Search

Recently Cameron and Storvick introduced and studied an operator valued function space integral related to the Feynman integral. The main theorems of their study establish the existence of the function space integral as a weak operator limit of operators defined at the first stage by finite-dimensional integrals. This paper provides a substantial strengthening of their existence theorem giving the function space integrals as strong operator limits rather than as weak operator limits.

The dynamic response of an infinite, elastic plate to a steadily moving transverse force is studied under the so-called improved plate theory. Analytic solutions for the subsonic and the intersonic cases are obtained using the method of integral transforms. The usual difficulty of the nonuniqueness of the steady-state solutions is circumvented by formulating the problems as initial value problems and then seeking the large time limits of the solutions of such initial value problems. The subsonic and intersonic solutions obtained here are similar to that obtained previously for the supersonic case. However, in contrast with the supersonic solution, which contains two families of Mach lines, the subsonic solution contains no Mach lines and the intersonic solution contains only one family of Mach lines.

Two lower bounds of Sperb for the first eigenvalue of an elastically attached membrane are proved in an elementary way.

In sections 2 and 3 two methods for proving the non existence of certain universal Banach spaces, are presented. In section 4 it is proved that every infinite-dimensional conjugate Banach space has a two-dimensional subspace whose unit cell is not a polygon.

n/a

n/a

It is shown that an arbitrary locally flat imbedding of one piecewise linear manifold into another of codimension greater than two can be changed to a piecewise linear imbedding by means of an arbitrarily small isotopy of the ambient manifold. Bibligraphy: 5 entries.

If BP denotes the space of smooth alternating p-forms on the C' manifold M, we are interested in finding the spaces of R-linear maps from iPP to be that commute with the diffeomorphisms of M. For a compact manifold M these spaces were found by R. S. Palais. In this note we find them for noncompact M. Let V, W be the spaces of sections of two tensor bundles over a connected Coo manifold M of dimension n. In a paper [3] appearing in Trans. Amer. Math. Soc. 92 (1959), R. S. Palais studied a number of the spaces 4 (V, W) of the R-linear maps V->W natural with respect to the action on V and W of the group G of diffeomorphisms on M. We sketch the definition of this action: If g-G we define R,:F(TM) -4F(TM) to be the differential of g, and Rg:r(T*M)-->F(T*M) to be the dual of the differential of g-1. For the other spaces of tensor fields Rg is defined as the obvious tensor product of these two R-linear isomorphisms. A map c: V->W is then called natural if cRg =R,c for all g G. Palais was particularly interested in the case when V and W are the space IP of p-forms and the space bq of q-forms respectively, 0 < p, q

It is the object of this paper to obtain theorems concerning the stability of the straight line solutions, and equilateral triangle solutions, respectively, in the problem of three bodies by means of the theorems and calculations of two preceding papers.t It is shown that the generalized theorems of Bohl can be applied to a neighborhood of the straight line solutions, with arbitrary masses, and to a neighborhood of the equilateral triangle solutions, if the masses are such that the characteristic exponents of the generating orbit are not all pure imaginaries. The mutual distances of the three masses are assumed constant on the generating orbit, in both cases.

Čebyšev centers have been studied extensively. In this paper an alternate concept of center for infinite bounded point sets is introduced. Some of the results in this paper for this new type of center are similar to previous results for Čebyšev centers.

The purpose of this paper is to prove a taming theorem for a codimension one manifold that is locally flat at some point and has 1-ULC complement. We also prove that any two sufficiently close locally flat embeddings of a codimension one manifold are ambient isotopic. Since this paper was first submitted, R. Daverman has shown that, given any point on a codimension one manifold with 1-ULC complement, some neighborhood of that point lies on a codimension one sphere that is locally flat at some points and has 1-ULC complement. Hence the two papers combined prove that a codimension one manifold is locally flat if and only if its complement is 1-ULC.

Let E, F be real analytic vector bundles over a real analytic compact manifold M and D: F(E)-+r(F) a real analytic polynomial differential operator satisfying D(0) = 0. Let s(i) be a parametrized family of cross sections of E, where t moves in some neighborhood of 0 in a euclidean space. We say that s(i) is a deformation of the solution 0 if s(0) = 0 and D(s(0) = 0. In the present paper, we will show the existence of deformations of the solution 0 under some conditions. Namely, let L be the linearized differential operator of D at 0 and assume: (1) the equation D(s) = 0 is involutive, (2) L is elliptic, (3) H(M9 0) = 0, where 0 is the solution sheaf of the equation L(s) = 0. Then we can prove that there is a deformation 5(0 which is complete at t = 0 in an appropriate sense. (Theorem 1, 2) We would like to point out the analogy between the above result and a theorem in [3] on the existence of deformations of complex structures. In fact, the arguments proceed along almost the same line as in [3]. In § 1, we prove some propositions which are needed in the later sections. In §2, we construct the deformation s(0 and prove its completeness in §3. Finally the author wishes to express his hearty gratitude to Professor N. Tanaka for his constant encouragement and valuable suggestions during the preparation of this paper.

