Compact Metrics Research Articles

Stability of Einstein metrics under Ricci flow

We prove dynamical stability and instability theorems for compact Einstein metrics under the Ricci flow. We give a nearly complete charactarization of dynamical stability and instability in terms of the conformal Yamabe invariant and the Laplace spectrum. In particular, we prove dynamical stability of some classes of Einstein manifolds for which it was previously not known. Additionally, we show that the complex projective space with the Fubini-Study metric is surprisingly dynamically unstable.

No-boundary prescriptions in Lorentzian quantum cosmology

We analyse the impact of various boundary conditions on the (minisuperspace) Lorentzian gravitational path integral. In particular we assess the implications for the Hartle-Hawking no-boundary wavefunction. It was shown recently that when this proposal is defined as a sum over compact metrics, problems arise with the stability of fluctuations. These difficulties can be overcome by an especially simple implementation of the no-boundary idea: namely to take the Einstein-Hilbert action at face value while adding no boundary term. This prescription simultaneously imposes an initial Neumann boundary condition for the scale factor of the universe and, for a Bianchi IX spacetime, Dirichlet conditions for the anisotropies. Another way to implement the no-boundary wavefunction is to use Robin boundary conditions. A sub-class of Robin conditions allows one to specify the Hubble rate on the boundary hypersurface, and we highlight the surprising aspect that specifying the final Hubble rate (rather than the final size of the universe) significantly alters the off-shell structure of the path integral. The conclusion of our investigations is that all current working examples of the no-boundary wavefunction force one to abandon the notion of a sum over compact and regular geometries, and point to the importance of an initial Euclidean momentum.

On products of quasi-F-compacta

A Hausdorff compact space is called a quasi-F-compactum if it admits a decomposition into a special well-ordered inverse system with almost fully closed neighboring projections. Each Fedorchuk compactum (or F-compactum) is a quasi-F-compactum. We prove that for any uncountable cardinal number λ there exists an F-compactum X of spectral height λ, all finite powers of which are F-compacta of the same spectral height. Thus, the analogue of the assertion on the anti-multiplicativity of the class of F-compacta of spectral height 3 is false for F-compacta of uncountable spectral height.The product of a quasi-F-compactum (F-compactum) onto a countable compactum is always a quasi-F-compactum (F-compactum). At the same time, the product of a quasi-F-compactum of the spectral height 3 to an uncountable metrizable compactum is never a quasi-F-compactum of a countable spectral height.We also prove a number of assertions about almost fully closed mappings of products of quasi-F-compacta to metrizable compacta.

Borel complexity up to the equivalence

We say that two classes of topological spaces are equivalent if each member of one class has a homeomorphic copy in the other class and vice versa. Usually when the Borel complexity of a class of metrizable compacta is considered, the class is realized as the subset of the hyperspace K([0,1]ω) containing all homeomorphic copies of members of the given class. We are rather interested in the lowest possible complexity among all equivalent realizations of the given class in the hyperspace.We recall that to every analytic subset of K([0,1]ω) there exists an equivalent Gδ subset. Then we show that up to the equivalence open subsets of the hyperspace K([0,1]ω) correspond to countably many classes of metrizable compacta. Finally we use the structure of open subsets up to equivalence to prove that to every Fσ subset of K([0,1]ω) there exists an equivalent closed subset.

Boundary regularity for asymptotically hyperbolic metrics with smooth Weyl curvature

In this paper, we study the regularity of asymptotically hyperbolic metrics with Einstein condition near boundary and Weyl curvature smooth enough in arbitrary dimension. Following Michael Anderson's method, we show that $C^{m,\alpha}$ conformally compact Riemannian metrics with Einstein equation vanishing to finite order near boundary have conformal compactifications that are $C^{m+2,\alpha}$ up to the boundary when Weyl curvature is in $C^{m,\alpha}$ and the boundary metric is in $C^{m+2,\alpha}$ where $m\geq 3.$

Compactifiable classes of compacta

We introduce the notion of compactifiable classes – these are classes of metrizable compact spaces that can be up to homeomorphic copies “disjointly combined” into one metrizable compact space. This is witnessed by so-called compact composition of the class. Analogously, we consider Polishable classes and Polish compositions. The question of compactifiability or Polishability of a class is related to hyperspaces. Strongly compactifiable and strongly Polishable classes may be characterized by the existence of a corresponding family in the hyperspace of all metrizable compacta. We systematically study the introduced notions – we give several characterizations, consider preservation under various constructions, and raise several questions.

