FRACTAL IMAGES OF GENERALIZED JULIA SETS

Fractals ◽  
1996 ◽  
Vol 04 (04) ◽  
pp. 533-541 ◽  
Author(s):  
KIM-KHOON ONG ◽  
AICHYUN SHIAH ◽  
ZDZISLAW E. MUSIELAK
Keyword(s):  
Real Axis ◽  
Complex Plane ◽  
Julia Sets ◽  
Real Numbers ◽  
Positive Real ◽  
The Real ◽  
Fractal Images ◽  
Zero Values ◽  
Principal Value ◽  
Iteration Function

The iteration function [Formula: see text], where both α and β are positive real numbers, is used to generate families of the generalized Julia sets, [Formula: see text]. The calculations are restricted to the principal value of zα + iβ and the obtained results demonstrate that classical Julia sets, [Formula: see text] are significantly deformed when non-zero values of β are considered. As a result of this deformation, the area of stable regions in the complex plane changes and a process of splitting and shifting takes place along the real axis. It is shown that this process is responsible for the formation of new fractal images of generalized Julia sets.

On certain expansions of the solutions of the general Lamé equation

1942 ◽  
Vol 38 (4) ◽  
pp. 364-367 ◽  
Author(s):  
A. Erdélyi
Keyword(s):  
Real Axis ◽  
Complex Plane ◽  
Recurrence Relations ◽  
Characteristic Exponent ◽  
Periodic Functions ◽  
The Real ◽  
Lamé Equation ◽  
Arbitrary Complex ◽  
Principal Value ◽  
Image Position

1. In this paper I shall deal with the solutions of the Lamé equationwhen n and h are arbitrary complex or real parameters and k is any number in the complex plane cut along the real axis from 1 to ∞ and from −1 to −∞. Since the coefficients of (1) are periodic functions of am(x, k), we conclude ](5), § 19·4] that there is a solution of (1), y0(x), which has a trigonometric expansion of the formwhere θ is a certain constant, the characteristic exponent, which depends on h, k and n. Unless θ is an integer, y0(x) and y0(−x) are two distinct solutions of the Lamé equation.It is easy to obtain the system of recurrence relationsfor the coefficients cr. θ is determined, mod 1, by the condition that this system of recurrence relations should have a solution {cr} for whichk′ being the principal value of (1−k2)½

FRACTAL IMAGES OF GENERALIZED MANDELBROT SETS

Fractals ◽  
1994 ◽  
Vol 02 (01) ◽  
pp. 111-121
Author(s):  
AICHYUN SHIAH ◽  
KIM-KHOON ONG ◽  
ZDZISLAW E. MUSIELAK
Keyword(s):  
Complex Plane ◽  
Transformation Function ◽  
Real Numbers ◽  
Mandelbrot Sets ◽  
Fractal Images ◽  
Principal Value ◽  
Generalized Mandelbrot Sets

The transformation function z ← zα+βi+c, with both α and β being either positive or negative integers or real numbers, is used to generate families of mostly new fractal images in the complex plane [Formula: see text]. The calculations are restricted to the principal value of zα+βi and the obtained fractal images are called the generalized Mandelbrot sets, ℳ (α, β). Three general classes of ℳ (α, β) are considered: (1) α ≠ 0 and β = 0; (2) α ≠ 0 and β ≠ 0; and (3) α = 0 and β ≠ 0. Our results demonstrate that the shapes of fractal images representing ℳ (α, 0) are usually significantly deformed when β ≠ 0, and that the size of either stable (α > 0) or unstable (α < 0) regions in the complex plane may increase as a result of non-zero β. It is also shown that fractal images of the generalized Mandelbrot sets ℳ (0, β) are significantly different than those obtained with a non-zero α.

scholarly journals The Complexity of Approximating the Matching Polynomial in the Complex Plane

2021 ◽  
Vol 13 (2) ◽  
pp. 1-37
Author(s):  
Ivona Bezáková ◽  
Andreas Galanis ◽  
Leslie Ann Goldberg ◽  
Daniel Štefankovič
Keyword(s):  
Real Axis ◽  
Complex Plane ◽  
Maximum Degree ◽  
Negative Real Axis ◽  
Positive Real ◽  
Bounded Degree ◽  
Matching Polynomial ◽  
Correlation Decay ◽  
Connective Constant ◽  
General Complex

