Equivalent Definitions Research Articles

Aceptabilidad lingüística en la comprensión de cuestionarios de valoración de servicios de salud en Iberoamérica

The use of questionnaires as measuring tools involves the use of language. Spanish intralinguistic variation entails differences in the ways of perceiving the world, analyzing events and behaving, which must be taken into account when designing and adapting questionnaires. The IA-PCAT (Primary Care Assessment Tools for Iberomerica) harmonization process differs from the cross-cultural adaptation scenarios described in the scientific literature, since it intends to obtain a single Spanish product suitable for multiple target populations, that emanates from questionnaires previously adapted in several populations. Hence, this work requires the use of new analysis categories. The aim of this methodological note is to propose discriminant definitions of conceptual equivalence, practical equivalence and linguistic acceptability as analysis categories of the data collected during cognitive interviews to assess the questionnaires’ comprehension, carried out with health professionals and users of healthcare services.

Open texture clarified

ABSTRACT ‘Open texture’ is the property of concepts or terms that they are not fully defined with regard to unexpected questions. Even though it is a ubiquitous phenomenon, there is no consensus on which concepts are subject to open texture. I present three equivalent definitions of open texture, connecting the concept to the notions of analyticity and algebraicity. I show that the claim that a concept is open-textured is equivalent to the claim that the class of analytic sentences using it can be changed, which in turn is equivalent to the claim that its definition is not algebraic. While this equivalence is interesting on its own, with it I can show the extent of open texture: all concepts that are not algebraically defined mathematical concepts are open textured. Finally, I show that this has unfortunate consequences for attempts to provide a foundation of mathematics.

Shortest and straightest geodesics in sub-Riemannian geometry

There are several different, but equivalent definitions of geodesics in a Riemannian manifold, based on two characteristic properties: geodesics as shortest curves and geodesics as straightest curves. They are generalized to sub-Riemannian manifolds, but become non-equivalent. We give an overview of different approaches to the definition, study and generalization of sub-Riemannian geodesics and discuss interrelations between different definitions. For Chaplygin transversally homogeneous sub-Riemannian manifold Q, we prove that straightest geodesics (defined as geodesics of the Schouten partial connection) coincide with shortest geodesics (defined as the projection to Q of integral curves (with trivial initial covector) of the sub-Riemannian Hamiltonian system). This gives a Hamiltonization of Chaplygin systems in non-holonomic mechanics.We consider a class of homogeneous sub-Riemannian manifolds, where straightest geodesics coincide with shortest geodesics, and give a description of all sub-Riemannian symmetric spaces in terms of affine symmetric spaces.

MBJ-neutrosophic filters of equality algebras

In this paper, we introduce the notion of MBJneutrosophic sub-algebra and MBJ-neutrosophic filter on equality algebras and investigate some equivalence definitions, properties and the relation between them. Also, by using the notion of MBJ-neutrosophic filter, we introduce a congruence relation on equality algebra and show that the quotient is an equality algebra.

Derivations in Gamma Rings with γ-Lie and γ-Jordan Structures

Let M be weak Nobusawa Γ-ring and γ be non-zero element of Γ. In this paper, we study the definition of γ-semi prime Γ-ring, and introduce definition of γ-semi simple Γ-ring and γ-Jordan ideal of M. We give equivalent definitions for γ-prime Γ-ring and γ-semi prime Γ-ring. Also by using relation between Γ-rings and rings, we give some commutativity conditions on Γ-rings.

Entropy and Time.

The idea that entropy is associated with the “arrow of time” has its roots in Clausius’s statement on the Second Law: “Entropy of the Universe always increases.” However, the explicit association of the entropy with time’s arrow arises from Eddington. In this article, we start with a brief review of the idea that the “increase in entropy” is somehow associated with the direction in which time increases. Then, we examine three different, but equivalent definitions of entropy. We find that none of these definitions indicate any hint of a relationship between entropy and time. We can, therefore, conclude that entropy is a timeless quantity. We also discuss the reasons as to why some scientists went astray in associating entropy with time’s arrow. Finally, we shall discuss Boltzmann’s H-Theorem, which is viewed by many as a proof of the Second Law of Thermodynamics.