Variational and Stability Properties of Constant Solutions to the NLS Equation on Compact Metric Graphs

We consider the nonlinear Schrodinger equation with pure power nonlinearity on a general compact metric graph, and in particular its stationary solutions with fixed mass. Since the the graph is compact, for every value of the mass there is a constant solution. Our scope is to analyze (in dependence of the mass) the variational properties of this solution, as a critical point of the energy functional: local and global minimality, and (orbital) stability. We consider both the subcritical regime and the critical one, in which the features of the graph become relevant. We describe how the above properties change according to the topology and the metric properties of the graph.

On uniqueness of conformally compact Einstein metrics with homogeneous conformal infinity

In this paper we show that for a Berger metric gˆ on S3, the non-positively curved conformally compact Einstein metric on the 4-ball B1(0) with (S3,[gˆ]) as its conformal infinity is unique up to isometries and it is the metric constructed by Pedersen [21]. In particular, since in [18], we proved that if the Yamabe constant of the conformal infinity Y(S3,[gˆ]) is close to that of the round sphere then any conformally compact Einstein manifold filled in must be negatively curved and simply connected, therefore if gˆ is a Berger metric on S3 with Y(S3,[gˆ]) close to that of the round metric, the conformally compact Einstein metric filled in is unique up to isometries.

Compactness of conformally compact Einstein manifolds in dimension 4

In this paper, we establish some compactness results of conformally compact Einstein metrics on 4-dimensional manifolds. Our results were proved under assumptions on the behavior of some local and non-local conformal invariants, on the compactness of the boundary metrics at the conformal infinity, and on the topology of the manifolds.

The BackMAP Python module: how a simpler Ramachandran number can simplify the life of a protein simulator.

Protein backbones occupy diverse conformations, but compact metrics to describe such conformations and transitions between them have been missing. This report re-introduces the Ramachandran number (ℛ) as a residue-level structural metric that could simply the life of anyone contending with large numbers of protein backbone conformations (e.g., ensembles from NMR and trajectories from simulations). Previously, the Ramachandran number (ℛ) was introduced using a complicated closed form, which made the Ramachandran number difficult to implement. This report discusses a much simpler closed form of ℛ that makes it much easier to calculate, thereby making it easy to implement. Additionally, this report discusses how ℛ dramatically reduces the dimensionality of the protein backbone, thereby making it ideal for simultaneously interrogating large numbers of protein structures. For example, 200 distinct conformations can easily be described in one graphic using ℛ (rather than 200 distinct Ramachandran plots). Finally, a new Python-based backbone analysis tool—BackMAP—is introduced, which reiterates how ℛ can be used as a simple and succinct descriptor of protein backbones and their dynamics.

Collapsing limits of the Kähler–Ricci flow and the continuity method

We consider the Kahler–Ricci flow on certain Calabi–Yau fibration, which is a Calabi–Yau fibration with one dimensional base or a product of two Calabi–Yau fibrations with one dimensional bases. Assume the Kahler–Ricci flow on total space admits a uniform lower bound for Ricci curvature, then the flow converges in Gromov–Hausdorff topology to the metric completion of the regular part of generalized Kahler–Einstein current on the base, which is a compact length metric space homeomorphic to the base. The analogue results for the continuity method on such Calabi–Yau fibrations are also obtained. Moreover, we show the continuity method starting from a suitable Kahler metric on the total space of a Fano fibration with one dimensional base converges in Gromov–Hausdorff topology to a compact metric on the base. During the proof, we show the metric completion of the regular part of a generalized Kahler–Einstein current on a Riemann surface is compact.

Not every metrizable compactum is the limit of an inverse sequence with simplicial bonding maps

It is shown that not every metrizable compactum can be written as the inverse limit of an inverse sequence of finite triangulated polyhedra with simplicial bonding maps. This result came from a correspondence between Sibe Mardešić and Leonard R. Rubin, who has provided the Introduction below, and who with some suggestions from Šime Ungar and Vera Tonić, has edited the proof that was communicated to him by Sibe Mardešić. It will be found as Corollary 2.2.

Interpolation for de-Dopplerisation

‘De-Dopplerisation’ is one aspect of a problem frequently encountered in experimental acoustics: deducing an emitted source signal from received data. It is necessary when source and receiver are in relative motion, and requires interpolation of the measured signal. This introduces error. In acoustics, typical current practice is to employ linear interpolation and reduce error by over-sampling. In other applications, more advanced approaches with better performance have been developed. Associated with this work is a large body of theoretical analysis, much of which is highly specialised. Nonetheless, a simple and compact performance metric is available: the Fourier transform of the ‘kernel’ function underlying the interpolation method. Furthermore, in the acoustics context, it is a more appropriate indicator than other, more abstract, candidates. On this basis, interpolators from three families previously identified as promising —— piecewise-polynomial, windowed-sinc, and B-spline-based —— are compared. The results show that significant improvements over linear interpolation can straightforwardly be obtained. The recommended approach is B-spline-based interpolation, which performs best irrespective of accuracy specification. Its only drawback is a pre-filtering requirement, which represents an additional implementation cost compared to other methods. If this cost is unacceptable, and aliasing errors (on re-sampling) up to approximately 1% can be tolerated, a family of piecewise-cubic interpolators provides the best alternative.

