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It is well known that metric spaces are an instance of categorical enrichment in a particular quantale. We show that in a categorically natural way a notion of Lipschitz norm arises in the context of an arbitrary diagram of quantales, instead of just one particular quantale. The generalised Lipschitz norm we present depends functorially on the diagram and is itself a functor to the indexing category of the diagram. The entire process is, in a way we make precise, an instance of a concrete Grothendieck construction.

The purpose of this paper is to establish some fixed point results for a pair of fuzzy dominated mappings satisfying contractive conditions on closed ball in multiplicative metric space. Some new fixed point results with graphic contraction on closed ball for a pair of fuzzy graph dominated mappings on multiplicative metric space have been established. Furthermore, we find a unique common solution for a system of non linear Voltera type integral equations and lastly we give an application to ensure the existence of common bounded solution of a functional equation in dynamic programming.

In this paper we give a new method of constructing partial order on the set of formal balls of a KM-fuzzy metric space, which overcomes a key defect that the partial order on the set of formal balls depends on the minimum t-norm ∧ in the case of [19] , [21] . Further, based on a continuous t-norm, we will establish a connection between KM-fuzzy metric space and domain theory, which is that a KM-fuzzy metric space is T-complete if and only if its poset of formal balls is a domain.

Gated sets in metric spaces play an important role in the study of convexity. In this paper, we investigate the notion of gated sets in the context of betweenness of fuzzy metric spaces. For this purpose, we analyze the validity of postulates presented by Huntington and Kline to the betweenness in GV fuzzy metric spaces. The postulates are different from the usual properties of metric betweenness, amounting to thirteen. Interestingly, the validity of most of these postulates to the betweenness need to be considered in fuzzy normed spaces. In particular, two of the postulates for this relation are valid when the space is 1-dimensional. Five of the postulates for this relation are centrally studied in connection with strict convexity. As a consequence, several characterizations on strict convexity in a GV fuzzy normed space under the minimum t-norm are established.

This manuscript was built to generalize Ekeland variational principle for mixed monotone functions in the setting of partially ordered complete metric spaces. The results obtained are applied to give different proofs for tripled fixed points of mixed monotone mappings in the mentioned space by using a variational technique. The results presented in our manuscripts generalize and expand many of the findings presented in the earlier period. For the sobriety and enhancement of our paper, two examples are given and the existence and uniqueness of the solution to a periodic boundary value problem are studied as applications.

Shape analogy is a key technique in analyzing time series . That is, time series are compared by how much they look alike. This concept has been applied for many years in geometry. Notably, none of the current techniques describe a time series as a geometric curve that is expressed by its relative location and form in space. To fill this gap, we introduce Shape-Sphere , a vector space where time series are presented as points on the surface of a sphere. We prove a pseudo-metric property for distances in Shape-Sphere. We show how to describe the average shape of a time series set using the pseudo-metric property of Shape-Sphere by deriving a centroid from the set. We demonstrate the effectiveness of the pseudo-metric property and its centroid in capturing the ‘ shape ’ of a time series set, using two important machine learning techniques , namely: Nearest Centroid Classifier and K-Means clustering, using 85 publicly available data sets. Shape-Sphere improves the nearest centroid classification results when the shape is the differentiating feature while keeping the quality of clustering equivalent to current state-of-the-art techniques.

We study existence, uniqueness, and regularity properties of the Dirichlet problem related to fractional Dirichlet energy minimizers in a complete doubling metric measure space $(X,d_X,\mu_X)$ satisfying a $2$-Poincar\'e inequality. Given a bounded domain $\Omega\subset X$ with $\mu_X(X\setminus\Omega)>0$, and a function $f$ in the Besov class $B^\theta_{2,2}(X)\cap L^2(X)$, we study the problem of finding a function $u\in B^\theta_{2,2}(X)$ such that $u=f$ in $X\setminus\Omega$ and $\mathcal{E}_\theta(u,u)\le \mathcal{E}_\theta(h,h)$ whenever $h\in B^\theta_{2,2}(X)$ with $h=f$ in $X\setminus\Omega$. We show that such a solution always exists and that this solution is unique. We also show that the solution is locally H\"older continuous on $\Omega$, and satisfies a non-local maximum and strong maximum principle. Part of the results in this paper extend the work of Caffarelli and Silvestre in the Euclidean setting and Franchi and Ferrari in Carnot groups., Comment: 42 pages, comments welcome, submitted. Revision to add crucial references and attributions to the introduction

