Double Measurements Research Articles

Box-counting measure of metric spaces

Abstract In this paper, we introduce a new notion called the box-counting measure of a metric space. We show that for a doubling metric space, an Ahlfors regular measure is always a box-counting measure; consequently, if E is a self-similar set satisfying the open set condition, then the Hausdorff measure restricted to E is a box-counting measure. We show two classes of self-affine sets, the generalized Lalley-Gatzouras type self-affine sponges and Barański carpets, always admit box-counting measures; this also provides a very simple method to calculate the box-dimension of these fractals. Moreover, among others, we show that if two doubling metric spaces admit box-counting measures, then the multi-fractal spectra of the box-counting measures coincide provided the two spaces are Lipschitz equivalent.

Some New Types of Fuzzy Closed Sets, SeparationAxioms, and Compactness via Double Fuzzy Topologies

In this article, we first defined a stronger form of (r,s)-generalized fuzzy semi-closed sets (briefly, (r,s)-gfsc sets) called (r,s)-g*fsc sets and investigated some of its features. Moreover, we showed that (r,s)-fsc set → (r,s)-g*fsc set → (r,s)-gfsc set, but the converse may not be true. In addition, we explored novel types of fuzzy generalized mappings between double fuzzy topological spaces (U, τ, τ*) and (V, η, η*), and the relationships between these classes of mappings were examined with the help of some illustrative examples. Thereafter, we introduced novel types of higher separation axioms called (r,s)-GFS-regular and (r,s)-GFS-normal spaces with the help of (r,s)-gfsc sets and discussed some topological properties of them. Finally, some novel types of compactness via (r,s)-gfso sets were defined and the relationships between them were introduced.

“They don’t mean to hurt”: Female gamers’ reluctance in recognizing and confronting sexism in gaming as an online-offline juxtaposition

Female gamers have long suffered from gender-based online abuse in the gaming community. Apart from commonly observed quitting and gender-masking behaviors from female gamers, this study explores what female gamers understand as sexism, how female gamers react to it, and why they choose certain reactions instead of others. Findings show that female gamers are keenly conscious of normalized sexism in gaming culture, and thus often prioritize preventing personal interaction with strangers online, resulting in their shared preference for gaming with trusted acquaintances, which makes gaming an online-offline juxtaposition. Shouldering gender norms in doubled dimensions of gaming and specific real-life relationships, female gamers thus become reluctant to recognize and confront less violent sexism from male acquaintances. Female gamers’ strategic self-protection, although gaining them relatively safer gaming spaces, also consolidates sexism in gaming, and further suggests gaming as a critical social space for reproducing broader gender inequalities.

N=(2,2) superfields and geometry revisited

Abstract We take a fresh look at the relation between generalised Kähler geometry and N = (2, 2) supersymmetric sigma models in two dimensions formulated in terms of (2, 2) superfields. Dual formulations in terms of different kinds of superfield are combined to give a formulation with a doubled target space and both the original superfield and the dual superfield. For Kähler geometry, we show that this doubled geometry is Donaldson’s deformation of the holomorphic cotangent bundle of the original Kähler manifold. This doubled formulation gives an elegant geometric reformulation of the equations of motion. We interpret the equations of motion as the intersection of two Lagrangian submanifolds (or of a Lagrangian submanifold with an isotropic one) in the infinite dimensional symplectic supermanifold which is the analogue of phase space. We then consider further extensions of this formalism, including one in which the geometry is quadrupled, and discuss their geometry.

