ideal $I$ of $C(X)$ are of the form ${I^q} \cap I,q \in \beta X - Z(I)$. If $A \subset \beta X$ is closed, ${I^A}$ is a finitely generated ideal iff $ A$ is open.
On generalized commuting order of automorphisms with quasi-discrete spectrum
Nobuo
Aoki
79-97
Existence and uniqueness of fixed-points for semigroups of affine maps.
Robert E.
Huff
99-106
Abstract: The Day fixed-point theorem is extended to include both existence and uniqueness. For uniqueness of fixed-points, continuity for pointwise limits of a semigroup of continuous affine maps is needed ; necessary and sufficient conditions for this are obtained and compared with the stronger condition of equicontinuity. The comparison is between, on the one hand, the above condition, separate continuity, and weak compactness, and, on the other hand, equicontinuity, joint continuity, and strong compactness. An extension of the Kakutani fixed-point theorem results. Also as a corollary, known necessary and sufficient conditions for continuity of the convolution operation are obtained.
A collection of sequence spaces
J. R.
Calder;
J. B.
Hill
107-118
Abstract: This paper concerns a collection of sequence spaces we shall refer to as ${d_\alpha }$ spaces. Suppose $ \alpha = ({\alpha _1},{\alpha _2}, \ldots )$ is a bounded number sequence and ${\alpha _i} \ne 0$ for some $i$. Suppose $ \mathcal{P}$ is the collection of permutations on the positive integers. Then ${d_\alpha }$ denotes the set to which the number sequence $x = ({x_1},{x_2}, \ldots )$ belongs if and only if there exists a number $ k > 0$ such that $\displaystyle h_\alpha(x) = \operatorname{lub}_{p \in \mathcal{P}} \sum\limits_{i = 1}^\infty \vert x_{F(i)} \alpha_i\vert < k.$ $h_\alpha$ is a norm on $d_\alpha$ and $ (d_\alpha, h_\alpha)$ is complete. We classify the $ {d_\alpha }$ spaces and compare them with ${l_1}$ and $m$. Some of the $ {d_\alpha }$ spaces are shown to have a semishrinking basis that is not shrinking. Further investigation of the bases in these spaces yields theorems concerning the conjugate space properties of ${d_\alpha }$. We characterize the sequences $\beta$ such that, given $\alpha ,{d_\beta }, = {d_\alpha }$. A class of manifolds in the first conjugate space of ${d_\alpha }$ is examined. We establish some properties of the collection of points in the first conjugate space of a normed linear space $S$ that attain their maximum on the unit ball in $S$. The effect of renorming ${c_0}$ and ${l_1}$ with $ {h_\alpha }$ and related norms is studied in terms of the change induced on this collection of functionals.
Jacobi's bound for the order of systems of first order differential equations
Barbara A.
Lando
119-135
Abstract: Let ${A_1}, \ldots ,{A_n}$ be a system of differential polynomials in the differential indeterminates ${y^{(1)}}, \ldots ,{y^{(n)}}$, and let $\mathcal{M}$ be an irreducible component of the differential variety $\mathcal{M}({A_1}, \ldots ,{A_n})$. If $\dim \mathcal{M} = 0$, there arises the question of securing an upper bound for the order of $\mathcal{M}$ in terms of the orders ${r_{ij}}$ of the polynomials $ {A_i}$ in ${y^{(j)}}$. It has been conjectured that the Jacobi number $\displaystyle J = J({r_{ij}}) = \max \{\sum\limits_{i = 1}^n {{r_{i{j_i}}}} :{j_1}, \ldots ,{j_n}{\text{ is a permutation of 1,}} \ldots ,n \}$ provides such a bound. In this paper $J$ is obtained as a bound for systems consisting of first order polynomials. Differential kernels are employed in securing the bound, with the theory of kernels obtained in a manner analogous to that of difference kernels as given by R. M. Cohn.
Flat modules over commutative noetherian rings
Wolmer V.
