Class Of Fields Research Articles

Modularity of Galois traces of ray class invariants

After Zagier’s significant work (in: Bogomolov and Katzarkov (eds) Motives, polylogarithms and hodge theory, part I, International Press, Somerville, 2002) on traces of singular moduli, Bruinier and Funke (J Reine Angew Math 594:1–33, 2006) generalized his result to the traces of singular values of modular functions on modular curves of arbitrary genus. In class field theory, the extended ring class field is a generalization of the ray class field over an imaginary quadratic field. By using Shimura’s reciprocity law, we construct primitive generators of the extended ring class fields by using Siegel functions of arbitrary level $$N\ge 2$$ and identify their Galois traces with Fourier coefficients of weight 3/2 harmonic weak Maass forms. This would extend the results of Jeon et al. (Math Ann 353:37–63, 2012) and Jung et al. (Modularity of Galois traces of Weber’s resolvents, under revision).

Growth of Fine Selmer Groups in Infinite Towers

AbstractIn this paper, we study the growth of fine Selmer groups in two cases. First, we study the growth of fine Selmer ranks in multiple $\mathbb{Z}_{p}$-extensions. We show that the growth of the fine Selmer group is unbounded in such towers. We recover a sufficient condition to prove the $\unicode[STIX]{x1D707}=0$ conjecture for cyclotomic $\mathbb{Z}_{p}$-extensions. We show that in certain non-cyclotomic $\mathbb{Z}_{p}$-towers, the $\unicode[STIX]{x1D707}$-invariant of the fine Selmer group can be arbitrarily large. Second, we show that in an unramified $p$-class field tower, the growth of the fine Selmer group is unbounded. This tower is non-Abelian and non-$p$-adic analytic.

World History on a Par with Chinese History? China's Search for World Power in Three Stages

ABSTRACT In 2011, the Department of Education in China made a policy change by elevating the study and teaching of world history to a first-class academic field. This article argues that this elevation reflects an unprecedented move by both the Chinese government and historical community to put world history on a par with Chinese history with regard to their importance as subjects of knowledge. It discusses the three periods where world history developed as a field of study in modern China, focusing on the latest one from the turn of the twenty-first century to today. During this period, world historians in China launched their search for a new, Chinese grand narrative for understanding and interpreting worldwide historical development. While inspired by the theory of modernization and advances in global history, the historians have also endeavoured to draw wisdom from China's rich cultural tradition and its recent successful experience in economic expansion. The elevation of world history to a first class field of academic study suggests a landmark development in the direction.

ОРГАНІЗАЦІЯ ПРАКТИКО-ТЕХНІЧНОЇ ПІДГОТОВКИ МАЙБУТНІХ ІТ-ФАХІВЦІВ ЗА ДОПОМОГОЮ ВЕБІНАРІВ

Аналіз сучасної наукової та навчально-методичної літератури дозволив дійти висновку, що сьогодні є недостатньою кількість професійно-орієнтованої навчальної літератури з використання технології вебінару в навчальному процесі. В даній статті подано комплексне дослідження, в якому вивчаються питання програмного забезпечення, організаційно-методичних вимог до проведення вебінарів, вибір платформи. Розглянуто передумови, описано процес створення, проаналізовано функціональні характеристики та результати функціонування системи навчання технічних дисциплін майбутніх фахівців з інформаційних технологій. Серед значної кількості відомих платформ для підтримки проведення вебінарів виокремлено платформу BigBlueButton, яка є безкоштовним і вільно поширюваним програмним забезпеченням з відкритим вихідним кодом. Описано та рекомендовано низку організаційно-методичних заходів для проведення успішного вебінару. Відзначено, що дотримання методичної, технічної та організаційної складових проведення вебінарів є надзвичайно важливим для досягнення позитивного результату у підготовці фахівців. Показано, що використання технології вебінарів значно розширює можливості організації освітнього процесу, сприяє формуванню у майбутніх ІТ-фахівців системи технічних понять, знань та умінь в галузі проведення онлайнових занять і, таким чином, надає можливість удосконалити практико-технічну підготовку студентів. Попри всі незручності, практично всі види аудиторних занять у традиційній освіті можуть бути реалізовані засобами вебінарів. Проте, ряд проблем такої технології як форми організації навчання не дають підстави для повної заміни традиційного аудиторного навчання. Її використання уможливлює значно покращити та вдосконалити освітній процес, перевести на інший рівень процес осмислення навчального матеріалу, урізноманітнити форми організації навчальної діяльності.

