Metric Spaces

and Florida, my path started in Cleveland, OH. I thank my family and friends in Cleveland, who supported and nurtured my ambitions to get into this field. Knowing that I have my sisters, brother, parents, cousins, and everyone else in my huge family there when I need them has been a great comfort throughout the dissertation process. To my little sister Nicole, I'm ready to watch you be the next one to be called "doctor". Just keep pushing yourself. You'll get there. And to my grand-aunt, Phyllis Robinson, I wish you could be around to watch me walk across the stage one more time. Thank you for everything you've done. I'll carry the legacy that started with you into the next phase of my life and onward. When I moved to SUNY Buffalo, in my mind, I was starting my own crusade. I started alone. I had a plan. But life had a different one. I ended up where I am at the time of this writing because of individuals who went out of their way to support me at every step along this journey. As my first research professor, Shambhu Upadhyaya, I thank you. Not only for introducing me to the world of research, but also for providing me with guidance and support both while at Buffalo and afterward. You were the first to ever tell me I should get a PhD. While I was too naïve to listen to you then, I eventually got around to it. I would also like to acknowledge other professors who were there for me after Buffalo, including Florian Buchholz, Brett Tjaden and Xunhua Wang. The time spent with you and lessons learned gave me the confidence and experience that I needed to pursue a PhD. Finally, thanks to my advisor, Michael Laszlo, my committee members and all the other professors and classmates I've interacted with at Nova Southeastern. I've been a dreamer my entire life, imagining so many ways of helping the world but knowing, at some level, that many things were impossible. But with the support and nurture from the all the staff at BBN, I learned that I could build the impossible. This would have never happened without the support and valuable lessons from many of the seniors at BBN, including Jack Marin, Joshua Edmison, Richard Burne, Carl Powell, as well as all of the staff that I've worked with over the years. You are all just as responsible for shaping me into a contributor to science as all my experiences in University. Dr. John Everett (currently at DARPA) deserves a great deal of thanks for initially supporting the idea of being on my committee, nominating me and serving as a true mentor for the 2015 DARPA Rising session, and being the program manager for the first of many research efforts that I hope to have my name on. I now stand on the shoulders of these giants. And to John-Francis Mergen, my mentor and one of my closest friends, I thank you and Lynne for always being there as family and for believing in me. While knowing you, I've seen you make the case for change in the world as only you can make it. In that time, you've taught me that I can do the same, and you can bet that I will. My lifelong friends deserve just as much acknowledgement for their support as everyone else mentioned here. So to Vera Neroni, Elizabeth Brown, Michelle Chu, Cyrus Chu, Douglas Campese, Jessica Lombardo, Scott Goldweber, and everyone else in the surrogate family that has joined me on my crusade, thank you for getting me to this point. Everyone on this page has contributed to the person I am today with a trust that I will strive to create a better tomorrow. I will not fail that trust. And I will not fail to try. Campbell 3 combinatorial optimization problems in graph theory . In all likelihood, this may have been because the size of data used for the problem was typically smaller, making the problem seem trivial while anything larger was deemed intractable. At the time of this writing, progress in practically solving the problem has only occurred in the last six decades. Much of the true scientific investigation started with Alfonso Shimbel, in his introduction of the all-pairs shortest path (APSP) problem . All possible path queries are automatically answered and stored for the APSP problem, while querying is done upon request for the PPSP problem. The solution to the PPSP problem requires an efficient computation of the shortest path between an arbitrary pair of nodes be established. Shortly after Shimbel, Edsger W. Dijkstra was credited with discovering the algorithm that, at the time of this writing, is the best, most well-known, commonly used, and simplest method of solving the shortest path algorithm in a graph . This algorithm is widely known as Dijkstra's algorithm. A decade after its creation, the A* search algorithm showed, by adding a heuristic that estimates distance, that it could run a shortest path query in significantly faster time than Dijkstra's algorithm . Fundamentally, the A* algorithm is Dijkstra's algorithm that takes into account a distance estimation heuristic derived from characteristics of the graph. While other algorithms have been developed in an attempt to contest them, these two greedy optimization algorithms serve as the basis for most modern day shortest path solutions. As researchers find more use for graph theory in the storage, retrieval, and analysis of big data, extremely fast solutions to problems such as the shortest path Campbell 4 problem are in great demand. However, not even Dijkstra's or the A* algorithm can solve the problem for massive datasets without a significant increase in their requirements for computational time and space. For this reason, modern research focuses on performing computations on the graph prior to allowing it to be queried for shortest path. The results of these computations are used to guide, narrow, or inform the search such that arbitrary queries can be performed significantly faster on graphs that represent huge data corpuses. Modern approaches typically exploit mathematical approximation techniques (Delling,

