scholarly journals CONNES DISTANCE BY EXAMPLES: HOMOTHETIC SPECTRAL METRIC SPACES

2012 ◽  
Vol 24 (09) ◽  
pp. 1250027 ◽  
Author(s):  
JEAN-CHRISTOPHE WALLET
Keyword(s):  
Dirac Operator ◽  
Harmonic Oscillator ◽  
Infinite Number ◽  
Metric Spaces ◽  
Connected Components ◽  
Spectral Triples ◽  
Metric Properties ◽  
Spectral Distance ◽  
Physics Literature ◽  
Moyal Plane

We study metric properties stemming from the Connes spectral distance on three types of non-compact non-commutative spaces which have received attention recently from various viewpoints in the physics literature. These are the non-commutative Moyal plane, a family of harmonic Moyal spectral triples for which the Dirac operator squares to the harmonic oscillator Hamiltonian and a family of spectral triples with the Dirac operator related to the Landau operator. We show that these triples are homothetic spectral metric spaces, having an infinite number of distinct pathwise connected components. The homothetic factors linking the distances are related to determinants of effective Clifford metrics. We obtain, as a by-product, new examples of explicit spectral distance formulas. The results are discussed in detail.

scholarly journals Spectral distance on Lorentzian Moyal plane

2020 ◽  
Vol 17 (06) ◽  
pp. 2050089
Author(s):  
Anwesha Chakraborty ◽  
Biswajit Chakraborty
Keyword(s):  
Noncommutative Geometry ◽  
Coherent States ◽  
Euclidean Case ◽  
Spectral Triples ◽  
Spectral Distance ◽  
Moyal Plane ◽  
Pure States ◽  
Math Phys ◽  
Distance Formula

We present here a completely operatorial approach, using Hilbert–Schmidt operators, to compute spectral distances between time-like separated “events”, associated with the pure states of the algebra describing the Lorentzian Moyal plane, using the axiomatic framework given by [N. Franco, The Lorentzian distance formula in noncommutative geometry, J. Phys. Conf. Ser. 968(1) (2018) 012005; N. Franco, Temporal Lorentzian spectral triples, Rev. Math. Phys. 26(8) (2014) 1430007]. The result shows no deformations of non-commutative origin, as in the Euclidean case, if the pure states are constructed out of Glauber–Sudarshan coherent states.

scholarly journals Two classes of metric spaces

2016 ◽  
Vol 17 (1) ◽  
pp. 57 ◽  
Author(s):  
Isabel Garrido ◽  
Ana S. Meroño
Keyword(s):  
Metric Spaces ◽  
Lipschitz Functions ◽  
The Other ◽  
Metric Properties ◽  
Other Hand ◽  
Common Framework

<p>The class of metric spaces (X,d) known as small-determined spaces, introduced by Garrido and Jaramillo, are properly defined by means of some type of real-valued Lipschitz functions on X. On the other hand, B-simple metric spaces introduced by Hejcman are defined in terms of some kind of bornologies of bounded subsets of X. In this note we present a common framework where both classes of metric spaces can be studied which allows us to see not only the relationships between them but also to obtain new internal characterizations of these metric properties.</p>

On the metric dimension and diameter of circulant graphs with three jumps

2018 ◽  
Vol 10 (01) ◽  
pp. 1850008
Author(s):  
Muhammad Imran ◽  
A. Q. Baig ◽  
Saima Rashid ◽  
Andrea Semaničová-Feňovčíková
Keyword(s):  
Positive Integer ◽  
Connected Graph ◽  
Metric Spaces ◽  
Metric Dimension ◽  
Resolving Set ◽  
Circulant Graphs ◽  
Infinite Class ◽  
Metric Properties ◽  
Minimum Number ◽  
Metric Basis

Let [Formula: see text] be a connected graph and [Formula: see text] be the distance between the vertices [Formula: see text] and [Formula: see text] in [Formula: see text]. The diameter of [Formula: see text] is defined as [Formula: see text] and is denoted by [Formula: see text]. A subset of vertices [Formula: see text] is called a resolving set for [Formula: see text] if for every two distinct vertices [Formula: see text], there is a vertex [Formula: see text], [Formula: see text], such that [Formula: see text]. A resolving set containing the minimum number of vertices is called a metric basis for [Formula: see text] and the number of vertices in a metric basis is its metric dimension, denoted by [Formula: see text]. Metric dimension is a generalization of affine dimension to arbitrary metric spaces (provided a resolving set exists). Let [Formula: see text] be a family of connected graphs [Formula: see text] depending on [Formula: see text] as follows: the order [Formula: see text] and [Formula: see text]. If there exists a constant [Formula: see text] such that [Formula: see text] for every [Formula: see text] then we shall say that [Formula: see text] has bounded metric dimension, otherwise [Formula: see text] has unbounded metric dimension. If all graphs in [Formula: see text] have the same metric dimension, then [Formula: see text] is called a family of graphs with constant metric dimension. In this paper, we study the metric properties of an infinite class of circulant graphs with three generators denoted by [Formula: see text] for any positive integer [Formula: see text] and when [Formula: see text]. We compute the diameter and determine the exact value of the metric dimension of these circulant graphs.

