Showing papers in "Transactions of the American Mathematical Society in 1980"

Homological algebra on a complete intersection, with an application to group representations

Abstract: Let R be a regular local ring, and let A = R/(x), where x is any nonunit of R. We prove that every minimal free resolution of a finitely generated A -module becomes periodic of period 1 or 2 after at most dim A steps, and we examine generalizations and extensions of this for complete intersections. Our theorems follow from the properties of certain universally defined endomorphisms of complexes over such rings. Let A be a commutative ring, and let x G A be a nonzero divisor. How does homological algebra over A/(x) = B differ from that over AI In this paper we will study a certain natural endomorphism / of complexes of free A / (x)-modu\es which seems to reflect some of the difference. For example, the (homotopic) triviality of t is an obstruction (closely related to the usual one in Ext2,) to the lifting of a complex of free 5-modules to a complex of free /I-modules. More generally, if x,, . . . , xn is an A -sequence, we study « natural endomorphisms /,,..., tn of complexes of free A/(xx, . . . , x")-modules, and try to use them to explain the way in which free resolutions over A/(xx, . . . , x") differ from free resolutions over A (the construction and elementary properties of these endomorphisms is given in §1). In this paper, we will study the case in which A is a regular local ring and B = A/(xx, . . . , x") is not regular. (It would also be very interesting to understand the case in which both A and A/(x) = B were regular-with, say, A of mixed characteristic and B ramified or of characteristic p.) In this case, the homological algebra over A is dominated, roughly speaking, by the fact that minimal /I-free resolutions are finite; we seek to understand the eventual behavior of minimal 5-free resolutions in terms of the tt. For example, if « = 1, so that B = A/(x), we prove that / is eventually an isomorphism, so that every minimal 5-free resolution becomes periodic of period 2 after at most 1 + dim B steps (§6). We also show that the 5-modules with periodic resolutions are the maximal Cohen-Macaulay modules without free direct summands. Since the periodic part of a periodic resolution over A/(x) (or more generally, over A/(xx, . . . , xn), if x,, . . . , x" is an A -sequence) is easy to describe explicitly (§5), this yields information on maximal Cohen- Macaulay modules.

Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms

Abstract: We introduce a class of nilpotent Lie groups which arise naturally from the notion of composition of quadratic forms, and show that their standard sublaplacians admit fundamental solutions analogous to that known for the Heisenberg group. By a theorem of Hormander [6], if XI, ... , X, are vector fields on a manifold N with the property that their commutators up to a certain order span the tangent space at every point, then the differential operator

On generalized harmonic analysis

Abstract: Motivated by Wiener's work on generalized harmonic analysis, we consider the Marcinkiewicz space 6XP(R) of functions of bounded upper averagep power and the space St(R) of functions of bounded upper p variation. By identifying functions whose difference has norm zero, we show that St(R), 1

Generation of analytic semigroups by strongly elliptic operators under general boundary conditions

Abstract: Strongly elliptic operators are shown to generate analytic semigroups of evolution operators in the topology of uniform convergence, when realized under general boundary conditions on (possibly) unbounded domains. An application to the existence and regularity of solutions to parabolic initial-boundary value problems is indicated. Introduction. Extending the results of a previous paper [19], we propose to prove a theorem on the generation of analytic semigroups by strongly elliptic operators A of order 2m in the topology of uniform convergence, under more general boundary conditions. As before, there is a direct application to parabolic initial-boundary value problems: we give an existence and uniqueness theorem for such problems, using the Kato-Tanabe theory for temporally inhomogeneous evolution equations du/dt + A(t)u = / The topology of uniform convergence in the space variables yields classical solutions of this parabolic problem which are analytic in time at each space point; furthermore, the initial values are assumed in the pointwise continuous sense. Since we treat general boundary conditions, our parabolic problem is roughly comparable to the one discussed in Arima [4], where existence and uniqueness are treated (using a fundamental solution of the parabolic problem), but not analyticity. The semigroup generation theorem for strongly elliptic operators is by now well established in the Lp spaces. The L2 case was treated by Browder in [5]; shortly thereafter Agmon, in [1], gave a method of proving the basic a priori estimate \\\\u\\\\ < (M/\\z\\)\\\\iA + z)u\\\\, |arg _| < \\m + e, (E) in Lp, and this method has been filled out with existence theorems and used by several authors, including Friedman [8], Higuchi [9], Lau [14], and Freeman and Schechter [7]. The work of Lau offers an attractive combination of full development and general hypotheses, so we shall borrow Lp results from [14]. We recall that the Kato-Tanabe theorem applied in the Lp topology gives solutions which are Received by the editors June 25, 1979. AMS (MOS) subject classifications (1970). Primary 35K35, 47D05. 'This work was supported in part by the U. S. Department of Energy under contract EY-76-C-02-0016. Accordingly, the U. S. Government retains a nonexclusive, royalty-free license to publish or reproduce the published form of this contribution, or allow others to do so, for U. S. Government purposes. © 1980 American Mathematical Society 0002-9947/80/0000-02 1 8/$04.00 299 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

