Showing papers in "Transactions of the American Mathematical Society in 1987"

Infinite-dimensional linear systems with unbounded control and observation: a functional analytic approach

Abstract: : The object of these notes is to develop a unifying framework for the functional analytic representation of infinite dimensional linear systems with unbounded input and output operators. On the basis of the general approach new results are derived on the wellposedness of feedback systems and on the linear quadratic control problem. The implications of the theory for large classes of functional and partial differential equations are discussed in detail. Keywords: Representation of infinite dimensional systems, semigroups, boundary control, feedback, linear quadratic control.

Area and Hausdorff dimension of Julia sets of entire functions

Abstract: We show the Julia set of A sin(z) has positive area and the action of A sin(z) on its Julia set is not ergodic; the Julia set of A exp(z) has Hausdorff dimension two but in the presence of an attracting periodic cycle its area is zero.

A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions

Abstract: We show that, typically, lower semicontinuous functions on a Banach space densely inherit lower subderivatives of the same degree of smoothness as the norm. In particular every continuous convex function on a space with a Gâteaux (weak Hadamard, Frechet) smooth renorm is densely Gâteaux (weak Hadamard, Frechet) differentiable. Our technique relies on a more powerful analogue of Ekeland's variational principle in which the function is perturbed by a quadratic-like function. This "smooth" variational principle has very broad applicability in problems of nonsmooth analysis.

Asymptotic behavior and traveling wave solutions for parabolic functional-differential equations

Abstract: On presente une generalisation de la theorie des equations KPP et de diffusion bistables non lineaires. On montre que des solutions onde de propagation existent pour des equations fonctionnelles paraboliques non lineaires

Braids and the Jones polynomial

Abstract: An important new invariant of knots and links is the Jones polynomia}, and the subsequent generalized Jones polynomial or two-variable polynomial. We prove inequalities relating the number of strands and the crossing number of a braid with the exponents of the variables in the generalized Jones polynomial which is associated to the link formed from the braid by connecting the bottom ends to the top ends. We also relate an exponent in the polynomial to the number of components of this link. In [J] Vaughan Jones introduced a new polynomial invariant of oriented knots and links in 3-space. Subsequently a number of researchers (Freyd and Yetter, Hoste, Lickorish and Millett, and Ocneanu; see [FYHLMO]) independently realized that thlis could be generalized to produce an invariant which is a Laurent polynomial of two variables and which specializes to give both the invariant of [J] and the classical Alexander polynomial (see [R]). More precisely, there is a function P from isotopy classes of oriented links to Z[x,x-1,y,y-l]. If L iS an oriented link we will write P(L) = j(x,y). By abuse of notation we will also write P(b) for b an oriented braid, meaning, of course, the invariant associated to the oriented link determined by closing up b in the usual way. Several equivalent ways of codifying this invariant are described in [FYHLMO]. We have chosen the approach of Lickorish and Millett. Hence our P(K) = j(X, Y) iS precisely their PK with x and y substituted for I and m respectively (see [FYHLMO] and [L-M]). There appears as yet to be no consensus on a name for this two-variable invariant. In order to avoid awkward circumlocutions and being unable to amalgamate the names of the six or eight individuals who actually discovered it, we have chosen to refer to it as the generalized Jones polynomial, or Jones polynomial for short, when there is no likelihood of confusion with the original Jones polynomial. This polynomial, which we will denote j(x, y), is characterized by Figure 1. The interpretation of this is as follows: Given a regular projection of a link K+ with a crossing pictured as below, one can form two new links K_ and KO which are identical to K+ except the one crossing is changed as shown. When this is done the Jones polynomials of the three links are related as in the formula. It is a remarkable fact that this together with the normalization that P (unknot) = 1, un+quely determines a Laurent polynomial j(x, y) and that this polynomial is a link invariant, i.e., is independent of the projection. This result can be found in [L-M]. This formula, in fact, gives a good method for recursively computing j(x, y). One Received by the editors January 13, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 57M25. Research supported in part by NSF Grant MCS 83

Global existence for 1D, compressible, isentropic Navier-Stokes equations with large initial data

Abstract: We prove the global existence of weak solutions of the Cauchy problem for the Navier-Stokes equations of compressible, isentropic flow of a polytropic gas in one space dimension. The initial velocity and density are assumed to be in L2 and L2 n BV respectively, modulo additive constants. In particular, no smallness assumptions are made about the intial data. In addition, we prove a result concerning the asymptotic decay of discontinuities in the solution when the adiabatic constant exceeds 3/2.

