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A new contraction condition for multivalued maps in metric spaces is introduced and then, based on this new condition, we prove two fixed point theorems for such contractions. The new condition uses the altering distance technique and a Pompeiu type metric on the family of nonempty and closed subsets of a metric space. Our results essentially compliments and generalizes some well known results. As application, we model a nonconvex Hammerstein type integral inclusion and prove an existence theorem for this problem. [ABSTRACT FROM AUTHOR]

We study universal consistency and convergence rates of simple nearest-neighbor prototype rules for the problem of multiclass classification in metric spaces. We first show that a novel data-dependent partitioning rule, named Proto-NN, is universally consistent in any metric space that admits a universally consistent rule. Proto-NN is a significant simplification of OptiNet, a recently proposed compression-based algorithm that, to date, was the only algorithm known to be universally consistent in such a general setting. Practically, Proto-NN is simpler to implement and enjoys reduced computational complexity. We then proceed to study convergence rates of the excess error probability. We first obtain rates for the standard k-NN rule under a margin condition and a new generalized-Lipschitz condition. The latter is an extension of a recently proposed modified-Lipschitz condition from Rd to metric spaces. Similarly to the modified-Lipschitz condition, the new condition avoids any boundness assumptions on the data distribution. While obtaining rates for Proto-NN is left open, we show that a second prototype rule that hybridizes between k-NN and Proto-NN achieves the same rates as k-NN while enjoying similar computational advantages as Proto-NN. However, as k-NN, this hybrid rule is not consistent in general. [ABSTRACT FROM AUTHOR]

Explicit expressions and computational approaches are given for the Fortet–Mourier distance between a positively weighted sum of Dirac measures on a metric space and a positive finite Borel measure. Explicit expressions are given for the distance to a single Dirac measure. For the case of a sum of several Dirac measures one needs to resort to a computational approach. In particular, two algorithms are given to compute the Fortet–Mourier norm of a molecular measure, i.e. a finite weighted sum of Dirac measures. It is discussed how one of these can be modified to allow computation of the dual bounded Lipschitz (or Dudley) norm of such measures. [ABSTRACT FROM AUTHOR]

Permutation-based Indexing (PBI) approaches have been proven to be particularly effective for conducting large-scale approximate metric searching. These methods rely on the idea of transforming the original metric objects into permutation representations, which can be efficiently indexed using data structures such as inverted files. The standard conceptualization of permutation associated with a metric object involves only the use of object distances and their relative orders from a set of anchors called pivots. In this paper, we generalized this definition in order to enlarge the class of permutation representations that can be used by PBI approaches. In particular, we introduced the concept of permutation induced by a space transformation and a sorting function, and we investigated which properties these transformations should possess to produce permutations that are effective for metric search. Furthermore, as a practical outcome, we defined a new type of permutation representation that is calculated using distances from pairs of pivots. This proposed technique allowed us to produce longer permutations than traditional ones for the same number of object-pivot distance calculations. The advantage lies in the fact that when longer permutations are employed, the use of inverted files built on permutation prefixes leads to greater efficiency in the search phase. • Permutation-based Indexing methods are effective for large-scale metric search. • Definition of permutations induced by a space transformation. • Study of space transformation properties to induce good permutations for metric search. • Novel permutations calculated using both object-pivot and pivot-pivot distances. [ABSTRACT FROM AUTHOR]

Let (X, d) be a metric space and Iso(X, d) the associated isometry group. We study the subadditivity of the minimal displacement function f : → Iso(X, d) R for different metric spaces. When (X, d) is ultrametric, we prove that the minimal displacement function is subadditive. We show, by a simple algebraic argument, that subadditivity does not hold for the direct isometry group of the hyperbolic plane. The same argument can be used for other metric spaces. [ABSTRACT FROM AUTHOR]

We give a permutation model in which Stone's theorem (every metric space is paracompact) is false and the Boolean Prime Ideal Theorem (every ideal in a Boolean algebra extends to a prime ideal) is true. The erring metric space in our model attains only rational distances and is not metacompact. Transfer theorems give the comparable independence in the Zermelo-Fraenkel setting, answering a question of Good, Tree, and Watson. [ABSTRACT FROM AUTHOR]

