Showing papers on "Space (mathematics) published in 1969"

Indices of Function Spaces and their Relationship to Interpolation

Abstract: A special case of the theorem of Marcinkiewicz states that if T is a linear operator which satisfies the weak-type conditions (p, p) and (q,q), then T maps Lr continuously into itself for any r with p < r < q. In a recent paper (5), as part of a more general theorem, Calderόn has characterized the spaces X which can replace Lr in the conclusion of this theorem, independent of the operator T. The conditions which X must satisfy are phrased in terms of an operator S(σ) which acts on the rearrangements of the functions in X. One of Calderόn's results implies that if X is a function space in the sense of Luxemburg (9), then X must be a rearrangement-invariant space.

BEST METHODS OF APPROXIMATION AND INTERPOLATION OF DIFFERENTIABLE FUNCTIONS IN THE SPACE $ C_{\lbrack-1,\,1\rbrack}$

Abstract: In this paper are computed the -diameter of the class of functions defined on in .This problem reduces to the variational problem ???whose solution is described in Theorem 1 of the paper.Bibliography 6 items.

Studies in visual perception of architectural spaces and rooms

Abstract: Garling, T. Studies in visual perception of architectural spaces and rooms. I. Judgment scales of open and closed space. Scand. J. Psychol., 1969, io, 250–256.—Eight observers judged open and closed space, viewing spaces from different spots along a street in a small town. The spaces were viewed and judged either as wholes or parts by using a ratio scaling procedure. The observers were found to make reliable judgments of closed and open space. The function relating these judgment to each other approximated, however, a complementary relation as expected for category ratings. Judgments of whole spaces might be predicted from averaged judgments of their parts.

Cell-like spaces

Abstract: In this note we give a characterization of those compacta which can be embedded in manifolds as cellular sets (the cell-like spaces). There are three conditions equivalent to cell likeness for a finitedimensional compactum X. One of these is that X have the Cechhomotopy-type of a point, as defined by Borsuk in [1]. Another is a technical condition which implies that McMillan's cellularity criterion [5] holds, not just for compact absolute retracts but for arbitrary cell-like spaces. It follows that most of the theorems of [5] hold for cell-like subsets of manifolds. DEFINITION. A subset X of the n-manifold N is said to be cellular in N if there exists a sequence Ql, Q2, * * * of topological n-cells in N, with XCQi+,CInt Qi for each i, such that x=n 1 Qi. Clearly cellularity is a property of the embedding of X in N. However, there is the corresponding intrinsic topological property, as follows: DEFINITION. A space X is cell-like if there is an embedding f of X into some manifold N such that f(X) is cellular in N. Cellularity was first defined by Brown in [3]. Since then, the idea has been important in the study of manifolds, and hence the problem of recognizing cellular sets has also been important. By far the best recognition criterion was given by McMillan in [5]. There he proved that, if X is a compact absolute retract in the interior of the combinatorial n-manifold N, n > 5, then X is cellular in N provided that the inclusion XCN has property (*) below.

Remarks on singularities of finite codimension in complex dynamical systems

Abstract: There is an interesting connection between the theory of algebraic functions and Artin’s braid theory: the space G n of nth-degree polynomials not having multiple roots is the space K(π ,1) for the group B(n) of braids on n strands:

A metrization theorem for linearly orderable spaces

Abstract: A topological space X is linearly orderable if there is a linear ordering of the set X whose open interval topology coincides with the topology of X It is known that if a linearly orderable space is semimetrizable then it is, in fact, metrizable [1] We will use this fact to give a particularly simple metrization theorem for linearly orderable spaces, namely that a linearly orderable space is metrizable if and only if it has a Gs diagonal This is an interesting analogue of the well-known metrization theorem which' states that a compact Hausdorff space is metrizable if it has a G5 diagonal

Metric structures and chiral symmetries

Abstract: Various aspects of nonlinear-group realizations are discussed and it is shown that the nonlinearSU3×SU3 chiral-invariant meson Lagrangian, and hence the meson-meson low-energy processes, can be expressed completely in terms of the canonical metric on the group space ofSU3.

Realcompactifications and Normal Bases

Abstract: In a recent paper (see [2]), Orrin Frink introduced a method to provide Hausdorff compactifications for Tychonoff or completely regular T1 spaces X. His method utilized the notion of a normal base. A normal base ℒ for the closed sets of a space X is a base which is a disjunctive ring of sets, disjoint members of which may be separated by disjoint complements of members of ℒ.

The discrete countable chain condition

Abstract: able. It is well known that a separable space satisfies the countable chain condition, and it is easy to find an example to show that the converse fails [6, p. 60]. That CCC implies DCCC is obvious from the definitions. An example of E. Michael [7] gives us a topological space that satisfies tl-he DCCC but not the CCC. This space is Lindelbf, hence satisfies the DCCC (Theorem 2.1), but has an uncountable subset with the discrete topology, hence does not satisfy the CCC.

Quasiregular two-distance sets

Abstract: Schoenberg investigated the sets of n + 2 points in Euclidean n -space R n for which the ratio of the largest and the smallest of its distances attains a minimum d n . He called these sets quasiregular, or most nearly equilateral. Schoenberg gave bounds for d n and conjectured that all quasiregular sets were two-distance sets. For any n , there exists a unique two-distance set of n + 2 points in R n for which the ratio >1 of its two distances attains a minimum. This chapter presents the application of linear algebra to the ( – 1, 1, 0) adjacency matrices of graphs.

Proof of the Symmetrization Postulate

Abstract: It is proved within the framework of nonrelativistic quantum mechanics that identical particles are either boson or fermions. The starting assumptions are: (a) if ψ(x1 … xn) is in the space H of allowed states, then so is Pψ for every permutation P; (b) |Pψ|2 = |ψ|2 for all ψ∈H, all allowed configurations (x1 … xn), and all ψ∈H; (c) H is a vector space (principle of superposition); (d) every ψ∈H is continuous along every path in the n‐particle configuration space C; and (e) there is at least one physical observable connecting each pair of irreducible components of C.

Symmetries of type N empty space-times possessing twisting geodesic rays

Abstract: It is shown that if the geodesic ray congruence associated with an empty space-time of type N is not hypersurface orthogonal (that is the geodesic rays are twisting) then the space-time has at most one symmetry.