Toeplitz operators on the Segal-Bargmann space

Abstract: In this paper, we give a complete characterization of those functions on 2n-dimensional Euclidean space for which the Berezin-Toeplitz quantizations admit a symbol calculus modulo the compact operators. The functions in question are characterized by a condition of "small oscillation at infinity" .

Hypergeometric functions over finite fields

Abstract: In this paper the analogy between the character sum expansion of a complex-valued function over GF(q) and the power series expansion of an analytic function is exploited in order to develop an analogue for hypergeometric series over finite fields. It is shown that such functions satisfy many summation and transformation formulas analogous to their classical counterparts.

A general theory of canonical forms

Abstract: Si G est un groupe de Lie compact et M une G-variete riemannienne d'orbites principales de codimension k alors une section ou une forme canonique pour M est une sous-variete fermee lisse a k dimensions de M qui rencontre toutes les orbites de M orthogonalement

Scalar curvature functions in a conformal class of metrics and conformal transformations

Abstract: On etudie le probleme de prescrire la courbure scalaire dans une classe conforme. Pour l'action du groupe conforme, on montre l'universalite des conditions d'integrabilite dues a J.L. Kazdan et F.W. Warner

_{+1}-free graphs: asymptotic structure and a 0-1 law

Abstract: Etude asymptotique de la structure des graphes etiquetes sans K l+1 . Etablissement de la l-colorabilite et de la loi (0, 1)

Special functions of matrix argument. I. Algebraic induction, zonal polynomials, and hypergeometric functions

Abstract: Hypergeometric functions of matrix argument arise in a diverse range of applications in harmonic analysis, multivariate statistics, quantum physics, molecular chemistry, and number theory. This paper presents a general theory of such functions for real division algebras. These functions, which generalize the classical hypergeometric functions, are defined by infinite series on the space S = S(n, F) of all n x n Hermitian matrices over the division algebra F. The theory depends intrinsically upon the representation theory of the general linear group G = GL(n, F) of invertible n x n matrices over F, and the theme of this work is the full exploitation of the inherent group theory. The main technique is the use of the method of "algebraic induction" to realize explicitly the appropriate representations of G, to decompose the space of polynomial functions on S, and to describe the "zonal polynomials" from which the hypergeometric functions are constructed. Detailed descriptions of the convergence properties of the series expansions are given, and integral representations are provided. Future papers in this series will develop the fine structure of these functions. 0. Introduction. We begin a series of articles in which we develop the fine structure of generalized hypergeometric functions of matrix argument. By "fine structure", we allude to the analogues of such classical results as series expansions, integral formulas, asymptotics, differential equations, summation formulas, addition theorems, composition formulas, and recurrence relations. This first paper, in which we simultaneously treat real, complex, and quaternionic analysis, is the result of our desire to present a complete theory of hypergeometric functions of matrix argument over real division algebras, not only as a framework for the body of detailed results to follow in later papers, but also to clarify the representationtheoretic foundation for such a theory. Although these hypergeometric functions are of interest on purely analytic grounds, they arise in a wide range of applications. Indeed, various classes of Received by the editors May 19, 1986. 1980 Malhema1zics S*ject Claszficain (1985 Ren). Primary 22E30, 22E45, 33A75, 43A85, 43A90, 62H10; Secondary 20G20, 32A07, 32M15, 44A10, 62E15. Key w and phrases. Generalized hypergeometric functions, zonal polynomials, representation theory, algebraic induction, multivariate statistics, general linear group, generalized gamma functions, Pochhammer symbols, Laplace transforms, maximal compact subgroup, invariant polynomials, positive cones, symmetric spaces, Schur functions, special functions of matrix argument. The first author is on leave from the University of Wyoming. The second author was partially supported by the National Science Foundation under Grant MCS-8403381, and by the Research Council of the University of North Carolina. (B1987 American Mathematical Society 0002-9947/87 $1.00 + $.25 per page

