Showing papers in "Transactions of the American Mathematical Society in 1986"

Definable sets in ordered structures. i

Abstract: This paper introduces and begins the study of a well-behaved class of linearly ordered structures, the ¢minimal structures. The definition of this class and the corresponding class of theories, the strongly dLminlmal theories, is made in analogy with the notions from stability theory of minimal structures and strongly minimal theories. Theorems 2.1 and 2.3, respectively, provide characterizations of 41-minimal ordered groups and rings. Several other simple results are collected in §3. The primary tool in the analysis of ¢;minimal structures is a strong analogue of ii forking symmetry," given by Theorem 4.2. This result states that any (parametri- cally) definable unary function in an Yminimal structure is piecewise either constant or an ore er-preserving or reversing bijection of intervals. The results that follow include the existence and uniqueness of prime models over sets (Theorem 5.1) and a characterization of all btO-categorical Aminimal structures (Theorem 6.1). 1. Introduction. The class of linearly ordered structures has long been an im- portant subject of concern to model theorists. Impressive results have been obtained in the study of models of several particular theories that extend the theory of linear order. Among those theories that have been approached successfully are Peano arithmetic, the theory of ordered abelian groups, that of real-closed fields, and that of linear order itself. Yet, very little has been done in the way of developing a general model theory for ordered structures. In this paper, we develop the model theory for a class of linearly ordered structures that we isolate by demanding that a structure in this class satisfy a condition whose effect is that the linear ordering and the algebraic part of the structure behave quite well with respect to one another. Let L be a finitary first-order language, and w an L-structure. A set of n-tuples A c X" is said to be parametrically definable if there is some L-formula

On the homology of associative algebras

Abstract: We present a new free resolution for k as an G-module, where G is an associative augmented algebra over a field k. The resolution reflects the combinatorial properties of G. Introduction. Let k be a field and let G be an associative augmented k-algebra. For many purposes one wishes to have a projective resolution of k as a G-module. The bar resolution is always easy to define, but it is often too large to use in practice. At the other extreme, minimal resolutions may exist, but they are often hard to write down in a way that is amenable to calculations. The main theorem of this paper presents a compromise resolution. Though rarely minimal, it is small enough to offer some bounds but explicit enough to facilitate calculations. As it relies heavily upon combinatorial constructions, it is best suited for analyzing otherwise tricky algebras given via generators and relations. Since several results we get as consequences of the main theorem have been obtained before through other means, this paper may be viewed as generalizing and unifying several seemingly unrelated ideas. In particular, we are generalizing Priddy's results on Koszul algebras [12], extending homology computations by Govorov [9] and Backelin [3], and complementing Bergman's methods regarding the diamond lemma [6]. Three results may be of interest. The homology of the mod p Steenrod algebra is given in terms of the homology of a new chain complex smaller than the A-algebra in Theorem 3.5. Formula (16) offers an efficient algorithm for the determination of Hilbert series, and Theorem 4.2 asserts the existence of new bounds on the torsion groups of commutative graded rings. 1. Definitions and the main theorem. Throughout this paper, k denotes any field and G is an associative k-algebra with unity. The field k embeds in G via 77: k -*G and we suppose that G has an augmentation, i.e., a k-algebra map s: G -? k for which 77 is a right inverse. S denotes a set of generators for G as a k-algebra and k(S) is the free associative k-algebra with unity on S. There is a canonical surjection f: k(S) -? G once S is chosen, and the augmentation E is determined once we know s(x) for each x E S. In particular, this means that k(S) may be augmented such that f becomes a map of augmented algebras. To S we associate a function e: S -? Z+ called a grading. In the absence of a more compelling choice we often take e to be grading by length, i.e., e(x) = 1 for Received by the editors May 23, 1983 and, in revised form, February 22, 1984. This paper was the subject of an invited one-hour address in Boulder, Colorado, during the week-long AMS summer program on Combinatorics and Algebra, June 1983. 1980 Mathematics Subject Classification (1985 Revision). Primary 16A62; Secondary 13D03, 55S10. (?)1986 American Mathematical Society 0002-9947/86 $1.00 + $.25 per page

