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Analysis and Mathematical Physics, 2023
The paper investigates two types of real trigonometric polynomial equations: \[ A(\theta)y'=B_1(\theta)+B_n(\theta)y^n \] and \[ A(\theta)y^{n-1}y'=B_1(\theta)+B_n(\theta)y^n \] The authors focus on the first equation and demonstrate that when $n\geq 4$, it has a maximum of 3 real trigonometric polynomial solutions if $n$ is even and 5 real ...
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The paper investigates two types of real trigonometric polynomial equations: \[ A(\theta)y'=B_1(\theta)+B_n(\theta)y^n \] and \[ A(\theta)y^{n-1}y'=B_1(\theta)+B_n(\theta)y^n \] The authors focus on the first equation and demonstrate that when $n\geq 4$, it has a maximum of 3 real trigonometric polynomial solutions if $n$ is even and 5 real ...
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Polynomial solutions to piezoelectric beams (I)—Several exact solutions
Applied Mathematics and Mechanics, 2005zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Ding, Hao-Jiang, Jiang, Ai-Min
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Solutions to Problems: Factorization of Polynomials
2021In this chapter, the problems of the ninth chapter are fully solved, in detail, step-by-step, and with different methods.
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Orthonormal Polynomials in Wavefront Analysis: Analytical Solution
Frontiers in Optics, 2006Zernike circle polynomials are in widespread use for wavefront analysis because of their orthogonality over a circular pupil and their representation of balanced classical aberrations. In recent papers, we derived closed-form polynomials that are orthonormal over a hexagonal pupil, such as the hexagonal segments of a large mirror. We extend our work to
Virendra N, Mahajan, Guang-ming, Dai
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FIXED DEGREE SOLUTIONS OF POLYNOMIAL EQUATIONS
IFAC Proceedings Volumes, 1992Abstract Given the linear polynomial equation AX+ BY=C we study the existence of solutionpairs X, Y having prespecified degrees, and parametrize all such pairs. This result proves useful in the analysis and design of linear control systems.
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Constant solutions of polynomial equations
International Journal of Control, 1991Abstract A necessary and sufficient condition is given for the equation AX + BY = C in polynomial matrices to have a constant solution pair X, Y and also for X to be non-singular. A sufficient condition is then established under which the equation with A and B fixed has a constant solution for each C from a given class. Applications to the construction
VLADIMÍR KUČERA, PETR ZAGALAK
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NUMERICAL SOLUTIONS OF POLYNOMIAL EQUATIONS
1968Abstract : The report describes the FORTRAN Subroutine POLYR and a related complete program BRL-RSSR for finding all roots (real and complex) of a real polynomial equation P(x) = Summation from i = o to i = N of ((A sub i)(x to the power (N-i)))=0. The method which is used combines the root squaring and the subresultant (extracting quadratic factors ...
Henry Wisniewski, Tadeusz Leser
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Minimal degree solutions of polynomial equations
1987Consider the general Bézout equation of the form \(A_ 1X_ 1+...+A_ rX_ r=C\) where C and the \(A_ i\) are from a polynomial ring R, and we are looking for a solution for the unknowns \(X_ i\) in the same ring. The case where R is the ring of polynomials in two variables over the real or complex field and \(C=1\) arises in multidimensional systems and ...
GENTILI, GRAZIANO, D. STRUPPA
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Polynomial Solutions of Transition Curves
Journal of Surveying Engineering, 2011The solutions of transition curves presented in this paper are new geometric solutions that can be used in various tasks related to road designing. Their basic advantage is that they form groups of transition curves. This paper is concerned with transition curves with the classical curvature diagram and so-called general transition curves. These can be
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Orthogonality of the Polynomial Solutions
2010In this section we consider the possible orthogonality of polynomials satisfying a three-term recurrence relation of the form (2.5.1). Hereby we use Favard’s theorem (see for instance (Chihara in An Introduction to Orthogonal Polynomials. Gordon and Breach, New York, 1978)):
Roelof Koekoek +2 more
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