KORELASI PENGUASAAN KOSAKATA BIDANG LINGKUNGAN HIDUP DAN KETERAMPILAN MENULIS TEKS EKSPLANASI SISWA KELAS XI SMK NEGERI 2 PADANG

ABSTRACT The purpose of this study is to describe the following matters. First, it describes the mastery of the vocabulary in the environmental field of class XI students of SMK Negeri 2 Padang. Second, describing the explanatory writing skills of the eleventh grade students of SMK Negeri 2 Padang. Third, explaining the correlation between the mastery of environmental vocabulary and the writing skills of explanatory text for grade XI students of SMK Negeri 2 Padang. This type of research is quantitative research with descriptive methods. Based on the results of the study, the following three things can be concluded. First, the vocabulary mastery of the environmental field of class XI students of SMK Negeri 2 Padang is in good qualifications (77.85). Second, the explanatory text writing skills in class XI students of SMK Negeri 2 Padang are in good qualifications (77.93). Third, there is a significant correlation between vocabulary mastery in the field of environment and writing skills in explanatory text for class XI students of SMK Negeri 2 Padang on the degree of freedom n-1 and a significant level of 95% so that it can be concluded that H0 is rejected and H1 is accepted because the test results prove that tcount is greater ttable 4.93> 1.68. Kata Kunci: Korelasi, Kosakata Bidang Lingkungan Hidup, Menulis Teks Eksplanasi

On Existence of Euclidean Ideal Classes in Real Cubic and Quadratic Fields with Cyclic Class Group

Lenstra introduced the notion of Euclidean ideal classes for number fields to study cyclicity of their class groups. In particular, he showed that the class group of a number field with unit rank greater than or equal to one is cyclic if and only if it has a Euclidean ideal class. The only if part in the above result is conditional on the extended Riemann hypothesis. Graves and Murty showed that one does not require the extended Riemann hypothesis if the unit rank of the number field is greater than or equal to four and its Hilbert class field is abelian over rationals. In this article, we study real cubic and quadratic fields with cyclic class groups and show that they have a Euclidean ideal class under certain conditions.

Emergent literacy instruction: ‘continua of biliteracy’ among newly immigrated adolescents

This article focuses on the instruction of recently immigrated adolescents with limited educational backgrounds who are developing emergent literacy. It is based on an ethnographic study conducted in a public Swedish language introductory class in 2017/2018. Its purpose is to investigate how the students engaged in literacy practices during the instruction, and how they were supported by their teachers. Empirical data, which included field notes, audio recordings, and interviews with students, were analyzed with the help of two dimensions of Hornberger’s continua of biliteracy, namely, the content and development of biliteracy. Analysis of the content of biliteracy indicated that students’ previous knowledge, as well as class field trips and tangible examples, served as important foundations for their instruction. Analysis of the development of biliteracy showed that the teachers’ engagement with the students’ diverse linguistic and other semiotic resources contributed to the students’ participation in literacy practices. While the framework applied to the data includes several dimensions of literacy, indispensable for research in this complex context, the analysis also illuminates the need for the inclusion of additional dimensions in order to account for the role of interpersonal relations.

On relative projectivity of lattices in Auslander-Reiten components for group rings

Let G be a finite group, Q a p-subgroup of G and P a Sylow p-subgroup of Let be a complete discrete valuation ring of characteristic zero with residue class field of characteristic p > 0. Suppose that the group ring is of infinite representation type and is sufficiently large. Let be a stable Auslander-Reiten component containing a Scott -lattice S(Q) with vertex Q. Then all the -lattices in have P as their vertices. Also, we show that there exists an indecomposable -lattice L with vertex P such that a kG-module is Q-projective.Communicated by J. Zhang

Class field theory, Diophantine analysis and the asymptotic Fermat's Last Theorem

Recent results of Freitas, Kraus, Şengün and Siksek, give sufficient criteria for the asymptotic Fermat's Last Theorem to hold over a specific number field. Those works in turn build on many deep theorems in arithmetic geometry. In this paper we combine the aforementioned results with techniques from class field theory, the theory of p-groups and p-extensions, Diophantine approximation and linear forms in logarithms, to establish the asymptotic Fermat's Last Theorem for many infinite families of number fields, and for thousands of number fields of small degree. For example, we prove the effective asymptotic Fermat's Last Theorem for the infinite family of fields Q(ζ2r)+ where r≥2.