Random recursive constructions: asymptotic geometric and topological properties

Abstract: We study some notions of \"random recursive constructions\" in Euclidean m-space which lead almost surely to a particular type of topological object; e.g., Cantor set, Sierpiriski curve or Menger curve. We demonstrate that associated with each such construction is a \"universal\" number a such that almost surely the random object has Hausdorff dimension a. This number is the expected value of the sum of some ratios which in the deterministic case yields Moran's formula. We introduce the notion of a \"random recursive construction\" and prove several basic facts about such constructions. We give specific examples which lead to random Cantor sets, Sierpinski curves or Menger universal curves. It is perhaps best to begin with a specific example of such a construction. To this end, let us make some notation. Let N be the set of positive integers and R the real numbers. If S is a set, let 5* be the set of all finite sequences of elements of S including 0, the empty sequence. If a = (ai,..., an) and ß — (bi,..., bm) are elements of S, then |q| = n, the length of a, and a * ß = (ai,... ,an,bi,... ,bm). Now, consider the following construction of a Cantor subset of [0,1], the unit interval. (Of course, by a Cantor set we mean a compact, perfect, 0-dimensional metric space.) Set J® = [0,1] and, by recursion, if Ja = [a,b], for a G {0,1}*, then set Jo-,0 = [a,a + x(b — a)] and JCT*i = [o + y(b — a), b], where the point (x, y) is chosen from the triangular region A = {(s,t) | 0 < s < t < 1} according to the uniform distribution. It follows from the results given in this paper that with probability one, the set K = f] (J ̂ n [ct6{0,1}TM is a Cantor set and the Hausdorff dimension of K, dimH(K), is (\\/Ï7 3)/2. The paper is organized into four sections. In §1, we define the notion of a random construction and prove a few basic facts concerning such a construction. We demonstrate that with each construction there is a number a such that with probability one the object constructed has Hausdorff dimension < a. In this section, we relate our results to some deterministic results of P. A. P. Moran [15]. In §2 (Theorem 2.1), we show that certain commonly occurring constructions have finite moments of all orders. This result is necessary for our proof that with probability one the Hausdorff dimension is a. Received by the editors December 27, 1984 and, in revised form, July 9, 1985. 1980 Mathematics Subject Classification. Primary 54H20, 60B05; Secondary 28C10.

Real hypersurfaces and complex submanifolds in complex projective space

Abstract: Let M be a real hypersurface in /"'(C), J be the complex structure and | denote a unit normal vector field on M. We show that M is (an open subset of) a homogeneous hypersurface if and only if M has constant principal curvatures and Jt, is principal. We also obtain a characterization of certain complex submanifolds in a complex projective space. Specifically, /""(C) (totally geodesic), Q", Pl(C) x P"(C). SU{5)/S(U{2) X (7(3)) and SO(10)/t/(5) are the only complex submani- folds whose principal curvatures are constant in the sense that they depend neither on the point of the submanifold nor on the normal vector.

Crossed products and inner actions of Hopf algebras

Abstract: This paper develops a theory of crossed products and inner (weak) actions of arbitrary Hopf algebras on noncommutative algebras. The theory covers the usual examples of inner automorphisms and derivations, and in addition is general enough to include \"inner\" group gradings of algebras. We prove that if 7r : H —► H is a Hopf algebra epimorphism which is split as a coalgebra map, then H is algebra isomorphic to A #„ H, a crossed product of H with the left Hopf kernel A of it. Given any crossed product A #CT H with H (weakly) inner on A, then A #CT if is isomorphic to a twisted product AT[H] with trivial action. Finally, if H is a finite dimensional semisimple Hopf algebra, we consider when semisimplicity or semiprimeness of A implies that of A #„ H; in particular this is true if the (weak) action of H is inner. Introduction. The purpose of this paper is to begin to lay the foundations of a general theory of actions of Hopf algebras on noncommutative algebras. The importance of such a theory derives from three special cases in which the Hopf algebra is a group algebra, an enveloping algebra of a Lie algebra, or the dual of a group algebra. These cases show that the theory encompasses the study of actions of groups as automorphisms of algebras, the study of actions of Lie algebras as derivations of algebras, and the study of group graded rings, respectively. Each of these areas of study have been quite active lately (see [16, 9, 5]). A fundamental concept in the first two cases has been the notion of inner action. One of the major purposes of this paper is to study inner actions of Hopf algebras; as a new example, we will investigate what is meant by an inner grading. Another important concept in the first two cases is that of a semidirect product (smash product): for group actions we have skew group rings and for Lie algebra actions we have differential polynomial rings. This notion is also defined for Hopf algebra actions. More generally one can consider crossed products A ffa H of an algebra A with a Hopf algebra H, where the multiplication of the copy of H in A ffa H is twisted by a cocycle o, and their study is the other major purpose of this paper. It turns out that these two concepts (inner actions and crossed products) are closely interrelated. Now inner actions and crossed products of Hopf algebras were both studied by Sweedler [24] in the context of the cohomology theory of Hopf algebras. The present paper owes a great debt to Sweedler's work, especially in §§1 and 4. However, his set-up was restricted at crucial places to the situation where H is a cocommutative Received by the editors December 12, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 16A24; Secondary 16A72, 16A03, 46L40. The third author was partially supported by NSF Grant DMS 8500959 and by a Guggenheim Memorial Foundation Fellowship. ©1986 American Mathematical Society 0002-9947/86 $1.00 + $.25 per page

The Fefferman metric and pseudo-Hermitian invariants

Abstract: C. Fefferman has shown that a real strictly pseudoconvex hypersurface in complex n-space carries a natural conformai Lorentz metric on a circle bundle over the manifold. This paper presents two intrinsic constructions of the metric, valid on an abstract CR manifold. One is in terms of tautologous differential forms on a natural circle bundle; the other is in terms of Webster's pseudohermitian invariants. These results are applied to compute the connection and curvature forms of the Fefferman metric explicitly.