Efficient reassembling of three-regular planar graphs

A reassembling of a simple graph $$G = (V,E)$$ is an abstraction of a problem arising in earlier studies of network analysis (Bestavros and Kfoury, in: Proceedings of IFIP working conference on domain-specific 1640 languages (DSL 2011), EPTCS volume 66, 2011; Kfoury, in: Proceedings of SBLP 2011: Brazilian symposium on programming; Kfoury, in Sci Comput Program 93(Part A):19–38; Soule et al., in: Proceedings of 4th international workshop on equation-based object oriented modeling languages and tools, Zurich). There are several equivalent definitions of graph reassembling; in this report we use a definition which makes it closest to the notion of graph carving. A reassembling is a rooted binary tree whose nodes are subsets of V and whose leaf nodes are singleton sets, with each of the latter containing a distinct vertex of G. The parent of two nodes in the reassembling is the union of the two children’s vertex sets. The root node of the reassembling is the full set V. The edge-boundary degree of a node in the reassembling is the number of edges in G that connect vertices in the node’s set to vertices not in the node’s set. A reassembling’s $$\alpha $$-measure is the largest edge-boundary degree of any node in the reassembling. A reassembling of G is $$\alpha $$-optimal if its $$\alpha $$-measure is the minimum among all $$\alpha $$-measures of G’s reassemblings. The problem of finding an $$\alpha $$-optimal reassembling of a simple graph in general was already shown to be NP-hard (Kfoury and Mirzaei, in J Comb Optim 33(3):1057–1089; Mirzaei and Kfoury, in: Efficient reassembling of graphs, Part 2: the balanced case. CoRR, arXiv:1602.02863; among others). In this report we present an algorithm which, given a 3-regular plane graph $$G = (V,E)$$ as input, returns a reassembling of G with an $$\alpha $$-measure independent of $$n = |\,V\,|$$ and upper-bounded by 2k, where k is the edge-outerplanarity of G. (Edge-outerplanarity is distinct but closely related to the usual notion of outerplanarity; as with outerplanarity, for a fixed edge-outerplanarity k, the number n of vertices can be arbitrarily large). Our algorithm runs in linear time $${{\mathcal {O}}}(n)$$. Moreover, we construct a class of 3-regular plane graphs for which this $$\alpha $$-measure is optimal, by proving that 2k is the lower bound on the $$\alpha $$-measure of any reassembling of a graph in that class.

RALPH HENSTOCK'S INFLUENCE ON THE INTEGRATION THEORY

Professor Ralph Henstock, a distinguished analyst who devoted most of his research to integration theory, died on January 7, 2007. It was my good fortune to know Ralph for almost 30 years, and to derive much inspiration from his pioneering work. Rather than commenting briefly on a variety of Ralph’s achievements, I will elaborate on that which has been the most influential: a Riemann type definition of the Denjoy-Perron integral . An early recognition that the powerful Lebesgue integral does not integrate the derivatives of all differentiable functions led to the development of the Denjoy-Perron integral. Three equivalent definitions of the Denjoy-Perron integral were available at the beginning of the last century: descriptive and constructive definitions presented by Denjoy [16, Chapter 8], and a definition based on approximations by majorants and minorants due to Perron [16, Chapter 6, Section 6]. These definitions differ widely, and establishing their equivalence is not easy — for a comprehensive treatment see [4, Chapter 11]. Neither definition is simple, and attempts to generalize any of them to higher dimensions were not successful. Indeed, none of the multidimensional integrals based on these definitions integrates partial derivatives of all differentiable functions. It was a major independent accomplishment of Henstock [5] and Kurzweil [10] to observe that a minor but ingenious change in the classical definition of the Riemann integral produces the Denjoy-Perron integral. The striking simplicity of their definition revitalized the efforts toward finding a multidimensional analog of the Denjoy-Perron integral. The initial impetus was further promoted by Henstock’s diligent work on the general properties of Riemann type integrals, summarized in monographs [6, 7, 8]. After introducing the Henstock-Kurzweil result without proof, I will discuss two separate but related topics:

