Equivalent Definitions Research Articles

A Novel Approach to the Fractional Laplacian via Generalized Spherical Means

Although at least ten equivalent definitions of the fractional Laplacian exist in unbounded domains, we introduce an additional equivalent definition based on the generalized spherical mean-value operator—a Fourier multiplier operator involving the normalized Bessel function. Specifically, we demonstrate that this new definition allows us to reduce any n-dimensional fractional Laplacian to a one-dimensional operator, which simplifies computation and enhances efficiency. We propose two methods for computing the generalized spherical means of a given function: one by solving standard wave equations and the other by solving Darboux’s equations.

Combinatorial Results on Barcode Lattices

AbstractA barcode is a finite multiset of intervals on the real line. Jaramillo-Rodriguez (2023) previously defined a map from the space of barcodes with a fixed number of bars to a set of multipermutations, which presented new combinatorial invariants on the space of barcodes. A partial order can be defined on these multipermutations, resulting in a class of posets known as combinatorial barcode lattices. In this paper, we provide a number of equivalent definitions for the combinatorial barcode lattice, show that its Möbius function is a restriction of the Möbius function of the symmetric group under the weak Bruhat order, and show its ground set is the Jordan-Hölder set of a labeled poset. Furthermore, we obtain formulas for the number of join-irreducible elements, the rank-generating function, and the number of maximal chains of combinatorial barcode lattices. Lastly, we make connections between intervals in the combinatorial barcode lattice and certain classes of matchings.

Distribution and prevalence of refractive error in Iranian adult population results of the PERSIAN eye cohort study PECS

The Persian Eye Cohort Study, a population-based cross-sectional study from 2015 to 2020, examined refractive error prevalence among 48,618 Iranian adults aged 31 to 70. The study encompassed six centers in Iran, employing random cluster sampling for demographic, medical, and socioeconomic data collection through interviews. Ophthalmic exams included visual acuity, automated and manual objective refraction, subjective refraction, slit lamp, and fundus examinations. Using the spherical equivalent definition, the sample population was categorized into groups. Results indicated a mean age of 49.52 ± 9.31 and a mean refractive error of 0.26 diopters (D) ± 1.6 SD (95% CI − 0.27 to -0.24), ranging from -26.1 to + 18.5 SD. Prevalence of myopia (< −0.5D) and hyperopia (> + 0.5D) was 22.6% (95% CI 22.2–23%) and 12.5% (95% CI 12.1–12.8%), respectively. Regarding different age groups, the prevalence of hyperopia and astigmatism exhibited a steady and significant rise with increasing age (p-value < 0.001 for both). The prevalence of Myopia, however, showed a distinctive pattern, initially increasing in adults under 45, declining in those aged 55–64, and rising again among individuals aged 60 and older. Female gender, older age, urban residency, higher education, higher income, and Fars ethnicity were significantly related to a higher prevalence of myopia (p-value < 0.001 for all). Female gender (p-value < 0.001), aging (p-value < 0.001), urban residency (p-value = 0.029), and lower-income (p-value = 0.005) were significantly related to higher prevalence of hyperopia. Astigmatism (> 1D) was prevalent in 25.5% of participants (95% CI 25.1–25.9%) and correlated with male gender, aging, urban residency, illiteracy, and higher income (p-value < 0.001, < 0.001, < 0.001, < 0.001, 0.014, respectively). The study’s comparison with regional and international surveys highlighted the increase in myopia among those over 65 due to higher nuclear cataract rates in older adults. Myopia positively related to education, income, and urban residency, while hyperopia did not exhibit such associations.

Checking equivalence of corecursive streams: An inductive procedure

In recent work, non-periodic streams have been defined corecursively, by representing them with finitary equational systems built on top of various operators, besides the standard constructor. When only the stream constructor is allowed in equations, only periodic streams can be represented, and the structures of periodic streams and infinite regular trees are isomorphic. Therefore, one can use the theory of regular trees to get a sound and complete procedure to decide whether two equational systems are equivalent, that is, define the same streams. However, such an isomorphism no longer exists if one allows other operators in equations; in particular, there exist systems of equations which have the same unique solution as streams, but not as regular trees. Hence, equality of regular trees becomes stronger then equality of streams, with a negative impact on termination of functions whose definition is based on the equivalence of the representation of streams as finitary equational systems. To overcome this problem, we provide a weaker definition of equivalence, and prove its soundness and relative completeness. This definition is coinductive, hence non-algorithmic. However, we show that it can be turned into an equivalent inductive procedure.