Rational singularities and almost split sequences

Abstract: The main aim of this paper is to relate almost split sequences to singularity theory by showing that the McKay quiver built from the finite-dimen- sional representations of a finite subgroup G of GL(2, C), where C is the complex numbers, is isomorphic to the AR quiver of the reflexive modules of the quotient singularity associated with G. Over the past decade, almost split sequences have been playing an increasingly important role in the representation theories of finite-dimensional algebras and classical orders (see (5 and 3, 8) for basic existence theorems in these contexts). While they have been known for some time to exist in higher-dimensional situations (3), it has not been at all clear how they related to singularity theory, if at all. The main aim of this paper is to relate almost split sequences to singularity theory by showing that the McKay quiver built from the fimte-dimensional complex represen- tations of a finite subgroup G of GL(2,C), where C is the complex numbers, is isomorphic to the AR quiver of the reflexive modules over the quotient singularity R associated with G. As in the case of finite-dimensional algebras, the AR quiver of reflexive R-modules is defined in terms of the almost split sequences of reflexive R-modules. In the case G c SL(2,C), McKay observed that the underlying graph of the McKay quiver, with the trivial module removed, is isomorphic to the desingulariza tion graph of the associated singularity. Various explanations of this phenomenon have been given by Knorrer, Gonzalez-Sprinberg-Verdier (6) and Artin-Verdier (1), which along the way have established, most explicitly in (1), a natural one-to-one correspondence between the indecomposable reflexive R-modules and the nodes of the desingularization graph. But why the almost split sequences describe the edge of the desingularization graph still remains to be explained. An effort has been made to make this paper as self-contained as possible. In particular, no prior knowledge of almost split sequences is required. Before describing the contents of the six sections of this paper we fix some notation. Throughout this paper G is a finite group, k an algebraically closed field of characteristic not dividing the order of G and V a two-dimensional k-representation of G. Setting S = k((X, Y)), the k-algebra of formal power series, the two-dimen- sional representation V gives a linear action of G on S as a group of k-algebra automorphisms. We denote by S(G) the skew group ring given by this action.

Chaotic functions with zero topological entropy

Abstract: On caracterise la classe M⊂C°(I,I) des applications chaotiques en ce sens. On montre que M contient certaines appliquations qui ont a la fois une entropie topologique nulle et des attracteurs infinis. On montre que le complement de M consiste en applications qui ont seulement des trajectoires approchables par des cycles

Nonlinear oblique boundary value problems for nonlinear elliptic equations

Abstract: On considere le probleme aux valeurs limites a derivee oblique non lineaire pour des equations uniformement elliptiques quasi-lineaires et totalement non lineaires d'ordre 2. Les operateurs elliptiques satisfont des conditions de structure naturelle. On donne des theoremes d'existence pour les solutions

Weighted inequalities for the one-sided Hardy-Littlewood maximal functions

Abstract: Let M+f(x) = SUPh , 0(l/h)Jx+h (t)1 Idt denote the one-sided maximal function of Hardy and Littlewood. For w(x) > 0 on R and 1 0 and a = v-ll(P -), then M+ is bounded from LP(v) to LP(w) if and only if [ [M (X,a)] w 0. The corresponding weak type inequality is also characterized. Further properties of A weights, such as A + A e and A p = (A +)(A1 -P, are established.

On integers free of large prime factors

Abstract: The number 'l'(x,y) of integers y has been given satisfactory estimates in the regions y exp{(log log X)5/3+6}. In the intermediate range, only very crude estimates have been obtained so far. We close this "gap" and give an expression which approximates l'(x, y) uniformly for x > y > 2 within a factor 1 + O((log y)/(log x) + (log y)/y). As an application, we derive a simple formula for Ti(cx, y)/Fl'(x, y), where 1 < c < y. We also prove a short interval estimate for l'(x, y).

On the Hausdorff dimension of some graphs

Abstract: Consider the functions Wb(x)= b-cn[1(bnX + On)--1(0n)] n=-oo where b > 1, 0 0 such that if b is large enough, then the Hausdorff dimension of the graph of Wb is bounded below by 2a (C/ ln b). We also show that if a function f is convex Lipschitz of order a, then the graph of f has a-finite measure with respect to Hausdorff's measure in dimension 2 a. The convex Lipschitz functions of order a include Zygmund's class A,. Our analysis shows that the graph of the classical van der WaerdenTagaki nowhere differentiable function has a-finite measure with respect to h(t)-=t/ In(1/t). We consider the Hausdorff dimension of the graphs of various continuous functions. We introduce a new geometric property of a function: convex Lipschitz of some order, and obtain an upper bound on the dimension of a graph with this property. In particular, our analysis includes functions of the form

The chromatic number of kneser hypergraphs

Abstract: On utilise une generalisation due a Barany, Shlosman et Szucs du theoreme de Borsuk-Ulam, pour etablir une conjecture d'Erlos relative a la coloration des r-sousensembles d'un ensemble a n elements et un theoreme de partition optimale

Quasilinear evolution equations and parabolic systems

Abstract: It is shown that general quasilinear parabolic systems possess unique maximal classical solutions for sufficiently smooth initial values, provided the boundary conditions are" time-independent". Moreover it is shown that, in the autonomous case, these equations generate local semi flows on appropriate Sobolev spaces. Our results apply, in particular, to the case of prescribed boundary values (Dirichlet boundary conditions).

