45 research outputs found

    Modified Theories of Gravity and Cosmological Applications

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    This reprint focuses on recent aspects of gravitational theory and cosmology. It contains subjects of particular interest for modified gravity theories and applications to cosmology, special attention is given to Einstein–Gauss–Bonnet, f(R)-gravity, anisotropic inflation, extra dimension theories of gravity, black holes, dark energy, Palatini gravity, anisotropic spacetime, Einstein–Finsler gravity, off-diagonal cosmological solutions, Hawking-temperature and scalar-tensor-vector theories

    Classification and computational search for planar functions in characteristic 3

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    Masteroppgave i informatikkINF399MAMN-PROGMAMN-IN

    Collected Papers (on Neutrosophic Theory and Its Applications in Algebra), Volume IX

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    This ninth volume of Collected Papers includes 87 papers comprising 982 pages on Neutrosophic Theory and its applications in Algebra, written between 2014-2022 by the author alone or in collaboration with the following 81 co-authors (alphabetically ordered) from 19 countries: E.O. Adeleke, A.A.A. Agboola, Ahmed B. Al-Nafee, Ahmed Mostafa Khalil, Akbar Rezaei, S.A. Akinleye, Ali Hassan, Mumtaz Ali, Rajab Ali Borzooei , Assia Bakali, Cenap Özel, Victor Christianto, Chunxin Bo, Rakhal Das, Bijan Davvaz, R. Dhavaseelan, B. Elavarasan, Fahad Alsharari, T. Gharibah, Hina Gulzar, Hashem Bordbar, Le Hoang Son, Emmanuel Ilojide, Tèmítópé Gbóláhàn Jaíyéolá, M. Karthika, Ilanthenral Kandasamy, W.B. Vasantha Kandasamy, Huma Khan, Madad Khan, Mohsin Khan, Hee Sik Kim, Seon Jeong Kim, Valeri Kromov, R. M. Latif, Madeleine Al-Tahan, Mehmat Ali Ozturk, Minghao Hu, S. Mirvakili, Mohammad Abobala, Mohammad Hamidi, Mohammed Abdel-Sattar, Mohammed A. Al Shumrani, Mohamed Talea, Muhammad Akram, Muhammad Aslam, Muhammad Aslam Malik, Muhammad Gulistan, Muhammad Shabir, G. Muhiuddin, Memudu Olaposi Olatinwo, Osman Anis, Choonkil Park, M. Parimala, Ping Li, K. Porselvi, D. Preethi, S. Rajareega, N. Rajesh, Udhayakumar Ramalingam, Riad K. Al-Hamido, Yaser Saber, Arsham Borumand Saeid, Saeid Jafari, Said Broumi, A.A. Salama, Ganeshsree Selvachandran, Songtao Shao, Seok-Zun Song, Tahsin Oner, M. Mohseni Takallo, Binod Chandra Tripathy, Tugce Katican, J. Vimala, Xiaohong Zhang, Xiaoyan Mao, Xiaoying Wu, Xingliang Liang, Xin Zhou, Yingcang Ma, Young Bae Jun, Juanjuan Zhang

    On the right nucleus of Petit algebras

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    Let D be division algebra over its center C, let σ be an endormorphism of D, let δ be a left σ-derivation of D, and let R=D[t; σ, δ] be a skew polynomial ring. We study the structure of a class of nonassociative algebras, denoted by Sf, whose construction canonically generalises that of the associative quotient algebras R/Rf where f ∈ R is right-invariant. We determine the structure of the right nucleus of Sf when the polynomial f is bounded and not right invariant and either δ = 0, or σ = idD. As a by-product, we obtain a new proof on the size of the right nuclei of the cyclic (Petit) semifields Sf. We look at subalgebras of the right nucleus of Sf, generalising several of Petit's results [Pet66] and introduce the notion of semi-invariant elements of the coefficient ring D. The set of semi-invariant elements is shown to be equal to the nucleus of Sf when f is not right-invariant. Moreover, we compute the right nucleus of Sf for certain f. In the final chapter of this thesis we introduce and study a special class of polynomials in R called generalised A-polynomials. In a differential polynomial ring over a field of characteristic zero, A-polynomials were originally introduced by Amitsur [Ami54]. We find examples of polynomials whose eigenring is a central simple algebra over the field C ∩ Fix(σ) ∩ Const(δ)

