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Register a new userRemarks to Glazeks results on n-ary groups
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Authors
Wieslaw A. Dudek
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It is a survey of the results obtained by K. Glazeks and his co-workers. We restrict our attention to the problems of axiomatizations of n-ary groups, classes of n-ary groups, properties of skew elements and homomorphisms induced by skew elements, constructions of covering groups, classifications and representations of n-ary groups. Some new results are added too.
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This note contains two remarks. The first remark concerns the extension of the well-known Cayley representation of rotation matrices by skew symmetric matrices to rotation matrices admitting -1 as an eigenvalue and then to all orthogonal matrices. We review a method due to Hermann Weyl and another method involving multiplication by a diagonal matrix whose entries are +1 or -1. The second remark has to do with ways of flipping the signs of the entries of a diagonal matrix, C, with nonzero diagonal entries, obtaining a new matrix, E, so that E + A is invertible, where A is any given matrix (invertible or not).
Dirichlet proves the general convergence of Fourier series, after pointing out errors in an earlier attempt by Cauchy. We transcribed from Crelles Journal (1829) with numerous typographical corrections, and added a completed bibliography. Dirichlet prouve la convergence generale de la series de Fourier, apr`es avoir montre des erreurs dans un essai par Cauchy. Nous avons transcrit de Crelles journal (1829) avec de nombreuses corrections typographiques, et avons ajoute une bibliographie compl`ete.
The twin group $T_n$ is a right angled Coxeter group generated by $n- 1$ involutions and having only far commutativity relations. These groups can be thought of as planar analogues of Artin braid groups. In this note, we study some properties of twin groups whose analogues are well-known for Artin braid groups. We give an algorithm for two twins to be equivalent under individual Markov moves. Further, we show that twin groups $T_n$ have $R_infty$-property and are not co-Hopfian for $n ge 3$.
In this paper, we define $omega$-derivations, and study some properties of $omega$-derivations, with its properties we can structure a new $n$-ary Hom-Nambu algebra from an $n$-ary Hom-Nambu algebra. In addition, we also give derivations and representations of $n$-ary Hom-Nambu algebras.
The growing amount of data available in modern-day datasets makes the need to efficiently search and retrieve information. To make large-scale search feasible, Distance Estimation and Subset Indexing are the main approaches. Although binary coding has been popular for implementing both techniques, n-ary coding (known as Product Quantization) is also very effective for Distance Estimation. However, their relative performance has not been studied for Subset Indexing. We investigate whether binary or n-ary coding works better under different retrieval strategies. This leads to the design of a new n-ary coding method, Linear Subspace Quantization (LSQ) which, unlike other n-ary encoders, can be used as a similarity-preserving embedding. Experiments on image retrieval show that when Distance Estimation is used, n-ary LSQ outperforms other methods. However, when Subset Indexing is applied, interestingly, binary codings are more effective and binary LSQ achieves the best accuracy.
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