Showing papers in "Transactions of the American Mathematical Society in 1981"

A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves

Abstract: Any graph-manifold can be obtained by plumbing according to some plumbing graph I\ A calculus for plumbing which includes normal forms for such graphs is developed. This is applied to answer several questions about the topology of normal complex surface singularities and analytic families of complex curves. For instance it is shown that the topology of the minimal resolution of a normal complex surface singularity is determined by the link of the singularity and even by its fundamental group if the singularity is not a cyclic quotient singularity or a cusp singularity. In this paper we describe a calculus for plumbed manifolds which lets one algorithmically determine when the oriented 3-manifolds M(TX) and A/fTj) ob- tained by plumbing according to two graphs Tx and T2 are homeomorphic (3-mani- folds are oriented 3-manifolds throughout this paper, and homeomorphisms of 3-manifolds are orientation preserving). We then apply the calculus to answer several questions about the topology of isolated singularities of complex surfaces and one-parameter families of complex curves. These results are described below. Since the class of 3-manifolds obtainable by plumbing is precisely the class of graph-manifolds, which were classified, with minor exceptions, by Waldhausen (24), a calculus for plumbing is in some sense implicit in Waldhausen's work. Moreover, it has been known for some time that the calculus can be put in a form like the one given here, but the details have never appeared in the literature. A related calculus for plumbing trees has been worked by Bonahon and Siebenmann (1) in order to classify their "algebraic knots". We describe the calculus in greater generality than is needed for the present applications, since this involves minimal extra work, and the calculus is needed elsewhere ((4), (14), and (15)). In particular, in an appendix we describe two generalizations of it. The calculus consists of a collection of moves one can do to a plumbing graph T without altering the plumbed manifold M(T). To see these moves are sufficient, we describe how they can be used to reduce any graph to a normal form which is

A perturbation method in critical point theory and applications

Abstract: This paper is concerned with existence and multiplicity results for nonlinear elliptic equations of the type -Au = |u|''_1u + h(x) in P», u = 0 on 3s. Here, s c R^ is smooth and bounded, and h e L2(Q) is given. We show that there exists pN > 1 such that for any p e (\,pN) and any h e L2(I2), the preceding equation possesses infinitely many distinct solutions. The method rests on a characterization of the existence of critical values by means of noncontractibility properties of certain level sets. A perturbation argument enables one to use the properties of some associated even functional. Several other applications of this method are also presented.

Nonsmooth analysis: differential calculus of nondifferentiable mappings

Abstract: A new approach to local analysis of nonsmooth mappings from one Banach space into another is suggested. The approach is essentially based on the use of set-valued mappings of a special kind, called fans, for local approximation. Convex sets of linear operators provide an example of fans. Generally, fans can be considered a natural set-valued extension of linear operators. The first part of the paper presents a study of fans; the second is devoted to calculus and includes extensions of the main theorems of classical calculus. Introduction. The idea to extend the framework of differential calculus so as to cover more general classes of functions and mappings is by no means new. Basically, it was the underlying idea for the differentiation theory connected with the Lebesgue integral and for the theory of distributions. Both theories deal essentially with what could be called nonlocal aspects of the calculus centered around the Newton-Leibniz and integration by parts formulae. The notion of the value of a derivative at a given point makes no sense in either of them. Nonsmooth analysis appeared in the 1970's just to carry out an extension of the local aspect of the calculus connected with the idea of (linear) approximation of a mapping about a given point. Certain separate ideas and results appeared of course much earlier (one could recall the Dini numbers for instance) but a systematic study began during the last decade when the natural development of the optimization theory made the need for such an extension very acute and, as often happens, practical and heuristic computations were initiated before an adequate theory appeared (see [52] and references therein). It is not surprising that the main impulse came from the optimization theory which has natural mechanisms generating nonsmoothness. But as a result, most of the efforts were applied to obtain more and more refined conditions for extrema with less interest in those aspects of analysis that are less immediately connected with this purpose. The only exception was perhaps the generalized gradients of Clarke whose analytical virtues were recognized from the beginning ([8]-[12], [17], [20]-[24], [26], [36], [38], [45], [46], [49], [50], [53], [61] and others). The original motivation for the present research was just to find a satisfactory extension of Clarke's approach to mappings in infinite-dimensional spaces and the first version of this paper (see [25] for a summary of results) was written completely along these lines. Later, however, it became more and more difficult to ignore a Received by the editors September 25, 1979 and, in revised form, July 18, 1980. 1980 Mathematics Subject Classification. Primary 26A96; Secondary 26A57. © 1981 American Mathematical Society 0002-9947/81/0000-0300/$ 15.00 1 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