Univalent harmonic functions

Abstract: Several families of complex-valued, univalent, harmonic functions are studied from the point of view of geometric function theory. One class consists of mappings of a simply-connected domain onto an infinite horizontal strip with a normalization at the origin. Extreme points and support points are determined, as well as sharp estimates for Fourier coefficients and distortion theorems. Next, mappings in Izi > 1 are considered that leave infinity fixed. Some coefficient estimates, distortion theorems, and covering properties are obtained. For such mappings with real boundary values, many extremal problems are solved explicitly.

Character table and blocks of finite simple triality groups

Abstract: Based on recent work of Spaltenstein [14] and the Deligne-Lusztig theory of irreducible characters of finite groups of Lie type, in this paper the character table of the finite simple groups iD4(q) is given. As an application we obtain a classification of the irreducible characters of iDi(q) into r-blocks for all primes r > 0. This enables us to verify Brauer's height zero conjecture, his conjecture on the bound of irreducible characters belonging to a give block, and the Alperin-McKay conjecture for the simple triality groups }D4(q). It also follows that for every prime r there are blocks of defect zero in }Di(q). Introduction. Let G„ -3D4(q) be a simple triality group defined over a finite field GF(q) with q p\" elements, where p > 0 is a prime number and « is a positive integer. In [14] N. Spaltenstein computed the values of the eight unipotent irreducible characters of G„. Using his results we determine the character table of G„ in §4. In Theorem 4.3 the nonunipotent irreducible characters of G„ are presented in the form of precise linear combinations of the virtual Deligne-Lusztig characters RT&, where © is a linear character of the a-fixed points of a a-stable maximal torus T of the corresponding algebraic group G. The values of the Deligne-Lusztig characters are given in Table 3.6. By Lusztig's Jordan form of the irreducible characters of a finite group of Lie type [11] each irreducible character x of G„ is of the form x = Xi?M> where t is a semisimple element of Ga and xu lS a unipotent irreducible character of the centralizer Cc(t) of t. The group theoretical structure of the centralizers Cc(t) of the semisimple elements t of Ga is given in Proposition 2.2, and of the 7 (up to Ga-conjugacy) maximal tori 7), 0 < /' < 6, in Proposition 1.2. It follows that Cc(t) has at most three unipotent irreducible characters, namely the trivial 1, the Steinberg character St or a unipotent character of degree either qs = q(q + 1) or qs' = q(q — 1). If t # 1 is regular, we write x, instead of x,,i, in all other cases xr,i> X/,st> X>1 Xi.qsor X/.ststA complete classification of the irreducible characters of Ga with their degrees is given in Table 4.4. On the set of conjugacy classes of semisimple elements t of Ga one can define an equivalence relation as follows. Two such conjugacy classes ip» and ip» are equivalent if and only if their centralizers Cc(i,) and Cc(t2) are G0-conjugate. If q is odd, there are 15 equivalence classes with representatives s¡, 1 < / < 15, where s^ = 1 Received by the editors February 13, 1985 and, in revised form, June 16, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 20C20. 39 ©1987 American Mathematical Society 0002-9947/87 $1.00 + $.25 per page License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 40 D. I, DERIZIOTIS AND G. O. MICHLER and s2 # 1 is the unique conjugacy class of involutions of G„. If q is even, the equivalence class of s2 does not exist, and we have only 14 equivalence classes. Using the first author's work on the Brauer complex [5] of Gg and the computer, we obtain in Table 4.4 the numbers of semisimple conjugacy classes of Ga belonging to a given equivalence class [s¡], 1 < ; < 15. Applying then Proposition 2.2 and Spaltenstein's characterization of the unipotent conjugacy classes of Ga [14], we can give in Proposition 2.3 a complete classification of all conjugacy classes of G0. In particular, we show that the number k(Ga) of all conjugacy classes of Ga is k(G„) = q4 + q3 + q2 + q + 5, if2|<7, and k(Ga) = q4 + q3 + q2 + q + 6, if 2 \\ q. By means of these results we determine in §5 the distribution of the irreducible characters of Ga into r-blocks, where r is a prime number dividing the group order \\Ga\\. If r = p, then by Humphrey's theorem [10] G„ has only the principal p-block B0 and a block B of defect zero consisting of the irreducible Steinberg character. For r ¥= p Theorem 5.9 asserts that each /--block B with defect group D determines, up to G0-conjugacy, a unique semisimple /-'-element s of G„ such that an irreducible character x,,„ of G„ belongs to B if and only if i is G„-conjugate to sy for some V G D, and xu is an