Convergence of Continuous Stochastic Processes on Compact Metric Spaces Converging in the Lipschitz Distance

We introduce a new distance, a Lipschitz-Prokhorov distance $d_{LP}$, on the set $\mathcal {PM}$ of isomorphism classes of pairs $(X, P)$ where $X$ is a compact metric space and $P$ is the law of a continuous stochastic process on $X$. We show that $(\mathcal {PM}, d_{LP})$ is a complete metric space. For Markov processes on Riemannian manifolds, we study relative compactness and convergence.

A Framework for Software Defect Prediction and Metric Selection

Automated software defect prediction is an important and fundamental activity in the domain of software development. However, modern software systems are inherently large and complex with numerous correlated metrics that capture different aspects of the software components. This large number of correlated metrics makes building a software defect prediction model very complex. Thus, identifying and selecting a subset of metrics that enhance the software defect prediction method’s performance are an important but challenging problem that has received little attention in the literature. The main objective of this paper is to identify significant software metrics, to build and evaluate an automated software defect prediction model. We propose two novel hybrid software defect prediction models to identify the significant attributes (metrics) using a combination of wrapper and filter techniques. The novelty of our approach is that it embeds the metric selection and training processes of software defect prediction as a single process while reducing the measurement overhead significantly. Different wrapper approaches were combined, including SVM and ANN, with a maximum relevance filter approach to find the significant metrics. A filter score was injected into the wrapper selection process in the proposed approaches to direct the search process efficiently to identify significant metrics. Experimental results with real defect-prone software data sets show that the proposed hybrid approaches achieve significantly compact metrics (i.e., selecting the most significant metrics) with high prediction accuracy compared with conventional wrapper or filter approaches. The performance of the proposed framework has also been verified using a statistical multivariate quality control process using multivariate exponentially weighted moving average. The proposed framework demonstrates that the hybrid heuristic can guide the metric selection process in a computationally efficient way by integrating the intrinsic characteristics from the filters into the wrapper and using the advantages of both the filter and wrapper approaches.

Gap results for compact quasi-Einstein metrics

In this paper, we work on compact quasi-Einstein metrics and prove several gap results. In the first part, we get a gap estimate for the first nonzero eigenvalue of the weighted Laplacian, by establishing a comparison theorem for the weighted heat kernel. In the second part, we establish two gap results for the Ricci curvature and the scalar curvature, based on which some rigid properties can be derived.

The paucity of universal compacta in cohomological dimension

Let C be a class of spaces. An element Z ∈ C is called universal for C if each element of C embeds in Z . It is well-known that for each n ∈ N , there exists a universal element for the class of metrizable compacta X of (covering) dimension dim ⁡ X ≤ n . The situation in cohomological dimension over an abelian group G , denoted dim G , is almost the opposite. Our results will imply in contradistinction that for each nontrivial abelian group G and for n ≥ 2 , there exists no universal element for the class of metrizable compacta X with dim G ⁡ X ≤ n .

Suprema of continuous functions on connected spaces

The technique of extension of compact spaces by means of adding suprema of continuous functions was introduced by Koszmider in the construction of an indecomposable Banach space of the form C(K). Generally, the behaviour of such extensions with respect to connectedness is delicate. We provide conditions under which any extension is connected, as well as showing that connectedness cannot be preserved indefinitely, i.e., that for every metrizable compactum K there exists a disconnected space L which is obtained by finitely many extensions starting with K.

Uniform Continuity of Continuous Functions on Compact Metric Spaces

A basic theorem asserts that a continuous function on a compact metric space with values in another metric space is uniformly continuous. The usual proofs based on a contradiction argument involving sequences or on the covering property of compact sets are quite sophisticated for students taking a first course on real analysis. We present a direct proof only using results that are established anyway in such an introductory course. Let f ∶ X → Y be a continuous function from the compact metric space (X, dX) into the metric space (Y , dY ). The function F ∶ X ×X → R given by F (x, y) ∶= dY (

G-fibrations and twisted products

We prove that the functor of twisted product G×H− takes H-fibrations to G-fibrations when G is a compact metrizable (not necessarily Lie) group and H is its closed subgroup. This result is applied to the study of strong G-fibrations. In particular, we show that every G-map E→G/H is a strong G-fibration provided that E is a G-fibrant space.