On type of metric spaces

Abstract: Families of finite metric spaces are investigated. A notion of metric type is introduced and it is shown that for Banach spaces it is consistent with the standard notion of type. A theorem parallel to the Maurey-Pisier Theorem in Local Theory is proved. Embeddings of Zp-cubes into the ¡i-cube (Hamming cube) are discussed.

Affine semigroups and Cohen-Macaulay rings generated by monomials

Abstract: We give a criterion for an arbitrary ring generated by monomials to be Cohen-Macaulay in terms of certain numerical and topological properties of the additive semigroup generated by the exponents of the monomials. As a consequence, the Cohen-Macaulayness of such a ring is dependent upon the characteristic of the ground field. Introduction. Let N denote the set of nonnegative integers. By an affine semigroup we mean a finitely generated submonoid S of the additive monoid N n, where n is some positive integer. Let k[S] denote the semigroup ring of S over a field k. Then one can identify k[S] with the subring of a polynomial ring k[t l , ... , tn] generated by the monomials IX = lil ••• t:n, x = (Xl"'" Xn) E S. Obviously, every subring of k[t l , ... , In] generated by a finite set of monomials is the semigroup ring of the affine semigroup in N n generated by the exponents of the monomials. So one has, up to isomorphisms, a one-to-one correspondence between affine semigroups and affine varieties which are given parametrically by finite sets of monomials. When illustrating problems of algebraic geometry one almost inevitably tends to choose varieties of this type. Even when dealing with a quite general variety, either its singularities or a certain blowup may well be defined in local coordinates by monomials [15, 19]. Moreover, one can also use rings generated by monomials to study solutions of linear equations in nonnegative integers or, equivalently, invariants of a torus acting linearly on a polynomial ring [11, 26]. Therefore, a criterion for such a ring to be Cohen-Macaulay in terms of the associated semigroup would be very useful. It should be mentioned that the first example of a non-Coh~n­ Macaulay domain (in modem language), given by F. S. Macaulay at the beginning of this century [17, p. 98], was the ring k[/t, Ith, Il/~, Ii] and that analyzing this example, Grobner [7] already posed the problem of classifying rings generated by monomials of the same degree with respect to their Cohen-Macaulayness. The first step toward such a criterion was taken by Hochster [11], who succeeded in characterizing normal rings generated by monomials in terms of the associated semigroups and showed that they are always Cohen-Macaulay. Although this result was motivated by a conjecture on rings of invariants of reductive linear algebraic groups, which was later settled [12], its proof deserved much attention. It suggested Received by the editors July 22, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 13RI0; Secondary 14M05. 145 ©1986 American Mathematical Society 0002-9947/86 $1.00 + $.25 per page License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 146 N. V. TRUNG AND L. T. HOA the use of topological techniques in studying rings generated by monomials. The next step was a criterion given by Goto et al. for the case that the ring has a system of parameters consisting of monomials [5]. These results inspired many other works and were re-proved many times by different techniques, such as rational resolution [15], Hilbert functions of graded algebras [25], homology of polyhedral complexes [13,26], and the Hodge algebra [10]. In both steps, one used the coincidence of an affine semigroup S with one of its extensions to indicate the Cohen-Macaulayness of k[S]. Inspired by this phenomenon, Goto and Watanabe [6] defined a suitable extension Sf of S (see below) and claimed that Sf = S is a necessary and sufficient condition for k[S] to be CohenMacaulay. Let Z and Q denote the sets of integers and rational numbers, respectively. Consider the elements of S as points in the space Qn. Let G denote the additive group in zn generated by S and put r = rankzG. Let Cs denote the convex rational polyhedral cone spanned by S in Qn. Then Cs is r-dimensional. Suppose that Fl , ... , Fm are the (r I)-dimensional faces of Cs. Let Sj denote the set of elements x E G such that x + yES for some element yES n F;, i = 1, ... , m. Then they define Sf = n?