Metric Spaces

We introduce the Symplectic Structure of Information Geometry based on Souriau's Lie Group Thermodynamics model, with a covariant definition of Gibbs equilibrium via invariances through co-adjoint action of a group on its momentum space, defining physical observables like energy, heat, and momentum as pure geometrical objects. Using Geometric (Planck) Temperature of Souriau model and Symplectic cocycle notion, the Fisher metric is identified as a Souriau Geometric Heat Capacity. Finally, in the framework of Lie Group Thermodynamics, an Euler-Poincaré equation is elaborated with respect to thermodynamic variables. We conclude with a new variational principal for thermodynamics built through an invariant Poincaré-Cartan-Souriau integral.

– Maurice Fréchet a introduit en 1939, la borne inférieure de la variance de tout estimateur statistique, donnée par l'inverse de la matrice de Fisher, cette dernière définissant la métrique de la « Géométrie de l'Information ». Le mathématicien Jean-Louis Koszul a montré que cette métrique pouvait être définie de façon plus générale comme le hessien de l'Entropie (transformée de Legendre du logarithme de la fonction caractéristique). De façon analogue, le physicien Roger Balian a élaboré une métrique de Fisher quantique comme le hessien de l'Entropie de von Neumann. Enfin, le physicien Jean-Marie Souriau a donné une nouvelle version covariante de cette métrique de Fisher pour les groupes dynamiques, où elle apparaît comme une capacité thermique géométrique via le cocycle symplectique lié au groupe et une température géométrique. Abstract-Maurice Fréchet introduced in 1939, the lower bound of any statistical estimator variance, given by the inverse of the Fisher matrix, the latter defining the " Information Geometry" metric. The mathematician Jean-Louis Koszul showed that this metric could be defined in a more general way as the hessian of the Entropy (Legendre transform of the characteristic function logarithm). In a similar way, the physicist Roger Balian developed a quantum Fisher metric as the hessian of the von Neumann Entropy. Finally, the physicist, Jean-Marie Souriau gave a new covariant version of this Fisher metric for the dynamic groups, where it appears as a geometrical heat capacity, via the symplectic cocycle linked to the group and a geometrical temperature.

In this paper we study on contribution of fixed point theorem in Metric spaces and Quasi Metric spaces. Definition: 1 (Metric Space) Let X be a non-empty set-A function XxX →R (the set of reals) such that p:XxX→R is called a metric or distance function (if ad only if) p satisfies the following conditions. (i) p (x,y) ≥ 0 for all x, y x (ii) p (x,y)= 0 if x=y (iii) p (x,y) = p (y,x) for all x,y x (iv) p (x,y) ≥ p (x,z)+ p (z ,y) for all x,y,z X If p is a metric for X, then the pair (X, p) is called a metric space. Definition: 2 (Cauchy sequence) Let (X,p) be a metric space .Then a sequence { xn} of points of X is said to be a Cauchy sequence if for each >0,there exists a positive integer n0 such that m,nn0 implies p(xm,xn) < . Definition: 3 (Completeness): A metric space(X,p) is said to be complete if every Cauchy sequence in X converges to point of X. Definition: 4 (Contraction mapping) Let (X, p) be a complete metric space. A mapping T:XX is said to be a contraction mapping if there exists a real number  with 0 << 1 such that, p{T(x) ,T(y)})  p (x,y) <p(x,y)  x, y,  X i.e In a contraction mapping, the distance between the images of any two points is less than the distance between the points.