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Most search methods in metric spaces assume that the topology of the object collection is reasonably regular. However, there exist nested metric spaces, where objects in the collection can be grouped into clusters or subspaces, in such a way that different dimensions or variables explain the differences between objects inside each subspace. This paper proposes a two levels index to solve search problems in spaces with this topology. The idea is to have a first level with a list of clusters, which are identified and sorted using Sparse Spatial Selection (SSS) and Lists of Clusters techniques, and a second level having an index for each dense cluster, based on pivot selection, using SSS. It is also proposed for future work to adjust the second level indexes through dynamic pivots selection to adapt the pivots according to the searches performed in the database.

Hybrid dynamic spatial approximation trees are recently proposed data structures for searching in metric spaces, based on combining the concepts of spatial approximation and pivot based algorithms. These data structures are hybrid schemes, with the full features of dynamic spatial approximation trees and able of using the available memory to improve the query time. It has been shown that they compare favorably against alternative data structures in spaces of medium difficulty. In this paper we complete and improve hybrid dynamic spatial approximation trees, by presenting a new search alternative, an algorithm to remove objects from the tree, and an improved way of managing the available memory. The result is a fully dynamic and optimized data structure for similarity searching in metric spaces.

Resumen El modelo de espacios métricos permite abstraer muchos de los problemas de búsqueda por proximidad. La búsqueda por proximidad tiene múltiples aplicaciones especialmente en el área de bases de datos multimedia. La idea es construir un ındice para la base de ...

Let f : X → Y be a perfect n-dimensional surjective map of paracompact spaces and Y a C-space. We consider the following property of continuous maps g : It is shown that all maps g ∈ C(X, I n+1 ) with the above property form a dense G δ -set in the function space C(X, I n+1 ) equipped with the source limitation topology. Moreover, for every n + 1 ≤ m ≤ ω the space C(X, I m ) contains a dense G δ -set of maps having this property.

Resumen El modelo de bases de datos métrico-temporal permite abordar aquellas situaciones en las que resulta necesario realizar búsquedas por similitud teniendo en cuenta también la componente temporal. En este artículo presentamos una mejora al índice métrico-temporal FHQT-Temporal, que soporta valores continuos de la función de distancia, manteniendo la eficiencia ante cambios de valores del radio de búsqueda e incrementos de los intervalos de tiempo. Además se muestran resultados de la verificación experimental de esta estructura para un conjunto de datos determinado.

A technical error invalidates the proof that the optimal rootmean-square deviation of vector sets induces a metric given in Kaindl & Steipe [Acta Cryst. (1997), A53, 809]. Nevertheless, the conclusions are correct and a revised proof is given in Steipe [Acta Cryst. (2002), A58, 506]. The author is indebted to Dr Xu Huafeng for pointing out the technical error in the previously published manuscript (Kaindl & Steipe, 1997).

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. We will synthetize the analogies between both Koszul and Souriau models, and will reduce their definitions to the exclusive Cartan "Inner Product". Interpreting Legendre transform as Fourier transform in (Min,+) algebra, we conclude with a definition of Entropy given by a relation mixing Fourier/Laplace transforms: Entropy = (minus) Fourier (Min,+) o Log o Laplace (+,X) .

