Analytic wave front sets for solutions of linear differential equations of principal type
Karl Gustav
Andersson
1-27
Abstract: The propagation of analyticity for solutions u of $P(x,D)u = f$ is studied, in terms of wave front sets, for a large class of differential operators $P = P(x,D)$ of principal type. In view of a theorem by L. Hörmander [9], the results obtained imply rather precise results about the surjectivity of the mapping $P:{C^\infty }(\Omega ) \to {C^\infty }(\Omega )$.
Some results on the length of proofs
R. J.
Parikh
29-36
Abstract: Given a theory T, let $\vdash _T^kA$ mean ``A has a proof in T of at most k lines". We consider a formulation $P{A^\ast}$ of Peano arithmetic with full induction but addition and multiplication being ternary relations. We show that ${ \vdash ^k}A$ is decidable for $P{A^\ast}$ and hence $P{A^\ast}$ is closed under a weak $ \omega$-rule. An analogue of Gödel's theorem on the length of proofs is an easy corollary.
Primary ideals in rings of analytic functions
R. Douglas
Williams
37-49
Abstract: Let A be the ring of all analytic functions on a connected, noncompact Riemann surface. We use the valuation theory of the ring A as developed by N. L. Alling to analyze the structure of the primary ideals of A. We characterize the upper and lower primary ideals of A and prove that every nonprime primary ideal of A is either an upper or a lower primary ideal. In addition we give some necessary and sufficient conditions for certain ideals of A to be intersections of primary ideals.
Reductions of ideals in commutative rings
James H.
Hays
51-63
Abstract: All rings considered in this paper are commutative, associative, and have an identity. If A and B are ideals in a ring, then B is a reduction of A if $B \subseteq A$ and if $B{A^n} = {A^{n + 1}}$ for some positive integer n. An ideal is basic if it has no reductions. These definitions were considered in local rings by Northcott and Rees; this paper considers them in more general rings. Basic ideals in Noetherian rings are characterized to the extent that they are characterized in local rings. It is shown that elements of the principal class generate a basic ideal in a Noetherian ring. Prüfer domains do not have the basic ideal property, that is, there may exist ideals which are not basic; however, a characterization of Prüfer domains can be given in terms of basic ideals. A domain is Prüfer if and only if every finitely generated ideal is basic.
Differentials on quotients of algebraic varieties
Carol M.
Knighten
65-89
Abstract: The relations between differentials invariant with respect to a finite group acting on a variety and the differentials on the quotient variety are studied. If the quotient map is unramified in codimension 1 we have an isomorphism for Zariski differentials, but not in general for Käahler differentials. Necessary and sufficient conditions for isomorphism of the Zariski differentials are given when the finite group acts linearly. Examples illustrate the scope of the theorems and some open problems.
Jacobson structure theory for Hestenes ternary rings
Robert Allan
Stephenson
91-98
Abstract: The principal results are an extension of the density theorem to Hestenes ternary rings and a characterization of primitive ternary rings.
Classes of automorphisms of free groups of infinite rank
Robert
Cohen
99-120
Abstract: This paper is concerned with finding classes of automorphisms of an infinitely generated free group F which can be generated by ``elementary'' Nielsen transformations. Two different notions of ``elementary'' Nielsen transformations are explored. One leads to a classification of the automorphisms generated by these transformations. The other notion leads to the subgroup B of $ {\operatorname{Aut}}(F)$ consisting of the ``bounded length'' automorphisms of F. We prove that the class of ``bounded 3-length'' automorphisms ${B_3}$ and the class of ``elementary simultaneous'' Nielsen transformations generate the same subgroup of ${\operatorname{Aut}}(F)$. We show that for the class T of automorphisms of ``2 occurring generators", the groups generated by $T \cap B$ and the ``elementary simultaneous'' Nielsen transformations are identical. These results lead to the conjecture that B is generated by the ``elementary simultaneous Nielsen transformations". A study is also made of the subgroup S of the ``triangular automorphisms'' of ${F_\infty }$, the free group on a countably infinite set of free generators. It is found that a ``triangular automorphism'' may be factored into ``splitting automorphisms'' of $ {F_\infty }$, which may be viewed as the ``elementary'' automorphisms of S.
Logic and invariant theory. I. Invariant theory of projective properties
Walter
Whiteley
121-139
Abstract: This paper initiates a series of papers which will reexamine some problems and results of classical invariant theory, within the framework of modern first-order logic. In this paper the notion that an equation is of invariant significance for the general linear group is extended in two directions. It is extended to define invariance of an arbitrary first-order formula for a category of linear transformations between vector spaces of dimension n. These invariant formulas are characterized by equivalence to formulas of a particular syntactic form: homogeneous formulas in determinants or ``brackets". The fuller category of all semilinear transformations is also introduced in order to cover all changes of coordinates in a projective space. Invariance for this category is investigated. The results are extended to cover invariant formulas with both covariant and contravariant vectors. Finally, Klein's Erlanger Program is reexamined in the light of the extended notion of invariance as well as some possible geometric categories.
