Showing papers in "Acta Mathematica in 1967"

Hypoelliptic second order differential equations

Abstract: that is, if u must be a C ~ function in every open set where Pu is a C ~ function. Necessary and sufficient conditions for P to be hypoelliptic have been known for quite some time when the coefficients are constant (see [3, Chap. IV]). I t has also been shown tha t such equations remain hypoelliptic after a perturbation by a \"weaker\" operator with variable coefficients (see [3, Chap. VIII) . Using pseudo-differential operators one can extend the class of admissible perturbations further; in particular one can obtain in that way many classes of hypoelliptic (differential) equations which are invariant under a change of variables (see [2]). Roughly speaking the sufficient condition for hypoelliptieity given in [2] means tha t the differential equations with constant coefficients obtained by \"freezing\" the arguments in the coefficients at a point x shall be hypoelliptie and not vary too rapidly with x . However, the sufficient conditions for hypoelliptieity given in [2] are far from being necessary. For example, they are not satisfied by the equation

The d-step conjecture for polyhedra of dimension d<6

Abstract: Two functions Δ and Δb, of interest in combinatorial geometry and the theory of linear programming, are defined and studied Δ(d, n) is the maximum diameter of convex polyhedra of dimensiond withn faces of dimensiond−1; similarly, Δb(d,n) is the maximum diameter of bounded polyhedra of dimensiond withn faces of dimensiond−1 The diameter of a polyhedronP is the smallest integerl such that any two vertices ofP can be joined by a path ofl or fewer edges ofP It is shown that the boundedd-step conjecture, ie Δb(d,2d)=d, is true ford≤5 It is also shown that the generald-step conjecture, ie Δ(d, 2d)≤d, of significance in linear programming, is false ford≥4 A number of other specific values and bounds for Δ and Δb are presented

On the closure of characters and the zeros of entire functions

Abstract: The problem to be studied in this paper concerns the closure properties on an interval of a set of characters {e~nx}~, where A = {2n}~ is a given set of real or complex numbers without finite point of accumulation. This problem is for obvious reasons depending on the distribution of zeros of certain entire functions of exponential type. The main problem of the paper is to determine the closure radius Q = Q(A)defined as the upper bound of numbers r such that (ei~x)~EA span the space L 2 ( r , r ) . The value of r does not change if a finite number of points are removed from or adjoined to A. Nor does Q(A) change if the metric in the previous definition is replaced by any other LV-metric, or by a variety of other topologies. I f A contains complex numbers we shall always assume (1)< 6~t ~ (0.1) 9 ~eA ~

On entire functions of exponential type and indicators of analytic functionals

Abstract: We shall be concerned with the indicatorp of an analytic functional μ on a complex manifoldU: $$p(\varphi ) = \overline {\mathop {\lim }\limits_{t \to + \infty } } \frac{l}{t}\log \left| {\mu (e^{t\varphi } )} \right|,$$ where ϕ is an arbitrary analytic function onU. More specifically, we shall consider the smallest upper semicontinuous majorantp J of the restriction ofp to a subspace £ of the analytic functions. An obvious problem is then to characterize the set of functionsp J which can occur as regularizations of indicators. In the case whenU=C n and £ is the space of all linear functions onC n , this set can be described more easily as the set of functions $$\mathop {\lim }\limits_{\theta \to \zeta } \overline {\mathop {\lim }\limits_{t \to + \infty } } \frac{l}{t}\log \left| {u(t\theta )} \right|$$ (0.1) ofn complex variables ζ∈C n whereu is an entire function of exponential type inC n . We hall prove that a function inC n is of the form (0.1) for some entire functionu of exponential type if and only if it is plurisubharmonic and positively homogeneous of order one (Theorem 3.4). The proof is based on the characterization given by Fujita and Takeuchi of those open subsets of complex projectiven-space which are Stein manifolds.

On the greatest prime factor of a quadratic polynomial

Abstract: as x ~ ~ , for which as for many other interesting results in the theory of numbers we are indebted to Chebyshev, has a t t racted the interest of several mathematicians. Revealed posthumously as little more than a fragment in one of Chebyshev's manuscripts, the theorem was first published and fully proved in a memoir by Markov in 1895 [6], while later in the same year a generalisation by Ivanov [4] appeared in which the polynomial n2§ 1 was replaced by n2+A for any positive A (an account of both Markov's and Ivanov ' s work is to be found in Paragraphs 147 and 149 of Landau's Primzahlen [5]). In 1921 Nagell [7] improved and further generalised Chebyshev's theorem by shewing that for any e log~ x, X