The narrow 2-class field tower of some real quadratic number fields

The narrow 2-class field tower of some real quadratic number fields

Generating ray class fields of real quadratic fields via complex equiangular lines

For certain real quadratic fields $K$ with sufficiently small discriminant we produce explicit unit generators for specific ray class fields of $K$ using a numerical method that arose in the study of complete sets of equiangular lines in $\mathbb{C}^d$ (known in quantum information as symmetric informationally complete measurements or SICs). The construction in low dimensions suggests a general recipe for producing unit generators in infinite towers of ray class fields above arbitrary real quadratic $K$, and we summarise this in a conjecture. There are indications [19,20] that the logarithms of these canonical units are related to the values of $L$-functions associated to the extensions, following the programme laid out in the Stark Conjectures.

The forgotten land use class: Mapping of fallow fields across the Sahel using Sentinel-2

Remote sensing-derived cropland products have depicted the location and extent of agricultural lands with an ever increasing accuracy. However, limited attention has been devoted to distinguishing between actively cropped fields and fallowed fields within agricultural lands, and in particular so in grass fallow systems of semi-arid areas. In the Sahel, one of the largest dryland regions worldwide, crop-fallow rotation practices are widely used for soil fertility regeneration. Yet, little is known about the extent of fallow fields since fallow is not explicitly differentiated within the cropland class in any existing remote sensing-based land use/cover maps, regardless of the spatial scale. With a 10 m spatial resolution and a 5-day revisit frequency, Sentinel-2 satellite imagery made it possible to disentangle agricultural land into cropped and fallow fields, facilitated by Google Earth Engine (GEE) for big data handling. Here we produce the first Sahelian fallow field map at a 10 m resolution for the baseline year 2017, accomplished by designing a remote sensing driven protocol for generating reference data for mapping over large areas. Based on the 2015 Copernicus Dynamic Land Cover map at 100 m resolution, the extent of fallow fields in the cropland class is estimated to be 63% (403,617 km2) for the Sahel in 2017. Similar results are obtained for five contemporary cropland products, with fallow fields occupying 57–62% of the cropland area. Yet, it is noted that the total estimated area coverage depends on the quality of the different cropland products. The share of cropped fields within the Copernicus cropland area is found to be higher in the arid regions (200–300 mm rainfall) as compared to the semi-arid regions (300–600 mm rainfall). The woody cover fraction within cropped and fallow fields is found to have a reversed pattern between arid (higher woody cover in cropped fields) and semi-arid (higher woody cover in fallow fields) regions. The method developed, using cloud-based Earth Observation (EO) data and computation on the GEE platform, is expected to be reproducible for mapping the extent of fallow fields across global croplands. Future applications based on multi-year time series is expected to improve our understanding of crop-fallow rotation dynamics in grass fallow systems being key in teasing apart how cropland intensification and expansion affect environmental variables, such as soil fertility, crop yields and local livelihoods in low-income regions such as the Sahel. The mapping result can be visualized via a web viewer (https://buwuyou.users.earthengine.app/view/fallowinsahel).

