Showing papers in "Bulletin of the American Mathematical Society in 1967"

Differentiable dynamical systems

Abstract: This is a survey article on the area of global analysis defined by differentiable dynamical systems or equivalently the action (differentiable) of a Lie group G on a manifold M. An action is a homomorphism G→Diff(M) such that the induced map G×M→M is differentiable. Here Diff(M) is the group of all diffeomorphisms of M and a diffeo- morphism is a differentiable map with a differentiable inverse. Everything will be discussed here from the C ∞ or C r point of view. All manifolds maps, etc. will be differentiable (C r , 1 ≦ r ≦ ∞) unless stated otherwise.

An inequality with applications to statistical estimation for probabilistic functions of Markov processes and to a model for ecology

Abstract: 1. Summary. The object of this note is to prove the theorem below and sketch two applications, one to statistical estimation for (proba-bilistic) functions of Markov processes [l] and one to Blakley's model for ecology [4]. 2. Result. THEOREM. Let P(x)=P({xij}) be a polynomial with nonnegative coefficients homogeneous of degree d in its variables {##}. Let x= {##} be any point of the domain D: ## §:(), ]pLi ## = 1, i = l, • • • , p, j=l, • • • , q%. For x= {xij} ££> let 3(#) = 3{##} denote the point of D whose i, j coordinate is (dP\\ \\ f « dP 3(*) Then P(3(x))>P(x) unless 3(x)=x. Notation, fi will denote a doubly indexed array of nonnegative integers: fx= {M#}> i = l> • • • >

Bounds for the fundamental solution of a parabolic equation

Abstract: possesses a fundamental solution provided that the coefficients are Holder continuous. Here x— (xi, • • • , xn) denotes a point in E n with n E= 1, t denotes a point on the real line, and we employ the convention of summation over repeated indices. The fundamental solution g(x, t; £, r) can be constructed by the classical parametrix method, and it satisfies the inequality O^g^Ky, where y is the fundamental solution of aAu = Ut for some constant a>0 and K>0 is a constant which depends upon the Holder norms of the coefficients ([4], [5]). Several authors have investigated the problem of bounding g from below. Il'in, Kalashnikov, and Oleinik [5] proved that gg^const (t—r)~ in the paraboloid |#—£| 2 ^const ( /—r) ; while Besala [3] and Friedman [4] have derived lower bounds for g which are valid when t—r is bounded away from zero. In the appendix to his important paper [6] on Holder continuity of solutions of parabolic and elliptic equations, Nash asserts the existence of global upper and lower bounds for the fundamental solution of the divergence structure parabolic equation

On rings of operators

Abstract: Let 5C be a complex Hubert Space, J3(3C) the ring of bounded operators on 3C, E an abelian symmetric subring of B(3Z) containing the identity which is closed in the weak operator topology, E\ the commutant of E, and suppose E\ has a cyclic vector £o which we normalize so that |£o| = 1 . Diximier [ l] has shown that E (respect. Ei), as a Banach space, is the dual of the Banach space R (respect. Ri) of all linear forms on E (respect. E\) that are continuous in the ultra-strong topology of E (respect. Ei). In this note we show that every TCzR is also continuous in the weak operator topology of E, from which it follows that a linear functional T on E is continuous in either the weak, ultraweak, strong, or ultrastrong topologies if and only if it is continuous in all four simultaneously. In the process, we obtain an integral representation for such T, which we later use in a theorem on centrally reducible positive functionals on E%. We denote the maximal ideal space of E by M, and for A, B, • • • £ E , we denote the corresponding Gel'fand transforms by a, 6, • • • . Then A—>a is an isometric isomorphism from £ onto C(M). Consequently, every bounded linear functional on E has the form

Nonlinear mappings of nonexpansive and accretive type in Banach spaces

Abstract: There are two important connections between the classes of nonexpansive and of accretive mappings which give rise to a strong connection between the fixed point theory of nonexpansive mappings and the mapping theory of accretive maps. These are: (1) If U is a nonexpansive mapping of D(U) into X, and if we set T — I--U, D(T)=D(U), then T is an accretive mapping of D{T) into X. (2) If { U(i)f t^O} is a semigroup of (nonlinear) mappings of X into X with infinitesimal generator T, then all the mappings U(t) are nonexpansive if and only if ( — 7") is accretive. For the special case when X is a Hubert space H (and the concept of an accretive mapping coincides with that of a monotone mapping), the writer in Browder [3], [4] used the observation (1) above and the theory of monotone mappings in Hubert space to prove the following fixed point theorem for nonexpansive maps: If C is a closed bounded convex subset of the Hubert space H, U a nonexpansive mapping of C into C which maps the boundary of C into C, then U has a fixed point in C. This line of argument has also been exploited to yield further results on the existence and calculation of fixed points of nonexpansive mappings in Hubert space and in the class of Banach spaces having weakly continuous duality mappings (like the spaces l,