Monotone classes beyond VNP

In this work, we study the natural monotone analogues of various equivalent definitions of VPSPACE: a well studied class (Poizat 2008, Koiran & Perifel 2009, Malod 2011, Mahajan & Rao 2013) that is believed to be larger than VNP. We observe that these monotone analogues are not equivalent unlike their non-monotone counterparts, and propose monotoneVPSPACE (mVPSPACE) to be defined as the monotone analogue of Poizat's definition. With this definition, mVPSPACE turns out to be exponentially stronger than mVNP and also satisfies several desirable closure properties that the other analogues may not.Our initial goal was to understand the monotone complexity of transparent polynomials, a concept that was recently introduced by Hrubeš & Yehudayoff (2021). In that context, we show that transparent polynomials with large support are hard for the monotone analogues of all the known definitions of VPSPACE, except for the one due to Poizat.

Fractional Sobolev spaces on Riemannian manifolds

This article studies the canonical Hilbert energy Hs/2(M)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^{s/2}(M)$$\\end{document} on a Riemannian manifold for s∈(0,2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s\\in (0,2)$$\\end{document}, with particular focus on the case of closed manifolds. Several equivalent definitions for this energy and the fractional Laplacian on a manifold are given, and they are shown to be identical up to explicit multiplicative constants. Moreover, the precise behavior of the kernel associated with the singular integral definition of the fractional Laplacian is obtained through an in-depth study of the heat kernel on a Riemannian manifold. Furthermore, a monotonicity formula for stationary points of functionals of the type E(v)=[v]Hs/2(M)2+∫MF(v)dV\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {E}}(v)=[v]^2_{H^{s/2}(M)}+\\int _M F(v) \\, dV$$\\end{document}, with F≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F\\ge 0$$\\end{document}, is given, which includes in particular the case of nonlocal s-minimal surfaces. Finally, we prove some estimates for the Caffarelli–Silvestre extension problem, which are of general interest. This work is motivated by Caselli et al. (Yau’s conjecture for nonlocal minimal surfaces, arxiv preprint, 2023), which defines nonlocal minimal surfaces on closed Riemannian manifolds and shows the existence of infinitely many of them for any metric on the manifold, ultimately proving the nonlocal version of a conjecture of Yau (Ann Math Stud 102:669–706, 1982). Indeed, the definitions and results in the present work serve as an essential technical toolbox for the results in Caselli et al. (Yau’s conjecture for nonlocal minimal surfaces, arxiv preprint, 2023).

Enhancing decision-making in breast cancer diagnosis for women through the application of nano beta open sets

Nano topology is a modern type of topology based on rough sets, introducing nano open sets and their properties. This study explores the new properties of nano-β-open sets and nano-closures in nano topological space, providing some equivalent definitions, theorems, and lemmas for illustration. The effectiveness of incorporating nano β-open sets in medical decision-making is demonstrated in diagnosing breast cancer in two cases for women. Results indicated that utilizing nano open sets enhanced the accuracy, sensitivity, and specificity of cancer diagnosis. Additionally, an algorithm has been developed to aid clinicians in diagnosing breast cancer infections. The overarching objective is to provide physicians with a valuable tool to make informed decisions and reach optimal conclusions in patient care.

Introduction to Haar Measure Tools in Quantum Information: A Beginner&amp;apos;s Tutorial

The Haar measure plays a vital role in quantum information, but its study often requires a deep understanding of representation theory, posing a challenge for beginners. This tutorial aims to provide a basic introduction to Haar measure tools in quantum information, utilizing only basic knowledge of linear algebra and thus aiming to make this topic more accessible. The tutorial begins by introducing the Haar measure with a specific emphasis on characterizing the moment operator, an essential element for computing integrals over the Haar measure. It also covers properties of the symmetric subspace and introduces helpful tools like tensor network diagrammatic notation, which aid in visualizing and simplifying calculations. Next, the tutorial explores the concept of unitary designs, providing equivalent definitions, and subsequently explores approximate notions of unitary designs, shedding light on the relationships between these different notions. Practical examples of Haar measure calculations are illustrated, including the derivation of well-known formulas such as the twirling of a quantum channel. Lastly, the tutorial showcases the applications of Haar measure calculations in quantum machine learning and classical shadow tomography.