This paper continues the study of the smoothness properties of (real) topological linear spaces. First, the smoothness results previously obtained about various important classes of locally convex spaces, such as Schwartz spaces, are improved. Then, following the ideas of Bonic and Frampton, we use these results to give sufficient conditions for the existence of smooth partitions of unity on manifolds modelled on topological linear spaces. 1. Preliminaries. In order to make this paper self-contained we include the definition of the two types of derivative used and the definition of an 8-category. We will employ the definitions of the derivative in topological linear spaces investigated in detail by Averbukh and Smolyanov [2], [31. See also [14], [15] and [191. Let TLS denote the class of all Hausdorff topological linear spaces over the real field R. Let 2 1(E, F) = 2(E, F) denote the set of all continuous linear maps from E into F, where E, F e TLS. We define by induction p(E, F) = 2(E, p 1(E, F)). Each ?1p(E, F) is given the topology of uniform convergence on bounded subsets of E. If X is a topological space, W(X) will denote the class of all open subsets of X. Let f: U -' V, where U e ((E), V E C(F), E e TLS and F c TLS. Then we say f is Fre6cbet differentiable at x e U, if there exists u e S(E, F) such that for each bounded subset B of E and for each o-neighbourhood W in F, there exists 8 > 0 such that f(x + th) f(x) u * th E tW, whenever b E B and ItL

In a previously published paper [9], the authors proved an approximation theorem for geodesically convex functions on Riemannian manifolds (Theorem 2 of [91). Applications of this theorem to complex function theory on noncompact K~ihler manifolds have been announced in [8, II and III] . The main result of the present paper is an application of this approximation theorem to Riemannian geometry proper. To describe this principal theorem, let M be a Riemannian manifold and let Z(M) be the closure of the set of all C ~176 subharmonic functions in C~ the algebra of continuous functions on M equipped with the compact open topology. (A C ~ function f is subharmonic if Af>O, where the sign

The subject of this paper is uniqueness and non-uniqueness in the Cauchy problem at simple characteristic points of the initial hypersurface. We only consider linear partial differential equations with analytic coefficients. We use the notation from the book [1] by H6rmander with the exception that D~ = t3/~xj, 1 0 in a neighbourhood of x °, and if H is of a type covered by [6, Theorem 3] then there is a neighbourhood V of x ° and a function u e C ~ ( V ) such that P ( x , D ) u = O , x ° ~ suppu C {x; ~0(x) =>0, x ~ V}." When we want to construct characteristic hypersurfaces that are not hyperplanes we have to rely on integration of Hamilton's equations. So we restrict ourselves to operators with real principal part and to simple characteristic points on the initial hypersurface. The three points of departure of this paper is the method to prove non-uniqueness by Malgrange [3], the uniqueness theorem [2, Theorem 8.1] by H6rmander, see Theorem 3 below, and the technique of proving uniqueness taken from Persson [5]. The part played by Theorem 3 is a passive one leaving to us to study the case when a priori the bicharacteristic curve does not leave the support of the solution. In [9] Zachmanoglou proves the following theorem.