The François Massieu 1869 idea to derive some mechanical and thermal properties of physical systems from " Characteristic Functions " , was developed by Gibbs and Duhem in thermodynamics with the concept of potentials, and introduced by Poincaré in probability. This paper deals with generalization of this Characteristic Function concept by Jean-Louis Koszul in Mathematics and by Jean-Marie Souriau in Statistical Physics. The Koszul-Vinberg Characteristic Function (KVCF) on convex cones will be presented as cornerstone of " Information Geometry " theory, defining Koszul Entropy as Legendre transform of minus the logarithm of KVCF, and Fisher Information Metrics as hessian of these dual functions, invariant by their automorphisms. In parallel, Souriau has extended the Characteristic Function in Statistical Physics looking for other kinds of invariances through co-adjoint action of a group on its momentum space, defining physical observables like energy, heat and momentum as pure geometrical objects. In covariant Souriau model, Gibbs equilibriums states are indexed by a geometric parameter, the Geometric (Planck) Temperature, with values in the Lie algebra of the dynamical Galileo/Poincaré groups, interpreted as a space-time vector, giving to the metric tensor a null Lie derivative. Fisher Information metric appears as the opposite of the derivative of Mean " Moment map " by geometric temperature, equivalent to a Geometric Capacity or Specific Heat. These elements has been developed by author in [10][11]. KOSZUL CHARACTERISTIC FUNCTION/ENTROPY BY LEGENDRE DUALITY We define Koszul-Vinberg Hessian metric on convex sharp cone, and observe that the Fisher information metric of Information Geometry coincides with the canonical Koszul Hessian metric (given by Koszul forms). We also observe, by Legendre duality (Legendre transform of minus Koszul characteristic function logarithm), that we are able to introduce a Koszul Entropy, that plays the role of generalized Shannon Entropy. Koszul [1][2][3][4] and E. Vinberg have introduced an affinely invariant hessian metric on a sharp convex cone Ω  through its characteristic function ψ. In the following, Ω is a sharp open convex cone in a vector space E of finite dimension on R (a convex cone is sharp if it does not contain any full straight line). In dual space *

—Automation of Enterprise Information Systems has resulted in several information security issues. There is a need to devise ways of measuring information security. Existing techniques mostly concentrate on finding ways of measuring specific attributes of security devices. This paper is an initial step towards the development of a formal methodology for measuring enterprise information system security. The proposed technique may also be used to compare the relative security of information systems.

This paper considers whether an analogy between distance and dissimilarlity supports the thesis that degree of dissimilarity is distance in a metric space. A straightforward way to justify the thesis would be to define degree of dissimilarity as a function of number of properties in common and not in common. But, infamously, this approach has problems with infinity. An alternative approach would be to prove representation and uniqueness theorems, according to which if comparative dissimilarity meets certain qualitative conditions, then it is representable by distance in a metric space. I will argue that this approach faces equally severe problems with infinity.

René Descartes has introduced (Cartesian) coordinates systems to solve geometric problems with analysis tool (analytic geometry)
(Fisher) Metric Space and (Matrix) Lie Group Theory are tools to avoid selection of any arbitrary coordinates system
For (Fisher) Metric Space, we only consider distance between points independently of their coordinates (Fisher Metric provides distance between densities of probabilities in their parameters space: the Fisher distance is invariant by reparameterization)
For (Matrix) Lie Group, we say that a manifold is homogeneous if a group acts transitively on it. In this case, we can express the geodesic equation between two point on the Manifold by Euler-Poincaré equation, independently of any systems of coordinates.

We investigate weighted Sobolev spaces on metric measure spaces (X,d,m). Denoting by rho the weight function, we compare the space W^{1,p}(X,d,rho m) (which always concides with the closure H^{1,p}(X,d,rho m) of Lipschitz functions) with the weighted Sobolev spaces W^{1,p}_{rho}(X,d,m) and H^{1,p}_{rho}(X,d,m) defined as in the Euclidean theory of weighted Sobolev spaces. Under mild assumptions on the metric measure structure and on the weight we show that W^{1,p}(X,d,rho m)=H^{1,p}_{rho}(X,d,m). We also adapt results by Muckenhoupt and a recent paper of Zhikov to the metric measure setting, considering appropriate conditions on rho that ensure the equality W^{1,p}_{rho}(X,d,m)=H^{1,p}_{rho}(X,d,m).

In this paper, we give some new definition of Compatible mappings of type (P), type (P-1) and type (P-2) in intuitionistic generalized fuzzy metric spaces and prove Common fixed point theorems for six mappings under the conditions of compatible mappings of type (P-1) and type (P-2) in complete intuitionistic fuzzy metric spaces. Our results intuitionistically fuzzify the result of Muthuraj and Pandiselvi [15] KEYWORDS Intuitionistic fuzzy metric spaces, compatible mappings of type (P), type (P-1) and type (P-2) , common fixed point .

