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Ring Theory
Published in Paul L. Goethals, Natalie M. Scala, Daniel T. Bennett, Mathematics in Cyber Research, 2022
It is easy to check that the set R[x] satisfies all six conditions given in Definition 15.2 under the usual operations of polynomial addition and polynomial multiplication. Consequently, R[x] is a ring, called the polynomial ring in the variable x with coefficients from R.
Mathematical Background
Published in Alfred J. Menezes, Paul C. van Oorschot, Scott A. Vanstone, Handbook of Applied Cryptography, 2018
Alfred J. Menezes, Paul C. van Oorschot, Scott A. Vanstone
Let ℤp be the finite field of order p. The theory of greatest common divisors and the Euclidean algorithm for integers carries over in a straightforward manner to the polynomial ring ℤp[x] (and more generally to the polynomial ring F[x], where F is any field).
Data-driven optimal tracking control of discrete-time linear systems with multiple delays via the value iteration algorithm
Published in International Journal of Systems Science, 2022
Longyan Hao, Chaoli Wang, Guang Zhang, Chonglin Jing, Yibo Shi
To transform the system with time delay, we first define the delay operator as (Garate-Garcia et al., 2011), such that where . If there is a polynomial ring , then the polynomial matrix can be written in the following form: where is the polynomial degree of , and are constant coefficient matrices with an appropriate dimension. The polynomial matrix defined in (6) satisfies the following addition and multiplication:
Robustness analysis of the feedback interconnection of discrete-time negative imaginary systems via integral quadratic constraints
Published in International Journal of Control, 2021
Qian Zhang, Liu Liu, Yufeng Lu
Denote , , and denotes the set of proper real-rational function matrices defined almost everywhere in . Let be the set of all -valued bounded holomorphic proper real-rational functions on . denotes the polynomial ring with real coefficients. A square polynomial matrix is called a unimodular matrix if is a non-zero constant.
General remainder theorem and factor theorem for polynomials over non-commutative coefficient rings
Published in International Journal of Mathematical Education in Science and Technology, 2020
A. Cuida, F. Laudano, E. Martinez-Moro
In this final section we will point to some known results for evaluating and dividing on skew polynomial rings. This will show the potential of the topic all around the mathematical instruction, note that also this approach has been used actually in applied mathematics as for example in coding theory (see for example Martínez-Penas, 2018). We will rectrict ourselves to skew polynomial rings over fields with no derivation but the same results can be stated easily for general skew polynomial rings over division rings (Lam & Leroy, 1988). Let F be a field and σ an automorphism of F. The skew polynomial ring is the set of polynomials over the field F where the addition is defined as the usual one in the polynomial ring and the multiplication is defined as follows Indeed it a non-commutative ring unless σ is the identity. One can see that the Euclidean algorithm holds for right division which implies that R is a left principal ideal domain (Lam & Leroy, 1988). So following the previous sections we can evaluate a (left) polynomial on . Note that the obviuous ‘evaluation’ is wrong since it does not take into account the action of σ, but if we define whe have the following Factor Theorem