_lSj' In this paper, we shall see that the condition Sf = S is not sufficient for the Cohen-Macaulayness of k[S], and that one has to add some topological condition on the convex cone Cs to get a correct criterion. To formulate this we need some more notation. Let [1, m] denote the set of the integers 1, ... , m. For every subset J of [1, m], set GJ = n Sj \ U Sj' i~J JEJ and let 'lTJ be the simplicial complex of nonempty subsets I of J with the property n j E IS n F; *" (0). Note that one calls 'lTJ acyclic if the reduced homology group Hi'ITJ ; k) vanishes for all q ~ O. MAIN THEOREM. Let S be an arbitrary affine semigroup. Then k[S] is a CohenMacaulay (resp. Gorenstein) ring if and only if the following conditions are satisfied: (i) Sf = S (resp. there exists an element x E G such that every element of G[l,m] is the difference of x by some element of S). (ii) For every nonempty proper subset J of [1, m], GJ = 0 or 'lTJ is acyclic. It will follow from some property of the Cousin complex of k[S] and from an explicit description of all local cohomology modules of k[Sf] in terms of GJ and 'lTJ • We will also give some simple methods for checking the above conditions. As a consequence, one immediately gets the abovementioned results of Hochster and Goto et al. Another application is a criterion for the Rees algebra (blowing-up) of a ring generated by monomials to be Cohen-Macaulay (resp. Gorenstein). In particular, as in the work of Reisner on polynomial rings modulo ideals generated by square-free monomials [20] where a similar link to topology is given, we will show that the Cohen-Macaulayness of k[S] is dependent upon the characteristic of the License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use AFFINE SEMIGROUPS AND COHEN-MACAULAY RINGS 147 field k. Moreover, using the main theorem, we have been able to solve Grabner's problem for some particular cases [31] (cf. [21, 29, and 30)). We would like to mention that Stanley [26] also obtained a similar result on modules associated with solutions of systems of linear diophantine equations, which overlaps with ours only in Hochster's normal case. This paper is organized in five sections. §1 gives a counterexample to the result of Goto and Watanabe via some consideration on the Rees algebras of affine semigroup rings. In §2, S' will be related to the Cousin complex of k[S] in order to show that S' = S if k[S] is a Cohen-Macaulay ring. §3 deals with the local cohomology modules of k[S']. Criteria for k[S] to be Cohen-Macaulay (resp. Gorenstein) are given in §4. There we will also deal with the Buchsbaumness of k[S]. The aim of §5 is to construct an affine semigroup ring whose local cohomology modules are just the reduced homology groups of a given finite simplicial complex. All notations introduced above will be used throughout. Moreover, if A and B are subsets of zn, G(A) denotes the additive group generated by A in zn, and A ± B is the set of elements a ± b with a E A and b E B. If x, y, ... are elements of zn, we will denote their components by Xi'Yi"'" i = 1, ... , n, respectively. For unexplained notations and standard facts in commutative algebra, algebraic topology, and local cohomology, we refer the reader to [18, 24, and 8]. ACKNOWLEDGMENT. The authors would like to thank S. Ikeda for pointing out that our earlier conclusion on the Cohen-Macaulayness of the Rees algebras of Cohen-Macaulay rings generated by monomials is false (see §1). This led us to check the result of [6, II]. Thanks are also due to S. Goto for encouraging our study, and to L. Robbiano for some useful suggestions. 1. Counterexamples to the result of Goto and Watanabe. Let S be an arbitrary affine semigroup in N n. Set S = {x E G; PX E S for some p > O}, S(i) = {x E S; Xi = A}, i = 1, ... ,n. Then we call S standard if the following conditions are satisfied: (1) S = G n N n , (2) S(i) =F S(j) for i =F j, (3) rankZG(S(i») = r 1, i = 1, ... , n. Geometrically, these conditions mean that Cs has exactly n (r 1 )-dimensional faces lying on the hyperplane Xi = 0, i = 1, ... , n. In this case, we may assume that S(i) = S n F; and Si = S S(i)' Goto and Watanabe [6, Theorem 3.3.3] claimed that if S is standard, then k[S] is a Cohen-Macaulay ring if and only if S' = S. We shall see that this is false. First, we have to remark that every affine semigroup can be transformed isomorphically onto a standard one by the following technique which is due to Hochster [11, p. 323]. HOCHSTER'S TRANSFORMATION. Let W denote the vector space generated by S in Qn. Then one can find m linear functionals 11"'" 1m from W to Q corresponding with Fl , ..• , Fm such that Cs = {x E W; li{x} ~ Oforalli}. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 148 N. V. TRUNG AND L. T. HOA See also [15, p. 6]. Let T denote the linear transformation which sends every element x E W to the element (ll(X), ... , Im(x» E Qm. By replacing Ii by a suitable positive integer multiple, one can assume that T(S) ~ N m. Hochster has shown that T(S) is isomorphic to S and that T(S) = G(T(S)) n N m. Obviously, T also induces an isomorphism between the semigroups S n F; and T(S){iJ' i = 1, ... , m. Since S n F; =1= S n Fj for i =1= j, and rankzG(S n F;) = r 1, we can conclude that T(S) is a standard affine semigroup in N m. Moreover, since T(S)i = T(S) T(S)(i) = T(S S n F;) = T(Si)' we also have T(S)' = T(S'). According to this transformation, one can omit the assumption on the standardness of S in the claim of Goto and Watanabe. The counterexample will also be constructed in the nonstandard case by using the following observation. Let A: Qn -+ Q be a linear functional such that A(S) ~ N, and, if xES and A(X) = 0, then x = O. Then we call the affine semigroup E>,.:= {(x, p) E N n+1 ; xES and A(X) > p} a blowing-up extension of S. The name stems from the