Some properties of measure and category

Abstract: Several elementary cardinal properties of measure and category on the real line are studied. For example, one property is that every set of real numbers of cardinality less than the continuum has measure zero. All of the properties are true if the continuum hypothesis is assumed. Several of the properties are shown to be connected with the properties of the set of functions from integers to integers partially ordered by eventual dominance. Several, but not all, combinations of these properties are shown to be consistent with the usual axioms of set theory. The main technique used is iterated forcing. Six properties of measure and category on the real line are studied. A(c) is the proposition that the union of fewer than continuum many meager sets is meager. B(c) says that the real line is not the union of fewer than continuum many meager sets. U(c) is the proposition that every subset of the real line of cardinality less than continuum is meager. A(m), B(m), and U(w) are defined analogously by replacing meager by measure zero. In the first section some equivalent forms of these properties are given, for example, it is shown that A(c) iff B(c) and every family of elements of to\" of cardinality less than the continuum is eventually dominated by an element of ww. Characterizations of U(c) and B(c) are also given. In the second section we prove some theorems about unions of closed sets of measure zero, small dominating families, and strong measure zero sets. In the remaining sections several combinations of these properties are shown to be consistent with ZFC. These consistency results are summarized in the third section. The last section contains some open problems. I would like to thank K. Kunen for several helpful discussions. 1. The properties and some of their equivalent forms. All the properties we consider are equivalent whether stated for 2\", w\", or the real line. For definiteness they will be stated for the Cantor space 2\", so we will begin by reviewing the usual product topology and measure on 2\" and also establish some standard terminology. For sets X and Y let Yx denote the set of functions from X into Y and \\X\\ denote cardinality of X. Let 2

Iteration and the solution of functional equations for functions analytic in the unit disk

Abstract: This paper considers the classical functional equations of Schroeder f o p = Xf, and Abel f o p = f + 1, and related problems of fractional iteration where 9p is an analytic mapping of the open unit disk into itself. The main theorem states that under very general conditions there is a linear fractional transformation 4) and a function a analytic in the disk such that ) a = oa p and that, with suitable normalization, 4) and a are unique. In particular, the hypotheses are satisfied if 4p is a probability generating function that does not have a double zero at 0. This intertwining relates solutions of functional equations for T to solutions of the corresponding equations for (D. For example, it follows that if 4p has no fixed points in the open disk, then the solution space of f o (p = Xf is infinite dimensional for every nonzero X. Although the discrete semigroup of iterates of T usually cannot be embedded in a continuous semigroup of analytic functions mapping the disk into itself, we find that for each z in the disk, all sufficiently large fractional iterates of p can be defined at z. This enables us to find a function meromorphic in the disk that deserves to be called the infinitesimal generator of the semigroup of iterates of p. If the iterates of (p can be embedded in a continuous semigroup, we show that the semigroup must come from the corresponding semigroup for 4), and thus be real analytic in t. The proof of the main theorem is not based on the well known limit technique introduced by Koenigs (1884) but rather on the construction of a Riemann surface on which an extension of 9p is a bijection. Much work is devoted to relating characteristics of (p to the particular linear fractional transformation constructed in the theorem.