Towards complexity in de Sitter space from the doubled-scaled Sachdev-Ye-Kitaev model

How can we define complexity in dS space from microscopic principles? Based on recent developments pointing towards a correspondence between a pair of double-scaled Sachdev-Ye-Kitaev (DSSYK) models/ 2D Liouville-de Sitter (LdS2) field theory/ 3D Schwarzschild de Sitter (SdS3) space in [1–3], we study concrete complexity proposals in the microscopic models and their dual descriptions. First, we examine the spread complexity of the maximal entropy state of the doubled DSSYK model. We show that it counts the number of entangled chord states in its doubled Hilbert space. We interpret spread complexity in terms of a time difference between antipodal observers in SdS3 space, and a boundary time difference of the dual LdS2 CFTs. This provides a new connection between entanglement and geometry in dS space. Second, Krylov complexity, which describes operator growth, is computed for physical operators on all sides of the correspondence. Their late time evolution behaves as expected for chaotic systems. Later, we define the query complexity in the LdS2 model as the number of steps in an algorithm computing n-point correlation functions of boundary operators of the corresponding antipodal points in SdS3 space. We interpret query complexity as the number of matter operator chord insertions in a cylinder amplitude in the DSSYK, and the number of junctions of Wilson lines between antipodal static patch observers in SdS3 space. Finally, we evaluate a specific proposal of Nielsen complexity for the DSSYK model and comment on its possible dual manifestations.

Fuzzy Double S – Hausdorff Spaces

This study introduces and explores the concept of fuzzy double s-Hausdorff spaces within the field of fuzzy topology. Fuzzy double s-Hausdorff spaces extend the classical Hausdorff condition by incorporating two distinct fuzzy topologies on a single set, allowing for a more nuanced approach to separation where uncertainty and gradation are involved. The research develops formal definitions and characterizations of these spaces, establishing key theorems that outline their properties and behavior. In this article, a meaning of FDS -T2 remains presented through covering the description of S- T2 introduce by Srivastava [3] and obtain result analogues towards the result of S- T2.

A necessary condition for the boundedness of the maximal operator on $$L^{p(\cdot)}$$ over reverse doubling spaces of homogeneous type

AbstractLet $$(X,d,\mu)$$ ( X , d , μ ) be a space of homogeneous type and $$p(\cdot) \colon X \to[1,\infty]$$ p ( · ) : X → [ 1 , ∞ ] be a variable exponent. We show that if the measure $$\mu$$ μ is Borel-semiregular and reverse doubling, then the condition $${ess\,inf}_{x\in X}p(x)>1$$ e s s i n f x ∈ X p ( x ) > 1 is necessary for the boundedness of the Hardy–Littlewood maximal operator $$M$$ M on the variable Lebesgue space $$L^{p(\cdot)}(X,d,\mu)$$ L p ( · ) ( X , d , μ ) .

Approximation schemes for Min-Sum [formula omitted]-Clustering

We consider the Min-Sum k-Clustering (k-MSC) problem. Given a set of points in a metric which is represented by an edge-weighted graph G=(V,E) and a parameter k, the goal is to partition the points V into k clusters such that the sum of distances between all pairs of the points within the same cluster is minimized.The k-MSC problem is known to be APX-hard on general metrics. The best known approximation algorithms for the problem obtained by Behsaz et al. (2019) achieve an approximation ratio of O(log|V|) in polynomial time for general metrics and an approximation ratio 2+ϵ in quasi-polynomial time for metrics with bounded doubling dimension. No approximation schemes for k-MSC (when k is part of the input) is known for any non-trivial metrics prior to our work. In fact, most of the previous works rely on the simple fact that there is a 2-approximate reduction from k-MSC to the balanced k-median problem and design approximation algorithms for the latter to obtain an approximation for k-MSC.In this paper, we obtain the first Quasi-Polynomial Time Approximation Schemes (QPTAS) for the problem on metrics induced by graphs of bounded treewidth, graphs of bounded highway dimension, graphs of bounded doubling dimensions (including fixed dimensional Euclidean metrics), and planar and minor-free graphs. We bypass the barrier of 2 for k-MSC by introducing a new clustering problem, which we call min-hub clustering, which is a generalization of balanced k-median and is a trade off between center-based clustering problems (such as balanced k-median) and pair-wise clustering (such as Min-Sum k-clustering). We then show how one can find approximation schemes for Min-hub clustering on certain classes of metrics.