A global theory of internal solitary waves in two-fluid systems

Abstract: : The study of single-crested progressing gravity waves was initiated over a century ago with the observations by Russell of what he termed solitary waves, which progressed without change of form over a considerable distance on the Glasgow-Edinburgh Canal. The mathematical analysis of this wave motion on the surface of water, begun in the nineteenth century, has undergone a rapid development in the last three decades, due to the scattering theory for the Korteweg-de Vries equation, which models the motion of long waves due to the development of techniques in nonlinear analysis allowing for the analysis of finite amplitude motions. The work on surfce waves has many parallels in the study of waves in fluids with variable density. In the case of a heterogeneous fluid with a free upper surface, gravity waves still occur, in analogy with surface waves in a fluid of constant density. What is distinctive about a fluid with density stratification, however, is the presence of waves which are predominantly due to the stratification and not to the free surface. These waves, called internal waves, exist in a heterogeneous fluid even when it is confined between horizontal boundaries, a configuration which precludes gravity waves in a fluid of constant density. This paper is concerned with progressing solitary gravity waves in a system consisting of two fluids of differing densities confined in a channel of unit depth and infinite horizontal extent.

On the Neumann problem for some semilinear elliptic equations and systems of activator-inhibitor type

Abstract: On deduit des estimations a priori pour des solutions positives du probleme de Neumann pour des systemes elliptiques semilineaires ainsi que pour des equations isolees semilineaires reliees a ces systemes

The blow-up boundary for nonlinear wave equations

Abstract: Consider the Cauchy problem for a nonlinear wave equation Du = F(u) in N space dimensions, N (x) with {x) continuously difierentiable. Introduction. Consider the nonlinear wave equation (0.1) Uu = u,t- Au = F(u) for x 6 R", i > 0, with the initial data (0.2) u(x,0)-/(x), ul(x,0) = g(x) for x G RN. It is well known that if F(u) is nonnegative and superlinear, then, in general, a solution cannot exist for all times. Furthermore, if T is the supremum of all times s such that a classical solution exists for all 0 < t < s, then sup |w(jc, t) | -* oo if t -* T.

Recursive labelling systems and stability of recursive structures in hyperarithmetical degrees

Abstract: We show that, under certain assumptions of recursiveness in 2t, the recursive structure 2t is AO-stable for ca w, we need to deal with limit ordinals, and we frame an analogous result [Proposition 2] for (-yn)-systems where (-yn) is an increasing sequence of ordinals. Propositions 1 and 2 are then established by the

Invariants of the Lusternik-Schnirelmann type and the topology of critical sets

Abstract: We introduce and study in detail generalizations of the notion of Lusternik-Schnirelmann category which give information about the topology of the critical set of a differentiable function We also improve a result of T Ganea about the equality of the strong category and the category (even in the classical case) The category cat(X) of a space X in the sense of Lusternik and Schnirelmann [14] is the smallest number k such that there exists a covering { X1,, Xk } of X (of a certain kind, cf 12(1)) for whicheach inclusion X c X is nullhomotopic The motivation for introducing this concept was that it gives a lower bound for the number of critical points of a function More precisely, if M is a closed differentiable manifold and f is a differentiable real function on M then the number of critical points of f is at least cat(M) We propose the following generalization: If _V is any class of spaces we replace the condition that Xj c X is nullhomotopic by requiring that it factors through some A E _ up to homotopy and we obtain the notion of -V-category, -V-cat(X) If _/ consists only of the one-point space, this is the classical cat(X) If _V is the class of q-connected spaces s/-cat is the "q-dimensional homotopy category" introduced by Fox in [7] Another interesting example is the class _V of q-dimensional spaces If f: M -+ R is as above then -V-cat(M) does not give any new information about the number of critical points, because it is less than or equal to cat(M) It does give, however, under certain conditions, some new information on the topological structure of the critical set Roughly, one can say that either there are at least -V-cat(M) critical values of f or there is one critical value -y of f such that the corresponding set of critical points K n f 1(y) is not of the homotopy type of any space in _V (cf ?2 for more details) Quite a number of papers have appeared on Lusternik-Schnirelmann category and related notions (cf [13] for a survey) In particular there has been a revival of interest in recent years We shall present a systematic theory for our more general Received by the editors November 25, 1985 1980 Mathematics Subject Classification (1985 Revision) Primary 55M30; Secondary 58E05, 55P50

Congruences on regular semigroups

Abstract: Let S be a regular semigroup and let p be a congruence relation on S. The kernel of p, in notation ker p, is the union of the idempotent eclasses. The trace of p, in notation trp, is the restriction of p to the set of idempotents of S. The pair (kerp, trp) is said to be the congruence pair associated with p. Congruence pairs can be characterized abstractly, and it turns out that a congruence is uniquely determined by its associated congruence pair. The triple ((p V £)/£, ker p, (p V R)/R) is said to be the congruence triple associated with p. Congruence triples can be characterized abstractly and again a congruence relation is uniquely determined by its associated triple. The consideration of the parameters which appear in the above-mentioned representations of congruence relations gives insight into the structure of the congruence lattice of S. For congruence relations p and 0, put pTz 0 [pTrr 0, pTo] if andonlyifpV: =0V: [pvR =0VR,trp=tr0]. ThenTz, Tr and T are complete congruences on the congruence lattice of S and T = Tl n T. Introduction and summary. After it was realized by Wagner that a congruence on an inverse semigroup S is uniquely determined by its idempotent classes, Preston provided an abstract characterization of such a family of subsets of S called the kernel normal system (see [2, Chapter 10]). This approach was the only usable means for handling congruences on inverse semigroups for two decades. A new approach to the problem of describing congruences on inverse semigroups was sparked by the work of Scheiblich [13] who described congruences in terms of kernels and traces. A systematic exposition of the achievements of this approach can be found in [10, Chapter III]. It was recognized by Feigenbaum [3] that every congruence p on a regular semigroup S is uniquely determined by its kernel, kerp, equal to elements tequivalent to idempotents, and its trace, trp, equal to the restriction of p to the set E(S) of idempotents of S. In the case of an inverse semigroup S, kerp and trp, as well as their mutual relationship, can be described abstractly by means of simple conditions on a subset of S and an equivalence on E(S) (see [10, Chapter III]). Following in the footsteps of Scheiblich, for orthordox and arbitrary regular semigroups, Feigenbaum [3] and lYotter [14] adopted the following approach: trp is characterized abstractly and to each such trp all matching kernels are described. This unbalances the symmetry of the kernel-trace approach by giving preference to the trace. Hence a balanced view relative to the kernel and the trace is evidently called for. The unqualified success of the kernel-trace approach for inverse semigroups, including its diverse ramifications, gave a certain hope that this may also turn out to be the case for regular semigroups. Judging by the complexity of regular semigroups and the attempts made for both orthodox and general regular semigroups, Received by the editors November 2, 1984. 1980 Mathematics Subject Classification. Primary 20M10; Secondary 08A30. (a)1986 American Mathematical Society 0002-9947/86 $1.00 + $.25 per page