Fredholm and invertible -tuples of operators. The deformation problem

Abstract: Using J. L. Taylor's definition of joint spectrum, we study Fredholm and invertible «-tuples of operators on a Hilbert space. We give the foundations for a "several variables" theory, including a natural generalization of Atkinson's theorem and an index which well behaves. We obtain a characterization of joint invertibility in terms of a single operator and study the main examples at length. We then consider the deformation problem and solve it for the class of almost doubly commuting Fredholm pairs with a semi-Fredholm coordinate. 1. Introduction. 1. Let T be a (bounded linear) operator on a Banach space %. T is said to be invertible if there exists an operator S on % such that TS = ST = 1%, the identity operator on 9C. By the Open Mapping Theorem, this is equivalent to ker T = (0) and R(T) = range of T = %. The last formulation does not rely upon the existence of an inverse for T, but rather on the action of the operator T. When T is replaced by an «-tuple of commuting operators, several definitions of nonsingular- ity exist. J. L. Taylor (19) has obtained one which reflects the actions of the operators, by considering the Koszul complex associated with the «-tuple. 2. In this paper we develop a general "several variables" theory on the basis of Taylor's work and study commuting and almost commuting (= commuting mod- ulo the compacts) «-tuples of operators on a Hilbert space %. We obtain a characterization of joint invertibility in terms of the invertibility of a single operator, which is essential for our approach. From that we get a number of corollaries which generalize nicely the known elementary results in "one variable". At the same time, the referred characterization allows us to define a continuous, invariant under compact perturbations, integer-valued index on the class of Fred- holm «-tuples (those almost commuting «-tuples which are invertible in the Calkin algebra). This index extends the classical one for Fredholm operators. We prove that an almost commuting «-tuple of essentially normal operators with all commu- tators in trace class has index zero (« > 2) and that a natural generalization of Atkinson's theorem holds for «-tuples.

Prime knots and tangles

Abstract: A study is made of a method of proving that a classical knot or link is prime. The method consists of identifying together the boundaries of two prime tangles. Examples and ways of constructing prime tangles are explored. Introduction. This paper explores the idea of the prime tangle that was briefly introduced in [KL]. It is here shown that summing together two prime tangles always produces a prime knot or link. Here a tangle is just two arcs spanning a 3-ball, and such a tangle is prime if it contains no knotted ball pair and its arcs cannot be separated by a disc. A few prototype examples of prime tangles are given (Figure 2), together with ways of using them to create infinitely many more (Theorem 3). Then, usage of the prime tangle idea becomes a powerful machine, nicely complementing other methods, in the production of prime knots and links. Finally it is shown (Theorem 5) that the idea of the prime tangle has a very natural interpretation in terms of double branched covers. The paper should be interpreted as being in either the P.L. or smooth category. With the exception of the section on double branched covers, all the methods used are the straightforward (innermost disc) techniques of the elementary theory of 3-manifolds. None of the proofs is difficult; the paper aims for significance rather than sophistication. Indeed various possible generalizations of the prime tangle have not been developed (e.g. n arcs, for n > 2, or arcs in a ball-with-holes meeting each boundary component in four points) in order to avoid unnecessary complication. The author wishes to record his gratitude to the University of California at Santa Barbara for providing opportunity, facilities and inspiration for the writing of this