Connected k-Center and k-Diameter Clustering

Motivated by an application from geodesy, we study the connected k-center problem and the connected k-diameter problem. The former problem has been introduced by Ge et al. (ACM Trans Knowl Discov Data 2(2):1–35, 2008. https://doi.org/10.1145/1376815.1376816) to model clustering of data sets with both attribute and relationship data. These problems arise from the classical k-center and k-diameter problems by adding a side constraint. For the side constraint, we are given an undirected connectivity graphG on the input points, and a clustering is now only feasible if every cluster induces a connected subgraph in G. Usually in clustering problems one assumes that the clusters are pairwise disjoint. We study this case but additionally also the case that clusters are allowed to be non-disjoint. This can help to satisfy the connectivity constraints. Our main result is an O(log2k)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(\\log ^2k)$$\\end{document}-approximation algorithm for the disjoint connected k-center and k-diameter problem. For Euclidean spaces of constant dimension and for metrics with constant doubling dimension, the approximation factor improves to O(1). Our algorithm works by computing a non-disjoint connected clustering first and transforming it into a disjoint connected clustering. We complement these upper bounds by several upper and lower bounds for variations and special cases of the model.

Essential norm and Schatten class of Hankel operators on doubling Fock spaces

In this paper, we obtain the essential norm of Hankel operators from a doubling Fock space Fφp to a weighted Lebesgue space Lφq for all possible 1≤p,q<∞. We also characterize the Schatten-h class of Hankel operators from Fφ2 to Lφ2.

Fixed point results in intuitionistic fuzzy pentagonal controlled metric spaces with applications to dynamic market equilibrium and satellite web coupling.

This manuscript contains several new spaces as the generalizations of fuzzy triple controlled metric space, fuzzy controlled hexagonal metric space, fuzzy pentagonal controlled metric space and intuitionistic fuzzy double controlled metric space. We prove the Banach fixed point theorem in the context of intuitionistic fuzzy pentagonal controlled metric space, which generalizes the previous ones in the existing literature. Further, we provide several non-trivial examples to support the main results. The capacity of intuitionistic fuzzy pentagonal controlled metric spaces to model hesitation, capture dual information, handle imperfect information, and provide a more nuanced representation of uncertainty makes them important in dynamic market equilibrium. In the context of changing market dynamics, these aspects contribute to a more realistic and flexible modelling approach. We present an application to dynamic market equilibrium and solve a boundary value problem for a satellite web coupling.

Multi-view clustering via double spaces structure learning and adaptive multiple projection regression learning

Multi-view clustering aims to group objects with high similarity into one group according to the heterogeneous features of different views. The graph-based clustering methods have obtained excellent results. However, there remain a few common drawbacks. For example, some methods do not consider graphs' high-order structure information. Thus, fuller data information cannot be obtained. In addition, some methods remove noise, outliers, and redundant information in the graph learning phase, resulting in the loss of graph information. Furthermore, using predefined graphs cannot exploit complementary information between views. A triple strategy-based multi-view clustering method is presented to solve the above issues. First, Laplacian graphs are used for fusion learning, and the underlying first-order and second-order structure information among views are explored simultaneously. Then, a label fusion scheme is designed to eliminate noise, outliers, and redundant information and to mine the intrinsic characteristics of data labels. Besides, the consistent label matrix in adaptive regression learning is used to explore complementary information between views in a mutually guided learning way. Finally, the objective function is solved by using an efficient iterative method. Six types of experiments are conducted on eleven real-world multi-view datasets, and the conclusions that can be drawn are: (1) the proposed algorithm achieves the best results in terms of clustering accuracy on ten datasets with an average accuracy improvement of 5.11% compared to other algorithms. Specifically, the accuracy improved by 9.05% on dataset HW and 10.95% on dataset Reuters compared to the second results; (2) The ablation experiments confirm that the different learning strategies included in the proposed algorithm allow it to achieve better clustering performance.

