Showing papers in "Transactions of the American Mathematical Society in 1971"

Entropy for group endomorphisms and homogeneous spaces

Abstract: Topological entropy há(T) is defined for a uniformly continuous map on a metric space. General statements are proved about this entropy, and it is calculated for affine maps of Lie groups and certain homogeneous spaces. We compare hd(T) with measure theoretic entropy h(T); in particular h(T) = hd(T) for Haar measure and affine maps Ton compact metrizable groups. A particular case of this yields the wellknown formula for h(T) when T is a toral automorphism. Introduction. We shall study topological entropy, concentrating on its relation to measure theoretic entropy and algebraic examples. Our topological entropy hd(T) is defined (in §2) for a uniformly continuous map F on a metric space (X, d). In [1] a topological entropy h(T) was defined for a continuous map on a compact topological space; if the space is compact metric then h(T)—hd(T). An essential part of this paper is the computation of hd(T) for certain maps on noncompact spaces. Suppose p is a Borel measure on p(X) = l, and p is F-invariant (i.e. p(T~x(A)) =p(A) for every Borel set A). One can then define a measure theoretic entropy hu(T) as follows: Call a={Au ..., Ar} a (finite) measurable partition of A\" if the A¡ are disjoint measurable subsets of X covering X. Now set Hja) = 2 -4mc\\ T-kAik) iogp(mn t-*a\\ Then the limit hß(T, a) = limm_00 (\\/m)HJa) exists and one defines hu(T) = sup {hu(T, a) : a is a finite measurable partition of X}. (See [6] for details about measure theoretic entropy.) Two points in X are separated by a = {A±,..., Ar} provided they lie in different ^i's. We shall use the following fact to compute entropy : Fact (see [6]). Let {ak}k = 0 be a sequence of measurable partitions of X satisfying the following property: If x, ye X are distinct there is an n(x, y) such that ak separates x and y whenever k S: n(x, y). Then hu(f) = supfc h(T, ak). As is generally known, if T: G -> G is a surjective endomorphism of a compact metrizable group, then F preserves Haar measure p. For such a F we show that the Received by the editors October 17, 1969 and, in revised form, February 2, 1970. AMS 1969 subject classifications. Primary 2870, 2875; Secondary 5482.

Necessary conditions for stability of diffeomorphisms

Abstract: S Smale has recently given sufficient conditions for a diffeomorphism to be Q-stable and conjectured the converse of his theorem The purpose of this paper is to give some limited results in the direction of that converse We prove that an in- stable diffeomorphism / has only hyperbolic periodic points and moreover that if p is a periodic point of period k then the Arth roots of the eigenvalues of dff are bounded away from the unit circle Other results concern the necessity of transversal intersection of stable and unstable manifolds for an fi-stable diffeomorphism Introduction We will say that a diffeomorphism /: M—s- M of a compact manifold is Cl-stable if (a) there is a neighborhood N(f) of/in the C1 topology such that g e N(f) implies there is a homeomorphism « from the nonwandering set of/, Q(f) to the nonwandering set of g, 0(g) which satisfies g-« = «•/; and (b) if p is a periodic point off then dim Ws(p;f) = dim Ws(h(p); g) Property (b) is not usually included in the definition of Q-stable (see (3)), but it is a weak condition which is very natural and is apparently necessary for the proof of one of our lemmas (22) In his paper (4), S Smale provides sufficient conditions for a diffeomorphism to be £2-stable One of his conditions is that the nonwandering set have a hyperbolic structure Recall that a closed invariant set A is said to have a hyperbolic structure if (a) There is continuous splitting of the restriction of the tangent bundle to A,

Bonded projections, duality, and multipliers in spaces of analytic functions

Abstract: Let q) and b be positive continuous functions on [0, 1) with T(r) -*0 as r -11 and f S 0(r) dr

Isolated invariant sets and isolating blocks

Abstract: Closed invariant sets of smooth flow on compact manifold involving homoclinic or heteroclinic point theory of Poincare