LOWER ASSOUAD-TYPE DIMENSIONS OF UNIFORMLY PERFECT SETS IN DOUBLING METRIC SPACES

In this paper, we are concerned with the relationship among the lower Assouad-type dimensions. For uniformly perfect sets in doubling metric spaces, we obtain a variational result between two different but closely related lower Assouad spectra. As an application, we show that the limit of the lower Assouad spectrum as [Formula: see text] tends to 1 is equal to the quasi-lower Assouad dimension, which provides an equivalent definition to the latter. On the other hand, although the limit of the lower Assouad spectrum as [Formula: see text] tends to 0 exists, there exist uniformly perfect sets such that this limit is not equal to the lower box-counting dimension. Moreover, by the example of Cantor cut-out sets, we show that the new definition of quasi-lower Assouad dimension is more accessible, and indicate that the lower Assouad dimension could be strictly smaller than the lower spectra and the quasi-lower Assouad dimension.

Percolation on an infinitely generated group

AbstractWe give an example of a long range Bernoulli percolation process on a group non-quasi-isometric with ℤ, in which clusters are almost surely finite for all values of the parameter. This random graph admits diverse equivalent definitions, and we study their ramifications. We also study its expected size and point out certain phase transitions.

On the structural dynamics of laminated composite plates and sandwich structures; a new perspective on damping identification

This paper presents the modelling and the dynamic characterisation of laminated composite plates and sandwich structures in terms of stiffness and damping. The developments used in this paper are based on the analytical multilayer model of Guyader and Lesueur (JSV, 1978). The model considers linear shear, membrane and bending effects in each layer. The characteristics of the structure are determined by means of an equivalent thin plate methodology. The first main novelty of this paper consists in adapting this methodology for laminated plates (orthotropic multilayers with arbitrary orthotropic angle per layer). An experimental validation of this adaptation is presented for a laminated composite plate. Concerning the modelling of the structural loss factor, a space domain definition based on the spatial attenuation of a plane wave is compared to an energetic method and an equivalent definition based on the thin plate theory. The results show that the equivalent definition overestimates the loss factor in high frequencies since the thin plate theory only considers the flexural behaviour of the structure. On the contrary, the space domain definition (which give similar results as compared to the energetic one for lightly damped structures) considers the frequency dependent variation of the dynamic behaviour of the structure by means of the ratio between the group and phase velocities. The latter approach is considered to be more correct. The second main novelty of this article is on the experimental validation of this space domain definition. The structural loss factors of two sandwich structures are identified from measurements using modal, energetic and spatial methods. The results using the space domain definition are in very good agreement with the analytical predictions and the estimations of the modal and energetic methods for both plates for a large frequency band (up to 20 kHz), demonstrating the validity of the approach developed in this paper.

Recognizing hyperelliptic graphs in polynomial time

Based on analogies between algebraic curves and graphs, Baker and Norine introduced divisorial gonality, a graph parameter for multigraphs related to treewidth, multigraph algorithms and number theory. Various equivalent definitions of the gonality of an algebraic curve translate to different notions of gonality for graphs, called stable gonality and stable divisorial gonality.We consider so-called hyperelliptic graphs (multigraphs of gonality 2, in any meaning of graph gonality) and provide a safe and complete set of reduction rules for such multigraphs. This results in an algorithm to recognize hyperelliptic graphs in time O(m+nlog⁡n), where n is the number of vertices and m the number of edges of the multigraph. A corollary is that we can decide with the same runtime whether a two-edge-connected graph G admits an involution σ such that the quotient G/〈σ〉 is a tree.

