Compact Metrics Research Articles

On countably saturated and pre-dyadic spaces

In this paper, we introduce a new class of spaces which extends the class of dyadic compacta. This permits to extend a classical theorem of Engelking and Pełczyński. Technically, our arguments are based on the concept of a countably saturated space. This is a space in which the closure of every countable subset is a metrizable compactum. The existence of a dense countably saturated subspace in compact Hausdorff spaces is shown below to play a crucial role.

On uniqueness and existence of conformally compact Einstein metrics with homogeneous conformal infinity. II

On uniqueness and existence of conformally compact Einstein metrics with homogeneous conformal infinity. II

The nonlinear stability of (n + 1)-dimensional FLRW spacetimes

We prove nonlinear Lyapunov stability of a family of “([Formula: see text])”-dimensional cosmological models of general relativity locally isometric to the Friedman–Lemaître–Robertson–Walker (FLRW) spacetimes including a positive cosmological constant. In particular, we show that the perturbed solutions to the Einstein–Euler field equations around a class of spatially compact FLRW metrics (for which the spatial slices are compact negative Einstein spaces in general and hyperbolic for the physically relevant [Formula: see text] case) arising from regular Cauchy data remain uniformly bounded and decay to a family of metrics with constant negative spatial scalar curvature. To accomplish this result, we employ an energy method for the coupled Einstein–Euler field equations in constant mean extrinsic curvature spatial harmonic (CMCSH) gauge. In order to handle Euler’s equations, we construct energy from a current that is similar to the one derived by Christodoulou [The Formation of Shocks in 3-dimensional Fluids, EMS Monographs in Mathematics, Vol. 2 (European Mathematical Society, 2007)] (and which coincides with Christodoulou’s current on the Minkowski space) and show that this energy controls the desired norm of the fluid degrees of freedom. The use of a fluid energy current together with the CMCSH gauge condition casts the Einstein–Euler field equations into a coupled elliptic–hyperbolic system. Utilizing the estimates derived from the elliptic equations, we first show that the gravity-fluid energy functional remains uniformly bounded in the expanding direction. Using this uniform boundedness property, we later obtain sharp decay estimates if a positive cosmological constant [Formula: see text] is included, which confirms that the accelerated expansion of the physical universe that is induced by the positive cosmological constant is sufficient to control the nonlinearities in the case of small data. A few physical consequences of this stability result are discussed.

Perturbation compactness and uniqueness for a class of conformally compact Einstein manifolds

Abstract In this paper, we establish compactness results for some classes of conformally compact Einstein metrics defined on manifolds of dimension d ≥ 4. In the special case when the manifold is the Euclidean ball with the unit sphere as the conformal infinity, the existence of such class of metrics has been established in the earlier work of C. R. Graham and J. Lee (“Einstein metrics with prescribed conformal infinity on the ball,” Adv. Math., vol. 87, no. 2, pp. 186–225, 1991). As an application of our compactness result, we derive the uniqueness of the Graham–Lee metrics. As a second application, we also derive some gap theorem, or equivalently, some results of non-existence CCE fill-ins.

Assessing drug safety by identifying the axis of arrhythmia in cardiomyocyte electrophysiology

Many classes of drugs can induce fatal cardiac arrhythmias by disrupting the electrophysiology of cardiomyocytes. Safety guidelines thus require all new drugs to be assessed for pro-arrhythmic risk prior to conducting human trials. The standard safety protocols primarily focus on drug blockade of the delayed-rectifier potassium current (IKr). Yet the risk is better assessed using four key ion currents (IKr, ICaL, INaL, IKs). We simulated 100,000 phenotypically diverse cardiomyocytes to identify the underlying relationship between the blockade of those currents and the emergence of ectopic beats in the action potential. We call that relationship the axis of arrhythmia. It serves as a yardstick for quantifying the arrhythmogenic risk of any drug from its profile of multi-channel block alone. We tested it on 109 drugs and found that it predicted the clinical risk labels with an accuracy of 88.1–90.8%. Pharmacologists can use our method to assess the safety of novel drugs without resorting to animal testing or unwieldy computer simulations.