High-harmonic generation by electric polarization, spin current, and magnetization

High-harmonic generation (HHG), a typical nonlinear optical effect, has been actively studied in electron systems such as semiconductors and superconductors. As a natural extension, we theoretically study HHG from electric polarization, spin current and magnetization in magnetic insulators under terahertz (THz) or gigahertz (GHz) electromagnetic waves. We use simple one-dimensional spin chain models with or without multiferroic coupling between spins and the electric polarization, and study the dynamics of the spin chain coupled to an external ac electric or magnetic field. We map spin chains to two-band fermions and invoke an analogy of semiconductors and superconductors. With a quantum master equation and Lindblad approximation, we compute the time evolution of the electric polarization, spin current, and magnetization, showing that they exhibit clear harmonic peaks. We also show that the even-order HHG by magnetization dynamics can be controlled by static magnetic fields in a wide class of magnetic insulators. We propose experimental setups to observe these HHG, and estimate the required strength of the ac electric field $E_0$ for detection as $E_0\sim100$kV/cm--1MV/cm, which corresponds to the magnetic field $B_0\sim0.1$T--1T. The estimated strength would be relevant also for experimental realizations of other theoretically-proposed nonlinear optical effects in magnetic insulators such as Floquet engineering of magnets.

The Lambda Invariants at CM Points

Abstract In this paper, we show that $\lambda (z_1) -\lambda (z_2)$, $\lambda (z_1)$, and $1-\lambda (z_1)$ are all Borcherds products on $X(2) \times X(2)$. We then use the big CM value formula of Bruinier, Kudla, and Yang to give explicit factorization formulas for the norms of $\lambda (\frac{d+\sqrt d}2)$, $1-\lambda (\frac{d+\sqrt d}2)$, and $\lambda (\frac{d_1+\sqrt{d_1}}2) -\lambda (\frac{d_2+\sqrt{d_2}}2)$, with the latter under the condition $(d_1, d_2)=1$. Finally, we use these results to show that $\lambda (\frac{d+\sqrt d}2)$ is always an algebraic integer and can be easily used to construct units in the ray class field of ${\mathbb{Q}}(\sqrt{d})$ of modulus $2$. In the process, we also give explicit formulas for a whole family of local Whittaker functions, which are of independent interest.

Generalized Jacobians and explicit descents

We develop a cohomological description of various explicit descents in terms of generalized Jacobians, generalizing the known description for hyperelliptic curves. Specifically, given an integer $n$ dividing the degree of some reduced effective divisor $\mathfrak{m}$ on a curve $C$, we show that multiplication by $n$ on the generalized Jacobian $J_\frak{m}$ factors through an isogeny $\varphi:A_{\mathfrak{m}} \rightarrow J_{\mathfrak{m}}$ whose kernel is naturally the dual of the Galois module $(\operatorname{Pic}(C_{\overline{k}})/\mathfrak{m})[n]$. By geometric class field theory, this corresponds to an abelian covering of $C_{\overline{k}} := C \times_{\operatorname{Spec}{k}} \operatorname{Spec}(\overline{k})$ of exponent $n$ unramified outside $\mathfrak{m}$. The $n$-coverings of $C$ parameterized by explicit descents are the maximal unramified subcoverings of the $k$-forms of this ramified covering. We present applications of this to the computation of Mordell-Weil groups of Jacobians.

Preface

We are pleased to introduce the Proceedings of the conference Personnel training and legal support for the implementation of research projects of the MegaScience class (June 20-21, 2019) organized by National Research Nuclear University MEPhI (Moscow Engineering Physics Institute) and the Kutafin Moscow State Law University (MSAL).The conference was attended by scientists conducting research on unique scientific facilities of the mega-science class. Participants of the conference presented both positive and negative experience in implementing scientific projects of the megascience class, made proposals on optimal organization of these projects. The problems of legal regulation of creation and functioning of unique scientific facilities of the megascience class, as well as the problems of international scientific cooperation were identified.The received papers were reviewed according to the standards of the Journal of Physics: Conference Series and to scientific criteria developed by organizers. This volume of journal includes 21 papers devoted to various aspects of cooperation in the field of megascience class projects.List of Conference Photograph are available in the pdf.