The logarithmic limit-set of an algebraic variety

Abstract: Let C be the field of complex numbers and V a subvariety of (C{O})n. To study the "exponential behavior of Vat infinity", we define V(a) as the set of limitpoints on the unit sphere Sn-1 of the set of real n-tuples (u, log I i u.,u log IXn ), where x e V and u, = (1 + (log lx,l)2) -2. More algebraically, in the case of arbitrary base-field k we can look at places "at infinity" on V and use the values of the associated valuations on X1, Xn to construct an analogous set V(b). Thirdly, simply by studying the terms occurring in elements of the ideal I defining V, we define another closely related set, V',). These concepts are introduced to prove a conjecture of A. E. Zalessky on the action of GL(n, Z) on k[X1 1, . . ., Xn 1], then studied further. It is shown among other things that V(b) = V(c) (when defined) V(a. If a certain natural conjecture is true, then equality holds where we wrote "-" and the common set V. Sn1 is a finite union of convex spherical polytopes. 1. A conjecture of Zalessky. Let k be a field, and k[X ]=k[X11,..., Xn l] the ring obtained by adjoining n commuting indeterminates and their inverses to k. This is the group algebra on the free abelian group of rank n, Zn, so GL(n, Z) has a natural action on it. Call a subgroup of Zn nontrivial if it is of infinite order and infinite index in Zn; and call an ideal I( k[X'] nontrivial if it is of infinite dimension (i.e., nonzero) and infinite codimension in k[X+] as k-vector spaces. A. E. Zalessky conjectures in [1, Problem V.9], and we shall here prove: THEOREM 1. Let I be a nontrivial ideal in k[X ], and Hc GL(n, Z) the stabilizer subgroup of I. Then H has a subgroup Ho offinite index, which stabilizes a nontrivial subgroup of Zn (equivalently, which can be put into block-triangular form

Algebras of iterated path integrals and fundamental groups

Abstract: A method of iterated integration along paths is used to extend deRham cohomology theory to a homotopy theory on the fundamental group level. For every connected C°° manifold 3JI with a base point p, we construct an algebra ■7r1 = 7T1(2Si,p) consisting of iterated integrals, whose value along each loop at p depends only on the homotopy class of the loop. Thus ir1 can be taken as a commutative algebra of functions on the fundamental group ^(SBi), whose multiplication induces a comultiplication w1 -*■ u-1 ® ir1, which makes w1 a Hopf algebra. The algebra w1 relates the fundamental group to analysis of the manifold, and we obtain some analytical conditions which are sufficient to make the fundamental group nonabelian or nonsolvable. We also show that w1 depends essentially only on the differentiable homotopy type of the manifold. The second half of the paper is devoted to the study of structures of algebras of iterated path integrals. We prove that such algebras can be constructed algebraically from the following data: (a) the commutative algebra A of C functions on 93); (b) the A-modale M of C" 1-forms on SD!; (c) the usual differentiation d: A -> M; and (d) the evaluation map at the base point p, s : A -*■ K, K being the real (or complex) number field. A path a: [0, 1] -*■ 3W will be understood to be piecewise smooth. For w, wx, w2,---eM, let Ja w be the usual integral, and define, for r> 1, [ HV ■ -Wr = J Í J f Wl •■ ■ HV-iWot(f), <*(*)) dt, where a1 denotes the restriction a|[0, /]. Such iterated path integrals can be taken as functions on a space of paths. The integral J wx ■ ■ ■ w„ taken to be a A'-valued function on the set of paths (resp. loops) from the base point p, will be written as Jp wx ■ ■ ■ wr (resp. §p wx ■ ■ ■ wr). The totality of such path integrals jp wx • • • wr (resp. §„ wx ■ ■ ■ wr), r ^ 1, together with the constant function 1 generate an algebra P (resp. Q). Actually Q is a Hopf algebra with a comultiplication CqQ^ Q <8> Q reflecting the multiplication of loops. Those elements of Q, whose value on each loop depends only on its homotopy class, form a Hopf subalgebra n1 of Q. Evidently n1 consists of Ä'-valued functions on the fundamental group 7^(9)7). Received by the editors March 15, 1970. AMS 1970 subject classifications. Primary 53C65; Secondary 57D99.