We study the problem of approximating the value of the matching polynomial on graphs with edge parameter γ, where γ takes arbitrary values in the complex plane. When γ is a positive real, Jerrum and Sinclair showed that the problem admits an FPRAS on general graphs. For general complex values of γ, Patel and Regts, building on methods developed by Barvinok, showed that the problem admits an FPTAS on graphs of maximum degree Δ as long as γ is not a negative real number less than or equal to −1/(4(Δ −1)). Our first main result completes the picture for the approximability of the matching polynomial on bounded degree graphs. We show that for all Δ ≥ 3 and all real γ less than −1/(4(Δ −1)), the problem of approximating the value of the matching polynomial on graphs of maximum degree Δ with edge parameter γ is #P-hard. We then explore whether the maximum degree parameter can be replaced by the connective constant. Sinclair et al. showed that for positive real γ, it is possible to approximate the value of the matching polynomial using a correlation decay algorithm on graphs with bounded connective constant (and potentially unbounded maximum degree). We first show that this result does not extend in general in the complex plane; in particular, the problem is #P-hard on graphs with bounded connective constant for a dense set of γ values on the negative real axis. Nevertheless, we show that the result does extend for any complex value γ that does not lie on the negative real axis. Our analysis accounts for complex values of γ using geodesic distances in the complex plane in the metric defined by an appropriate density function.

scholarly journals Pseudo-hermitian random matrix theory: a review

2021 ◽  
Vol 2038 (1) ◽  
pp. 012009
Author(s):  
Joshua Feinberg ◽  
Roman Riser
Keyword(s):  
Real Axis ◽  
Complex Plane ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Matrix Theory ◽  
Complex Conjugate ◽  
Indefinite Metric ◽  
The Real ◽  
New Type ◽  
Diagrammatic Method

Abstract We review our recent results on pseudo-hermitian random matrix theory which were hitherto presented in various conferences and talks. (Detailed accounts of our work will appear soon in separate publications.) Following an introduction of this new type of random matrices, we focus on two specific models of matrices which are pseudo-hermitian with respect to a given indefinite metric B. Eigenvalues of pseudo-hermitian matrices are either real, or come in complex-conjugate pairs. The diagrammatic method is applied to deriving explicit analytical expressions for the density of eigenvalues in the complex plane and on the real axis, in the large-N, planar limit. In one of the models we discuss, the metric B depends on a certain real parameter t. As t varies, the model exhibits various ‘phase transitions’ associated with eigenvalues flowing from the complex plane onto the real axis, causing disjoint eigenvalue support intervals to merge. Our analytical results agree well with presented numerical simulations.

scholarly journals Global parameterization of $$\pi \pi $$ scattering up to 2 $${\mathrm {\,GeV}}$$

2019 ◽  
Vol 79 (12) ◽  
Author(s):  
J. R. Pelaez ◽  
A. Rodas ◽  
J. Ruiz de Elvira
Keyword(s):  
Experimental Data ◽  
Partial Wave ◽  
Real Axis ◽  
Complex Plane ◽  
Dispersion Relations ◽  
The Real ◽  
Partial Waves

AbstractWe provide global parameterizations of $$\pi \pi \rightarrow \pi \pi $$ππ→ππ scattering S0 and P partial waves up to roughly 2 GeV for phenomenological use. These parameterizations describe the output and uncertainties of previous partial-wave dispersive analyses of $$\pi \pi \rightarrow \pi \pi $$ππ→ππ, both in the real axis up to 1.12 $${\mathrm {\,GeV}}$$GeV and in the complex plane within their applicability region, while also fulfilling forward dispersion relations up to 1.43 $${\mathrm {\,GeV}}$$GeV. Above that energy we just describe the available experimental data. Moreover, the analytic continuations of these global parameterizations also describe accurately the dispersive determinations of the $$\sigma /f_0(500)$$σ/f0(500), $$f_0(980)$$f0(980) and $$\rho (770)$$ρ(770) pole parameters.

scholarly journals Plummeting and blinking eigenvalues of the Robin Laplacian in a cuspidal domain

2019 ◽  
Vol 150 (6) ◽  
pp. 2871-2893 ◽  
Author(s):  
Sergei A. Nazarov ◽  
Nicolas Popoff ◽  
Jari Taskinen
Keyword(s):  
Dimension Reduction ◽  
Discrete Spectrum ◽  
Spectral Theory ◽  
Real Axis ◽  
Complex Plane ◽  
Lipschitz Domain ◽  
High Rate ◽  
The Real ◽  
Robin Laplacian ◽  
Asymptotic Forms

We consider the Robin Laplacian in the domains Ω and Ωε, ε > 0, with sharp and blunted cusps, respectively. Assuming that the Robin coefficient a is large enough, the spectrum of the problem in Ω is known to be residual and to cover the whole complex plane, but on the contrary, the spectrum in the Lipschitz domain Ωε is discrete. However, our results reveal the strange behaviour of the discrete spectrum as the blunting parameter ε tends to 0: we construct asymptotic forms of the eigenvalues and detect families of ‘hardly movable’ and ‘plummeting’ ones. The first type of the eigenvalues do not leave a small neighbourhood of a point for any small ε > 0 while the second ones move at a high rate O(| ln ε|) downwards along the real axis ℝ to −∞. At the same time, any point λ ∈ ℝ is a ‘blinking eigenvalue’, i.e., it belongs to the spectrum of the problem in Ωε almost periodically in the | ln ε|-scale. Besides standard spectral theory, we use the techniques of dimension reduction and self-adjoint extensions to obtain these results.