We study the Bounded-Degree Light Approximate Shortest-path Tree problem: Given a metric space ( V , d ) on n points, a root r ∈ V and a degree bound b , the goal is to find a tree T spanning V with maximum degree at most b , total weight comparable to the weight of a Minimum Spanning Tree of ( V , d ) , and low stretch of tree distances from r to every vertex compared to input distances. Our main result is an algorithm that given a metric ( V , d ) with doubling dimension k , returns a tree T rooted at r spanning V with degree b and stretch max { O ( k ) / log b , O ( 1 ) } having weight within a constant factor the weight of a MST of ( V , d ) . On the negative side, we show that the trade-off between degree and stretch is unavoidable (up to constant factors in the Big- O ) even when the weight of the tree is not taken into account.

In this work new global dynamic bifurcation theorems are established for local semiflows on complete metric spaces. These theorems are applied to the nonlinear evolution equation u t + A u = f λ ( u ) in a Banach space X, where A is a sectorial operator with compact resolvent, and f λ ( 0 ) = 0 for all λ ∈ R . In particular, if f λ takes the form f λ ( u ) = λ u + f ( u ) , it is shown that the global dynamic bifurcation branch of a bifurcation point ( 0 , λ 0 ) is necessarily unbounded without assuming the “crossing odd-multiplicity” condition. It has long been recognized that dynamical methods can be helpful in solving static problems. As an example, the bifurcation of elliptic equations on bounded domains associated with homogenous Dirichlet boundary condition is addressed. By considering the corresponding nonclassical parabolic flows and applying the global dynamic bifurcation theorems given here for local semiflows, some new results with global features are derived.

For a given metric space X, we consider the set of all normal fuzzy sets on X, denoted by F ( X ) . In this work, we study expanding, positively expansive and weakly positively expansive dynamical systems ( X , f ) and how they are reflected in the dynamical system ( F ( X ) , f ˆ ) , where f ˆ is the Zadeh's extension of f and F ( X ) has one of the following metrics: the levelwise metric, the endograph metric, the sendograph metric and the Skorokhod metric. We mainly show that if we consider the following conditions: (i) ( X , f ) is positively expansive (resp. expanding); (ii) ( K ( X ) , f ‾ ) is positively expansive (resp. expanding); (iii) ( F ∞ ( X ) , f ˆ ) is positively expansive (resp. expanding); (iv) ( F 0 ( X ) , f ˆ ) is positively expansive (resp. expanding). Then (iv)⇒ (iii) ⇔ (ii) ⇒ (i). For expanding dynamical systems, we present a compact metric space and a locally compact metric space to show that (i) ⇏ (ii) and (iii) ⇏ (iv), respectively. For positively expansive dynamical systems, there is a compact metric space satisfying that (i) ⇏ (ii), but we don't know if (iii) ⇒ (iv).

This paper mainly discusses three points. First of all, four kinds of intuitionistic similarities based on implication operator and the corresponding logical metric spaces are constructed. Secondly, from the perspective of perturbation parameters, different intuitionistic similarities are compared and analyzed by means of topological method. Finally, the applications of similarities are shown. On the one hand, the constructed similarities are used to solve a pattern recognition problem. On the other hand, the robustness of the Triple I method of intuitionistic reasoning is analyzed by adopting the constructed similarity as the perturbation parameter.