Littlewood-Paley and multiplier theorems on weighted ^{} spaces

Abstract: The Littlewood-Paley operator y(f), for functions f defined on RX, is shown to be a bounded operator on certain weighted LP spaces. The weights satisfy an AP condition over the class of all n-dimensional rectangles with sides parallel to the coordinate axes. The necessity of this class of weights demonstrates the 1-dimensional nature of the operator. Results for multipliers are derived, including weighted versions of the Marcinkiewicz Multiplier Theorem and Hormander's Multiplier Theorem.

Chaotic behavior in piecewise continuous difference equations

Abstract: A class of piecewise continuous mappings with positive slope, mapping the unit interval into itself is studied. Families of 1-1 mappings depending on some parameter have periodic orbits for most parameter values, but have an infinite invariant set which is a Cantor set for a Cantor set of parameter values. Mappings which are not 1-1 exhibit chaotic behavior in that the asymptotic behavior as measured by the rotation number covers an interval of values. The asymptotic behavior depends sensitively on initial data in that the rotation number is either a nowhere continuous function of initial data, or else it is a constant on all but a Cantor set of the unit interval.

The asymptotic behavior of gas in an -dimensional porous medium

Abstract: Consider the flow of gas in an «-dimensional porous medium with initial density u0(x) > 0. The density u(x, l) then satisfies the nonlinear degenerate parabolic equation u, = Awm where m > 1 is a physical constant. Assuming that I = S \"o(x)0), (1.1) u(x, 0) = u0(x) (x E R\"). (1.2) The function u represents the density of a gas in a porous medium and m is a physical constant, m > 1. We assume that u0(x) is continuous, u0(x) > 0, u0(x) ^ 0, u0(x) < M, u0 e Ll(R\") n L2(7T) (M constant), (1.3) and set I = f u0(x)dx. (1.4) J R\" By a weak solution of (1.1), (1.2) we mean a function u satisfying, for any T > 0, fT f \\(u(x,t))2+\\Vxum(x,t)\\2]dxdt<<* J0 JRn 0; (1.5) JRn Received by the editors August 27, 1979. AMS (MOS) subject classifications (1970). Primary 35K55, 76S05; Secondary 35K15. 'This work is partially supported by National Science Foundation Grant MCS-7817204 and AFOSR Grant 78-3602. © 1980 American Mathematical Society 0002-9947/80/0000-0 565/$04.2 S

A strong Stieltjes moment problem

Abstract: This paper is concerned with double sequences of complex numbers C = [cn}^x and with formal Laurent series Lq(C) = 2f c_mzm and /.„(C) = 2o°cmz~m generated by them. We investigate the following related problems: (1) Does there exist a holomorphic function having L0(C) and LX(C) as asymptotic expansions at z — 0 and z = oo, respectively? (2) Does there exist a real-valued bounded, monotonically increasing function (f) with infinitely many points of increase on [0, oo) such that, for every integer n, c„ = / 5°(-0\" dty(t)l The latter problem is called the strong Stieltjes moment problem. We also consider a modified moment problem in which the function ¡f/(t) has at most a finite number of points of increase. Our approach is made through the study of a special class of continued fractions (called positive 7\"-fractions) which correspond to L0(C) at z = 0 and LX(C) at z = oo. Necessary and sufficient conditions are given for the existence of these corresponding continued fractions. It is further shown that the even and odd parts of these continued fractions always converge to holomorphic functions which have Lq(C) and LX(C) as asymptotic expansions. Moreover, these holomorphic functions are shown to be represented by Stieltjes integral transforms whose distributions ^°\\t) and ^'\"'(O solve the strong Stieltjes moment problem. Necessary and sufficient conditions are given for the existence of a solution to the strong Stieltjes moment problem. This moment problem is shown to have a unique solution if and only if the related continued fraction is convergent. Finally it is shown that the modified moment problem has a unique solution if and only if there exists a terminating positive T-fraction that corresponds to both Lq(C) and ¿„(C). References are given to other moment problems and to investigations in which negative, as well as positive, moments have been used.