scholarly journals The spectra of compact operators in Hilbert spaces

1965 ◽  
Vol 7 (1) ◽  
pp. 34-38
Author(s):  
T. T. West
Keyword(s):  
Hilbert Space ◽  
Unit Circle ◽  
Real Axis ◽  
Complex Plane ◽  
Hilbert Spaces ◽  
Compact Operators ◽  
Conformal Mappings ◽  
The Real ◽  
Finite Set

In [2] a condition, originally due to Olagunju, was given for the spectra of certain compact operators to be on the real axis of the complex plane. Here, by using conformal mappings, this result is extended to more general curves. The problem divides naturally into two cases depending on whether or not the curve under consideration passes through the origin. Discussion is confined to the prototype curves C0 and C1. The case of C0, the unit circle of centre the origin, is considered in § 3; this problem is a simple one as the spectrum is a finite set. In § 4 results are given for C1 the unit circle of centre the point 1, and some results on ideals of compact operators, given in § 2, are needed. No attempt has been made to state results in complete generality (see [2]); this paper is kept within the framework of Hilbert space, and particularly simple conditions may be given if the operators are normal.

scholarly journals Integrals of subharmonic functions and their differences with weight over small sets on a ray

2020 ◽  
Vol 54 (2) ◽  
pp. 162-171
Author(s):  
B.N. Khabibullin
Keyword(s):  
Characteristic Function ◽  
Lebesgue Measure ◽  
Real Axis ◽  
Complex Plane ◽  
Measurable Subset ◽  
Positive Part ◽  
Subharmonic Functions ◽  
The Real ◽  
The Difference

Let $E$ be a measurable subset in a segment $[0,r]$ in the positive part of the real axis in the complex plane, and $U=u-v$ be the difference of subharmonic functions $u\not\equiv -\infty$ and $v\not\equiv -\infty$ on the complex plane. An integral of the maximum on circles centered at zero of $U^+:=\sup\{0,U\} $ or $|u|$ over $E$ with a function-multiplier $g\in L^p(E) $ in the integrand is estimated, respectively, in terms of the characteristic function $T_U$ of $U$ or the maximum of $u$ on circles centered at zero, and also in terms of the linear Lebesgue measure of $E$ and the $ L^p$-norm of $g$. Our main theorem develops the proof of one of the classical theorems of Rolf Nevanlinna in the case $E=[0,R]$, given in the classical monograph by Anatoly A. Goldberg and Iossif V. Ostrovsky, and also generalizes analogs of the Edrei\,--\,Fuchs Lemma on small arcs for small intervals from the works of A.\,F.~Grishin, M.\,L.~Sodin, T.\,I.~Malyutina. Our estimates are uniform in the sense that the constants in these estimates do not depend on $U$ or $u$, provided that $U$ has an integral normalization near zero or $u(0)\geq 0$, respectively.

Quaternions: The hypercomplex number system

2008 ◽  
Vol 92 (525) ◽  
pp. 431-436 ◽  
Author(s):  
Sandra Pulver
Keyword(s):  
Real Axis ◽  
Imaginary Axis ◽  
Vertical Axis ◽  
Number System ◽  
Real Coefficient ◽  
Complex Numbers ◽  
Square Root ◽  
Real Numbers ◽  
The Real ◽  
Hypercomplex Number

Are there solutions of the equation x2 + 1 = 0 ? Carl Fredrich Gauss (1777–1855) conjectured that there was a solution and that it was the square root of - 1 . But since the squares of all real numbers, positive or negative, are positive, Gauss introduced a fanciful idea. His solution to this equation was , which he named i. He integrated i with the real numbers to form a set known as , the complex numbers, where each element in that set was of the form a + bi, where a, . Gauss illustrated this on a graph, the horizontal axis became the real axis and represented the real coefficient, while the vertical axis became the imaginary axis and represented the imaginary coefficient.

scholarly journals On the Result of Invariance of the Closure Set of the Real Projections of the Zeros of an Important Class of Exponential Polynomials

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
J. M. Sepulcre
Keyword(s):  
Important Class ◽  
Exponential Polynomial ◽  
Critical Interval ◽  
Real Numbers ◽  
Positive Real ◽  
Exponential Polynomials ◽  
The Real ◽  
Linearly Independent ◽  
Pointwise Characterization

We provide the proof of a practical pointwise characterization of the setRPdefined by the closure set of the real projections of the zeros of an exponential polynomialP(z)=∑j=1ncjewjzwith real frequencieswjlinearly independent over the rationals. As a consequence, we give a complete description of the setRPand prove its invariance with respect to the moduli of thecj′s, which allows us to determine exactly the gaps ofRPand the extremes of the critical interval ofP(z)by solving inequations with positive real numbers. Finally, we analyse the converse of this result of invariance.

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