Finite metric spaces are an essential tool in discrete mathematics and have applications in several areas including computational biology, image analysis, speech recognition, and information retrieval. Given any such metric D on a finite set X , an important problem is to find appropriate ways to realize D by weighting the edges in some graph G containing X in its vertex set such that D ( x , y ) equals the length of a shortest path from x to y in G for all x , y ∈ X . Here we focus on realizations with minimum total edge weight, called optimal realizations. By considering the 2-connected components and bridges in any optimal realization G of D we obtain an additive decomposition of D into simpler metrics. We show that this decomposition, called the block decomposition, is canonical in that it only depends on D and not on G , and that the decomposition can be computed in O ( | X | 3 ) time. As well as providing a fundamental new way to decompose any finite metric space, we expect that the block decomposition will provide a useful preprocessing tool for deriving metric realizations.

Similarity searching has become widely utilized in many online services processing unstructured and complex data, e.g., Google Images. Metric spaces are often applied to model and organize such data by their mutual similarity. As top-k queries provide only a local view on data, a data analyst must pose multiple requests to observe the entire dataset. Thus, group-by operators for metric data have been proposed. These operators identify groups by respecting a given similarity constraint and produce a set of objects per group. The analyst can then tediously browse these sets directly, but representative members may provide better insight. In this paper, we focus on concise representations of metric datasets. We propose a novel concept of a metric hull which encompasses a given set by selecting a few objects. Testing an object to be part of the set is then made much faster. We verify this concept on synthetic Euclidean data and real-life image and text datasets and show its effectiveness and scalability. The metric hulls provide much faster and more compact representations when compared with commonly used ball representations.

In this work, we show that two distance functions independently defined on the space of subcopulas are topological equivalent. In this process, we also defined another distance function equivalent to the first two distances. Moreover, all metric spaces of subcopulas with fixed domain under the supremum distance are metric subspaces under this new distance function. We also prove that the Sklar correspondence can be viewed as a Lipschitz map under these metrics. Thus, the rate of convergence of empirical subcopulas can be computed directly from the rate of convergence of empirical distributions. The same holds for other statistics results.

The red-blue median problem considers a set of red facilities, a set of blue facilities, and a set of clients located in some metric space. The goal is to open k r red facilities and k b blue facilities such that the sum of the distance from each client to its nearest opened facility is minimized, where k r , k b ≥ 0 are two given integers. Designing approximation algorithms for this problem remains an active area of research due to its applications in various fields. However, in many applications, the existence of noisy data poses a big challenge for the problem. In this paper, we consider the prize-collecting red-blue median problem, where the noisy data can be removed by paying a penalty cost. The current best approximation guarantee for the prize-collecting red-blue median problem is a ratio of 24, which was obtained by LP-rounding. We deal with this problem using a local search algorithm. We construct a layered structure of the swap pairs, which yields a ( 9 + ϵ ) -approximation for the prize-collecting red-blue median problem. Our techniques generalize to a more general prize-collecting τ-color median problem, where the facilities have τ different types, and give a ( 4 τ + 1 + ϵ ) -approximation for the case where τ is a constant.

The main goal of this paper is to study a new dispersive order in an arbitrary metric space. More precisely, given two random variables on a metric space, the order decides which one has a more concentrated distribution. It is motivated by the wish to be able to make new comparisons that the currently defined orderings cannot make. Related to this problem, several statistical parameters of a random variable, giving different kinds of information about the distribution, are studied. It is proved that the new dispersive order also compares some of these parameters. Special attention is paid to Riemannian manifolds, a natural class of metric spaces. Here, the new order behaves in a distinguished way: the differentiability axioms allow obtaining several equations which facilitate the decision. The principal properties of the order are presented, as well as several applications. Finally, the independence of this order from those already known is proved.

[EN] We present a McShane-Whitney extension theorem for real-valued fuzzy Lipschitz maps defined between fuzzy metric spaces. Motivated by the potential applications of the obtained results, we generalize the mathematical theory of extensions of Lipschitz maps to the fuzzy context. We develop the problem in its full generality, explaining the similarities and differences with the classical case of extensions on metric spaces., The first and the last authors gratefully acknowledge the support of the Ministerio de Ciencia, Innovacion y Universidades, Agencial Estatal de Investigacion and FEDER under grant MTM2016-77054-C2-1-P. The second author acknowledges financial support from Ministerio de Ciencia, Innovacion y Universidades, Agencial Estatal de Investigacion and FEDER under grant PGC2018-095709-B-C21.