Vortex rings: existence and asymptotic estimates

Abstract: The existence of a family of steady vortex rings is established by a variational principle. Further, the asymptotic behavior of the solutions is obtained for limiting values of an appropriate parameter X; as A —» oo the vortex ring tends to a torus whose cross-section is an infinitesimal disc. 0. Introduction. The study of steady vortex rings in an ideal fluid has been the subject of many investigations (see, for example, [3], [19] and the references given there). The classical examples are Helmholtz's rings of small cross-section [17] and Hill's spherical vortex [18]. A general existence theorem for vortex rings was first established by Fraenkel and Berger [13] (see also the very recent work [5], [20] with a similar approach); this paper also contains an excellent survey of the subject. The approach in [13] is based on a variational principle for the stream function. More recently Benjamin [4] developed a new approach based on a variational principle for the vorticity. This approach is more natural since (i) the vorticity has compact support (whereas the stream function does not) and (ii) the quantities involved in the variational principle have direct physical significance. In this paper we establish the existence of vortex rings by a new method. As in [4] we formulate the problem in a variational form for the kinetic energy as a functional of the vorticity. We take the admissible functions to vary in the set S^ of functions f(x) satisfying: f (x) = f (r, z) = f (r, z) where x = (r, 0, z), (0.1) i , , j r2$(x) dx = I, j$(x)dx0] (c > 0), (0.3) fit) = c(t + )p (c>0,B>0). The method of solving our variational problem is in some sense an adaptation of the method of Auchmuty [1] and Auchmuty and Beals [2] (see also [14]-[16]) who Received by the editors May 22, 1980. AMS (MOS) subject classifications (1970). Primary 35J20, 76G05; Secondary 31A15, 35J05. 1 The first author is partially supported by National Science Foundation Grant MCS-781 7204. © 1981 American Mathematical Society 0002-9947/81/0000-0500/$10.25 1 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 2 AVNER FRIEDMAN AND BRUCE TURKINGTON dealt with a variational problem for self-gravitating rotating fluids. There are, however, several differences, the most serious one being the nature of the constraints in (0.1). Another object of this paper is to derive asymptotic estimates on the solution, which we shall denote by f = £A, as X —> oo. Denoting the support of fA(r, z) by Bx, we prove (0.4) E(Sx) = (1/8V2 7r2)log X + 0(1) (E($x) = kinetic energy), (0.5) c/VX < diameter^) < C/VX (0 < c < C < oo) as X -» oo, and (0.6) Bx is asymptotically a disc about (V2 , 0) with radius 1/ (ttVIX ). In §1 we give the physical background of the problem. In §2 we state the existence theorems in variational form for the vorticity, in cases (0.2), (0.3). We also give an account of the relevant existence theorems in the literature. The existence of a vortex ring for the vorticity function (0.3) is obtained in §4. It is preceded by various estimates and some crucial energy identities which are derived in §3. In §5 we establish the existence of a vortex ring with the vorticity function (0.2) by considering it as a limit case of (0.3) with B —> 0; we were unable to treat the case (0.2) more directly because of the nature of the constraints in (0.1). In §§6-8 asymptotic estimates are derived for A—»oo; we specialize here, for simplicity, to the case (0.2). In §6 crude estimates are obtained on both E(^x) and on the support of £A. The precise estimates (0.4), (0.5) are established in §7, using a capacity method recently developed by Caffarelli and Friedman [10]. Finally, (0.6) is proved in §8. Capacity methods have been recently introduced also by Berger and Fraenkel [6], [7]. Results of the form (0.4)-(0.6) have been proved by Fraenkel [11], [12] for vortex ring solutions defined by solving an integral equation with parameter X. It is not known whether our solution, which is obtained by a variational principle, coincides with the solution of Fraenkel. In any case, the methods of Fraenkel and ours are entirely different. 1. Physical background. In this section we describe the equations governing the motion of a steady vortex ring in an ideal fluid and express the physical quantities involved in the form needed for the variational principle of the subsequent section. Throughout the sequel we shall use the following notations: x = (r, 9, z) denotes the cylindrical coordinates of x G R3; {L, ie, iz) represents the associated standard orthonormal frame; dx = r dr dO dz denotes the volume element. Also, we shall write dx for the measure 2-rrr dr dz on the half-plane H = {(r, z); 0 < r < oo, -oo < z < oo}. The vortex ring is assumed to be steady, symmetric about the z-axis, and propagating with constant speed W in the positive z-direction. With respect to axes fixed in the ring, the velocity field \\(x), x e R3, has the form (1.1) y(x) = vr(r, z)L + v\\r, z)\\z, License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

Multivariate rearrangements and Banach function spaces with mixed norms

Abstract: Multivariate nonincreasing rearrangement and averaging functions are defined for functions defined over product spaces. An investigation is made of Banach function spaces with mixed norms and using multivariate rearrangements. Particular emphasis is given to the L(P, Q; *) spaces. These are Banach function spaces which are in terms of mixed norms, multivariate rearrangements and the Lorentz L(p, g) spaces. Embedding theorems are given for the various function spaces. Several well-known theorems are extended to the L(P, Q; *) spaces. Principal among these are the Strong Type (Riesz-Thorin) Interpolation Theorem and the Convolution (Young's inequality) Theorem.