We prove that the functor P of Radon probability measures transforms any open map between completely metrizable spaces into a soft map. This result is applied to establish some properties of Milyutin maps between completely metrizable spaces.

On this paper, with the help of some known results on cone metric spaces, we find a metric with respect to which a class of non-contractive functions on a given metric turn to be contractive. Namely, if we see that for a given mapping, there is a cone metric space in which the mapping is contractive, than we can construct a metric with respect to which the mapping will be contractive as well.

In this paper, we consider the robust interpretation of metric temporal logic (MTL) formulas over timed sequences of states. For systems whose states are equipped with nontrivial metrics, such as continuous, hybrid, or general metric transition systems, robustness is not only natural, but also a critical measure of system performance. In this paper, we define robust, multi-valued semantics for MTL formulas, which capture not only the usual Boolean satisfiability of the formula, but also topological information regarding the distance, ε, from unsatisfiability. We prove that any other timed trace which remains ε-close to the initial one also satisfies the same MTL specification with the usual Boolean semantics. We derive a computational procedure for determining an under-approximation to the robustness degree ε of the specification with respect to a given finite timed state sequence. Our approach can be used for robust system simulation and testing, as well as form the basis for simulation-based verification.

Bourgain's theorem is actually more general, and holds for any lp. (We present a proof for the � 1 case due to Fakcharoenphol, Rao and Talwar (2003) since it has been useful in subsequent developments, as we will see.) Furthermore, examining the proof of the theorem one can derive an efficient algorithm to produce such a low-distortion embedding. (Aside, this is not always the case with mathematical proofs. Later in the course we shall encounter proofs of existence for combinatorial objects that do not entail efficient algorithms to construct these objects.) We start with some notation we will use throughout this scribe. Denote the original metric by (X, d), where X is a set of |X| = n elements and d : X × X � →� + a distance function. Denote by

Querying large datasets by proximity, using a distance under the metric space model, has a large number of applications in multimedia, pattern recognition, statistics, etc. There is an ever growing number of indexes and algorithms for proximity querying, however there is only a handful of indexes able to perform well without user intervention to select parameters. One of such indexes is the Distal Spatial Approximation Tree (DiSAT) which is parameter-less and has demonstrated to be very efficient outperforming other approaches. The main drawback of the DiSAT is its static nature, that is, once built, it is difficult to add or to remove new elements. This drawback prevents the use of the DiSAT for many interesting applications. In this paper we overcome this weakness. We use a standard technique, the Bentley and Saxe algorithm, to produce a new index which is dynamic while retaining the simplicity and appeal for practitioners of the DiSAT. In order to improve the DiSAF performance, we do not attempt to directly apply the Bentley and Saxe technique, but we enhance its application by taking advantage of our deep knowledge of the DiSAT behavior.

The Distal Spatial Approximation Tree (DiSAT) is one of the most competitive indexes for exact proximity searching. The absence of parameters, the most salient feature, makes the index a suitable choice for a practitioner. The most serious drawback is the static nature of the index, not allowing further insertions once it is built. On the other hand, there is an old approach from Bentley and Saxe (BS) allowing the dynamization of decomposable data structures. The only requirement is to provide a decomposition operation. This is precisely our contribution, we define a decomposition operation allowing the application of the BS technique. The resulting data structure is competitive against the static counterparts.

The Dynamic Spatial Approximation Tree (DSAT) is a data structure specially designed for searching in metric spaces. It has been shown that it compares favorably against alternative data structures in spaces of high dimension or queries with low selectivity. The DSAT supports insertion and deletions of elements. However, it has been noted that eliminations degrade the structure over time. is proposed a method to handle deletions over the DSAT, which shown to be superior to the former in the sense that it permits controlling the expected deletion cost as a proportion of the insertion cost. In this paper we propose and study a new deletion method, based on the deletions strategies presented in , which has demonstrated to be better. The outcome is a fully dynamic data structure that can be managed through insertions and deletions over arbitrarily long periods of time without any reorganization.

