Search

The paper contains a survey of several results from the author's papers and joint papers by the author and Zakharyan, both on the zero existence and approximation for single-valued and multivalued (α, β)-search functionals, and also on the zero existence preservation for parametric family of such functionals, under the parameter changing. Some corollaries of these results in the fixed point and coincidence theory of single-valued and multivalued mappings of metric spaces are also given. The comparison is provided with some known results by other authors. In the concluding part of the paper, we investigate the problem on the existence of a parameter-continuous single-valued branch of zeros for a parametric family of search functionals. A theorem on the existence of a solution of this problem is proved. [ABSTRACT FROM AUTHOR]

Recently, the fixed-point theorem for fuzzy contractive mappings has been investigated within the framework of intuitionistic fuzzy metric-like spaces. This interesting topic was explored through the utilization of G-Cauchy sequences as defined by Grabiec. The aim of this study is to enhance the aforementioned results in a few aspects. Initially, the proof of the fixed-point theorem is simplified and condensed, allowing for potential generalization to papers focusing on similar fixed-point analyses. Furthermore, instead of G-Cauchy sequences, the classical Cauchy sequences proposed by George and Veeramani are examined, incorporating an additional condition on the fuzzy metric. Within this context, a solution to an old unresolved question posed by Gregory and Sapena is provided. The findings are reinforced by relevant examples. Finally, the introduced fuzzy metrics are applied to the field of image processing. [ABSTRACT FROM AUTHOR]

The goal of this paper is to investigate complete noncompact warped product gradient Ricci solitons. Nonexistence results, estimates for the warping function and for its gradient are proven. When the soliton is steady or expanding these nonexistence results generalize to a broader context certain estimates and rigidity obtained when studying warped product Einstein manifolds. When the soliton is shrinking, it is presented as a nonexistence theorem with no counterpart in the Einstein case, which is proved using properties of the first eigenvalue of a weighted Laplacian. [ABSTRACT FROM AUTHOR]

A sublinear bilipschitz equivalence (SBE) between metric spaces is a map from one space to another that distorts distances with bounded multiplicative constants and sublinear additive error. Given any sublinear function, the associated sublinearly Morse boundaries are defined for all geodesic proper metric spaces as a quasi‐isometrically invariant and metrizable topological space of quasi‐geodesic rays. In this paper, we prove that sublinearly‐Morse boundaries of proper geodesic metric spaces are invariant under suitable SBEs. A tool in the proof is the use of sublinear rays, that is, sublinear bilispchitz embeddings of the half line, generalizing quasi‐geodesic rays. As an application, we distinguish a pair of right‐angled Coxeter groups brought up by Behrstock up to SBE. We also show that under mild assumptions, generic random walks on countable groups are sublinear rays. [ABSTRACT FROM AUTHOR]

Deep neural networks have made tremendous gains in emulating human-like intelligence, and have been used increasingly as ways of understanding how the brain may solve the complex computational problems on which this relies. However, these still fall short of, and therefore fail to provide insight into how the brain supports strong forms of generalization of which humans are capable. One such case is out-of-distribution (OOD) generalization - successful performance on test examples that lie outside the distribution of the training set. Here, we identify properties of processing in the brain that may contribute to this ability. We describe a two-part algorithm that draws on specific features of neural computation to achieve OOD generalization, and provide a proof of concept by evaluating performance on two challenging cognitive tasks. First we draw on the fact that the mammalian brain represents metric spaces using grid cell code (e.g., in the entorhinal cortex): abstract representations of relational structure, organized in recurring motifs that cover the representational space. Second, we propose an attentional mechanism that operates over the grid cell code using determinantal point process (DPP), that we call DPP attention (DPP-A) - a transformation that ensures maximum sparseness in the coverage of that space. We show that a loss function that combines standard task-optimized error with DPP-A can exploit the recurring motifs in the grid cell code, and can be integrated with common architectures to achieve strong OOD generalization performance on analogy and arithmetic tasks. This provides both an interpretation of how the grid cell code in the mammalian brain may contribute to generalization performance, and at the same time a potential means for improving such capabilities in artificial neural networks. [ABSTRACT FROM AUTHOR]

