Algebraic Field Research Articles

Spectral properties of Killing vector fields of constant length

This paper is devoted to the study of properties of Killing vector fields of constant length on Riemannian manifolds. If g is a Lie algebra of Killing vector fields on a given Riemannian manifold (M,g), and X∈g has constant length on (M,g), then we prove that the linear operator ad(X):g→g has a pure imaginary spectrum. More detailed structural results on the corresponding operator ad(X) are obtained. Some special examples of Killing vector fields of constant length are constructed.

On Direct Product of a Fuzzy Subgroup with an Anti-fuzzy Subgroup

We have introduced and analysed some new refreshing concepts in the field of fuzzy abstract algebra. The main contributions of this paper are fivefold: (1) we have introduced the notion of dual-fuzzy subgroup, (2) we have defined the direct product of a fuzzy subgroup with an anti-fuzzy subgroup, (3) Furthermore, we have defined mixed level subset and mixed level subgroup, (4) we have also developed some new theories as well as propositions based on these newly defined notions and lastly (5) we have redefined these notions using general T-norm and T* conorm.

Representations of Lie algebras of vector fields on affine varieties

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, which admit a compatible action of the algebra of functions. Gauge modules are generalizations of modules of tensor densities whose construction was inspired by non-abelian gauge theory. Rudakov modules are generalizations of a family of induced modules over the Lie algebra of derivations of a polynomial ring studied by Rudakov [23]. We prove general simplicity theorems for these two types of modules and establish a pairing between them.

Matematik Öğretmen Adaylarının, 6. Sınıf Öğrencilerinin Cebirsel Örüntüleri Genellemelerine İlişkin Farkındalıkları

Matematik öğretmeninin sahip olması gereken bilgi, temel konu ve kavramları bilmenin ötesinde ayrıcalıklı bir bilgi türüdür. Bu bilginin bileşenlerinden birisi de öğrencinin matematiksel düşüncesini farketmektir. Özel olarak cebir alanında öğrencilerin örüntü genelleme süreçlerini anlamlandırmak ve hata kaynaklarını belirlemek için öğretmenin bu türde bir farkındalığa sahip olması gereklidir. Bu çalışmanın amacı matematik öğretmen adaylarının öğrencilerin örüntü genelleme süreçlerine ilişkin farkındalığını araştırmaktır. Bu amaca yönelik veri toplama aracı olarak farklı tipte yaklaşım ve stratejilerden oluşan gerçek öğrenci yanıtları kullanılmıştır. Nitel araştırma yöntemlerinin benimsendiği çalışmanın katılımcıları 2015-2016 eğitim-öğretim yılında bir devlet üniversitesinin matematik öğretmenliği bölümüne devam eden ve “cebirsel kavramlar ve öğretimi dersi”ni almakta olan 35 öğretmen adayıdır. Araştırma bulgularına göre, öğretmen adayları öğrencilerin görünür stratejilerini fark etme konusunda başarılı sayılabilirler. Ne var ki, adayların öğrencilerin hatalarının altında yatan nedenleri belirlemeye yönelik açıklamaları yeterli bulunmamıştır.

On the Lie 2-algebra of sections of an [formula omitted]-groupoid

In this work we introduce the category of multiplicative sections of an LA-groupoid. We prove that this category carries a natural strict Lie 2-algebra structure, which is Morita invariant. As applications, we study the algebraic structure underlying multiplicative vector fields on a Lie groupoid and in particular vector fields on differentiable stacks. We also introduce the notion of geometric vector field on the quotient stack of a Lie groupoid, showing that the space of such vector fields is a Lie algebra. We describe the Lie algebra of geometric vector fields in several cases, including classifying stacks, quotient stacks of regular Lie groupoids and in particular orbifolds, and foliation groupoids.

Algebraic field theory operads and linear quantization

We generalize the operadic approach to algebraic quantum field theory (arXiv:1709.08657) to a broader class of field theories whose observables on a spacetime are algebras over any single-colored operad. A novel feature of our framework is that it gives rise to adjunctions between different types of field theories. As an interesting example, we study an adjunction whose left adjoint describes the quantization of linear field theories. We also analyze homotopical properties of the linear quantization adjunction for chain complex valued field theories, which leads to a homotopically meaningful quantization prescription for linear gauge theories.