Suppose that f : S 1 → R 2 f:{S^1} \to {R^2} is an immersion, i.e., a C 1 {C^1} map such that f ′ f’ is never zero. We call f normal if there are only finitely many self-intersections and these are transverse double points. A normal immersion f can be topologically determined by a finite number of combinatorial invariants. Using these invariants it is possible to give considerable information about extensions of f to D 2 {D^2} . In this paper we give a new set of invariants, inspired by the work of S. Blank, to solve several problems concerning the existence of certain kinds of extensions. The problems solved are as follows: (1) When does f have a light open extension F : D 2 → R 2 F:{D^2} \to {R^2} ? (Recall that light means F − 1 ( y ) {F^{ - 1}}(y) is totally disconnected for all y and open means F maps open sets of the interior of D 2 {D^2} to open sets of R 2 {R^2} .) Because of the work of Stoïlow, the question is equivalent to the following: when does there exist a homeomorphism h : S 1 → S 1 h:{S^1} \to {S^1} , such that fh has an analytic extension to D 2 {D^2} ? (2) Suppose that F : D 2 → R 2 F:{D^2} \to {R^2} is light, open, sense preserving, and, at each point of S 1 {S^1} , F is a local homeomorphism. At each point of the interior of D 2 {D^2} , F is locally topologically equivalent to the power mapping z m {z^m} on D 2 , m ≥ 1 {D^2},m \geq 1 . Points where m > 1 m > 1 are called branch points and m − 1 m - 1 is the multiplicity of the point. There are only a finite number of branch points. The problem is to discover the minimum number of branch points of any properly interior extension of f. Also we can ask what multiplicities can arise for extensions of a given f. (3) Given a normal f, find the maximum number of properly interior extensions of f that are pairwise inequivalent. Since each immersion of the disk is equivalent to a local homeomorphism, the problem of immersion extensions is a special case of this. It is Blank’s solution of the immersion problem that prompted this paper.

In this paper smooth maps /: Mn —> Nn with a zerodimensional critical set are considered. The singularities of these maps in the case n — 2 are known to be points where / is locally topologically equivalent to z -»zd {d~ 2,3, •••)• Originally these singularities were studied in connection with the regularity of Douglas' solution to Plateau's problem. In § 1, an Euler characteristic formula is developed which generalizes both the Riemann-Hurwitz equation from complex analysis and the usual Euler characteristic formula for covering maps. Section 2 is devoted to several technical lemmas while § 3 applies these lemmas to the case where M is the disc (with holes) and N is a compact orientable 2-manifold. It is shown that for the existence of such a map there is a lower limit depending upon the genus of N and on the number of holes of M. The singularities of these maps have been characterized by Church and Timourian [2]. In the case n — 2 these maps are locally topologically equivalent to z —-> zd (d = 2, 3, ) and for n > 2, these maps are covering projections. For n — 2, the singularities and maps are special cases of branch points and branched immersions. In this case, the Euler characteristic formula represents a generalization of the classical Riemann-Hurwitz equation for light interior transformations on 2-manifolds. When f'^dN) = dM, the formula produced here reduces to the Riemann-Hurwitz equation. For n > 2, the maps are covering projections and the Euler characteristic formula reduces to the usual equation. The mappings considered in this paper are not, in general, interior transformations on the boundary of M. These considered here, however, appear to have more applications with regard to questions which have arisen from Plateau's problem. Heinz [6] and Gulliver, Osserman, and Royden [5] have shown that these maps are of much more interest than solely in the context of minimal surfaces. A very readable account of the Riemann-Hurwitz formula may be found in Whyburn [10]. The authors wish to express their gratitude to the referee for his many helpful suggestions.

In this paper we give necessary conditions for the local solubility of linear differential and pseudodifferential equations with smooth coefficients. Our results generalize those of H. Lewi, Hormander, Nirenberg, Treves, Egorov, and others, described in the survey paper [16] (see also [17], [18]).

In this paper we are concerned with certain nonlinear type optimization problems related to the estimation of the universal constant KG appearing in Grothendieck's fundamental Theorem in the metric theory of tensor products. Let KG,k be the smallest constant such that Grothendieck's Theorem remains valid for m x m matrices, m≤k. In this paper we show KG,2=KG,3=√2.

In this paper the mathematical methods of quantum field theory are applied to some problems that arise in the statistical description of mechanical systems with very many (in the idealized case, infinitely many) degrees of freedom. This application is based on a graphical representation of the individual terms of the formal perturbation series in powers of the coupling constant in the form of Feynman diagrams. A variety of properties of such diagrams makes it possible to sum partially the perturbation series with a view to obtaining closed integral equations that contain the required quantities as unknowns. The approach is treated in more detail in connection with the statistical hydrodynamics of a developed turbulent flow, which is similar to the theory of a quantum Bose field with strong interaction. The functional formulation of statistical hydrodynamics makes it possible to obtain integral equations of turbulence theory, which can also be derived by means of diagram methods. At the end of the paper, some closed equations of statistical hydrodynamics are considered.

The proof of divergence of Fourier series which was constructed in the authors' earlier paper [Bull. Austral. Math. Soc. 8 (1973), 289–304] is not complete. The object of this paper is to show that the unestimated terms do not disturb the divergence.

