Context Of Algebras Research Articles

On Lie Homomorphisms of Complex IntuitionisticFuzzy Lie Algebras

In this paper, we investigate the properties of complex intuitionistic fuzzy Lie subalgebras and ideals under Lie algebra homomorphisms, focusing on both images and inverse images. We provide detailed proofs for several new results concerning the preservation of complex intuitionistic fuzzy structures through homomorphisms, and we introduce additional homomorphism-related properties for these structures. This work extends known results to the context of complex intuitionistic fuzzy Lie algebras, contributing new insights into their behavior under algebraic mappings.

Missed encounters: what may be relevant for an AI is not for a human being

Abstract The World Wide Web has been a fundamental part of our daily lives for years. Its algorithmic framework ensnares our online journeys in an “endless recurrence” of the “same” by creating multiple filter bubbles. Digital algorithms establish a precise “order of discourse,” leaving little to no room for deviation. Functioning as a colossal machinic apparatus, the web embodies the culmination of Artificial Intelligence (AI), transforming every piece of posted content into a database that profiles our online behavior and activities. This article explores whether approaches that describe everyday human communication, such as the theories of relevance developed by Alfred Schutz, or by Dan Sperber and Deirdre Wilson, can be applied to the realm of digital algorithmic grammars governing the web, and to the semiospheres that animate it. The conclusions raise some doubts: a gap exists between algorithmic language and its historical counterparts, characterized by a disparity between logical-mathematical grammar and other linguistic-historical-natural ones. While these two types of languages coexist within the digital landscape, their relevance differs; what may be “relevant” in an algebraic context may not necessarily translate to our live conversational exchanges, and vice versa. Although merged in the digital sphere, these languages operate distinctly, and proficiency in one does not guarantee adeptness in interpreting the other accurately.

Development of Algebra Test Using the Item Response Theory Approach for Junior High School Students

This research aims to develop valid and reliable measuring tool for students' algebraic abilities that can be used in schools and the general public. The research follows a structured test development design, including stages such as preparing test specifications and items, field testing, revising items, and test development. The questions are aligned with the 2013 curriculum syllabus, ensuring relevance to educational standards. The test was given to 662 junior high school students in Kendari City, Indonesia, and their responses were analyzed using the item response theory (IRT) model with two logistic parameters: item difficulty level and item discriminatory power. The BILOG MG program was employed to estimate item and ability parameters. Before conducting item analysis with IRT, essential assumption tests were conducted, including unidimensional and model fit tests. The results of the development process, based on item analysis using BILOG MG, yielded 15 items covering various aspects of algebraic abilities. These items were derived from indicators such as recognizing algebraic forms, identifying elements within these forms, performing addition, subtraction, multiplication, and division operations on algebraic forms, presenting and solving real-world problems in algebraic contexts, and addressing contextual problems involving algebraic operations. The items demonstrated good fit with the model and exhibited an appropriate level of item difficulty and discriminatory power, making them suitable for use as a reliable assessment tool. Consequently, these developed tests are deemed effective for measuring students' foundational algebraic abilities.

A positive/tropical critical point theorem and mirror symmetry

Call a Laurent polynomial W ‘complete’ if its Newton polytope is full-dimensional with zero in its interior. Suppose W is a Laurent polynomial with coefficients in the positive part of the field of (generalised) Puiseaux series. Here a Puiseaux or generalised Puiseux series (with exponents in R) is called ‘positive’ if the coefficient of its leading term is in R>0. We show that W has a unique positive critical point pcrit, i.e. all of whose coordinates are positive, if and only if W is complete. For any complete, positive Laurent polynomial W in r variables we also obtain from its positive critical point pcrit a canonically associated ‘tropical critical point’ dcrit∈Rr by considering the valuations of the coordinates of pcrit. Moreover we give a finite recursive construction of dcrit in terms of a generalisation of the Newton polytope that we call the ‘Newton datum’ of W.We show that this result is compatible with a general form of mutation, so that it can be applied in a cluster varieties setting. We also show that our theorem carries over to the case where the exponents of monomials appearing in W are not integral but in R, even though W is then no longer Laurent.Finally, we describe applications to both algebraic and symplectic toric geometry inspired by mirror symmetry. On the one hand, in the algebraic context of a complete toric variety XΣ we apply our results to obtain for any divisor class [D] satisfying a certain integrality property, a canonical choice of torus-invariant representative. This generalises the standard toric boundary divisor of XΣ to divisor classes other than the anti-canonical class. On the other hand, our result generalises a result of [11] and relates to the construction of canonical non-displaceable Lagrangian tori for toric symplectic orbifolds using [13,37].