To a metric space X we associate a compact topological space ν ′ (X) called the corona of X. Then a coarse map f : X → Y between metric spaces is mapped to a continuous map ν ′ (f) : ν ′ (X) → ν ′ (Y) between coronas. Sheaf cohomology assigned to a coarse metric space is preserved and reflected by the corona functor. This work reveals new tools to analyze the Higson corona. [ABSTRACT FROM AUTHOR]

A large number of real-world optimization problems can be formulated as Mixed Integer Linear Programs (MILP). MILP solvers expose numerous configuration parameters to control their internal algorithms. Solutions, and their associated costs or runtimes, are significantly affected by the choice of the configuration parameters, even when problem instances have the same number of decision variables and constraints. On one hand, using the default solver configuration leads to suboptimal solutions. On the other hand, searching and evaluating a large number of configurations for every problem instance is time-consuming and, in some cases, infeasible. In this study, we aim to predict configuration parameters for unseen problem instances that yield lower-cost solutions without the time overhead of searching-and-evaluating configurations at the solving time. Toward that goal, we first investigate the cost correlation of MILP problem instances that come from the same distribution when solved using different configurations. We show that instances that have similar costs using one solver configuration also have similar costs using another solver configuration in the same runtime environment. After that, we present a methodology based on Deep Metric Learning to learn MILP similarities that correlate with their final solutions' costs. At inference time, given a new problem instance, it is first projected into the learned metric space using the trained model, and configuration parameters are instantly predicted using previously-explored configurations from the nearest neighbor instance in the learned embedding space. Empirical results on real-world problem benchmarks show that our method predicts configuration parameters that improve solutions' costs by up to 38% compared to existing approaches. [ABSTRACT FROM AUTHOR]

This paper studies presentations of the Sierpinski gasket as a final coalgebra for a functor on three categories of metric spaces with additional designated points. The three categories which we study differ on their morphisms: one uses short (non-expanding) maps, the second uses continuous maps, and the third uses Lipschitz maps. The functor in all cases is very similar to what we find in the standard presentation of the gasket as an attractor. It was previously known that the Sierpinski gasket is bilipschitz equivalent (though not isomorhpic) to the final coalgebra of this functor in the category with short maps, and that final coalgebra is obtained by taking the completion of the initial algebra. In this paper, we prove that the Sierpiniski gasket itself is the final coalgebra in the category with continuous maps, though it does not occur as the completion of the initial algebra. In the Lipschitz setting, we show that the final coalgebra for this functor does not exist. [ABSTRACT FROM AUTHOR]

In computational neuroscience, recurrent neural networks are widely used to model neural activity and learning. In many studies, fixed points of recurrent neural networks are used to model neural responses to static or slowly changing stimuli, such as visual cortical responses to static visual stimuli. These applications raise the question of how to train the weights in a recurrent neural network to minimize a loss function evaluated on fixed points. In parallel, training fixed points is a central topic in the study of deep equilibrium models in machine learning. A natural approach is to use gradient descent on the Euclidean space of weights. We show that this approach can lead to poor learning performance due in part to singularities that arise in the loss surface. We use a reparameterization of the recurrent network model to derive two alternative learning rules that produce more robust learning dynamics. We demonstrate that these learning rules avoid singularities and learn more effectively than standard gradient descent. The new learning rules can be interpreted as steepest descent and gradient descent, respectively, under a non-Euclidean metric on the space of recurrent weights. Our results question the common, implicit assumption that learning in the brain should be expected to follow the negative Euclidean gradient of synaptic weights. [ABSTRACT FROM AUTHOR]

The present study introduces fractal interpolation function as an attractor of the iterated function system involving a finite number of Reich contractions on a complete metric space. Some amendments are made concerning the results of Reich contractions in the existing literature, with respect to Hausdorff metric, by imposing certain additional constraints on the Reich contractivity factors. Moreover, a non-smooth version of classical interpolant is constructed with Reich contractions as an analogue to the α -fractal function. A fractal operator is proposed to approximate any given continuous function using the newly generated Reich-type fractal function. Apart from the construction, Reich-fractal functions are employed to reconstruct the Lorenz attractor, and the results are graphically compared with fractal functions generated using other contraction mappings. [ABSTRACT FROM AUTHOR]

