Showing papers on "Space (mathematics) published in 1972"
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01 Jan 1972
TL;DR: In this paper, the authors define the notion of potentials and their basic properties, including the capacity and capacity of a compact set, the properties of a set of irregular points, and the stability of the Dirichlet problem.
Abstract: 1. Spaces of measures and signed measures. Operations on measures and signed measures (No. 1-5).- 2. Space of distributions. Operations on distributions (No. 6-10)..- 3. The Fourier transform of distributions (No. 11-13).- I. Potentials and their basic properties.- 1. M. Riesz kernels (No. 1-3).- 2. Superharmonic functions (No. 4-5).- 3. Definition of potentials and their simplest properties (No. 6-9)...- 4. Energy. Potentials with finite energy (No. 10-15).- 5. Representation of superharmonic functions by potentials (No. 16-18).- 6. Superharmonic functions of fractional order (No. 19-25).- II. Capacity and equilibrium measure.- 1. Equilibrium measure and capacity of a compact set (No. 1-5).- 2. Inner and outer capacities and equilibrium measures. Capacitability (No. 6-10).- 3. Metric properties of capacity (No. 11-14).- 4. Logarithmic capacity (No. 15-18).- III. Sets of capacity zero. Sequences and bounds for potentials.- 1. Polar sets (No. 1-2).- 2. Continuity properties of potentials (No. 3-4).- 3. Sequences of potentials of measures (No. 5-8).- 4. Metric criteria for sets of capacity zero and bounds for potentials (No. 9-11).- IV. Balayage, Green functions, and the Dirichlet problem.- 1. Classical balayage out of a region (No. 1-6).- 2. Balayage for arbitrary compact sets (No. 7-11).- 3. The generalized Dirichlet problem (No. 12-14).- 4. The operator approach to the Dirichlet problem and the balayage problem (No. 15-18).- 5. Balayage for M. Riesz kernels (No. 19-23)...- 6. Balayage onto Borel sets (No. 24-25).- V. Irregular points.- 1. Irregular points of Borel sets. Criteria for irregularity (No. 1-6)...- 2. The characteristics and types of irregular points (No. 7-8)...- 3. The fine topology (No. 9-11).- 4. Properties of set of irregular points (No. 12-15).- 5. Stability of the Dirichlet problem. Approximation of continuous functions by harmonic functions (No. 16-22).- VI. Generalizations.- 1. Distributions with finite energy and their potentials (No. 1-5)...- 2. Kernels of more general type (No. 6-11).- 3. Dirichlet spaces (No. 12-15).- Comments and bibliographic references.
1,885 citations
TL;DR: In this paper, the first variation of the k-dimensional area integral is studied and a regularity theorem for weakly defined k dimensional surfaces in Riemannian manifolds whose first variation is summable to a power greater than k is given.
Abstract: Suppose M is a smooth m dimensional Riemannian manifold and k is a positive integer not exceeding m. Our purpose is to study the first variation of the k dimensional area integrand in M. Our main result is a regularity theorem for weakly defined k dimensional surfaces in M whose first variation of area is summable to a power greater than k. A natural domain for any k dimensional parametric integral in M, among which the simplest is the k dimensional area integral, is the space of k dimensional varifolds in M introduced by Almgren in [AF 1]. Such a varifold is defined to be a Radon measure on the bundle over M whose fiber at each point p of M is the Grassmann manifold of k dimensional linear subspaces of the tangent space to M at p. If V is a varifold in M, let I I V I I be the Radon measure on M obtained from V by ignoring the fiber variables. Naturally injected in the space of k dimensional varifolds in M is the set of k dimensional rectifiable subsets of M, which includes the set of k dimensional submanifolds of M as well as more general k dimensional surfaces in M. A k dimensional varifold in M is said to be rectifiable (integral) if it can be strongly approximated by a positive real (integral) linear combination of varifolds corresponding to continuously differentiable k dimensional submanifolds of M. To any classical k dimensional geometric object in M there corresponds a k dimensional integral varifold in M. If N is a smooth Riemannian manifold and F: M-e N is smooth, then F induces in a natural way a strongly continuous mapping F# of the k dimen-
962 citations
TL;DR: In this article, the authors study controllability, observability, and realization theory for a particular class of systems for which the state space is a differentiable manifold which is simultaneously a group or, more generally, a coset space.
