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Coding Theory
Published in Leslie Hogben, Richard Brualdi, Anne Greenbaum, Roy Mathias, Handbook of Linear Algebra, 2006
Joachim Rosenthal, Paul Weiner
Let 𝔽[x] be the polynomial ring in one indeterminate over 𝔽. Let 〈xn − 1〉 be the ideal generated by xn − 1. The quotient ring 𝔽[x]/〈xn − 1〉 is denoted Rn. The polynomials of Rn are identified with vectors in 𝔽n via a0+a1x+a2x2+⋯an−1xn−1↦[a0a1a2⋯an−1].
Algebraic Structures and Applications
Published in Richard L. Shell, Ernest L. Hall, Handbook of Industrial Automation, 2000
If I is an ideal of a ring R, then the quotient ring or factor ring is given by R/I={I+a|a∈R}with addition andmultiplication
Algebraic Structures I (Matrices, Groups, Rings, and Fields)
Published in R. Balakrishnan, Sriraman Sridharan, Discrete Mathematics, 2019
R. Balakrishnan, Sriraman Sridharan
In the above example, what are all the elements that are mapped to 3? They are all the numbers n ≡ 3(mod 5) in Z. We denote this set by [3], where [3]={…,−12,−7,−2,3,8,13,…}. We call this set the residue class modulo 5 defined by 3 in Z. Note that [3] = [8], etc. Hence Z5={[0],[1],[2],[3],[4]}, where [m] = [n] iff m ≡ n(mod 5) for any integers m and n iff m − n ∈ (5), the ideal generated by 5 in Z. The ring Z5 is often referred to as the residue class ring or quotient ring modulo the ideal (5).
Polynomial semantics for modal logics
Published in Journal of Applied Non-Classical Logics, 2019
Juan C. Agudelo-Agudelo, Santiago Echeverri-Valencia
The ring of boolean polynomials on, denoted by , is the quotient ring , where . Defining as being the set , it is easy to check that is a Gröbner basis for , with respect to the graded lexicographic order. Thus, we can identify elements of with their canonical representatives module , which are the polynomials in whose variables have exponents at most 1. The ring of boolean polynomials onX, denoted by , can then be seen as the union of all (i.e. and the addition and the product of two polynomials are defined as the respective operations in , where n is the least positive natural number such that ).
Fully homomorphic encryption: a general framework and implementations
Published in International Journal of Parallel, Emergent and Distributed Systems, 2020
Let A be an Euclidean domain having unique quotients and remainders, and the secret encryption key m be a fixed element of A. The latter is known only to the owner of the data X and stored safely locally. Then m generates the ideal Am consisting of all multiples with and therefore the quotient ring A/Am consisting of all residual classes of elements of A modulo m.
On the uniform algebraic observability of multi-switching linear systems
Published in International Journal of Control, 2021
Laura Menini, Corrado Possieri, Antonio Tornambè
The results given in this paper can be extended to the case of linear systems having time-varying and switching dynamical matrices. Namely, consider the system where and are time-varying polynomial matrices. Thus, define for . Hence, by using the tools detailed in Section 4, define the ideal and let . Hence, coercing into , define the quotient ring , and let be the field of the rational functions in . Thus, fix as ambient ring, coerce into and consider the ideals By the reasoning given in Section 4, these ideals are principal, i.e. there exist non-constant polynomials such that , . If , then such functions can be readily used to estimate the state of system (21); see Menini et al. (2019) for further details. However, in this case, it is not easy to find a lower bound on L that guarantees ‘generic’ existence of linear 's.