Consider the problem of finding a point furthest from x for each point x in a metric space ([ n ] , d) , where [ n ] = { 1 , 2 , ... , n }. We prove this problem to have a deterministic O (n) -time 1 / 3 -approximation algorithm. As a corollary, the maximum spanning tree problem in metric spaces has a deterministic O (n) -time 1 / 3 -approximation algorithm. We also give a Monte Carlo O (n 1. 5 log n) -time algorithm outputting, for each x ∈ [ n ] , a point y x ∈ [ n ] satisfying d (x , y x) ≥ Δ / 2 , where Δ ≡ max x , y ∈ [ n ] d (x , y). As a corollary, we have a Monte Carlo O (n 1. 5 log n) -time algorithm for finding a spanning tree of weight at least (1 / 2 − o (1)) Δ n in ([ n ] , d). [ABSTRACT FROM AUTHOR]

We show that coarse maps between countable metric spaces of bounded geometry induce natural transformations of sufficiently good endofunctors of C^{*}-algebras and prove that this correspondence is invariant with respect to coarse homotopies. [ABSTRACT FROM AUTHOR]

This article presents a new proof of a theorem concerning bounds of the spectrum of the product of unitary operators and a generalization for differentiable curves of this theorem. The proofs involve metric geometric arguments in the group of unitary operators and the sphere where these operators act. [ABSTRACT FROM AUTHOR]

We introduce Poincaré-type inequalities based on rearrangement invariant spaces in the setting of metric measure spaces and analyze when they imply the doubling condition on the underline measure. [ABSTRACT FROM AUTHOR]

Let (K,d) be a connected compact metric space and p\in (1, \infty). Under the assumption of Kigami [ Conductive homogeneity of compact metric spaces and construction of p-energy , Memoirs of the European Mathematical Society, vol. 5, Europea Mathematical Society (EMS), Berline, 2023, Assumption 2.15] and the conductive p-homogeneity, we show that \mathcal {W}^p(K)\subset C(K) holds if and only if p>\operatorname {dim}_{AR}(K,d), where \mathcal {W}^p(K) is Kigami's (1,p)-Sobolev space and \operatorname {dim}_{AR}(K,d) is the Ahlfors regular dimension. [ABSTRACT FROM AUTHOR]

In the framework of the first-order differential structure introduced by Gigli, we obtain a Gauss–Green formula on regular bounded open sets in doubling metric measure spaces supporting a weak Poincaré inequality, valid for BV functions and vector fields with integrable divergence. Then, we study least gradient functions in metric measure spaces and provide an Euler–Lagrange-type formulation of the least gradient problem, using this formula as the main tool. [ABSTRACT FROM AUTHOR]

For A ⊆ ω , the coarse similarity class of A, denoted by [ A ] , is the set of all B ⊆ ω such that the symmetric difference of A and B has asymptotic density 0. There is a natural metric δ on the space of coarse similarity classes defined by letting δ ([ A ] , [ B ]) be the upper density of the symmetric difference of A and B. We study the metric space of coarse similarity classes under this metric, and show in particular that between any two distinct points in this space there are continuum many geodesic paths. We also study subspaces of the form { [ A ] : A ∈ } where is closed under Turing equivalence, and show that there is a tight connection between topological properties of such a space and computability-theoretic properties of . We then define a distance between Turing degrees based on Hausdorff distance in the metric space (, δ). We adapt a proof of Monin to show that the Hausdorff distances between Turing degrees that occur are exactly 0, 1 / 2 , and 1, and study which of these values occur most frequently in the senses of Lebesgue measure and Baire category. We define a degree a to be attractive if the class of all degrees at distance 1 / 2 from a has measure 1, and dispersive otherwise. In particular, we study the distribution of attractive and dispersive degrees. We also study some properties of the metric space of Turing degrees under this Hausdorff distance, in particular the question of which countable metric spaces are isometrically embeddable in it, giving a graph-theoretic sufficient condition for embeddability. Motivated by a couple of issues arising in the above work, we also study the computability-theoretic and reverse-mathematical aspects of a Ramsey-theoretic theorem due to Mycielski, which in particular implies that there is a perfect set whose elements are mutually 1-random, as well as a perfect set whose elements are mutually 1-generic. Finally, we study the completeness of (, δ) from the perspectives of computability theory and reverse mathematics. [ABSTRACT FROM AUTHOR]