Schur correlation functions on S3 \xd7 S1

The Schur limit of the superconformal index of four-dimensional mathcal{N} = 2 superconformal field theories has been shown to equal the supercharacter of the vacuum module of their associated chiral algebra. Applying localization techniques to the theory suitably put on S3 × S1, we obtain a direct derivation of this fact. We also show that the localization computation can be extended to calculate correlation functions of a subset of local operators, namely of the so-called Schur operators. Such correlators correspond to insertions of chiral algebra fields in the trace-formula computing the supercharacter. As a by-product of our analysis, we show that the standard lore in the localization literature stating that only off-shell supersymmetrically closed observables are amenable to localization, is incomplete, and we demonstrate how insertions of fermionic operators can be incorporated in the computation.

Affine Killing vector fields on homogeneous surfaces with torsion

Many extensions of general relativity are based on considering metric and affine structures as independent properties of spacetime. This leads to the possibility of introducing torsion as an independent degree of freedom. In this article we examine the effects of torsion on the affine Killing vectors of two-dimensional manifolds. We give a complete description of the Lie algebras of affine Killing vector fields on homogeneous surfaces. This can be used in the search of non-metrizable surfaces of interest.

Coupled Sasaki-Ricci solitons

Motivated by the study of coupled K\"ahler-Einstein metrics by Hultgren and Witt Nystr\"om and coupled K\"ahler-Ricci solitons by Hultgren, we study in this paper coupled Sasaki-Einstein metrics and coupled Sasaki-Ricci solitons. We first show an isomorphism between the Lie algebra of all transverse holomorphic vector fields and certain space of coupled basic functions related to coupled twisted Laplacians for basic functions, and obtain extensions of the well-known obstructions to the existence of K\"ahler-Einstein metrics to this coupled case. These results are reduced to coupled K\"ahler-Einstein metrics when the Sasaki structure is regular. Secondly we show the existence of toric coupled Sasaki-Einstein metrics when the basic first Chern class is positive extending the work of Hultgren.

Minoration de la hauteur de Weil dans un compositum de corps de rayon

The study of Property (B) starts as a special case of Lehmer's conjecture. An algebraic field is said to satisfy Property (B) if there exists a positive constant bounding by below the height of every point of infinite order.In this paper we prove that, under certain uniformity conditions, the compositum of a familly of fields satisfies Property (B).

Holomorphic vector fields and Hodge numbers of compact pseudo-Kähler nilmanifolds

In this paper, we consider relations of the Lie algebra of holomorphic vector fields and Hodge numbers of a compact pseudo-Kähler nilmanifold.

Isomorphism and classification for countable structures

We introduce a topology on the space of all isomorphism types represented in a given class of countable models, and use this topology as an aid in classifying the isomorphism types. This mixes ideas from effective descriptive set theory and computable structure theory, extending concepts from the latter beyond computable structures to examine the isomorphism problem on arbitrary countable structures. We give examples using specific classes of fields and of trees, illustrating how the new concepts can yield classifications that reveal differences between seemingly similar classes. Finally, we use a computable homeomorphism to define a measure on the space of isomorphism types of algebraic fields, and examine the prevalence of relative computable categoricity under this measure.

Steps Towards Redesigning Cryptosystems by a Non-associative Algebra of IP-Loops

In block ciphers, the substitution boxes (S-boxes) are used with the purpose to induce confusion in cryptosystems. For the last three decades most of algebraic and chaotic S-boxes are based on the associative algebras of Galois fields. However, recently a little but remarkable work on the designing of S-boxes over some local associative algebras have been published, which is in fact a mock of the associative algebras of Galois fields. The main purpose of these establishments is to hide data and improve the security levels of crypto algorithms. In this study we redesign S-boxes over the elements of inverse property loop. The fascinating features of this structure are; it’s non-associativity and the existence of the inverse of zero element. These properties not only increase the availability of the number of structures but also provide easy approach in constructing S-boxes. The newly designed S-boxes are assessed by various analyses, which include nonlinearity, strict avalanche criterion, bit independence criterion, linear approximation probabilities, and differential approximation probabilities. Majority logic criterion is also performed to evaluate its application in various encryption systems.

Toward a modular precision ecosystem for high-performance computing

With the memory bandwidth of current computer architectures being significantly slower than the (floating point) arithmetic performance, many scientific computations only leverage a fraction of the computational power in today’s high-performance architectures. At the same time, memory operations are the primary energy consumer of modern architectures, heavily impacting the resource cost of large-scale applications and the battery life of mobile devices. This article tackles this mismatch between floating point arithmetic throughput and memory bandwidth by advocating a disruptive paradigm change with respect to how data are stored and processed in scientific applications. Concretely, the goal is to radically decouple the data storage format from the processing format and, ultimately, design a “modular precision ecosystem” that allows for more flexibility in terms of customized data access. For memory-bounded scientific applications, dynamically adapting the memory precision to the numerical requirements allows for attractive resource savings. In this article, we demonstrate the potential of employing a modular precision ecosystem for the block-Jacobi preconditioner and the PageRank algorithm—two applications that are popular in the communities and at the same characteristic representatives for the field of numerical linear algebra and data analytics, respectively.