أثر بَرنامج تَدرِيبي مُستند إِلى نَموذج (TPACK) في تَحسين الرَغبة المُنتجَة لَدى مُعلمات المَرحَلة الأساسية في الأُردُن

The study aimed to investigate the effectiveness of a training program based on the (TPACK) model in improving productive disposition among primary school mathematics teachers, the guide for the training program and study tools was prepared, ensured their validity and stability. The guideline included the definition of the (TPACK) model with algebraic context. An available sample was selected from the schools of Jerash Kasbah of the Directorate of Education of Jerash Governorate, consisting of (36) primary mathematics teachers, divided into two groups: one experimental consisted of (18) teachers trained according to the (TPACK) model, the other was a control group consisted of (18) teachers who did not undergo training. The results showed that training according to (TPACK) model improved the level of performance of the experimental group mathematics teachers on the productive disposition scale, and raised Their performance to an expert level, in the light of the results presented the change of the recommendations of the training programs with the application of specific educational content and integration models.

ANALYSIS OF STUDENTS' DIFFICULTIES IN SOLVING ALGEBRAIC OPERATION PROBLEMS BASED ON PROBLEM-SOLVING SKILLS IN GRADE VII AT SMP NEGERI 8 TAMBUN SELATAN

This study aims to analyze the difficulties faced by students when solving algebraic operation problems based on the criteria of problem solving ability. This study was conducted to understand the factors that master students' difficulties in solving algebraic problems. This research used descriptive method with qualitative approach. The research subjects consisted of 35 students of class VII-G at SMP N 8 Tambun Selatan. Data were collected through observation, tests, questionnaires, and documentation. Data analysis used the Miles and Huberman data analysis model. Data validity was checked through method tringulation. The results showed that of the 35 students studied: 1) 61% of students had difficulty in understanding the problem, 2) 57% of students had difficulty in planning the solution, 3) 73% of students had difficulty in carrying out the plan, and 4) 86% of students had difficulty in checking back. The main factors that influence these difficulties are students' understanding of basic algebraic concepts, difficulties in applying mathematical concepts to the algebraic context and underdeveloped problem solving skills. In understanding math problems and formulating the steps needed to solve them, students often face difficulties.

Reasoning productively across algebraic contexts: Students develop coordinated notions of inverse

The concept of inverse is threaded throughout K-16 mathematics. Scholars frequently advocate for students to understand the underlying structure: combining an element and its inverse through the binary operations yields the relevant identity element. This ‘coordinated’ way of reasoning is challenging for students to employ; however, little is known about how students might reason en route to developing it. In this study, we analyze a teaching experiment with two beginning abstract algebra students through the lens of three ways of reasoning about inverse: inverse as an undoing, inverse as a manipulated element, and inverse as a coordination of the binary operation, identity, and set. In particular, we examine the implications of these ways of reasoning as students work to develop inverse as a coordination. We identify pedagogical tools and facets of instructional design that appeared to support students’ development of inverse as a coordination. We further suggest that all three ways of reasoning can support productive activity with inverses.

Examining the concept of inverse: Theory-building via a standalone literature review

Inverse is a critical topic throughout the K–16 mathematics curriculum where students encounter the notion of mathematical inverse across many contexts. The literature base on inverses is substantial, yet context-specific and compartmentalized. That is, extant research examines students’ reasoning with inverses within specific algebraic contexts. It is currently unclear what might be involved in productively reasoning with inverses across algebraic contexts, and whether the specific ways of reasoning from the literature can be abstracted to more general ways of reasoning about inverse. To address this issue, we conducted a standalone literature review to explicate and exemplify three cross-context ways of reasoning that, we hypothesize, can support students’ productive engagement with inverses in a variety of algebraic contexts: inverse as an undoing, inverse as a manipulated element, and inverse as a coordination of the binary operation, identity, and set. Findings also include explicating affordances and constraints for each of these ways of reasoning. Finally, we reflect on when and how standalone literature reviews can serve the purpose of unifying fragmented and obscured insights about key mathematical ideas.