Projectively equivalent 2-dimensional superintegrable systems with projective symmetries

This paper combines two classical theories, namely metric projective differential geometry and superintegrability. We study superintegrable systems on 2-dimensional geometries that share the same geodesics, viewed as unparametrized curves. We give a definition of projective equivalence of such systems, which may be considered the projective analog of (conformal) Stäckel equivalence (coupling constant metamorphosis). Then, we discuss the transformation behavior for projectively equivalent superintegrable systems and find that the potential on a projectively equivalent geometry can be reconstructed from a characteristic vector field. Moreover, potentials of projectively equivalent Hamiltonians follow a linear superimposition rule. The techniques are applied to several examples. In particular, we use them to classify, up to Stäckel equivalence, the superintegrable systems on geometries with one, non-trivial projective symmetry.

Paraunitary matrices, entropy, algebraic condition number and Fourier computation

The Fourier Transform is one of the most important linear transformations used in science and engineering. Cooley and Tukey's Fast Fourier Transform (FFT) from 1964 is a method for computing this transformation in time O(nlog⁡n). From a lower bound perspective, relatively little is known. Ailon shows in 2013 an Ω(nlog⁡n) bound for computing the normalized Fourier Transform assuming only unitary operations on two coordinates are allowed at each step, and no extra memory is allowed. In 2014, Ailon then improved the result to show that, in a κ-well conditioned computation, Fourier computation can be sped up by no more than O(κ). The main conjecture is that Ailon's result can be exponentially improved, in the sense that κ-well condition cannot admit ω(log⁡κ) speedup.The main result here is that ‘algebraic’ κ-well condition cannot admit ω(κ) speedup. One equivalent definition of algebraic condition number is related to the degree of polynomials naturally arising as the computation evolves. Using the maximum modulus theorem from complex analysis, we show that algebraic condition number upper bounds standard condition number, and equals it in certain cases. Algebraic condition number is an interesting measure of numerical computation stability in its own right, and provides a novel computational lens. Moreover, based on evidence from other recent related work, we believe that the approach of algebraic condition number has a good chance of establishing an algebraic version of the main conjecture.

Elasticity [formula omitted]-tensors and the strong ellipticity condition

In this paper, we establish two sufficient conditions for the strong ellipticity of any fourth-order elasticity tensor and investigate a class of tensors satisfying the strong ellipticity condition, the elasticity M-tensor. The first sufficient condition is that the strong ellipticity holds if the unfolding matrix of this fourth-order elasticity tensor can be modified into a positive definite one by preserving the summations of some corresponding entries. Second, an alternating projection algorithm is proposed to verify whether an elasticity tensor satisfies the first condition or not. Besides, the elasticity M-tensor is defined with respect to the M-eigenvalues of elasticity tensors. We prove that any nonsingular elasticity M-tensor satisfies the strong ellipticity condition by employing a Perron-Frobenius-type theorem for M-spectral radii of nonnegative elasticity tensors. Other equivalent definitions of nonsingular elasticity M-tensors are also established.

Convergence & rates for Hamilton–Jacobi equations with Kirchoff junction conditions

We investigate rates of convergence for two approximation schemes of time-independent and time-dependent Hamilton–Jacobi equations with Kirchoff junction conditions. We analyze the vanishing viscosity limit and monotone finite-difference schemes. Following recent work of Lions and Souganidis, we impose no convexity assumptions on the Hamiltonians. For stationary Hamilton–Jacobi equations, we obtain the classical \(\epsilon ^{\frac{1}{2}}\) rate, while we obtain an \(\epsilon ^{\frac{1}{7}}\) rate for approximations of the Cauchy problem. In addition, we present a number of new techniques of independent interest, including a quantified comparison proof for the Cauchy problem and an equivalent definition of the Kirchoff junction condition.