Integrable generators of Lie algebras of vector fields on ℂ n

Abstract There exist three vector fields with complete polynomial flows on ℂ n {\mathbb{C}^{n}} , n ≥ 2 {n\geq 2} , which generate the Lie algebra generated by all algebraic vector fields on ℂ n {\mathbb{C}^{n}} with complete polynomial flows. In particular, the flows of these vector fields generate a group that acts infinitely transitively. The analogous result holds in the holomorphic setting.

Children’s Capacity for Algebraic Thinking in the Early Grades

Patterns pervade our everyday life and are natural and familiar to children. Exciting new research in the field of Early Algebra describes the development of young children’s algebraic thinking through the meaningful study of patterns. This paper contributes to this research by reporting on a case study of two primary schools in South Africa and looking at the capacity of learners for algebraic thinking in the early grades. Data was gathered using Grade 3 learner tests and focus group interviews based on pattern tasks. The data analysis process examined learners’ pattern matching and building explanations related to each data collection instrument using Blanton, Brizuela, Gardiner, Sawrey and Newman-Owens’s (2015. A learning trajectory in 6-year-old’s thinking about generalising functional relationships. Journal for Research in Mathematics Education, 46, 511–558) levels of sophistication in learner’s thinking about functional relationships. The results of the study suggest that young learners have the potential to think algebraically when offered opportunities to do so, particularly in terms of recursive and functional thinking when solving pattern problems. Traditionally instructional practice emphasises lengthy exposure to recursive thinking as a first route to algebraic thinking: this recent research suggests that young children also have the capacity to engage in functional thinking. Thus, we may need to rethink our approach to algebraic thinking through the development of a pattern-based approach to algebra.

Multisymplectic structures and invariant tensors for Lie systems

A Lie system is a non-autonomous system of differential equations describing the integral curves of a non-autonomous vector field taking values in a finite-dimensional Lie algebra of vector fields, a so-called Vessiot–Guldberg Lie algebra. This work pioneers the analysis of Lie systems admitting a Vessiot–Guldberg Lie algebra of Hamiltonian vector fields relative to a multisymplectic structure: the multisymplectic Lie systems. Geometric methods are developed to consider a Lie system as a multisymplectic one. By attaching a multisymplectic Lie system via its multisymplectic structure with a tensor coalgebra, we find multisymplectic and coalgebra methods to derive superposition rules, constants of motion, and invariant tensor fields relative to its evolution. Our results are illustrated with examples occurring in physics and mathematics such as Schwarz equations, control systems, or diffusion-type equations. This also shows the interest of multisymplectic techniques in ordinary differential equations, which represents a not much investigated field of application of multisymplectic geometry.

Coloring Operads for Algebraic Field Theory

AbstractIn these proceedings we summarize previous work where we formalize a general concept of algebraic field theories using operads. After giving a gentle reminder of algebraic quantum field theory, operads and their algebras, we construct field theory operads, whose algebras are exactly algebraic field theories. Specifically, they satisfy a suitable version of the Einstein causality axiom. From this construction we get adjunctions between different types of field theories, including adjunctions related to local‐to‐global extensions and the time‐slice axiom, and a quantization functor for linear field theories that is compatible with these structures. We also take first steps towards a derived linear quantization functor.

Geometric realization of algebraic conformal field theories

We explore new connections between the fields and local observables in two dimensional chiral conformal field theory. We show that in a broad class of examples, the von Neumann algebras of local observables (a conformal net) can be obtained from the fields (a unitary vertex operator algebra) via a continuous geometric interpolation procedure involving Graeme Segal's functorial definition of conformal field theory, and that the conformal net may be thought of as a boundary value of the Segal CFT. In particular, we construct conformal nets from these unitary vertex operator algebras by showing that ‘geometrically mollified’ versions of the fields yield bounded, local observables on the Hilbert space completion of the vertex algebra. These are the first results which unite the three major definitions of chiral conformal field theory. This work is inspired by Henriques' picture of conformal nets arising from degenerate Riemann surfaces.

Interactions between Algebraic Geometry and Noncommutative Algebra

The workshop presented current developments exploring the boundaries and intersections between the fields of noncommutative algebra and representation theory on the one hand, and algebraic geometry on the other hand, with particular emphasis on topics where such interactions have led to substantial recent progress.