We study (p -harmonic) singular functions, defined by means of upper gradients, in bounded domains in metric measure spaces. It is shown that singular functions exist if and only if the complement of the domain has positive capacity, and that they satisfy very precise capacitary identities for superlevel sets. Suitably normalized singular functions are called Green functions. Uniqueness of Green functions is largely an open problem beyond unweighted R n , but we show that all Green functions (in a given domain and with the same singularity) are comparable. As a consequence, for p -harmonic functions with a given pole we obtain a similar comparison result near the pole. Various characterizations of singular functions are also given. Our results hold in complete metric spaces with a doubling measure supporting a p -Poincaré inequality, or under similar local assumptions. [ABSTRACT FROM AUTHOR]

Introduction/purpose: This paper establishes some new results of Piri–Kumam–Dung-type mappings in a complete metric space.Тhe goal was to improve the already published results. Methods: Using the property of a strictly increasing function as well as the known Lemma formulated in (Radenović et al, 2017), the authors have proved that a Picard sequence is a Cauchy sequence. Results: New results were obtained concerning the F – contraction mappings of S in a complete metric space. To prove it, the authors used only property (W1). Conclusion: The authors believe that the obtained results represent a significant improvement of many known results in the existing literature. [ABSTRACT FROM AUTHOR]

As composites of constant, finite (co)product, identity, and powerset functors, Kripke polynomial functors form a relevant class of Set -functors in the theory of coalgebras. The main goal of this paper is to expand the theory of limits in categories of coalgebras of Kripke polynomial functors to the context of quantale-enriched categories. To assume the role of the powerset functor we consider "powerset-like" functors based on the Hausdorff V -category structure. As a starting point, we show that for a lifting of a Set -functor to a topological category X over Set that commutes with the forgetful functor, the corresponding category of coalgebras over X is topological over the category of coalgebras over Set and, therefore, it is "as complete" but cannot be "more complete". Secondly, based on a Cantor-like argument, we observe that Hausdorff functors on categories of quantale-enriched categories do not admit a terminal coalgebra. Finally, in order to overcome these "negative" results, we combine quantale-enriched categories and topology à la Nachbin. Besides studying some basic properties of these categories, we investigate "powerset-like" functors which simultaneously encode the classical Hausdorff metric and Vietoris topology and show that the corresponding categories of coalgebras of "Kripke polynomial" functors are (co)complete. [ABSTRACT FROM AUTHOR]

In this manuscript, we introduce the notion of admissible hybrid Geraghty contraction and we investigate the existence of fixed points of such mappings in the setting of complete metric spaces. Our results not only extend and generalize several results in the fixed point theory literature, but also unify most of them. We give some corollaries to illustrate the novelty of the main result. [ABSTRACT FROM AUTHOR]

Nearest neighbor search is a fundamental and essential operation in applications from many domains, such as databases, machine learning, multimedia, and computer vision. Because exact searching results are not efficient for a high-dimensional space, a lot of efforts have turned to approximate nearest neighbor search. Although many algorithms have been continuously proposed in the literature each year, there is no comprehensive evaluation and analysis of their performance. In this paper, we conduct a comprehensive experimental evaluation of many state-of-the-art methods for approximate nearest neighbor search. Our study (1) is cross-disciplinary (i.e., including 19 algorithms in different domains, and from practitioners) and (2) has evaluated a diverse range of settings, including 20 datasets, several evaluation metrics, and different query workloads. The experimental results are carefully reported and analyzed to understand the performance results. Furthermore, we propose a new method that achieves both high query efficiency and high recall empirically on majority of the datasets under a wide range of settings. [ABSTRACT FROM AUTHOR]

Triangular fuzzy normed space and demonstration of fixed point theorems in normed space unescorted by any analytical structure is studied in this paper. Now we study space with a linear system but non-linear mapping is considered. We reduce our consideration to normed linear space our leading conclusions expanded to common locally convex space. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present a generalization of Nadler's fixed point principle for the case of multi-valued graph contractions. Also, a strict fixed point theorem is obtained. Connections to some variational analysis concepts are discussed and applications to generalized coupled fixed point problems are given. [ABSTRACT FROM AUTHOR]