Scattering theory and polynomials orthogonal on the real line

Abstract: The techniques of scattering theory are used to study polynomials orthogonal on a segment of the real line. Instead of applying these techniques to the usual three-term recurrence formula, we derive a set of two two-term re- currence formulas satisfied by these polynomials. One of the advantages of these new recurrence formulas is that the Jost function is related, in the limit as n -» oo, to the solution of one of the recurrence formulas with the boundary conditions given at n = 0. In this paper we investigate the properties of the Jost function and the spectral function assuming the coefficients in the recurrence formulas converge at a particular rate. of the more familiar three-term recurrence formula satisfied by polynomials or- thogonal on the real line (PORL). These two two-term recurrence formulas have several interesting properties. For example, the Jost function, which has been shown to be so useful in the theory of orthogonal polynomials (4), (7), is the limit of a sequence of polynomials satisfying one of the recurrence formulas with the boundary condition given at n = 0. It is natural to ask whether such a system of recurrence relations exists for PORL. In this paper we develop the theory of PORL along a line that parallels the theory of POUC and delve deeper into the consequences of applying scattering theory to PORL. In §11 we define the polynomials and derive the familiar three-term recurrence formula. Now, in analogy with POUC, a set of two two-term recurrence formulas is derived. These formulas plus the appropriate boundary conditions are taken as the fundamental equations defining the polynomials. From them the Christoffel- Darboux formula and Wronskian theorem are derived. In §111 the Jost function is defined and is shown to be the limit of a sequence of polynomials satisfying one of the recurrence formulas with the boundary condi- tions given at n = 0. Some of the properties of the Jost function are investigated. Since we have started with the recurrence relations it is necessary to show that the polynomials are indeed orthogonal. This is done in §IV. We also show how one can calculate the Jost function directly from the weight function.

Splitting criteria for -modules induced from a parabolic and the Berňsteĭn-Gel’fand-Gel’fand resolution of a finite-dimensional, irreducible -module

Abstract: Let g be a finite dimensional, complex, semisimple Lie algebra and let V be a finite dimensional, irreducible g-module. By computing a certain Lie algebra cohomology we show that the generalized versions of the weak and the strong Bemstein-Gelfand-Gelfand resolutions of V obtained by H. Garland and J. Lepowsky are identical. Let G be a real, connected, semisimple Lie group with finite center. As an application of the equivalence of the generalized Bernstein-Gelfand-Gelfand resolutions we obtain a complex in terms of the degenerate principal series of G, which has the same cohomology as the de Rham complex. Introduction. Let g be a finite dimensional, complex, semisimple Lie algebra and let V be a finite dimensional, irreducible g-module. In [2, Theorem 9.9], I. N. Bernstein, I. M. Gelfand and S. I. Gelfand constructed a resolution of V by certain a-modules with Verma composition series. In the same paper another resolution of V is constructed which resolves V by direct sums of Verma modules, improving Theorem 9.9. (Cf. [2, Theorem 10.1'].) In their work, Bernstein, Gelfand and Gelfand study systematically a certain category of g-modules known as the category 0. Both the weak and the strong resolutions were generalized by H. Garland and J. Lepowsky in the papers [9] and [14] where generalized Verma modules play the same role as Verma modules in [2]. We will refer to these resolutions as the generalized weak BGG resolution and the generalized strong BGG resolution. In this paper we prove that the two generalized BGG resolutions are identical. Our main theorem, Theorem 9.3 shows that each g-module in the generalized weak BGG resolution splits into a direct sum of generalized Verma modules. Consequently each such g-module is isomorphic to the g-module at the corresponding level in the generalized strong BGG resolution. One of our methods consists in studying a certain category of a-modules and later using such category as a framework in order to obtain a key lemma, Lemma 9.1. We would like to point out that, in the light of Yoneda's interpretation of cohomology, Lemma 9.1 implies vanishing theorems on Lie algebra cohomology. (See the remark following the proof of Lemma 9.1.) Received by the editors November 21, 1978 and, in revised form, May 21, 1979. 1980 Mathematics Subject Classification. Primary 22E47, 17B 10, 17B35, 22E46. ? 1980 American Mathematical Society 0002-9947/80/0000-0551 /$09.00

A noncommutative generalization and $q$-analog of the Lagrange inversion formula

Abstract: The Lagrange inversion formula is generalized to formal power series in noncommutative variables. A g-analog is obtained by applying a linear operator to the noncommutative formula before substituting commuting variables.