Oracle inequalities and upper bounds for kernel conditional U-statistics estimators on manifolds and more general metric spaces associated with operators

ABSTRACT U-statistics represent a fundamental class of statistics that emerge from modelling quantities of interest defined by multi-subject responses. These statistics generalize the empirical mean of a random variable X to summations encompassing all distinct k-tuples of observations drawn from X . A significant advancement was made by Stute [ConditionalU-statistics. Ann. Probab. 19 (1991), pp. 812–825], who introduced conditional U-statistics as a generalization of the Nadaraya–Watson estimates for regression functions. Stute demonstrated their robust pointwise consistency towards the conditional function r ( k ) ( φ , t ~ ) = E ( φ ( Y 1 , … , Y k ) ∣ ( X 1 , … , X k ) = t ~ ) , for t ~ ∈ R pk , where φ is a measurable function. In this investigation, we develop oracle inequalities and upper bounds for kernel-based estimators of conditional U-statistics of general order, applicable across a wide range of metric spaces associated with operators. Our analysis specifically targets doubling measure metric spaces, incorporating a non-negative self-adjoint operator characterized by Gaussian regularity in its heat kernel. Remarkably, our study achieves an optimal convergence rate in certain cases. To derive these results, we explore the regression function within a general framework, introducing several novel insights. These findings are established under sufficiently broad conditions on the underlying distributions. The theoretical results serve as essential tools for advancing the field of general-valued data, with potential applications including the examination of conditional distribution functions, relative-error prediction, the Kendall rank correlation coefficient, and discrimination problems – areas of significant independent interest.

MapReduce algorithms for robust center-based clustering in doubling metrics

Clustering is a pivotal primitive for unsupervised learning and data analysis. A popular variant is the (k,ℓ)-clustering problem, where, given a pointset P from a metric space, one must determine a subset S of k centers minimizing the sum of the ℓ-th powers of the distances of points in P from their closest centers. This formulation covers the well-studied k-median (ℓ=1) and k-means (ℓ=2) clustering problems. A more general variant, introduced to deal with noisy pointsets, features a further parameter z and allows up to z points of P (outliers) to be disregarded when computing the sum. We present a distributed coreset-based 3-round approximation algorithm for the (k,ℓ)-clustering problem with z outliers, using MapReduce as a computational model. An important feature of our algorithm is that it obliviously adapts to the intrinsic complexity of the dataset, captured by its doubling dimension D. Remarkably, for D=O(1), our algorithm requires sublinear local memory per reducer, and yields a solution whose approximation ratio is an additive term O(γ) away from the one achievable by the best known sequential (possibly bicriteria) algorithm, where γ can be made arbitrarily small. To the best of our knowledge, no previous distributed approaches were able to attain similar quality-performance tradeoffs for metrics with constant doubling dimension.

Characterizations of some new classes of four-dimensional matrices and dual spaces on the double spaces of Cesàro means

The main purpose in this study is to investigate some topological and algebraic properties of the absolutely double series spaces |C_{1,1}|_{k} defined by combining the first order Cesàro means with the concept of absolute summability k≥1. Beside this, we determine the α-dual of the space |C_{1,1}|₁ and the β(bp)- and γ-duals of the spaces |C_{1,1}|_{k} for k≥1. Finally, we characterize some new four-dimensional matrix classes (|C_{1,1}|_{k},υ), (|C_{1,1}|₁,υ), (|C_{1,1}|₁,L_{k}), (|C_{1,1}|_{k},L_{u}), (L_{u},|C_{1,1}|_{k}) and (L_{k},|C_{1,1}|₁), where υ∈{M_{u},C_{bp}} for 1≤k

Variation in psychopathological terminology

Abstract Representing specialized knowledge in the medical domain implies considering the dynamism of scientific and technological progress. The advancement of knowledge on diseases goes hand in hand with the reconceptualization processes undertaken by experts with consequent conceptual evolutions and possible variations of the terms used to designate medical concepts. Sometimes term variation is the result of a desire to avoid or overcome negative connotations anchored in medical terms, and leads to the creation of less evaluatively charged terms that carry a diminished ideological load. This study illustrates the case of Body Dysmorphic Disorder (BDD), a relatively under-/misdiagnosed medical condition which has been the object of multiple reconceptualizations by experts. We focus on the analysis of the conceptual evolution of BDD and the consequent variation occurring at the linguistic level. We adopt the theoretical assumption that terminology has a double dimension – conceptual and linguistic. Following on this assumption, the terminologist must examine both the experts’ conceptualizations of a given domain and the discourses produced by them in order to effectively represent the specialized knowledge of a specific subject field. To complete the analysis, we present how information about BDD is disseminated to non-experts through the analysis of a corpus of mass-media articles.