We provided a generalization of the existence and uniqueness of fixed points in partially ordered metric spaces for a monotone map. We applied the major results in the investigation of coupled fixed points for ordered pairs of two maps that met various monotone features, which included a mixed monotone property or a total monotone property. To ascertain necessary requirements for the existence and uniqueness of solutions to systems of matrix equations, the results regarding coupled fixed points for ordered pairs of maps were utilized. These results are illustrated with numerical examples. Some of the known results are a consequence of the results we obtained. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish some fixed point results for enriched Kannan-type mappings in convex metric spaces. We first give an affirmative answer to a recent Berinde and Păcurar's question (Remark 2.3) [J. Comput. Appl. Math., 386 (2021), 113217]. Furthermore, we establish the existence and uniqueness of fixed points for Suzuki-enriched Kannan-type mappings in the setting of convex metric spaces. Finally, we present an application to approximate the solution of the Volterra integral equations to support our results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of unified interpolative contractions of the Reich–Rus–Ćirić type and give some results about the fixed points for these mappings in the framework of relational metric spaces. We present examples where the results of some previous research are not relevant. Also, we apply our results to solving problems related to nonlinear matrix equations, emphasizing their practical importance. [ABSTRACT FROM AUTHOR]

In this work, we introduce a new type of multivalued fuzzy proximal quasi-contraction. These are generalized contractions which are a hybrid of H -contractive mappings and quasi-contractions. Furthermore, we establish the best proximity point results for newly introduced fuzzy contractions in the context of fuzzy b-metric spaces. Fuzzy b-metric spaces are more general than fuzzy metric spaces and are linked with the cosine distance, which is used in various contexts of artificial intelligence to measure the similarity between elements of a vector space. [ABSTRACT FROM AUTHOR]

We show that capacity can be computed with locally Lipschitz functions in locally complete and separable metric spaces. Further, we show that if (X,d,\mu) is a locally complete and separable metric measure space, then continuous functions are dense in the Newtonian space N^{1,p}(X). Here the measure \mu is Borel and is finite and positive on all metric balls. In particular, we don't assume properness of X, doubling of \mu or any Poincaré inequalities. These resolve, partially or fully, questions posed by a number of authors, including J. Heinonen, A. Björn and J. Björn. In contrast to much of the past work, our results apply to locally complete spaces X and dispenses with the frequently used regularity assumptions: doubling, properness, Poincaré inequality, Loewner property or quasiconvexity. [ABSTRACT FROM AUTHOR]

In this paper, we prove the fractional Hardy inequality on polarisable metric measure spaces. The integral Hardy inequality for 1 < p ≤ q < ∞ 1

In this paper, we shall deal with the long‐time dynamics of a nonlinear system of wave equation with a fractional damping and a small delay. Inspired by the works of Chueshov and Lasiecka on the property of quasi‐stability of dynamic systems, we show first the quasi‐stability property holds for the problem considered here and then prove the existence of global and exponential attractors. We shall also prove the upper semicontinuities of global attractors with respect to the fractional exponents. Finally, we shall show the continuities of global attractors with respect to a small parameter multiplying the delay term in a residual dense set and their upper semicontinuities in a complete metric space. [ABSTRACT FROM AUTHOR]

This article focuses on studying some fixed-point results via Ϝ-contraction of Hardy–Rogers type in the context of supermetric space and ordered supermetric space. We also introduced rational-type z-contraction on supermetric space. For authenticity, some illustrative examples and applications have been included. [ABSTRACT FROM AUTHOR]

