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Response surface analysis provides a mechanism to probe processes with problems that are either computationally impractical to solve directly, or have an intrinsic form of complexity (e.g., system-wide irreducible uncertainty, or emergent effects mediated by interaction dynamics). In virtually all fields of engineering, the method has been successfully applied to solve diverse problems. However; within the context of probing complex systems processes, we find considerably limited application. Additionally, the literature of non-parametric methods is not as well developed as its parametric counterpart. Given non-applicability of differentiability, or well-structured optimization, in some problem formulations; the problem of optimal configuration search becomes both conceptually and computationally challenging. We extend the surface response method to probe persistent features of filtrations constructed from network representations. We take an approach that treats networks as geometrical objects, and study their topological properties as a means to define topological equivalence classes via a continuous deformation transformation. Following developments in network based non-parametric profiling of response surfaces for multi-attribute heterogeneous value-type data; we present a procedure to automate network homotopy variation detection–for cases where the underlying kernel transformation is mediated by a network representation that is endowed with an intrinsic metric. We apply the Forman-Ricci curvature flow to detect the variation of homotopy as viewed from a filtration perspective. The resulting scheme enables empirical probing of optimal mean-variance properties of a yield quantity. The result is a fully non-parametric method that is not dependent on model parameter learning. We present this approach as a means to take into account the fundamental uncertainty that is associated with bandwidth selection when reflexive properties, or operational risk factors, are intrinsic properties in the creation and processing of the underlying data.

Let K be a convex body in $${\mathbb {R}} ^d$$, with $$d = 2,3$$. We determine sharp sufficient conditions for a set E composed of 1, 2, or 3 points of $$\mathrm{bd}K$$, to contain at least one endpoint of a diameter of K. We extend this also to convex surfaces, with their intrinsic metric. Our conditions are upper bounds on the sum of the complete angles at the points in E. We also show that such criteria do not exist for $$n\ge 4$$ points.

We introduce a new intrinsic metric in subdomains of a metric space and give upper and lower bounds for it in terms of well-known metrics. We also prove distortion results for this metric under quasiregular maps., Comment: 12 pages, 2 figures. arXiv admin note: text overlap with arXiv:1903.12475

Anadolu Üniversitesi ve Bilecik Şeyh Edebali Üniversitesi tarafından ortak yürütülen program. Frakteller doğayla olan ilişkisinden dolayı son zamanların popüler araştırma konularından biridir. Bütün fraktellerin ortak özelliklerinden biri kendine benzerliktir. Kendine benzer kümeler kuvvetli kendine benzer, zayıf kendine benzer ve rastgele kendine benzer kümeler gibi farklı sınıflarda incelenebilir. Kuvvetli kendine benzer kümelerin klasik modellerinin bazıları Cantor kümesi, Cantor toz bulutu, Sierpinski üçgeni, Sierpinski halısı, Koch eğrisi, kutu fraktalı, Sierpinski dörtyüzlüsü ve Menger süngeridir. Zayıf kendine benzer küme modelleri olarak Sierpinski pervanesi ve komşu Sierpisnki üçgeni verilebilir. Julia kümeleri ve Mandelbrot kümesi ise rastgele kendine benzer küme modelleridir. Kendine benzer kümeler yapılarından dolayı farklı kodlamalara ve bnu kodlamalarla uyumlu olan içsel metriklere sahiptir. Bu tezde ilk olarak klasik fraktaller üzerindeki kodlamalar araştırılacaktır. Literatürde verilen Sierpinski üçgeninin kod kümesi üzerindeki içsel metrik yapısı incelendikten sonra zayıf kendine benzer bir küme örneği olan Sierpinski pervanesi ve komşu Sierpinski üçgeni üzerinde içsel metrik formülleri inşa edilecektir. Son olarak ekli Sierpinski üçgeni üzerin de tanımlanan içsel metrik formülü kullanılarak iki veya daha fazla en kısa yola sahip noktalar sınıflandırılacaktır. Ayrıca jeodeziklerinin sayısı n=0,1,2,3,... için 2^n,3.2^n+n ve sonsuz olan noktaların olduğu gösterilecek ve jeodezik sayısına göre bu noktaların bazı kod temsilleri ifade edilecektir., Fractals are one of the most popular research topics of recent times because of their relationship with the nature. One common feature of all fractals is self-similarity. Self-similar sets can be studied in different classes; strong self-similar sets, weak self-similar sets and randomly self-similar sets. Some classical models of strong self-similar sets are Cantor set, Cantor dust, Sierpinski triangle, Sierpinski carpet, Koch curve, Box fractal, Sierpinski tetrahedron and Menger sponge. The Sierpinski propeller and adjecent Sierpinski triangle can be given as examples of weak sel-similar sets.Julia sets and Mandelbrot set are some examples of randomly self-similar sets. Self-similar sets have different codings and the intrinsic metrics that are compatible these codings due to their structures. In this thesis, firstly the codings of classical fractals will be investigated. After examining the intrinsic metric formula defined on the code set of Sierpinski gasket in the literature, the intrinsic metric formulas will be constructed on the Sierpinski propeller and adjecent Sierpinski triangle which are some example of weak self-similar set. Finally, by using the intrinsic metric formula on the added Sierpinski triangle the points which have two or more shortest paths will be classified. The points which have for n=0,1,2,3,... the number of geodesics 2^n,3.2^n+n and infinity have shown and some code representations of these points have been expressed according to the number of geodesics.