Abstract: The purpose of this paper is to study questions regarding controllability, observability, and realization theory for a particular class of systems for which the state space is a differentiable manifold which is simultaneously a group or, more generally, a coset space. We show that it is possible to give rather explicit expressions for the reachable set and the set of indistinguishable states in the case of autonomous systems. We also establish a type of state space isomorphism theorem. Our objective is to reduce all questions about the system to questions about Lie algebras generated from the coefficient matrices entering in the description of the system and in that way arrive at conditions which are easily visualized and tested.
473 citations
TL;DR: The main result of as mentioned in this paper is that if a Banach space admits a sequentially weakly continuous duality function, then a condition introduced by Opial to characterize weak limits by means of the norm is satisfied and the space has normal structure in the sense of Brodskii-Milm an.
Abstract: The main result of this paper asserts that if a Banach space admits a sequentially weakly continuous duality function, then a condition introduced by Opial to characterize weak limits by means of the norm is satisfied and the space has normal structure in the sense of Brodskii-Milm an. This result of geometric nature allows some unification in the fixed point theory for both single-valued and multi-valued non-expansive mappings.
228 citations
TL;DR: In this paper, a theory of analog resonances is reviewed which makes use of projection operators, and various processes which contribute to the escape amplitude of the analog state are classified, and some numerical estimates are given.
Abstract: A theory of analog resonances is reviewed which makes use of projection operators. The Hilbert space is divided into three parts: a continuum or open-channel space, an analog-state space, and a compound space. The phenomena are discussed in terms of the dynamical coupling of these spaces. The parameterization of the $T$ matrix is discussed in detail, and equations are presented for various cross sections. The commutator [$H$, ${T}_{\ensuremath{-}}$], where ${T}_{\ensuremath{-}}$ is the isospin-lowering operator, plays an important role in the theory, and the various terms which contribute to this commutator are discussed. The energy splitting of the isospin multiplet, i.e., the Coulomb displacement energy, is discussed in detail. The importance of the analog resonance phenomena for the extraction of spectroscopic information is stressed, and it is shown how such information may be obtained. Various processes which contribute to the escape amplitude of the analog state are classified, and some numerical estimates are given. For several regions of the Periodic Table, graphs are presented for the various theoretical escape amplitudes, continuum energy shifts, asymmetry phases, and optical phase shifts, etc. Spectroscopic factors are calculated and compared with those obtained in other experiments.
180 citations
109 citations
TL;DR: In this paper, it is shown that if M is complete and simply connected, then it possesses a global coordinate system whose coordinate vectors are unit-length asymptotic vectors.
Abstract: Let M be a connected ^-dimensional space form isometrically immersed in a simply connected (2w-l)-dimensional space form of strictly larger curvature. If M is minimal, it is proven that it must be a piece of the flat Clifford torus in the (2w-l)-sphere. If M is complete and simply connected, it is proven that M possesses a global coordinate system whose coordinate vectors are unit-length asymptotic vectors.
101 citations
101 citations
01 Jan 1972
TL;DR: In this article, the conditions générales d'utilisation (http://www.numdam.org/conditions) are defined, i.e., toute utilisation commerciale ou impression systématique is constitutive d'une infraction pénale.
Abstract: © Mémoires de la S. M. F., 1972, tous droits réservés. L’accès aux archives de la revue « Mémoires de la S. M. F. » (http://smf. emath.fr/Publications/Memoires/Presentation.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
TL;DR: A space is weakly θ-refinable if every open cover U of X has an open refinement V = ∪{V(n): n⩾ 1} such that given xϵX, one of the collections V(n) has finite, positive order at x as discussed by the authors.
Abstract: A space is weakly θ-refinable if every open cover U of X has an open refinement V = ∪{V(n): n⩾ 1} such that given xϵX, one of the collections V(n) has finite, positive order at x. Several equivalent properties of a space are given and are used to prove that: (a) if X is weakly θ-refinable and has closed sets Gδ then X is subparacompact; (b) any quasi-developable space (in the sense of Bennett) is weekly θ-refinable; (c) a space is quasi-developable if and only if it has a θ-base, (d) a linearly ordered topological space is paracompact if and only if it is weakly θ-refinable. Examples are given which show that weak θ-refinability is strictly weaker than the notion of θ-refinability introduced by Worrell and Wicke.
TL;DR: In this article, the authors considered an extension of the following inequality given in the book Inequalities by Hardy, Littlewood and Polya; if f is real-valued, twice differentiable on [0, ∞) and such that f and f are both in the space fn,∞), then f′ is in L,2(0,∆) and the extension consists in replacing f′ by M[f] wherechoosing f so that m and m are in L 2(∆, ∆).