irreducible unipotent character of CG(sy) such that syxu belongs to an /--block B of Cc(sy) with defect group D satisfying B = BG. This result can be considered to be an analogue of the Fong-Srinivasan characterization [8] of the /--blocks of the general linear and unitary groups. In Corollary 5.11 we show that for all primes r > 0 and all /--blocks B of G„ with defect group 8(B) = GD the number of all irreducible characters of G„ belonging to B is bounded by k(B) < \\D\\. This verifies a well-known conjecture of R. Brauer, see [7], in the case of the simple triality groups. He also conjectured that an /--block B of a finite group G has only irreducible characters of height zero if and only if its defect group 8(B) = CD is abelian. In case G = G0 this is shown for all primes r in Corollary 5.10. Let k0(B) be the number of irreducible characters of an /--block B of G with height zero. If 8(B) = CD denotes the defect group of D, H = NC(D), and 5j is the Brauer correspondent of B in H, then the Alperin-McKay conjecture asserts that k0(B) = k0(B¡). In the case of G = G0, we verify it for all primes r; see Corollary 5.12. Another application of Table 4.4 yields that in G0 there are /--blocks B of defect zero for every prime r > 0; see Corollary 5.1. Concerning the notation and terminology we refer to the books by Carter [2], Deriziotis [4], Feit [7], and Lusztig [11]. 1. Notations and known results on 3D4(q). Let G be a simple simply connected algebraic group of Dynkin diagram type D4 over the algebraic closure K of the prime field GF(/>) = Fp, p > 0. Let q = pm for some positive integer m, and let GF(<7) = F^ be the field with q elements. F* denotes the multiplicative group of every field F. Let T be a maximal torus of G, 0 the set of roots of G relative to T, X = Hom(7\\ K*)—the group of rational characters of T, Y = Uom(K*,T)—the License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use CHARACTER TABLE AND BLOCKS OF 3D4( hx and r(h2) = h2. Then t induces an isometry on V which again is denoted by t. The triality automorphism a = rq of G is induced by t times the field automorphism z -» z' of /C The simple group 3D4(q) = Ga = {g G G|a(g) = g} is called the Steinberg-Tits triality. Its order \\Ga\\ = 912(98 + q4 + l)(qb l)(q2 1). The torus T is a-stable. The restriction of a = qj onto T induces a linear transformation of V, again denoted by a. Let h: Hom(X, K*) -> T be defined as follows. For every x G Hom(X,K*), h(x) = t G T, where x(^) = A(i) for all X g X Then h is an isomorphism. Let A,, A2, A3, and A4 be the fundamental weights in X. Each element h(x) g 7 can uniquely be written as A(x)-n*(x*,.„), 1-1 where Xa,,z(a) = zMh,) for /g $, z g A'*, and where x(^,) = z, for 1 < i < 4. Let W be the Weyl group generated by all reflections wr at the hyperplanes of V orthogonal to the coroots hr, r g $. Then a acts on W by o(u') = awo'1 = twt\"1. In particular, o(wr) = wT(r). IF acts also on 7 by wh(x), where ( wx )( A ) = x(w_1(A)) for all A g X. Furthermore, wi±j denotes the reflection at the hyperplane of V orthogonal to the coroot £, + t}. Let r0 be the highest root of 0, and  = AU{-r0). Let J be an arbitrary r-invariant proper subset of Ä, and W the Weyl group of the torus T. The normalizer of J in W is denoted by Í2,. It is a a-stable subgroup of W. Two elements wx, w2 G S2y are called a-équivalent if Wj = ww2o(w~x) for some w g ay. The a-equivalence class of w g ay is denoted by [w], and //'(a^y) is the set of all a-equivalence classes [w] of Í2,. The possibilities of J and ß, are given in Table 1.0, up to IF-conjugacy. Table 1.0 J •'0 — {rl> r2' r3< r4l J\\ = {'V3,/-4,-/-( •/2= i'\"l^3^4} ¿S = {»2.-•\"<>} •/4= {-/■()} y5 = 0 a, a7 = i Œ/, = <^i+4^2 + 3) X (wi-awi+a) = (Z2) ®J2= = Z2 ßy, = Oi -3^2 + 4^2-4) = z2 Ûy4 = <^l-2> X X (w3 + 4> = (Z2)3 a, = if License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 42 D. I. deriziotis and g. o. michler Let ^j be the collection of all a-stable G-conjugates of Cc(x) where x is a semisimple element of G with r(x) = 1 for all re/, Then the group Ga acts on #y by conjugation. If 7 = 0 is the empty set, then Q0 = W, and x is a regular element of Ga. There is a one-to-one correspondence between the G0-orbits of a-stable maximal tori of G and the classes of Hl(a, W), see [1, p. 186]. It is known for the triality Ga =3D4(q) that \\Hl(a, W)\\ = 7; cf. [14]. Let T be a a-stable maximal torus of G, with Weyl group W = NC(T)/T. If 7\" is a a-stable maximal torus of G, then there is a unique class [w.] g rY^a, IF) with je {0,1,..., 6} such that T'„ is G-conjugate to 77. = T 0 = {i g r | H>ya(í) = i}. In particular, the element h(x) — nf=1 h(x>,, z,) g T belongs to T¡ if and only if 4 A(x) = Wjoh(x) = El a(Xwt(a,),z,í)/ = i For the sake of simplicity, each element h(x) Y\\4=xh(xh ,z) & T is denoted by A(x) = (zlt z2> z3> z4)With this notation we can parametrize all the elemen