The geodesics in various spherical Rindler frames are investigated. A display of some kinematical quantities of the spacetime is given. The constant acceleration from the metric acts as the surface gravity of the horizon r = 0. The radial geodesics are computed both for the Balasubramanian et al. form of the spherical Rindler space and for the non-diagonal metric of Huang and Sun. [ABSTRACT FROM AUTHOR]

The set of equivalence classes of cobounded actions of a group G$G$ on different hyperbolic metric spaces carries a natural partial order. Following Abbott–Balasubramanya–Osin, the group G$G$ is H$\mathcal {H}$‐accessible if the resulting poset has a largest element. In this paper, we prove that every nongeometric 3‐manifold has a finite cover with H$\mathcal {H}$‐inaccessible fundamental group and give conditions under which the fundamental group of the original manifold is H$\mathcal {H}$‐inaccessible. We also prove that every Croke–Kleiner admissible group (a class of graphs of groups that generalizes fundamental groups of three‐dimensional graph manifolds) has a finite index subgroup that is H$\mathcal {H}$‐inaccessible. [ABSTRACT FROM AUTHOR]

This manuscript aims to study the existence and uniqueness of solutions to a new system of differential equations. This system is a mixture of fractional operators and stochastic variables. The study has been completed under nonlocal functional boundary conditions. In the study, we used the fixed-point method to examine the existence of a solution to the proposed system, mainly focusing on the theorems of Leray, Schauder, and Perov in generalized metric spaces. Finally, an example has been provided to support and underscore our results. [ABSTRACT FROM AUTHOR]

After the initiation of Jachymski's contraction principle via digraph, the area of metric fixed point theory has attracted much attention. A number of outcomes on fixed points in the context of graph metric space employing various types of contractions have been investigated. The aim of this paper is to investigate some fixed point theorems for a class of nonlinear contractions in a metric space endued with a transitive digraph. The outcomes presented herewith improve, extend and enrich several existing results. Employing our findings, we describe the existence and uniqueness of a singular fractional boundary value problem. [ABSTRACT FROM AUTHOR]

In this paper, a large class of contractions is studied that contains Banach and Matkowski maps as particular cases. Sufficient conditions for the existence of fixed points are proposed in the framework of b-metric spaces. The convergence and stability of the Picard iterations are analyzed, giving error estimates for the fixed-point approximation. Afterwards, the iteration proposed by Kirk in 1971 is considered, studying its convergence, stability, and error estimates in the context of a quasi-normed space. The properties proved can be applied to other types of contractions, since the self-maps defined contain many others as particular cases. For instance, if the underlying set is a metric space, the contractions of type Kannan, Chatterjea, Zamfirescu, Ćirić, and Reich are included in the class of contractivities studied in this paper. These findings are applied to the construction of fractal surfaces on Banach algebras, and the definition of two-variable frames composed of fractal mappings with values in abstract Hilbert spaces. [ABSTRACT FROM AUTHOR]

The Roe algebra C * (X) is a noncommutative C * -algebra reflecting metric properties of a space X, and it is interesting to understand the correlation between the Roe algebra of X and the (uniform) Roe algebra of its discretization. Here, we perform a minor step in this direction in the simplest non-trivial example, namely X = R , by constructing a continuous field of C * -algebras over [ 0 , 1 ] , with the fibers over non-zero points constituting the uniform C * -algebra of the integers, and the fibers over 0 constituting a C * -algebra related to R. [ABSTRACT FROM AUTHOR]

In this work, we develop an interpolation theory for metric spaces inspired by the real method of interpolation. These interpolation spaces preserve Lipschitz operators under certain conditions. We also show that this method, valid in metrics spaces, still holds in normed spaces without any algebraic structure required. Furthermore, this interpolation method for metric spaces when applied to normed spaces is equivalent to the K-method, which has been widely studied in the literature. As an application, we interpolate Fréchet sequence spaces. [ABSTRACT FROM AUTHOR]

