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Some new families of definite polynomials and the composition conjectures | ||
| Computational Methods for Differential Equations | ||
| مقاله 3، دوره 5، شماره 3، مهر 2017، صفحه 214-223 اصل مقاله (345.95 K) | ||
| نوع مقاله: Research Paper | ||
| نویسندگان | ||
| Razie Shafeii Lashkarian* 1؛ Dariush Behmardi Sharifabad2 | ||
| 1Department of Basic science, Hashtgerd Branch, Islamic Azad University, Alborz, Iran | ||
| 2Department of Mathematics, Alzahra university, Tehran, Iran | ||
| چکیده | ||
| The planar polynomial vector fields with a center at the origin can be written as an scalar differential equation, for example Abel equation. If the coefficients of an Abel equation satisfy the composition condition, then the Abel equation has a center at the origin. Also the composition condition is sufficient for vanishing the first order moments of the coefficients. The composition conjecture and the moment vanishing problem ask for that the composition condition is a necessary condition to have the center or vanishing the moments. It is not known that if there exist examples of polynomials that satisfy the double moment conditions but don't satisfy the composition condition. In this paper we consider some composition conjectures and give some families of definite polynomials for which vanishing of the moments and the composition condition are equivalent. Our methods are based on a decomposition method for continuous functions. We give an orthogonal basis for the family of continuous functions and study the conjecture in terms of this decomposition. | ||
| کلیدواژهها | ||
| Abel equation؛ composition condition؛ composition conjecture؛ definite polynomial؛ moment | ||
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