Counting cycles in permutations by group characters, with an application to a topological problem

Abstract: The character theory of the symmetric group is used to derive properties of the number of permutations, with k cycles, which are expressible as the product of a full cycle with an element of an arbitrary, but fixed, conjugacy class. For the conjugacy class of fixed point free involutions, this problem has application to the analysis of singularities in surfaces. 1. Introduction. For nonnegative integers k and N, and a partition \p of N, let ef denote the number of permutations w on N symbols such that it has exactly k cycles, and such that it can be expressed as a product of an arbitrary, but fixed, cycle of length N and a permutation in the conjugacy class indexed by \p. The purpose of this paper is to derive the generating function for these numbers, and to obtain some of their properties. The method makes direct use of combinatorial and algebraic properties of the group algebra of the symmetric group. A special case of this problem is of particular interest. Let e(p)(n) denote the number e\ when \p indexes the conjugacy class of permutations on pn symbols, with n cycles of length p. The matter of calculating ek2 ) arose in connection with work by Harris and Morrison (4) on singularities in surfaces. It has also occurred indirectly in the work of Gross (2) on graph embeddings. Harer and Zagier (3) have shown, by an independent method, that the sequence e(k2)(n) for k, n > 1 satisfies a three-term linear recurrence equation with coefficients which are polynomials in n. To fix ideas, note that for n = 2 the permissible permutations are

Representations of crossed products by coactions and principal bundles

Abstract: The main purpose of this paper is to establish a covariant representation theory for coactions of locally compact groups on C*-algebras (including a notion of exterior equivalence), to show how these results extend the usual notions for actions of groups on C*-algebras, and to apply these ideas to classes of coactions termed pointwise unitary and locally unitary to obtain a complete realization of the isomorphism theory of locally trivial principal G-bundles in this context. We are also able to obtain all (Cartan) principal G-bundles in this context, but their isomorphism theory remains elusive.