Az alsó középiskolában használt angol tankönyvek szerepe a kommunikatív nyelvoktatásban két Sunshine-ból tanító alsó középiskolai tanár esettanulmánya alapján

A Kairyudo kiadó gondozásában megjelenő Sunshine a Japán Oktatásügyi, Kulturális, Sport- valamint Tudomány- és Technológiaügyi Minisztérium (MEXT) által jóváhagyott alsó középiskolai tankönyvcsalád. Jelen esettanulmány két Iwate Prefektúrában a Sunshine tankönyvekből tanító pedagógus munkáját megfigyelve a következő kérdésekre keresi a választ: mely tulajdonságokban felel meg a tankönyv a Nemzeti Alaptanterv szempontjainak, a két pedagógus hogyan értelmezi és építi be a tankönyv egyes jellemzőit mindennapos tanári tevékenységébe, illetve milyen módszereket alkalmaznak a tankönyv kiegészítésére. A kutatáshoz a Sunshine tankönyvek hivatalos tájékoztató kiadványa és kiegészítő brosúrája, óralátogatások, óralátogatási jegyzőkönyvek, félig strukturált interjúk, tanteremi oktatási eszközök szolgáltattak alapot. A kutatási eredmények szerint a Sunshine tankönyvcsalád általánosságban megfelel a kommunikatív megközelítésnek, de a tanároknak saját ötleteikkel kell kiegészíteniük a tankönyvet.

SIC-POVMs and the Stark Conjectures

Abstract The existence of $d^2$ pairwise equiangular complex lines [equivalently, a symmetric informationally complete positive operator-valued measure (SIC-POVM)] in $d$-dimensional Hilbert space is known only for finitely many dimensions $d$. We prove that, if there exists a set of real units in a certain ray class field (depending on $d$) satisfying certain algebraic properties, a SIC-POVM exists, when $d$ is an odd prime congruent to 2 modulo 3. We give an explicit analytic formula that we expect to yield such a set of units. Our construction uses values of derivatives of zeta functions at $s=0$ and is closely connected to the Stark conjectures over real quadratic fields. We verify numerically that our construction yields SIC-POVMs in dimensions 5, 11, 17, and 23, and we give the first exact SIC-POVM in dimension 23.

Composition law and complex multiplication

Let K be an imaginary quadratic field of discriminant dK, and let n be a nontrivial integral ideal of K in which N is the smallest positive integer. Let QN(dK) be the set of primitive positive definite binary quadratic forms of discriminant dK whose leading coefficients are relatively prime to N. We adopt an equivalence relation ∼n on QN(dK) so that the set of equivalence classes QN(dK)/∼n can be regarded as a group isomorphic to the ray class group of K modulo n. We further establish an explicit isomorphism of QN(dK)/∼n onto Gal(Kn/K) in terms of Fricke invariants, where Kn denotes the ray class field of K modulo n. This would be a certain extension of the classical composition theory of binary quadratic forms, originated and developed by Gauss and Dirichlet.

PENGEMBANGAN PERANGKAT PEMBELAJARAN KONSEP BAKTERI DENGAN MODEL INKUIRI PADA SISWA SMA

Biology learning process that occurs in SMAN 2 Muara Teweh still using devices based learning and student handbooks with the curriculum SBC devices are arranged in the Subject Teachers Council (MGMP). Student handbook based curriculum SBC used as basis for the preparation of such devices in general. Learning with the inquiry model of bacterial material has never been used by teachers in order to improve the learning spirit of students to learn. The material of bacteria is one of the materials related to the scientific attitude. The aim of this study is to produce a software development learning inquiry model with the concept of high school students of class X bacteria valid and practical and can study the implementation of the concept of bacteria using a model of an effective inquiry. Samples were individual trials that test 3 and test experts legibility on 6 students, small groups of test class numbered 19 students and class field tests totaling 24 students. The instrument of this research is the syllabus, lesson plans, worksheets, assessment sheets, key pieces of assessment, teaching materials, process observation sheet, psychomotor skills, critical thinking, character, social behavior and the activities of teachers and students. Data retrieved student learning outcomes through pretest and posttest in learning the concept of bacteria using inquiry model. Activity observation sheet obtained under observation while the activities of the skills assessed by sheets of skills. It can be concluded that Practicality learning device has been reached. the enforceability of RPP and either categorized student activity, student responses almost all the students are very happy. The effectiveness of the learning device has also been met, it is based on the cognitive learning products and cognitive processes has reached mastery that students get the value of N-gain with high category, the assessment results psychomotor skills, assessment of social attitudes and attitude of the character, as well as critical thinking has been included in both categories, research can produce development tools through inquiry model valid, practical and effective.