Abstract: This paper considers an extension of the following inequality given in the book Inequalities by Hardy, Littlewood and Polya; let f be real-valued, twice differentiable on [0, ∞) and such that f and f are both in the space fn, ∞), then f′ is in L,2(0, ∞) andThe extension consists in replacing f′ by M[f] wherechoosing f so that f and M[f] are in L2(0, ∞) and then seeking to determine if there is an inequality of the formwhere K is a positive number independent of f.The analysis involves a fourth-order differential equation and the second-order equation associated with M.A number of examples are discussed to illustrate the theorems obtained and to show that the extended inequality (*) may or may not hold.
TL;DR: A Note on Price Systems in Infinite Dimensional Space Author(s): Edward C. Prescott and Robert E. Lucas, Jr as discussed by the authors The paper was published by Blackwell Publishing for the Economics Department of the University of Pennsylvania and Institute of Social and Economic Research -Osaka University Stable URL: http://www.jstor.org/stable/2526035
Abstract: A Note on Price Systems in Infinite Dimensional Space Author(s): Edward C. Prescott and Robert E. Lucas, Jr. Source: International Economic Review, Vol. 13, No. 2 (Jun., 1972), pp. 416-422 Published by: Blackwell Publishing for the Economics Department of the University of Pennsylvania and Institute of Social and Economic Research -Osaka University Stable URL: http://www.jstor.org/stable/2526035 Accessed: 21/06/2010 20:10
31 Dec 1972
TL;DR: In this article, the authors describe geometric properties of differential operators, such as spherical functions on symmetric spaces, Conical distributions on the space of horocycles, central eigendistributions and characters.
Abstract: Introduction Some geometric properties of differential operators Spherical functions on symmetric spaces Conical distributions on the space of horocycles Central eigendistributions and characters Bibliography.
TL;DR: In this paper, a notion of linear dependence of one stochastic process upon another is introduced, and studied in the case of symmetric stable processes, and the problem of imbedding one Lq space into another is shown to be related to this study.
TL;DR: In this paper, a theory of existence and regularity of solutions of the partial differential equation (L_p) with boundary conditions satisfying general boundary conditions is given. But this theory is not applicable to the case where the boundary conditions are arbitrary.
Abstract: An $L_p $ theory $(1 < p < \infty )$ of existence and regularity of solutions of the partial differential equation $(1 - \gamma \mathcal{M}(t)){{\partial u} / {\partial t}}) - \mathcal{L}(t)u = f$ satisfying general boundary conditions is given. For each t, $\mathcal{M}(t)$ is a linear elliptic partial differential operator in the space variables, $\mathcal{L}(t)$ is a linear differential operator whose order does not exceed that of $\mathcal{M}(t)$ and $\gamma $ is a nonzero complex constant.
TL;DR: In this article, the authors give an example of a Banach space X such that X * is isometric to a subspace of C(ωθ) and X is not isomorphic to a complemented subspace in any C(K) space.
Abstract: We give an example of a Banach spaceX such that (i)X
* is isometric tol
1, (ii)X is isometric to a subspace ofC(ωθ) and (iii)X is not isomorphic to a complemented subspace of anyC(K) space.
[...]
TL;DR: In this paper, a general technique, algebraic quantization, is provided for going from classical (c) paradigms, typically discrete logical structures, to q analogs.
Abstract: Quantum concepts can be applied to space-time processes to make a quantum (q) theory that is free of the possibility of divergencies inherent in classical continuum theories, yet causal, Lorentz-invariant, and asymptotically Poincar\'e-invariant for large times. A general technique, algebraic quantization, is provided for going from classical (c) paradigms, typically discrete logical structures, to q analogs. Applied to the two-dimensional c checker-board, algebraic quantization gives a q theory of time and space asymptotic to the four-dimensional Minkowski c theory in the limit of large time. Applied to the simplest dynamics on such a checkerboard, a piece that makes the same move again and again, algebraic quantization gives a q dynamics asymptotic to a massless spin-\textonehalf{} two-component dynamics in the same limit. The quantum of time, if it exists, must have spin \textonehalf{}. Some features of general relativity such as curvature seem plausible consequences of a quantum theory of space-time processes.
TL;DR: In this article, the authors considered the problem of maximizing the expectation of the discounted total reward in Markovian decision processes with arbitrary state space and compact action space varying with the state.