Metric space searching is an emerging technique to address the problem of similarity searching in many applications. In order to efficiently answer similarity queries, the database must be indexed. In some interesting real applications dynamism is an indispensable property of the index. There are very few actually dynamic indexes that support not only searches, but also insertions and deletions of elements. The dynamic spatial approximation tree (DSAT) is a data structure specially designed for searching in metric spaces, which compares favorably against other data structures in high dimensional spaces or queries with low selectivity. Insertions are efficient and easily supported in DSAT, but deletions degrade the structure over time. Several methods are proposed to handle deletions over the DSAT. One of them has shown to be superior to the others, in the sense that it permits controlling the expected deletion cost as a proportion of the insertion cost and searches does not overly degrade after several deletions. In this paper we propose and study a new alternative deletion method, based on the better existing strategy. The outcome is a fully dynamic data structure that can be managed through insertions and deletions over arbitrarily long periods of time without any significant reorganization.

A diferencia de la aproximación para bases de datos tradicionales, la comunidad de Recuperación de Información siempre ha considerado los resultados de las búsquedas como una lista rankeada de objetos. Dada una consulta, algunos objetos son más relevantes a la especificación de la consulta que otros y los usuarios habitualmente están interesados en los objetos más relevantes, es decir los objetos cuyo ranking es más alto. Este paradigma de búsqueda recientemente se ha generalizado en un modelo en el cual ...

En general, es difícil tanto para los usuarios que intentan recuperar información multimedia poder especificar claramente sus intereses ya través de una consulta bien definida, como para aquéllos que diseñan el sistema decidir cuáles de las características de los objetos multimedia pueden resultar relevantes. La forma en que los datos multimedia se representan, como ellos se almacenan y el costo de transferirlos, entre distintos niveles de la jerarquía de memoria o sobre una red, afectan directamente las respuestas del sistema ...

Among the similarity queries in metric spaces, there are one that obtains the k-nearest neighbors of all the elements in the database (All-k-NN). One way to solve it is the naı̈ve one: comparing each object in the database with all the other ones and returning the k elements nearest to it (k-NN). Another way to do this is by preprocessing the database to build an index, and then searching on this index for the k-NN of each element of the dataset. Answering to the All-k-NN problem allows to build the k-Nearest Neighbor graph (kNNG). Given an object collection of a metric space, the Nearest Neighbor Graph (NNG) associates each node with its closest neighbor under the given metric. If we link each object to their k nearest neighbors, we obtain the k Nearest Neighbor Graph (kNNG).The kNNG can be considered an index for a database, which is quite efficient and can allow improvements. In this work, we propose a new technique to solve the All-k-NN problem which do not use any index to obta...

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. From the few dynamic indexes, even fewer work well in secondary memory. That is, most of them need the index in main memory in order to operate efficiently. In this paper we introduce a secondary-memory variant of the Dynamic Spatial Approximation Tree with Clusters (DSACL-tree) which has shown to be competitive in main memory. The resulting index handles well the secondary memory scenario and is competitive with the state of the art. The resulting index is a much more practical data structure that can be useful in a wide range of database applications.

Many computational applications need to look for informa- tion in a database. Nowadays, the predominance of non- conventional databases makes the similarity search (i.e., searching elements of the database that are "similar" to a given query) becomes a preponderant concept. The Spatial Approximation Tree has been shown that it compares favorably against alternative data structures for similarity searching in metric spaces of medium to high di- mensionality ("difficult" spaces) or queries with low selec- tivity. However, for the construction process the tree root has been randomly selected and the tree ,in its shape and performance, is completely determined by this selection. Therefore, we are interested in improve mainly the searches in this data structure trying to select the tree root so to re- flect some of the own characteristics of the metric space to be indexed. We regard that selecting the root in this way it allows a better adaption of the data structure ...

