Notion Of Equivalence Research Articles

Program Equivalence in the Erlang Actor Model

This paper presents the formal semantics of concurrency in Core Erlang, an intermediate language for Erlang, along with a notion of program equivalence (based on barbed bisimulation) that is able to model equivalence between programs that have different communication structures but the same observable behaviour. The novelty in our formalisation is its extent: it includes semantics for messages and exit and link signals, in addition to most of Core Erlang’s sequential features. Furthermore, unlike previous studies, this work formalises message receipt using primitive operations, consistent with the standard as of Erlang/OTP 23. In this novel formalisation, we show some generally applicable program equivalences (such as process identifier renaming and silent evaluation) and present a practical case study featuring the equivalence of sequential and concurrent list processing.

Extended derivations of algebras

We study the concept of extended derivations of algebras which expands diverse definitions of generalized derivations given in the literature. We concentrate on the family of the anti-commutative algebras and classify such spaces of derivations by considering a suitable notion of equivalence when we restrict attention to sucssh algebras. Afterwards, we investigate a link between the well-known ( α , β , γ ) -derivations and some new extended derivations obtained in our classification. Finally, we present applications of extended derivations to study degenerations of algebras.

P-Permutation equivalences between blocks of group algebras

We extend the notion of a p-permutation equivalence between two p-blocks A and B of finite groups G and H, from the definition in [5] to a virtual p-permutation bimodule whose components have twisted diagonal vertices. It is shown that various invariants of A and B are preserved, including defect groups, fusion systems, and Külshammer-Puig classes. Moreover it is shown that p-permutation equivalences have additional surprising properties. They have only one constituent with maximal vertex and the set of p-permutation equivalences between A and B is finite (possibly empty). The paper uses new methods: a consequent use of module structures on subgroups of G×H arising from Brauer constructions which in general are not direct product subgroups, the necessary adaptation of the notion of tensor products between bimodules, and a general formula (stated in these new terms) for the Brauer construction of a tensor product of p-permutation bimodules.

Morita invariance of unbounded bivariant K-theory

We introduce a notion of Morita equivalence for non-selfadjoint operator algebras equipped with a completely isometric involution (operator ∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$*$$\\end{document}-algebras). We then show that the unbounded Kasparov product by a Morita equivalence bimodule induces an isomorphism between equivalence classes of twisted spectral triples over Morita equivalent operator ∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$*$$\\end{document}-algebras. This leads to a tentative definition of unbounded bivariant K-theory and we prove that this bivariant theory is related to Kasparov’s bivariant K-theory via the Baaj-Julg bounded transform. Moreover, the unbounded Kasparov product provides a refinement of the usual interior Kasparov product. We illustrate our results by proving C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^1$$\\end{document}-versions of well-known C∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^*$$\\end{document}-algebraic Morita equivalences in the context of hereditary subalgebras, conformal equivalences and crossed products by discrete groups.

On invariant means and pre-syndetic subgroups

Beyond the locally compact case, equivalent notions of amenability diverge, and some properties no longer hold, for instance amenability is not inherited by topological subgroups. This investigation is guided by some amenability-type properties of groups of paths and loops. It is shown that a version of amenability called skew-amenability is inherited by pre-syndetic subgroups in the sense of Basso and Zucker (in particular, by co-compact subgroups). It follows that co-compact subgroups of amenable topological groups whose left and right uniformities coincide are amenable. We discuss a version of amenability belonging to P. Malliavin and M.-P. Malliavin: the existence of a mean on bounded Borel functions that is invariant under the left action of a dense subgroup. We observe that this property is in general strictly stronger than amenability, and establish for it Reiter- and Følner-type criteria. Finally, there is a review of open problems.

Divergent Tandem Acyl Carrier Proteins Necessitate In-Series Polyketide Processing in the Leinamycin Family.

