Equivalent Definitions Research Articles

Noncommutative field theory from angular twist

We consider a noncommutative field theory in three space-time dimensions, with space-time star commutators reproducing a solvable Lie algebra. The $\ensuremath{\star}$-product can be derived from a twist operator and it is shown to be invariant under twisted Poincar\'e transformations. In momentum space the noncommutativity manifests itself as a noncommutative $\ensuremath{\star}$-deformed sum for the momenta, which allows for an equivalent definition of the $\ensuremath{\star}$-product in terms of twisted convolution of plane waves. As an application, we analyze the $\ensuremath{\lambda}{\ensuremath{\phi}}^{4}$ field theory at one loop and discuss its UV/IR behavior. We also analyze the kinematics of particle decay for two different situations: the first one corresponds to a splitting of space-time where only space is deformed, whereas the second one entails a nontrivial $\ensuremath{\star}$-multiplication for the time variable, while one of the three spatial coordinates stays commutative.

Phillips symmetric operators and their extensions

This article is devoted to the investigation of self-adjoint (and, more generally, proper) extensions of Phillips symmetric operators (PSO). A closed densely defined symmetric operator with equal defect numbers is considered a Phillips symmetric operator if its characteristic function is a constant on C+. We present equivalent definitions of PSO and prove that proper extensions with real spectra of a given PSO are similar to each other. Our results imply that one-point interaction of the momentum operator iddx+αδ(x−y) leads to unitarily equivalent self-adjoint operators with Lebesgue spectra. Self-adjoint operators with nontrivial spectral properties can be obtained as a result of more complicated perturbations of the momentum operator. In this way, we study special classes of perturbations which can be characterized as one-point interactions defined by the nonlocal potential γ∈L2(R).

Soft (1,2)*-Omega Separation Axioms and Weak Soft (1,2)*-Omega Separation Axioms in Soft Bitopological Spaces

In the present paper we introduce and study new classes of soft separation axioms in soft bitopological spaces, namely, soft (1,2)*-omega separation axioms and weak soft (1,2)*-omega separation axioms by using the concept of soft (1,2)*-omega open sets. The equivalent definitions and basic properties of these types of soft separation axioms also have been studied.

Rortex and comparison with eigenvalue-based vortex identification criteria

Most of the current Eulerian vortex identification criteria, including the Q criterion and the λci criterion, are exclusively determined by the eigenvalues of the velocity gradient tensor or the related invariants and thereby can be regarded as eigenvalue-based criteria. However, these criteria will be plagued with two shortcomings: (1) these criteria fail to identify the swirl axis or orientation; (2) these criteria are prone to severe contamination by shearing. To address these issues, a new vector named Rortex which represents the local fluid rotation was proposed in our previous work. In this paper, an alternative eigenvector-based definition of Rortex is introduced. The direction of Rortex, which represents the possible axis of the local rotation, is determined by the real eigenvector of the velocity gradient tensor. And then the rotational strength obtained in the plane perpendicular to the possible axis is used to define the magnitude of Rortex. This new equivalent definition allows a much more efficient implementation. Furthermore, a systematic interpretation of scalar, vector, and tensor versions of Rortex is presented. By relying on the tensor interpretation, the velocity gradient tensor is decomposed to a rigid rotation part and a non-rotational part including shearing, stretching, and compression, different from the traditional symmetric and anti-symmetric tensor decomposition. It can be observed that shearing always manifests its effect on the imaginary part of the complex eigenvalues and consequently contaminates eigenvalue-based criteria, while Rortex can exclude the shearing contamination and accurately quantify the local rotational strength. In addition, in contrast to eigenvalue-based criteria, not only the iso-surface of Rortex but also the Rortex vectors and the Rortex lines can be applied to investigate vortical structures. Several comparative studies on simple examples and realistic flows are studied to confirm the superiority of Rortex.

The transformation between the Galois NLFSRs and the Fibonacci NLFSRs via semi-tensor product of matrices

In this paper, a novel way is proposed to investigate the transformation between Galois nonlinear feedback shift registers (NLFSRs) and Fibonacci NLFSRs. First, the Galois NLFSRs and Fibonacci NLFSRs are regarded as two Boolean networks (BNs), and the corresponding algebraic equations are obtained based on semi-tensor product (STP) of matrices. Then, the definition of absolute equivalence is given to investigate the transformation between these two kinds of NLFSRs. Furthermore, some interesting results are presented to achieve the transformation. Finally, an example is provided to illustrate the effectiveness of obtained results.