Configurations of surfaces in 4-manifolds

Abstract: We consider collections of surfaces { F,} smoothly embedded, except for a finite number of isolated singularities, self-intersections, and mutual intersections, in a 4-manifold M. A small 3-sphere about each exceptional point will intersect these surfaces in a link. If [Fi] E H2(M) are linearly dependent modulo a prime power, we find lower bounds for I genus (Fi) in terms of the [F;], and invariants of the links that describe the exceptional points. 0. Introduction. The following special case of our main theorem is easy to state. THEOREM 0.1. Let M be a closed smooth 4-manifold and {Fi) a collection of n smoothly embedded surfaces in general position. Let xi = [Fi] E H2(M). Suppose U Fi is connected and E aixi = p'r where p is a prime, 0 2y( xiy) x x sign M 1, b > 0 and (a, b) = 1. Let # be the total number of intersection points of F1 and F2. Then we have

The Topology on the Primitive Ideal Space of Transformation Group C # - Algebras and C.C.R. Transformation Group C # -Algebras

Abstract: If (G, 8) is a second countable transformation group and the stability groups are amenable then C*(G, 8) is C.C.R. if and only if the orbits are closed and the stability groups are C.C.R. In addition, partial results relating closed orbits to C.C.R. algebras are obtained in the nonseparable case. In several cases, the topology of the primitive ideal space is calculated explicitly. In particular, if the stability groups are all contained in a fixed abelian subgroup H, then the topology is computed in terms of H and the orbit structure, provided C*(G, 8) and C*(H, 8) are EH-regular. These conditions are automatically met if G is abelian and (G, 8) is second countable.

A class of extremal functions for the Fourier transform

Abstract: We determine a class of real valued, integrable functions f(x) and corresponding functions M$x) such that f(x) 1, and the value of MfO) is miimi. Several applications of these functions to number theory and analysis are given.

An effective version of Dilworth’s theorem

Abstract: 0. Introduction. Loosely speaking, a subset A of the natural numbers is recursive iff there exists an algorithm (i.e., a finite computer program) which upon input of a natural number n outputs " 1" if n E A and "0" otherwise. Similarly a partial ordering (A, A) is recursive iff there is an algorithm which upon input of an ordered pair of natural numbers (a, b) outputs "1" if a < b and "0" otherwise. For a more careful definition of recursive relations see [R]. One of the attractions of linite combinatorics over infinite combinatorics is its explicit constructions. One never has to consider whether a finite object "really" exists. This paper is part of a program to enlarge the domain of finite combinatorics to certain infinite structures while preserving the explicit constructions of the smaller domain. We shall consider a recursive combinatorics whose domain is the recursive structures. Since finite structures are trivially recursive this domain does indeed extend that of finite combinatorics. Moreover each structure in this domain is explicitly exhibited by some finite computer program. Questions from the graph theory of this combinatorics have been studied by Bean [B], [Bi], Kierstead [K], and Schmerl [S], [Si]. Generally their results relate the chromatic number of a recursive graph with certain properties to its recursive chromatic number. The following example illustrates these ideas. Dilworth's theorem [D] asserts that any partial ordering of finite width n can be covered by n chains. If the partial ordering is finite, then one can actually exhibit these chains (by trial and error, if by no other method). The following easy argument demonstrates how Dilworth's theorem for countably infinite partial orderings follows from Dilworth's theorem for finite partial orderings. Let P = {pi: i E N). We show by induction that for all i E N there exist chains CO,..., C,:'-1 such that: (i) ifj < i and k < n then Ck C k (ii)pi E CO+l U ... U Cn_1; (iii) if Q is a finite subset of P then Co, . . ., C,'1 can be extended to chains that cover Q.

Arithmetic of elliptic curves upon quadratic extension

Abstract: This paper is a study of variations in the rank of the Mordell-Weil group of an elliptic curve E defined over a number field F as one passes to quadratic extensions K of F. Let S(K) be the Selmer group for multiplication by 2 on E(K). In analogy with genus theory, we describe S(K) in terms of various objects defined over F and the local norm indices <" = dimF2£(Ft))/Norm{£(AH,)} for each completion Fv of F. In particular we show that dim S(K) + dim E(K)2 has the same parity as Zi". We compute i" when E has good or multiplicative reduction modulo v. Assuming that the 2-primary component of the Tate-Shafare- vitch group U1(K) is finite, as conjectured, we obtain the parity of rank E(K). For semistable elliptic curves defined over Q and parametrized by modular functions our parity results agree with those predicted analytically by the conjectures of Birch and Swinnerton-Dyer. 1. Introduction. Let E be an elliptic curve defined over a number field F. Our motivating question is this: What can be said about variations in the rank of the Mordell-Weil group E{K) over quadratic extensions K = F(dl/2)1 Let E(d) denote the twist of E which becomes isomorphic to E over K but not over F. Concretely, if we choose for E a model over F of the form y2 = f(x) then a model for £,(d) is given by dy2 = f(x). If a denotes the generator of Gal(K/F), then E(F) can be identified with the ( + l)-eigenspace and E(d\F) with the (-l)-eigenspace of a acting on E(K). It follows that rank E(K) = rank E(F) + rank E(d\F). An equiv- alent question therefore is to describe changes in the rank of E(d\F) as d varies. For certain specific curves defined over Q this question has been discussed for