Modeling Metallic Fatigue Data Using the Birnbaum–Saunders Distribution

This work employs the Birnbaum–Saunders distribution to model the fatigue-life of metallic materials under cyclic loading and compares it with the normal distribution. Fatigue-limit models are fitted to three datasets of unnotched specimens of 75S-T6 aluminum alloys and carbon laminate with different loading types. A new equivalent stress definition that accounts for the effect of the experiment type is proposed. The results show that the Birnbaum–Saunders distribution consistently outperforms the normal distribution in fitting the fatigue data and provides more accurate predictions of fatigue-life and survival probability.

Maximal resolving sets in a graph

Let G be a connected graph. A subset [Formula: see text] of [Formula: see text] is called a resolving set of G if the code of any vertex [Formula: see text] with respect to S is different from the code of any other vertex where code of u with respect to S denoted by [Formula: see text] is defined as [Formula: see text]. Resolving set was earlier studied in the name of locating set by Slater and Harary and Melter too studied this concept. The minimum cardinality of a resolving set is called the metric dimension (locating number). A vertex [Formula: see text] in a connected graph G is said to resolve two vertices [Formula: see text] if [Formula: see text] Clearly, x resolves [Formula: see text] A subset S of [Formula: see text] is a resolving set of G if for any two distinct vertices [Formula: see text] there exists a vertex [Formula: see text] such that x resolves [Formula: see text] Motivated by this equivalent definition, a study of resolving chain and maximal resolving set is initiated in this paper. Also, study of total resolving sets is initiated.

Equivalence of Generalised Feistel Networks

This paper focuses on equivalences between Generalised Feistel Networks (GFN) of type-II. We introduce a new definition of equivalence which captures the concept that two GFNs are identical up to re-labelling of the inputs/outputs, and give a procedure to test this equivalence relation. Such two GFNs are therefore cryptographically equivalent for several classes of attacks. It induces a reduction o the space of possible GFNs: the set of the (k!)2 possible even-odd GFNs with 2k branches can be partitioned into k! different classes.This result can be very useful when looking for an optimal GFN regarding specific computationally intensive properties, such as the minimal number of active S-boxes in a differential trail. We also show that in several previous papers, many GFN candidates are redundant as they belong to only a few classes. Because of this reduction of candidates, we are also able to suggest better permutations than the one of WARP: they reach 64 active S-boxes in one round less and still have the same diffusion round that WARP. Finally, we also point out a new family of permutations with good diffusion properties.

Isomorphism classes of Drinfeld modules over finite fields

We study isogeny classes of Drinfeld A-modules over finite fields k with commutative endomorphism algebra D, in order to describe the isomorphism classes in a fixed isogeny class. We study when the minimal order A[π] of D generated by the Frobenius π occurs as an endomorphism ring by proving when it is locally maximal at π, and show that this happens if and only if the isogeny class is ordinary or k is the prime field. We then describe how the monoid of fractional ideals of the endomorphism ring E of a Drinfeld module ϕ up to D-linear equivalence acts on the isomorphism classes in the isogeny class of ϕ, in the spirit of Hayes. We show that the action is free when restricted to kernel ideals, of which we give three equivalent definitions, and determine when the action is transitive. In particular, the action is free and transitive on the isomorphism classes in an isogeny class which is either ordinary or defined over the prime field, yielding a complete and explicit description in these cases, which can be implemented as a computer algorithm.

Internalizing Indistinguishability with Dependent Types

In type systems with dependency tracking, programmers can assign an ordered set of levels to computations and prevent information flow from high-level computations to the low-level ones. The key notion in such systems is indistinguishability : a definition of program equivalence that takes into account the parts of the program that an observer may depend on. In this paper, we investigate the use of dependency tracking in the context of dependently-typed languages. We present the Dependent Calculus of Indistinguishability (DCOI), a system that adopts indistinguishability as the definition of equality used by the type checker. DCOI also internalizes that relation as an observer-indexed propositional equality type, so that programmers may reason about indistinguishability within the language. Our design generalizes and extends prior systems that combine dependency tracking with dependent types and is the first to support conversion and propositional equality at arbitrary observer levels. We have proven type soundness and noninterference theorems for DCOI and have developed a prototype implementation of its type checker.

Equivalence testing for multiple groups

SummaryTesting for equivalence, rather than testing for a difference, is an important component of some scientific studies. While the focus of the existing literature is on comparing two groups for equivalence, real‐world applications arise regularly that require testing across more than two groups. This paper reviews the existing approaches for testing across multiple groups and proposes a novel framework for multigroup equivalence testing under a Bayesian paradigm. This approach allows for a more scientifically meaningful definition of the equivalence margin and a more powerful test than the few existing alternatives. This approach also allows a new definition of equivalence based on future differences.

