Notion Of Equivalence Research Articles

Polynomial and horizontally polynomial functions on Lie groups

We generalize both the notion of polynomial functions on Lie groups and the notion of horizontally affine maps on Carnot groups. We fix a subset S of the algebra {mathfrak {g}} of left-invariant vector fields on a Lie group {mathbb {G}} and we assume that S Lie generates {mathfrak {g}}. We say that a function f:{mathbb {G}}rightarrow {mathbb {R}} (or more generally a distribution on {mathbb {G}}) is S-polynomial if for all Xin S there exists kin {mathbb {N}} such that the iterated derivative X^k f is zero in the sense of distributions. First, we show that all S-polynomial functions (as well as distributions) are represented by analytic functions and, if the exponent k in the previous definition is independent on Xin S, they form a finite-dimensional vector space. Second, if {mathbb {G}} is connected and nilpotent, we show that S-polynomial functions are polynomial functions in the sense of Leibman. The same result may not be true for non-nilpotent groups. Finally, we show that in connected nilpotent Lie groups, being polynomial in the sense of Leibman, being a polynomial in exponential chart, and the vanishing of mixed derivatives of some fixed degree along directions of {mathfrak {g}} are equivalent notions.

Deformations of Lagrangian submanifolds in log-symplectic manifolds

This paper is devoted to deformations of Lagrangian submanifolds contained in the singular locus of a log-symplectic manifold. We prove a normal form result for the log-symplectic structure around such a Lagrangian, which we use to extract algebraic and geometric information about the Lagrangian deformations. We show that the deformation problem is governed by a DGLA, we discuss whether the Lagrangian admits deformations not contained in the singular locus, and we give precise criteria for unobstructedness of first order deformations. We also address equivalences of deformations, showing that the gauge equivalence relation of the DGLA corresponds with the geometric notion of equivalence by Hamiltonian isotopies. We discuss the corresponding moduli space, and we prove a rigidity statement for the more flexible equivalence relation by Poisson isotopies.

A fine-grained computational interpretation of Girard’s intuitionistic proof-nets

This paper introduces a functional term calculus, called pn, that captures the essence of the operational semantics of Intuitionistic Linear Logic Proof-Nets with a faithful degree of granularity, both statically and dynamically. On the static side, we identify an equivalence relation on pn-terms which is sound and complete with respect to the classical notion of structural equivalence for proof-nets. On the dynamic side, we show that every single (exponential) step in the term calculus translates to a different single (exponential) step in the graphical formalism, thus capturing the original Girard’s granularity of proof-nets but on the level of terms. We also show some fundamental properties of the calculus such as confluence, strong normalization, preservation of β-strong normalization and the existence of a strong bisimulation that captures pairs of pn-terms having the same graph reduction.

Lexical-Semantic Mainstreaming in the Context of Crossculturalcomparisons

The article addresses the issue of re-thinking lexical meanings concerning a wide range of ideologically biased words currently developing in media space. The author studies the contexts, which compare equivalent notions in the realm of ideological confrontation between Russia and the West. New meanings are revealed attributed to foreign equivalents in response to the facts of public life. The article shows that having been set against traditional Russian understanding of foreign equivalent, Russian words actualize the meanings associated with realities of life and culture in today’s Russia.

Exponentially Handsome Proof Nets and Their Normalization

Handsome proof nets were introduced by Retor\'e as a syntax for multiplicative linear logic. These proof nets are defined by means of cographs (graphs representing formulas) equipped with a vertices partition satisfying simple topological conditions. In this paper we extend this syntax to multiplicative linear logic with units and exponentials. For this purpose we develop a new sound and complete sequent system for the logic, enforcing a stronger notion of proof equivalence with respect to the one usually considered in the literature. We then define combinatorial proofs, a graphical proof system able to capture syntactically the proof equivalence, for the cut-free fragment of the calculus. We conclude the paper by defining the exponentially handsome proof nets as combinatorial proofs with cuts and defining an internal normalization procedure for this syntax.

Sparse regular variation

AbstractRegular variation provides a convenient theoretical framework for studying large events. In the multivariate setting, the spectral measure characterizes the dependence structure of the extremes. This measure gathers information on the localization of extreme events and often has sparse support since severe events do not simultaneously occur in all directions. However, it is defined through weak convergence, which does not provide a natural way to capture this sparsity structure. In this paper, we introduce the notion of sparse regular variation, which makes it possible to better learn the dependence structure of extreme events. This concept is based on the Euclidean projection onto the simplex, for which efficient algorithms are known. We prove that under mild assumptions sparse regular variation and regular variation are equivalent notions, and we establish several results for sparsely regularly varying random vectors.