Hyperspaces with a countable character of closed subsets

For a regular space X, the hyperspace (CL(X),τF) (resp., (CL(X),τV)) is the space of all nonempty closed subsets of X with the Fell topology (resp., Vietoris topology). In this paper, we give the characterization of the space X such that the hyperspace (CL(X),τF) (resp., (CL(X),τV)) with a countable character of closed subsets. We mainly prove that (CL(X),τF) has a countable character on each closed subset if and only if X is compact metrizable, and (CL(X),τF) has a countable character on each compact subset if and only if X is locally compact and separable metrizable. Moreover, we prove that (K(X),τV) have the compact-Gδ-property if and only if X have the compact-Gδ-property and every compact subset of X is metrizable.

A computationally efficient method for assessing the impact of an active viral cyber threat on a high-availability cluster

The field of computer science, like its sub-field of cyber threat modelling, is rapidly evolving. The prerequisites for key changes can be summarized as follows: cyber threats are evolving; there are leaks of special services tools; agile development methodology is being introduced everywhere; the boundaries of the object of protection are blurred; the scope of application of artificial intelligence is expanding; potentially vulnerable API integrations are increasingly being used. These factors lead to the fact that the processes of analysis of cyber threats, analysis of protective measures, generalization of data, and development of protective tools should now be considered continuous, not discrete. At the same time, the cost of cybersecurity increases like an avalanche in an attempt to avoid reputational and information losses. The only way to avoid this tendency is to apply a rational, scientific, accurate method of cognition to these processes. Thus, the creation of mathematical models of processes in the field of cybersecurity is now more relevant than ever. The article is devoted to the investigation of the process of the influence of an active viral cyber threat on a high-availability cluster in the paradigm of the provisions of the theory of Markov processes, graph theory and the theory of mathematical analysis. The main contribution of the research is a formalized computationally efficient method of approximate estimation of the average number of affected elements of the target high-availability cluster under the influence of an active viral cyber threat. Also, a criterion that allows estimating the trend of the quantitative parameter of the metric of the model of the studied process at medium and long time intervals is proposed. To obtain the declared scientific result, the authors: - formulated a Markov model of the process of the influence of an active viral cyber threat on a high availability cluster; - substantiated a compact metric for accurate assessment of the average number of cluster elements affected by an active viral cyber threat at any time; - formulated a computationally efficient method of approximate estimation of the parameter of the mentioned metric for the model of the target studied process; - proposed a criterion that allows researchers to evaluate the trend of the parameter of the mentioned metric for the model of the target researched process at medium and long intervals of time. The adequacy of the formulated method has been proven empirically.

The Heat Kernel on the Diagonal for a Compact Metric Graph

We analyze the heat kernel associated with the Laplacian on a compact metric graph, with standard Kirchhoff–Neumann vertex conditions. An explicit formula for the heat kernel as a sum over loops, developed by Roth and Kostrykin–Potthoff–Schrader, allows for a straightforward analysis of small-time asymptotics. We show that the restriction of the heat kernel to the diagonal satisfies a modified version of the heat equation. This observation leads to an “edge” heat trace formula, expressing the a sum over eigenfunction amplitudes on a single edge as a sum over closed loops containing that edge. The proof of this formula relies on a modified heat equation satisfied by the diagonal restriction of the heat kernel. Further study of this equation leads to explicit formulas for graphs which are symmetric about each vertex.

Static near-horizon geometries and rigidity of quasi-Einstein manifolds

Static vacuum near horizon geometries are solutions $(M,g,X)$ of a certain quasi-Einstein equation on a closed manifold $M$, where $g$ is a Riemannian metric and $X$ is a closed 1-form. It is known that when the cosmological constant vanishes, there is rigidity: $X$ vanishes and consequently $g$ is Ricci flat. We study this form of rigidity for all signs of the cosmological constant. It has been asserted that this rigidity also holds when the cosmological constant is negative, but we exhibit a counter-example. We show that for negative cosmological constant if $X$ does not vanish identically, it must be incompressible, have constant norm, and be nontrivial in cohomology, and $(M,g)$ must have constant scalar curvature and zero Euler characteristic. If the cosmological constant is positive, $X$ must be exact (and vanishing if $\dim M=2$). Our results apply more generally to a broad class of quasi-Einstein equations on closed manifolds. We extend some known results for quasi-Einstein metrics with exact 1-form $X$ to the closed $X$ case. We consider near horizon geometries for which the vacuum condition is relaxed somewhat to allow for the presence of a limited class of matter fields. An appendix contains a generalization of a result of Lucietti on the Yamabe type of quasi-Einstein compact metrics (with arbitrary $X$).