In the present paper, we obtain three results on the existence of a fixed point for nonexpansive mappings. Two of them are generalizations of the result for F-contraction, while third one is a generalization of a recent result for set-valued contractions. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence and uniqueness of common fixed point of tower type contractive mappings in complete metric (pseudo)modular spaces involving the theoretic relation. However, the newly introduced contraction in this paper further characterize and includes in their full strength several existing results in metrical fixed point theory. Some nontrivial supportive examples were given to justify our result. Our results generalize, improve, and unify some existing results. [ABSTRACT FROM AUTHOR]

For a semigroup S$S$ whose universal right congruence is finitely generated (or, equivalently, a semigroup satisfying the homological finiteness property of being type right‐FP1$FP_1$), the right diameter of S$S$ is a parameter that expresses how 'far apart' elements of S$S$ can be from each other, in a certain sense. To be more precise, for each finite generating set U$U$ for the universal right congruence on S$S$, we have a metric space (S,dU)$(S,d_U)$ where dU(a,b)$d_U(a,b)$ is the minimum length of derivations for (a,b)$(a,b)$ as a consequence of pairs in U$U$; the right diameter of S$S$ with respect to U$U$ is the diameter of this metric space. The right diameter of S$S$ is then the minimum of the set of all right diameters with respect to finite generating sets. We develop a theoretical framework for establishing whether a semigroup of transformations or partitions on an arbitrary infinite set X$X$ has a finitely generated universal right/left congruence, and, if it does, determining its right/left diameter. We apply this to prove results such as the following. Each of the monoids of all binary relations on X$X$, of all partial transformations on X$X$, and of all full transformations on X$X$, as well as the partition and partial Brauer monoids on X$X$, have right diameter 1 and left diameter 1. The symmetric inverse monoid on X$X$ has right diameter 2 and left diameter 2. The monoid of all injective mappings on X$X$ has right diameter 4, and its minimal ideal (called the Baer–Levi semigroup on X$X$) has right diameter 3, but neither of these two semigroups has a finitely generated universal left congruence. On the other hand, the semigroup of all surjective mappings on X$X$ has left diameter 4, and its minimal ideal has left diameter 2, but neither of these semigroups has a finitely generated universal right congruence. [ABSTRACT FROM AUTHOR]

Consider a mapping f : X → Y between two metric measure spaces. We study generalized versions of the local Lipschitz number Lip ⁡ f , as well as of the distortion number H f that is used to define quasiconformal mappings. Using these numbers, we give sufficient conditions for f being a BV mapping f ∈ BV loc ⁢ (X ; Y) or a Newton–Sobolev mapping f ∈ N loc 1 , p ⁢ (X ; Y) , with 1 ≤ p < ∞ . [ABSTRACT FROM AUTHOR]

We characterise Geometric Property (T) by the existence of a certain projection in the maximal uniform Roe algebra Cu,max∗(X)$C_{u,\max }^*(X)$, extending the notion of Kazhdan projection for groups to the realm of metric spaces. We also describe this projection in terms of the decomposition of the metric space into coarsely connected components. [ABSTRACT FROM AUTHOR]

The Π‐operator, also known as Ahlfors–Beurling transform, plays an important role in solving the existence of locally quasiconformal solutions of Beltrami equations. In this paper, we first construct the Π‐operator on a general Clifford–Hilbert space. This Π‐operator is also an L2 isometry. Further, it can also be used for solving certain Beltrami equations when the Hilbert space is the L2 space of a measure space. Then, we show that this technique can be applied to construct the classical Π‐operator in the complex plane and some other examples on some conformally flat manifolds, which are constructed by U/Γ, where U is a simply connected subdomain of either ℝn+1 or 핊n−1, and Γ is a Kleinian group acting discontinuously on U. The Π‐operators on those manifolds also preserve the isometry property in certain L2 spaces, and their Lp norms are bounded by the Lp norms of the Π‐operators on ℝn+1 or 핊n−1, depending on where U lies. The applications of the Π‐operator to solutions of the Beltrami equations on those conformally flat manifolds are also discussed. At the end, we investigate the Π‐operator theory in the upper‐half space with the hyperbolic metric. [ABSTRACT FROM AUTHOR]