Abstract: We consider the problem of maximizing the expectation of the discounted total reward in Markovian decision processes with arbitrary state space and compact action space varying with the state. We get the existence theorem for a $(p, \epsilon)$-optimal stationary policy, and the relation between the optimality of a policy and the optimality equation. Assuming the action space is a compact subset of $n$-dimensional Euclidean space, the existence of an optimal stationary policy is established, and an algorithm is obtained for finding the optimal policy. The last two facts are based on the Borel implicit function lemma given in this paper.
TL;DR: In this paper, the authors consider a Boltzmann gas which fills all of space and is under the influence of a field of conservative external force whose potential is bounded from below, and prove existence and uniqueness for the general solution of the nonlinear Maxwell-Boltzmann equation at least in a finite interval of time.
Abstract: We consider a Boltzmann gas which fills all of space and is under the influence of a field of conservative external force whose potential is bounded from below. Assuming the intermolecular force has a cut-off, we prove existence and uniqueness for the general solution of the nonlinear Maxwell-Boltzmann equation at least in a finite interval of time. The solution can be constructed by the method of successive approximations in corresponding complete spaces. Definitions of these spaces are connected with an exponential function of the total energy of the molecule. Some indications of future generalizations and investigations are given.
TL;DR: In this paper, new criteria in the multiplier form are presented for the input-output stability in the L 2 -space of a linear system with a time-varying element k(t) in a feedback loop.
Abstract: New criteria in the multiplier form are presented for the input-output stability in the L 2 -space of a linear system with a time-varying element k(t) in a feedback loop. These are sufficient conditions for the system stability and involve conditions on the shifted imaginary-axis behavior of the multipliers. The criteria permit the use of noncausal multipliers, and it is shown that this necessitates dk/dt to be bounded from above as well as from below. The method of derivation draws on the theory of positivity of compositions of operators and time-varying gains, and the results are shown to be more general than the existing criteria.
TL;DR: The space H 1(R n) of integrable functions whose Riesz transforms are integrables was introduced in this article, where the authors considered the problem of preserving singular integral operators in the Lp theory of partial differential equations.
Abstract: It is well-known that the space L 1(Rn ) of integrable functions on Euclidean space fails to be preserved by singular integral operators. As a result the rather large Lp theory of partial differential equations also fails for p = 1. Since L 1 is such a natural space, many substitute spaces have been considered. One of the most interesting of these is the space we will denote by H 1(R n) of integrable functions whose Riesz transforms are integrable.
TL;DR: In this paper, it was shown that the complex recurrent space-times are of Petrov type D or N and the spaces of type D are non-empty and decomposable into the product of two 2-dimensional spaces of arbitrary curvature, the recurrence vector being necessarily real.
Abstract: It is shown that the complex recurrent space-times are of Petrov type D or N. The spaces of type D are non-empty and decomposable into the product of two 2-dimensional spaces of arbitrary curvature, the recurrence vector being necessarily real. It is established that the problem of determining the type N complex recurrent spaces is equivalent to that of determining the spaces of type N admitting a recurrent null vector field. A coordinate system is given in which the metric of such spaces is determined up to three arbitrary functions and a constant. The spaces fall into four invariant classes according to the permitted algebraic form of the Ricci tensor. The Einstein-Maxwell equations are solved, the solutions obtained providing examples for three of the classes. It seems that the remaining class cannot contain any physically reasonable solutions of Einstein’s equations. The general metric of the complex recurrent space of Petrov type N with recurrence vector proportional to the principal null vector of the Weyl tensor is also given and some of its properties are discussed. The space of plane gravitational waves admitting a singular electromagnetic field is the only physically significant space-time of this class. The conformally recurrent spaces and the conformal symmetric spaces are also determined.
TL;DR: In this paper, it was shown that for test fields of spin-2 in vacuum spaces, solutions of the propagation equation are restricted to constant multiples of the Weyl spinor.
Abstract: The propagation of a zero rest-mass test field of arbitrary spins>1 through curved space-time is found to be subject to strong constraints. A null test field is shown to be possible only in a restricted class of spaces previously introduced by Kundt and Thompson. This result is in fact a simultaneous generalization of the theorems of Robinson and of Goldberg and Sachs. For test fields of spin-2 in vacuum spaces, solutions of the propagation equation are restricted, save in a few exceptional cases, to constant multiples of the Weyl spinor. The exceptional cases are discussed, and appear to be physically uninteresting.