On a theorem of Chern
J. H.
Sampson
141-153
Abstract: A new proof is given for Chern's theorem showing that the Laplace operator for differential forms commutes with decomposition of forms associated with G-structures admitting a suitable connection. An analogous result is proved for symmetric tensor fields, and an application is made to determine all harmonic symmetric fields on a compact space of constant negative curvature. Vector-valued forms are also discussed.
Asymptotic properties of Gaussian random fields
Clifford
Qualls;
Hisao
Watanabe
155-171
Abstract: In this paper we study continuous mean zero Gaussian random fields $ X(p)$ with an N-dimensional parameter and having a correlation function $\rho (p,q)$ for which $1 - \rho (p,q)$ is asymptotic to a regularly varying (at zero) function of the distance ${\text{dis}}\;(p,q)$ with exponent $0 < \alpha \leq 2$. For such random fields, we obtain the asymptotic tail distribution of the maximum of $X(p)$ and an asymptotic almost sure property for $ X(p)$ as $\vert p\vert \to \infty $. Both results generalize ones previously given by the authors for $N = 1$.
Isomorphism of simple Lie algebras
B. N.
Allison
173-190
Abstract: Let $\mathcal{L}$ and $ \mathcal{L}$ and $ \mathcal{L}$ is also studied. In particular, a result about this kernel in the rank one reduced case is proved. This result is then used to prove a conjugacy theorem for the simple summands of the anisotropic kernel in the general reduced case. The results and methods of this paper are rational in the sense that they involve no extension of the base field.
Closed countably generated structures in $C(X)$
B.
Roth
191-197
Abstract: Let $C(X)$ be the space of continuous real or complex valued functions on a compact space X with the sup norm topology. In the present paper, the subalgebras, vector lattices, and vector lattice ideals of $ C(X)$ which are closed and countably generated are characterized.
Oriented and weakly complex bordism algebra of free periodic maps
Katsuyuki
Shibata
199-220
Abstract: Free cyclic actions on a closed oriented (weakly almost complex, respectively) manifold which preserve the orientation (weakly complex structure) are considered from the viewpoint of equivariant bordism theory. The author gives an explicit presentation of the oriented bordism module structure and multiplicative structure of all orientation preserving (and reversing) free involutions. The odd period and weakly complex cases are also determined with the aid of the notion of formal group laws. These results are applied to a nonexistence problem for certain equivariant maps.
Self-dual axioms for many-dimensional projective geometry
Martinus
Esser
221-236
Abstract: Proposed and compared are four equivalent sets R, S, T, D of self-dual axioms for projective geometries, using points, hyperplanes and incidence as primitive elements and relation. The set R is inductive on the number of dimensions. The sets S, T, D all include the axiom ``on every n points there is a plane", the dual of this axiom, one axiom on the existence of a certain configuration, and one or several axioms on the impossibility of certain configurations. These configurations consist of $(n + 1)$ points and $(n + 1)$ planes for sets S, T, but of $ (n + 2)$ points and $ (n + 2)$ planes for set D. Partial results are obtained by a preliminary study of self-dual axioms for simplicial spaces (spaces which may have fewer than 3 points per line).
Interpolation between consecutive conjugate points of an $n$th order linear differential equation
G. B.
Gustafson
237-255
Abstract: The interpolation problem ${x^{(n)}} + {P_{n - 1}}{x^{(n - 1)}} + \cdots + {P_0}x = 0$, $ {x^{(i)}}({t_j}) = 0,i = 0, \cdots ,{k_j} - 1,j = 0, \cdots ,m$, is studied on the conjugate interval $ [a,{\eta _1}(a)]$. The main result is that there exists an essentially unique nontrivial solution of the problem almost everywhere, provided $ {k_1} + \cdots + {k_m} \geq n$, and cer tain other inequalities are satisfied, with $a = {t_0} < {t_1} < \cdots < {t_m} = {\eta _1}(a)$. In particular, this paper corrects the results of Azbelev and Caljuk (Mat. Sb. 51 (93) (1960), 475-486; English transl., Amer. Math. Soc. Transl. (2) 42 (1964), 233-245) on third order equations, and shows that their results are correct almost everywhere.
Fatou theorems for eigenfunctions of the invariant differential operators on symmetric spaces
H. Lee
Michelson
257-274
Abstract: On a Riemannian symmetric space of noncompact type we introduce a generalization of the Poisson kernel which may be used to generate simultaneous eigenfunctions of the invariant differential operators with eigenvalues not necessarily zero. We investigate the boundary behavior of our generalized Poisson integrals, extending to them many of the Fatou-type theorems known for harmonic functions.
