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Primitive and Almost Primitive Elements of Schreier Varieties
Journal of Mathematical Sciences, 2019According to the classical monograph by \textit{P. M. Cohn} [Universal Algebra. New York etc.: Harper and Row (1965; Zbl 0141.01002)], a variety of universal algebras is a class of all algebras that satisfy all identities from a given set of identities.
Artamonov, V. A. +3 more
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Tame Almost Primitive Elements
Results in Mathematics, 2000Let \(G\) be a finitely generated group, \(V\) a set of laws, and \(\mathcal V\) the variety defined by \(V\). An element \(\omega\in G\) is \(\mathcal V\)-generic in \(G\) if \(\omega\in V(G)\) but \(\omega\notin V(K)\) for any proper subgroup ...
Konieczny, Jochen +2 more
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2007
Let n be a positive integer. A nonzero element γ of the finite field F of order q = 2n is said to be "strongly primitive" if every element (aγ+b)/(cγ+d), with a, b, c, d in {0, 1} and ad-bc not zero, is primitive in the usual sense. We show that the number N of such strongly primitive elements is asymptotic to θθ′ ċ q where θ is the product of (1-1/p ...
Daniel Goldstein, Alfred W. Hales
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Let n be a positive integer. A nonzero element γ of the finite field F of order q = 2n is said to be "strongly primitive" if every element (aγ+b)/(cγ+d), with a, b, c, d in {0, 1} and ad-bc not zero, is primitive in the usual sense. We show that the number N of such strongly primitive elements is asymptotic to θθ′ ċ q where θ is the product of (1-1/p ...
Daniel Goldstein, Alfred W. Hales
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Primitive Elements in Symmetric Algebras
Canadian Journal of Mathematics, 1974Let-R be a commutative ring with 1, and let be the symmetric algebra of an R-module M. We regard the isomorphisms S 0(M) ≅ R and S
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Subgroups of free groups and primitive elements
Journal of Group Theory, 2010Let \(F\) be a free group on the basis \(X=\langle x_1,x_2,\dots,x_n\rangle\) and \(H\) be a finitely generated subgroup of \(F\). A question is [see \url{http://www.grouptheory.info/}, Question F39b]: Is it possible to decide if \(H\) contains a primitive element of \(F\)?
Clifford, A., Goldstein, R. Z.
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Primitive Constituent Elements of Cryptographic Protocols
2018 IEEE SmartWorld, Ubiquitous Intelligence & Computing, Advanced & Trusted Computing, Scalable Computing & Communications, Cloud & Big Data Computing, Internet of People and Smart City Innovation (SmartWorld/SCALCOM/UIC/ATC/CBDCom/IOP/SCI), 2018Many cryptographic protocols have been proposed, and many studies of them have been done. However, there is no study to identify constituent elements of cryptographic protocols that are elements of the protocols consist of. The constituent elements can be used for the basis of classification of already proposed cryptographic protocols, the basis of ...
Sho Ishibashi +3 more
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1992
Abstract You will recall that in our example of GF(16) we introduced ‘logarithms’ for non-zero field elements that could be used like conventional logarithms to convert multiplication into addition. This is certainly of practical significance, since addition modulo 15 is easily implemented on a chip, while polynomial multiplication ...
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Abstract You will recall that in our example of GF(16) we introduced ‘logarithms’ for non-zero field elements that could be used like conventional logarithms to convert multiplication into addition. This is certainly of practical significance, since addition modulo 15 is easily implemented on a chip, while polynomial multiplication ...
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1999
In the previous chapter, we introduced basic geometrical elements like points, lines, polygons, conics, etc. In this chapter we will describe how simple polyhedra like “boxes” (parallelepipedums), prisms, cylinders, pyramids, cones, and frustums, etc., are implemented in OPEN GEOMETRY. We also describe our “virtual camera.”
Georg Glaeser, Hellmuth Stachel
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In the previous chapter, we introduced basic geometrical elements like points, lines, polygons, conics, etc. In this chapter we will describe how simple polyhedra like “boxes” (parallelepipedums), prisms, cylinders, pyramids, cones, and frustums, etc., are implemented in OPEN GEOMETRY. We also describe our “virtual camera.”
Georg Glaeser, Hellmuth Stachel
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Generalized Primitive Elements
2004Let K be a field, char K ≠ 2, and let X be a finite set, X = {x1, ... , x n }. In what follows, F = F(X) denotes the free K-algebra without the unity element on the set X of free generators of one of the following varieties of algebras over a field K: the variety of all algebras, the variety of Lie algebras, varieties of color Lie superalgebras, the ...
Alexander A. Mikhalev +2 more
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PRIMITIVITY OF FINITE SEMIFIELDS WITH 64 AND 81 ELEMENTS
International Journal of Algebra and Computation, 2007A finite semifield D is a finite nonassociative ring with identity such that the set D* = D \{0} is a loop under the product. Wene conjectured in [1] that any finite semifield is either right or left primitive, i.e. D* is the set of right (or left) principal powers of an element in D. In this paper we study the primitivity of finite semifields with 64
Irvin Roy Hentzel, Ignacio F. Rúa
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