Metric Geometry Research Articles

Useful topologies and separable systems

<p>Let X be an arbitrary set. A topology <em>t</em> on X is said to be useful if every continuous linear preorder on X is representable by a continuous real valued order preserving function. Continuous linear preorders on X are induced by certain families of open subsets of X that are called (linear) separable systems on X. Therefore, in a first step useful topologies on X will be characterized by means of (linear) separable systems on X. Then, in a second step particular topologies on X are studied that do not allow the construction of (linear) separable systems on X that correspond to non representable continuous linear preorders. In this way generalizations of the Eilenberg Debreu theorems which state that second countable or separable and connected topologies on X are useful and of the theorem of Estévez and Hervés which states that a metrizable topology on X is useful, if and only if it is second countable can be proved.</p><p> </p>

Extending Baire Property by countably many sets

We prove that if ZFC is consistent so is ZFC + any sequence (An) of subsets of a Polish space (X, r) there exists a separable metrizable topology T' on X with B (X, T) C B (X, T'), MGR(X, T' ) n B (X, T) = MGR(X, T) n B(X, T) and An Borel in T' for all n. This is a category analogue of a theorem of Carlson on the possibility of extending Lebesgue measure to any countable collection of sets. A uniform argument is presented, which gives a new proof of the latter as well. Some consequences of these extension properties are also studied.

Caractérisations de certains espaces de suites de réels

Here we give a characterization for topological properties of subspaces of the product space ℝ N of sequences of reals. For analytic ideals, i.e. analytic vector subspaces X ⊆ ℝ N which verify: [xX and ∀n, ¦ y(n)¦≤ ¦x(n)¦] ⇒ yX, we have the following dichotomy: X admits a polish vector space topology stronger than the product topology of ℝ N or one can embed into X in a strong sense the space of finite sequences c 00 or the space ℓ ∞. If X admits such a Polish topology we find special functions which define this topology and have a simple form; we name them “evaluations functions” and with this notion we specify the descriptive complexity of X and we give some properties of ideals which only admit a complete metrizable topology.

Random Partitions by Semigroup Methods

of all partitions of {1,2,3,...}, or rather its distribution. There is a natural compact metrizable topology on P taking care of measurability questions. With respect to the maximum operation P becomes an abelian semigroup, and our first theorem characterizes random partitions as normalized positive definite functions on the subsemigroup of partitions "with finite support". We then present a new proof of Kingman's theorem stating that the exchangeable random partitions form a simplex whose extreme points are the so-called "paint-box distributions". An interesting moment problem which is still open arises in this connection.

Ideal Spaces of Banach Algebras

The ideal space is either a compact, metrizable topology, or it is neither Hausdorff nor first countable. TAF-algebras are shown to exhibit the first type of behaviour. Applications to Banach bundles (which motivate the study), and to PI-Banach algebras, are given.

Basic intervals in the partial order of metrizable topologies

For a set X, let Σ m ( X) denote the set of metrizable topologies on X, partially ordered by inclusion. We investigate the nature of intervals in this partial order, with particular emphasis on basic intervals (in other words, intervals in which the topology changes at at most one point). We show that there are no nontrivial finite intervals in Σ m ( X) (indeed, every such interval contains a copy of P (ω)/fin). We show that although not all intervals in Σ m ( X) are lattices, all basic intervals in Σ m are lattices. In the case where X is countable, we show that there are at least two isomorphism classes of basic intervals in Σ m ( X), and assuming the Continuum Hypothesis there are exactly two such isomorphism classes.

Nouvelles Phonologies

Along with an introduction to the present issue of Langages, the paper offers an analysis of the contemporary field of phonology. From the seventies up to now, different phonological approaches and frameworks are examined : autosegmental and metrical phonology, natural, lexical, dependency, government and geometry feature models are analysed as are more contemporary theories such as connectionism, harmonic phonology and optimality theory. The configurational and dynamical hypotheses that organise current phonological research are presented and discussed.

Estimates of the total gravitational radiation in the head-on black hole collision.

We report on calculations of the total gravitational energy radiated in the head-on black hole collision, where we use the geometry of the Robinson-Trautman metrics.

Multiple-Edge XAS Studies of Synthetic Iron-Copper Bridged Molecular Assemblies Relevant to Cytochrome c Oxidase. Structure Determination Using Multiple-Scattering Analysis with Statistical Evaluation of Errors.

