The Algebraic and Geometric Classification of Four Dimensional Superalgebras

10.26686/wgtn.16915645 ◽

2021 ◽

Author(s):

◽

Aaron Armour

Keyword(s):

Classification Problem ◽

Associative Algebras ◽

New Variety ◽

Graded Algebras ◽

Finite Representation ◽

Algebraic Classification ◽

Irreducible Components ◽

Module Algebras ◽

Sweedler’S Hopf Algebra

The algebraic and geometric classification of k-algbras, of dimension fouror less, was started by Gabriel in “Finite representation type is open” [12]. Several years later Mazzola continued in this direction with his paper “Thealgebraic and geometric classification of associative algebras of dimensionfive” [21]. The problem we attempt in this thesis, is to extend the resultsof Gabriel to the setting of super (or Z2-graded) algebras — our main effortsbeing devoted to the case of superalgebras of dimension four. Wegive an algebraic classification for superalgebras of dimension four withnon-trivial Z2-grading. By combining these results with Gabriel’s we obtaina complete algebraic classification of four dimensional superalgebras. This completes the classification of four dimensional Yetter-Drinfeld modulealgebras over Sweedler’s Hopf algebra H4 given by Chen and Zhangin “Four dimensional Yetter-Drinfeld module algebras over H4” [9]. Thegeometric classification problem leads us to define a new variety, Salgn —the variety of n-dimensional superalgebras—and study some of its properties. The geometry of Salgn is influenced by the geometry of the varietyAlgn yet it is also more complicated, an important difference being thatSalgn is disconnected. While we make significant progress on the geometricclassification of four dimensional superalgebras, it is not complete. Wediscover twenty irreducible components of Salg4 — however there couldbe up to two further irreducible components.

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Structure and classification of Hom-associative algebras

Acta et Commentationes Universitatis Tartuensis de Mathematica ◽

10.12697/acutm.2020.24.06 ◽

2020 ◽

Vol 24 (1) ◽

pp. 79-102

Author(s):

Abdenacer Makhlouf ◽

Ahmed Zahari

Keyword(s):

Deformation Theory ◽

Matrix Algebra ◽

Algebraic Varieties ◽

Associative Algebras ◽

Irreducible Components

The purpose of this paper is to study the structure and the algebraic varieties of Hom-associative algebras. We characterize multiplicative simple Hom-associative algebras and give some examples deforming the 2 × 2-matrix algebra to simple Hom-associative algebras. We provide a classification of n-dimensional Hom-associative algebras for n ≤ 3. Then we study irreducible components using deformation theory.

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Hilbert space methods in the theory of Jordan algebras. I

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100051690 ◽

1975 ◽

Vol 78 (2) ◽

pp. 293-300 ◽

Cited By ~ 7

Author(s):

C. Viola Devapakkiam

Keyword(s):

Hilbert Space ◽

Classification Problem ◽

Jordan Algebras ◽

Inner Product ◽

Uniqueness Problem ◽

Associative Algebras ◽

Infinite Dimensional

In this paper, we study the structure of certain infinite-dimensional Jordan algebras admitting an inner product. These algebras, called J*-algebras in the sequel, have already been considered in (4) in connexion with the norm uniqueness problem for non-associative algebras. We deal here with the structure and classification of these algebras. Existence of self-adjoint idempotents plays a central role in the classification problem.

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On the homomorphisms of sequences

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100076283 ◽

1952 ◽

Vol 48 (4) ◽

pp. 521-532 ◽

Cited By ~ 1

Author(s):

W. H. Cockcroft

Keyword(s):

Homotopy Theory ◽

Free Groups ◽

Algebraic Theory ◽

Classification Problem ◽

Two Dimensional ◽

Homotopy Classification ◽

Algebraic Classification ◽

Crossed Modules

The algebraic classification problem with which I shall be concerned in this paper is suggested by the algebraic theory of homomorphisms of free crossed modules whose groups of operators are free groups. This theory arises in the study of the homotopy theory of two-dimensional C.W. complexes*. It is shown in (3) that the homotopy theory of such complexes, including the homotopy classification of mappings, is equivalent to this purely algebraic theory.

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Varieties of nilpotent Lie superalgebras of dimension ≤ 5

Forum Mathematicum ◽

10.1515/forum-2019-0244 ◽

2020 ◽

Vol 32 (3) ◽

pp. 641-661 ◽

Cited By ~ 3

Author(s):

María Alejandra Alvarez ◽

Isabel Hernández

Keyword(s):

Arbitrary Dimension ◽

Lie Superalgebras ◽

Algebraic Classification ◽

Irreducible Components

AbstractIn this paper, we study the varieties of nilpotent Lie superalgebras of dimension {\leq 5}. We provide the algebraic classification of these superalgebras and obtain the irreducible components in every variety. As a byproduct, we construct rigid nilpotent Lie superalgebras of arbitrary dimension.

