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Number Theory and Cryptographic Hardness Assumptions
Published in Jonathan Katz, Yehuda Lindell, Introduction to Modern Cryptography, 2020
At this point, it may be helpful to see some examples. Example 9.10A set may be a group under one operation, but not another. For example, the set of integers ℤ is an abelian group under addition: the identity is the element 0, and every integer g has inverse −g. On the other hand, it is not a group under multiplication since, for example, the integer 2 does not have a multiplicative inverse in the integers. ◊Example 9.11The set of real numbers ℝ is not a group under multiplication, since 0 does not have a multiplicative inverse. The set of nonzero real numbers, however, is an abelian group under multiplication with identity 1.◊
Mathematical Foundations
Published in Chintan Patel, Nishant Doshi, Internet of Things Security, 2018
A group is called abelian group or commutative group if it satisfies the commutative property also. An abelian group is denoted by (S, △). So if a, b, c, a−1 ∈ , i is the identity of abelian group then a△b ∈ (Closure property)a△b = b△a (Commutative)a△(b△c) = (a△b)△c (Associative property)a△i = a (Identity element)a△a−1 = i (Inverse element)
Elements of Galois Fields
Published in Erchin Serpedin, Thomas Chen, Dinesh Rajan, Mathematical Foundations for SIGNAL PROCESSING, COMMUNICATIONS, AND NETWORKING, 2012
Examples of groups include the set of integers Z under integer addition (which is an abelian group), and the set of all non-zero real numbers R\{0} under multiplication (also an abelian group). Nonsingular (square) matrices with real elements under matrix multiplication form a non-abelian group as matrix multiplication does not commute. On the other hand, the set of non-zero integers under multiplication or the set of non-negative integers under addition do not form a group (in both cases not all elements of the set have an inverse).
LDPC Codes Based on Rational Functions
Published in IETE Journal of Research, 2021
Mohammad Gholami, Akram Nassaj
Define . Clearly, is a non-abelian group with the following product operation: Now, for a given exponent matrix E of a RLDPC code with the elements belong to , by an extension of E, we mean the exponent matrix derived by multiplying the rows or columns of E by some elements of . By Lemma 3.1, it can be seen easily that the RLDPC code with the exponent matrix E is equivalent with an APM-LDPC code if and only if there is an extension of E, say , such that all of the elements of belong to the set . In continue, type-I and type-II RLDPC codes are precedented which are and are not equivalent with APM-LDPC codes,respectively.
Geometry of symplectic partially hyperbolic automorphisms on 4-torus
Published in Dynamical Systems, 2020
Consider standard torus as the factor group of the abelian group with respect to its discrete subgroup of integer vectors. Denote the related group homomorphism being simultaneously a smooth covering map. The coordinates in the space will be denoted by Let A be an unimodular matrix with integer entries. Since the linear map , generated by the matrix A, transforms the subgroup onto itself, such a matrix generates diffeomorphism of the torus called the automorphism of the torus [1,2,11]. Topological properties of such maps are the classical object of research (see, for example, [1,11,18,23]). Because the torus automorphism also preserves the standard volume element on the torus carried over from , then its ergodic properties have also been the subject of research in many works [5,12,26]. The following classical Halmos theorem holds for automorphisms of a torus [12].
Topological machine learning for multivariate time series
Published in Journal of Experimental & Theoretical Artificial Intelligence, 2022
Chengyuan Wu, Carol Anne Hargreaves
The th chain group of a simplicial complex is the free abelian group with basis the set of oriented -simplices. The boundary operator is defined on an oriented simplex by