On the principal series of ${\rm Gl}\sb{n}$ over$p$-adic fields
Roger E.
Howe
275-286
Abstract: The entire principal series of $G = G{l_n}(F)$, for a p-adic field F, is analyzed after the manner of the analysis of Bruhat and Satake for the spherical principal series. If K is the group of integral matrices in $G{l_n}(F)$, then a ``principal series'' of representations of K is defined. It is shown that precisely one of these occurs, and only once, in a given principal series representation of G. Further, the spherical function algebras attached to these representations of K are all shown to be abelian, and their explicit spectral decomposition is accomplished using the principal series of G. Computation of the Plancherel measure is reduced to MacDonald's computation for the spherical principal series, as is computation of the spherical functions themselves.
On the character of Weil's representation
Roger E.
Howe
287-298
Abstract: The importance of certain representations of symplectic groups, usually called Weil representations, for the general problem of finding representations of certain group extensions is made explicit. Some properties of the character of Weil's representation for a finite symplectic group are given and discussed, again in the context of finding representations of group extensions. As a by-product, the structure of anisotropic tori in symplectic groups is given.
Krull dimension in power series rings
Jimmy T.
Arnold
299-304
Abstract: Let R denote a commutative ring with identity. If there exists a chain ${P_0} \subset {P_1} \subset \cdots \subset {P_n}$ of $n + 1$ prime ideals of R, where ${P_n} \ne R$, but no such chain of $n + 2$ prime ideals, then we say that R has dimension n. The power series ring $R[[X]]$ may have infinite dimension even though R has finite dimension.
A second quadrant homotopy spectral sequence
A. K.
Bousfield;
D. M.
Kan
305-318
Abstract: For each cosimplicial simplicial set with basepoint, the authors construct a homotopy Spectral sequence generalizing the usual spectral sequence for a second quadrant double chain complex. For such homotopy spectral sequences, a uniqueness theorem and a general multiplicative pairing are established. This machinery is used elsewhere to show the equivalence of various unstable Adams spectral sequences and to construct for them certain composition pairings and Whitehead products.
Pairings and products in the homotopy spectral sequence
A. K.
Bousfield;
D. M.
Kan
319-343
Abstract: Smash and composition pairings, as well as Whitehead products are constructed in the unstable Adams spectral sequence; and these pairings and products are described homologically on the ${E_2}$. level. In the special case of the Massey-Peterson spectral sequence, the composition action is given homologically by the Yoneda product, while the Whitehead product vanishes. It is also shown that the unstable Adams spectral sequence over the rationals, with its Whitehead products, is given by the primitive elements in the rational cobar spectral sequence.
$k$-parameter semigroups of measure-preserving transformations
Norberto Angel
Fava
345-352
Abstract: An individual ergodic theorem is proved for semigroups of measure-preserving transformations depending on k real parameters, which generalizes N. Wiener's ergodic theorem.
A fixed point theorem-free approach to weak almost periodicity
William A.
Veech
353-362
Abstract: In this paper we present a generalization of the Eberlein, de Leeuw and Glicksberg decomposition theorem for weakly almost periodic functions which does not rely on any fixed point theorem for its proof. A generalization of the Ryll-Nardzewski fixed point theorem is given.
Comparison of eigenvalues for linear differential equations of order $2n$
Curtis C.
Travis
363-374
Abstract: An abstract eigenvalue comparison theorem is proven for $ {u_0}$-positive linear operators in a Banach space equippped with a cone of ``nonnegative'' elements. This result is then applied to certain linear differential equations of order 2n in order to obtain eigenvalue comparison theorems of an ``integral type."
Induced flows
Karl
Petersen;
Leonard
Shapiro
375-390
Abstract: The construction of induced transformations is considered in the setting of topological dynamics. Sufficient conditions are given for induced flows to be topologically weakly mixing, and it is proved that Toeplitz flows and certain Sturmian flows satisfy these conditions and give rise to new and easily constructed classes of flows which have entropy zero and are uniquely ergodic, minimal, and topologically weakly mixing. An example is given of a weakly mixing minimal flow which is not topologically strongly mixing.
Positive approximants
Richard
Bouldin
391-403
Abstract: Let $T = B + iC$ with $B = {B^\ast},C = {C^\ast}$ and let $\delta (T)$ denote the the distance of T to the set of nonnegative operators. We find upper and lower bounds for $ \delta (T)$. We prove that if P is any best approximation for T among nonnegative operators then $P \leq B + {({(\delta (T))^2} - {C^2})^{{\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}}}$. Provided $B \geq 0$ or T is normal we characterize those T which have a unique best approximation among the nonnegative operators. If T is normal we characterize its best approximating nonnegative operators which commute with it. We characterize those T for which the zero operator is the best approximating nonnegative operator.
$k$-congruence orders for $E\sb{k}$
Grattan P.
