Class Of Fields Research Articles

Quadratic vector fields in class I

In [Ye et al., Theory of Limit Cycles, 1986], quadratic systems are classified into three different normal forms (I, II and III) with increasing number of parameters. The simplest family is I and even several subfamilies of it have been studied, and some global attempts have been done, up to this paper, the full study was still undone. In this article, we make an interdisciplinary global study of Class I. Since the family has four parameters, we have studied it using the same technique that has already been used in several papers with similar systems which is based on the algebraic invariants of the Sibirskii's school. The bifurcation diagram for this class, done in the adequate parameter space which is the 3-dimensional real projective space, is quite rich in its complexity and yields 261 subsets with 49 different phase portraits for Class I (2 of them corresponding to linear systems), 7 of which have limit cycles. The phase portraits are always represented in the Poincaré disc. The bifurcation set is formed by an algebraic set of bifurcations of singularities, finite or infinite and by an analytic set of curves corresponding to phase portraits which have separatrix connections. Algebraic invariants were needed to construct the algebraic part of the bifurcation set, symbolic computations to deal with some quite complex invariants and numerical calculations to determine the position of the analytic bifurcation set of connections.

Concept model and realization of object orientied system with metaprogramming

The purpose of the research is to develop a conceptual model of an object oriented system and implement it using a functional language with metaprogramming support. Methods. A conceptual model of an object oriented system was developed based on a dynamic object system with message passing and dynamic dispatch. The developed object oriented system includes functions for creating and deleting classes, creating and deleting object instances of a class, functions for working with object properties, functions for adding methods to a class and passing messages. Inheritance is also possible, when a child class will have the properties of the parent class and its own properties. Results. The object oriented system was implemented using a set of macros and helper functions. The basic data structure was a hash table with key access capabilities, which was also implemented using macros. When a class is created, an instance constructor is automatically generated, setting the initial values of the object’s fields. Inheritance is implemented as the inclusion of fields of a parent class in a child object. Polymorphism is implemented as dynamic dispatch: when processing a message, a handler method is searched. The resulting object oriented system, together with the hash table module, occupies 97 lines. Conclusion. As a result of the work, a conceptual model of an object oriented system was created and implemented, which includes the main functions of the object oriented programming paradigm. It has been shown that macros can be used to create a compact object oriented system without the need to add any new functions or capabilities to the functional language interpreter.

Approximated harmonic maps with tension fields in Zygmund class

Suppose that u is a map from D8 to a compact smooth Riemannian manifold N with bounded energy. We show that there exists a constant λ>0 which depends only on N and E(u,D8) such that if the tension field τ belongs to Zygmund class Llnλ⁡L(D8), then the Hopf differential of u belongs to the Zygmund class Lln3⁡L(D1) and the norm ‖h‖Lln3⁡L(D1) depends only on N,E(u,D8) and ‖τ‖Llnλ⁡L(D8). As a direct corollary, we obtain the energy identity and necklessness of a blow-up sequence un with bounded energy E(un) and bounded τ(un) in Llnλ⁡L(D8).

ON abelian extensions and Kronecker’s jugendtraum in algebraic number theory

Abstract Each subdivision within the cyclotomic domain constitutes an Abelian numerical field, and Kronecker established the reverse of this statement. His aspirations extended beyond the realm of rational numbers, delving into the Abelian extensions of imaginary quadratic numerical fields and introducing the renowned jugendtraum. It wasn’t until the emergence of class field theory in the 20th century that this youthful dream was confirmed as valid. This concept also maintains a close connection to Hilbert’s 12th problem, which remains unresolved to this day.

DEVELOPMENT OF CONTEXTUAL-BASED MODULES ON LINES AND ANGLES FOR SEVENTH GRADE STUDENTS AT SMP NEGERI 1 ULUSUSUA

This research aims to develop a module focused on lines and angles to enhance students' contextual skills. The study employs a Research and Development (R&D) approach using the 4D model (define, design, development, disseminate), with data collected through validation, questionnaires, and tests, analyzed both qualitatively and quantitatively. Based on the findings, the lines and angles module is proven to be valid, practical, and effective. Subject matter experts rated it as valid (75%), media experts also rated it valid (74.07%), and language experts rated it as very valid (75%). In limited class trials, practicality was rated very high (average 4.57), and similarly high in field trials (average 4.43). Effectiveness in improving learning outcomes was moderately good in limited class trials (average 0.65) and field trials (average 0.68). The developed module is intended to support independent learning among students and assist teachers in effectively teaching mathematics.