We work in the setting of metric spaces endowed with a ω0-distance. Thanks to two suitable families of functions, we introduce a new type of contraction which we call ω-contraction. We use the ω-contractions in order to establish new and more general results of existence and uniqueness of fixed point. In particular, we stress that as applications of our main result, we get the existence and uniqueness of fixed point for cyclic mappings and mappings that satisfy a contractive condition of integral type. [ABSTRACT FROM AUTHOR]

We prove that two common best proximity point theorems proved by Sadiq Basha [4] and by Mongkolkeha and Kumam [10] can be regarded as a direct consequence of Browder's fixed point theorem [5]. Moreover, the assumptions imposed in their results can be relaxed. We also present some supplementary results and some examples. [ABSTRACT FROM AUTHOR]

In this paper, we extend the results of Wardowski by applying some new conditions on the self map on a complete metric space, concerning the F-contractions defined by Wardowski. We present some fixed point results of Wardowski type. An example is given to demonstrate the novelty of our work. [ABSTRACT FROM AUTHOR]

In this paper, we have investigated one of the basic topological properties, called Metrizability in multiset topological space. Metrizable spaces are those topological spaces which are homeomorphic to a metric space. So, we first give the notion of metric between two multi-points in a finite multiset and studied some significant properties of a multiset metric space. The notion of metrizability is then studied by using this metric. Besides, the Urysohn's lemma which is considered to be one of the important tools in studying some metrization theorems in topology is also discussed in context with multisets. [ABSTRACT FROM AUTHOR]

We continue the work of [4, 2, 3], in which we discuss published assertions concerning fixed points in digital topology - assertions that are incorrect or incorrectly proven; that are severely limited or reduce to triviality; or that we improve upon. [ABSTRACT FROM AUTHOR]

We study on the Rhoades' question concerning the discontinuity problem at fixed point for a self-mapping T of a metric space. We obtain a new solution to this question. Our result generalizes some recent theorems existing in the literature and implies the uniqueness of the fixed point. However, there are also cases where the fixed point set of a self-mapping contains more than 1 element. Therefore, by a geometric point of view, we extend the Rhoades' question to the case where the fixed point set is a circle. We also give a solution to this extended version. [ABSTRACT FROM AUTHOR]

In this paper, we prove new fixed-circle (resp. fixed-disc) results using the bilateral type contractions on a metric space. To do this, we modify some known contractive conditions called the Jaggi-type bilateral contraction and the Dass-Gupta type bilateral contraction. We give some examples to show the validity of our obtained results. Also, we construct an application to rectified linear units activation functions used in the neural networks. This application shows the importance of studying "fixed-circle problem". [ABSTRACT FROM AUTHOR]

We introduce an intrinsic filtration to the magnitude chain complex of a metric space, and study basic properties of the associated spectral sequence of the magnitude homology. As an application, the third magnitude homology of the circle is computed. [ABSTRACT FROM AUTHOR]

Aiming at the quantification of data privacy and data utility in privacy protection, based on the basic principles of metric space and norm, this paper proposed a privacy and data utility metric model. First, it gave the data numerical processing method. And converted the data into a matrix for calculation. Secondly, it introduced a privacy preference function to measure the change of sensitive attributes over time. Then, it analyzed the privacy protection model and quantified the data changes generated by the privacy protection technology. Finally, this paper built a metric space, and gave privacy amount, data utility and privacy protection calculations. Simulation experiments show that the established metric model can effectively reflect the amount of private information. [ABSTRACT FROM AUTHOR]

Let (M, d) be a metric space, .... In this paper, we will give sufficient conditions ensuring that a Frum-Ketkov operator is weakly Picard. Some generalized Frum-Ketkov operators are also studied. [ABSTRACT FROM AUTHOR]