In this paper I argue that the analysis of natural properties as convex subsets of a metric space in which the distances are degrees of dissimilarity is incompatible with both the definition of degree of dissimilarity as number of natural properties not in common and the definition of degree of dissimilarity as proportion of natural properties not in common, since in combination with either of these definitions it entails that every property is a natural property, which is absurd. I suggest it follows that we should think of the convex class analysis of natural properties as a variety of resemblance nominalism.

We introduce a new concept of generalized metric spaces and extend some well-known related fixed point theorems including Banach contraction principle, Ćirić's fixed point theorem, a fixed point result due to Ran and Reurings, and a fixed point result due to Nieto and Rodríguez-López. This new concept of generalized metric spaces and extend some well-known related fixed point theorems recover various topological spaces including standard metric spaces, b-metric spaces, dislocated metric spaces, and modular spaces.

2-metric space is an interesting nonlinear generalization of the classical one of metric space. In this paper we established fixed point theorems in 2-Metric Spaces by using some new extensions of Kannan fixed point theorem obtained by Kannan [1,2]. The result can be considered as an extension and generalization of fixed point theorems on 2-metric spaces and kannan fixed point theorems, given by many well known authors announced in the available literature.

In this paper, we prove tripled coincidence and common fixed point theorems for F : X × X × X → X and g : X → X satisfying almost generalized contractions in partially ordered metric spaces. The presented results generalize the theorem of Berinde and Borcut [Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces, Nonlinear Anal. 74 4889-4897]. Also, some examples are presented.

In 1944, McKinsey and Tarski proved that S4 is the logic of the topological interior and closure operators of any separable dense-in-itself metric space. Thus, the logic of topological interior and closure over arbitrary metric spaces coincides with the logic of the real line, the real plane, and any separable dense-in-itself metric space; it is finitely axiomatisable and PSpace-complete. Because of this result S4 has become a logic of prime importance in Qualitative Spatial Representation and Reasoning in Artificial Intelligence. And in Logic this result has triggered the investigation of a number of variants and extensions of S4 designed for reasoning about qualitative aspects of metric spaces.

Motivated by an example in [Mag], we study, inside a separable metric space (X, d), the relations between centered and non centered m-dimensional densities of a Radon measure µ in X and their relations with spherical and centered spherical mdimensional Hausdorff measures. Eventually we give an application to finite perimeter sets in Carnot groups.

We investigate the expressive power and computational properties of two different types of languages intended for speaking about distances. First, we consider a first-order language FM the two-variable fragment of which turns out to be undecidable in the class of distance spaces validating the triangular inequality as well as in the class of all metric spaces. Yet, this two-variable fragment is decidable in various weaker classes of distance spaces. Second, we introduce a variable-free 'modal' language MS which, when interpreted in metric spaces, has the same expressive power as the two-variable fragment of FM. We determine natural and expressive fragments of MS which are decidable in various classes of distance spaces validating the triangular inequality, in particular, the class of all metric spaces.

Let T : D ⊂ X → X be an iteration function in a complete metric space X . In this paper we present some new general complete convergence theorems for the Picard iteration x n+1 = Tx n with order of convergence at least r ≥ 1. Each of these theorems contains a priori and a posteriori error estimates as well as some other estimates. A central role in the new theory is played by the notions of a function of initial conditions of T and a convergence function of T . We study the convergence of the Picard iteration associated to T with respect to a function of initial conditions E: D → X . The initial conditions in our convergence results utilize only information at the starting point x 0 . More precisely, the initial conditions are given in the form E(x 0 ) ∈ J, where J is an interval on R + containing 0. The new convergence theory is applied to the Newton iteration in Banach spaces. We establish three complete ω-versions of the famous semilocal Newton-Kantorovich theorem as well as a complete version of the famous semilocal α-theorem of Smale for analytic functions.