Whitney stratified chains and cochains

Abstract: This paper contains the technical constructions necessary for a "geometric cycle" definition of cohomology and homology in the context of Whitney stratifications. Cup and cap products are interpreted as the transverse intersection of geometric cocycles and cycles.

On the convergence of closed-valued measurable multifunctions

Abstract: In this paper we study the convergence almost everywhere and in measure of sequences of closed-valued multifunctions. We first give a number of criteria for the convergence of sequences of closed subsets. These results are used to obtain various characterizations for the convergence of measurable multifunctions. In particular we are interested in the convergence properties of (measurable) selections.

Uniqueness of invariant means for measure-preserving transformations

Abstract: For some compact abelian groups X (e.g. T', n > 2, and HII.1 Z2), the group G of topological automorphisms of X has the Haar integral as the unique G-invariant mean on L.,(X, Ax). This gives a new characterization of Lebesgue measure on the bounded Lebesgue measurable subsets /8 of Rn, n > 3; it is the unique normalized positive finitely-additive measure on ,8 which is invariant under isometries and the transformation of Rn: (xI, ... ., Xn) (xI + x2, x2, ... I x"). Other examples of, as well as necessary and sufficient conditions for, the uniqueness of a mean on L,Q(X, /3, p), which is invariant by some group of measure-preserving transformations of the probability space (X, /3, p), are described. 0. Introduction. Let ,8 be the ring of bounded Lebesgue measurable sets in Rn or in Sn, the n-dimensional unit sphere in Rn+ , and let Xn be the Lebesgue measure on /8 normalized by Xn(J") = 1, where J = [0, 1], or by Xn(Sn) = 1 respectively. The classical characterization by Lebesgue of ? is that it is the unique positive real-valued functionf on ,8 which satisfies these three conditions. (a)f(Jn) = 1 orf(Sn) = 1 respectively, (b) f is invariant under isometries, (c) f is countably-additive. In 1923, Banach [1] studied the question of Ruziewicz as to whetherf is still unique when (c) is replaced by (c0) f is finitely-additive. Banach gave a negative answer to this question for R 1, R2 and S1; but for R , n > 3, or sn, n > 2, this question is still unanswered. In this paper we will prove a theorem which comes close to a solution of this problem for Rn, n > 3. THEOREM 3.10. If f satisfies (a), (b), and (c0), and if f is invariant under the transformation of R n, n > 3, given by (xI, X2, ... , X,) F(XI + x2, x2,9. . , xn), then f= xn. Our method of proof will give several other theorems similar to this one. In Theorem 3.10, we do not need the full strength of (b); we need only know that f is absolutely continuous with respect to X, i.e. for N E /3, f(N) = 0 whenever &(N) = 0. The fact that for Rn, n > 3, or sn, n > 2, (a), (b), and (c0) imply this absolute continuity of f is an observation of Tarski which is proved like this. Received by the editors March 3, 1980; presented to the Society, January 10, 1981. AMS (MOS) subject classifications (1970). Primary 28A70, 43A07, 43A40. ? 1981 American Mathematical Society 0002-9947/81 /0000-0268/$04.50