Isometries for the legendre-fenchel transform

Abstract: It is shown that on the space of lower semicontinuous convex functions defined on R', the conjugation map-the Legendre-Fenchel transform-is an isometry with respect to some metrics consistent with the epi-topology. We also obtain isometries for the infinite dimensional case (Hilbert space and reflexive Banach space), but this time they correspond to topologies finer than the Mosco- epi-topology.

Index filtrations and the homology index braid for partially ordered Morse decompositions

Abstract: : On a Morse decomposition of an invariant set in a flow there are partial orderings defined by the flow. These are called admissable orderings of the Morse decomposition. The index filtrations for a total ordering of a Morse decomposition are generalized in this paper with the definition and proof of existence of index filtrations for admissable partial orderings of a Morse decomposition. It is shown that associated to an index filtration there is a collection of chain complexes and chain maps called the chain complex braid of the index filtration. The homology index braid of the corresponding admissable ordering of the Morse decomposition is obtained by passing to homology in the chain complex braid. Keywords: Morse decomposition; Conley index; Index filtration.

Simply-connected 4-manifolds with a given boundary

Abstract: Let M be a closed, oriented, connected 3-manifold. For each bilinear, symmetric pairing (Zn, L), our goal is to calculate the set VL(M) of all oriented homeomorphism types of compact, 1-connected, oriented 4manifolds with boundary M and intersection pairing isomorphic to (Zn, L). For each pair (Zn, L) which presents H. (M), we construct a double coset space BL (M) and a function Ct: VL(M) BL (M). The set Bt (M) is the quotient of the group of all link-pairing preserving isomorphisms of the torsion subgroup of H1 (M) by two naturally occuring subgroups. When (Zn, L) is an even pairing, we construct another double coset space BL(M), a function 6L: VL(M) -B fL(M) and a projection P2: BL(M) BL (M) such that P2 C ^L = Ct Our main result states that when (Zn, L) is even the function C^L is injective, as is the function C X A: VL(M) -* BL(M) x Z/2 when (Z', L) is odd. Here A is a Kirby-Siebenmann obstruction to smoothing. It follows that the sets VL(M) are finite and of an order bounded above by a constant depending only on H1(M). We also show that when Hi (M;Q) 0 and (Z',L) is even, Ct = P2 C 6L is injective. It seems likely that via the functions cL x A and 6L, the sets BL(M) x Z/2 and BL(M) calculate VL(M) when (Zn, L) is respectively odd and even. We verify this in several cases, most notably when H1 (M) is free abelian. The results above are based on a theorem which gives necessary and sufficient conditions for the existence of a homeomorphism between two 1connected 4-manifolds extending a given homeomorphism of their boundaries. The theory developed is then applied to show that there is an m > 0, depending only on H1(M), such that for any self-homeomorphism f of M, fm extends to a self-homeomorphism of any 1-connected, compact 4-manifold with boundary M. Introduction. In M. Freedman's fundamental paper [Fr], he-classified closed, simply-connected, oriented 4-manifolds up to orientation preserving homeomorphism. He showed that modulo the Kirby-Siebenmann invariant, these manifolds are in bijective correspondence through their intersection pairings with the set of unimodular, bilinear, symmetric pairings over Z (see Theorem (1.5) of [Fr] and Corollary (2.2.3) of [Q]). In this paper we begin a classification of simply connected 4-manifolds with connected boundary. Our results with regard to this problem are based on a theorem which gives necessary and sufficient conditions for the existence of a homeomorphism between 1-connected 4-manifolds extending a given homeomorphism of their boundaries. Received by the editors September 19, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 57N15. Research supported by an NSERC (Canada) post-doctoral fellowship. (@)1986 American Mathematical Society 0002-9947/86 $1.00 + $.25 per page

Interaction de deux chocs pour un système de deux lois de conservation, en dimension deux d'espace

