Showing papers on "Space (mathematics) published in 1974"

Second-order scalar-tensor field equations in a four-dimensional space

Abstract: Lagrange scalar densities which are concomitants of a pseudo-Riemannian metric-tensor, a scalar field and their derivatives of arbitrary order are considered. The most general second-order Euler-Lagrange tensors derivable from such a Lagrangian in a four-dimensional space are constructed, and it is shown that these Euler-Lagrange tensors may be obtained from a Lagrangian which is at most of second order in the derivatives of the field functions.

Indefinite Inner Product Spaces

Abstract: I. Inner Product Spaces without Topology.- 1. Vector Spaces.- 2. Inner Products.- 3. Orthogonality.- 4. Isotropic Vectors.- 5. Maximal Non-degenerate Subspaces.- 6. Maximal Semi-definite Subspaces.- 7. Maximal Neutral Subspaces.- S. Projections of Vectors on Subspaces.- 9. Ortho-complemented Subspaces.- 10. Dual Pairs of Subspaces.- 11. Fundamental Decompositions.- Notes to Chapter I.- II. Linear Operators in Inner Product Spaces without Topology.- 1. Linear Operators in Vector Spaces.- 2. Isometric Operators.- 3. Symmetric Operators.- 4. Cayley Transformations.- 5. Principal Vectors of Cayley Transforms.- 6. Pairs of Inner Products: Semi-boundedness.- 7. Pairs of Inner Products: Sign.- 8. Plus-operators.- 9. Pesonen Operators.- 10. Fundamental Projectors.- 11. Fundamental Symmetries. Angular Operators.- Notes to Chapter II.- III. Partial Majorants and Admissible Topologies on Inner Product Spaces.- 1. Locally Convex Topologies on Vector Spaces.- 2. Partial Majorants. The Weak Topology.- 3. Metrizable Partial Majorants.- 4. The Polar of a Normed Partial Majorant.- 5. Admissible Topologies.- 6. Orthogonal Companions and Admissible Topologies.- 7. Projections and Admissible Topologies.- 8. Intrinsic Topology.- 9. Projections and Intrinsic Topology.- Notes to Chapter III.- IV. Majorant Topologies on Inner Product Spaces.- 1. Majorants.- 2. Majorants and Metrizable Partial Majorants.- 3. Orthonormal Systems.- 4. Minimal Majorants.- 5. Majorants and Decomposability.- 6. Decomposition Majorants.- 7. Invariant Properties of E+ and E-.- 8. Subspaces of Spaces with a Hilbert Majorant.- Notes to Chapter IV.- V. The Geometry of Krein Spaces.- 1. Krein Spaces.- 2. Krein Spaces as Completions.- 3. Subspaces.- 4. Maximal Semi-definite Subspaces.- 5. Uniformly Definite Subspaces.- 6. Non-uniformly Definite Subspaces.- 7. Maximal Uniformly Definite Subspaces.- 8. Regular and Singular Subspaces.- 9. Alternating Pairs.- 10. Dissipative Operators in Hilbert Space.- Notes to Chapter V.- VI. Unitary and Selfadjoint Operators in Krein Spaces.- 1. Preliminaries.- 2. The Adjoint of an Operator.- 3. Isometric Operators.- 4. Unitary and Rectangular Isometric Operators.- 5. Spectral Properties of Unitary Operators.- 6. Selfadjoint Operators.- 7. Cayley Transformations.- 8. Unitary Dilations.- Notes to Chapter VI.- VII. Positive Operators and Plus-operators in Krein Spaces.- 1. Positive Operators.- 2. Operators of the Form T*T.- 3. Uniformly Positive Operators.- 4. Plus-operators.- 5. Strict Plus-operators.- 6. Doubly Strict Plus-operators.- Notes to Chapter VII.- VIII. Invariant Semi-definite Subspaces of Linear Operators in Krein Spaces.- 1. Fundamentally Reducible Operators.- 2. Invariant Positive Subspaces of Plus-operators.- 3. Invariant Semi-definite Subspaces of Unitary and Selfadjoint Operators.- 4. Quadratic Pencils of Operators in Hilbert Space.- 5. Quadratic Operator Equations in I-Iilbert Space.- 6. Spectral Functions.- Notes to Chapter VIII.- IX. Pontrjagin Spaces and Their Linear Operators.- 1. The Spaces ?k* Positive Subspaces.- 2. Closed Subspaces.- 3. Isometric Operators: Continuity.- 4. Isometric and Symmetric Operators: Number and Length of Jordan Chains.- 5. Proof of Theorem 4.3.- 6. Regular Symmetric Extensions.- 7. Invariant Positive Subspaces: Existence.- 8. Invariant Positive Subspaces: Uniqueness.- 9. Common Invariant Positive Subspaces for Commuting Operators.- Notes to Chapter IX.- Index of Terms.- Index of Symbols.

