Algebraic Field Research Articles

Computability in infinite Galois theory and algorithmically random algebraic fields

AbstractWe introduce a notion of algorithmic randomness for algebraic fields. We prove the existence of a continuum of algebraic extensions of that are random according to our definition. We show that there are noncomputable algebraic fields which are not random. We also partially characterize the index set, relative to an oracle, of the set of random algebraic fields computable relative to that oracle.In order to carry out this investigation of randomness for fields, we develop computability in the context of the infinite Galois theory (where the relevant Galois groups are uncountable), including definitions of computable and computably enumerable Galois groups and computability of Haar measure on the Galois groups.

Cyclotomic Generating Functions

It is a remarkable fact that for many statistics on finite sets of combinatorial objects, the roots of the corresponding generating function are each either a complex root of unity or zero. These and related polynomials have been studied for many years by a variety of authors from the fields of combinatorics, representation theory, probability, number theory, and commutative algebra. We call such polynomials cyclotomic generating functions (CGFs). With Konvalinka, we have studied the support and asymptotic distribution of the coefficients of several families of CGFs arising from tableau and forest combinatorics. In this paper, we survey general CGFs from algebraic, analytic, and asymptotic perspectives. We review some of the many known examples of CGFs in combinatorial representation theory; describe their coefficients, moments, cumulants, and characteristic functions; and give a variety of necessary and sufficient conditions for their existence arising from probability, commutative algebra, and invariant theory. As a sample result, we show that CGFs are "generically" asymptotically normal, generalizing a result of Diaconis on $q$-binomial coefficients using work of Hwang-Zacharovas. We include several open problems concerning CGFs.

The Hadamard condition on a Cauchy surface and the renormalized stress-energy tensor

Given a Cauchy surface in a curved spacetime and a suitably defined quantum state on the CCR algebra of the Klein-Gordon quantum field on that surface, we show, by expanding the squared spacetime geodesic distance and the 'U' and 'V' Hadamard coefficients (and suitable derivatives thereof) in sufficiently accurate covariant Taylor expansions on the surface that the renormalized expectation value of the quantum stress-energy tensor on the surface is determined by the geometry of the surface and the first 4 time derivatives of the metric off the surface, in addition to the Cauchy data for the field's two-point function. This result has been anticipated in and is motivated by a previous investigation by the authors on the initial value problem in semiclassical gravity, for which the geometric initial data corresponds, a priori, to the spatial metric on the surface and up to 3 time derivatives off the surface, but where it was argued that the fourth derivative can be obtained with aid of the field equations on the initial surface.

A novel study on the structure of left almost hypermodules

The concept of left almost hypermodule evolves as a novel extension in the field of abstract algebra, specifically within the broader framework of hypermodules. The left almost hypermodule is characterized by a set endowed with two operations, evincing properties that extends across traditional module theory and hypermodules. This abstract intents to provide a succinct overview of salient attributes and prospective implications of the left almost hypermodule, stimulating further exploration of its properties and applications. The paper provides a new definition of hypermodule that acts on the left almost hyperring, referred to as left almost hypermodule (abbreviated as LA-hypermodule), and provides some examples of this new structure. We further examine the variations between hypermodules and left almost hypermodules. By using the concept of left almost polygroups, we explore the transition from left almost polygroup to a left almost hypermodule over left almost hyperring. Lastly, we observe the outcomes in connection to homomorphism and regular relations on left almost hypermodules.

Twisted tensor products of field algebras

Let \mathcal{A} be a C*-algebra, {\boldsymbol h} a Hilbert space, and {\mathcal{C}}_{\boldsymbol h} the CAR algebra over {\boldsymbol h} . We construct a twisted tensor product of \mathcal{A} by {\mathcal{C}}_{\boldsymbol h} such that the two factors are not necessarily one in the relative commutant of the other. The resulting C*-algebra may be regarded as a generalized CAR algebra constructed over a suitable Hilbert \mathcal{A} -bimodule. As an application, we exhibit a class of fixed-time models where a free Dirac field (giving rise to the {\mathcal{C}}_{\boldsymbol h} factor) in general is not relatively local to a free scalar field (which yields the \mathcal{A} factor). In some of the models, gauge-invariant combinations of the two (not relatively local) fields form a local observable net.