Vasconcelos
137-143
Abstract: In this work we study flat modules over commutative noetherian rings under two kinds of restriction: that the modules are either submodules of free modules or that they have finite rank. The first ones have nicely behaved annihilators: they are generated by idempotents. Among the various questions related to flat modules of finite rank, emphasis is placed on discussing conditions implying its finite generation, as for instance, (i) over a local ring, a flat module of constant rank is free, and (ii) a flat submodule of finite rank of a free module is finitely generated. The rank one flat modules already present special problems regarding its endomorphism ring; in a few cases it is proved that they are flat over the base ring. Finally, a special class of flat modules--unmixed--is discussed, which have, so to speak, its source of divisibility somewhat concentrated in the center of its endomorphism ring and thus resemble projective modules over flat epimorphic images of the base ring.
Compact directed spaces
L. E.
Ward
145-157
Abstract: A directed space is a partially ordered topological space in which each two elements have a common predecessor. It is a consequence of a theorem of A. D. Wallace that a compact directed space is acyclic if each of its principal ideals is acyclic. This result is extended by considering the situation where at most finitely many principal ideals are not acyclic. It turns out that some of the elements which generate nonacyclic principal ideals must be maximal and that the $p$th cohomology group of the space must contain the $ p$th cohomology group of such a principal ideal as a direct summand. In the concluding sections it is shown that these spaces can be made acyclic by dividing out a closed ideal which contains all of the nonacyclic principal ideals, and some results on the acyclicity properties of minimal partial orders on compact spaces are proved.
Geometry associated with semisimple flat homogeneous spaces
Takushiro
Ochiai
159-193
Abstract: Our object is Cartan connections with semisimple flat homogeneous spaces as standard spaces. We study these from the viewpoint of $G$-structures of second order. This allows us especially to treat classical projective and conformal connections in the unifying manner. We also consider its application to the problem of certain geometric transformations.
Weak convergence of conditioned sums of independent random vectors.
Thomas M.
Liggett
195-213
Abstract: Conditions are given for the weak convergence of processes of the form $ ({{\mathbf{X}}_n}(t)\vert{{\mathbf{X}}_n}(1) \in {E^n})$ to tied-down stable processes, where $ ({{\mathbf{X}}_n}(t)$ is constructed from normalized partial sums of independent and identically distributed random vectors which are in the domain of attraction of a multidimensional stable law. The conditioning events are defined in terms of subsets ${E^n}$ of ${R^d}$ which converge in an appropriate sense to a set of measure zero. Assumptions which the sets $ {E^n}$ must satisfy include that they can be expressed as disjoint unions of ``asymptotically convex'' sets. The assumptions are seen to hold automatically in the special case in which $ {E^n}$ is taken to be a ``natural'' neighborhood of a smooth compact hypersurface in ${R^d}$.
The spectrum of partial differential operators on $L\sp{p}\,(R\sp{n})$
Franklin T.
Iha;
C. F.
Schubert
215-226
Abstract: The purpose of this paper is to prove that if the polynomial $ P(\xi )$ associated with a partial differential operator $P$ on $ {L^p}({R^n})$, with constant coefficients, has the growth property, $\vert P(\xi ){\vert^{ - 1}} = O(\vert\xi {\vert^{ - r}}),\vert\xi \vert \to \infty $ for some $r > 0$, then the spectrum of $ P$ is either the whole complex plane or it is the numerical range of $ P(\xi )$; and if $ P(\xi )$ has some additional property (all the coefficients of $ P(\xi )$ being real, for example), then the spectrum of $P$ is the numerical range for those $ p$ sufficiently close to 2. It is further shown that the growth property alone is not sufficient to ensure that the spectrum of $ P$ is the numerical range of $P(\xi )$.
On the generalized Blaschke condition
P. S.
Chee
227-231
Abstract: In an earlier paper, the author has shown that the generalized Blaschke condition is satisfied by bounded holomorphic functions in the polydisc or the ball in ${C^N}$. By essentially the same method, it is shown in the present paper that the same condition is satisfied by larger classes of functions, the Nevanlinna classes.
The local limit theorem and some related aspects of super-critical branching processes
Krishna B.