Relative perinormality and relative global perinormality in a new pullback construction and idealization

If (S, J) is a zero-dimensional local ring, K = S / J its residue class field and D an integral domain with quotient field K, we show that twin notions of relative perinormality (N. Epstein, J. Shapiro, Perinormality in pullbacks Journal of Commutative Algebra 11 (3):341–362, 2019) and relative global perinormality (H. Klawa, Globally perinormal domains and graded perinormality, Doctoral Dissertation, George Mason University, 2023) are preserved by a pullback construction R = D × K S . As a byproduct, we recover the preservation of these notions in a new pullback construction of Chang et al. and idealization.

Linking invariants for valuations and orderings on fields

The mod-2 arithmetic Milnor invariants, introduced by Morishita, provide a decomposition law for primes in canonical Galois extensions of Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Q}$$\\end{document} with unitriangular Galois groups and contain the Legendre and Rédei symbols as special cases. Morishita further proposed a notion of mod-q arithmetic Milnor invariants, where q is a prime power, for number fields containing the qth roots of unity and satisfying certain class field theory assumptions. We extend this theory from the number field context to general fields, by introducing a notion of a linking invariant for discrete valuations and orderings. We further express it as a Magnus homomorphism coefficient and relate it to Massey product elements in Galois cohomology.

Local Non Abelian Class Field Theory

The “local class field theory”, which can be defined as the description of the extensions of a given local field K in terms of the algebraic and analytic objects depending ony on the base K is one of the central problems of modern number theory. The theory developed for the abelian extensions, around the fundamental works of Artin and Hasse in the first quarter of the 20th century. It is natural to ask if one could construct this theory including the non-abelian extensions of the base field. There are two approches to this problem. One approach is based on the ideas of Langlands, and the other on Koch. Koch’s method was later generalized by Fesenko and Koch-de Shalit for some non-abelian extensions of the base field. On the other hand, Ikeda and Serbest extended Fesenko’s works to construct a non-abelian local class field theory. But there is a restriction for the base field K. In this study, we extended Ikeda-Serbest’s construction of the local reciprocity map for a certain local field to any local field. Also we have shown that the extended map satisfies the certain functoriality and ramification theoretic properties.

Ring class fields and a result of Hasse

For squarefree d>1, let M denote the ring class field for the order Z[−3d] in F=Q(−3d). Hasse proved that 3 divides the class number of F if and only if there exists a cubic extension E of Q such that E and F have the same discriminant. Define the real cube roots v=(a+bd)1/3 and v′=(a−bd)1/3, where a+bd is the fundamental unit in Q(d). We prove that E can be taken as Q(v+v′) if and only if v∈M. As byproducts of the proof, we give explicit congruences for a and b which hold if and only if v∈M, and we also show that the norm of the relative discriminant of F(v)/F lies in {1,36} or {38,318} according as v∈M or v∉M. We then prove that v is always in the ring class field for the order Z[−27d] in F. Some of the results above are extended for subsets of Q(d) properly containing the fundamental units a+bd.

Class field theory, Hasse principles and Picard–Brauer duality for two-dimensional local rings

Class field theory, Hasse principles and Picard–Brauer duality for two-dimensional local rings

Visual aspects of Gaussian periods and analogues

Gaussian periods have been studied for centuries in the realms of number theory, field theory, cryptography, and elsewhere. However, it was only within the last decade or so that they began to be studied from a visual perspective. By plotting Gaussian periods in the complex plane, various interesting and insightful patterns emerge, leading to various conjectures and theorems about their properties. In this paper, we offer a description of Gaussian periods, along with examples of the structure that can occur when plotting them in the complex plane. In addition to this, we offer two ways in which this study can be generalized to other situations — one relating to supercharacter theory, the other relating to class field theory — along with discussions and visual examples of each.