The dependence of the generalized Radon transform on defining measures

Abstract: ABsmAcr. Guillemin proved that the generalized Radon transform R and its dual R' are Fourier integral operators and that R'R is an elliptic pseudodifferential operator. In this paper we investigate the dependence of the Radon transform on the defining measures. In the general case we calculate the symbol of R'R as a pseudodifferential operator in terms of the measures and give a necessary condition on the defining measures for R'R to be invertible by a differential operator. Then we examine the Radon transform on points and hyperplanes in RX with general measures and we calculate the symbol of R'R in terms of the defining measures. Finally, if R'R is a translation invariant operator on RI then we prove that R'R is invertible and that our condition is equivalent to (R'R)' being a differential operator.

Two approaches to supermanifolds

Abstract: The problem of supplying an analogue of a manifold whose sheaf of functions contains anticommuting elements has been approached in two ways. Either one extends the sheaf of functions formally, as in the category of graded manifolds [3], [8], or one mimicks the usual definition of a manifold, having replaced Eucidean space with a suitable product of the odd and even parts of an exterior algebra as in the category of supermanifolds [6]. This paper establishes the equivalence of the category of supermanifolds with the category of graded manifolds. Introduction. Supermanifolds or graded manifolds were defined to provide a "space" whose "functions" would include anticommuting elements. There have been two approaches to the definition of these objects, emerging from two approaches to the study of the geometry of a manifold. Traditionally, the differential geometer regards the space itself as the primary object, but it is also possible to take the algebraic geometer's point of view and study the geometry of the space through the algebraic structure of its sheaf of functions. Supermanifolds as defined by de Witt [6] for example, follows the first approach, while graded manifolds, as defined by Kostant [8] are inspired by the algebraic geometer's approach. The purpose of this paper is to establish the equivalence of the category of supermanifolds with the category of graded manifolds. In the first section, basic definitions are given and the main theorem is stated. Smooth maps on "super-Euclidean space" are constructed in ?2, and the proof of the equivalence theorem is given in ?3. I would like to thank Professor S. Stemnberg for bringing de Witt's work to my attention, and for subsequent discussions, and also Carolyn Schroeder for her help in finding the "right" definition for smooth maps on super-Euclidean space. 1. Definitions and results. All algebras and vector spaces are over the real numbers, although similar constructions could be carried out using complex numbers. All (ordinary) manifolds are considered to be real, smooth, Hausdorff paracompact manifolds. 1.1. DEFINITION. An algebra A is called a Z2-graded algebra (or simply a graded algebra) if A can be written as a direct sum of linear subspaces A = AO @1 A1 such that AiAj c Ai+j(mod2). Received by the editors March 28, 1979. AMS (MOS) subject classifications (1970). Primary 58A05.

The product of two countably compact topological groups

Abstract: We use MA (= Martin's Axiom) to construct two countably compact topological groups whose product is not countably compact. To this end we first use MA to construct an infinite countably compact topological group which has no nontrivial convergent sequences.

The Minakshisundaram-Pleijel coefficients for the vector-valued heat kernel on compact locally symmetric spaces of negative curvature