scholarly journals Pseudo-Riemannian spectral triples and the harmonic oscillator

2013 ◽  
Vol 73 ◽  
pp. 37-55 ◽  
Author(s):  
Koen van den Dungen ◽  
Mario Paschke ◽  
Adam Rennie
Keyword(s):  
Harmonic Oscillator ◽  
Spectral Triples

scholarly journals Matrix geometries emergent from a point

2014 ◽  
Vol 26 (09) ◽  
pp. 1450017
Author(s):  
Francesco D'Andrea ◽  
Fedele Lizzi ◽  
Pierre Martinetti
Keyword(s):  
Equivalence Classes ◽  
Unitary Equivalence ◽  
Categorical Approach ◽  
Spectral Triples ◽  
Moyal Plane

We describe a categorical approach to finite noncommutative geometries. Objects in the category are spectral triples, rather than unitary equivalence classes as in other approaches. This enables us to treat fluctuations of the metric and unitary equivalences on the same footing, as representatives of particular morphisms in this category. We then show how a matrix geometry (Moyal plane) emerges as a fluctuation from one point, and discuss some geometric aspects of this space.

scholarly journals Metric spaces and ideals of functions

1989 ◽  
Vol 32 (3) ◽  
pp. 395-400
Author(s):  
R. Kaufman
Keyword(s):  
Banach Space ◽  
Metric Spaces ◽  
Compact Metric Spaces ◽  
Metric Properties ◽  
Maximal Ideals ◽  
Convergence Of Sequences ◽  
State Of Affairs ◽  
Convergence In Norm ◽  
Invertible Elements ◽  
Descriptive Set

In each metric space (X, d) there is defined the space Lip X of complex-valued, bounded, and uniformly Lipschitzian functions. In the algebra Lip X, it is natural to ask for ideals closed in various notions of convergence, and also to identify the invertible elements. In particular, are the invertible elements exactly those with no zero in X? Wiener's Tauberian Theorem in Fourier analysis is the first and most remarkable example of this harmonious state of affairs. A moment's reflection confirms that, for the algebra Lip X, this is true only for compact metric spaces X, the trivial examples in our investigation. We therefore introduce a type of convergence weaker than convergence in norm; it has already proved useful in some problems in descriptive set theory and reflects in a subtle way the metric properties of X. A sequence (fn) in Lip X converges strongly to g, written s – limfn=g, if ∥fn∥≦C in the Banach space Lip X and lim fn(x)=g(x) for each element x of X. In Section 3 we explain how this is really a type of convergence in the dual space of a certain Banach space . This brings us to the edge of some recondite questions about iterated (or even transfinite) limits, and we have adhered to the notion of strong limits to avoid these questions. To illustrate the differences between these two approaches, we mention this problem: which maximal ideals of Lip X are closed with respect to strong convergence of sequences? This is not the problem studied in Section 1.

scholarly journals AN AMBIGUOUS STATEMENT CALLED THE "TETRAD POSTULATE" AND THE CORRECT FIELD EQUATIONS SATISFIED BY THE TETRAD FIELDS

2005 ◽  
Vol 14 (12) ◽  
pp. 2095-2150 ◽  
Author(s):  
WALDYR A. RODRIGUES ◽  
QUINTINO A. G. SOUZA
Keyword(s):  
Wave Equation ◽  
Dirac Operator ◽  
Wave Equations ◽  
Point Of View ◽  
Field Equations ◽  
Hodge Laplacian ◽  
Clifford Bundle ◽  
Physics Literature ◽  
Laplacian Operators ◽  
Detailed Derivation