A combinatorial model for series-parallel networks

Abstract: The category of pregeometries with basepoint is defined and explored. In this category two important operations are extensively characterized: the series connection S(G, H), and the parallel connection P(G, H) = S(G, H); and the latter is shown to be the categorical direct sum. For graphical pregeometries, these notions coincide with the classical definitions. A pregeometry Fis a nontrivial series (or parallel) connection relative to a basepoint p iff the deletion F\\p (contraction F/p) is separable. Thus both connections are zz-ary symmetric operators with identities and generate a free algebra. Elements of the subalgebra A[C2l generated by the two point circuit are defined as series-parallel networks, and this subalgebra is shown to be closed under arbitrary minors. Nonpointed series-parallel networks are characterized by a number of equivalent conditions: 1. They are in A[C2] relative to some point. 2. They are in A[C2] relative to any point. For any connected minor K of three or more points: 3. K is not the four point line or the lattice of partitions of a four element set. 4. K or K is not a geometry. 5. For any point e in K, K\\e or K/e is separable. Series-parallel networks can also be characterized in a universally constructed ring of pregeometries generalized from previous work of W. Tutte and A. Grothendieck. In this Tutte-Grothendieck ring they are the pregeometries for which the Crapo invariant equals one. Several geometric invariants are directly calculated in this ring including the complexity and the chromatic polynomial. The latter gives algebraic proofs of the two and three color theorems.

Weighted norm inequalities for singular and fractional integrals

Abstract: Inequalities of the form 11 lxl cTf 1,q < C 11 lxl af 11, are proved for certain well-known integral transforms, T, in En. The transforms considered include Calder6n-Zygmund singular integrals, singular integrals with variable kernel, fractional integrals and fractional integrals with variable kernel.

The convergence with vanishing viscosity of nonstationary Navier-Stokes flow to ideal flow in ₃

Abstract: It is shown here that a unique solution to the Navier-Stokes equations exists in R3 for a small time interval independent of the viscosity and that the solutions for varying viscosities converge uniformly to a function that is a solution to the equations for ideal flow in R3. The existence of the solutions is shown by transforming the Navier-Stokes equations to an equivalent system solvable by applying fixed point methods with estimates derived from using semigroup theory. Introduction. We wish to find a solution, local in time, to the Cauchy problem for the Navier-Stokes equations for viscous incompressible flow in R3 and show that the solutions of the Navier-Stokes equations for various viscosities converge, as the viscosity goes to zero, to a function that is a solution to the Euler equations for an ideal (inviscid) fluid. The Navier-Stokes equations are (E') av/lt + (v * grad) v-vAv = -grad P + B, VV = v O, with constraints lim v(x, t) = 0 and v(x, 0) = Cx), Ixl +o where x = (x1, X2, X3) is a point in R3; t is in some time interval [0, T]; the velociy v(x, t)=(v1(x, t), v2(x, t), v3(x, t)); the pressure is P(x, t); the force is B(x, t) = (B1(x, t), B2(x, t), B3(x, t)); and the constant v>0 is the viscosity (the coefficient of kinematic viscosity). The Euler equations for ideal flow differ from the Navier-Stokes equations (E') only in that the viscosity term vAv does not occur in the Euler equations. Uniqueness and existence of a solution to the Navier-Stokes equations in R3 has been shown for both bounded and unbounded domains: in both cases existence has been shown only for a sufficiently small time interval. The first results are those of C. W. Oseen [11] and Jean Leray [8]. The time interval where the solution is shown to exist must be small enough to satisfy a condition of form T? Kv, where K is an appropriate constant and v is the viscosity. Thus the length of the time interval Received by the editors January 28, 1970 and, in revised form, June 25, 1970. AMS 1968 subject classifications. Primary 7635, 7646; Secondary 3536, 4750.

A maximal function characterization of the class

Abstract: Let u be harmonic in the upper half-plane and O 0, 0 0, in the upper half-plane Im z> 0 if it is defined there and sup | IF(x + iy) IP dx o 0 00 If f(z) is any function defined in the upper half-plane, its nontangential maximal function is given by Na(f)(x) = sup If(z)I ZGra(x) where F(x) is the cone {z=s+iy Is-xl 0, if

Constructive polynomial approximation on spheres and projective spaces.