In this chapter, neutrosophic triplet partial g - metric spaces are obtained. Then, some definitions and examples are given for neutrosophic triplet partial g - metric space. Based on these definitions, new theorems are given and proved. In addition, it is shown that neutrosophic triplet partial g - metric spaces are different from the classical g - metric spaces, neutrosophic triplet metric spaces. Thus, we add a new structure in neutrosophic triplet theory. Also, thanks to neutrosophic triplet partial g - metric space, researchers can obtain new fixed point theorems for neutrosophic triplet theory. [ABSTRACT FROM AUTHOR]

Ordinal data analysis is an interesting direction in machine learning. It mainly deals with data for which only the relationships ' < ', ' = ', ' > ' between pairs of points are known. We do an attempt of formalizing structures behind ordinal data analysis by introducing the notion of ordinal spaces on the base of a strict axiomatic approach. For these spaces we study general properties as isomorphism conditions, connections with metric spaces, embeddability in Euclidean spaces, topological properties etc. [ABSTRACT FROM AUTHOR]

A G-process is briefly a process ([A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, Vol. 182 (Springer, 2013)], [C. M. Dafermos, An invariance principle for compact processes, J. Differential Equations9 (1971) 239–252], [P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, Vol. 176 (Amer. Math. Soc., 2011)]) for which the role of evolution parameter is played by a general topological group G. These processes are broad enough to include the G -actions (characterized as autonomous G -processes) and the two-parameter flows (where G = ℝ). We endow the space of G -processes with a natural group structure. We introduce the notions of orbit, pseudo-orbit and shadowing property for G -processes and analyze the relationship with the G -processes group structure. We study the equicontinuous G -processes and use it to construct nonautonomous G -processes with the shadowing property. We study the global solutions of the G -processes and the corresponding global shadowing property. We study the expansivity (global and pullback) of the G -processes. We prove that there are nonautonomous expansive G -processes and characterize the existence of expansive equicontinuous G -processes. We define the topological stability for G -processes and prove that every expansive G -process with the shadowing property is topologically stable. Examples of nonautonomous topologically stable G -processes are given. [ABSTRACT FROM AUTHOR]

Triangular fuzzy normed space and demonstration of fixed-point theorems in normed space unescorted by any analytical structure is studied in this paper. Now we study space with a linear system but non-linear mapping is considered. We reduce our consideration to normed linear space our leading conclusions expanded to common locally convex space. [ABSTRACT FROM AUTHOR]

The aim of this paper is to initiate the study of best proximity point and optimal coincidence point results of some a Pata-proximal admissible contraction of type-I and type-II in the framework of complete metric space. Some examples are presented to support the results obtained herein. Our results unify, extend and generalize various existing results in literature. [ABSTRACT FROM AUTHOR]

In this paper, by weakening the conditions on the gauge function 𝜓, some new fixed point and common fixed point (common coupled fixed point, common tripled fixed point) theorems for nonlinear mappings with a gauge function 𝜓 in Menger probabilistic metric spaces are established. An example is given to illustrate our theory. [ABSTRACT FROM AUTHOR]

In this paper, we investigate one of the classical problems of the metric fixed point theory: Whether there is a contraction condition which does not force the mapping to be continuous at the fixed point. We propose a contraction conditions in rational form that has a unique fixed point but not necessarily continuous at the given fixed point. [ABSTRACT FROM AUTHOR]

We introduce the notion of generalized b-metric, as a b-metric which can take infinite values, and prove the existence and uniqueness of the completion of some particular b-metric spaces (called generalized strong b-metric spaces). Some fixed point results in b-metric spaces and their counterparts in generalized b-metric spaces are proved. [ABSTRACT FROM AUTHOR]

The canonical generalizations of two classical norms on Besov spaces are shown to be equivalent even in the case of nonlinear Besov spaces, that is, function spaces consisting of functions taking values in a metric space and equipped with some Besov-type topology. The proofs are based on atomic decomposition techniques and metric embeddings. Additionally, we provide embedding results showing how nonlinear Besov spaces embed in nonlinear p-variation spaces, and vice versa. We emphasize that we assume neither the unconditional martingale difference property of the involved spaces nor their separability. [ABSTRACT FROM AUTHOR]