A global lifting for finite-dimensional Lie algebras of complete vector fields

Motivated by the so-called Lifting and Approximation Theorem by Rothschild and Stein, we consider a set of vector fields [Formula: see text] on a manifold [Formula: see text], and we study the problem of obtaining a global lifting of the [Formula: see text]’s to a system of generators of the Lie algebra [Formula: see text] of a Lie group [Formula: see text]. By assuming that the Lie algebra [Formula: see text] generated by [Formula: see text] is finite-dimensional and all of the [Formula: see text]’s are complete vector fields, but without the assumption that they satisfy Hörmander’s rank condition, we reduce the lifting problem to a result of Palais on integrability. This proves that any [Formula: see text] in [Formula: see text] is [Formula: see text]-related to a left invariant vector field [Formula: see text], where [Formula: see text] is a smooth map resulting from a right action [Formula: see text] of [Formula: see text] on [Formula: see text]. Both the lifting map [Formula: see text] and the lifting vector fields [Formula: see text] are globally defined, and our result generalizes the global Lifting obtained by Folland in the special case of dilation-invariant vector fields. According to Palais’ Integrability, the map [Formula: see text] is obtained via the flow of suitable vector fields on [Formula: see text] and the image set of [Formula: see text], namely, [Formula: see text], is the Sussmann orbit of [Formula: see text] through [Formula: see text]. Examples are provided, showing that a germ of [Formula: see text], obtained through the integration of [Formula: see text] and depending only on the Baker–Campbell–Hausdorff formula, is often sufficient to get the global lifting without the need of abstract results.

The $J$-invariant over splitting fields of Tits algebras

We describe the $J$-invariant of a semisimple algebraic group $G$ over a generic splitting field of a Tits algebra of $G$ in terms of the $J$-invariant over the base field. As a consequence we prove a 10-year-old conjecture of Quéguiner-Mathieu, Semenov, and Zainoulline on the $J$-invariant of groups of type $\mathrm {D}_n$. In the case of type $\mathrm {D}_n$ we also provide explicit formulas for the first component and in some cases for the second component of the $J$-invariant.

The gaussian sequence 3th order in finite groups

Recent research examines Gaussian numbers in third-order recurrent sequences in connection with finite groups in the field of Abstract Algebra. Thus, some definitions were introduced aiming to transform infinite sequences into finite ones. In this regard, the Fourier transform was used as a visualization technique, explored in Google Colab. The mathematical theorems presented were established with the purpose of examining the patterns of these sequences and their corresponding cycles. As a future perspective, it is intended to investigate other mathematical theorems to generalize the sequences into finite groups.

The Impact of GeoGebra in Analytic Algebra: A Bibliometric Review

GeoGebra is a software that is often used in mathematics learning. GeoGebra is an electronic program that contains a set of tools that equip students with mathematical skills. This research aims to look at research trends related to algebra learning that integrates GeoGebra. This research is literature review research, there are 95 publications collected from the Scopus database which are then analyzed using the bibliometric analysis method assisted by the Vos viewer application. Data taken from the Scopus database was refined in 4 stages, namely identification, filtering, eligibility, and inclusion, resulting in 95 publications. The research results show that Spain and Austria are the most influential countries and have high cooperation with other countries in this field. The focus of research related to GeoGebra on algebra material is, 1) geometry and students; 2) GeoGebra and algebra; 3) elementary geometry and computational theory; 4) dynamic geometry and computer algebra. The new themes in this research are symbolic computation, quantifier elimination and real quantifier elimination. This research not only enriches the scientific literature in the fields of GeoGebra and algebra but also provides practical guidance for educators and researchers to improve mathematics teaching and learning methods in the future.

Short Note: Exploring Ideals in Graph Theory

We are re-evaluating the novel concept of 'ideal' within the context of graph theory. The concept of an 'ideal' is well-established in the fields of topology and algebra, having been extensively researched by numerous scholars due to its mathematical significance (see references) [2-6]. Tree-width is a well-known graph width parameter. In our discussion, we explore the relationship between tree-width and ideal. Overall, this exploration contributes to a deeper understanding of these concepts and their relevance in various mathematical and logical contexts.