Numerical methods for post-formed timber gridshells: Simulation of the forming process and assessment of R-Funicularity

A nonlinear staged construction analysis is adopted here to simulate the forming process of timber gridshell. In this respect, particular attention is paid to the modelling of the cylindrical joints connecting the orthogonal laths. Cylindrical joints enable the initially flat lattice to be formed into a doubly curved gridshell shape; therefore, accurate modelling of the joints’ mechanical behaviour is a crucial aspect when simulating the forming process of timber gridshells.Another important parameter influencing the gridshell shape is the orientation of laths making up the initially flat lattice. Using an equivalent continuum definition, the final geometry of the gridshells, obtained from different laths orientations, is analysed in here to assess how laths orientation affects the structural shape in terms of funicularity. The Relaxed Funicularity Ellipse method presented in Gabriele et al. (2018) is used for such purpose.

Pointwise properties of <inline-formula><tex-math id="M1">\begin{document}$ L^p $\end{document}</tex-math></inline-formula>-viscosity solutions of uniformly elliptic equations with quadratically growing gradient terms

<p style='text-indent:20px;'>We prove various pointwise properties of <inline-formula><tex-math id="M2">\begin{document}$ L^p $\end{document}</tex-math></inline-formula>-viscosity solutions of fully nonlinear uniformly elliptic equations with unbounded measurable terms and possibly quadratically growing gradient terms. These include differentiability properties and "pointwise maximum principle". In particular we discuss an equivalent pointwise definition of <inline-formula><tex-math id="M3">\begin{document}$ L^p $\end{document}</tex-math></inline-formula>-viscosity solution which, together with pointwise properties, allows to operate on <inline-formula><tex-math id="M4">\begin{document}$ L^p $\end{document}</tex-math></inline-formula>-viscosity solutions almost as if they were strong solutions and move them freely from one equation to another. Such results were proved before in [<xref ref-type="bibr" rid="b6">6</xref>,<xref ref-type="bibr" rid="b8">8</xref>,<xref ref-type="bibr" rid="b30">30</xref>] for equations with linearly growing gradient terms, however it appears that they have not been widely used. Here we generalize pointwise properties of <inline-formula><tex-math id="M5">\begin{document}$ L^p $\end{document}</tex-math></inline-formula>-viscosity solutions to a larger class of equations and show that they create a very powerful set of tools which can be used for instance to prove regularity results for equations and differential inequalities.

Two Generalized Classes of Exponentially Dichotomous Linear Differential Systems on the Time Half-Line without Uniform Estimates for the Solution Norms. I

We consider two generalizations $$A{\cal E}_n$$ and $$W{\cal E}_n$$ of the class $${\cal E}_n$$ of linear $$n$$-dimensional exponentially dichotomous systems on the half-line. The definition of these classes differs from the classical definition of exponential dichotomy in terms of estimates for the norms of solutions in that we allow these estimates to hold starting from some time depending on the solution (the class $$A{\cal E}_n)$$ or the constant factors in the estimates to depend on the solution (the class $$W{\cal E}_n)$$. We give an equivalent definition of the class $$W{\cal E}_n$$ in the language of Bohl exponents and prove that the proper inclusions $${\cal E}_n\subset A{\cal E}_n \subset W{\cal E}_n$$ hold for $$n\geq 2$$.

Non-linear functionals, deficient topological measures, and representation theorems on locally compact spaces

We study non-linear functionals, including quasi-linear functionals, p-conic quasi-linear functionals, d-functionals, r-functionals, and their relationships to deficient topological measures and topological measures on locally compact spaces. We prove representation theorems and show, in particular, that there is an order-preserving, conic-linear bijection between the class of finite deficient topological measures and the class of bounded p-conic quasi-linear functionals. Our results imply known representation theorems for finite topological measures and deficient topological measures. When the space is compact we obtain four equivalent definitions of a quasi-linear functional and four equivalent definitions of functionals corresponding to deficient topological measures.