Non-Hamiltonian approach to conformal quantum field theory

Abstract: A requirement of completeness of the operator set involved in the theory at small distances is formulated which replaces the unitarity condition of the S- matrix in the usual theory Explicit expressions are obtained for the contribution of the intermediate state with a definite symmetry in the Wightman function Together with the localization'' condition the completeness condition leads to a set of algebraic equations for the anomalous dimensionalities and interaction constants which may be regarded as the sum rule for these quantities The approximate solutions of the equations found in 4- epsilon dimension space yield results which are equivalent to the Hamiltonian approach (auth)

Geometric sources of redundancy in intensity data and their use for phase determination

Abstract: Linear equations are derived in direct space, which express the relation between the electron densities of crystals built from the same molecule, but with different lattices or several identical subunits in their asymmetric units. They are shown to be equivalent to the most general 'molecular-replacement' phase equations in reciprocal space. The solution of these phase equations by the method of successive projections is discussed. This algorithm, best implemented in direct space by averaging operations, is shown to be convergent for over-determined problems, and to be equivalent to a least-squares solution of the phase equations.

Self-Adjoint Extensions of Symmetric Subspaces

Abstract: A theory of self-adjoint extensions of closed symmetric linear manifolds beyond the original space is presented. It is based on the Cayley transform of linear manifolds. Resolvent and spectral families of such extensions are characterized. These extensions are also determined by means of analytic contractions between the "deficiency spaces" of the original symmetric linear manifold.

Anosov flows with gibbs measures are also Bernoullian

Abstract: We consider a special flowS t over a shift in the space of sequences (X, μ) constructed using a continuousf with {fx380-1} We formulate a condition for μ such that theK-flowS t is aB-flow.

Black holes in static electrovac space--times

Abstract: The theorem of Israel which characterizes the Reissner-Nordstrom solutions as the only well behaved asymptotically flat electrovac spaces with a simple regular horizon is extended by weakening the assumptions. Critical points of the gravitational potential are not a priori excluded and the topology of the eguipotential surfaces is not restricted. The regularity of the horizon is formulated in terms of bounds for certain geometrical quantities and the assumption of existence, in some extension, of a bifurcation surface for the horizons is not made. The possibilities of non-static or non-conservative electromagnetic fields in a static space-time are discussed and excluded by physical arguments.

On compact* spaces and compactifications

Abstract: The space 83X of Z-ultrafilters on X with the standard filter space topology is shown to be compact*. Without considering the reflection associated with compact* spaces, we also prove that products of compact* spaces are compact*, in response to a request for a direct proof. Introduction. Compact* spaces were defined by W. W. Comfort [2] as completely regular Hausdorff spaces X for which every maximal ideal in C*(X) is fixed. He proved without the axiom of choice that every completely regular Hausdorff space X can be densely C *-embedded in a compact* space OX and deduced that products of compact* spaces are compact*. The problem of proving directly the productivity of compactness* was raised and left open. In ?1 of this note we establish a one-to-one correspondence between the maximal ideals of C*(X) and the Z-ultrafilters on X without the axiom of choice and show that X is compact* if and only if every Z-ultrafilter on X converges. We then have a topological method for the study of compactness*. We use the above method to show in ?2 that the space f3X of Z-ultrafilters on X [3] is compact* and that the classical characterizations of OX [31 hold independently of the axiom of choice. Finally, in ?3 we prove directly that products of compact* spaces are compact* and that closed subspaces of compact* spaces are compact*. The method of proof differs from that of ?2 in that it involves a consideration of maximal ideals in rings of real valued bounded continuous functions. W. W. Comfort's theorem referred to above is a consequence of the results of this section. An alternative construction of fX has recently been given by R. E. Chandler [1]. Received by the editors October 26, 1972 and, in revised form, February 11, 1973. AMS (MOS) subject classifications (1970). Primary 54C45, 54D30; Secondary 04A25.