Brumer–Stark units and explicit class field theory

Brumer–Stark units and explicit class field theory

Multipole expansion of the gravitational field in a general class of fourth-order theories of gravity and the application in gyroscopic precession

A viable weak-field and slow-motion approximation method is constructed in F(R, RμνRμν, Rμν ρσRμν ρσ ) gravity, a general class of fourth-order theories of gravity. By applying this method, the metric, presented in the form of the multipole expansion, outside a spatially compact source up to 1/c 3 order is provided, and the closed-form expressions for the source multipole moments are all presented explicitly. The metric consists of the massless tensor part, the massive scalar part, and the massive tensor part, where the former is exactly the metric in General Relativity, and the latter two are the corrections to it. It is shown that the corrections bear the Yukawa-like dependence on the two massive parameters and predict the appearance of six additional sets of source multipole moments, which indicates that up to 1/c 3 order, there exist six degrees of freedom beyond General Relativity within F(R, RμνRμν, Rμν ρσRμν ρσ ) gravity. By means of the metric, for a gyroscope moving around the source without experiencing any torque, the multipole expansions of its spin's angular velocities of the Thomas precession, the geodetic precession, and the Lense-Thirring precession are derived, and from them, the corrections to the angular velocities of the three types of precession in General Relativity can be read off. These results indicate that differently from f(R) or f(R,𝒢) gravity, the most salient feature of the general F(R, RμνRμν, Rμν ρσRμν ρσ ) gravity is that it gives the nonvanishing correction to the gyroscopic spin's angular velocity of the Lense-Thirring precession in General Relativity.

Nurturing Preservice Mathematics Teachers’ Reasoning about Measures of Central Tendency and Variability

This study aimed to examine the development of preservice mathematics teachers’ reasoning about measures of central tendency and variability. To this end, a classroom teaching experiment enriched by four compelling learning activities was conducted with 14 preservice teachers. The activities in the teaching experiment were designed to promote classroom discussion and collaboration. The study’s data comprised the video recordings of class sessions, field notes, and participants’ artifacts. The findings revealed the ideas about statistical measures generated by the preservice teachers based on their communal reasoning in the classroom. When using appropriate statistical measures for the context, they made sense of the measures of central tendency as indicators of certainty in situations including uncertainty. In other words, they could integrate central tendency and variability measures rather than just focusing on them independently. This study has valuable implications for teacher education in developing preservice teachers’ statistical reasoning on central tendency and variability measures.

A decomposition theorem for 0-cycles and applications to class field theory

We describe the abelian {e}tale fundamental group with modulus in terms of 0-cycles on a class of smooth quasi-projective varieties over finite fields which may not admit smooth compactifications. Using a limit process, this yields a cycle-theoretic analogue of the $K$-theoretic reciprocity theorem of Kato-Saito for such varieties. The proof is based on a decomposition theorem for the cohomological Chow group of 0-cycles on the double of a quasi-projective $R_1$-scheme over a field along a closed subscheme in terms of the Chow groups with and without modulus of the scheme. This also extends the decomposition theorem of Binda-Krishna to smooth schemes over imperfect fields.

Tempered perfect lattices in the binary case

A new algorithm for computing Hecke operators for SLn was introduced in [14]. The algorithm uses tempered perfect lattices, which are certain pairs of lattices together with a quadratic form. These generalize the perfect lattices of Voronoi [17]. The present paper is the first step in characterizing tempered perfect lattices. We obtain a complete classification in the plane, where the Hecke operators are for SL2(Z) and its arithmetic subgroups. The results depend on the class field theory of orders in imaginary quadratic number fields.

Identifying the Impact of Classes of the Kazakh National Sports Togyzkumalak Game on Increasing Educational and Cognitive Activity of Primary Class Students

The effectiveness of the impact of the Kazakh national sports Togyzkumalak game on increasing the educational and cognitive activity of primary school students in the educational and cognitive process is considered. Based on the analysis, it can be noted the need for further development and systematization of the use of knowledge from various fields in classes on the Togyzkumalak game, as well as in other academic subjects, such as mathematics, language, physical education and others. This involves creating themed challenges, stories, and other forms of playful interaction. This approach sees the path to the effective formation of the personality of a student with multifaceted needs and the development of their educational and cognitive abilities through individualization and activation of educational and cognitive activities. Classes on the Togyzkumalak game based on an interdisciplinary approach can contribute to the diversified development of students’ personality, which is confirmed by the results of the study. Regardless of the individual characteristics of the students, this approach creates a positive basis for their neuropsychic development.