The aim of this paper is to develop a differential calculus for functions with values in a quasilinear metric space. Using only the metric on a quasilinear metric space and without using the Hukuhara difference, a new concept of derivative is introduced for functions with values in a quasilinear metric space. This new derivative concept generalizes known concepts of derivatives for set-valued and fuzzy value functions.

The ‘the’ in the title hides a subtlety. A metric space induces not one but four topologies - by means of open sets, closed sets, closure, and interior - they just so happen to coincide. The agreement between these four structures arising from a metric function d : X × X → [ 0 , ∞ ] is due to a combination of the metric axioms and the lattice structure of [ 0 , ∞ ] . Further motivation materializes from Lawvere's observation from 1973 to the effect that a (slightly generalized) metric space is a category enriched in [ 0 , ∞ ] . Metric spaces taking values in structures other than [ 0 , ∞ ] are relevant for generalizations of metric spaces and find a natural home in Lawvere's categorical setting. In particular, in recent years quantales emerged as structures occupying an important niche in between [ 0 , ∞ ] and arbitrary monoidal categories. Since a category enriched in a quantale Q is the same thing as a metric space taking values in Q one may ask whether such a thing belongs to algebra or geometry. Further, does the quadruplet of topologies associated to a Q-valued space/category still consist of identical siblings? We propose a litmus test for the geometricity of Q-valued spaces as we investigate these issues.

In this paper, the existence and uniqueness of a fixed point in a soft cone metric space are discussed for a single self-mapping using expanding and comparison function in the setting of cone metric space for random operator. These established results improve and modify some existing results in the literature. The obtained results are very useful in mechanical engineering and digital electronics. To find the results the fundamental result Banach contraction principal, is used. The established results satisfied the previous well known results.

We consider the dispersed p → -neighbor k -supplier problem. In the classical k -supplier problem, we have to select k suppliers in a metric space such that the maximum distance between a customer and its closest supplier is minimized. Here, we generalize this problem to the case where each customer possibly needs service from more than one supplier. Moreover, the selected suppliers should not be too close to each other, i.e., they need to be dispersed. For the classical k -supplier problem, and its special case the k -center problem, there is a 3- and a 2-approximation respectively. We show that these guarantees can also be given in the case when customers need service from multiple suppliers, without imposing dispersion constraints. If we generalize the problem to the dispersed case, without imposing neighboring constraints, we get inapproximability results depending on the measure of dispersion. We also show (almost) matching upper bounds. Finally, we show that adding both the neighbor requirement and the dispersion requirement leads to an inapproximable problem.

In this paper, we establish a new Class category of non expansive mappings in metric space which is wider than the class category of mappings satisfying contractive condition (CC). Special properties and various fixed point results for these mappings in metric space are obtained here. Also we discussed the results and examples of the convergence of some iteration schemes to the fixed point in metric space. Obtained results are useful in material research as well as in image processing.

The concise and reliable representation of a given source dataset is available for data based learning and decision-making, which can be obtained through a controllable data fusion process within a formatted space. Based on the density matrix representation of a given source weather dataset, this paper first calculates the Hellinger distances and affinities between different density matrices, and then takes them to construct a two-dimensional metric space using the longest Hellinger metric and affinity paths. Within this space, the given source data units are converted into the corresponding rectangle nodes which are determined by the minimum horizontal and vertical distances between different data units. According to a predefined detection size in this space, the basic blocks centred by different rectangle nodes are classified into different subset blocks for fusion. Each subset block is jointed by the basic blocks with overlapped areas. During the detection process, the detection size keeps increasing until a predefined reference variable, such as the relative density for all subset blocks, reaches a predefined turning point. The fusion of the rectangle nodes in a subset block depends on the distances between the included rectangle nodes’ positions and this subset block’s centre position which is calculated according to the included rectangle nodes’ weights and positions. The experimental analysis shows that the proposed weather data fusion method is controllable and stable and can obtain concise and reliable fusion results for learning inputs and decision-making.