Euclidean Distance-Based Skeletons: A Few Notes on Average Outward Flux and Ridgeness

Among the various existing and mathematically equivalent definitions of the skeleton, we consider the set of critical points of the Euclidean distance transform of the shape. The problem of detecting these points and using them to generate a skeleton that is stable , thin and homotopic to the shape has been the focus of numerous papers. Skeleton branches correspond to ridges of the distance map, i.e. continuous lines of points that are local maxima of the distance in at least one direction. Extracting these ridges is a non-trivial task on a discrete grid. In this context, the average outward flux, used in the Hamilton-Jacobi skeleton [43], and the ridgeness measure [28] have been proposed as ridge detectors. We establish the mathematical relation between these detectors and, extending the work in [18], we study various local shape configurations, on which closed-form expressions or approximations of the average outward flux and ridgeness can be derived. In addition , we conduct experiments to assess the accuracy of skeletons generated using these measures, and study the influence of their respective parameters.

SenseDefs: a multilingual corpus of semantically annotated textual definitions

Definitional knowledge has proved to be essential in various Natural Language Processing tasks and applications, especially when information at the level of word senses is exploited. However, the few sense-annotated corpora of textual definitions available to date are of limited size: this is mainly due to the expensive and time-consuming process of annotating a wide variety of word senses and entity mentions at a reasonably high scale. In this paper we present SenseDefs, a large-scale high-quality corpus of disambiguated definitions (or glosses) in multiple languages, comprising sense annotations of both concepts and named entities from a wide-coverage unified sense inventory. Our approach for the construction and disambiguation of this corpus builds upon the structure of a large multilingual semantic network and a state-of-the-art disambiguation system: first, we gather complementary information of equivalent definitions across different languages to provide context for disambiguation; then we refine the disambiguation output with a distributional approach based on semantic similarity. As a result, we obtain a multilingual corpus of textual definitions featuring over 38 million definitions in 263 languages, and we publicly release it to the research community. We assess the quality of SenseDefs’s sense annotations both intrinsically and extrinsically on Open Information Extraction and Sense Clustering tasks.

Equivalence of formulations of the MKP hierarchy and its polynomial tau-functions

We show that a system of Hirota bilinear equations introduced by Jimbo and Miwa defines tau-functions of the modified KP (MKP) hierarchy of evolution equations introduced by Dickey. Some other equivalent definitions of the MKP hierarchy are established. All polynomial tau-functions of the KP and the MKP hierarchies are found. Similar results are obtained for the reduced KP and MKP hierarchies.

Bisimulation Equivalence of Discrete-Time Stochastic Linear Control Systems

In this paper, we propose a definition of equivalence via stochastic bisimulation for the class of discrete-time stochastic linear control systems with possibly degenerate normally distributed disturbances. The notion is inspired by the notion of probabilistic bisimulation for probabilistic chains. Geometric necessary and sufficient conditions for checking this notion are derived. Model reduction via Kalman-like decomposition is also proposed. Connections with stochastic reachability are discussed and with finite horizon stochastic safety problems established. A discussion on the use of stochastic bisimulation equivalence for control design is given, and an application to optimal control problems with stochastic reachability specifications is finally presented.

Genetic programming with semantic equivalence classes

Abstract In this paper, we introduce the concept of semantics-based equivalence classes for symbolic regression problems in genetic programming. The idea is implemented by means of two different genetic programming systems, in which two different definitions of equivalence are used. In both systems, whenever a solution in an equivalence class is found, it is possible to generate any other solution in that equivalence class analytically. As such, these two systems allow us to shift the objective of genetic programming: instead of finding a globally optimal solution, the objective is now to find any solution that belongs to the same equivalence class as a global optimum. Further, we propose improvements to these genetic programming systems in which, once a solution that belongs to a particular equivalence class is generated, no other solution in that class is accepted in the population during the evolution anymore. We call these improved versions filtered systems. Experimental results obtained via seven complex real-life test problems show that using equivalence classes is a promising idea and that filters are generally helpful for improving the systems' performance. Furthermore, the proposed methods produce individuals with a much smaller size with respect to geometric semantic genetic programming. Finally, we show that filters are also useful to improve the performance of a state-of-the-art method, not explicitly based on semantic equivalence classes, like linear scaling.

Partially symmetric nonnegative rectangular tensors and copositive rectangular tensors

In this paper, we prove a maximum property of the largest H-singular value of a partially symmetric nonnegative rectangular tensor, and establish some bounds for this singular value. Then we give the definition of copositive rectangular tensors. This concept extends from the concept of copositive square tensors. Partially symmetric nonnegative rectangular tensors and positive semi-definite rectangular tensors are examples of copositive rectangular tensors. We establish some necessary conditions and some sufficient conditions for a real partially symmetric rectangular tensor to be a copositive rectangular tensor. We also give an equivalent definition of strictly copositive rectangular tensors. Moreover, some further properties of copositive rectangular tensors are discussed.