In this paper, we improve on the results of two previous papers (8, 9) by establishing a general existence theorem (section 1·3, below) for a class of periodic oscillations of a wide class of non-linear differential equations of the second order in the real domain which are perturbations of the autonomous differential equationwhere g(x) is strictly non-linear. We then, by way of illustrating the power of the theorem, apply it to the problems which Morris (section 2·2 below), Shimuzu (section 2·3 below) and Loud (section 2·5 below) set themselves on the existence of periodic oscillations of certain differential equations which are perturbations of equations of the form (1·1·1).

This paper deals with the question of the existence of classical solutions for the equations $$\frac{{\partial ^{2} u}{\partial t^{2} }} + \sum_{\begin{subarray}{l} |\alpha| \leqslant m \\ | \beta | \leqslant m \end{subarray}} D^{\alpha} (A_{\alpha \beta } (x,t) D^{\beta} u) = f (t,x,u)$$ on [0,T] × G. G is a bounded or unbounded domain; the differential operator in the space variables is elliptic; the initial values of u are prescribed and Dαu (t,x) vanishes for (t,x) ∈ [0,T] × ∂G, |α|≤ m−1. First we develop a method for solving regularly linear wave equations. In contrast to the usual compatibility conditions, our method requires less differentiability in t but imposes some boundary conditions on f(t). It allows some applications to nonlinear problems which will be treated in the second part of this paper and which e.g. enable us to solve ∂2 u/∂t2−A(t)u+u3=f.

Let R be a metrization with distance xy of an open convex set D in the 2-dimensional real affine plane such that xy + yz = xz whenever x, y, z lie on an affine line with y between x and z and such that all the balls px ≤ ρ are compact. The study of such metrics, called open plane projective metrics falls under the topic of Hilbert's Problem IV of his famous mathematical problems. In this paper it is proved that if in R the sets of points equidistant from lines lie again on lines then D must be the entire affine plane and the distance must in fact be a norm. The paper contributes to and gives extensions of similar results proved earlier. The novel features of the present result are that in the space collinearity of points x, y, z is taken only as a sufficient condition for the equality xy + yz = xz. Consequently the solution encompasses all normed linear planes, that is, norms whose unit circles are not necessarily strictly convex are also admitted.

The object of this paper is to find the solutions of some dual equations involving the inverses of certain Mellin type transforms that were first introduced by D. Naylor in his paper [1]. Because these transforms are relatively unknown we shall begin by defining them and making a note of some of their properties. The main result is contained in the following theorem.

In this paper a study is made of continuous collections of decomposable continua on a spherical surface. Properties of the decomposition spaces of such collections filling up continua are established, and a characterization of the decomposition spaces of such collections filling up a spherical surface is obtained. The results of the present paper are related to certain results obtained by R. D. Anderson. He has shown [2 ] that there is a continuous collection of nondegenerate continua filling up the plane which is with respect to its elements homeomorphic to the plane, and that there is a continuous collection of continua filling up a plane onedimensional continuous curve which is with respect to its elements homeomorphic to the plane. On the other hand, he has obtained [1 I a characterization of the plane decomposition spaces of continuous collections of nondegenerate continuous curves filling up a plane continuum. Such decomposition spaces are special types of hereditary continuous curves. The results of the present paper show that the decomposition spaces of continuous collections of decomposable continua filling up compact plane continua are special types of hereditary continuous curves, and consequently that even under this weaker hypothesis such collections are dimension reducing if they fill up twodimensional continua and are dimension preserving if they fill up one-dimensional continua. It might be noted that there do exist such collections filling up one-dimensional continua, but that Theorem 2 of this paper gives a strong restriction about the nature of such collections. Theorems about continuous collections of continua in a compact metric space can be stated equivalently in terms of monotone interior transformations (p. 130 of reference [10]). It is understood throughout this paper that space is compact and metric. DEFINITIONS. If H is a point set and e is a positive number, V(e, H) denotes the set of all points P such that some point of H is at a distance from P of less than e. If H and K are point sets, d(H, K) will be used to denote the greatest lower bound of the set of all positive numbers e such that K intersects V(e, H). For two

In 1949 I published in the Acta Mathematica (vol. 81) a rather long paper: “L'intégrale de Riemann-Liouville et le problème de Cauchy.” This work will be quoted in the sequel as Acta paper. Only minor local references to this paper will be made here, and knowledge of it is not required for the reading of the present article. The notations used here are slightly different from those used in my former paper.In the Acta paper I introduce multiple integrals and of the Riemann- Liouville type depending on a parameter α and converging for sufficiently large values of α. I give the solution of the Cauchy problem for the wave equation in a unique formula, the same for space-time of odd or even dimensions, implying an analytic continuation with respect to the parameter α.