Enumeration of Latin squares with conjugate symmetry

AbstractA Latin square has six conjugate Latin squares obtained by uniformly permuting its (row, column, symbol) triples. We say that a Latin square has conjugate symmetry if at least two of its six conjugates are equal. We enumerate Latin squares with conjugate symmetry and classify them according to several common notions of equivalence. We also do similar enumerations under additional hypotheses, such as assuming the Latin square is reduced, diagonal, idempotent or unipotent. Our data corrected an error in earlier literature and suggested several patterns that we then found proofs for, including (1) the number of isomorphism classes of semisymmetric idempotent Latin squares of order equals the number of isomorphism classes of semisymmetric unipotent Latin squares of order , and (2) suppose and are totally symmetric Latin squares of order . If and are paratopic then and are isomorphic.

Orbit equivalence of higher-rank graphs

We study the notion of continuous orbit equivalence of finitely-aligned higher-rank graphs. We show that there is a continuous orbit equivalence between two finitely-aligned higher-rank graphs that preserves the periodicity of boundary paths if and only if the boundary path groupoids are isomorphic. We also study eventual one-sided conjugacy of finitely-aligned higher-rank graphs and two-sided conjugacy of row-finite higher-rank graphs.

Tietze Equivalences as Weak Equivalences

A given monoid usually admits many presentations by generators and relations and the notion of Tietze equivalence characterizes when two presentations describe the same monoid: it is the case when one can transform one presentation into the other using the two families of so-called Tietze transformations. The goal of this article is to provide an abstract and geometrical understanding of this well-known fact, by constructing a model structure on the category of presentations, in which two presentations are weakly equivalent when they present the same monoid. We show that Tietze transformations form a pseudo-generating family of trivial cofibrations and give a proof of the completeness of these transformations by an abstract argument in this setting.

Investigating the exceptionality of scattered polynomials

Scattered polynomials over a finite field Fqn have been introduced by Sheekey in 2016, and a central open problem regards the classification of those that are exceptional. So far, only two families of exceptional scattered polynomials are known. Very recently, Longobardi and Zanella weakened the property of being scattered by introducing the notion of L-qt-partially scattered and R-qt-partially scattered polynomials, for t a divisor of n. Indeed, a polynomial is scattered if and only if it is both L-qt-partially scattered and R-qt-partially scattered. In this paper, by using techniques from algebraic geometry over finite fields and function fields theory, we show that the property which is the hardest to be preserved is the L-qt-partially scattered one. We investigate a large family F of R-qt-partially scattered polynomials, containing examples of exceptional R-qt-partially scattered polynomials, which turn out to be connected with linear sets of so-called pseudoregulus type. We introduce two different notions of equivalence preserving the property of being R-qt-partially scattered. Many polynomials in F are inequivalent and geometric arguments are used to determine their equivalence classes under the action of ΓL(2n/t,qt).

Apples and oranges? Establishing equivalence in comparative sport policy research

ABSTRACT The notion of equivalence is important in the context of comparative studies, such as those that compare performance across sporting nations or those that compare good governance across different sport organisations. However, despite its importance, the concept has been interpreted and employed in different ways, resulting in the term being misunderstood or conflated. This article examines the concept of equivalence, discusses how issues of non-equivalence can arise, and identifies potential strategies that can be employed by researchers to ensure it is more appropriately addressed. We examine and apply three main types of equivalence (construct, sample and functional) to two empirical cases, (1) the SPLISS analysis of critical success factors in elite sport policy and (2) Play the Game’s Sport Governance Observer to demonstrate how researchers attempt to overcome or at least mitigate the problems of equivalence and how, despite these efforts, there remain equivalence-related problems that limit the reliability and credibility of comparative elements of the study. We conclude our paper by discussing the implications for comparative sport research and specifically how future comparative sport research may be enhanced.