Cyclic forms on DG-Lie algebroids and semiregularity

Given a transitive DG-Lie algebroid (\mathcal{A},\rho) over a smooth separated scheme X of finite type over a field \mathbb{K} of characteristic zero, we define a notion of connection \nabla\colon\mathbf{R}\Gamma(X,\operatorname{Ker}\rho)\to\mathbf{R}\Gamma(X,\Omega_X^1[-1]\otimes\operatorname{Ker}\rho) and construct an L_\infty -morphism between DG-Lie algebras f\colon \mathbf{R}\Gamma(X,\operatorname{Ker}\rho) \rightsquigarrow\mathbf{R}\Gamma(X,\Omega_X^{\leq1}[2]) associated to a connection and to a cyclic form on the DG-Lie algebroid. In this way, we obtain a lifting of the first component of the modified Buchweitz–Flenner semiregularity map in the algebraic context, which has an application to the deformation theory of coherent sheaves on X admitting a finite locally free resolution. Another application is to the deformations of (Zariski) principal bundles on X .

$C^*$-Algebras

Operator algebras form a very active area of mathematics which, since its inception in the 1940s, has always been driven by interactions with other fields of mathematics and physics. The scope of these interactions is very wide, ranging over dynamical systems, (non-commutative) geometry, functional analysis, (geometric) group theory, topology, random matrices, harmonic analysis and quantum information theory. The goals of this workshop were to stimulate new collaborations across these fields of mathematics, to disseminate recent progress by giving participants a global view on the subject and to specially focus on two important developments: the solution of the Connes embedding problem by methods from quantum information theory and the progress on noncommutative dynamical systems, especially in the topological C^* -algebra context.

Detecting and describing ramification for structured ring spectra

AbstractJohn Rognes developed a notion of Galois extension of commutative ring spectra, and this includes a criterion for identifying an extension as unramified. Ramification for commutative ring spectra can be detected by relative topological Hochschild homology and by topological André–Quillen homology. In the classical algebraic context, it is important to distinguish between tame and wild ramification. Noether’s theorem characterizes tame ramification in terms of a normal basis, and tame ramification can also be detected via the surjectivity of the trace map. For commutative ring spectra, we suggest to study the Tate construction as a suitable analog. It tells us at which integral primes there is tame or wild ramification, and we determine its homotopy type in examples in the context of topological K-theory and topological modular forms.

Differential Antisymmetric Infinitesimal Bialgebras, Coherent Derivations and Poisson Bialgebras

We establish a bialgebra theory for differential algebras, called differential antisymmetric infinitesimal (ASI) bialgebras by generalizing the study of ASI bialgebras to the context of differential algebras, in which the derivations play an important role. They are characterized by double constructions of differential Frobenius algebras as well as matched pairs of differential algebras. Antisymmetric solutions of an analogue of associative Yang-Baxter equation in differential algebras provide differential ASI bialgebras, whereas in turn the notions of O-operators of differential algebras and differential dendriform algebras are also introduced to produce the former. On the other hand, the notion of a coherent derivation on an ASI bialgebra is introduced as an equivalent structure of a differential ASI bialgebra. They include derivations on ASI bialgebras and the set of coherent derivations on an ASI bialgebra composes a Lie algebra which is the Lie algebra of the Lie group consisting of coherent automorphisms on this ASI bialgebra. Finally, we apply the study of differential ASI bialgebras to Poisson bialgebras, extending the construction of Poisson algebras from commutative differential algebras with two commuting derivations to the context of bialgebras, which is consistent with the well constructed theory of Poisson bialgebras. In particular, we construct Poisson bialgebras from differential Zinbiel algebras.

Bipartite entanglement and the arrow of time

We provide a different perspective on the close relationship between entanglement and time. Our main focus is on bipartite entanglement, where this connection is foreshadowed both in the positive partial transpose criterion due to Peres [A. Peres, Phys. Rev. Lett. 77, 1413 (1996)] and in the classification of quantum within more general nonsignaling bipartite correlations [M. Frembs and A. D\"oring, Phys. Rev. A 106, 062420 (2022)]. Extracting the relevant common features, we identify a necessary and sufficient condition for bipartite entanglement in terms of a compatibility condition with respect to time orientations in local observable algebras, which express the dynamics in the respective subsystems. As an outlook and a program for future work, we discuss the relevance of the latter in the broader context of von Neumann algebras and the thermodynamical notion of time naturally arising within the latter.

Grounding Game Semantics in Categorical Algebra

I present a formal connection between algebraic effects and game semantics, two important lines of work in programming languages semantics with applications in compositional software verification. Specifically, the algebraic signature enumerating the possible side-effects of a computation can be read as a game, and strategies for this game constitute the free algebra for the signature in a category of complete partial orders (cpos). Hence, strategies provide a convenient model of computations with uninterpreted side-effects. In particular, the operational flavor of game semantics carries over to the algebraic context, in the form of the coincidence between the initial algebras and the terminal coalgebras of cpo endofunctors. Conversely, the algebraic point of view sheds new light on the strategy constructions underlying game semantics. Strategy models can be reformulated as ideal completions of partial strategy trees (free dcpos on the term algebra). Extending the framework to multi-sorted signatures would make this construction available for a large class of games.