Since the work of Claude Shannon, Entropy has been introduced axiomatically. The recent extension of information geometry to Lie groups via symplectic models of statistical mechanics, initiated by Jean-Marie Souriau (Lie groups thermodynamics), makes it possible to give a purely geometric "constructive" definition of the Entropy. Starting from the symmetry group which acts on the data or the system, the Entropy is constructed as an invariant Casimir function on the symplectic foliation generated by the co-adjoint orbits of the group (co-adjoint orbits constructed by the action of the group on the moment map, where the moment map is a geometrization of Noether's theorem). The level sets of Entropy thus appear as the leaves of this symplectic foliation generated by the co-adjoint action. In the case of the symmetry group of Euclidean space, we find the classical Shannon Entropy, which is therefore a special case of this model. We therefore move from Shannon's axiomatic definition to a geometric “constructive” definition of Entropy. Symmetry generates Entropy via moment map.
In the theory of representations of Lie groups introduced by Alexandre Kirillov, we associate with these co-adjoint orbits a symplectic manifold via a 2 KKS form (Kirillov- Kostant -Souriau) and an associated Riemannian metric. As part of the extension of Information geometry to Lie groups, this metric appears exactly like the Fisher-Koszul metric . Thanks to the cohomology of Lie algebra and the Souriau cocycle , we can deal with cases where the co-adjoint operator is not equivariant (case of non-null cohomology) .
This model makes it possible to extend the notion of Gaussian probability density for an informational measurement on Lie groups or for homogeneous spaces on which a Lie group acts, by considering the Gibbs density at Maximum Entropy, which has the property of being covariant under the action of the group. We will illustrate this fact by giving the definition of a Gaussian with maximum entropy for the half-plane and the Poincaré disk via the group SL(2,R) and SU(1,1), and for the covariance matrices symmetric positive definite SPD via the group SL( n,R ) and SU( n,n ) (the SDP matrices being considered as the pure imaginary axis of the Siegel half-space).
This symplectic foliation structure, level sets of Entropy, also makes it possible to interpret the 2nd principle of thermodynamics by studying its transverse structures. Initiated by Charles Ehresmann and Georges Reeb , the theory of foliations makes it possible to give the geometry structure of thermodynamics, made up of a symplectic foliation (describing non-dissipative dynamics) and a transverse structure given by a Riemannian foliation (describing dissipative dynamics with creation of entropy). The metric which appears on the transverse Riemannian foliation is the dual metric of the Fisher metric, that is to say the Hessian of Entropy. We thus have 2 transverse foliations associated with the 2 dual metrics of information geometry. The dynamics on these 2 transverse foliations is given by the equation of the “metriplectic” flow with a Poisson bracket to move on the symplectic leaves (non-dissipative flow because we move on the Entropy level sets; l entropy being an invariant function of Casimir in coadjoint representation on symplectic leaves) and a metric bracket to move on the transverse Riemannian leaves (production of Entropy at constant Energy; Energy is a function of Casimir for the metric bracket on these leaves which are therefore the level sets of Energy).
Recently, it was shown that the metriplectic flow was compatible with Onsager relations in non-equilibrium thermodynamics. The metriplectic flow is used in machine learning for TINNs ( Thermodynamics-Informed Neural Network) and to stabilize  bosonic CAT and GKP Qubits of quantum computers (the dissipative Linblad equation is a linear approximation of the metriplectic equation).
This model is studied within the framework of 2 European actions, COST CaLISTA ( https://site.unibo.it/calista/en ) and MSCA CaLIGOLA ( https://site.unibo.it/caligola/en ), and also in the “North Paris Basin” seminar on “geometric structures of dissipation”.
References :
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[2] Barbaresco, F. (2023). Symplectic Foliation Transverse Structure and Libermann Foliation of Heat Theory and Information Geometry. In: Nielsen, F., Barbaresco, F. ( eds ) Geometric Science of Information. GSI 2023. Lecture Notes in Computer Science, vol 14072. Springer, Cham. https://doi.org/10.1007/978-3-031-38299-4_17 ; https://link.springer.com/chapter/10.1007/978-3-031-38299-4_17 
[3] Barbaresco F. (2022). Symplectic Foliation Structures of Non-Equilibrium Thermodynamics as Dissipation Model: Application to Metriplectic Non-Linear Lindblad Quantum Master Equation, MDPI special Issue "Geometric Structure of Thermodynamics: Theory and Applications", 2022, Entropy 2022, 24(11), 626; https://doi.org/10.3390/e24111626
[4] Barbaresco F. (2024). Symplectic Foliation Model of Sadi Carnot's Thermodynamics: from Carathéodory's seminal idea to Souriau's Lie Groups Thermodynamics, Celebration of 200 years since Sadi Carnot's Reflections on the motive power of Fire, 1824–2024 & the Carnots , https://carnotlille2024.sciencesconf.org/ resource/page/id/2
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Although the success rate of handwritten character recognition using a nearest neighbour technique together with edit distance is satisfactory, the exhaustive search is expensive. Some fast methods as AESA and LAESA have been proposed to find nearest neighbours in metric spaces. The average number of distances computed by these algorithms is very low and does not depend on the number of prototypes in the training set. In this paper, we compare the behaviour of these algorithms when string and tree edit distances are used.