Abstract: This paper contains constructive generalizations to functions defined on spheres and projective spaces of the Jackson theorems on polynomial approxima- tion. These results, (3.3) and (4.6), give explicit methods of constructing uniform approximations to smooth functions on these spaces by polynomials, together with error estimates based on the smoothness of the function and the degree of the poly- nomial. The general method used exploits the fact that each space considered is the orbit of some compact subgroup, G, of an orthogonal group acting on a Euclidean space. For such homogeneous spaces a general result (2.1) is proved which shows that a G-invariant linear method of polynomial approximation to continuous functions can be modified to yield a linear method which produces better approximations to k-times differentiable functions. Jackson type theorems (3.4) are also proved for functions on the unit ball (which is not homogeneous) in a Euclidean space. Introduction. We previously extended the Jackson theorems to any smooth compact submanifold M of a Euclidean space E (see (12)). The proofs of these theorems were not really constructive and made use of a rather ad hoc extension of a function from M to some ball in E. In the present paper we show how, in case M is a sphere or projective space, constructive versions of these theorems can be proved which do not require us to extend functions of M. We begin in ?1 by defining differentiability and other smoothness properties for functions on a homogeneous manifold M in terms of the homogeneous structure of M. Then in ?2 we show how the homogeneity of M together with the Jackson theorems for C(M) can be combined to prove the Jackson estimates for Ck(M). These general results are applied to prove the Jackson estimates for spheres and balls in ?3 and for projective spaces over the reals, complexes and quaternions in ?4. A general reference for the differential geometry of homogeneous spaces of compact Lie groups which we use is Helgason (4). However we have tried in ??3 and 4 to be reasonably explicit so that the reader without a background in Lie

On the classification of symmetric graphs with a prime number of vertices

Abstract: We determine all the symmetric graphs with a prime number of vertices. We also determine the structure of their groups.

Convex hulls of some classical families of univalent functions

Abstract: Let S denote the functions that are analytic and univalent in the open unit disk and satisfy /(0) = 0 and/'(0) = 1 Also, let K, St, SR, and C be the subfamilies of 5 consisting of convex, starlike, real, and close-to-convex mappings, respectively The closed convex hull of each of these four families is determined as well as the extreme points for each Moreover, integral formulas are obtained for each hull in terms of the probability measures over suitable sets The extreme points for each family are particularly simple; for example, the Koebe functions f(z) = z/(l-xz)2, \x\ = 1, are the extreme points of cl co St These results are applied to discuss linear extremal problems over each of the four families A typical result is the following: Let J be a "nontrivial" continuous linear functional on the functions analytic in the unit disk The only functions in St that satisfy Re/(/) = max {ReJ(g) : geSt} are Koebe functions and there are only a finite number of them Introduction We shall be concerned with the closed convex hulls of various families of functions that are analytic and univalent in the open unit disk A={z eC : \z\< 1} For each family considered we obtain integral representations for the closed convex hull, and we determine all the extreme points In each case the extreme points are strikingly simple and familiar functions Thus we obtain a powerful tool for solving linear extremum problems over such families Let us establish some notation and outline our main results We shall let A denote the set of all functions analytic in A With the natural topology of uniform convergence on compact subsets of A, A is a locally convex linear topological space (15, p 150) Let S be the subset of A consisting of functions / that are univalent (one-to-one) in A and satisfy/(0)=0,/'(0) = l It is well known (7, p 217) that S is compact in A, or, equivalently, that S is closed and locally uniformly bounded (On each compact subset of A there is a common bound for all the functions in S) We shall be particularly interested in the following subfamilies of S K={fe S : /(A) is convex}, St={fe S : /(A) is starlike with respect to 0},

On branch loci in Teichmüller space

Abstract: The branch locus of the ramified covering of the space of moduli of Fuchsian groups with fixed presentation by the corresponding Teichmiller space is decomposed into a union of Teichmiuller spaces, each characterised by a description of the action of the conformal self-mappings admitted by the underlying Riemann surfaces. Equivalence classes of subloci under the action of the modular group are studied, and counted in certain simple cases. One may compute as a result the number of conjugacy classes of elements of prime order in the mapping class group of closed surfaces.