Note on the group of vertical diffeomorphisms of a principal bundle & its relation to the Frölicher-Nijenhuis bracket

The group of vertical diffeomorphisms of a principal bundle forms the action Lie groupoid associated to the bundle. The former is generated by the group of maps with value in the structure group, which is also the group of bisections of the groupoid. The corresponding Lie algebra of general vertical vector fields is generated by maps with value in the Lie algebra of the structure group. The bracket on these maps, induced by the bracket of vertical vector fields, is an “extended” bracket on gauge parameters: it has been introduced heuristically in physics, notably in the study of asymptotic symmetries of gravity. Seeing the set of Lie algebra-valued maps as sections of the action Lie algebroid associated to the bundle, the extended bracket is understood to be a Lie algebroid bracket on those sections.Here, we highlight that this bracket can also be seen to arise from the Frölicher-Nijenhuis bracket of vector-valued differential forms. The benefit of this viewpoint is to insert this extended bracket within the general framework of derivations of forms on a bundle. Identities relating it to the usual operations of Cartan calculus — inner product, exterior and (Nijenhuis-) Lie derivative — are immediately read as special cases of general results. We also consider the generalised gauge transformations induced by vertical diffeomorphisms, and discuss their peculiar features. In particular, locally, and contrary to standard gauge transformations arising from vertical bundle automorphisms, they are distinguishable from local gluings when iterated. Yet, the gauge principle still holds.

Polarization-encoded photonic quantum-to-quantum Bernoulli factory based on a quantum dot source.

A Bernoulli factory is a randomness manipulation routine that takes as input a Bernoulli random variable, outputting another Bernoulli variable whose bias is a function of the input bias. Recently proposed quantum-to-quantum Bernoulli factory schemes encode both input and output variables in qubit amplitudes. This primitive could be used as a subroutine for more complex quantum algorithms involving Bayesian inference and Monte Carlo methods. Here, we report an experimental implementation of a polarization-encoded photonic quantum-to-quantum Bernoulli factory. We present and test three interferometric setups implementing the basic operations of an algebraic field (inversion, multiplication, and addition), which, chained together, allow for the implementation of a generic quantum-to-quantum Bernoulli factory. These in-bulk schemes are validated using a quantum dot-based single-photon source featuring high brightness and indistinguishability, paired with a time-to-spatial demultiplexing setup to prepare input resources of up to three single-photon states.

Hybrid analysis of radiative corrections to neutron decay with current algebra and effective field theory

We introduce a useful framework for high-precision studies of the neutron beta decay by merging the current algebra description and the fixed-order effective field theory calculation of the electroweak radiative corrections to the neutron axial form factor. We discuss the advantages of this hybrid method and show that it only requires a minimal amount of lattice QCD inputs to achieve a 10−4 theory accuracy for the Standard Model prediction of the neutron lifetime and the axial-to-vector coupling ratio λ, both important to the search for physics beyond the Standard Model.

Representations Of Lie Algebras of Vector Fields on Pairing Between Gauge Modules and Rudakov Modules

For an irreducible affine variety X over an algebraically closed field of characteristic zero, we define two new classes of modules over the Lie algebra of vector fields on X: gauge modules and Rudakov modules. These modules admit a compatible action of the algebra of functions on X. We prove general simplicity theorems for these two types of modules, demonstrating their irreducibility under specific conditions. Additionally, we establish a pairing between gauge modules and Rudakov modules, highlighting the connections and interactions between these two classes of modules. We have established that gauge modules and Rudakov modules, corresponding to simple glN -modules, remain irreducible as modules over the Lie algebra of vector fields unless they appear in the de Rham complex. Additionally, we have studied the irreducibility of tensor products of Rudakov modules, providing a comprehensive understanding of these module structures and their applications.

The effectiveness of the Madrasati platform according to the Needham constructivist model in increasing achievement of second-year intermediate female students in the TIMSS tests (algebra field) and the teacher’s point of view

رصدت تقارير هيئة تقويم التعليم بالمملكة العربية السعودية متوسط أداء طلبة الصف الثاني متوسط في مشاركات الاختبارات الدولية TIMSS في الأعوام الثلاثة المتتالية 2011,2015,2019 لتشير التقارير إلى انخفاض في مستوى أداء الطلبة والحصول على مراكز متأخرة حيث احتلت المملكة العربية السعودية المرتبة (37) من أصل (39) دولة مشاركة عام 2019، واستثمارًا لجهود وزارة التعليم في تطوير التعليم سعت الدراسة الحالية إلى تدريب طالبات الصف الثاني متوسط على تطوير مهارات التفكير لحل مسائل محاكية لأسئلة الاختبارات الدوليةTIMSS(مجال الجبر) باستخدام منصة مدرستي وفق نموذج نيدهام البنائي، وقد نهجت الدراسة النهج المختلط على عينة قصدية قوامها (20) طالبة من طالبات الصف الثاني متوسط تم اختبارهن اختبار قبلي تبعه برنامج تعليمي مقترح باستخدام استراتيجية نيدهام البنائية من خلال بطاقات مقترحة بتسلسل يتوافق مع نموذج نيدهام بمراحله الخمس، ثم تبع مرحلة التدريب الاختبار البعدي للطالبات العينة. أظهرت نتائج تحليل البيانات الكمية والوصفية الأثر الإيجابي للتدريب وفق البرنامج التعليمي المقترح، ومدى فاعلية منصة مدرستي وفق نموذج نيدهام البنائي في زيادة تحصيل طالبات الصف الثاني متوسط في اختبارات الدولية TIMSS (مجال الجبر).