Although the success rate of handwritten character recognition using a nearest neighbour technique together with edit distance is satisfactory, the exhaustive search is expensive. Some fast methods as AESA and LAESA have been proposed to find nearest neighbours in metric spaces. The average number of distances computed by these algorithms is very low and does not depend on the number of prototypes in the training set. In this paper, we compare the behaviour of these algorithms when string and tree edit distances are used.

Pivot-based algorithms are effective tools for proximity searching in metric spaces. They allow trading space overhead for number of distance evaluations performed at query time. With additional search structures (that pose extra space overhead) they can also reduce the amount of side computations. We introduce a new data structure, the Fixed Queries Array (FQA), whose novelties are (1) it permits

We present an efficient method for approximate search in a combination of several metric spaces-which are a generalization of low level image features-using an inverted index. Our approximation gives very high recall with subsecond response time on a real data set of one million images extracted from Flickr. We further exploit the inverted index to improve efficiency of the query processing by combining our search in metric features with search in associated textual metadata.

Mustafa and Sims [12] introduced the notion of G-metric as a possible generalization of usual notion of a metric space. The author generalized the notion of G-metric to more than three variables and introduced the concept of Generalized n-metric spaces [10]. In this paper, We prove Banach fixed point theorem and a Suzuki-type fixed point theorem in Generalized n-metric spaces. We also discuss applications to certain functional equations arising in dynamic programming.

In high-dimensional and complex metric spaces, determining the nearest neighbor (NN) of a query object Õ can be a very expensive task, because of the poor partitioning operated by index structures-the so-called "curse of dimensionality". This also affects approximately correct (AC) algorithms, which return as result a point whose distance from Õ is less than´½ •¯µ times the distance between Õ and its true NN. In this paper we introduce a new approach to approximate similarity search, called PAC-NN queries, where the error bound¯can be exceeded with probability AE and both¯and AE parameters can be tuned at query time to trade the quality of the result for the cost of the search. We describe sequential and index-based PAC-NN algorithms that exploit the distance distribution of the query object in order to determine a stopping condition that respects the error bound. Analysis and experimental evaluation of the sequential algorithm confirm that, for moderately large data sets and suitableā nd AE values, PAC-NN queries can be efficiently solved and the error controlled. Then, we provide experimental evidence that indexing can further speed-up the retrieval process by up to 1-2 orders of magnitude without giving up the accuracy of the result.