An extension of the Weyl-von Neumann theorem to normal operators

Abstract: We prove that a normal operator on a separable Hubert space can be written as a diagonal operator plus a compact operator. If, in addition, the spectrum lies in a rectifiable curve we show that the compact operator can be made HilbertSchmidt. In 1909 Hermann Weyl proved [3] that each bounded Hermitian operator on a separable Hubert space can be written as the sum of a diagonal operator and a compact operator. J. von Neumann proved in 1935 [2] that the compact operator can actually be made Hilbert-Schmidt and also observed that boundedness is unnecessary. P. R. Halmos has inquired in [1] whether the Weyl result, and, in his 1969 Brazilian lectures, whether the von Neumann result can be extended to normal operators. We prove here that Weyl's result can be so extended. That is, each normal operator can be written as the sum of a diagonal operator and a compact operator. If, in addition, the spectrum lies in a rectifiable curve, von Neumann's result holds. That is, the compact operator can actually be made Hilbert-Schmidt. In each of these cases the bounded case is the crucial one; the extension to unbounded operators is easy. It is also easy to see that, as in the Weyl and von Neumann results, the norm of the compact operator or the Hilbert-Schmidt norm of the Hilbert-Schmidt operator respectively may be made arbitrarily small. We conjecture that, in general, for normal operators, the compact operator cannot be made Hilbert-Schmidt. Indeed, we conjecture that the operator given by multiplication by z on L2[7x7] cannot be so decomposed; the positive 2dimensional Lebesgue measure of the purely continuous spectrum of this operator seems to prevent the compact operator from being Hilbert-Schmidt. We were first made acquainted with these questions by a lecture of P. R. Halmos at the regional conference sponsored by the NSF at Texas Christian University in May 1970. We gratefully acknowledge valuable discussion with J. Dyer, P. Halmos and H. Porta. We use standard terminology. However, for economy of notation we will allow ourselves the looseness of speaking of an unbounded operator \"on\" 77. We use Received by the editors October 8, 1970. AMS 1970 subject classifications. Primary 47B15, 47B05, 47B10.

Automorphisms of a free associative algebra of rank 2. II

Abstract: Let R be a commutative domain with 1. R(x, y) stands for the free associative algebra of rank 2 over R; R[x, y>] is the polynomial algebra over R in the commuting indeterminates x' and y. We prove that the map Ab: Aut (R(x, y)) Aut (R[x, y ]) induced by the abelianization functor is a monomorphism. As a corollary to this statement and a theorem of Jung [51, Nagata [71 and van der Kulk [81* that describes the automorphisms of FR, y ] (F a field) we are able to conclude that every automorphism of F(x, y) is tame (i.e. a product of elementary automorphisms). R stands for a commutative domain with 1. R(x, y) is the free associative algebra of rank 2 over R on the free generators x and y; R[x, "-1 is the polynomial algebra over R on the commuting indeterminates x and y. We will prove here that the answer to the following conjecture [3, p. 1971 is in the affirmative: If F is a field then the group of automorphisms of F(x, y) is generated by the elementary automorphisms (defined below) of F(x, y) (i.e. every automorphism of F(x, y) is tame). In fact, we are going to prove here that, if R is as above, the map Ab: Aut (R (x, y)) Aut (R[x y71) induced by the abelianization functor is a monomorphism and as a consequence of this statement and a theorem of Jung, Nagata and van der Kulk* that says that every automorphism of F[x, y 1 is tame (for F a field) we will be able to give a complete description of Aut (F(x, y)). The proof is a generalization of the proof of the main theorem of [41; in fact the algorithm we use here to solve a system of equations in R(x, y) is essentially the same we used in the previous paper. We will refer to [41 for additional details in the proofs. I am indebted to G. M. Bergman for making the observation that the tameness result is not true in the generality claimed in our previous paper [41 and announced in the Bulletin of the AMS in November 1971 [Automorphisms of a free associative algebra of rank 2, Bull. Amer. Math. Soc. 77(1971), 992-9941, since the corresponding tameness theorem for the abelian case (i.e. the theorem of Jung, Received by the editors August 5, 1971. AMS 1970 subject classifications. Primary 16AO6, 16A72; Secondary 20F55, 16A02.