An X-ray absorption spectroscopy study has been carried out at the Fe and Cu K-edges for two bridged molecular assemblies, both of which contain an Fe-X-Cu (X = O(2)(-), OH(-)) bridge unit, some of whose features are relevant to the binuclear site of cytochrome c oxidase. The two complexes [(OEP)Fe-O-Cu(Me(6)tren)](1+) and [(OEP)Fe-(OH)-Cu(Me(5)tren)(OClO(3))](1+) have similar structural fragments around the metal centers except that they differ significantly in the bridge structure (the former contains a linear oxo bridge while the latter has a bent hydroxo bridge). We report a comparative study of these complexes using multiple-scattering (MS) EXAFS analysis and the program package GNXAS. It is found that there is a dramatic increase in the amplitude of the Fe-X-Cu MS pathway as the bridge unit approaches linearity. Full EXAFS MS analysis enables accurate quantitation of bridge metrical details and geometry for both complexes. These studies were done with an expanded version of GNXAS, which allows for simultaneous multiple-edge fitting. Such multiple-edge analysis (using both Fe and Cu edge data) allows common pathways (in this case involving the Fe-X-Cu bridge) to be constrained to be the same, thus improving the observation/variable ratio and enhancing sensitivity for determination of the bridge structure. The accuracy of the structural determination for the bridge units is evaluated by a statistical analysis methodology in which correlations among fitting parameters are identified and contour plots are used to determine random error. The overall error in the EXAFS structural determination is found by establishing the variance with the crystallographically determined values: for the EXAFS-determined parameters at distances below 4 Å, distances and angles deviated on average from crystallographic values by 0.014 Å and 1.5 degrees, respectively. It is also established that structural features in the Fe absorption preedge are diagnostic of oxo vs hydroxo ligation. The relevance of this study to the structural definition of binuclear bridged sites in cytochrome c oxidase and other metalloenzymes is considered.

Recursive Dynamic Programming: Heuristic Rules, Bounding and State Space Reduction

An alternative approach towards dynamic programming (DP) is presented: Recursions. A basic deterministic model and solution function are defined and the model is generalized to include stochastic processes, with the traditional stochastic DP model as a special case. Heuristic rules are included in the models simply as restrictions on decision spaces and a small slaughter pig marketing planning example illustrates the potential state space reductions induces by such rules.

Towards an Open and Bounded Space

Towards an Open and Bounded Space

Connectivity of Knight's Graphs

This paper is concerned with connectivity of graphs associated with generalized knight's moves. The minimal connected generalized knight's graphs are shown to be planar or toroidal, which was unexpected in view of the complex nature of the graph generated by knight's moves on a chess-board. Each move of a knight in chess takes the knight two steps parallel to one side of the board and one step parallel to the other side. Using moves of this kind a knight can move between any two positions on a standard 8 by 8 chess-board. Indeed it can move between any two positions on a 3 by 4 board, but not on any smaller board. A metrical geometry on the digital plane determined by standard knight's moves has been studied recently [1,6], and further metrics associated with generalized knight's moves have been noted [2]. The first step is to determine which knight's moves are transitive on an unbounded board. It is then natural to try to determine for each of these the smallest board on which it is transitive. It is convenient to express the problem in terms of knight's graphs in which the vertices are points in the plane with integer coefficients and the edges correspond to the knight's moves. An edge in the graph of a {p, ^}-knight joins two vertices where the difference between one of their coordinates is p and the difference between the other is q. Transitivity of a knight's moves on a board corresponds to connectivity of a knight's graph. The problem on the unrestricted set of vertices is easily solved. It is shown in §2 that the {p, ^}-knight's graph on 2 is connected if and only if p and q are mutually prime and their sum is odd. This is a special case of more general results on knight's graphs on Z which have been studied by G. A. Jones [3]. The problem on restricted sets of vertices is more difficult. In this paper it is shown that if the {p, q)-knight's graph is connected on 2 and p<q then the {p, q)-knight's graph is connected on a board of size X by Y if and only if minfA', Y}^p + q and max{X, Y} 5= 2 max{/?, q). The necessity of the condition is established in § 3. For {1, 2n}- and {n, n + l}-knight's graphs the sufficiency of the condition follows from an analysis of the connected components of these graphs on 2n + 1 by 2n + 1 blocks of vertices. This is established in § 4. In order to deal with the other cases we introduce in § 5 a cyclic ordering of the lines, and cyclic orderings of the odd and even vertices within each line. It is easily seen that most vertices on a line are connected in the graph to the next vertex in order on that line. The sufficiency of the condition is proved in § 6 for the case where q 5= 2p — 1 and in § 7 for the case wherep + Kq <2p - 1. The proofs depend on showing that on certain lines the other vertices are also connected in the graph to the next vertex in order on that line, and that this implies that the whole graph is connected. The cyclic orderings of the lines and of vertices within each line are valid when p = 1 and when q = p + 1, but the calculations in § 6 and § 7 break