The leinamycin family of polyketides are promising antitumour antibiotics, yet several aspects of their biosynthesis remain elusive. All leinamycin family members bear a sulfur-containing moiety which is essential for the anticancer activity exhibited by leinamycin. The key building blocks required for the incorporation of these functionalities are introduced in the final module of the polyketide synthase (PKS), which elegantly combines β-branching and thiocysteine incorporation to generate a diverse library of sulfur-based molecular scaffolds. Two acyl carrier proteins (ACPs) form a key didomain component of this module, but their amino acid sequence divergence has brought into question the common notion of functional equivalence. Here, we provide unprecedented functional evidence that these tandem ACPs play distinct roles in the final module of polyketide assembly. Using the weishanmycin biosynthetic pathway as a template, the in vitro reconstitution of key polyketide chain extension and β-branching steps in this module has revealed strict functional selectivity for a single ACP. Furthermore, we propose a cryptic transacylation step must occur prior to polyketide off-loading and cyclization. Altogether, these mechanistic investigations suggest that an atypical in-series mechanism underpins sulfur incorporation in the leinamycin family, and provides significant progress towards delineating their late-stage assembly.

Non-Abelian Transport Distinguishes Three Usually Equivalent Notions of Entropy Production

We extend entropy production to a deeply quantum regime involving noncommuting conserved quantities. Consider a unitary transporting conserved quantities (“charges”) between two systems initialized in thermal states. Three common formulas model the entropy produced. They respectively cast entropy as an extensive thermodynamic variable, as an information-theoretic uncertainty measure, and as a quantifier of irreversibility. Often, the charges are assumed to commute with each other (e.g., energy and particle number). Yet quantum charges can fail to commute. Noncommutation invites generalizations, which we posit and justify, of the three formulas. The noncommutation of charges, we find, breaks the equivalence of the formulas. Furthermore, different formulas quantify different physical effects of the noncommutation of charges on entropy production. For instance, entropy production can signal contextuality—true nonclassicality—by becoming nonreal. This work opens up stochastic thermodynamics to noncommuting—and so particularly quantum—charges. Published by the American Physical Society 2024

On p-permutation equivalences between direct products of blocks

We extend the notion of a p-permutation equivalence to an equivalence between direct products of block algebras. We prove that a p-permutation equivalence between direct products of blocks gives a bijection between the factors and induces a p-permutation equivalence between corresponding blocks.

Bisimulation between base argumentation and premise-conclusion argumentation

The structured argumentation system that represents arguments by premise-conclusion pairs is called premise-conclusion argumentation (PA) and the one that represents arguments by their premises is called base argumentation (BA). To assess whether BA and PA have the same ability in argument evaluation by extensional semantics, this paper defines the notion of extensional equivalence between BA and PA. It also defines the notion of bisimulation between BA and PA and shows that bisimulation implies extensional equivalence. To illustrate how base argumentation, bisimulation and extensional equivalence can contribute to the study of PA, we prove some new results about PA by investigating the extensional properties of a base argumentation framework and exporting them to two premise-conclusion argumentation frameworks via bisimulation and extensional equivalence. We show that there are essentially three kinds of extensions in these frameworks and that the extensions in the two premise-conclusion argumentation frameworks are identical.

Strong equivalence of graded algebras

We introduce the notion of a strong equivalence between graded algebras and prove that any partially-strongly-graded algebra by a group G is strongly-graded-equivalent to the skew group algebra by a product partial action of G. As to a more general idempotent graded algebra B, we point out that the Cohen-Montgomery duality holds for B, and B is graded-equivalent to a global skew group algebra. We show that strongly-graded-equivalence preserves strong gradings and is nicely related to Morita equivalence of product partial actions. Furthermore, we prove that any product partial group action α is globalizable up to Morita equivalence; if such a globalization β is minimal, then the skew group algebras by α and β are graded-equivalent; moreover, β is unique up to Morita equivalence. Finally, we show that strongly-graded-equivalent partially-strongly-graded algebras with orthogonal local units are stably isomorphic as graded algebras.