Theory of random evolutions with applications to partial differential equations

Abstract: The selection from a finite number of strongly continuous semigroups by means of a finite-state Markov chain leads to the new notion of a random evolution. Random evolutions are used to obtain probabilistic solutions to abstract systems of differential equations. Applications include one-dimensional first order hyperbolic systems. An important special case leads to consideration of abstract telegraph equations and a generalization of a result of Kac on the classical 7¡-dimensional telegraph equation is obtained and put in a more natural setting. In this connection a singular perturbation theorem for an abstract telegraph equation is proved by means of a novel application of the classical central limit theorem and a representation of the solution for the limiting equation is found in terms of a transformation formula involving the Gaussian distribution. In a little-known section of an out-of-print book [7], Mark Kac has considered a particle which moves on a line at speed v, and reverses direction according to a Poisson process with intensity a. After showing that such a motion is governed by a pair of partial differential equations, Kac comments, \"The amazing thing is that these two equations can be combined into a hyperbolic equation\"—namely, the telegraph equation, 1W =

A construction of Lie algebras from a class of ternary algebras

Abstract: A class of algebras with a ternary composition and alternating bilinear form is defined. The construction of a Lie algebra from a member of this class is given, and the Lie algebra is shown to be simple if the form is nondegenerate. A characterization of the Lie algebras so constructed in terms of their structure as modules for the three-dimensional simple Lie algebra is obtained in the case the base ring contains 1/2. Finally, some of the Lie algebras are identified; in particular, Lie algebras of type E8 are obtained. A construction of Lie algebras from Jordan algebras discovered independently by J. Tits [7] and M. Koecher [4] has been useful in the study of both kinds of algebras. In this paper, we give a similar construction of Lie algebras from a ternary algebra with a skew bilinear form satisfying certain axioms. These ternary algebras are a variation on the Freudenthal triple systems considered in [1]. Most of the results we obtain for our construction are parallel to those for the TitsKoecher construction (see [3, Chapter VIII]). In ?1, we define the ternary algebras, derive some basic results about them, and give two examples of such algebras. In ?2, the Lie algebras are constructed and shown to be simple if and only if the skew bilinear form is nondegenerate. In ?3, we give a characterization, in the case the base ring contains 1/2, of the Lie algebras obtained by our construction in terms of their structure as modules for the threedimensional simple Lie algebra. Finally, in ?4, we identify some of the simple Lie algebras obtained by our construction from the examples of ?1. In particular, we show that we can construct a Lie algebra of type E8 from a 56-dimensional space which is a module for a Lie algebra of type E7. A similar construction was given by H. Freudenthal in [2]. 1. A class of ternary algebras. We shall be interested in a module TM over an arbitrary commutative associative ring D with 1 which possesses an alternating bilinear form and a ternary product which satisfy (TI) = + z for x,y, ze M; (T2) = + x for x, y, z e 9; (T3) , w> = , z> + for x, y, z, w e 9; Received by the editors March 12, 1970 and, in revised form, June 19, 1970. AMS 1969 subject classifications. Primary 1730.

The structure of pseudocomplemented distributive lattices. II. Congruence extension and amalgamation

Abstract: This paper continues the examination of the structure of pseudocomplemented distributive lattices. First, the Congruence Extension Property is proved. This is then applied to examine properties of the equational classes A,-1, n<, which is a complete list of all the equational classes of pseudocomplemented distributive lattices (see Part I). The standard semigroups (i.e., the semigroup generated by the operators H, S, and P) are described. The Amalgamation Property is shown to hold iff n< 2 or n = w. For 3< n < co, does not satisfy the Amalgamation Property; the deviation is measured by a class Amal (,kn) (C 2n The finite algebras in Amal (2n) are determined. 0. Introduction. This paper continues the examination of the structure of pseudocomplemented distributive lattices begun in Part I, [8]. Using the description of congruences given in Part I, we verify the Congruence Extension Property in ?1. This, in effect, states that a *-congruence on a subalgebra can be extended to a *-congruence on the algebra. This property is applied in ??2 and 3. In ?2 we determine the \"standard semigroups\" of the equational classes of pseudocomplemented distributive lattices, which is, roughly speaking, the semigroup generated by the operators H, S, and P in the sense of [5]. In ?3 it is shown that the Amalgamation Property holds in 4iOn (notation of Part I) if and only if n = -1, 0, 1, 2, or co, and that the subalgebra theorem for free products of B. Jonsson [7] holds for exactly the same equational classes. Since the Amalgamation Property fails to hold for 3, 4, . . ., we introduce a concept attempting to measure the extent of this failure. This concept is the amalgamation class of M Amal (Xk). The Amalgamation Property holds in -X if and only if Amal ( =) = ?4 contains results on Amal(n) for 2< n < c; in particular, we determine the finite algebras in Amal (?J.) 1. The Congruence Extension Property. A class Xk of algebras is said to satisfy the Congruence Extension Property if, given any algebra B and subalgebra A, both in V and any congruence E) on A, there is a congruence 0 on B such that the Received by the editors July 8, 1970. AMS 1970 subject classifications. Primary 06A35; Secondary 08A25, 18A20, 18A30, 18C05.

