An example concerning infinite factorizations of transcendental entire functions

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Permutable entire functions and their Julia sets

In 1922-23, Julia and Fatou proved that any 2 rational functions f and g of degree at least 2 such that f(g(z)) = g(f(z)), have the same Julia set. Baker then asked whether the result remains true for nonlinear entire functions. In this paper, we shall show that the answer to Baker's question is true for almost all nonlinear entire functions. The method we use is useful for solving functional equations. It actually allows us to find out all the entire functions g which permute with a given f which belongs to a very large class of entire functions. © 2001 Cambridge Philosophical Society.published_or_final_versio

On Briot-Bouquet differential equations

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Fermat functional equations revisited

Invited Speakerpublished_or_final_versio

A companion matrix approach to the study of zeros and critical points of a polynomial

In this paper, we introduce a new type of companion matrices, namely, D-companion matrices. By using these D-companion matrices, we are able to apply matrix theory directly to study the geometrical relation between the zeros and critical points of a polynomial. In fact, this new approach will allow us to prove quite a number of new as well as known results on this topic. For example, we prove some results on the majorization of the critical points of a polynomial by its zeros. In particular, we give a different proof of a recent result of Gerhard Schmeisser on this topic. The same method allows us to prove a higher order Schoenberg-type conjecture proposed by M.G. de Bruin and A. Sharma. © 2005 Elsevier Inc. All rights reserved.preprin

Periodic solutions of a derivative nonlinear Schrödinger equation: Elliptic integrals of the third kind

International Conference on Engineering and Computational Mathematics, The Hong Kong Polytechnic University, Hong Kong, 27–29 May 2009The nonlinear Schrödinger equation (NLSE) is an important model for wave packet dynamics in hydrodynamics, optics, plasma physics and many other physical disciplines. The 'derivative' NLSE family usually arises when further nonlinear effects must be incorporated. The periodic solutions of one such member, the Chen-Lee-Liu equation, are studied. More precisely, the complex envelope is separated into the absolute value and the phase. The absolute value is solved in terms of a polynomial in elliptic functions while the phase is expressed in terms of elliptic integrals of the third kind. The exact periodicity condition will imply that only a countable set of elliptic function moduli is allowed. This feature contrasts sharply with other periodic solutions of envelope equations, where a continuous range of elliptic function moduli is permitted. © 2011 Elsevier B.V. All rights reserved.postprin

Finiteness of fixed equilibrium configurations of point vortices in the plane with a background flow

For a dynamic system consisting of n point vortices in an ideal plane fluid with a steady, incompressible and irrotational background flow, a more physically significant definition of a fixed equilibrium configuration is suggested. Under this new definition, if the complex polynomial w that determines the aforesaid background flow is non-constant, we have found an attainable generic upper bound $\frac{(m+n-1)!}{(m-1)!\,n_1!\cdots n_{i_0}!}$ for the number of fixed equilibrium configurations. Here, m = deg w, i0 is the number of species, and each ni is the number of vortices in a species. We transform the rational function system arising from fixed equilibria into a polynomial system, whose form is good enough to apply the BKK theory (named after Bernshtein (1975 Funct. Anal. Appl. 9 183–5), Khovanskii (1978 Funct. Anal. Appl. 12 38–46) and Kushnirenko (1976 Funct. Anal. Appl. 10 233–5)) to show the finiteness of its number of solutions. Having this finiteness, the required bound follows from Bézout's theorem or the BKK root count by Li and Wang (1996 Math. Comput. 65 1477–84).postprin

Computation of the in-wheel switched reluctance motor inductance using finite element method

Author name used in this publication: K. W. E. ChengAuthor name used in this publication: X. D. XueVersion of RecordPublishe

Density estimates on composite polynomials

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Dosimetric evaluation of the interplay between LINAC movement and tumor motion in respiratory gated VMAT of lung cancer

This journal suppl. entitled: Proceedings of the American Society for Radiation Oncology 54th Annual MeetingConference Theme: Advancing Patient Care Through InnovationPURPOSE/OBJECTIVE(s): Respiratory gated radiation therapy of lung cancer helps to minimize the treated volume and hence treatment side effects. VMAT can reduce the treatment time while producing a highly conformed dose distribution. However, in gated VMAT delivery, the interplay effect between the LINAC movement (MLC and gantry) and tumor motion may result in undesirable hot and cold spots jeopardizing tumor coverage. In this study we investigated the possible dosimetric errors caused by the interplay between the tumor motion and the LINAC movement for gated VMAT lung cancer treatment. MATERIALS/METHODS: We studied 2 …published_or_final_versio