The term Stone-type duality often refers to a dual equivalence between a category of lattices or other partially ordered structures on one side and a category of topological structures on the other. This paper is part of a larger endeavour that aims to extend a web of Stone-type dualities from ordered to metric structures and, more generally, to quantale-enriched categories. In particular, we improve our previous work and show how certain duality results for categories of \([0,1]\)-enriched Priestley spaces and \([0,1]\)-enriched relations can be restricted to functions. In a broader context, we investigate the category of quantale-enriched Priestley spaces and continuous functors, with emphasis on those properties which identify the algebraic nature of the dual of this category. published

We propose and study the MpUFLP (universal facility location problem in the p-th power of metric space) in this paper, where the universal facility location problem (UFLP) extends several classical facility location problems like the uncapacitated facility location, hard-capacitated facility location, soft-capacitated facility location, incremental-cost facility location, concave-cost facility location, etc. In UFLP, a set of facilities, a set of clients, as well as the distances between them are given. Each facility has its specific cost function w.r.t. the amount of clients assigned to that facility. The goal is to assign the clients to facilities such that the sum of facility cost and service cost is minimized. In traditional facility location problems, the unit service cost is proportional to the distance between the client and its assigned facility and thus metric. However, in our work, this assumption is removed and a generalized version of universal facility location problem is proposed, which is the so-called MnUFLP. When p = 2 , it is also known as l 2 2 measure considered by Jain and Vazirani [J. ACM'01] and Fernandes et al. [Math. Program.'15]. Particularly in this case, we extend their work to include the aforementioned variants of facility location and a local search based ( 11.18 + e ) -approximation algorithm is proposed. Furthermore, the reanalysis of the proposed algorithm gives a p-related performance guarantee for general p.

Funding: This research was supported by ERDF “CyberSecurity, CyberCrime and Critical Information Infrastructures Center of Excellence” (No.CZ.02.1.01/0.0/0.0/16 019/0000822) and by ESRC “Administrative Data Research Centres 2018” (No. ES/S007407/1). Scalable similarity search in metric spaces relies on using the mathematical properties of the space in order to allow efficient querying. Most important in this context is the triangle inequality property, which can allow the majority of individual similarity comparisons to be avoided for a given query. However many important metric spaces, typically those with high dimensionality, are not amenable to such techniques. In the past convex transforms have been studied as a pragmatic mechanism which can overcome this effect; however the problem with this approach is that the metric properties may be lost, leading to loss of accuracy. Here, we study the underlying properties of such transforms and their effect on metric indexing mechanisms. We show there are some spaces where certain transforms may be applied without loss of accuracy, and further spaces where we can understand the engineering tradeoffs between accuracy and efficiency. We back these observations with experimental analysis. To highlight the value of the approach, we show three large spaces deriving from practical domains whose dimensionality prevents normal indexing techniques, but where the transforms applied give scalable access with a relatively small loss of accuracy. Postprint

We correct a mistake in our paper “Poincare inequalities and Newtonian Sobolev functions on noncomplete metric spaces” (Bjorn and Bjorn, 2019 [2] ).

Aiming at the simplest possible representation of maxmin copulas in terms of two generators Kosir and Omladic recently introduced the class of so-called reflected maxmin (RMM) copulas in [10] . All results for RMM copulas can easily be translated to results on maxmin copulas. Building upon the afore-mentioned article we provide alternative simplified (mainly Markov kernel based) proofs for some of their theorems and, more importantly, derive numerous new results. In particular, we show that the family of all RMM copulas is a compact subset of the metric space of all copulas (with the standard uniform metric), characterize absolutely continuous RMM copulas, and show that the limit of absolutely continuous RMM copulas is absolutely continuous too. Finally, we determine Kendall's τ and Spearman's ρ of RMM copulas, prove some first inequalities for τ and ρ, and state a conjecture on sharp inequalities. Some examples illustrating the main results round off the paper.