On uniqueness and existence of conformally compact Einstein metrics with homogeneous conformal infinity

In this paper we show that for a generalized Berger metric $${\hat{g}}$$ on $${\mathbb {S}}^3$$ close to the round metric, the conformally compact Einstein (CCE) manifold (M, g) with $$({\mathbb {S}}^3, [{\hat{g}}])$$ as its conformal infinity is unique up to isometries. For the high-dimensional case, we show that if $${\hat{g}}$$ is an $$\text {SU}(k+1)$$ -invariant metric on $${\mathbb {S}}^{2k+1}$$ for $$k\ge 1$$ , the non-positively curved CCE metric on the $$(2k+1)$$ -ball $$B_1(0)$$ with $$({\mathbb {S}}^{2k+1}, [{\hat{g}}])$$ as its conformal infinity is unique up to isometries. In particular, since in Li (Trans Amer Math Soc 369(6): 4385–4413, 2017), we proved that if the Yamabe constant of the conformal infinity $$Y({\mathbb {S}}^{2k+1}, [{\hat{g}}])$$ is close to that of the round sphere then any CCE manifold filled in must be negatively curved and simply connected, therefore if $${\hat{g}}$$ is an $$\text {SU}(k+1)$$ -invariant metric on $${\mathbb {S}}^{2k+1}$$ which is close to the round metric, the CCE metric filled in is unique up to isometries. Using the continuity method, we prove an existence result of the non-positively curved CCE metric with prescribed conformal infinity $$({\mathbb {S}}^{2k+1}, [{\hat{g}}])$$ when the metric $${\hat{g}}$$ is $$\text {SU}(k+1)$$ -invariant.

Li-Yorke chaos and topological distributional chaos in a sequence

We study here the topological notion of Li-Yorke chaos defined for uniformly continuous self-maps defined on uniform Hausdorff spaces, which are not necessarily compact metrizable. We prove that a weakly mixing uniformly continuous self-map defined on a second countable Baire uniform Hausdorff space without isolated points is Li-Yorke chaotic. Further, we define and study the notion of topological distributional chaos in a sequence for uniformly continuous self-maps defined on uniform Hausdorff spaces. We prove that Li-Yorke chaos is equivalent to topological distributional chaos in a sequence for uniformly continuous self-maps defined on second countable Baire uniform Hausdorff space without isolated points. As a consequence, we obtain that Devaney chaos implies topological distributional chaos in a sequence.

Critical metrics on 4-manifolds with harmonic anti-self dual Weyl tensor

In this paper, we study 4-dimensional simply connected, compact critical metric of the volume functional with harmonic anti-self dual Weyl tensor. We show that a 4-dimensional simply connected, compact critical metric of the volume functional with harmonic anti-self dual Weyl tensor and satisfying a suitable pinching condition is isometric to a geodesic ball in a simply connected space form R4, H4 or S4.

On boundedly compact metrics and UC metrics

Let $P_1$ and $P_2$ be potential properties of a metric for which each of $P_1 \wedge P_2, P_1 \wedge \neg P_2, \neg P_1 \wedge P_2$ and $\neg P_1 \wedge \neg P_2$ is possible. We call such a pair \textit{strongly independent} if whenever a metrizable space admits a metric for which either $P_1$ or $\neg P_1$ holds and separately a metric for which either $P_2$ or $\neg P_2$ holds, then there must exist a compatible metric for which both conditions at once hold. We show that boundedly compactness and UC-ness form a strongly independent pair and so do boundedness and UC-ness. Metrizable spaces that admit a boundedly compact metric when equipped with a non-UC metric have been recently studied with respect to the lineability of the non-uniformly continuous real-valued functions defined on them within $C(X)$ [4].