Verbally prime T-ideals and graded division algebras

Let F be an algebraically closed field of characteristic zero and let G be a finite group. We consider graded Verbally prime T-ideals in the free G-graded algebra. It turns out that equivalent definitions in the ordinary case (i.e. ungraded) extend to nonequivalent definitions in the graded case, namely verbally prime G-graded T-ideals and strongly verbally prime T-ideals. At first, following Kemer's ideas, we classify G-graded verbally prime T-ideals. The main bulk of the paper is devoted to the stronger notion. We classify G-graded strongly verbally prime T-ideals which are T-ideal of affine G-graded algebras or equivalently G-graded T-ideals that contain a Capelli polynomial. It turns out that these are precisely the T-ideal of G-graded identities of finite dimensional G-graded, central over F (i.e. Z(A)e=F) which admit a G-graded division algebra twisted form over a field k which contains F or equivalently over a field k which contains enough roots of unity (e.g. a primitive n-root of unity where n=ord(G)).

Remarks on Pickands’ theorem

REMARKS ON PICKANDS’ THEOREMIn this article we present the Pickands theorem and his double sum method. We follow Piterbarg’s proof of this theorem. Since his proof relies on general lemmas, we present a complete proof of Pickands’ theorem using the Borell inequality and Slepian lemma. The original Pickands’ proof is rather complicated and is mixed with upcrossing probabilities for stationary Gaussian processes. We give a lower bound for Pickands constant. Moreover, we review equivalent definitions, simulations and bounds of Pickands constant.

A novel equivalent definition of Caputo fractional derivative without singular kernel and superconvergent analysis

In this article, we present a new second-order finite difference discrete scheme for a fractal mobile/immobile transport model based on equivalent transformative Caputo formulation. The new transformative formulation takes the singular kernel away to make the integral calculation more efficient. Furthermore, this definition is also effective where α is a positive integer. Besides, the T-Caputo derivative also helps us to increase the convergence rate of the discretization of the α-order(0 < α < 1) Caputo derivative from O(τ2−α) to O(τ3−α), where τ is the time step. For numerical analysis, a Crank-Nicolson finite difference scheme to solve the fractal mobile/immobile transport model is introduced and analyzed. The unconditional stability and a priori estimates of the scheme are given rigorously. Moreover, the applicability and accuracy of the scheme are demonstrated by numerical experiments to support our theoretical analysis.

The effects of a revised operational dose quantity on the response characteristics of neutron survey instruments

The ICRU is considering revising the definition of ambient dose equivalent. This paper investigates the impacts of the proposed change on four designs of neutron survey instrument, the GNU, HSREM, LB6411 and Studsvik 2202D, in terms of their respective energy dependences of response and their performances in realistic workplace fields. In some circumstances the current designs of instrument still produce acceptable characteristics, but in general they may need to be re-optimized to better match the requirements of the new operational quantity; to that end, a simple retrofit solution for the GNU is demonstrated. The performance criteria against which instruments are judged may also need to be revised to reflect the proposed change.

Different definitions of conic sections in hyperbolic geometry

In classical Euclidean geometry, there are several equivalent definitions of conic sections. We show that in the hyperbolic plane, the analogues of these same definitions still make sense, but are no longer equivalent, and we discuss the relationships among them.

Back-and-forth systems for fuzzy first-order models

This paper continues the study of model theory for fuzzy logics by addressing the fundamental issue of classifying models according to their first-order theory. Three different definitions of elementary equivalence for fuzzy first-order models are introduced and separated by suitable counterexamples. We propose several back-and-forth conditions, based both on classical two-sorted structures and on non-classical structures, that are useful to obtain elementary equivalence in particular cases as we illustrate with several examples.

An infinite family of links with critical bridge spheres

A closed, orientable, splitting surface in an oriented $3$-manifold is a topologically minimal surface of index $n$ if its associated disk complex is $(n-2)$-connected but not $(n-1)$-connected. A critical surface is a topologically minimal surface of index $2$. In this paper, we use an equivalent combinatorial definition of critical surfaces to construct the first known critical bridge spheres for nontrivial links.

Regular fuzzy equivalences on two - mode fuzzy Networksi

The notion of social roles is a centerpiece of most sociological theoretical considerations. Regular equivalences were introduced by White and Reitz in [15] as the least restrictive among the most commonly used definitions of equivalence in social network analysis. In this paper we consider a generalization of this notion to a bipartite case. We define a pair of regular equivalences on a two-mode social network and we provide an algorithm for computing the greatest pair of regular equivalences.

The linear canonical wavelet transform on some function spaces

It is well known that the domain of Fourier transform (FT) can be extended to the Schwartz space [Formula: see text] for convenience. As a generation of FT, it is necessary to detect the linear canonical transform (LCT) on a new space for obtaining the similar properties like FT on [Formula: see text]. Therefore, a space [Formula: see text] generalized from [Formula: see text] is introduced firstly, and further we prove that LCT is a homeomorphism from [Formula: see text] onto itself. The linear canonical wavelet transform (LCWT) is a newly proposed transform based on the convolution theorem in LCT domain. Moreover, we propose an equivalent definition of LCWT associated with LCT and further study some properties of LCWT on [Formula: see text]. Based on these properties, we finally prove that LCWT is a linear continuous operator on the spaces of [Formula: see text] and [Formula: see text].