Extensions of measures and the von Neumann selection theorem

Abstract: Let (X, Bx) be a Blackwell space, where BX is the a-algebra of Borel sets. Then if a is a finite measure defined on a countably generated sub-a-algebra BG Bx, a can be extended to a Borel measure T. Equivalently, if X and Y are Blackwell andf: X-* Y is Borel, and It is a Borel measure carried on f(X)c Y, then there exists a Borel measure T on X with Tf =o, where T'(E)=T(f1 (E)). We characterize {TI Tf = a} if f is semischlicht. Let BX denote the Borel sets of a topological space X. We consider the following measure extension (or equivalently restriction) problem: given a measure (we will always mean finite measure) a defined on a or-algebra Bc Bx, can or be extended to all of Bx, i.e., does there exist a Borel measure isuch that i-(E)=a(E) for all E e B? It is well known (see [1, p. 71], for details) that if B1 and B2 are c-algebras, and B2 is generated by B1 and finitely many additional sets, then any measure on B1 can be extended to B2. The result is not known for countably generated extensions. We show below (Theorem 5) that if X is a Blackwell space and B is a countably generated sub-a-algebra of Bx, then any measure on B extends to Bx. A Blackwell space is a measure space (X, Bx), where X is an analytic subset of a complete separable metric space (c.s.m.). A subset A of a c.s.m. is analytic iff A is the continuous image of a c.s.m. We note that the analytic sets form a proper subset of Ux, the set of absolutely measurable subsets of X, where E e Ux iff E is f-measurable for all finite Borel measures 4u, where "'7" denotes the completion of It, i.e., given j, there exist El, E2-e Bx such that El c Ec E2 and M(E2-E1)=O. A function g is said to be absolutely measurable if g-1(V) E Ux for all open V. Details may be found in [3], [4], or [5]. We note that if Xc S, X analytic, S a c.s.m., then Bx={Er)XjEceBS}, so elements of BX are topologically analytic, and not necessarily Borel in S. We begin by considering a special class of sub-cr-algebras of Bx. Let f:X-Y be Borel measurable, and let Bf={f-l(E)JE e Bx}. Given a Borel Presented to the Society, April 20, 1973; received by the editors January 30, 1973 and, in revised form, June 4, 1973. AMS (MOS) subject classifications (1970). Primary 28A05, 28A60, 28-00. 1 Partially supported by NSF GP-38265. ? American Mathematical Society 1974

Stability properties of finite meromorphic operator functions. III

Abstract: Let A be a meromorphic function with values in the space of bounded linear operators between two Banach spaces X and Y , and assume that the coefficients of the principal part of the Laurent expansion of A at a certain point λ 0 are degenerate operators In this paper it is shown that under rather general hypotheses the null spaces (resp ranges) of A ( λ ) converge in the gap topology to a certain subspace of X (resp Y ) as λ approaches λ 0 Further, under slightly stronger conditions, the null spaces (resp ranges) of A ( λ ) have a fixed complementary subspace in X (resp Y ) for all λ in some deleted neighbourhood of λ 0 The hypotheses of these stability theorems are fulfilled if A is Fredholm at λ 0 or has values in the set of degenerate operators

Existence of a basis in the space of functions analytic in the disk, and some properties of franklin's system

Abstract: It is shown that there is a basis in the space of functions analytic in the unit disk and continuous in the closed unit disk. This answers a question posed by Banach. It is further shown that the Franklin system is an unconditional basis in the spaces Lp(0, 1) for 1 < p < ∞. Bibliography: 8 items.