Polynomial Representations of the Witt Lie Algebra

Abstract The Witt algebra ${\mathfrak{W}}_{n}$ is the Lie algebra of all derivations of the $n$-variable polynomial ring $\textbf{V}_{n}=\textbf{C}[x_{1}, \ldots , x_{n}]$ (or of algebraic vector fields on $\textbf{A}^{n}$). A representation of ${\mathfrak{W}}_{n}$ is polynomial if it arises as a subquotient of a sum of tensor powers of $\textbf{V}_{n}$. Our main theorems assert that finitely generated polynomial representations of ${\mathfrak{W}}_{n}$ are noetherian and have rational Hilbert series. A key intermediate result states polynomial representations of the infinite Witt algebra are equivalent to representations of $\textbf{Fin}^{\textrm{op}}$, where $\textbf{Fin}$ is the category of finite sets. We also show that polynomial representations of ${\mathfrak{W}}_{n}$ are equivalent to polynomial representations of the endomorphism monoid of $\textbf{A}^{n}$. These equivalences are a special case of an operadic version of Schur–Weyl duality, which we establish.

A category equivalence on the Lie algebra of polynomial vector fields

For any positive integer n, let Wn=Der(C[t1,…,tn]). The subspaces hn=Span{t1∂∂t1,…,tn∂∂tn} and Δn=Span{∂∂t1,…,∂∂tn} are two abelian subalgebras of Wn. We show that a full subcategory Ω1 of the category of Wn-modules M which are locally finite over Δn is equivalent to some full subcategory of weight Wn-modules M which are cuspidal modules when restricted to the subalgebra sln+1 of Wn.

Automorphisms of projective structures

We study the problem of classifying local projective structures in dimension two having non trivial Lie symmetries. In particular we obtain a classification of foliated projective structures having positive dimensional Lie algebra of projective vector fields.

An efficient polynomial-based verifiable computation scheme on multi-source outsourced data

With the development of cloud computing, users are more inclined to outsource complex computing tasks to cloud servers with strong computing capacity, and the cloud returns the final calculation results. However, the cloud is not completely trustworthy, which may leak the data of user and even return incorrect calculations on purpose. Therefore, it is important to verify the results of computing tasks without revealing the privacy of the users. Among all the computing tasks, the polynomial calculation is widely used in information security, linear algebra, signal processing and other fields. Most existing polynomial-based verifiable computation schemes require that the input of the polynomial function must come from a single data source, which means that the data must be signed by a single user. However, the input of the polynomial may come from multiple users in the practical application. In order to solve this problem, the researchers have proposed some schemes for multi-source outsourced data, but these schemes have the common problem of low efficiency. To improve the efficiency, this paper proposes an efficient polynomial-based verifiable computation scheme on multi-source outsourced data. We optimize the polynomials using Horner’s method to increase the speed of verification, in which the addition gate and the multiplication gate can be interleaved to represent the polynomial function. In order to adapt to this structure, we design the corresponding homomorphic verification tag, so that the input of the polynomial can come from multiple data sources. We prove the correctness and rationality of the scheme, and carry out numerical analysis and evaluation research to verify the efficiency of the scheme. The experimental indicate that data contributors can sign 1000 new data in merely 2 s, while the verification of a delegated polynomial function with a power of 100 requires only 18 ms. These results confirm that the proposed scheme is better than the existing scheme.

Emergent Spinor Fields from Exotic Spin Structures

Abstract The classification of emergent spinor fields according to modified bilinear covariants is scrutinized in space-times with nontrivial topology, which induce inequivalent spin structures. Extended Clifford algebras, constructed by equipping the underlying space-time with an extended bilinear form with additional terms coming from the nontrivial topology, naturally yield emergent extended algebraic spinor fields and their subsequent extended bilinear covariants, which are contrasted to the classical spinor classification. An unexpected duality between the standard and the exotic spinor field classes is therefore established, showing that a complementary fusion process among the spinor field classes sets in, when extended Clifford bundles are addressed in multiply connected space-times.