By using current methods of best proximity point theory we can not establish and prove best proximity point results in modular metric spaces. For this purpose, we introduce the notion of P D − modular metric spaces and then we establish some best proximity point theorems for certain new contraction mappings. As an application of the obtained results we establish some best proximity point results in fuzzy metric spaces. Examples are furnished to demonstrate the validity of the obtained results.

In this work, a metric is presented on the set of boundedly-compact pointed metric spaces that generates the Gromov-Hausdorff topology. A similar metric is defined for measured metric spaces that generates the Gromov-Hausdorff-Prokhorov topology. This extends previous works which consider only length spaces or discrete metric spaces. Completeness and separability are also proved for these metrics. Hence, they provide the measure theoretic requirements to study random (measured) boundedly-compact pointed metric spaces, which is the main motivation of this work. In addition, we present a generalization of the classical theorem of Strassen which is of independent interest. This generalization proves an equivalent formulation of the Prokhorov distance of two finite measures, having possibly different total masses, in term of approximate coupling. A Strassen-type result is also proved for the Gromov-Hausdorff-Prokhorov metric for compact spaces., Comment: 36 pages, 1 figures. This paper is part of the previous version arXiv:1812.03760 which is now split into two articles

The aim of this paper is to study the theory of weighted (pseudo-quasi-)metrics based on L-fuzzy sets. We first prove that the category of weighted pointwise quasi-metric spaces is isomorphic to that of L-partial metric spaces. Then we introduce the remote-neighborhood mapping and use it to characterize the L-partial pseudo-quasi-metric. Finally, we investigate properties of the L-topology induced by the L-partial pseudo-quasi-metrics.

This paper shows that there exists a contraction whose Zadeh's extension is not a contraction under the Skorokhod metric, answering negatively Problems 5.8 and 5.12 posted in Jardon, Sanchez, and Sanchis (2019) [5] .

Based on a ternary betweenness relation, Coxeter introduced the notion of a line in ordered geometries. Later, Chen and Chvatal observed that in particular the distance function of a finite metric space induces a betweenness relation. They conjecture that in every metric space on n points either all points lie on one line or there exist n mutually distinct lines. A weaker version conjectures that this holds for every finite graph on n vertices with the graph-theoretic distance. We prove this conjecture for ( q , q − 4 ) -graphs.

Cubical complexes are metric spaces constructed by gluing together unit cubes in an analogous way to the construction of simplicial complexes. We construct Brownian motion on such spaces, define random walks, and prove that the transition kernels of the random walks converge to that for Brownian motion. The proof involves pulling back onto the complex the distribution of Brownian sample paths on a single cube, combined with a distribution on walks between cubes. The main application lies in analysing sets of evolutionary trees: several tree spaces are cubical complexes and we briefly describe our results and applications in this context.

Motivated by obtaining more concise and reliable fusion results for relevant applications, such as, in sector monitoring, risk managing, early-warning, dynamic prediction, this paper explores the detection and fusion models of weather data from multiple perspectives. We represent the weather data as the collection of quantum states, and study construction of the quantum metric space in which the related weather data fusion method is developed. Since weather data usually contain raw (image or sensor) data, feature data, or decision data, the weather data fusion belongs to normal three levels of data fusion, i.e., pixel level, feature level or decision level. In this paper we represent weather data units as density matrixes, and take the quantified data units as the basis for constructing the two-dimensional quantum metric space consisting of Hellinger and Bures metrics. All the density matrix units in this space are transformed into the rectangle nodes for detection and fusion. Then, according to the neighborhood relationships between rectangle nodes and the density of rectangle nodes in a specified rectangle, the source weather dataset is divided into different subsets. The key to the fusion of the rectangle nodes in a subset is to calculate the presupposing central node in this subset's corresponding rectangle. And the corresponding object node depends on the areas of the rectangles including rectangle nodes and the central node. The experimental evaluation demonstrates that, compared with the already developed fusion methods, the proposed weather data fusion method can obtain more concise and reliable fusion results for decision applications.