Abstract: On resout le probleme de Cauchy dans le cas ou la solution ne presente que des chocs, en dimension d'espace n=2. On ramene la construction des chocs issus d'une meme surface initiale a la resolution d'un probleme de Goursat non lineaire, a frontiere libre. On etudie par les memes techniques l'interaction ou le croisement de deux chocs

Embedding strictly pseudoconvex domains into balls

Abstract: Every relatively compact strictly pseudocc)nvex domain D with C2 boundary in a Stein manifold can be embedded as a closed complex submanifold of a finite dimensional ball. However, for each n > 2 there exist bounded strictly pseudoconvex domains D in Cn with real-analytic boundary such that no proper holomorphic map from D into any finite dimensional ball extends smoothly to D. 0. Introduction. In this paper we study the representations of bounded strictly pseudoconvex domains D c cn. If the boundary of D is of class Ck, k E {2, 3, . . ., oo}, then, by a theorem of Fornaess [8] and Khenkin [13], D can be mapped biholomorphically onto the intersection xn Q of a bounded strictly convex domain Q c CN with Ck boundary and a closed complex submanifold X defirXed in a neighborhood of Q in CN, X intersecting the boundary of Q transversally. Moreover, the map f: D X n Q extends to a holomorphic map on a neighborhood of D. The convex domain Q depends on D; hence a natural question is whether a similar result holds with Q replaced by the unit ball B = (z = (Z1L, * , ZN) E CN ||Z||2 = E |Zjl2 n that intersects bBN transversally such that D is biholomorphically equivalent to xn E3ff? This question has been mentioned by Lempert [14], Pinduk [20], Bedford [2] and others. Our main result is that the answer to this question is negative in general. If D is as above, then every biholomorphism of D onto XnE3N extends smoothly to D according to [3]. However, we will show that not all such domains D admit a proper holomorphic map into a finite dimensional ball that is smooth on D (Theorem 1.1). A similar local result was obtained independently by Faran [7]. We shall show that the answer to the question (Q) is positive if we allow the intersections of complex submanifolds with strictly convex domains Q c CN with real-analytic boundaries (Theorem 1.2). ThetheoremsofFornaess [8] andKhenkin [13] onlygiveanQwith smooth boundary. Received by the editors April 4t 1985. 1980 Mathematics Subject Classification. Primary 32H05. 1Research supported by a fellowship from the Alfred P. Sloan Foundation. (t)1986 American Mathematical Society 0002-9947/86 $1 .00 + $ .25 per page

Spectral theory of the linearized Vlasov-Poisson equation

Abstract: Etude de la theorie spectrale de l'equation de Vlasov-Poisson linearisee dont la solution se comporte comme une somme d'ondes planes pour de grandes valeurs du temps. Prolongement analytique de la resolvante de l'equation

Trellises formed by stable and unstable manifolds in the plane

Abstract: A trellis is the figure formed by the stable and unstable manifolds of a hyperbolic periodic point of a diSeomorphism of a 2-manifold. This paper describes and classifies some trellises. The set of homoclinic points is linearly ordered as a subset of the stable manifold and again as a subset of the unstable manifold. Each homoclinic point is assigned a type number which is constant on its orbit. Combinatorial properties of trellises are studied using type numbers and the pair of linear orderings. Trellises are important because their closures in some cases are strange attractors and in other cases are ergodic zones. Introduction. In Chapter 33 of New Methods of Celestial Mechanics, Poincare describes the figure formed by the stable and unstable manifolds of a hyperbolic fixed point of a transformation of the upper half plane as follows: "When we try to represent the figure formed by these two curves and their intersections, each of which corresponds to a doubly asymptotic solution, these intersections form a type of trellis, tissue, or grid with infinitely serrated mesh. Neither of the two curves must ever 6ut across itself again, but must bend back upon itself in a very complex manner in order to cut across all of the meshes in the grid an infinite number of times. The complexity of this figure will be striking, and I shall not even try to draw it. Nothing is more suitable for providing us with an idea of the complex nature of the three body problem, and of all the problems of dynamics in general...." The aim of this paper is to understand figures of the type Poincare described so well. If p is a hyperbolic periodic point of a diffeomorphism of a 2-manifold, then the figure formed by its stable and unstable manifolds will be called the trellis of p whenever these manifolds have nonempty intersection. Figure 1 illustrates the beginning development of such a trellis. Similarly the figure formed by the stable and unstable manifolds of a cycle of hyperbolic periodic points is called the trellis of the cycle. A cycle of hyperbolic periodic points of a diffeomorphism is a finite collection P of periodic points having a cyclic permutation : P > P such that the unstable manifold of p intersects the stable manifold of (p) for each p G P. It is natural to conjecture that in some cases "ergodic zones" are the closures of trellises. Anosov transformations of the torus provide examples where this is the case [1]. One may also conjecture that "strange attractors" are sometimes the closures of the unstable manifolds of cycles of hyperbolic periodic points. For certain values of the parameters a and b the Henon Map of the plane defined by T(x, y) = (1 + y ax, by) may provide examples. Received by the editors May 7, 1985. 1980 Mathematics Subject Classification. Primary 58F13; Secondary 58F05, 58F11. (<)1986 American Mathematical Society 0002-9947/86 $1.00 + $.25 per page