The Structure of σ-Ideals of Compact Sets

Abstract: Motivated by problems in certain areas of analysis, like measure theory and harmonic analysis, where σ-ideals of compact sets are encountered very often as notions of small or exceptional sets, we undertake in this paper a descriptive set theoretic study of σ-ideals of compact sets in compact metrizable spaces. In the first part we study the complexity of such ideals, showing that the structural condition of being a σ-ideal imposes severe definability restrictions. A typical instance is the dichotomy theorem, which states that σ-ideals which are analytic or coanalytic must be actually either complete coanalytic or else G_δ. In the second part we discuss (generators or as we call them here) bases for σ-ideals and in particular the problem of existence of Borel bases for coanalytic non-Borel σ-ideals. We derive here a criterion for the nonexistence of such bases which has several applications. Finally in the third part we develop the connections of the definability properties of σ-ideals with other structural properties, like the countable chain condition, etc.

$p$-ranks and automorphism groups of algebraic curves

Abstract: Let X be an irreducible complete nonsingular curve of genus g over an algebraically closed field k of positive characteristic p. If 9 > 2, the automorphism group Aut(X) of X is known to be a finite group, and moreover its order is bounded from above by a polynomial in g of degree four (Stichtenoth). In this paper we consider the Frank zy of X and investigate relations between zy and Aut(X). We show that zy affects the order of a Sylow psubgroup of Aut(X) (§3) and that an inequality IAut(X) | < 84(9-1)9 holds for an ordinary (i.e. zy = g) curve X (§4).

An operator-theoretic formulation of asynchronous exponential growth

Abstract: A strongly continuous semigroup of bounded linear operators T(t), t > O, in the Banach space X has asynchronous exponential growth with intrinsic growth constant A0 provided that there is a nonzero finite rank operator P0 in X such that limr OOe-X°tT(t)= Po. Necessary and sufficient cvonditions are established for T(t), t > O, to have asynchronous exponential growth. Applications are made to a maturity-time model of cell population growth and a transition probability model of cell population growth.

Invariant subspaces in Banach spaces of analytic functions

Abstract: We study the invariant subspace structure of the operator of multiplication by z, Mz, on a class of Banach spaces of analytic functions. For operators on Hilbert spaces our class coincides with the adjoints of the operators in the CowenDouglas class 1(Q). We say that an invariant subspace X satisfies cod X = 1 if zX has codimension one in X. We give various conditions on invariant subspaces which imply that codX= 1. In particular, we give a necessary and sufficient condition on two invariant subspaces X, X with cod X = cod X= 1 so that their span again satisfies cod(X v X) = 1. This result will be used to show that any invariant subspace of the Bergman space L#, p > 1, which is generated by functions in L2P, must satisfy cod X = 1. For an invariant subspace X we then consider the operator S = M* 1Xl . Under some extra assumption on the domain of holomorphy we show that the spectrum of S coincides with the approximate point spectrum iff cod X = 1. Finally, in the last section we obtain a structure theorem for invariant subspaces with cod X = 1. This theorem applies to Dirichlet-type spaces.