An infinite dimensional version of the Kronecker index and its relation with the Leray–Schauder degree

Let \mathfrak{f} be a compact vector field of class C^{1} on a real Hilbert space \mathbb{H} . Denote by \mathbb{B} the open unit ball of \mathbb{H} and by \mathbb{S}= \partial\mathbb{B} the unit sphere. Given a point q \notin \mathfrak{f}(\mathbb{S}) , consider the self-map of \mathbb{S} defined by \mathfrak{f}_{q}^{\partial}(p)= \frac{\mathfrak{f}(p)-q}{\|\mathfrak{f}(p)-q\|}, \quad p \in \mathbb{S}. If \mathbb{H} is finite dimensional, then \mathbb{S} is an orientable, connected, compact differentiable manifold. Therefore, the Brouwer degree, \deg_{\mathrm{Br}}(\mathfrak{f}_{q}^{\partial}) is well defined, no matter what orientation of \mathbb{S} is chosen, assuming it is the same for \mathbb{S} as domain and codomain of \mathfrak{f}_{q}^{\partial} . This degree may be considered as a modern reformulation of the Kronecker index of the map \mathfrak{f}_{q}^{\partial} . Let \deg_{\mathrm{Br}}(\mathfrak{f},\mathbb{B},q) denote the Brouwer degree of \mathfrak{f} on \mathbb{B} with target q . It is known that one has the equality \deg_{\mathrm{Br}}(\mathfrak{f},\mathbb{B},q) = \deg_{\mathrm{Br}}(\mathfrak{f}_{q}^{\partial}). Our purpose is an extension of this formula to the infinite dimensional context. Namely, we will prove that \deg_{\mathrm{LS}}(\mathfrak{f},\mathbb{B},q) = \deg_{\mathrm{bf}}(\mathfrak{f}_{q}^{\partial}), where \deg_{\mathrm{LS}}(\cdot) denotes the Leray–Schauder degree and \deg_{\mathrm{bf}}(\cdot) is the degree earlier introduced by M. Furi and the first author, which extends, to the infinite dimensional case, the Brouwer degree and the Kronecker index. In other words, here, we extend to the Leray–Schauder degree the boundary dependence property which holds for the Brouwer degree in the finite dimensional context.

Rekonstruksi Distribusi Soal Asesmen Kompetensi Minimum Kelas XI Bidang Literasi Membaca Berdasarkan Bentuk Soal

<p><em>Current learning should be able to equip students to have the skills to use life skills in order to develop the potential of students. The goal of national education is to develop the potential of students to become a democratic and responsible Pancasila student profile. One attempt to raise the profile of critical and democratic students is through literacy activities. Kemendikbud organizes assessment activities entitled National Assessment as an evaluation of the education system. The National Assessment is directed at mapping the quality of input, process, and learning outcomes. (AKM). AKM consists of literacy reading and numeration. Previous research has shown that AKM is a tool to measure student skills, not only in mastering content, but also in improving students' ability to think critically and solve problems. In the context of reading literacy, this research will focus on reconstruction of AKM components in reading literature in high school. This research uses a qualitative research method approach to research on the conditions of natural objects. The source of the data in the research is about the AKM field of literacy reading class IX. Data is obtained through observations and interviews with informants. Data analysis techniques by means of data collection, data reduction, data presentation, and drawing conclusions. The results of the research showed that based on the form of the question, the AKM question consists of five forms of the matter: double choice, double choice complex, ridiculous, short filling and description </em></p>

Computing a group action from the class field theory of imaginary hyperelliptic function fields

We explore algorithmic aspects of a simply transitive commutative group action coming from the class field theory of imaginary hyperelliptic function fields. Namely, the Jacobian of an imaginary hyperelliptic curve defined over Fq acts on a subset of isomorphism classes of Drinfeld modules. We describe an algorithm to compute the group action efficiently. This is a function field analog of the Couveignes-Rostovtsev-Stolbunov group action. We report on an explicit computation done with our proof-of-concept C++/NTL implementation; it took a fraction of a second on a standard computer. We prove that the problem of inverting the group action reduces to the problem of finding isogenies of fixed τ-degree between Drinfeld Fq[X]-modules, which is solvable in polynomial time thanks to an algorithm by Wesolowski. We give asymptotic complexity bounds for all algorithms presented in this paper.