Lie algebras associated with generalized Cartan matrices

Abstract: 2. Preliminaries. In this note, * will always denote a field of characteristic zero. An integral square matrix satisfying M l , M2, and M3 will be called a generalized Cartan matrix, or g.c.m. for short. Z will denote the integers, and in any Lie algebra we will use the symbol [lu h, , In] to denote the product [ • • • [[hh] • • • ]L]. 1 These results were obtained in my dissertation at the University of Toronto under the supervision of Professor M.J. Wonenburger.

Deformations of homomorphisms of Lie groups and Lie algebras

Abstract: The purpose of this note is to announce several results on deformations of homomorphisms of Lie groups and Lie algebras. Our main theorems are precise analogues of two basic theorems on deformations of complex analytic structures on compact manifolds, the rigidity theorem of Frölicher-Nijenhuis [3] and the local completeness theorem of Kuranishi fs]. In our results, sheaf cohomology is replaced by the cohomology of Lie groups and Lie algebras. Our proofs rely heavily on the theory of deformations in graded Lie algebras (GLA's) developed in [9]. Our results on Lie algebra homomorphisms follow immediately from the results given there, once the appropriate GLA is defined. Detailed proofs of the results on Lie group homomorphisms (which are only outlined here) will appear elsewhere.

Sums of ultrafilters

Abstract: By a space we mean a separated uniformizable topological space; and Z stands for the product of any constant family {z\\a^A } such that the cardinal of A is m. In our examples the spaces X in A and F^° in B are not pseudocompact. An exhibition of A and B with countably compact replaced by pseudocompact is done in [3]; it does not require Theorem C. Trivial examples of spaces with properties in A and B do not seem to be available. Observe the proof of A and B reduces to the following.

Nonlinear equations of evolution and nonlinear accretive operators in Banach spaces

Abstract: where (assuming that the conjugate space X* of X is strictly convex), / is the mapping of X into X* which assigns to each x of X the bounded linear functional w = J(x) such that (w, x) = \\\\w\\\\'\\\\x\\\\ and IMHWI. I t is our object in the present note to present some new and sharper results on two related topics: (1) The existence theory of solutions for the initial value problem for nonlinear equations of evolution of the form (2) du/di + T(t)u(t) = ƒ(/, u(t)) (t è 0)

The obstruction to fibering a manifold over a circle

Abstract: Let M be a compact, connected, smooth manifold whose dimension is greater than five, and let / be a continuous map from M to the circle, which we denote by S. Suppose that / restricted to the boundary of M9 denoted by bM9 is a smooth fibration. We note that a map h from a smooth manifold N to S is a smooth fibration if h is smooth, and for each point x of N the derivative of h maps the tangent plane to N at x onto the tangent plane to S at h(x). We wish to address ourselves in this talk to the following problem.

The virial theorem and its application to the spectral theory of Schrödinger operators

Abstract: e"1 g((l + e)x) q{x) | g q0(x) G Qt(Rr) holds for 0 < e < e 0 and some/3>0; in particular we have rqr(x) èqo(x); hence rqr£.Qp(R ). Under these conditions we shall prove in §2 a very general form of the Virial Theorem of quantum mechanics. In §§3 and 4 this theorem will be used to deduce some results on the spectrum of H. Let L2(R ) be the Hubert space of functions which are squaresummable over R; the inner product in this space will be denoted by ( • , • ), the norm by | • | . From condition (I) one can conclude (e.g. Ikebe-Kato [2]): (1) The operator H with domain D(H)=H2(R ) is selfadjoint in L2(R ) (H2(R ) is the closure of Co(R) with respect to the norm k | 2 = { Z ; , * Idtyidxfixà^+^sldu/dxjlt+lul*}"*). (2) For uGD(H) and qGQa(R ) we have quEL2(R ). (3) For u, vED(H) we have Au, AvEL2(R ) and (Au, v) = (u, Av).

On the Hauptvermutung for manifolds

Abstract: The "Hauptvermutung" is the conjecture that homeomorphic (finite) simplicial complexes have isomorphic subdivisions, i.e. homeomorphic implies piecewise linearly homeomorphic. I t was formulated in the first decade of this century and seems to have been inspired by the question of the topological invariance of the Betti and torsion numbers of a finite simplicial complex. The Hauptvermutung is known to be true for simplicial complexes of dimension 4 (Milnor, 1961). The Milnor examples, K and L, have two notable properties: (i) K and L are not manifolds, (ii) K and L are not locally isomorphic. Thus it is natural to restrict the Hauptvermutung to the class of piecewise linear w-manifolds, simplicial complexes where each point has a neighborhood which is piecewise linearly homeomorphic to Euclidean space R or Euclidean half space R\. We assume that Hz(M, Z) has no 2-torsion.