In this paper we introduce a new class of mappings, obtained by merging the concepts of almost contraction and convex contraction and called almost convex contractions. This general class includes both the almost contractions and convex contractions. Existence fixed point theorems for almost convex contractions of order 2 and of order p are established. [ABSTRACT FROM AUTHOR]

An important property of metric spaces is the existence and uniqueness of completion. In this paper, we prove existence and uniqueness of completion of a complex valued strong b-metric space. [ABSTRACT FROM AUTHOR]

In this paper, the equivalence between multi-valued maps satisfying the Mizoguchi-Takahashi's uniformly locally contractive condition and multi-valued maps satisfying the Nadler's uniformly locally contractive condition is obtained on metrically convex space. We have provided examples to illustrate that this equivalence need not be true on any arbitrary metric space. [ABSTRACT FROM AUTHOR]

In this paper, introduce the notion of an interpolative Hardy-Rogers type Z -contraction and we revisit the renowned Hardy-Rogers contraction in the framework of interpolation. We investigate the existence of fixed points for such mappings in the context of metric spaces and list the immediate consequences that cover some existing results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, our aim is to obtain new fixed-disc results on metric spaces. To do this, we present a new approach using the set of simulation functions and some known fixed-point techniques. We do not need to have some strong conditions such as completeness or compactness of the metric space or continuity of the self-mapping in our results. Taking only one geometric condition, we ensure the existence of a fixed disc of a new type contractive mapping. [ABSTRACT FROM AUTHOR]

In this paper we continue our earlier work about topological first passage percolation and answer certain questions asked in our previous paper. Notably, we prove that apart from trivialities, in the generic configuration there exists exactly one geodesic ray, in the sense that we consider two geodesic rays distinct if they only share a finite number of edges. Moreover, we show that in the generic configuration any not too small and not too large convex set arises as the limit of a sequence B (t n) t n for some t n → ∞ . Finally, we define topological Hilbert first passage percolation, and amongst others we prove that certain geometric properties of the percolation in the generic configuration guarantee that we consider a setting linearly isomorphic to the ordinary topological first passage percolation. [ABSTRACT FROM AUTHOR]

Limit of sequences is a basic concept in mathematical analysis. In this paper we study it in more details using another basic concept of analysis, measure on sets of positive integers. A key role in our considerations is played by the concept of a degree of convergence of a given sequence to a given point with respect to a particular measure on the set of positive integers, as a number in interval [ 0 , 1 ]. We study its properties depending on properties of the chosen measure. It appears that standard limits and their known generalizations (convergence with respect to a filter or ideal) are extremal special cases in our approach. [ABSTRACT FROM AUTHOR]

The purpose of this article, we consider the existence of a unique best proximity point x*ξ A such that d(x*; Tx*) = dist(A;B) for generalized '-weak contraction mapping T : A → B; where A;B(6=;) are subsets of a metric space (X; d). [ABSTRACT FROM AUTHOR]

Let (K , d) be a compact metric space, 0 < α ≤ 1 and Lip α K the space of the Lipschitz functions on K. It is known that the Banach algebra Lip α K is a BSE-algebra. In this paper, for a commutative unital semisimple Banach algebra A , we prove that the Banach algebra Lip α (K , A) of the A -valued Lipschitz functions is a BSE-algebra if and only if A is so. [ABSTRACT FROM AUTHOR]

In this paper we define the notions of a norm and a prenorm on BL-algebras and we give some methods for constructing them. We introduce (pre)metrics on BL-algebras and find some connections between them and (pre)norms. Using prenorms, we find some conditions that a BL-algebra become a metric space. Finally, we show that if there exists a prenorm on a BL-algebra A, then for any filter F in A, there are topologies U and V on A and on A/F respectively such that (A , U) and (A / F , V) are (semi)topological BL-algebras. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study a Lipschitz version of the Bishop–Phelps–Bollobás property (Lip-BPB property). This property deals with the possibility of making a uniformly simultaneous approximation of a Lipschitz map F and a pair of points at which F almost attains its norm by a Lipschitz map G and a pair of points such that G strongly attains its norm at the new pair of points. We first show that if M is a finite pointed metric space and Y is a finite-dimensional Banach space, then the pair (M , Y) has the Lip-BPB property, and that both finiteness assumptions are needed. Next, we show that if M is a uniformly Gromov concave pointed metric space (i.e. the molecules of M form a set of uniformly strongly exposed points), then (M , Y) has the Lip-BPB property for every Banach space Y. We further prove that this is the case for finite concave metric spaces, ultrametric spaces, and Hölder metric spaces. The extension of the Lip-BPB property from (M , R) to some Banach spaces Y and some results for compact Lipschitz maps are also discussed. [ABSTRACT FROM AUTHOR]