This paper aims to develop a metric indexing method that uses users’ queries for reducing the search cost of similarity search systems and for avoiding the insertion cost in dynamic data sets. We have proposed an indexing method which is able to improve its structure based on users’ queries. The proposed method, called I-Clusters, is a metric clustering based method, extended from the List of Clusters method. This method decreases the construction costs, and it improves the search cost after the execution of queries. The I-Clusters method allows solving the trade-off between the construction cost and the searching cost, and it also allows indexing dynamic datasets without additional cost of objects insertion. The experiment results show that the I-Clusters method significantly reduces the search cost based on queries execution, and the search performance of the proposed method can reach that of List of Clusters. Keywords: Content-based retrieval, Similarity search, Data structures, Indexing, Nearest neighbours, Range query

In this paper we have generalizeda certain and adequate circumstance nether which gluings of H- metric spaces on weak outwardly hyperconvex subsets are hyperconvex. Our results are the extension of the results obtained by B. Miesch and M. Pavon (J. Topo. Anal., volume 9, issue 3, September 2017), to the case of intersection property for gluing on weak outwardly hyperconvex subsets gives rise to H- spaces

The point of this articleis to acquire culmination and uniqueness of “random fixed point theorem”on“b-metric space”

This paper concerns on some new theorems on intuitionistic fuzzy metric spaces for four mappings using integral inequality. Here we have derived common fixed point formappings using E.A property

In this work, we define a distance function on the set of bivariate subcopulas to generate a compact metric space. Moreover, the copula space equipped with the uniform distance is essentially a metric subspace of this subcopula space. We also characterize the convergence in this space, and provide the interrelationship with the convergence of the distribution functions.

This paper contains the coupled soft fixed theorems with themonotone and alpha monotone rule in soft metric space

Following his discovery that finite metric spaces have injective envelopes naturally admitting a polyhedral structure, Isbell, in his pioneering work on injective metric spaces, attempted a characterization of cellular complexes admitting the structure of an injective metric space. A bit later, Mai and Tang confirmed Isbell's conjecture that a simplicial complex is injectively metrizable if and only if it is collapsible. Considerable advances in the understanding, classification and applications of injective envelopes have since been made by Dress, Huber, Sturmfels and collaborators, and most recently by Lang. Unfortunately a combination theory for injective polyhedra is still unavailable. Here we expose a connection to the duality theory of cubings -- simply connected non-positively curved cubical complexes -- which provides a more principled and accessible approach to Mai and Tang's result, providing one with a powerful tool for systematic construction of locally-compact injective metric spaces: Main Theorem. Any complete pointed Gromov--Hausdorff limit of locally-finite piecewise-$\ell_\infty$ cubings is injective. This result may be construed as a combination theorem for the simplest injective polytopes, $\ell_\infty$-parallelopipeds, where the condition for retaining injectivity is the combinatorial non-positive curvature condition on the complex. Thus it represents a first step towards a more comprehensive combination theory for injective spaces. In addition to setting the earlier work on injectively metrizable complexes within its proper context of non-positively curved geometry, this paper is meant to provide the reader with a systematic review of the results~ ---~ otherwise scattered throughout the geometric group theory literature~ ---~ on the duality theory and the geometry of cubings, which make this connection possible., Significantly expanded and revised. 40 pages, 6 figures, to appear in Expositiones Mathematicae (electronic https://doi.org/10.1016/j.exmath.2018.06.001)

We discuss the smallness of the set of attractors for iterated function systems. This paper is an attempt to measure the difference between the family of IFS attractors and a broader family, the set of attractors for the weak iterated function systems. We prove that the IFS attractors form a σ-porous subset of K([0, 1]d), the family of compact subsets of a metric space [0, 1]d. We also show that weak IFS attractors form a first category set.