The l2-boundedness of pseudodifferential operators

Abstract: On donne une nouvelle demonstration du theoreme de Calderon-Vaillancourt. On obtient la L 2 -continuite de a(x,D) si le symbole a(x,ξ) satisfait a certaines conditions

Stationary configurations of point vortices

Abstract: The motion of point vortices in a plane of fluid is an old problem of fluid mechanics, which was given a Hamiltonian formulation by Kirchhoff. Stationary configurations are those which remain self-similar throughout the motion. Results of two types are presented. Configurations which are in equilibrium or which translate uniformly are counted using methods of algebraic geometry, which establish necessary and sufficient conditions for existence. Relative equilibria (rigidly rotating configurations) which lie on a line are studied using a topological construction applicable to other power-law systems. Upper and lower bounds for such configurations are found for vortices with mixed circulations. Arrangements of three vortices which collide in finite time are well known. One-dimensional families of such configurations are shown to exist for more than three vortices. Stationary configurations of four vortices are examined in detail. The motion of point vortices in a plane of fluid is an old problem of fluid mechanics, which was given a Hamiltonian formulation by Kirchhoff. Stationary configurations are those which remain self-similar throughout the motion, and are of considerable physical interest. It is known that a configuration of vortices in equilibrium must have total vortex angular momentum 0. A converse is proved, namely that for almost every choice of circulations with zero vortex angular momentum, there are exactly (n 2)! equilibrium configurations. A similar statement is proved for rigidly translating configurations with total circulation zero. The proofs involve ideas from algebraic geometry. Relative equilibria (rigidly rotating configurations) were studied by Palmore in the case of positive circulations. Upper and lower bounds for the number of collinear relative equilibria for arbitrary circulations are obtained by means of a topological construction which is applicable to other power-law systems. Arrangements of three vortices which collide in finite time are well known. One-dimensional families of such configurations are shown to exist for n vortices, n > 3. Stationary configurations of four vortices are examined in detail. CHAPTER 0. INTRODUCTION The motion of point vortices in the plane is an old problem in fluid mechanics. It was first given a Hamiltonian formulation by Kirchhoff (1876), who proved that n vortices with distinct positions (xi, Yi) (i = 1,. . ., n) in the (x, y) plane, and Received by the editors July 31, 1985 and, in revised form, June 13, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 76C05; Secondary 70H05. t1987 American Mathematical Society 0002-9947/87 $1.00 + $.25 per page

-theory and multipliers of stable *-algebras

Abstract: The main theorem is that if A is a C*-algebra with a countable approximate identity consisting of projections, then the unitary group of M(A ? K) is contractible. This gives a realization, via the index map, of Ko (A) as components in the set of Fredholm operators on HA. For a C*-algebra A let HA = {(ai) I ai E A and Eai*ai converges in norm). Then HA becomes a right A-module under the action (ai)b = (aib). With the A-valued inner product ((ai), (b1)) = Ea*bi, HA is a Hilbert A-module. Y(HA) is the C *-algebra of operators on HA that have an adjoint; >k(HA) is an ideal of Y(HA), called the compact operators on HA. Set [11, ?2, Definition 4]. Operators in ??(HA) invertible modulo 3f(HA) are called Fredholm and have an index that takes its values in KO(A). The paper is divided into two parts. In Part 1 we show that index: [A] -* KO(A) is an isomorphism, where [t,] denotes the set of path components of the set of Fredholm operators. In Part 2 we show that the unitary group of M(A ? K) is contractible when A has a countable approximate identity consisting of projections. As 2?(HA)_ M(A ? K), this in particular implies that the unitary group of S?(HA) is connected when A is unital. This latter result is used to prove the isomorphism mentioned above. The results of this paper were announced in [14] (see Remark 2.5(2)). This research is based on the author's doctoral work done at Dalhousie University under the supervision of Professor P. A. Fillmore. The author also wishes to thank W. J. Phillips for many useful suggestions. PART 1. THE INDEX OF A FREDHOLM OPERATOR 1.1. We describe the index map from Fredholm operators on Hilbert C*-modules to the corresponding K-groups. The principal ingredients of this construction have been discussed by several authors: Kasparov [12], Miscenko and Fomenko [15], and Pimsner, Popa, and Voiculescu [17]. However, as we shall need refinements and variations of these constructions, we will give a detailed account of the index map. Throughout this paper K or )Y(H) will denote the C*-algebra of compact operators on a separable infinite dimensional Hilbert space H. For any two C*-algebras A and B, A ? B will denote the completion of the algebraic tensor product AOB in the spatial or minimal C *-norm (see e.g. Effros and Lance [8, ?2]). Received by the editors July 28, 1983 and, in revised form December 27, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 46L05, 46M20. ?01987 American Mathematical Society 0002-9947/87 $1.00 + $.25 per page

