Julia sets of Newton’s method for a class of complex-exponential function F(z)=P(z)e Q(z)

Abstract

In this paper, we analyze the theory of the Julia set (J set) of Newton's method, construct the Julia sets of Newton's method of function F (z) = ze z w (w ∈ C) through iteration method, and analyze the attracting region of the two fixed points 0 and ∞ when w are different values. Consequently, we draw the following conclusions: (1) When the judge conditions for the iterative algorithm are changed to |N(z n ) − z n | ≤ EOF, the properties of the figures in our experiments are contrary to the conclusions in (Wegner and Peter son, Fractal Creations, pp. 168-231, 1991); (2) The attracting regions of the fixed points 0 and ∞ for w = 2n (n = 0, ±2, ±4, . . .) are symmetrical about x-axis and y-axis; select the main argument to be in [−π, π), for arbitrary w = α (α ∈ C), the attracting regions of the fixed points 0 and ∞ are symmetrical about the x-axis;(3) The attracting regions of the two fixed points 0 and ∞ of J set for w = ±η have rotational symmetry of η times; (4) If w = −4.7, k = 0.8, then the attracting regions of different magnifications display a startling similarity, J set holds infinite self-similar structures; (5) When w is a complex number, because the selection of main argument θ z in the negative x-axis is not continuous, the fault and rupture of the attracting regions of the two fixed points 0 and ∞ appear only in the negative x-axis.

An Overview of the Insects Antifeedants

“…A presença ou ausência de atividade tem sido justificada pelas diferenças estruturais observadas. Estudos têm revelado que alguns fatores como características moleculares, substituição, ponto de ebulição, lipofilia, conformação, distribuição eletrônica, geometria, energias HOMO e LUMO, podem predizer a capacidade de "antifeedants" [53].…”

“…De acordo com Wang et al [53] os "antifeedants" mais desenvolvidos e aplicados são Terpenóides, neste sentido esses autores analisaram uma série de compostos Terpenóides, sintetizados, para verificar estatisticamente a relação entre estrutura química e atividade "antifeedants" em relação a Lipaphis erysimi (Kalt.) (Hemiptera: Aphididae) e constatou que fatores como teor de oxigênio, volume molecular, energia HOMO e LUMO e carga positiva total são fatores químicos que interferem na atividade relacionada.…”

“…A síntese química de terpenoides "antifeedants",é uma alternativaà extração, masé problemática devidoà complexidade da estrutura molecular eà presença de centros quirais que tornam sua síntese cara e demorada. Como uma possível solução, os terpenoides provenientes da planta podem ser quimicamente modificados [53,26].…”

Synthesis of Multi‐substituted Dihydropyrazoles by Copper‐Mediated [4+1] Cycloaddition Reaction of N‐Sulfonylhydrazones and Sulfoxonium Ylides

“…[2c] Very recently, Fang, Wang and co-workers developed a catalyst-free [4 + 1]-annulation of α-halo hydrazones with fluorinated sulfur ylides to produce 5-(trifluoromethyl)dihydropyrazoles. [7] In the meantime, Shao and co-workers demonstrated the substrate-controlled reactions to construct spirocyclopropylpyrazolones and bicyclic 4,5-dihydropyrazoles from 1,2-diaza-1,3-dienes and sulfur ylides. [8] In 2013, Zhang's group reported a copper mediated tandem CÀ N and NÀ N bond formation to synthesize 1,2,3-triazoles from readily available N-tosylhydrazones and amines.…”

Electronic and optical properties of pyrrole and thiophene oligomers: A density functional theory study

“…conducting polymer, density functional theory, polarizability, pyrrole, thiophene 1 | I N TR ODU C TI ON Due to their low dielectric constants, high nonresonant electric susceptibilities, high laser damage thresholds and other desirable properties, conducting polymers are widely used in signal transducers, [1] batteries, [2][3][4] field-effect transistors, [5][6][7] solar cells, [8][9][10] photovoltaic cells, [11][12][13][14][15] sensors, [16][17][18] and artificial muscles. [19][20][21] Polypyrrole (PPy) and polythiophene (PTh) are typical examples of five-membered heterocycle p-conjugated polymers.…”