Natural Language Processing (NLP) systems learn harmful societal biases that cause them to amplify inequality as they are deployed in more and more situations. To guide efforts at debiasing these systems, the NLP community relies on a variety of metrics that quantify bias in models. Some of these metrics are intrinsic, measuring bias in word embedding spaces, and some are extrinsic, measuring bias in downstream tasks that the word embeddings enable. Do these intrinsic and extrinsic metrics correlate with each other? We compare intrinsic and extrinsic metrics across hundreds of trained models covering different tasks and experimental conditions. Our results show no reliable correlation between these metrics that holds in all scenarios across tasks and languages. We urge researchers working on debiasing to focus on extrinsic measures of bias, and to make using these measures more feasible via creation of new challenge sets and annotated test data. To aid this effort, we release code, a new intrinsic metric, and an annotated test set focused on gender bias in hate speech., Comment: In Proceedings of ACL 2021, 9 pages

This thesis is devoted to some applications of cone-manifolds and discrete curvature to problems in 3-dimensional hyperbolic geometry. First, we prove a realization and rigidity result for a specific family of hyperbolic cone-3-manifolds. This allows us to give a new variational proof of the existence and uniqueness of a hyperbolic cone-metric on S_g with prescribed curvature in a given discrete conformal class. Here S_g is a closed orientable surface of genus g > 1. This also provides a new proof of the fact that every hyperbolic cusp-metric on S_g can be uniquely realized as a convex surface in a Fuchsian manifold. A Fuchsian manifold is a hyperbolic manifold homeomorphic to S_g ×[0; +∞) with geodesic boundary Sg × {0}. They are known as toy cases for studying geometry of non-compact hyperbolic 3-manifolds and hyperbolic 3-manifolds with boundary. Second, we consider compact Fuchsian manifolds with boundary, i.e., hyperbolic manifolds homeomorphic to S_g × [0; 1] with geodesic boundary S_g × {0}. We use cone-manifolds to prove that a compact Fuchsian manifold with convex boundary is uniquely determined by the induced metric on S_g × {1}. It is distinguishing that except convexity we do not put any other condition on the boundary, so it may be neither smooth nor polyhedral.

Tree metrics are encountered throughout pure and applied mathematics. Their simple structure makes them a convenient choice of metric in many applications from machine learning and computer science. At the same time, there is an elegant theory of harmonic analysis with respect to tree metrics that parallels the classical theory. A basic question in this field, which is of both theoretical and practical interest, is how to design efficient algorithms for building trees with good metric properties. In particular, given a finite metric space, we seek a random family of dominating tree metrics approximating the underlying metric in expectation. For general metrics, this problem has been solved: on the one hand, there are finite metric spaces that cannot be approximated by trees without incurring a distortion logarithmic in the size of the space, while the tree construction of Fakcharoenphol, Rao, and Talwar (FRT, 2003) shows how to achieve such a logarithmic error for arbitrary metrics. Since a distortion that grows even logarithmically with the size of the set may be too large for practical use in many settings, one naturally asks if there is a more restricted class of metrics where one can do better. The main result of this paper is that certain random family of trees already studied in the computer science literature, including the FRT trees, can be used to approximate snowflake metrics (metrics raised to a power less than 1) with expected distortion bounded by its doubling dimension and the degree of snowflaking. We also show that without snowflaking, the metric distortion can be bounded by a term logarithmic in the distance being approximated and linear in the dimension. We also present an optimal algorithm for building a single FRT tree, whose running time is bounded independently of all problem parameters other than the number of points. We conclude by demonstrating our theoretical results on a numerical example, and applying them to the approximation of the Earth Mover's Distance between probability distributions.