Riccati techniques and variational principles in oscillation theory for linear systems

Abstract: On etablit des conditions suffisantes pour que toutes les solutions soient oscillatoires. On utilise les techniques de Riccati et les principes variationnels

Periodic points and automorphisms of the shift

Abstract: On etudie le groupe d'automorphisme d'un decalage de Markov topologique a l'aide des points periodiques et des ensembles instables

On the central limit theorem for dynamical systems

Abstract: Etant donne un systeme dynamique aperiodique (X,T,μ), il y a un f∈L 2 (μ) avec ∫fdμ=0 satisfaisant le theoreme de la limite centrale

A classification of simple Lie modules having a 1-dimensional weight space

Abstract: Let L denote a simple Lie algebra over the complex numbers. In this paper, we classify and construct all simple L modules which may be infinite dimensional but have at least one 1-dimensional weight space. This completes the study begun earlier by the authors for the case of L = A,,. The approach presented here relies heavily on the results of Suren Fernando whose dissertation dealt with simple weight modules and their weight systems. 0. Introduction. Let L be a finite-dimensional simple Lie algebra over the complex field C having a Cartan subalgebra H and denote by C(L) the centralizer of H in the universal enveloping algebra U of L. If X: H -C is a weight function of a 1-dimensional weight space Mx in a simple L module M then q: C(L) -> C, defined by q(c)v = cv for v E Mx and c E C(L), is an algebra homomorphism called a mass function of M. Clearly X restricted to H is equal to X. Conversely, given any algebra homomorphism q: C(L) -C one can construct a unique simple L module which admits X as a mass function [4, 10]. In [4] the authors determined all algebra homomorphisms q: C(L) -C for the simple Lie algebras L of type An and using these classified all "pointed" An modules (where we call a module pointed if it is simple and has at least one 1-dimensional weight space). In this paper we complete the classification of all pointed L modules for arbitrary simple Lie algebras. The collection of all pointed L modules clearly includes the highest weight L modules and is included in the collection of all Harish-Chandra L modules relative to the Cartan subalgebra H (-i.e. simple L modules having a weight space decomposition and finite-dimensional weight spaces relative to H). The latter inclusion is strict since there exist examples of Harish-Chandra A2 modules in which every weight space is two dimensional. In the special case of A1 modules, every Harish-Chandra A1 module is pointed. Our approach to this problem makes heavy use of the results from Fernando's thesis [8]. In particular, from Fernando's results we see that in place of determining all algebra homomorphisms q: C(L) -C it suffices to find only those algebra homomorphisms X which are associated with so-called pointed "torsion free" L Received by the editors May 23, 1984. This paper was presented at the 1984 Canadian Mathematical Society's Summer Seminar on Lie algebras and related topics held at the University of Windsor. 1980 Mathematics Subject Classification. Primary 17B10.

A remark on the blowing-up of solutions to the Cauchy problem for nonlinear Schrödinger equations

Abstract: on considere des solutions de iu t =Δu+|u| p−1 u, u(o)=u 0 , ou x appartient a un domaine lisse ΩCR N , et on demontre que sous de bonnes conditions sur p, N et u 0 ∈H 2 (Ω)∩H 0 1 (Ω),∥⊇u(t)∥ L 2 eclate en un temps fini

Scalar curvature and warped products of Riemann manifolds

Abstract: We establish the relationship between the scalar curvature of a warped product M Xf N of Riemann manifolds and those ones of M and N. Then we search for weights f to obtain constant scalar curvature on M Xf N when M is compact.