    New classes of nonassociative divison algebras and MRD codes

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    In the first part of the thesis, we generalize a construction by J Sheekey that employs skew polynomials to obtain new nonassociative division algebras and maximum rank distance (MRD) codes. This construction contains Albert’s twisted fields as special cases. As a byproduct, we obtain a class of nonassociative real division algebras of dimension four which has not been described in the literature so far in this form. We also obtain new MRD codes. In the second part of the thesis, we study a general doubling process (similar to the one that can be used to construct the complex numbers from pairs of real numbers) to obtain new non-unital nonassociative algebras, starting with cyclic algebras. We investigate the automorphism groups of these algebras and when they are division algebras. In particular, we obtain a generalization of Dickson’s commutative semifields. We are using methods from nonassociative algebra throughout

    Notes in Pure Mathematics & Mathematical Structures in Physics

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    These Notes deal with various areas of mathematics, and seek reciprocal combinations, explore mutual relations, ranging from abstract objects to problems in physics.Comment: Small improvements and addition

    The eigenspaces of twisted polynomials over cyclic field extensions

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    Let KK be a field and σ\sigma an automorphism of KK of order nn. We study the eigenspace of a bounded skew polynomial f∈K[t;σ]f\in K[t;\sigma], with emphasis on the case of a cyclic field extension K/FK/F of degree nn, where σ\sigma generates the Galois group. We obtain lower bounds on its dimension, and compute it in special cases.Comment: Rewritten and streamlined new version, some results are improve

    Quaternion Algebras

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    This open access textbook presents a comprehensive treatment of the arithmetic theory of quaternion algebras and orders, a subject with applications in diverse areas of mathematics. Written to be accessible and approachable to the graduate student reader, this text collects and synthesizes results from across the literature. Numerous pathways offer explorations in many different directions, while the unified treatment makes this book an essential reference for students and researchers alike. Divided into five parts, the book begins with a basic introduction to the noncommutative algebra underlying the theory of quaternion algebras over fields, including the relationship to quadratic forms. An in-depth exploration of the arithmetic of quaternion algebras and orders follows. The third part considers analytic aspects, starting with zeta functions and then passing to an idelic approach, offering a pathway from local to global that includes strong approximation. Applications of unit groups of quaternion orders to hyperbolic geometry and low-dimensional topology follow, relating geometric and topological properties to arithmetic invariants. Arithmetic geometry completes the volume, including quaternionic aspects of modular forms, supersingular elliptic curves, and the moduli of QM abelian surfaces. Quaternion Algebras encompasses a vast wealth of knowledge at the intersection of many fields. Graduate students interested in algebra, geometry, and number theory will appreciate the many avenues and connections to be explored. Instructors will find numerous options for constructing introductory and advanced courses, while researchers will value the all-embracing treatment. Readers are assumed to have some familiarity with algebraic number theory and commutative algebra, as well as the fundamentals of linear algebra, topology, and complex analysis. More advanced topics call upon additional background, as noted, though essential concepts and motivation are recapped throughout

    Discrete Mathematics and Symmetry

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    Some of the most beautiful studies in Mathematics are related to Symmetry and Geometry. For this reason, we select here some contributions about such aspects and Discrete Geometry. As we know, Symmetry in a system means invariance of its elements under conditions of transformations. When we consider network structures, symmetry means invariance of adjacency of nodes under the permutations of node set. The graph isomorphism is an equivalence relation on the set of graphs. Therefore, it partitions the class of all graphs into equivalence classes. The underlying idea of isomorphism is that some objects have the same structure if we omit the individual character of their components. A set of graphs isomorphic to each other is denominated as an isomorphism class of graphs. The automorphism of a graph will be an isomorphism from G onto itself. The family of all automorphisms of a graph G is a permutation group

    Solvable crossed product algebras revisited

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    For any central simple algebra over a field F which contains a maximal subfield M with non-trivial automorphism group G = AutF (M), G is solvable if and only if the algebra contains a finite chain of subalgebras which are generalized cyclic algebras over their centers (field extensions of F) satisfying certain conditions. These subalgebras are related to a normal subseries of G. A crossed product algebra F is hence solvable if and only if it can be constructed out of such a finite chain of subalgebras. This result was stated for division crossed product algebras by Petit, and overlaps with a similar result by Albert which, however, was not explicitly stated in these terms. In particular, every solvable crossed product division algebra is a generalized cyclic algebra over F
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