On Best Simultaneous Approximation in Normed Linear Spaces

Abstract: Let S be a non-empty family of real valued continuous functions on [a, b]. Diaz and McLaughlin [1], [2], and Dunham [3] have considered the problem of simultaneously approximating two continuous functions f1 and f2 by elements of S. If || • || denotes the supremum norm, then the problem is to find an element * ∈ S if it exists, for which

Kolmogorov's Extension Theorem for Infinite Measures

Abstract: We consider the extension problem of a self-consistent family of infinite measures to a completely additive measure. For probability measures, Kolmogorov's extension theorem assures that the extension is uniquely possible. Our results are as follows: (a) For CT-finite measures, we can reduce the problem to the case of probability measures, so that the extension is uniquely possible. As an application, on an infinite dimensional vector space we can construct such a measure that is invariant both under rotations and homotheties with respect to the origin. It is obtained as the limit of ndimensional measure: Also we shall discuss about the Lorentz invariant measure on an infinite dimensional space. (b) If measures are not a-finite, under the additional condition (EG) in §6, the extension is possible but not unique. We shall mention about the largest and the smallest extension. As an application, we can consider the symbolic representation of a flow {7f} defined on an infinite measure space X, namely constructing an appropriate product space VVR and an appropriate measure on IVR, T$ on X is represented by a shift St on jj/R. w(.)_w(.+t)m

Extreme measurable selections

Abstract: . The extreme points of the set of measurable selections for a set-valued mapping are characterized. As a corollary, the extreme points of the unit ball of the space of "vectorvalued LP functions" are characterized, thus generalizing results of Sundaresan.

The Born-Green equation for liquid metals

Abstract: Conflicting reports have appeared in the literature concerning the usefulness of the Born-Green equation to determine effective interionic pair potentials in liquid metals from measured structure factors. To elucidate the problem, numerical investigations of the Born-Green equation have been made. These are based on recent structure factor data for sodium and potassium. Using the linearized simultaneous equation method of Waseda and Suzuki (1973), it is found that the equations in real space are ill conditioned. Hence a unique solution cannot be found numerically. However, in Fourier space unique solutions can be determined, but these solutions are temperature dependent. It is concluded that the temperature independent pair potentials for liquid metals which have been obtained from the Born-Green equation are not reliable.

Spoked wheels to deploy large surfaces in space

Abstract: Extensible booms were used to deploy and support solar cell arrays of varying areas. Solar cell array systems were built with one or two booms to deploy and tension a blanket with attached cells and bussing. A segmented and hinged rim supported by spokes joined to a common hub is described. This structure can be compactly packaged and deployed.

The Characterization of the Weak Closure of Certain Sets of Integrable Functions

Abstract: Let K be a set of integrable functions from a Hausdorff metrizable topological compact space T with a given regular Borel measure on it into an Euclidean space E. We assume that K has the following property (P) : for each measurable subset A of T and any two functions $u_1 ,u_2 $ from K, the function \[\chi _A u_1 + \chi _{T\backslash A} u_2 \in K,\] where $\chi _A $ is the characteristic function of the set A.In this paper we give the characterization of the closure $\bar K$ of set K having property (P) in the weak topology of the conjugate space $C^ * $ of the space C of continuous functions from T into E. In particular, we also obtain conditions under which the set K is closed in the weak topology $w(L_1 ,C)$.

Weights on Spaces

Abstract: Boolean σ-algebras and probability measures arise in the study of the logic of propositions associated with a single experiment, as was pointed out by Kolmogoroy [15]. Recently [4, 16, 17] a program has been initiated to generate models for the logic of propositions associated with multiple experiments. These investigations and others on the quantum logic approach to quantum mechanics (e.g.,10,12,14) have motivated us to study generali zed probability measures, called weights on a space (X,ξ).