The names tetrad, tetrads, cotetrads have been used with many different meanings in the physics literature, not all of them equivalent from the mathematical point of view. In this paper, we introduce unambiguous definitions for each of those terms, and show how the old miscellanea made many authors introduce in their formalism an ambiguous statement called the "tetrad postulate," which has been the source of much misunderstanding, as we show explicitly by examining examples found in the literature. Since formulating Einstein's field equations intrinsically in terms of cotetrad fields θa, a = 0, 1, 2, 3 is a worthy enterprise, we derive the equation of motion of each θausing modern mathematical tools (the Clifford bundle formalism and the theory of the square of the Dirac operator). Indeed, we identify (giving all details and theorems) from the square of the Dirac operator some noticeable mathematical objects, namely, the Ricci, Einstein, covariant D'Alembertian and the Hodge Laplacian operators, which permit us to show that each θasatisfies a well-defined wave equation. Also, we present for completeness a detailed derivation of the cotetrad wave equations from a variational principle. We compare the cotetrad wave equation satisfied by each θawith some others appearing in the literature, and which are unfortunately in error.

scholarly journals Limits of multiplicative inhomogeneous random graphs and Lévy trees: limit theorems

2021 ◽  
Author(s):  
Nicolas Broutin ◽  
Thomas Duquesne ◽  
Minmin Wang
Keyword(s):  
Random Graphs ◽  
Metric Spaces ◽  
Companion Paper ◽  
Explicit Construction ◽  
Convergence Condition ◽  
Asymptotic Distributions ◽  
Connected Components ◽  
Inhomogeneous Random Graphs ◽  
The Masses ◽  
The Continuum

AbstractWe consider a natural model of inhomogeneous random graphs that extends the classical Erdős–Rényi graphs and shares a close connection with the multiplicative coalescence, as pointed out by Aldous (Ann Probab 25:812–854, 1997). In this model, the vertices are assigned weights that govern their tendency to form edges. It is by looking at the asymptotic distributions of the masses (sum of the weights) of the connected components of these graphs that Aldous and Limic (Electron J Probab 3:1–59, 1998) have identified the entrance boundary of the multiplicative coalescence, which is intimately related to the excursion lengths of certain Lévy-type processes. We, instead, look at the metric structure of these components and prove their Gromov–Hausdorff–Prokhorov convergence to a class of (random) compact measured metric spaces that have been introduced in a companion paper (Broutin et al. in Limits of multiplicative inhomogeneous random graphs and Lévy trees: the continuum graphs. arXiv:1804.05871, 2020). Our asymptotic regimes relate directly to the general convergence condition appearing in the work of Aldous and Limic. Our techniques provide a unified approach for this general “critical” regime, and relies upon two key ingredients: an encoding of the graph by some Lévy process as well as an embedding of its connected components into Galton–Watson forests. This embedding transfers asymptotically into an embedding of the limit objects into a forest of Lévy trees, which allows us to give an explicit construction of the limit objects from the excursions of the Lévy-type process. The mains results combined with the ones in the other paper allow us to extend and complement several previous results that had been obtained via model- or regime-specific proofs, for instance: the case of Erdős–Rényi random graphs obtained by Addario-Berry et al. (Probab Theory Relat Fields 152:367–406, 2012), the asymptotic homogeneous case as studied by Bhamidi et al. (Probab Theory Relat Fields 169:565–641, 2017), or the power-law case as considered by Bhamidi et al. (Probab Theory Relat Fields 170:387–474, 2018).

scholarly journals Refinements of topological invariants of flows

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Tomoo Yokoyama
Keyword(s):  
Metric Spaces ◽  
Orbit Space ◽  
Singular Points ◽  
Connected Components ◽  
Topological Spaces ◽  
Topological Invariants ◽  
Orbit Spaces ◽  
Compact Metric Spaces ◽  
Hamiltonian Flows ◽  
Closed Orbits

<p style='text-indent:20px;'>We construct topological invariants, called abstract weak orbit spaces, of flows and homeomorphisms on topological spaces. In particular, the abstract weak orbit spaces of flows on topological spaces are refinements of Morse graphs of flows on compact metric spaces, Reeb graphs of Hamiltonian flows with finitely many singular points on surfaces, and the CW decompositions which consist of the unstable manifolds of singular points for Morse flows on closed manifolds. Though the CW decomposition of a Morse flow is finite, the intersection of the unstable manifold and the stable manifold of closed orbits need not consist of finitely many connected components. Therefore we study the finiteness. Moreover, we consider when the time-one map reconstructs the topology of the original flow. We show that the orbit space of a Hamiltonian flow with finitely many singular points on a compact surface is homeomorphic to the abstract weak orbit space of the time-one map by taking an arbitrarily small reparametrization and that the abstract weak orbit spaces of a Morse flow on a compact manifold and the time-one map are homeomorphic. In addition, we state examples whose Morse graphs are singletons but whose abstract weak orbit spaces are not singletons.</p>

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