Degenerations of $K3$ surfaces of degree $4$

Abstract: A generic K3 surface of degree 4 may be embedded as a nonsingular quartic surface in P3. Let /: X -» Spec C[[t]] be a family of quartic surfaces such that the generic fiber is regular. Let ¿\\), 2%, 24 be respectively a nonsingular quadric in P3, a cone in P3 over a nonsingular conic and a rational, ruled surface in P9 which has a section with self intersection —4. We show that there exists a flat, projective morphism /': X' —► Spec C[[i]] and a map p: Spec C[[<]] -» Spec Q[r]] such that (i) the generic fiber of /' and the generic fiber of the pull-back of / via p are isomorphic, (ii) the fiber X¿ of /' over the closed point of Spec C[[/]] has only insignificant limit singularities and (iii) Xg is either a quadric surface or a double cover of 2^,, 2° or 24. The theorem is proved using the geometric invariant theory. The purpose of this paper is to prove projective analog of the Kulikov-PerssonPinkham theorem [7], [11] via the geometric invariant theory in a special case. We recall that a nonsingular, projective surface, V, over C is called a Tí 3 surface if Hl(V, ov) = 0 and the canonical divisor class of the surface is trivial. It is called a AT3 surface of degree n if V carries a line bundle L with L • L = n. V is said to be generic if the rank of its Néron-Severi group is equal to one. If L is a line bundle on a generic ÄT3 surface V such that L ■ L = 4, then, the linear system \\L\\ has no fixed components and embeds V into P3 as a quartic surface [8]. Conversely, a nonsingular quartic surface is a 7c3 surface of degree 4. Let 5 denote Spec C[[/]]. A family of surfaces over 5 is a flat, projective morphism, f : X -» S such that the generic geometric fiber of f is a nonsingular, connected surface. A family of surfaces, f: A\" —» S is called a modification of the family f : X -» S if there exists a map p: S -» 5 such that the generic fiber of f and the generic fiber of the pull-back of f via p are isomorphic. We emphasize that a modification also is a projective morphism. Let 2n = a nonsingular quadric surface in P,, S\" = a cone over a nonsingular conic in P3, and S4 = a rational, ruled surface in P9 which has a section whose selfintersection is equal to -4. We prove Theorem 1. Let f: .Y—> S be a family of surfaces such that the generic geometric fiber of f is isomorphic to a quartic surface. Then, there exists a (projective) Received by the editors November 12, 1979. 1980 Mathematics Subject Classification. Primary 14J10, 14J25; Secondary 14C30.

Residually small varieties with modular congruence lattices

Abstract: We focus on varieties T of universal algebras whose congruence lattices are all modular. No further conditions are assumed. We prove that if the variety \"{ is residually small, then the following law holds identically for congruences over algebras in tV: ß[S, S] < [ß,S]. (The symbols in this formula refer to lattice operations and the commutator operation defined over any modular variety, by Hagemann and Herrmann.) We prove that a finitely generated modular variety \"V is residually small if and only if it satisfies this commutator identity, and in that case \"i is actually residually < n for some finite integer n. It is further proved that in a modular variety generated by a finite algebra A the chief factors of any finite algebra are bounded in cardinality by the size of A, and every simple algebra in the variety has a cardinality at most that of A. By a variety we mean a class of similar algebras closed under the formation of subalgebras, homomorphic images and direct products. A variety is locally finite if every finitely generated algebra in it is finite. A variety V(A) = HSP(A) generated by a finite algebra A is locally finite. If A is an infinite locally finite subdirectly irreducible algebra, then HS (A ) includes finite subdirectly irreducible algebras with no finite bound on their cardinality. (A theorem of Quackenbush [17].) Thus for a locally finite variety T there are exactly two possibilities: either there is an integer n such that every finite subdirectly irreducible algebra in T has size at most n, and then every subdirectly irreducible algebra in T has size at most n, or else T has finite subdirectly irreducible algebras with no finite bound on their size. In the latter case there are still many possibilities for the spectrum of subdirectly irreducible algebras in T. There may or may not be an infinite subdirectly irreducible algebra. A variety is called residually < k (where k is a cardinal, possibly finite) if every subdirectly irreducible algebra in it has cardinality < k. The only known example of a locally finite, residually < co (i.e. residually finite) variety that is not residually < n for any n < co is due to Baldwin and Berman [2]-their variety has infinitely many basic operations and is not generated by any one finite algebra. Received by the editors January 10, 1980. Presented to the Society on March 28 and 29, 1980 under the titles Varieties with modular congruences and Narrow varieties, and a characterization of finitely generated, decidable, modular varieties. 1980 Mathematics Subject Classification. Primary 08A15, 06A30; Secondary 20D99.