The Pseudo-metric spaces which have the property that all continuous real valued functions are uniformly continuous have been studied. It is proved that the following three conditions on pseudo-metric space X are equivalent a] Every continuous real valued function on X is uniformly continuous. b] Every sequence {x n } in X with lim d(x n) = 0 has a convergent subsequence. c] Set A is compact and for every 1 > 0, there is 2 > 0 such that d(x, A) > 1 implies d(x) > 2. Here A = set of all limit points of X and d(x) = d(x, X-{x}) Further it is proved that in a pseudo-metric space X, a subset E of X is compact if and only if every continuous function f:E → R is uniformly continuous and for every > 0 the set {x E / d(x) > } is finite.

Topology may be considered as an abstract study of the limit point concept. As such, it stems in part from recognition of the fact that many important mathematical topics depend entirely upon the properties of limit points. This study shows that compactness implies limit point compactness but not conversely and every compact space is locally compact but not conversely. This study also shows that compactness, limit point compactness and sequentially compactness are equivalent in metrizable spaces and the product of finitely many compact spaces is a locally compact space. This study introduce it here as an interesting application of the Tychonoff theorem.

The talk will explore the interconnectedness between language, mathematics, and physics, drawing parallels between concepts such as metric spaces in mathematics and the fabric of spacetime in physics, as well as their analogies in linguistic structures. It will address how languages are viewed as relational schemes, emphasizing the importance of understanding the relationships between symbols rather than their individual content.

The discussion of topics such as code-switching and translanguaging, respectively in the field of [visual] multi-language poetry and asemic writing, aims to highlight how these practices may challenge conventional notions of linguistic boundaries and signification, ultimately influencing the perception of the world through the synthesis of competing linguistic structures.

In this article, we introduce a new and simple approach to coupled and tripled coincidence point theory. By using our method, we establish coupled coincidence point results of Lakshmikantham and Ćirić, Binayak et al., Alotaibi and Alsulami without any type of commutativity condition on F and g. We also use our technique to prove tripled coincidence point results of Borcut and Berinde without commutativity of maps. Also, we give a supporting example of non-commuting, non-compatible mappings where the above mentioned results can not be applied. Mathematics Subject Classification: Primary, 47H10; Secondary, 54H25; 34B15.

In this paper, we introduce fuzzy generalized β-F-contractions as a generalization of fuzzy F-contractions with admissible mappings. We deduce sufficient conditions for the existence and uniqueness of fixed points for fuzzy generalized β-F-contractions in complete strong fuzzy metric spaces. Our results generalize several fixed-point results from the literature. We present an application of our main result.

According to the recent definition of efficiency via improvement sets, the aim of this paper is to characterize the set of optimal points for a set. New existence results are proved in multicriteria situations, and their novelty is illustrated via several examples. Moreover, the study of an economic model is provided as an example of application of our achievements.

According to the recent definition of efficiency via improvement sets, the aim of this paper is to characterize the set of optimal points for a set. New existence results are proved in multicriteria situations, and their novelty is illustrated via several examples. Moreover, the study of an economic model is provided as an example of application of our achievements.

The purpose of this paper is to prove some fixed point theorems satisfying rational type of contractive condition due to Jaggi [1] in the setting of dislocated-quasi metric spaces. Also for A-Contraction, a general class of contraction defined by Akram et al. [2], established a common fixed point theorem for a pair of mappings in dislocated metric space.

In this paper, we establish some stability results for the Jungck-Mann, Jungck-Krasnoselskij and Jungck iteration processes in arbitrary Banach spaces. These results are proved for a pair of nonselfmappings using the Jungck-Mann, Jungck-Krasnoselskij and Jungck iterations. Our results are generalizations and extensions to a multitude of stability results in literature including those of Imoru and Olatinwo [8], Jungck [10], Berinde [1] and many others.