Abstract: We use harmonic analysis on semisimple Lie groups to determine the Minakshisundaram-Pleijel asymptotic expansion for the trace of the heat kernel on natural vector bundles over compact, locally symmetric spaces of strictly negative curvature. Introduction. Let G be a connected, real semisimple Lie group of rank one with finite center. Let G = K* A * N be an Iwasawa decomposition of G and let M be the centralizer of A in K. Denote by g = t ) a @ n the corresponding Iwasawa decomposition of g, the Lie algebra of G. We use the G-invariant Riemannian metric on G/K induced by a1* Bg (a1 is a convenient constant and Bg is the Killing form of g). If (T, VT) is an irreducible representation of K we form the homogeneous vector bundle ET = G X VT -> G/K. There is unique G-invariant connection on ET such that if s is a C cross-section, X E p, a: G -> G/K is the canonical projection and a. designates the differential of a at e (the identity of G), then V ,7 (x)(s) = dt=O s(exp(tX)K). We denote by V2 the connection Laplacian on (ET, V). We consider now a discrete, torsion free subgroup F of G such that F \ G is compact. We give to X = F \ G/K the push down Riemannian metric. Then X is the most general compact locally symmetric space of negative curvature. Also, D = _V2 pushes down to a nonnegative, elliptic differential operator on F \ G x VT--> X. Let {An} = spec(D) be the spectrum of D. As it is well known, e -SD exists and is trace-class for s > 0. Moreover 4T(s) = tr(e sD) e-S? has an asymptotic expansion OT(S) _ Sd/2( E2 ais') as sO i=O (C = 2 dim(G/K)) and the coefficients ai are local Riemannian invariants of X (see [ABP], [BGM], [MS] and [MP]). In this paper we will use harmonic analysis on G Received by the editors May 10, 1978 and, in revised form, March 22, 1979. AMS (MOS) subject classifications (1970). Primary 43A65, 58G99. 'The results of this article are part of the doctoral thesis of the author, written under the direction of Dr. N. R. Wallach at Rutgers University. The author wishes to thank Dr. Wallach for his help and encouragement. Thanks are also due to the referee for useful suggestions. ? 1980 American Mathematical Society 0002-9947/80/0000-0300/$09.25 1 This content downloaded from 157.55.39.203 on Thu, 20 Oct 2016 04:09:04 UTC All use subject to http://about.jstor.org/terms

A maximal function characterization of ^{} on the space of homogeneous type

Abstract: Let 4#0(x) E 5S(R') and let fR4'o(y) dy4# 0. Forf E S'(R'), x E Rn and M > 0, let f + (x) = sup I * 40t(x) t>O and letf*M(x) = sup{If * 4'1(x)I: t > 0, 4#(y) E S(Rn), supp 4 c {y E Rn: IYI 0. Then there exists M(p, n), depending only on p and n, such that if M > M(p, n), then CIU' JII < lifW |1LP < Cl|f ' ILP for any f E S, '(R n), where c and C are positive constants depending only on 4'0, p, M and n. We investigate this on the space of homogeneous type with certain assumptions.

Equivariant homotopy theory and Milnor’s theorem

Abstract: The foundations of equivariant homotopy and cellular theory are examined; an equivariant Whitehead theorem is proved, and the classical results by Milnor about spaces with the homotopy-type of a CW complex are generalized to the equivariant case. The ambient group G is assumed compact Lie. Further results include equivariant cellular approximation and the procedure for replacement of an arbitrary G-space by a G-CW complex. This is the first of a series of three papers based on the author's thesis [Wal], the object of which is to discuss equivariant homotopy theory in general, and equivariant fibrations in particular, culminating in classification theorems for the various categories of equivariant fibrations and bundles. In the present paper, we discuss the foundations of equivariant homotopy theory and cellular theory and prove an equivariant version of Milnor's theorem on spaces having the homotopy type of CW complexes, where we allow a compact Lie group G to act everywhere. The second paper in this series, Equivariant fibratons and transfer sets up the background for the study of (J-fiber spaces and equivariant stable homotopy theory and contains a description of the equivariant transfer for equivariant fibrations with compact fiber. In the third paper, Equivariant classifying spaces and fibrations, the geometric bar construction is used to construct explicit classifying spaces for equivariant bundles and fibrations, these results depending heavily on the equivariant cellular theory presented here. Also in preparation is a fourth paper which will sequel the present series and will deal with the classificaton of oriented G-spherical fibrations and bundles [Wa2]. The three papers are divided as follows: Equivariant homotopy theory and Milnor's Theorem 1. Notations and definitions 2. Equivariant homotopy groups 3. Equivariant cellular theory 4. Milnor's Theorem 5. Approximation of G-CW complexes by G-simplicial complexes 6. Finite dimensional G-simplicial complexes are G-equilocally convex 7. Reasonable GELC spaces are dominated by G-CW complexes Received by the editors September 7, 1978 and, in revised form, January 15, 1979. AMS (MOS) subject classifications (1970). Primary 54H15.