Dirichlet spaces and strong Markov processes

Abstract: We show that there exists a suitable strong Markov process on the underlying space of each regular Dirichlet space. Potential theoretic concepts due to A. Beurling and J. Deny are then described in terms of the associated strong Markov process. The proof is carried out by developing potential theory for Dirichlet spaces and symmetric Ray processes and by using a method of transformation of underlying spaces. Introduction. This paper is a continuation of (10). We will use those notions and terminologies adopted in (10). Let (X, m, Y, &) be a D-space. We define (aor-) capacity of an open set A c X by Cap (A) = inf &'1o(u, u) if YA = 0,

Improbability of collisions in Newtonian gravitational systems

Abstract: It is shown that the set of initial conditions leading to a collision in finite time has measure zero. 1. It is well known that binary collisions of point masses in a Newtonian gravitational system are improbable in the sense that the set of initial conditions leading to this catastrophe at some finite time has (Lebesgue volume) measure zero. One would expect the same to be true for multiple collisions if only for some sort of aesthetic reasoning—there seems to be a binary collision contained within a multiple collision. What is shown here is that this is indeed the case; that is, the set of initial conditions leading to collision in finite time has measure zero. This problem has gained attention in recent years with its inclusion in J. E. Littlewood's list of problems [2, Problem 13]. It must be emphasized that the fact that the force law is the inverse square force law plays a crucial role in the proof of this result. To see that this is not true for all force laws, let 21= 2 mKrf, where m¡ is the mass and r{ the position vector of the ith mass relative to the center of mass of the systems. It is well known (see [3], [6], [8], for example) that in the inverse p force law I=i3-p)T+ip-l)h. Here T is the kinetic energy, A is the total energy of the system and the dots denote differentiation with respect to time. Note that if p^3, l-¿ip-\\)h. Integration yields IHp-i)ht2/2 + IiO)t + IiO). It follows that all initial conditions possessing negative A have the property that in finite time /—>0. Hence if the solution lasts long enough then the system will suffer a complete collapse. In particular, if n = 2 and A<0, there will be a collision. But the set of initial conditions yielding negative A has measure greater than zero. (Of course in the inverse square law the above is not true as for « = 2, negative energy leads in general to elliptic motion.) The reason that binary collisions are improbable in the inverse square law is that the system retains its analytic dependence on initial conditions at collision. (This comes from the fact that a binary collision can be regularized. See, for Received by the editors September 4, 1970 and, in revised form, January 19, 1971. AMS 1969 subject classifications. Primary 7034; Secondary 3440, 8500.

Cubes with knotted holes

Abstract: : The 3-dimensional Poincare conjecture is that a compact, connected, simply connected 3-manifold without boundary is topologically a 3-sphere S sup 3. Despite efforts to prove the conjecture, it has withstood attack. It is known that every orientable 3-manifold may be obtained by removing a collection of disjoint solid tori from S sup 3 and sewing them back differently. In this paper the author examine some of the possibilities for constructing a counterexample to the Poincare conjecture by removing a single solid torus from S sup 3 and sewing it back differently. Actually, they examine not only this process but one analogous to it which they call 'attaching a pillbox to a cube with a knotted hole.' (Author)