Equivalent Laplacian and adjacency quantum walks on irregular graphs

The continuous-time quantum walk is a particle evolving by Schr\"odinger's equation in discrete space. Encoding the space as a graph of vertices and edges, the Hamiltonian is proportional to the discrete Laplacian. In some physical systems, however, the Hamiltonian is proportional to the adjacency matrix instead. It is well known that these quantum walks are equivalent when the graph is regular, i.e., when each vertex has the same number of neighbors. If the graph is irregular, however, the quantum walks evolve differently. In this paper, we show that for some irregular graphs, if the particle is initially localized at a certain vertex, the probability distributions of the two quantum walks are identical, even though the amplitudes differ. We analytically prove this for a graph with five vertices and a graph with six vertices. By simulating the walks on all 1 018 689 568 simple, connected, irregular graphs with 11 vertices or less, we found 64 graphs with this notion of equivalence. We also give eight infinite families of graphs supporting these equivalent walks.

Equivalences of (co)module algebra structures over Hopf algebras

We introduce the notion of support equivalence for (co)module algebras (over Hopf algebras), which generalizes in a natural way (weak) equivalence of gradings. We show that for each equivalence class of (co)module algebra structures on a given algebra A, there exists a unique universal Hopf algebra H together with an H-(co)module structure on A such that any other equivalent (co)module algebra structure on A factors through the action of H. We study support equivalence and the universal Hopf algebras mentioned above for group gradings, Hopf-Galois extensions, actions of algebraic groups and cocommutative Hopf algebras. We show how the notion of support equivalence can be used to reduce the classification problem of Hopf algebra (co)actions. We apply support equivalence in the study of the asymptotic behaviour of codimensions of H-identities and, in particular, to the analogue (formulated by Yu. A. Bahturin) of Amitsur's conjecture, which was originally concerned with ordinary polynomial identities. As an example we prove this analogue for all unital H-module structures on the algebra $F[x]/(x^2)$ of dual numbers.

Exact and approximate role assignment for multi-layer networks

Abstract The concept of role equivalence has been applied in social network analysis for decades. Early definitions recognized two social actors as role equivalent, if they have identical relationships to the same other actors. Although this rather strong equivalence requirement has been relaxed in different ways, it is often challenging to detect interesting, non-trivial role equivalences, especially for social networks derived from empirical data. Multi-layer networks (MLNs) are increasingly gaining popularity for modelling collective adaptive systems, for example, engineered cyber-physical systems or animal collectives. Multiplex networks, a special case of MLNs, transparently and compactly describe such complex interactions (social, biological, transportation), where nodes can be connected by links of different types. In this work, we first propose a novel notion of exact and approximate role equivalence for multiplex MLNs. Then, we implement and experimentally evaluate the algorithm on a suite of real-world case studies. Results demonstrate that our notion of approximate role assignment not only obtains non-trivial partitions over nodes and layers as well, but it provides a fine-grained hierarchy of role equivalences, which is impossible to obtain by (combining) the existing role detection techniques. We demonstrate the latter by interpreting in detail the case study of Florence families, a classical benchmark from literature.

Les moyens lexicaux de la (re)création des effets sonores dans l’original et dans la traduction: le cas du roman de J. M. G. Le Clézio Ritournelle de la faim

Discussion of the translation process is complex and fraught with pitfalls, especially when it concerns the notions of translation equivalence and adequacy. This issue is of crucial importance concerning the translation of literary works. This article is dedicated to the study of the fundamental problem which translators of all times and all countries encounter, that is, to convey the creative idea of the author of the original work without perverting it, to preserve, throughout the translation process, all the semantic and stylistic nuances which form its content. This is the question around which our research problem revolves. We study here the approaches chosen by the Ukrainian translator Yarema Kravets, in order to recreate in his translation, the sound image that J. M. G. Le Clézio succeeded in creating in his novel Ritournelle de la faim. In order to render this research more usable, we have adopted the following methodological approach, founded on these two axes: the collection of lexical units from the source text which serve to create the sound image; and the comparative analysis of the original text with its translation, of the novel by J. M. G. Le Clézio, Ritournelle de la faim, with the goal of recreating the sound of the source text by identifying its means of transmission. The anticipated result of the given study, which is dedicated to the recreation of the sound effects in the translation of the J. M. G. Le Clézio’s novel Ritournelle de la faim, will be either to confirm or refute our primary hypothesis: that the translation proposed by Y. Kravets respects the spirit and the letter of the source text.