Applying the Bonk-Kleiner characterization of Ahlfors 2-regular quasispheres, we show that a metric two-sphere $X$ is a quasisphere if and only if $X$ is linearly locally connected and carries a weak metric doubling measure, i.e., a measure that deforms the metric on $X$ without much shrinking., Comment: 15 pages

[EN] In this paper we develop a new technique for constructing fuzzy metric spaces, in the sense of George and Veeramani, from metric spaces and by means of the Lukasievicz t-norm. In particular such a technique is based on the use of metric preserving functions in the sense of J. Dobos. Besides, the new generated fuzzy metric spaces are strong and completable, and if we add an extra condition, they are principal. Appropriate examples of such fuzzy metric spaces are given in order to illustrate the exposed technique., Valentín Gregori acknowledges the support of the Spanish Ministry of Economy and Competitiveness under Grant MTM2015-64373-P (MINECO/FEDER, UE). Oscar Valero acknowledges the support of the Spanish Ministry of Economy and Competitiveness under Grants TIN2014-56381-REDT (LODISCO), TIN2016-81731-REDT (LODISCO II) and AEI/FEDER, UE funds.

We prove the following localized version of a classical ellipsoid characterization: Let $B\subset\mathbb R^3$ be convex body with a smooth strictly convex boundary and 0 in the interior, and suppose that there is an open set of planes through 0 such that all sections of $B$ by these planes are linearly equivalent. Then all these sections are ellipses and the corresponding part of $B$ is a part of an ellipsoid. We apply this to differential geometry of Finsler surfaces in normed spaces and show that in certain cases the intrinsic metric of a surface imposes restrictions on its extrinsic geometry similar to implications of Gauss' Theorema Egregium. As a corollary we construct 2-dimensional Finsler metrics that do not admit local isometric embeddings to dimension~3., Comment: v2: minor clarification and cleanup

In this paper, we obtain a unique common coupled fixed point theorem by using $(\psi , \alpha , \beta )$ -contraction in ordered partial metric spaces. We give an application to integral equations as well as homotopy theory. Also we furnish an example which supports our theorem.

In the clustering of shapes, which is a longstanding challenge in the framework of geometric morphometrics and shape analysis, is crucial the selection and application of a suitable and appropriate measurement of distance among observations (i.e. individuals). The aim of this study is to model shapes from complex systems using Information Geometry tools. It is well-known that the Fisher information endows the statistical manifold, defined by a family of probability distributions, with a Riemannian metric, called the Fisher–Rao metric. With respect to this, geodesic paths are determined, minimizing information in Fisher sense. The geodesic distance induced by the Fisher–Rao metric can be used to define a shape metric which enables us to quantify differences between shapes. The discriminative power of the proposed Fisher–Rao distance is tested in the context of shapes clustering on both simulated and real data sets. Results show a better ability in recovering the true cluster structure with respect to the standard Kendall’s shape metric.

In this paper, we show an existence and uniqueness of fixed point for contractive mappings of integral type using altering distance functions in fuzzy metric spaces. Moreover, we give examples to support our results. Our results generalize corresponding results given in the literature.

On the aim and properties of - algebra, in this paper we introduce the two concepts algebra and algebra metric space as well as introduce concept convergent and Cauchy sequence in space and to study the existence of fixed point theorems with contraction condition and algebra expansion on this space.

Let $$M_p(X,T)$$ denote the Markov type p constant at time T of a metric space X, where $$p \ge 1$$ . We show that $$M_p(Y,T) \le M_p(X,T)$$ in each of the following cases: (a) X and Y are geodesic spaces and Y is covered by X via a finite-sheeted locally isometric covering, (b) Y is the quotient of X by a finite group of isometries, (c) Y is the $$L^p$$ -Wasserstein space over X. As an application of (a) we show that all compact flat manifolds have Markov type 2 with constant 1. In particular the circle with its intrinsic metric has Markov type 2 with constant 1. This answers the question raised by S.-I. Ohta and M. Pichot. Parts (b) and (c) imply new upper bounds for Markov type constants of the $$L^p$$ -Wasserstein space over $${\mathbb {R}}^d$$ . These bounds were conjectured by A. Andoni, A. Naor and O. Neiman. They imply certain restrictions on bi-Lipschitz embeddability of snowflakes into such Wasserstein spaces.