Principal 2-blocks of the simple groups of Ree type

Abstract: The decomposition numbers in characteristic 2 of the groups of Ree type are determined, as well as the Loewy and socle series of the indecomposable projective modules. Moreover, we describe the Green correspondents of the simple modules. As an application of this and similar works on other simple groups with an abelian Sylow 2-subgroup, all of which have been classified apart from those considered in the present paper, we show that the Loewy length of an indecomposable projective module in the principal block of any finite group with an abelian Sylow 2-subgroup of order 2\" is bounded by max{2n + 1, 2\"}. This bound is the best possible. Introduction. The purpose of this paper is to determine the algebra structure of the principal 2-blocks B of the simple groups R(q) of Ree type of order \\R(q)\\ = (q3 + \\)q\\q 1), where q = 32n+1, and n = 1, 2,_ Let (F, R, S) be a splitting 2-modular system for R(q), where F has characteristic 2, and S and R have characteristic zero. The character table of the groups R(q) was determined by Ward [15] up to a few but very essential values missing because of the incomplete classification of these groups. Ward [15] also showed that B contains eight ordinary irreducible characters £„ all of height zero, and five nonisomorphic simple FR(q)-moàxx\\cs op,, i = 1,2,..., 5, where

Quadratic forms and the Birman-Craggs homomorphisms

Abstract: Let Mg be the mapping class group of a genus g orientable surface M, and gg the subgroup of those maps acting trivially on the homology group H,(M, Z). Birman and Craggs produced homomorphisms from 5g to Z2 via the Rochlin invariant and raised the question of enumerating them; in this paper we answer their question. It is shown that the homomorphisms are closely related to the quadratic forms on H,(M, Z2) which induce the intersection form; in fact, they are in 1-1 correspondence with those quadratic forms of Arf invariant zero. Furthermore, the methods give a description of the quotient of 5. by the intersection of the kernels of all these homomorphisms. It is a Z2-vector space isomorphic to a certain space of cubic polynomials over H1(M, Z2). The dimension is then computed and found to be (39) + (2s). These results are also extended to the case of a surface with one boundary component, and in this situation the linear relations among the various homomorphisms are also determined.

Undecidability and definability for the theory of global fields

Abstract: We prove that the theory of global fields is essentially undecidable, using predicates based on Hasse's Norm Theorem to define valuations. Polynomial rings or the natural numbers are uniformly defined in all global fields, as well as Godel functions encoding finite sequences of elements. We will prove that the elementary theory of global fields is essentially undecidable. While this is a negative logical result, our proofs have the positive consequence that a great variety of number-theoretic objects, from rings of integers and valuations, to zeta-functions and adele rings, can be discussed in the theory of global fields. It is our hope that the theory may eventually be a vehicle for applying logical methods in number theory. Our main theorems are as follows. I. There is a finite collection of predicates which define every valuation, archimedean and nonarchimedean, of every global field, in terms of parameters. II. There is a sentence which distinguishes number fields from function fields. III. Given a global field K: If K is a number field, the theory of number fields defines its algebraic integers, the rational integers, and the natural numbers. If K is a function field, the theory of function fields defines its field of constants F, and for an arbitrary nonconstant x E K, defines the polynomial ring F[x] and a model of N given by the powers of x. Godel functions encoding all finite sequences of elements of K exist for each of these models of N. This investigation was motivated by the papers of Julia Robinson [9], [10] and Ershov [4], who used the Hasse-Minkowski theorem on quadratic forms to prove the undecidability of a given number field, or field of rational functions over a finite field of odd characteristic. Its original goal was to show the remaining algebraic function fields were undecidable, completing the parallel between number fields and function fields. Then Simon Kochen observed that the proof showed Received by the editors February 26, 1979 and, in revised form, June 25, 1979. AMS (MOS) subject classifications (1970). Primary 02G05; Secondary lOL05, 12N05.