Definition of information in computer science

Purpose of this study is to formulate a working definition of information to meet the needs of computer science. There is currently no strict definition of this term. There is a methodological contradiction: the development and application of information technologies requires accuracy and rigor, but at the same time the development is based on a vague, intuitive concept. Materials and methods. The materials for the study are existing classical approaches to understanding information, and the main method is the analysis of these approaches. The proposed definition is constructed taking into account two mathematical transformations: the selection of a certain subset and the mapping between sets. To formalize the allocation procedure, it is used apparatus of fuzzy sets. Results. A definition of information is proposed as the result of a mapping in which the selection of a subset from a set of prototypes leads to the selection of a corresponding subset from a set of images. The selected subset can be understood as fuzzy, then an equivalent definition of information is acceptable as a result of mapping, in which an increase in the heterogeneity of the distribution of the presence indicator on the set of prototypes leads to an increase in the heterogeneity of the distribution of the corresponding indicator on the set of images. The essence of the new definition is demonstrated using models of population dynamics in discrete time. The significance of the proposed approach for information technology is revealed using the example of the numerical method of multi-extremal optimization. It is shown that the proposed definition makes it possible to formulate effective stopping conditions for the numerical method of stochastic optimization, which guarantees the receipt of a given amount of information. Conclusion. The proposed understanding of information allows us to overcome the shortcomings of previous approaches to understanding the essence of information, retains all the advantages of the classical approach and is consistent with other well-known approaches in the field of computer science. This definition can be used to improve numerical optimization methods, as well as other information technology tools.

Non-lightlike framed rectifying curves in Minkowski 3-space

&lt;p&gt;In this paper, we define non-lightlike framed rectifying curves. They may have singularities. We give equivalent definitions and a construction method for non-lightlike framed rectifying curves. Moreover, we also study the relationship between the non-lightlike framed rectifying curves and the non-lightlike framed helices, as well as the properties of the centrode of non-lightlike framed rectifying curves.&lt;/p&gt;

Constructing Symmetric Equality Algebras

In this paper, we introduce the notion of strong fuzzy filter on hyper equality algebras and investigate some equivalence definitions of it. Then by using this notion we constructed a symmetric equality algebra and define a special form of classes. By using these, we define the concept of a fuzzy hyper congruence relation on hyper equality algebra and we prove that the quotient is made by it is an equality algebra. Also, by using a fuzzy equivalence relation on hyper equality, we introduce a fuzzy hyper congruence relation and prove that this fuzzy hyper congruence is regular and finally we prove that the quotient structure that is made by it is a symmetric hyper equality algebra.

On the symmetric 2-adic complexity of periodic binary sequences

Symmetric 2-adic complexity is a better measure for assessing the strength of a periodic binary sequence against the rational approximation attack. We study the symmetric 2-adic complexity of the sequences with known 2-adic complexity. To this end, we propose an equivalent definition of the symmetric 2-adic complexity and two sufficient conditions for a periodic sequence to achieve the maximum symmetric 2-adic complexity. As a first application, several types of sequences with maximum 2-adic complexity are easily shown to have the maximum symmetric 2-adic complexity. In addition, with the new definition, by analysing the algebraic structure of the sequences and applying the method for the calculation of their 2-adic complexity, we determine the symmetric 2-adic complexity of the Ding-Helleseth-Martinsen sequence and the sequence constructed by interleaving a pair of Legendre sequences.

Non-lightlike framed rectifying curves in Minkowski 3-space

&lt;p&gt;In this paper, we define non-lightlike framed rectifying curves. They may have singularities. We give equivalent definitions and a construction method for non-lightlike framed rectifying curves. Moreover, we also study the relationship between the non-lightlike framed rectifying curves and the non-lightlike framed helices, as well as the properties of the centrode of non-lightlike framed rectifying curves.&lt;/p&gt;

On a Bivariate Katz's Distribution Constructed by the Trivariate Reduction Method

In this paper, we propose the bivariate distribution of the Katz distribution [1] constructed by the trivariate reduction method, method developed in [6] and used in [5] to give an equivalent definition of the bivariate Poisson distribution [7, 8]. The constructed distribution includes, in particular, the bivariate Poisson distribution [5] and has interesting properties. For the estimation of the parameters, we used two methods: the method of moments and the maximum likelihood method using the EM algorithm. An application to concrete data has been made in order to carry out a comparative study between bivariate Poisson and Katz distributions, and we discuss the likelihood-ratio test, which assesses the goodness of fit of two competing statistical models.