A result showed by M. Gursky in [4] ensures that any metric g on the 4-dimensional sphere S4 satisfying Ricg=3g and injg(S4)≥π34 is isometric to the round metric. In this note, we prove that there exists a universal number i0 such that any metric g on the 4-dimensional sphere S4 satisfying Ricg=3g and injg(S4)≥π34−i0 is isometric to the round metric. Moreover, there exists a universal e0>0 such that any metric g on the 4-dimensional sphere S4 with nonnegative sectional curvature, Ricg=3g and 89π2−e0≤Volg(S4) is isometric to the round metric. This last result slightly improves a rigidity theorem also proved in [4].

In this paper, we topologically study the partial metric space, which may be seen as a new sub-branch of the pure asymmetric topology. We show that many familiar topological properties and principles still hold in certain partial metric spaces, although some results might need some advanced assumptions. Various properties, including separation axioms, countability, connectedness, compactness, completeness and Ekeland's variation principle, are discussed.

In this paper, we first establish a monotonicity formula for vector bundle-valued f -harmonic p -forms on a smooth metric measure space provided 〈 ∇ f , ∇ r 〉 is less that an explicit constant. As applications, we get some vanishing theorems for L 2 f -harmonic forms on concrete geometric models. In the second part, for a metric measure space with nonnegative ∞ -Bakry–Emery–Ricci curvature and with moderate volume growth, we prove that any bounded f -harmonic 1 -form must be parallel. Moreover, some vanishing theorems under nonnegative m -Bakry–Emery–Ricci curvature assumption are also proved. Finally, we consider smooth metric measure spaces with weighted Poincare inequality and show some vanishing theorems for L q f -harmonic 1 -forms.

The paper is concerned with local approximative and geometric properties of sets, with particular emphasis on strict solarity of such sets under certain constraints on the continuity of metric projections. A partial answer is given to the question due to B. Brosowski and F. Deutsch as to when the class of strict protosuns (Kolmogorov sets) coincides with the class of sets with outer radially continuous metric projection. The lower semi-continuous metric projection with monotone path-connected values is shown to have a continuous selection. A number of related results is obtained.

Far East Journal of Mathematical Sciences (FJMS), Volume 102, Number 5, 2017, Pages 965-978

In this paper, the concept of \(b_v(s)\)-metric space is introduced as a generalization of metric space, rectangular metric space, b-metric space, rectangular b-metric space and v-generalized metric space. We next give proofs of the Banach and Reich contraction principles in \(b_v(s)\)-metric spaces. Using a new result, we provide short proofs which are different from of the original ones in metric spaces. The results we obtain generalize many known results in fixed point theory. We also provide a solution to an open problem.

Choice of a distance metric is a key for the success in many machine learning and data processing tasks. The distance between two data samples traditionally depends on the values of their attribute...

To examine molecular geometry, we ask two questions: (1) What are the metric properties of a finite set of points in space, i.e., what are the relations of the distances between the points? (2) What is the symmetry or point group of the point set? These questions are answered by applying algebraic and algorithmic tools. Results from distance geometry are used to describe the metric. In combination with a so-called minimizing algorithm, distance geometry is also used to develop a procedure that allows generation of the symmetry group without referring to geometric intuition. The general methods presented here are applied to cyclohexane, facilitating a complete geometrical analysis of all its conformers.

The aim of this paper is to prove some results on the existence and uniqueness of elements of best approximation and continuity of the metric projection in metric spaces. For a subset M of a metric space (X; d), the nature of set of those points of X which have at most one best approximation in M has been discussed. Some equivalent conditions under which an M-space is strictly convex have also been given in this paper.

We study optimal approximations of sets in various metric spaces with sets of balls of equal radius. We consider an Euclidean plane, a sphere, and a plane with a special non-uniform metric. The main component in our constructions of coverings are optimal Chebyshev n-networks and their generalizations. We propose algorithms for constructing optimal coverings based on partitioning a given set into subsets and finding their Chebyshev centers in the Euclidean metric and their counterparts in non-Euclidean ones. Our results have both theoretical and practical value and can be used to solve problems arising in security, communication, and infrastructural logistics.

A pseudo-Finsleroid metric function F of the two-axes structure that involves the vertical axis and the horizontal axis is proposed assuming constancy of the curvature of indicatrix. The curvature is negative and the signature of the Finslerian metric tensor is exactly $$(+-\cdots )$$ . The function F endows the tangent space with the geometry which possesses many interesting Finslerian properties. The use of the angle representation is the underlying method which has been conveniently and successfully applied. The appearance of the positive-definite Finsleroid metric function in the horizontal sections of the tangent space is established.