A Radon transform on spheres through the origin in ⁿ and applications to the Darboux equation

Abstract: On domain C (Rf) we invert the Radon transform that maps a function to its mean values on spheres containing the origin. Our inversion formula implies that if f E C '(Rf) and its transform is zero on spheres inside a disc centered at 0, then f is zero inside that disc. We give functions f a C '(R f) whose transforms are identically zero and we give a necessary condition for a function to be the transform of a rapidly decreasing function. We show that every entire function is the transform of a real analytic function. These results imply that smooth solutions to the classical Darboux equation are determined by the data on any characteristic cone with vertex on the initial surface; if the data is zero near the vertex then so is the solution. If the data is entire then a real analytic solution with that data exists. In 1917 Radon inverted the first "Radon transform" [18]. This transform, R, maps a function on Rn to a function on the set of hyperplanes in Rn. If f is a continuous function of compact support on R" then Rf evaluated on a hyperplane is the integral of f over that hyperplane in its natural measure. The case n = 2 has many applications in science, engineering, and medicine [2], [3], [15], [21] and the transform on Rn (n arbitrary) has many applications to partial differential equations [13], [14]. Generalizations of this Radon transform to integrations over certain spheres and ellipsoids have been studied by John and others [13], [19] again in connection with partial differential equations. Moreover these examples are all special cases of the generalized Radon transform: given smooth manifolds X, Y, and a class of submanifolds of X, { Hy I y E Y), one specifies smooth measures on each Hy. The generalized Radon transform R from Co'(X) to functions on Y takes f E C0?(X) to the integrals of f over the manifolds Hy in the measures /, [7]. In many cases restrictions on the support of Rf imply restrictions on the support of f [10]; this fact is useful in applications to partial differential equations [11], [14]. In this article we define a Radon transform over spheres passing through the origin in Rn. 1ff E C(Rn), the transform f evaluated on a sphere containing 0 is the mean value of f over that sphere in its natural measure. Our main result, Theorem 1, is an inversion formula for this transform: if f E C (R n) then f(x) is determined by the values of f on spheres that lie inside the disc of radius Ixi Received by the editors August 17, 1979; presented to the Society, October 19, 1979 at Howard University. AMS (MOS) subject classifications (1970). Primary 44A05, 35Q05.

Free modular lattices

Abstract: It is shown that the word problem for the free modular lattice on five generators is recursively unsolvable. In this paper we show that the word problem for the free modular lattice on five generators, FM(5), is unsolvable. That is, there is no algorithm which can decide for arbitrary lattice terms u and o in five variables if u = v holds identically in all modular lattices. In fact, we show that there is a fixed lattice term u0 in five variables such that there is no algorithm for deciding if u0 = v holds in all modular lattices for an arbitrary five variable lattice term v.2 The free modular lattice on three generators, which is finite, was described in 1900 by R. Dedekind [7] and G. Birkhoff observed that FM(4) was infinite [2]. Interest in the word problem for free modular lattices greatly increased after P. Whitman's solution of the word problem for free lattices appeared [36]. Some partial results were obtained by K. Takeuchi [34], [35] (see Whitman's article [39]). Positive solutions were announced by Schutzenberger [33] and Gluhov [17] but these were refuted by Jonsson (see [38]) and Herrmann [20]. Interest in this problem was renewed in the late sixties partly because the wide applicability of Whitman's results was becoming apparent. Perhaps the most important result was that of G. Hutchinson and, independently, L. Lipshitz that there is a finitely presented modular lattice with unsolvable word problem [24], [41]. Hutchinson went on to show that this presentation could have five generators and one relation [25]. Moreover his results apply to many subvarieties of the variety of all modular lattices. Certain positive results were also described. R. Wille characterized those partially ordered sets P such that the modular lattice freely generated by P is finite [40]. C. Herrmann and A. Huhn have shown that for certain varieties of modular lattices generated by submodule lattices the word problem for the free lattices in these varieties is solvable [22]. Interestingly there is a nonempty intersection of these varieties and those to which Hutchinson's results apply. In particular the variety é£ generated by all subgroup lattices of abelian groups has a finitely Received by the editors December 4, 1978 and, in revised form, November 27, 1979; presented to the Society, October 19, 1978 under the title Recent results in modular lattice theory. AMS (MOS) subject classifications (1970). Primary 06A30; Secondary 02F47. 'This research was partially supported by NSF Grant No. MCS77-01933. 2The mathematically precise formulation of these results should be in terms of recursive functions [5]. Since our proof reduces the problem to showing that there is a finitely presented group with unsolvable word problem, it is not necessary for us to give a formal definition of a recursive function. © 1980 American Mathematical Society 0002-9947/80/0000-0403/$03.75 81 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use