scholarly journals A solver for General Unilateral Polynomial Matrix Equation with Second-Order Matrices Over Prime Finite Fields

2018 ◽  
Vol 973 ◽  
pp. 012067
Author(s):  
Filipp Burtyka
Keyword(s):  
Finite Fields ◽  
Matrix Equation ◽  
Polynomial Matrix ◽  
Second Order

scholarly journals A relativistic basis of the quantum theory.—II

1934 ◽  
Vol 145 (855) ◽  
pp. 645-656 ◽  
Keyword(s):  
Quantum Mechanics ◽  
Wave Equation ◽  
Quantum Theory ◽  
General Expression ◽  
Matrix Equation ◽  
Covariant Derivative ◽  
Riemannian Geometry ◽  
Order Equation ◽  
Second Order ◽  
Space Curvature

The paper is a continuation of the last paper communicated to these 'Proceedings.' In that paper, which we shall refer to as the first paper, a more general expression for space curvature was obtained than that which occurs in Riemannian geometry, by a modification of the Riemannian covariant derivative and by the use of a fifth co-ordinate. By means of a particular substitution (∆ μσ σ = 1/ψ ∂ψ/∂x μ ) it was shown that this curvature takes the form of the second order equation of quantum mechanics. It is not a matrix equation, however but one which has the character of the wave equation as it occurred in the earlier form of the quantum theory. But it contains additional terms, all of which can be readily accounted for in physics, expect on which suggested an identification with energy of the spin.

scholarly journals On the Parametric Solution to the Second-Order Sylvester Matrix EquationEVF2−AVF−CV=BW

2007 ◽  
Vol 2007 ◽  
pp. 1-16 ◽  
Author(s):  
Wang Guo-Sheng ◽  
Lv Qiang ◽  
Duan Guang-Ren
Keyword(s):  
Control Systems ◽  
Matrix Equation ◽  
Second Order ◽  
Diagonal Form ◽  
Sylvester Matrix Equation ◽  
Parametric Solution ◽  
Design Problems ◽  
Analysis And Design ◽  
Sylvester Matrix ◽  
The Matrix

This paper considers the solution to a class of the second-order Sylvester matrix equationEVF2−AVF−CV=BW. Under the controllability of the matrix triple(E,A,B), a complete, general, and explicit parametric solution to the second-order Sylvester matrix equation, with the matrixFin a diagonal form, is proposed. The results provide great convenience to the analysis of the solution to the second-order Sylvester matrix equation, and can perform important functions in many analysis and design problems in control systems theory. As a demonstration, an illustrative example is given to show the effectiveness of the proposed solution.

scholarly journals Solvability Theory and Iteration Method for One Self-Adjoint Polynomial Matrix Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Zhigang Jia ◽  
Meixiang Zhao ◽  
Minghui Wang ◽  
Sitao Ling
Keyword(s):  
Iterative Algorithm ◽  
Perturbation Analysis ◽  
Matrix Equation ◽  
Iteration Method ◽  
Numerical Experiments ◽  
Polynomial Matrix ◽  
Sufficient Conditions ◽  
Positive Definite ◽  
Solvability Theory ◽  
Adjoint Polynomial

The solvability theory of an important self-adjoint polynomial matrix equation is presented, including the boundary of its Hermitian positive definite (HPD) solution and some sufficient conditions under which the (unique or maximal) HPD solution exists. The algebraic perturbation analysis is also given with respect to the perturbation of coefficient matrices. An efficient general iterative algorithm for the maximal or unique HPD solution is designed and tested by numerical experiments.

SOME FOURTH-ORDER LINEAR DIVISIBILITY SEQUENCES

2011 ◽  
Vol 07 (05) ◽  
pp. 1255-1277 ◽  
Author(s):  
H. C. WILLIAMS ◽  
R. K. GUY
Keyword(s):  
Elliptic Curves ◽  
Large Class ◽  
Finite Fields ◽  
Fourth Order ◽  
Spanning Trees ◽  
Perfect Matchings ◽  
Second Order ◽  
Order Linear ◽  
Curves Over Finite Fields ◽  
Open Question

We extend the Lucas–Lehmer theory for second-order divisibility sequences to a large class of fourth-order sequences, with appropriate laws of apparition and of repetition. Examples are provided by the numbers of perfect matchings, or of spanning trees, in families of graphs, and by the numbers of points on elliptic curves over finite fields. Whether there are fourth-order divisibility sequences not covered by our theory is an open question.

Linear Recurring Sequences and Explicit Factors of x2nd−1 in 𝔽q[x]

2020 ◽  
Vol 27 (03) ◽  
pp. 563-574
Author(s):  
Manjit Singh
Keyword(s):  
Field Theory ◽  
Positive Integer ◽  
Finite Field ◽  
Finite Fields ◽  
Second Order ◽  
Order Linear ◽  
Linear Recurring Sequences

Let 𝔽q be a finite field of odd characteristic containing q elements, and n be a positive integer. An important problem in finite field theory is to factorize xn − 1 into the product of irreducible factors over a finite field. Beyond the realm of theoretical needs, the availability of coefficients of irreducible factors over finite fields is also very important for applications. In this paper, we introduce second order linear recurring sequences in 𝔽q and reformulate the explicit factorization of [Formula: see text] over 𝔽q in such a way that the coefficients of its irreducible factors can be determined from these sequences when d is an odd divisor of q + 1.

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