Structural Complexities in Sodium Ion Conductive Antiperovskite Revealed by Cryogenic Transmission Electron Microscopy.

We use low-dose cryogenic transmission electron microscopy (cryo-TEM) to investigate the atomic-scale structure of antiperovskite Na2NH2BH4 crystals by preserving the room-temperature cubic phase and carefully monitoring the electron dose. Via quantitative analysis of electron beam damage using selected area electron diffraction, we find cryogenic imaging provides 6-fold improvement in beam stability for this solid electrolyte. Cryo-TEM images obtained from flat crystals revealed the presence of a new, long-range-ordered supercell with a cubic phase. The supercell exhibits doubled unit cell dimensions of 9.4 Å × 9.4 Å as compared to the cubic lattice structure revealed by X-ray crystallography of 4.7 Å × 4.7 Å. The comparison between the experimental image and simulated potential map indicates the origin of the supercell is a vacancy ordering of sodium atoms. This work demonstrates the potential of using cryo-TEM imaging to study the atomic-scale structure of air- and electron-beam-sensitive antiperovskite-type solid electrolytes.

Routing on heavy path WSPD spanners

In this article, we present a construction of a spanner on a set of n points in Rd that we call a heavy path WSPD spanner. The construction is parameterized by a constant s>2 called the separation ratio. The size of the graph is O(sdn) and the spanning ratio is at most 1+2/s+2/(s−1). We also show that this graph has a hop spanning ratio of at most 2lg⁡n+1.We present a memoryless local routing algorithm for heavy path WSPD spanners. The routing algorithm requires a vertex v of the graph to store O(deg⁡(v)log⁡n) bits of information, where deg⁡(v) is the degree of v. The routing ratio is at most 1+4/s+1/(s−1) and at least 1+4/s in the worst case. The number of edges on the routing path is bounded by 2lg⁡n+1.We then show that the heavy path WSPD spanner can be constructed in metric spaces of bounded doubling dimension. These metric spaces have been studied in computational geometry as a generalization of Euclidean space. We show that, in a metric space with doubling dimension λ, the heavy path WSPD spanner has size O(sλn) where s is the separation ratio. The spanning ratio and hop spanning ratio are the same as in the Euclidean case.Finally, we show that the local routing algorithm works in the bounded doubling dimension case. The vertices require the same amount of storage, but the routing ratio becomes at most 1+(2+ττ−1)/s+1/(s−1) in the worst case, where τ≥11 is a constant related to the doubling dimension.

More on doubled Hilbert space in double-scaled SYK

We continue our study of the doubled Hilbert space formalism of double-scaled SYK model initiated in Okuyama (2024) [7]. We show that the 1-particle Hilbert space introduced by Lin and Stanford (2023) [6] is related to the doubled Hilbert space H0⊗H0 of the 0-particle Hilbert space H0 by some linear isomorphism C. It turns out that the entangler and disentangler appeared in our previous study are given by the dual (or adjoint) of C−1 and C.

Double resolvability parameters of fosmidomycin anti-malaria drug and exchange property

The practical and theoretical significance of the resolvability parameter makes it an important factor, particularly in the context of network analysis. Its significance is seen in various applications and consequences: Network security, efficient routing, social network analysis, facility location, and site selection. This article finds the double resolvability parameters of the fosmidomycin anti-malaria drug. Resolvability parameters like double metric, double edge metric, and double mixed metric dimensions of fosmidomycin anti-malaria drug also hold exchange properties in the molecular graph of fosmidomycin. We convert the molecular structures of fosmidomycin into molecular graphs and then find some resolvability parameters.