Similarity search in high-dimensional metric spaces is a key operation in many applications, such as multimedia databases, image retrieval, object recognition, and others. The high dimensionality of the data requires special index structures to facilitate the search. A problem regarding the creation of suitable index structures for highdimensional data is the relationship between the geometry of the data and the organization of an index structure. In this paper, we study the performance of a new index structure, called Divisive-Agglomerative Hierarchical Clustering tree (DAHC-tree), which reduces the effects imposed by the above liability. DAHC-tree is constructed by dividing and grouping the data set into compact clusters. We perform a rigorous experimental design and analyze the trade-offs involved in building such an index structure. Additionally, we present extensive experiments comparing our method against state-of-the-art of exact and approximate solutions. The conducted analysis and the reported comparative test results demonstrate that our technique significantly improves the performance of similarity queries.

We show that if f : X → Y is a quasisymmetric mapping between Ahlfors regular spaces, then dim H f (E) ≤ dim H E for " almost every " bounded Ahlfors regular set E ⊆ X. If additionally, X and Y are Loewner spaces then dim H f (E) = dim H E for " almost every " Ahlfors regular set E ⊂ X. The precise statements of these results are given in terms of Fuglede's modulus of measures. As a corollary of these general theorems we show that if f is a quasiconformal map of R N , N ≥ 2, then for Lebesgue a.e. y ∈ R N we have dim H f (y + E) = dim H E. A similar result holds for Carnot groups as well. For planar quasiconformal maps, our general estimates imply that if E ⊂ R is Ahlfors d-regular, d < 1, then some component of f (E × R) has dimension at most 2/(d + 1), and we construct examples to show this bound is sharp. In addition, we show there is a 1-dimensional set S ⊆ R and planar quasiconformal map f such that f (R × S) contains no rectifi-able sub-arcs. These results generalize work of Balogh et al. (J Math Pures Appl (2)99:125–149, 2013) and answer questions posed in Balogh et al. (J Math Pures Appl (2)99:125–149, 2013) and Capogna et al. (Mapping theory in metric spaces. http://aimpl.org/mappingmetric, 2016).

General local convergence theorems with order of convergence r ≥ 1 are provided for iterative processes of the type x n+1 = Tx n , where T : D ⊂ X → X is an iteration function in a metric space X . The new local convergence theory is applied to Newton iteration for simple zeros of nonlinear operators in Banach spaces as well as to Schröder iteration for multiple zeros of polynomials and analytic functions. The theory is also applied to establish a general theorem for the uniqueness ball of nonlinear equations in Banach spaces. The new results extend and improve some results of [K. Dočev, Über Newtonsche Iterationen, C. R. Acad. Bulg. Sci. 36 (1962) 695-701; J.F. Traub, H. Woźniakowski, Convergence and complexity of Newton iteration for operator equations, J.

Given a>0, we construct a weighted Lebesgue measure on R^n for which the family of non constant curves has p-modulus zero for p\leq 1+a but the weight is a Muckenhoupt A_p weight for p>1+a. In particular, the p-weak gradient is trivial for small p but non trivial for large p. This answers an open question posed by several authors. We also give a full description of the p-weak gradient for any locally finite Borel measure on the real line.

We show if a metric measure space admits a differentiable structure then porous sets have measure zero and hence the measure is pointwise doubling. We then give a construction to show if we only require an approximate differentiable structure the measure need no longer be pointwise doubling.

In the paper some types of equivalences over resemblance measures and some basic results about them are given. Based on induced partial orderings on the set of unordered pairs of units a dissimilarity between two resemblance measures over finite sets of units can be defined. As an example, using this dissimilarity standard association coefficients between binary vectors are compared both theoretically and computationally.

Recently, permutation based indexes have attracted interest in the area of similarity search. The basic idea of permutation based indexes is that data objects are represented as appropriately generated permutations of a set of pivots (or reference objects). Similarity queries are executed by searching for data objects whose permutation representation is similar to that of the query. This, of course assumes that similar objects are represented by similar permutations of the pivots. In the context of permutation-based indexing, most authors propose to select pivots randomly from the data set, given that traditional pivot selection strategies do not reveal better performance. However, to the best of our knowledge, no rigorous comparison has been performed yet. In this paper we compare five pivots selection strategies on three permutation-based similarity access methods. Among those, we propose a novel strategy specifically designed for permutations. Two significant observations emerge from our tests. First, random selection is always outperformed by at least one of the tested strategies. Second, there is not a strategy that is universally the best for all permutation-based access methods; rather different strategies are optimal for different methods.