Athreya;
Peter
Ney
233-251
Abstract: Let $\{ {Z_n}:n = 0,1,2, \ldots \}$ be a Galton-Watson branching process with offspring p.g.f. $ f(s) = \Sigma _0^\infty {p_j}{s^j}$. Assume (i) $\Sigma _1^\infty {j^2}{p_j} < \infty$ and (iii) ${Z_n}{m^{ - n}}$ with ${Z_0} = 1,{w^{(i)}}(x)$ the $i$-fold convolution of $ w(x),{P_n}(i,j) = P({Z_n} = j\vert{Z_0} = i),{\delta _0} = (\log \gamma _0^{ - 1}){(\log m)^{ - 1}}$ and ${\beta _0} = {m^{{\delta _0}/(3 + {\delta _0})}}$. Then for any $0 < \beta < {\beta _0}$ and $i$ we can find a constant $C = C(i,\beta )$ such that $\displaystyle \vert{m^n}{P_n}(i,j) - {w^{(i)}}({m^{ - n}}j)\vert \leqq C[\beta _0^{ - n}{({m^{ - n}}j)^{ - 1}} + {\beta ^{ - n}}]$ for all $j \geqq 1$. Applications to the boundary theory of the associated space time process are also discussed.
From immersions to embeddings of smooth manifolds
Francis X.
Connolly
253-271
Existence and convergence of probability measures in Banach spaces.
Alejandro D.
de Acosta
273-298
Abstract: Theorems of the Bochner-Sazonov type are proved for Banach spaces with a basis. These theorems give sufficient conditions of a topological nature under which a positive definite function is the characteristic functional of a probability measure. The conditions are, in a certain natural sense, best possible. Central limit theorems of the Lindeberg type for triangular systems of random variables taking values in a Banach space with a basis are obtained. Applications to ${l_p}$ and $C[0,1]$ are given.
Logarithmic convexity, first order differential inequalities and some applications
Howard Allen
Levine
299-320
Abstract: Let, for $t \in [0,T)(T < \infty ),D(t)$ be a dense linear subspace of a Hilbert space $H$, and let $M(t)$ and $N(t)$ be linear operators (possibly unbounded) mapping $ D(t)$ into $H$. Let $f:[0,T) \times H \to H$. We give sufficient conditions on $M,N$ and $f$ in order to insure uniqueness and stability of solutions to $\displaystyle (1)\quad M(t)du/dt = N(t)u + f(t,u),\quad u(0)\;$given$\displaystyle .$ This problem is not in general well posed in the sense of Hadamard. We cite some examples of (1) from the literature. We also give some examples of the problem $\displaystyle (2)\quad M(t)\frac{{{d^2}u}}{{d{t^2}}} = N(t)u + f\left( {t,u,\frac{{du}}{{dt}}} \right),\quad u(0),\frac{{du}}{{du}}(0)\;$prescribed$\displaystyle ,$ for which questions of uniqueness and stability were discussed in a previous paper.
Some strict inclusions between spaces of $L\sp{p}$-multipliers
J. F.
Price
321-330
Abstract: Suppose that the Hausdorff topological group $G$ is either compact or locally compact abelian and that ${C_c}$ denotes the set of continuous complex-valued functions on $G$ with compact supports. Let $L_p^q$ denote the restrictions to $ {C_c}$ of the continuous linear operators from ${L^p}(G)$ into ${L^q}(G)$ which commute with all the right translation operators. When $1 \leqq p < q \leqq 2$ or $2 \leqq q < p \leqq \infty$ it is known that $\displaystyle (1)\quad L_p^p \subset L_q^q.$ The main result of this paper is that the inclusion in (1) is strict unless $G$ is finite. In fact it will be shown, using a partly constructive proof, that when $G$ is infinite $\displaystyle \bigcup\limits_{1 \leqq q < p} {L_q^q \subsetneqq } L_p^p \subsetneqq \bigcap\limits_{p < q \leqq 2} {L_q^q}$ for $1 < p < 2$, with the first inclusion remaining strict when $p = 2$ and the second inclusion remaining strict when $p = 1$. (Similar results also hold for $2 \leqq p \leqq \infty $.) When $ G$ is compact, simple relations will also be developed between idempotent operators in $L_p^q$ and lacunary subsets of the dual of $ G$ which will enable us to find necessary conditions so that inclusion (1) is strict even if, for example, $L_p^p$ and $L_q^q$ are replaced by the sets of idempotent operators in $L_p^p$ and $L_q^q$ respectively.