The interpolation of quadratic norms

Abstract: The theory of the interpolation of Banach spaces has been widely developed in recent years by a number of authors. I t is natural to expect tha t the interpolation theory for Hilbert spaces should have a particularly simple character; that this is in fact the case is shown in the present study where a complete description of exact quadratic interpolation norms and exact quadratic interpolation methods is given. Since the literature on interpolation theory has been characterized by an expert as impenetrable [11] it has seemed worthwhile to make the exposition as complete and detailed as possible. Our arguments depend in an essential way on the beautiful theory of monotone matr ix functions and Cauchy interpolation problems discovered by Loewner in 1934, and our theory may be regarded as a natural application of Loewner's results. The description of the exact quadratic interpolation methods, given by our Theorem 2, has already been found by Foils and Lions [6] who establish a corresponding result under somewhat stronger hypotheses. I t should be emphasized tha t our definition of interpolation norms and interpolation methods differs only superficially from tha t regularly used in the literature [3]. We should also remark tha t the functions k(2) which give rise to the exact quadratic interpolations are the positive functions, concave of infinite order on the unit interval. This class has been studied by Kranss [7] and also Bendat and Sherman [4]. Let V be a linear space over the complex numbers upon which there is defined a pair of norms IIx]]0 and ]lxlll. We shall usually assume tha t those norms are compatible, tha t is to say, tha t any sequence {xk} in V which is simultaneously Cauchy for both norms,

Foundations of the theory of Dirichlet series

Abstract: 1. A modern reader, familiar with the methods of functional analysis, is struck with the conviction tha t the classical theory of Dirichlet series [3, 5, 6] must have content expressible in more congenial language. Harald Bohr recognized the analogy between harmonic series and the Fourier series of functions on the circle; later his theory of almostperiodic functions was shown to be par t of a theory of Fourier series on compact abelian groups tha t embraced the classical case of the circle group as well. Various generalizations t reat the spaces L ~, and it is fairly clear by now how much of Fourier series can be developed in the more general setting. Nevertheless another par t of the theory of Dirichlet series, to which Bohr himself contributed a great deal, does not seem to be harmonic analysis. This is the par t depending on the Dirichlet condition

Dual representations of Banach algebras

Abstract: The irepresentation theory for Banach algebras has three main branches that are only rather loosely connected with each other. The Gelfand representation of a commutative algebra represents the given algebra by continuous complex valued functions on a space built from the multiplicativc linear functionals on the algebra. A Banach star algebra is represented by operators on a Hilbert space, the Hilbert space being built by means of positive I.iermitian funetionals on the algebra. Finally, for general non-commutat ive Banach algebras, an extension of the Jacobson theory of representations of rings is available. In this general theory, the representations are built in terms of irreducible operator representations on Banach spaces, and, on the face of it, no par t is played by the linear functionals on the algebra. There is some evidence that the concepts involved in the general theory are not sufficiently strong to exploit to the full the Banach algebra situation. The purpose of the present paper is to develop a new unified general representation theory that is more closely related than the Jaeobson theory to the special theories for commutat ive and star algebra s. The central concept is that of a dual representation on a pair of Banach spaces in normed duality. I t is found tha t each continuous linear functional on a Banach algebra gives rise to a dual representation of the algebra, and thus the dual space of the Mgebra enters representation theory in a natural way. One may ask of a dual representation that it be irreducible on each of the pair of spaces in duality, and thus obtain a concept of irreducibility stronger than the classical one. Correspondingly one obtains a stronger concept of density. For certain pairs of spaces in duality, topological irreducibility on one of the spaces implies topological irreducibility on the other. However, we show tha t this is very far from being the case in general. We also consider a further concept of irreducibility, namely uniform strict transitivity, which is stronger than strict irreducibility. For certain pairs of spaces in duality, uniform strict transit ivity on one of the spaces implies

Functors whose domain is a category of morphisms

Abstract: Connected sequences of functors whose domain, is the category of morphisms of an arbitrary abelian categoryA and whose range categoryB is also abelian are compared with the composition functors of Eckmann and Hilton acting between the same categories Sequences of functors of both types are obtained from any half-exact functorA→B ifA has enough injectives and projectives.

Microbundles and S-duality

Abstract: This paper contains a generalization of the S-duality theorem proved independently by Atiyah, Bott and A. Shapiro, [1]. The S-duality theorem may be stated as follows: Let X be a compact di/]erentiable mani]old and let ~1, ~s be vector bundles over X such that T(X)O~I| is J-equivalent to a trivial vector bundle. Then the Thorn spaces o/~1 and ~ are S-duals in the sense o/Spanier and Whitehead. Our generalization goes in two directions. First we drop the differentiabflity condition on X and allow ~1, ~, to be microbundles (z(X) now meaning the tangent microbundle). Secondly we drop the compactness condition on X and compensate this by working with relativized Thom spaces over compact pairs. More precisely we show that if (A, B) is any compact pair in X, sufficiently nicely embedded, and (U, V) is a sufficiently nice open neighbourhood of (A, B), then