Autonomous partners: asymmetry and masculinity in Amazonian river trade

AbstractTrade on the Iriri River, in the eastern Brazilian Amazonia, is structured around a credit‐barter system between clients and bosses known as aviamento in Portuguese. Nowadays, bosses are river traders born in the riversides who offer goods on credit to riverside dwellers, who later pay these debts with fish and products they collect from the forest. While the system, found in the Amazon basin since colonial times, is perhaps most known for fostering debt‐peonage during the Rubber Boom (1870‐1912), it can also create relations of mutual dependency that promote the autonomy of trade partners within an asymmetrical exchange. When that is the case, this trade is central to how clients and bosses define themselves as hard‐working, reliable, and transparent, personifying qualities they deem essential to masculine honour. In such partnerships (and more dramatically when they break down), the asymmetry of the relation can be challenged through a notion of male equivalence in which men must defend their honour and adhere to a code of conduct that ultimately excludes women and children from trade.

The Simplicial Coalgebra of Chains Under Three Different Notions of Weak Equivalence

Abstract We study the simplicial coalgebra of chains on a simplicial set with respect to three notions of weak equivalence. To this end, we construct three model structures on the category of reduced simplicial sets for any commutative ring $R$. The weak equivalences are given by: (1) an $R$-linearized version of categorical equivalences, (2) maps inducing an isomorphism on fundamental groups and an $R$-homology equivalence between universal covers, and (3) $R$-homology equivalences. Analogously, for any field ${\mathbb{F}}$, we construct three model structures on the category of connected simplicial cocommutative ${\mathbb{F}}$-coalgebras. The weak equivalences in this context are (1′) maps inducing a quasi-isomorphism of dg algebras after applying the cobar functor, (2′) maps inducing a quasi-isomorphism of dg algebras after applying a localized version of the cobar functor, and (3′) quasi-isomorphisms. Building on a previous work of Goerss in the context of (3)–(3′), we prove that, when ${\mathbb{F}}$ is algebraically closed, the simplicial ${\mathbb{F}}$-coalgebra of chains defines a homotopically full and faithful left Quillen functor for each one of these pairs of model categories. More generally, when ${\mathbb{F}}$ is a perfect field, we compare the three pairs of model categories in terms of suitable notions of homotopy fixed points with respect to the absolute Galois group of ${\mathbb{F}}$.

All Visualisations Have Crooked Tales/Tails

AbstractThe breadth of architectural typologies is waning, or so Perry Kulper, Professor of Architecture at Taubman College, University of Michigan, tells us. Kulper's work has for decades focused on the development of dexterous, agile and creative methodologies for speculating about the possibilities of architecture(s) explored through notions of equivalence, remixing, reinterpreting and reimagining. Developed through seven spatial spectral strategies for an installation at the 2023 Chicago Architecture Biennial, familiar landmarks of the city are rescaled and combined to become other implements, adornments and spaces, bringing an uncanny familiarity to their propositions.

Gravity with torsion as deformed BF theory *

We study a family of (possibly non topological) deformations of BF theory for the Lie algebra obtained by quadratic extension of so(3,1) by an orthogonal module. The resulting theory, called quadratically extended General Relativity (qeGR), is shown to be classically equivalent to certain models of gravity with dynamical torsion. The classical equivalence is shown to promote to a stronger notion of equivalence within the Batalin–Vilkovisky formalism. In particular, both Palatini–Cartan gravity and a deformation thereof by a dynamical torsion term, called (quadratic) generalised Holst theory, are recovered from the standard Batalin–Vilkovisky formulation of qeGR by elimination of generalised auxiliary fields.