The EPI-Distance Topology: Continuity and Stability Results with Applications to Convex Optimization Problems

Let Γ(X) denote the proper, lower semicontinuous, convex functions on a Banach space X, equipped with the completely metrizable topology of uniform convergence of distance functions on bounded sets. We show, subject to some standard constraint qualifications, that the operations of addition and restriction are continuous. These results are applied to convex well-posed optimization problems, as well as to convergence of approximated solutions in infinite dimensional convex programming, to linear functionals exposing convex sets, and to metric projections. Also, for any given function in Γ(X), we obtain results regarding the convergence of its inf-convolution with smoothing kernels.

The EPI-Distance Topology

Let I X denote the proper, lower semicontinuous, convex functions on a Banach space X , equipped with the completely metrizable topology of uniform convergence of distance functions on bounded sets...

Convex Optimization and the Epi-Distance Topology

Let $\Gamma (X)$ denote the proper, lower semicontinuous, convex functions on a Banach space $X$, equipped with the completely metrizable topology $\tau$ of uniform convergence of distance functions on bounded sets. A function $f$ in $\Gamma (X)$ is called well-posed provided it has a unique minimizer, and each minimizing sequence converges to this minimizer. We show that well-posedness of $f \in \Gamma (X)$ is the minimal condition that guarantees strong convergence of approximate minima of $\tau$-approximating functions to the minimum of $f$. Moreover, we show that most functions in $\langle \Gamma (X),{\tau _{aw}}\rangle$ are well-posed, and that this fails if $\Gamma (X)$ is topologized by the weaker topology of Mosco convergence, whenever $X$ is infinite dimensional. Applications to metric projections are also given, including a fundamental characterization of approximative compactness.

The algebra of piecewise differentiable currents on smooth manifolds

A deRham current on a smooth manifold is called piecewise differentiable if its distributional components are partial derivatives of smooth densities supported on closed embedded smooth simplices. Such currents arise naturally in the differential geometry of piecewise differentiable Riemannian metrics. The main purpose of this paper is to construct a natural graded differential algebra containing the piecewise differentiable currents as a subcomplex, and to show that the inclusion of the complex of deRham forms induces monomorphism in homology. This result enables us to show, in the next note, that the Chern-Weil construction can still be used to extract the characteristic classes from (discontinuous) piecewise differentiable metrics on smooth manifolds.

Dynamical invariants of rigid motions on the hyperbolic plane

This paper is devoted to the investigation of the geometry of left invariant metrics on the isometry group of the hyperbolic plane determined by the kinetic energy of a rigid particle system.

Essentially unbounded chains in compact sets

We discuss the possible cofinalities of ⊆ *-chains in ℕ which are ⊆ *-unbounded in their topological closures.1. The problem. Throughout this paper, the power set ℕ of the set ℕ of natural numbers will be given its usual compact metrizable topology corresponding to the product topology of {0,1}N. We say that a ⊆ * b if a/b is finite.

Natural convection from point source embedded in Darcian porous medium

The buoyancy induced convection arising from a point heat source in a fluid saturated Darcy porous medium bounded by an adiabatic conical surface, of apex angle 2 θ ω , has been studied. The point heat source is located at the apex and the axis of the conical surface coincides with the direction of bouyant force vector. The self-similar equations are obtained for 0 < θ ω < π and Rayleigh number Ra from zero to infinity. The numerical solutions of self-similar equations have been obtained for θ ω (free plume), 3π,/4, π/2 (plume bounded by horizontal surface), and π/4. It is shown that for a given Ra, the centre line velocity and temperature increases as π ω (the bounding space) decreases. A perturbation solution for Ra → 0 also been studied to present closed form solution.

Quantization in curvilinear coordinates

Two prescriptions often used to find Hermitian operators corresponding to classical quantities can be removed. Components of momentum are of three types, linear momentumP〈q〉k canonical momentumP(q)k and generalized momentumPqk. Using metrical geometry, their mutual relations are established. The operatorsP〈q〉k andP(q)k are given by substituting quantum commutation brackets for classical Poisson brackets. The relations among classical quantities are divided into two types according to whether they have physical meaning. Those which have physical meaning go over into the corresponding operator relations.