An analytic Hochschild-Kostant-Rosenberg theorem

Let R be a Banach ring. We prove that the category of chain complexes of complete bornological R-modules (and several related categories) is a derived algebraic context in the sense of Raksit. We then use the framework of derived algebra to prove a version of the Hochschild-Kostant-Rosenberg Theorem, which relates the circle action on the Hochschild algebra to the de Rham-differential-enriched-de Rham algebra of a simplicial, commutative, complete bornological algebra. This has a geometric interpretation in the language of derived analytic geometry, namely, the derived loop stack of a derived analytic stack is equivalent to the shifted tangent stack. Using this geometric interpretation we extend our results to derived schemes.

Implicational tonoid semilinear logics: Prelinear logics and their relational semantics

This paper deals with semilinear extensions of implicational tonoid and partial Galois logics. To this end, first the class of implicational tonoid prelinear logics is defined and it is verified that these logics are semilinear in an algebraic context, namely an implicational tonoid logic is semilinear if it is complete over linearly ordered matrices. Next, a relational semantics is introduced for finitary implicational tonoid prelinear logics and it is proved that these logics are complete on the semantics. Thirdly the term “semilinear” is generalized to a notion to be applied in a set-theoretic context and it is verified that finitary implicational tonoid prelinear logics are semilinear in this context. Finally some extensions satisfying abstract Galois, dual Galois properties are introduced together with similar relational semantics for them and it is shown that these logics are semilinear in both contexts.

Some Implicational Semilinear Gaggle Logics: (Dual) Residuated-Connected Logics

Implicational partial Galois logics and some of their semilinear extensions, such as semilinear extensions satisfying abstract Galois and dual Galois connection properties, have been introduced together with their relational semantics. However, similar extensions satisfying residuated, dual residuated connection properties have not. This paper fills the gaps by introducing those semilinear extensions and their relational semantics. To this end, the class of implicational (dual) residuated-connected prelinear gaggle logics is defined and it is verified that these logics are semilinear. In particular, associated with the contribution of this work, we note the following two: One is that implications can be introduced by residuated connection in semilinear logics. This shows that the residuated, dual residuated connection properties are important and so need to be investigated in semilinear logics. The other is that set-theoretic relational semantics can be provided for semilinear logics. Semilinear logics have been dealt with extensively in algebraic context, whereas they have not yet been performed in the set-theoretic one.

TAMARKIN–TSYGAN CALCULUS AND CHIRAL POISSON COHOMOLOGY

AbstractWe construct and study some vertex theoretic invariants associated with Poisson varieties, specializing in the conformal weight $0$ case to the familiar package of Poisson homology and cohomology. In order to do this conceptually, we sketch a version of the calculus, in the sense of [12], adapted to the context of vertex algebras. We obtain the standard theorems of Poisson (co)homology in this chiral context. This is part of a larger project related to promoting noncommutative geometric structures to chiral versions of such.

Cohomological Blowups of Graded Artinian Gorenstein Algebras along Surjective Maps

Abstract We introduce the cohomological blowup of a graded Artinian Gorenstein algebra along a surjective map, which we term BUG (blowup Gorenstein) for short. This is intended to translate to an algebraic context the cohomology ring of a blowup of a projective manifold along a projective submanifold. We show, among other things, that a BUG is a connected sum, that it is the general fiber in a flat family of algebras, and that it preserves the strong Lefschetz property. We also show that standard graded compressed algebras are rarely BUGs, and we classify those BUGs that are complete intersections. We have included many examples throughout this manuscript.

Formula omitted] decomposition of denominator formulae of some BKM Lie superalgebras

We study a family of Siegel modular forms that are constructed using Jacobi forms that arise in Umbral moonshine. All but one of them arise as the Weyl-Kac-Borcherds denominator formula of some Borcherds-Kac-Moody (BKM) Lie superalgebras. These Lie superalgebras have a sl(2)ˆ subalgebra which we use to study the Siegel modular forms. We show that the expansion of the Umbral Jacobi forms in terms of sl(2)ˆ characters leads to vector-valued modular forms. We obtain closed formulae for these vector-valued modular forms. In the Lie algebraic context, the Fourier coefficients of these vector-valued modular forms are related to multiplicities of roots appearing on the sum side of the Weyl-Kac-Borcherds denominator formulae.