Murphy
405-412
Abstract: This paper generalizes the notion of congruence order for metric spaces to k-metric (k-dimensional metric) spaces. The k-congruence order of $ {E_k}$ with respect to the class of oriented semi k-metric spaces is determined. An example shows that this result is sharp.
An invariance principle for a class of $d$-dimensional polygonal random functions
Luis G.
Gorostiza
413-445
Abstract: A class of random functions is formulated, which represent the motion of a point in d-dimensional Euclidean space $(d \geq 1)$ undergoing random changes of direction at random times while maintaining constant speed. The changes of direction are determined by random orthogonal matrices that are irreducible in the sense of not having an almost surely invariant nontrivial subspace if $d \geq 2$, and not being almost surely nonnegative if $d = 1$. An invariance principle stating that under certain conditions a sequence of such random functions converges weakly to a Gaussian process with stationary and independent increments is proved. The limit process has mean zero and its covariance matrix function is given explicitly. It is shown that when the random changes of direction satisfy an appropriate condition the limit process is Brownian motion. This invariance principle includes central limit theorems for the plane, with special distributions of the random times and direction changes, that have been proved by M. Kac, V. N. Tutubalin and T. Watanabe by methods different from ours. The proof makes use of standard methods of the theory of weak convergence of probability measures, and special results due to P. Billingsley and B. Rosén, the main problem being how to apply them. For this, renewal theoretic techniques are developed, and limit theorems for sums of products of independent identically distributed irreducible random orthogonal matrices are obtained.
Probabilistic recursive functions
Irwin
Mann
447-467
Abstract: The underlying question considered in this paper is whether or not the purposeful introduction of random elements, effectively governed by a probability distribution, into a calculation may lead to constructions of number-theoretic functions that are not available by deterministic means. A methodology for treating this question is developed, using an effective mapping of the space of infinite sequences over a finite alphabet into itself. The distribution characterizing the random elements, under the mapping, induces a new distribution. The property of a distribution being recursive is defined. The fundamental theorem states that recursive distributions induce only recursive distributions. A function calculated by any probabilistic means is called $\psi$-calculable. For a class of such calculations, these functions are recursive. Relative to Church's thesis, this leads to an extension of that thesis: Every $\psi$-effectively calculable function is recursive. In further development, a partial order on distributions is defined through the concept of ``inducing.'' It is seen that a recursive atom-free distribution induces any recursive distribution. Also, there exist distributions that induce, but are not induced by, any recursive distribution. Some open questions are mentioned.
$\theta $-modular bands of groups
C.
Spitznagel
469-482
Abstract: The class of $ \theta$-modular bands of groups is defined by means of a type of modularity condition on the lattice of congruences on a band of groups. The main result characterizes $\theta $-modularity as a condition on the multiplication in the band of groups. This result is then applied to the classes of normal bands of groups and orthodox bands of groups.
Boundary representations on $C\sp{\ast} $-algebras with matrix units
Alan
Hopenwasser
483-490
Abstract: Let $\mathcal{A}$ be a ${C^\ast}$-algebra with unit, let $\mathcal{S}$ be a linear subspace of $\mathcal{A} \otimes {M_n}$ which contains the natural set of matrix units and which generates $\mathcal{A}$ as a ${C^\ast}$-algebra. Let $ \mathcal{J}$ be the subset of $\mathcal{A}$ consisting of entries of matrices in $ \mathcal{S}$. Then the boundary representations of $\mathcal{A} \otimes {M_n}$ relative to $\mathcal{S}$ are parametrized by the boundary representations of $ \mathcal{A}$ relative to $\mathcal{J}$. Also, a nontrivial example is given of a subalgebra of a ${C^\ast}$-algebra which possesses exactly one boundary representation.
Sets of formulas valid in finite structures
Alan L.
Selman
491-504
Abstract: A function $\mathcal{V}$ is defined on the set of all subsets of $ \omega$ so that for each set K, the value, ${\mathcal{V}_K}$, is the set of formulas valid in all structures of cardinality in K. An analysis is made of the dependence of ${\mathcal{V}_K}$ on K, For any set K, let $ {\text{d}}(K)$ be the Kleene-Post degree to which K belongs. It is easily seen that for all infinite sets K, ${\text{d}}({\mathcal{V}_{K \vee J}}) = {\text{d}}({\mathcal{V}_K}) \vee {\text{d}}({\mathcal{V}_J})$, and use this to prove that, for any two degrees a and b, ${\text{d}}(K) = {\text{a}}$ and $ {\text{d}}({\mathcal{V}_K}) = b$. Various similar results are also included.
Erratum to: ``A general class of factors of $E\sp{4}$''
Leonard R.
Rubin
505
Erratum to: ``Functional analytic properties of extremely amenable semigroups''
Edmond E.
Granirer
507