—We introduce a new probabilistic proximity search algorithm for range and K-nearest neighbor (K-NN) searching in both coordinate and metric spaces. Although there exist solutions for these problems, they boil down to a linear scan when the space is intrinsically high dimensional, as is the case in many pattern recognition tasks. This, for example, renders the K-NN approach to classification rather slow in large databases. Our novel idea is to predict closeness between elements according to how they order their distances toward a distinguished set of anchor objects. Each element in the space sorts the anchor objects from closest to farthest to it and the similarity between orders turns out to be an excellent predictor of the closeness between the corresponding elements. We present extensive experiments comparing our method against state-of-the-art exact and approximate techniques, both in synthetic and real, metric and nonmetric databases, measuring both CPU time and distance computations. The experiments demonstrate that our technique almost always improves upon the performance of alternative techniques, in some cases by a wide margin.

In 1944, McKinsey and Tarski proved that S4 is the logic of the topological interior and closure operators of any separable dense-in-itself metric space. Thus, the logic of topological interior and closure over arbitrary metric spaces coincides with the logic of the real line, the real plane, and any separable dense-in-itself metric space; it is finitely axiomatisable and PSpace-complete. Because of this result S4 has become a logic of prime importance in Qualitative Spatial Representation and Reasoning in Artificial Intelligence. And in Logic this result has triggered the investigation of a number of variants and extensions of S4 designed for reasoning about qualitative aspects of metric spaces.

Nearest-neighbour (NN) and k-nearest-neighbours (k-NN) techniques are widely used in many pattern recognition classification tasks. The linear approximating and eliminating search algorithm (LAESA) is a fast NN algorithm which does not assume that the prototypes are defined in a vector space; it only makes use of some of the distance properties (mainly the triangle inequality) in order to avoid distance computations.

We investigate the expressive power and computational properties of two different types of languages intended for speaking about distances. First, we consider a first-order language F M the two-variable fragment of which turns out to be undecidable in the class of distance spaces validating the triangular inequality as well as in the class of all metric spaces. Yet, this two-variable fragment is decidable in various weaker classes of distance spaces. Second, we introduce a variable-free 'modal' language MS which, when interpreted in metric spaces, has the same expressive power as the two-variable fragment of F M. We determine natural and expressive fragments of MS which are decidable in various classes of distance spaces validating the triangular inequality, in particular, the class of all metric spaces.

In this paper, we introduce a new algorithm for the public key transferring which is based upon functional analysis and metric spaces with basic properties of circles. Furthermore, we introduce an algorithm to make the changes in keys without meeting. The algorithms presented in this paper promises a high level of security as it uses metric distances, in spite of the large prime numbers used in the present algorithms. The security analysis of the new scheme is also discussed.

We study classes of continuous functions on R n that can be approximated in various degree by uniformly continuous ones (uniformly approachable functions). It was proved in [BDP1] that no polynomial function can distinguish between them. We construct examples that distinguish these classes (answering a question from [BDP1]) and we offer appropriate forms of uniform approachability that enable us to obtain a general theorem on coincidence in the class of all continuous functions.

We develop a general framework to analyze the convergence of linear-programming approximations for Markov control processes in metric spaces. The approximations are based on aggregation and relaxation of constraints, as well as inner approximations of the decision variables. In particular, conditions are given under which the control problem's optimal value can be approximated by a sequence of finite-dimensional linear programs.

We prove that if (X, d) is a metric space, C is a closed subset of X and x ∈ X, then the distance of x to R ∩ S agrees with the maximum of the distances of x to R and S, for every closed subsets R, S ⊂ C such that C = R ∪ S, if and only if C is x-boundedly connected.

The state of the art of searching for non-text data (e.g., images) is to use extracted metadata annotations or text, which might be available as a related information. However, supporting real content-based audio-visual search, based on similarity search on features, is significantly more expensive than searching for text. Moreover, such search exhibits linear scalability with respect to the data set size, so parallel query execution is needed.