Logical systems containing knowledge and know-how modalities have been investigated in several recent works. Independently, epistemic modal logics in which every knowledge modality is labeled with a degree of uncertainty have been proposed. This article combines these two research lines by introducing a bimodal logic containing knowledge and know-how modalities, both labeled with a degree of uncertainty. The main technical results are soundness, completeness, and incompleteness of the proposed logical system with respect to two classes of semantics. [ABSTRACT FROM AUTHOR]

Topology is a branch of mathematic that deals with the specific definitions given for spatial structure concepts, compares different definitions and explores the connections between the structures described on the sets. Also, neutrosophic triplet metric and neutrosophic triplet topology are a new concept in neutrosophy and they are completely different from classical structures. In this paper, we firstly study neutrosophic triplet metric topology. Furthermore, we give some new definitions and properties for neutrosophic triplet metric space, neutrosophic triplet topology and neutrosophic triplet metric topology. Thus, we obtain neutrosophic triplet metric topology using the neutrosophic triplet metric and neutrosophic triplet topology. Also, we show relationship between neutrosophic triplet metric space and neutrosophic triplet topology. [ABSTRACT FROM AUTHOR]

In this paper, we are revisiting Meir-Keeler type fixed operators in the setting of Branciari distance space. The presented results improve and generalize several existing results in the literature. We consider an example to illustrate our result. [ABSTRACT FROM AUTHOR]

We study ordinal approximation algorithms for maximum-weight bipartite matchings. Such algorithms only know the ordinal preferences of the agents/nodes in the graph for their preferred matches, but must compete with fully omniscient algorithms which know the true numerical edge weights (utilities). Ordinal approximation is all about being able to produce good results with only limited information. Because of this, one important question is how much better the algorithms can be as the amount of information increases. To address this question for forming high-utility matchings between agents in X and Y , we consider three ordinal information types: when we know the preference order of only nodes in X for nodes in Y , when we know the preferences of both X and Y , and when we know the total order of the edge weights in the entire graph, although not the weights themselves. We also consider settings where only the top preferences of the agents are known to us, instead of their full preference orderings. We design new ordinal approximation algorithms for each of these settings, and quantify how well such algorithms perform as the amount of information given to them increases. [ABSTRACT FROM AUTHOR]

Evaluation criteria play a key role in assessing the performances of hydrological models. Most previous criteria are based on the standard least square method, which assumes model residuals to be homoscedastic and is, therefore, not suitable for assessing cases with heteroscedastic residuals. Here, we compared a heteroscedastic and symmetric efficiency (HSE) criterion with the Nash--Sutcliffe efficiency (NSE) and the heteroscedastic maximum-likelihood estimator (HMLE) by running a monthly water balancemodel with four parameters (i.e., the abcd model) in 138 basins located in the continental United States derived from the Model Parameter Estimation Experiment dataset. The results show that compared to the NSE, the HSE and HMLE are both more effective for stabilizing variance and producing more uniform performances with flow magnitude, and the latter is slightly more effective than the former on stabilizing the residual heteroscedasticity, with the aid of an additional parameter. [ABSTRACT FROM AUTHOR]

In this work, we are interested to prove a general fixed point theorem for a mapping satisfying a new type of implicit relation in G-metric spaces. The results in this paper generalize the results obtained in [3], [7], [8], [10] and obtain other particular results. [ABSTRACT FROM AUTHOR]