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An aggregate method for thorax diseases classification

Scientific Reports ◽

10.1038/s41598-021-81765-9 ◽

2021 ◽

Vol 11 (1) ◽

Author(s):

Bayu Adhi Nugroho

Keyword(s):

Network Architecture ◽

Classification Problem ◽

Calculation Algorithm ◽

Training Pattern ◽

Deep Network ◽

Network Training ◽

Chest X Ray ◽

Medical Image Classification ◽

Positive Pattern

AbstractA common problem found in real-word medical image classification is the inherent imbalance of the positive and negative patterns in the dataset where positive patterns are usually rare. Moreover, in the classification of multiple classes with neural network, a training pattern is treated as a positive pattern in one output node and negative in all the remaining output nodes. In this paper, the weights of a training pattern in the loss function are designed based not only on the number of the training patterns in the class but also on the different nodes where one of them treats this training pattern as positive and the others treat it as negative. We propose a combined approach of weights calculation algorithm for deep network training and the training optimization from the state-of-the-art deep network architecture for thorax diseases classification problem. Experimental results on the Chest X-Ray image dataset demonstrate that this new weighting scheme improves classification performances, also the training optimization from the EfficientNet improves the performance furthermore. We compare the aggregate method with several performances from the previous study of thorax diseases classifications to provide the fair comparisons against the proposed method.

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Classification of unlabeled online media

Scientific Reports ◽

10.1038/s41598-021-85608-5 ◽

2021 ◽

Vol 11 (1) ◽

Author(s):

Sakthi Kumar Arul Prakash ◽

Conrad Tucker

Keyword(s):

Social Media ◽

Real World ◽

Graphical Model ◽

Ground Truth ◽

Classification Problem ◽

Machine Learning Algorithms ◽

Social Media Networks ◽

Online Social Media ◽

Wide Range

AbstractThis work investigates the ability to classify misinformation in online social media networks in a manner that avoids the need for ground truth labels. Rather than approach the classification problem as a task for humans or machine learning algorithms, this work leverages user–user and user–media (i.e.,media likes) interactions to infer the type of information (fake vs. authentic) being spread, without needing to know the actual details of the information itself. To study the inception and evolution of user–user and user–media interactions over time, we create an experimental platform that mimics the functionality of real-world social media networks. We develop a graphical model that considers the evolution of this network topology to model the uncertainty (entropy) propagation when fake and authentic media disseminates across the network. The creation of a real-world social media network enables a wide range of hypotheses to be tested pertaining to users, their interactions with other users, and with media content. The discovery that the entropy of user–user and user–media interactions approximate fake and authentic media likes, enables us to classify fake media in an unsupervised learning manner.

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VirusTaxo: Taxonomic classification of virus genome using multi-class hierarchical classification by k-mer enrichment

10.1101/2021.04.29.442004 ◽

2021 ◽

Author(s):

Rajan Saha Raju ◽

Abdullah Al Nahid ◽

Preonath Shuvo ◽

Rashedul Islam

Keyword(s):

Genome Sequence ◽

Rna Viruses ◽

Hierarchical Classification ◽

Classification Problem ◽

Virus Genome ◽

Taxonomic Classification ◽

Dna Viruses ◽

Dna And Rna ◽

Full Length Genome

AbstractTaxonomic classification of viruses is a multi-class hierarchical classification problem, as taxonomic ranks (e.g., order, family and genus) of viruses are hierarchically structured and have multiple classes in each rank. Classification of biological sequences which are hierarchically structured with multiple classes is challenging. Here we developed a machine learning architecture, VirusTaxo, using a multi-class hierarchical classification by k-mer enrichment. VirusTaxo classifies DNA and RNA viruses to their taxonomic ranks using genome sequence. To assign taxonomic ranks, VirusTaxo extracts k-mers from genome sequence and creates bag-of-k-mers for each class in a rank. VirusTaxo uses a top-down hierarchical classification approach and accurately assigns the order, family and genus of a virus from the genome sequence. The average accuracies of VirusTaxo for DNA viruses are 99% (order), 98% (family) and 95% (genus) and for RNA viruses 97% (order), 96% (family) and 82% (genus). VirusTaxo can be used to detect taxonomy of novel viruses using full length genome or contig sequences.AvailabilityOnline version of VirusTaxo is available at https://omics-lab.com/virustaxo/.

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Fields of Definition for Representations of Associative Algebras

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091518000391 ◽

2018 ◽

Vol 62 (1) ◽

pp. 291-304

Author(s):

Dave Benson ◽

Zinovy Reichstein

Keyword(s):

Finite Extension ◽

Representation Type ◽

Essential Dimension ◽

Extension Field ◽

Associative Algebras ◽

Separable Extension ◽

Finite Representation ◽

Field Of Definition ◽

Finite Dimensional ◽

A Finite Group

AbstractWe examine situations, where representations of a finite-dimensionalF-algebraAdefined over a separable extension fieldK/F, have a unique minimal field of definition. Here the base fieldFis assumed to be a field of dimension ≼1. In particular,Fcould be a finite field ork(t) ork((t)), wherekis algebraically closed. We show that a unique minimal field of definition exists if (a)K/Fis an algebraic extension or (b)Ais of finite representation type. Moreover, in these situations the minimal field of definition is a finite extension ofF. This is not the case ifAis of infinite representation type orFfails to be of dimension ≼1. As a consequence, we compute the essential dimension of the functor of representations of a finite group, generalizing a theorem of Karpenko, Pevtsova and the second author.

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Algebraic Classification of Field Equations

The General Theory of Relativity ◽

10.1007/978-1-4614-3658-4_7 ◽

2012 ◽

pp. 465-536

Author(s):

Anadijiban Das ◽

Andrew DeBenedictis

Keyword(s):

Field Equations ◽

Algebraic Classification

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