Compactness of conformally compact Einstein 4-manifolds II

In this paper, we establish a number of compactness results for some class of conformally compact Einstein 4-manifolds. In the first part of the paper, we improve the earlier compactness results obtained by Chang-Ge [15]. In the second part of the paper, as applications, we derive some further compactness results under the perturbation condition when the L2 norm of the Weyl curvature is small. We also derive the global uniqueness of conformally compact Einstein metrics on the 4-Ball constructed in the earlier work of Graham-Lee [31].

THE COMPLEXITY OF HOMEOMORPHISM RELATIONS ON SOME CLASSES OF COMPACTA

AbstractWe prove that the homeomorphism relation between compact spaces can be continuously reduced to the homeomorphism equivalence relation between absolute retracts, which strengthens and simplifies recent results of Chang and Gao, and Cieśla. It follows then that the homeomorphism relation of absolute retracts is Borel bireducible with the universal orbit equivalence relation. We also prove that the homeomorphism relation between regular continua is classifiable by countable structures and hence it is Borel bireducible with the universal orbit equivalence relation of the permutation group on a countable set. On the other hand we prove that the homeomorphism relation between rim-finite metrizable compacta is not classifiable by countable structures.

INDUCTIVE LIMITS OF C*-ALGEBRAS AND COMPACT QUANTUM METRIC SPACES

AbstractGiven a unital inductive limit of C*-algebras for which each C*-algebra of the inductive sequence comes equipped with a Rieffel compact quantum metric, we produce sufficient conditions to build a compact quantum metric on the inductive limit from the quantum metrics on the inductive sequence by utilizing the completeness of the dual Gromov–Hausdorff propinquity of Latrémolière on compact quantum metric spaces. This allows us to place new quantum metrics on all unital approximately finite-dimensional (AF) algebras that extend our previous work with Latrémolière on unital AF algebras with faithful tracial state. As a consequence, we produce a continuous image of the entire Fell topology on the ideal space of any unital AF algebra in the dual Gromov–Hausdorff propinquity topology.

Poincaré-Lovelock metrics on conformally compact manifolds

An important tool in the study of conformal geometry, and the AdS/CFT correspondence in physics, is the Fefferman-Graham expansion of conformally compact Einstein metrics. We show that conformally compact metrics satisfying a generalization of the Einstein equation, Poincaré-Lovelock metrics, also have Fefferman-Graham expansions. Moreover we show that conformal classes of metrics that are near that of the round metric on the n-sphere have fillings into the ball satisfying the Lovelock equation, extending the existence result of Graham-Lee for Einstein metrics.

Standard homogeneous C*-algebras as compact quantum metric spaces

Given a compact metric space $X$ and a unital C*-algebra $A$, we introduce a family of seminorms on the C*-algebra of continuous functions from $X$ to $A$, denoted by $C(X, A)$, induced by classical Lipschitz seminorms that produce compact quantum metrics

Some applications of conditional expectations to convergence for the quantum Gromov–Hausdorff propinquity

We prove that all the compact metric spaces are in the closure of the class of full matrix algebras for the quantum Gromov–Hausdorff propinquity. We also show that given an action of a compact metrizable group $G$ on a quasi-Leibniz quantum compact metric

Contraction, Lebesgue and Common Fixed Point Property of Fuzzy Metric Spaces

In this paper, we using contraction and contraction functions in complete fuzzy metric space and establish sequential characterization properties of Lebesgue fuzzy metric space and common fixed on it. Then first introduce a new type of Lebesgue fuzzy metric space, which is generalization of fuzzy metric space, second we study the topological properties of Lebesgue fuzzy metric space, third a relation between Lebesgue and weak G-complete, compact fuzzy metrics and Lebesgue integral mappings finally established characterization properties on it. We prove the existence of common fixed point and contraction mapping in fuzzy metric space using the property of Lebesgue fuzzy metric space and integral type of mappings. On the basis of these properties we are getting common fixed point of two mappings, three mappings and four mappings in a easy way as compared to old method like Banach contraction fixed point. Also coincidence fixed point theorem for two mapping, three mappings and four mappings using Lebesgue fuzzy metric space and integral type of mappings. Also contraction mappings property in fuzzy metric space is helpful to determine common fixed point in Lebesgue fuzzy metric space. We also discuss the Lebesgue property of several well-known fuzzy metric spaces in this paper and conclude uniqueness of common fixed point.