We consider the problem of partitioning, in a highly accurate and highly efficient way, a set of n documents lying in a metric space into k non-overlapping clusters. We augment the well-known furthest-point-first algorithm for k-center clustering in metric spaces with a filtering scheme based on the triangular inequality. We apply this algorithm to Web snippet clustering, comparing it against strong baselines consisting of recent, fast variants of the classical k-means iterative algorithm. Our main conclusion is that our method attains solutions of better or comparable accuracy, and does this within a fraction of the time required by the baselines. Our algorithm is thus valuable when, as in Web snippet clustering, either the real-time nature of the task or the large amount of data make the poorly scalable, traditional clustering methods unsuitable.

We consider the problem of embedding finite metrics with slack: we seek to produce embeddings with small dimension and distortion while allowing a (small) constant fraction of all distances to be arbitrarily distorted. This definition is motivated by recent research in the networking community, which achieved striking empirical success at embedding Internet latencies with low distortion into low-dimensional Euclidean space, provided that some small slack is allowed.

Multilayer Perceptrons (MLPs) use scalar products to compute weighted activation of neurons providing decision borders using combinations of soft hyperplanes. The weighted fun-in activation function corresponds to Euclidean distance functions used to compute similarities between input and weight vector. Replacing the fan-in activation function by non-Euclidean distance function offers a natural generalization of the standard MLP model, providing more flexible decision borders. An alternative way leading to similar results is based on renormalization of the input vectors using non-Euclidean norms in extended feature spaces. Both approaches influence the shapes of decision borders dramatically, allowing to reduce the complexity of MLP networks.

The state of the art of searching for non-text data (e.g., images) is to use extracted metadata annotations or text, which might be available as a related information. However, supporting real content-based audio-visual search, based on similarity search on features, is significantly more expensive than searching for text. Moreover, such search exhibits linear scalability with respect to the data set size, so parallel query execution is needed.

Nearest neighbour search is a widely used technique in pattern recognition. During the last three decades a large number of fast algorithms have been proposed. In this work we are interested in algorithms that can be used with any dissimilarity function provided that it fits the mathematical notion of distance.

In a multi-armed bandit problem, an online algorithm chooses from a set of strategies in a sequence of n trials so as to maximize the total payoff of the chosen strategies. While the performance of bandit algorithms with a small finite strategy set is quite well understood, bandit problems with large strategy sets are still a topic of very active investigation, motivated by practical applications such as online auctions and web advertisement. The goal of such research is to identify broad and natural classes of strategy sets and payoff functions which enable the design of efficient solutions.

The traditional problem of similarity search requires to find, within a set of points, those that are closer to a query point q, according to a distance function d. In this paper we introduce the novel problem of metric information filtering (MIF): in this scenario, each point x i comes with its own distance function d i and the task is to efficiently determine those points that are close enough, according to d i , to a query point q. MIF can be seen as an extension of both the similarity search problem and of approaches currently used in content-based information filtering, since in MIF user profiles (points) and new items (queries) are compared using arbitrary, personalized, metrics. We introduce the basic concepts of MIF and provide alternative resolution strategies aiming to reduce processing costs. Our experimental results show that the proposed solutions are indeed effective in reducing evaluation costs.

Metric space searching is an emerging technique to address the problem of efficient similarity searching in many applications, including multimedia databases and other repositories handling complex objects. Although promising, the metric space approach is still immature in several aspects that are well established in traditional databases. In particular, most indexing schemes are not dynamic, that is, few of them tolerate insertion of elements at reasonable cost over an existing index and only a few work efficiently in secondary memory.

In the last decade, the notion of metric embeddings with small distortion received wide attention in the literature, with applications in combinatorial optimization, discrete mathematics and bio-informatics. The notion of embedding is, given two metric spaces on the same number of points, to find a bijection that minimizes maximum Lipschitz and bi-Lipschitz constants. One reason for the popularity of the notion is that algorithms designed for one metric space can be applied to a different metric space, given an embedding with small distortion. The better the distortion, the better is the effectiveness of the original algorithm applied to a new metric space.