The effectiveness of aggregation functions used in fuzzy local contrast constructions

In this paper, we explore the concept of local contrast of a fuzzy relation, which can be perceived as a measure for distinguishing the degrees of membership of elements within a defined region of an image. We introduce four distinct methods for constructing fuzzy local contrast: one uses a similarity measure, the second relies on the aggregation of similarity, the third is based on the aggregation of restricted equivalence, and the fourth utilizes the notion of equivalence. We further divide the constructions using similarity measures into two categories based on the two known definitions of similarity: distance-based similarity and aggregation function-based similarity. These construction methods also incorporate fuzzy implications and negations. Aggregation functions, which can be manipulated to enhance the effectiveness of the constructed fuzzy local contrast, play a significant role in most of our proposed constructions. For each construction method, several examples of fuzzy local contrasts are provided. The usefulness of the new fuzzy local contrasts is examined by applying them in image processing for salient region detection.

Equivalent Notions in the context of compatible Endo-Lie Algebras

In this article, we introduce a notion of compatibility between two Endo-Lie algebras defined on the same linear space. Compatibility means that any linear combination of the two structures always induces a new Endo-Lie algebras structure. In this case of compatibility, we show that the notions of bialgebras, standard Manin triples and matched pairs are equivalent. We find this equivalence for the case of compatible Lie algebras since this is a particular case of compatible Endo-Lie algebras.

Decorated Discrete Conformal Maps and Convex Polyhedral Cusps

Abstract We discuss a notion of discrete conformal equivalence for decorated piecewise Euclidean surfaces (PE-surface), that is, PE-surfaces with a choice of circle about each vertex. It is closely related to inversive distance and hyperideal circle patterns. Under the assumption that the circles are non-intersecting, we prove the corresponding discrete uniformization theorem. The uniformization theorem for discrete conformal maps corresponds to the special case that all circles degenerate to points. Our proof relies on an intimate relationship between decorated PE-surfaces, canonical tessellations of hyperbolic surfaces and convex hyperbolic polyhedra. It is based on a concave variational principle, which also provides a method for the computation of decorated discrete conformal maps.

A Unified View on Forgetting and Strong Equivalence Notions in Answer Set Programming

Answer Set Programming (ASP) is a prominent rule-based language for knowledge representation and reasoning with roots in logic programming and non-monotonic reasoning. The aim to capture the essence of removing (ir)relevant details in ASP programs led to the investigation of different notions, from strong persistence (SP) forgetting, to faithful abstractions, and, recently, strong simplifications, where the latter two can be seen as relaxed and strengthened notions of forgetting, respectively. Although it was observed that these notions are related, especially given that they have characterizations through the semantics for strong equivalence, it remained unclear whether they can be brought together. In this work, we bridge this gap by introducing a novel relativized equivalence notion, which is a relaxation of the recent simplification notion, that is able to capture all related notions from the literature. We provide the necessary and sufficient conditions for relativized simplifiability, which shows that the challenging part is for when the context programs do not contain all the atoms to remove. We then introduce an operator that combines projection and a relaxation of SP-forgetting to obtain the relativized simplifications. We furthermore provide complexity results that complete the overall picture.

Towards a geometrical equivalence of norms

Angular equivalence of norms, introduced by Kikianty and Sinnamon (2017), is a notion of norm equivalence that is more attuned to the geometry of the norms. For certain geometrical properties and two angularly equivalent norms, it is the case that if one of the norms has a property, then so does the other. In this paper, we show further results in this direction, namely angularly equivalent norms share the property of non-squareness; and that the exposed points of the unit balls are in the same direction, under the condition that these points are assumed to be smooth with respect to both norms. A discussion on (the angular equivalence of) the dual norms of angularly equivalent norms is also given, giving a partial answer to an open problem as stated in the paper by Kikianty and Sinnamon (2017).

Equivalence for flag codes

Given a finite field Fq and a positive integer n, a flag is a sequence of nested Fq-subspaces of a vector space Fqn and a flag code is a nonempty collection of flags. The projected codes of a flag code are the constant dimension codes containing all the subspaces of prescribed dimensions that form the flags in the flag code.In this paper we address the notion of equivalence for flag codes and explore in which situations such an equivalence can be reduced to the equivalence of the corresponding projected codes. In addition, this study leads to new results concerning the automorphism group of certain families of flag codes, some of them also introduced in this paper.