Poincaré-Bendixson theory for leaves of codimension one

Abstract: The level of a local minimal set of a C2 codimension-one foliation of a compact manifold is a nonnegative integer defined inductively, level zero corresponding to the minimal sets in the usual sense. Each leaf of a local minimal set at level k is at level k. The authors develop a theory of local minimal sets, level, and how leaves at level k asymptotically approach leaves at lower level. This last generalizes the classical Poincare-Bendixson theorem and provides information relating growth, topological type, and level, e.g. if L is a totally proper leaf at level k then L has exactly polynomial growth of degree k and topological type k

The C. Neumann problem as a completely integrable system on an adjoint orbit

Abstract: It is shown by purely Lie algebraic methods that the C. Neumann problem-the motion of a material point on a sphere under the influence of a quadratic potential-is a completely integrable system of Euler-Poisson equations on a minimal-dimensional orbit of a semidirect product of Lie algebras. 1. The C. Neumann problem. The motion of a point on the sphere S`1 under the influence of a quadratic potential U(x) = 'Ax * x, x E R', A = diag(al,... , an) is a completely integrable Hamiltonian system. For n = 3 this has been shown by C. Neumann in 1859 [12] and for arbitrary n by K. Uhlenbeck [16], R. Devaney [3], J. Moser [10], [11], M. Adler, and P. van Moerbeke [2]. In this paper we show how this problem fits naturally in the framework of Euler-Poisson equations [41, [5], [14], [17] proving that the C. Neumann problem is a Hamiltonian system on a minimaldimensional adjoint orbit in a semidirect product of Lie algebras. Thus its complete integrability will follow entirely from Lie algebraic considerations. The equations of motion are i= -ax1 + AXx, i = 1,...,n, (1.1) where the Lagrange multiplier A = Ax' x 11x112 is chosen such that x E Sn-I during the motion. Set x = y and get the equivalent system to (1.1) xi = yi, yi = -a1xi + (Ax. x 11y112)xi, llxll = 1, x y = 0. (1.2) The following crucial remark that motivated the present investigation is due to K. Uhlenbeck and can be verified without any difficulties. LEMMA 1.1. Put X = (xixj), P = (yixj xiyj). System (1.2) is equivalent to X=[P,X, P =[X,A], lIxl = 1, x y = O. (1.3) Remark that if one replaces X and A by X Id/n and A (Tr(A))Id/n respectively, where Id is the n X n identity matrix, equations (1.3) remain unchanged. From now on we shall assume that in (1.3) this change has been made so that X, P, A E sl(n). The next section gives a Lie algebraic interpretation to these equations. Received by the editors March 12, 1980. 1980 Mathematics Subject Classification. Primary 58F07, 70H05; Secondary 53C15, 17B99.

Preservation of convergence of convex sets and functions in finite dimensions

Abstract: We study a convergence notion which has particular relevance for convex analysis and lends itself quite naturally to successive approximation schemes in a variety of areas. Motivated particularly by problems in optimization subject to constraints, we develop technical tools necessary for systematic use of this convergence in finite-dimensional settings. Simple conditions are established under which this convergence for sequences of sets, functions and subdifferentials is preserved under various basic operations, including, for example, those of addition and infimal convolution in the case of functions.

Nonseparable approximate equivalence

Abstract: This paper extends Voiculescu's theorem on approximate equivalence to the case of nonseparable representations of nonseparable C*-algebras. The main result states that two representations / and g are approximately equivalent if and only if rank/(.x) = rank g{x) for every x. For representations of separable C*-algebras a multiplicity theory is developed that characterizes approximate equivalence. Thus for a separable C*-algebra, the space of representations modulo approximate equivalence can be identified with a class of cardinal-valued functions on the primitive ideal space of the algebra. Nonseparable extensions of Voiculescu's reflexivity theorem for subalgebras of the Calkin algebra are also obtained.

A partition theorem for the infinite subtrees of a tree

Abstract: We prove a generalization for infinite trees of Silver's partition theorem. This theorem implies a version for trees of the Nash-Williams partition theorem.