We study stochastic processes on the Wasserstein space, together with their infinitesimal generators. One of these processes is modeled after Brownian motion and plays a central role in our work. Its infinitesimal generator defines a partial Laplacian on the space of Borel probability measures, and we use it to define heat flow on the Wasserstein space. We verify a distinctive smoothing effect of this flow for a particular class of initial conditions. To this end, we will develop a theory of Fourier analysis and conic surfaces in metric spaces. We note that the use of the infinitesimal generators has been instrumental in proving various theorems for Mean Field Games, and we anticipate they will play a key role in future studies of viscosity solutions of PDEs in the Wasserstein space.

A topological space X is called resolvable if it contains a dense subset with dense complement. Using only basic principles, we show that whenever the space X has a resolving subset that can be written as an at most countably infinite union of subsets, in such a way that a given vector lattice of (not necessarily bounded) continuous functions on X separates every point outside the resolving subset from each of its constituents, then the order continuous dual of this lattice is trivial. In order to apply this result in specific cases, we show that several spaces have resolving subsets that can be written as at most countably infinite unions of closed nowhere dense subsets. An appeal to the main result then yields, for example, that, under appropriate conditions, vector lattices of continuous functions on separable spaces, metric spaces, and topological vector spaces have trivial order continuous duals if they separate points and closed nowhere dense subsets. Our results in this direction extend known results in the literature. We also show that, under reasonably mild separation conditions, vector lattices of continuous functions on locally connected T 1 Baire spaces without isolated points have trivial order continuous duals. A discussion of the relation between our results and the non-existence of non-zero normal measures is included.

We obtain a Sobolev type embedding result for Besov spaces defined on a doubling measure metric space.

Working in the context of $\mu$-abstract elementary classes ($\mu$-AECs) - or, equivalently, accessible categories with all morphisms monomorphisms - we examine the two natural notions of size that occur, namely cardinality of underlying sets and internal size. The latter, purely category-theoretic, notion generalizes e.g. density character in complete metric spaces and cardinality of orthogonal bases in Hilbert spaces. We consider the relationship between these notions under mild set-theoretic hypotheses, including weakenings of the singular cardinal hypothesis. We also establish preliminary results on the existence and categoricity spectra of $\mu$-AECs, including specific examples showing dramatic failures of the eventual categoricity conjecture (with categoricity defined using cardinality) in $\mu$-AECs., Comment: 27 pages

Under certain initial conditions, we prove the existence of set-valued selectors of univariate compact-valued multifunctions of bounded (Jordan) variation when the notion of variation is defined taking into account only the Pompeiu asymmetric excess between compact sets from the target metric space. For this, we study subtle properties of the directional variations. We show by examples that all assumptions in the main existence result are essential. As an application, we establish the existence of set-valued solutions $X(t)$ of bounded variation to the functional inclusion of the form $X(t)\subset F(t,X(t))$ satisfying the initial condition $X(t_0)=X_0$., 32 pages, LaTeX, uses elsarticle

In the present paper some fixed-point theorems are proved in complete 2-metric space, which are extension of Banach contraction principle defined for complete metric spaces. Some more results are also established for common fixed point for two mappings .Fixed point theory plays an important role to obtained the solution of differential equation whether the equations are linear or nonlinear . The integral equation can also be solved by this theory. The obtained results are useful in materials research as well as in image processing and phase transition studies. The Results for 2- metric can also be generalized for fuzzy-2 metric spaces and fuzzy-3 metric spaces which will be very useful in decision making problem and uncertainty problem.

In this paper, we prove a common fixed point theorem for four pairs of mappings which satisfy the E . A - property in b - metric spaces. The proposed theorem expands and generalizes several well-known comparable results in the literature. Example is given showing that our result is proper extensions of the existing ones. Obtained results are useful in solving Mechanical problems as well as in image processing and deep learning. The established results satisfy the previous well known results.