Hierarchical planning in a supervisory control context with compositional abstraction

Hierarchy is a tool that has been applied to improve the scalability of solving planning problems modeled using Supervisory Control Theory. In the work of Hill and Lafortune (2016), the notion of cost equivalence was employed to generate an abstraction of the supervisor that, with additional conditions, guarantees that an optimal plan generated on the abstraction is also optimal when applied to the full supervisor. Their work is able to improve their abstraction by artificially giving transitions zero cost based on the sequentially-dependent ordering of events. Here, we relax the requirement on a specific ordering of the dependent events, while maintaining the optimal relationship between upper and lower levels of the hierarchy. This present paper also extends the authors’ work (Vilela and Hill 2020) where we developed a new notion of equivalence based on cost equivalence and weak bisimulation that we term priced-observation equivalence. This equivalence allows the supervisor abstraction to be generated compositionally. This helps to avoid the explosion of the state space that arises from having to first synthesize the full supervisor before the abstraction can be applied. Here, we also show that models with artificial zero-cost transitions can be created compositionally employing the new relaxed sequential dependence definition. An example cooperative robot control application is used to demonstrate the improvements achieved by the compositional approach to abstraction proposed by this paper.

The paradox of intertextuality and dilemmas in English translations of traditional Chinese medicine texts: terminology as a case study

ABSTRACT Structuralists use the concept and methods of intertextuality to locate and fix literary meaning, yet poststructuralists employ the same term to disrupt notions of meaning, making it difficult to navigate through this paradox in textual and translation studies. When this term was incorporated into Traditional Chinese Medicine (TCM) translation studies, the paradoxical meanings of the term were also adopted without resolving these paradoxes. This paper therefore focuses on the translation of a few key TCM terms, and attempts to clarify the following translation dilemmas that arise when attempting to apply the concept of intertextuality: 1) Intertextuality deconstructs clear and stable meanings and this leads to untranslatability; 2) It is impossible to reproduce the intertextual relations of the original; 3) The notion of equivalence in translation is therefore an illusion; 4) Translating becomes an intertextual practice itself, and the target text is just intertextually related to the source text. The paper probes into the potential usefulness and significance of intertextuality for TCM translation and finally proves that all dilemmas of translation can only be resolved through the very act of translating.

Rational homotopy equivalences and singular chains

Bousfield and Kan's $\mathbb{Q}$-completion and fiberwise $\mathbb{Q}$-completion of spaces lead to two different approaches to the rational homotopy theory of non-simply connected spaces. In the first approach, a map is a weak equivalence if it induces an isomorphism on rational homology. In the second, a map of connected and pointed spaces is a weak equivalence if it induces an isomorphism between fundamental groups and higher rationalized homotopy groups; we call these maps $\pi_1$-rational homotopy equivalences. In this paper, we compare these two notions and show that $\pi_1$-rational homotopy equivalences correspond to maps that induce $\Omega$-quasi-isomorphisms on the rational singular chains, i.e. maps that induce a quasi-isomorphism after applying the cobar functor to the dg coassociative coalgebra of rational singular chains. This implies that both notions of rational homotopy equivalence can be deduced from the rational singular chains by using different algebraic notions of weak equivalences: quasi-isomorphism and $\Omega$-quasi-isomorphisms. We further show that, in the second approach, there are no dg coalgebra models of the chains that are both strictly cocommutative and coassociative.

Mercurial Signatures for Variable-Length Messages

Abstract Mercurial signatures are a useful building block for privacy-preserving schemes, such as anonymous credentials, delegatable anonymous credentials, and related applications. They allow a signature σ on a message m under a public key pk to be transformed into a signature σ′ on an equivalent message m′ under an equivalent public key pk′ for an appropriate notion of equivalence. For example, pk and pk′ may be unlinkable pseudonyms of the same user, and m and m′ may be unlinkable pseudonyms of a user to whom some capability is delegated. The only previously known construction of mercurial signatures suffers a severe limitation: in order to sign messages of length ℓ, the signer’s public key must also be of length ℓ. In this paper, we eliminate this restriction and provide an interactive signing protocol that admits messages of any length. We prove our scheme existentially unforgeable under chosen open message attacks (EUF-CoMA) under a variant of the asymmetric bilinear decisional Diffie-Hellman assumption (ABDDH).

Morita equivalence of pseudogroups

We take advantage of the correspondence between pseudogroups and inverse quantal frames, and of the recent description of Morita equivalence for inverse quantal frames in terms of biprincipal bisheaves, to define Morita equivalence for pseudogroups and to investigate its applications. In particular, two pseudogroups are Morita equivalent if and only if their corresponding localic étale groupoids are. We explore the clear analogies between our definition of Morita equivalence for pseudogroups and the usual notion of strong Morita equivalence for C⁎-algebras and these lead to a number of concrete results.