The Cauchy integral for differential forms

Abstract: In a previous note, we introduced a certain determinant of differential forms (cf. formula (1.3) of [3]), which led to an elementary proof of the Cauchy integral formula for holomorphic functions of several complex variables. We now propose to amplify this procedure in order to obtain some integral representations for exterior differential forms. We shall be considering certain mappings ^(f, z) and ƒ(£\", z) of an open set V(Z C X C into O , where \\f/ is of class C and ƒ is holomorphic. Writing yp=fyi, • • • , ^«) and ƒ = (fh • • • , ƒ„), we set tyi / ) = ^ i / i + • * • +&nfnHenceforth, we shall always assume that (i/s jO^O a t t n e points under consideration. Now, instead of merely considering a single smooth mapping \\j/, we take n such mappings, /̂(D^ ^(2)? . . . , ^(\"). Each of these will be regarded as a column in terms of its components. We shall, furthermore, use the vectorvalued differential forms d^= X)*-i^Fy^?y a n d d ^ = 2j?-i#*/^yWith this notation, we look at the nXn determinant

Residually finite one-relator groups

Abstract: Let £ be the class of those groups (/, m) satisfying \l\ 5 ^ 1 ^ \m\ , /m^O, and / and m relatively prime. Furthermore, let 9iïl be the class of these groups (If m\ t) satisfying the conditions imposed above on I and m, and in addition the extra two conditions t> 1, and l, m and t relatively prime in pairs. The point of our initial remark is that 9ÏI looks more complicated than £. Actually £ is quite a nasty class of groups. Indeed the main result of [ l ] is that every group in £ is isomorphic to one of its proper factor groups, i.e. nonhopfian. Since finitely generated residually finite groups are hopfian (A. I. Mal'cev [2]) no group in £ is residually finite. Our contribution to Conjecture A is that the groups in 9TC are residually finite.

Poisson boundaries and envelopes of discrete groups

Abstract: In [4] we defined the Poisson boundaries for semisimple Lie groups. These spaces play a role in the theory of generalized harmonic functions on the Lie group similar to that played by the boundary of the unit disc in the classical theory of harmonic functions on the unit disc. I t is not hard to extend these notions to all separable, locally compact groups, and, in particular, they make sense for countable discrete groups. In this form we shall show that these ideas provide a useful tool for answering certain purely algebraic questions. Namely, we raise the following question. Let G be a connected Lie group, Y a discrete subgroup for which G/Y has finite (left-) invariant measure. To what extent is G determined by a knowledge of Y as an abstract group, and conversely, what is the influence of G on the structure of T? To make this question precise, let us say that G is an envelope of r if an isomorphic copy Y' of Y occurs as a discrete subgroup of G, and G = DY', where D is a subset of G with finite left-invariant Haar measure. Our question may now be stated in this way. How different can two connected Lie groups G\\ and G2 be if they both envelop the same countable group T? We shall be discussing a rather restricted version of this question. We suppose that G\\ and G% are semisimple and have no compact components, and that G\\ and G2 envelop the same group Y. Does it follow that Gi and G2 are isomorphic? (Without the hypothesis that Gi and G2 have no compact components we could always take G2 = GiXa compact group.) Our guess is that this is the case. However all we can prove is the following:

On the radial projection in normed spaces

Abstract: Let X be a real normed space with norm || ||, T the radial projection mapping defined by \( Tx = x,\quad {\text{if}}\left\| x \right\| \leqq 1,\quad {\text{and}}\quad Tx = x/\left\| x \right\|,\quad {\text{if}}\left\| x \right\| \geqq 1. \)

Quantization and representations of solvable Lie groups

Abstract: Introduction. In this note, we will announce a characterization of a connected, simply connected Type I solvable Lie group, G, and present a complete description of the set of all unitary equivalence classes of irreducible unitary representations of G together with a construction of an irreducible representation in each equivalence class. This result subsumes the results previously obtained on nilpotent Lie groups and solvable Lie groups of exponential type of Kirillov [3] and Bernât [2], respectively. Our result is made possible by a merging of a new general geometric approach to representation theory, based on the use of symplectic manifolds and quantization, of the second author with a detailed analysis of the Mackey inductive procedure which augments the results in [ l ] .