We study a family of invariants of compact metric spaces that combines the Curvature Sets defined by Gromov in the 1980 s with Vietoris–Rips Persistent Homology. For given integers k ≥ 0 and n ≥ 1 we consider the dimension k Vietoris–Rips persistence diagrams of all subsets of a given metric space with cardinality at most n. We call these invariants persistence sets and denote them as D n , k VR . We first point out that this family encompasses the usual Vietoris–Rips diagrams. We then establish that (1) for certain range of values of the parameters n and k, computing these invariants is significantly more efficient than computing the usual Vietoris–Rips persistence diagrams, (2) these invariants have very good discriminating power and, in many cases, capture information that is imperceptible through standard Vietoris–Rips persistence diagrams, and (3) they enjoy stability properties analogous to those of the usual Vietoris–Rips persistence diagrams. We precisely characterize some of them in the case of spheres and surfaces with constant curvature using a generalization of Ptolemy's inequality. We also identify a rich family of metric graphs for which D 4 , 1 VR fully recovers their homotopy type by studying split-metric decompositions. Along the way we prove some useful properties of Vietoris–Rips persistence diagrams using Mayer–Vietoris sequences. These yield a geometric algorithm for computing the Vietoris–Rips persistence diagram of a space X with cardinality 2 k + 2 with quadratic time complexity as opposed to the much higher cost incurred by the usual algebraic algorithms relying on matrix reduction. [ABSTRACT FROM AUTHOR]

In the online facility assignment problem OFA (k , { c i } i = 1 k) , there exist k servers s 1 , ... , s k on a metric space where each s i has an integer capacity c i and a request arrives one-by-one. The task of an online algorithm is to irrevocably match a current request with one of the servers with vacancies before the next request arrives. As special cases for OFA (k , { c i } i = 1 k) , we consider OFA (k , { c i } i = 1 k) on a line , which is denoted by OFAL (k , { c i } i = 1 k) and OFAL eq (k , { c i } i = 1 k) , where the latter is the case of OFAL (k , { c i } i = 1 k) with equidistant servers. In this paper, we perform the competitive analysis for the above problems. As a natural generalization of the greedy algorithm grdy, we introduce a class of algorithms called MPFS (Most Preferred Free Servers) and show that any MPFS algorithm has the capacity-insensitive property, i.e., for any MPFS algorithm alg for OFA (k , { c i } i = 1 k) , if alg is c -competitive when c 1 = ⋯ = c k = 1 , then alg is c -competitive for general { c i } i = 1 k . By applying the capacity-insensitive property of the greedy algorithm grdy, we derive the matching upper and lower bounds 4 k − 5 on the competitive ratio of grdy for OFAL eq (k , { c i } i = 1 k). To investigate the capability of MPFS algorithms, we show that the competitive ratio of any MPFS algorithm alg for OFAL eq (k , { c i } i = 1 k) is at least 2 k − 1. Then, we propose a new MPFS algorithm idas (Interior Division for Adjacent Servers) for OFAL (k , { c i } i = 1 k) and show that the competitive ratio of idas for OFAL eq (k , { c i } i = 1 k) is at most 2 k − 1 , i.e., idas for OFAL eq (k , { c i } i = 1 k) is best possible in all the MPFS algorithms. We also give numerical experiments to investigate the performance of idas and grdy and show that idas performs better than grdy for distribution of request sequences with locality. [ABSTRACT FROM AUTHOR]

In this work, the main discussion centres on avoiding the use of triangle inequality while proving Cauchy sequence to establish best proximity point theorems for single valued as well as multivalued non-self mappings. We prove such best proximity point theorems in the setting of non-triangular metric spaces and elaborate through examples. In this process host of the existing best proximity results are generalized and improved. To arouse further interest in the subject, we connect this work in solving a specific type of partial differential equation problem. [ABSTRACT FROM AUTHOR]

Neutrosophic set is a generalization of classical sets, fuzzy set, intuitionistic sets etc. A mathematical notion neutrosophic set dealing issues containing inconsistent, indeterminate and imprecise data. In this manuscript, we toss the notion of Neutrosophic b-metric-like spaces and obtain some fixed point results in the sense of Neutrosophic b-metric-like spaces. Our results are improvements of recent results in the existing literature. For the validity of these results some non-trivial examples are imparted. [ABSTRACT FROM AUTHOR]

In this paper, we prove embedding theorems for the Möbius invariant space \mathcal {Q}_p on the open unit ball of \mathbb {C}^n into logarithmic tent spaces in the Bergman metric. [ABSTRACT FROM AUTHOR]

The greedy spanner in a low-dimensional Euclidean space is a fundamental geometric construction that has been extensively studied over three decades, as it possesses the two most basic properties of a good spanner: constant maximum degree and constant lightness. Recently, Eppstein and Khodabandeh [28] showed that the greedy spanner in \(\mathbb {R}^2\) admits a sublinear separator in a strong sense: Any subgraph of k vertices of the greedy spanner in \(\mathbb {R}^2\) has a separator of size \(O(\sqrt {k})\). Their technique is inherently planar and is not extensible to higher dimensions. They left showing the existence of a small separator for the greedy spanner in \(\mathbb {R}^d\) for any constant d ≥ 3 as an open problem. In this article, we resolve the problem of Eppstein and Khodabandeh [28] by showing that any subgraph of k vertices of the greedy spanner in \(\mathbb {R}^d\) has a separator of size \(O(k^{1-1/d})\). We introduce a new technique that gives a simple criterion for any geometric graph to have a sublinear separator that we dub τ-lanky: A geometric graph is τ-lanky if any ball of radius r cuts at most τ edges of length at least r in the graph. We show that any τ-lanky geometric graph of n vertices in \(\mathbb {R}^d\) has a separator of size \(O(\tau n^{1-1/d})\). We then derive our main result by showing that the greedy spanner is O(1)-lanky. We indeed obtain a more general result that applies to unit ball graphs and point sets of low fractal dimensions in \(\mathbb {R}^d\). Our technique naturally extends to doubling metrics. We use the τ-lanky criterion to show that there exists a (1+ε)-spanner for doubling metrics of dimension d with a constant maximum degree and a separator of size \(O(n^{1-\frac{1}{d}})\) ; this result resolves an open problem posed by Abam and Har-Peled [1] a decade ago. We then introduce another simple criterion for a graph in doubling metrics of dimension d to have a sublinear separator. We use the new criterion to show that the greedy spanner of an n-point metric space of doubling dimension d has a separator of size \(O((n^{1-\frac{1}{d}}) + \log \Delta)\) where Δ is the spread of the metric; the factor log (Δ) is tightly connected to the fact that, unlike its Euclidean counterpart, the greedy spanner in doubling metrics has unbounded maximum degree. Finally, we discuss algorithmic implications of our results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce two new classes of mixed (S,T)-α-admissible mappings and interspersed (S, g,T)-α-admissible mappings and study the sufficient conditions for the existence and uniqueness of a common fixed point of generalized (αs, ξ, ħ, τ)-Geraghty contractive mapping in the framework of partial b-metric spaces. We also provide two examples to show the applicability and validity of our results. Moreover, we present an application to the existence of solutions to an integral equation by means of one of our results. [ABSTRACT FROM AUTHOR]

This research investigated different types of convex contractions in the setting of extended b-metric spaces from the point of view of the existence and uniqueness of their fixed points. The assumptions imposed on involved mappings refer to convexity of order 2, two-sided convexity or Ćirić-type convexity, which also fulfill a continuity type condition. An example was provided to emphasize the usability of the results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of elliptic-valued b-metric spaces, extending the notions of elliptic-valued metric spaces and complex-valued metric spaces. We present several fixed-point results that involve rational and product terms within this novel space framework. To support our main findings, we offer numerical examples. Additionally, we demonstrate an application of Urysohn integral equations. [ABSTRACT FROM AUTHOR]