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If f and g are transcendental entire functions, then so are f o g and g o f. We discuss here the dynamics of various types of escaping sets and also non-escaping sets of composition of transcendental entire functions f o g and g o f and also its relations with regard to its factors f and g.

We study the possible Hausdorff limits of the Julia sets and filled Julia sets of subsequences of the sequence of dual Chebyshev polynomials of a non-polar compact set K⊂ C and compare such limits to K. Moreover, we prove that the measures of maximal entropy for the sequence of dual Chebyshev polynomials of K converges weak* to the equilibrium measure on K.

We study the possible Hausdorff limits of the Julia sets and filled Julia sets of subsequences of the sequence of dual Chebyshev polynomials of a non-polar compact set K⊂ C and compare such limits to K. Moreover, we prove that the measures of maximal entropy for the sequence of dual Chebyshev polynomials of K converges weak* to the equilibrium measure on K.

We study the possible Hausdorff limits of the Julia sets and filled Julia sets of subsequences of the sequence of dual Chebyshev polynomials of a non-polar compact set K⊂ C and compare such limits to K. Moreover, we prove that the measures of maximal entropy for the sequence of dual Chebyshev polynomials of K converges weak* to the equilibrium measure on K.

For a probability measure with compact and non-polar support in the complex plane we relate dynamical properties of the associated sequence of orthogonal polynomials {P n } to properties of the support. More precisely we relate the Julia set of P n to the outer boundary of the support, the filled Julia set to the polynomial convex hull K of the support, and the Green’s function associated with P n to the Green’s function for the complement of K.

For a probability measure with compact and non-polar support in the complex plane we relate dynamical properties of the associated sequence of orthogonal polynomials {P n } to properties of the support. More precisely we relate the Julia set of P n to the outer boundary of the support, the filled Julia set to the polynomial convex hull K of the support, and the Green’s function associated with P n to the Green’s function for the complement of K.

For a probability measure with compact and non-polar support in the complex plane we relate dynamical properties of the associated sequence of orthogonal polynomials fPng to properties of the support. More precisely we relate the Julia set of Pn to the outer boundary of the support, the lled Julia set to the polynomial convex hull K of the support, and the Green's function associated with Pn to the Green's function for the complement of K.

This paper explores new types of fractals created by iteration of the functions xn+1 = f1(xn, yn) and yn+1 = f2(xn, yn) in a general plane, rather than in the complex plane. Iteration of such functions generates orbits with novel fractal patterns. Especially interesting are N-th order polynomials, raised to a positive or negative integer power, p. Such functions create novel fractal patterns, including budding, spiked, striped, dragon head, and bat-like forms. The present faculty working paper shows how to create a rich variety of complex and fascinating fractals using this generalized approach, which is accessible to students with high school level skills in mathematics and coding.

Novel fractal forms can be created by iteration of higher order polynomials of the complex variable, z, with both positive and negative exponents, followed by optional integer power transformation, k zz . Such functions lead to an expanded universe of fascinating fractal patterns that can be incorporated into computer generated art.

We study the geometric structure of the boundary of Herman rings in a model family of Blaschke products of degree 3 (up to quasiconformal deformation). Shishikura’s quasiconformal surgery relates the Herman ring to the Siegel disk of a quadratic polynomial. By studying the regularity properties of the maps involved, we transfer McMullen’s results on the fine local geometry of Siegel disks to the Herman ring setting.

This paper presents an improvement of the sixth-order method of Chun and Neta as a class of three-step iterations with optimal efficiency index, in the sense of Kung-Traub conjecture. Each member of the presented class reaches the highest possible order using four functional evaluations. Error analysis will be studied and numerical examples are also made to support the theoretical results. We then present results which describe the dynamics of the presented optimal methods for complex polynomials. The basins of attraction of the existing optimal methods and our methods are presented and compared to illustrate their performances.

This paper presents an improvement of the sixth-order method of Chun and Neta as a class of three-step iterations with optimal efficiency index, in the sense of Kung-Traub conjecture. Each member of the presented class reaches the highest possible order using four functional evaluations. Error analysis will be studied and numerical examples are also made to support the theoretical results. We then present results which describe the dynamics of the presented optimal methods for complex polynomials. The basins of attraction of the existing optimal methods and our methods are presented and compared to illustrate their performances.

Let f and g be two permutable transcendental entire functions. Assume that f has the form f(z) = p(z) + p(1)(z)e(q1(z)) + p(2)(z)e(q2(z)). We shall investigate the dynamical properties of f and g and show that they have the same Julia sets and Fatou sets, i.e., J(f) = J(g). This relates to an open question due to Baker.

Let f and g be two permutable transcendental entire functions. Assume that f has the form f(z) = p(z) + p(1)(z)e(q1(z)) + p(2)(z)e(q2(z)). We shall investigate the dynamical properties of f and g and show that they have the same Julia sets and Fatou sets, i.e., J(f) = J(g). This relates to an open question due to Baker.

Let f and g be two permutable transcendental entire functions. Assume that f has the form f(z) = p(z) + p(1)(z)e(q1(z)) + p(2)(z)e(q2(z)). We shall investigate the dynamical properties of f and g and show that they have the same Julia sets and Fatou sets, i.e., J(f) = J(g). This relates to an open question due to Baker.

In 1958, Baker posed a question that if f and g are two permutable transcendental entire functions, must their Julia sets be the same? Since then several classes of entire functions have been exhibited to support an affirmative answer to the question. In this paper, we shall complement or improve some of these results by exhibiting some new classes of entire functions.

In 1958, Baker posed a question that if f and g are two permutable transcendental entire functions, must their Julia sets be the same? Since then several classes of entire functions have been exhibited to support an affirmative answer to the question. In this paper, we shall complement or improve some of these results by exhibiting some new classes of entire functions.

In 1958, Baker posed a question that if f and g are two permutable transcendental entire functions, must their Julia sets be the same? Since then several classes of entire functions have been exhibited to support an affirmative answer to the question. In this paper, we shall complement or improve some of these results by exhibiting some new classes of entire functions.

In this paper first we prove that if f and g are two permutable transcendental entire functions satisfying f = f(1) (h) and g = g(1) (h), for some transcendental entire function h, rational function f(1) and a function g(1), which is analytic in the range of h, then F(g) subset of F(f). Then as an application of this result, we show that if f(z) = p(z)e(q(z)) + c, where c is a constant, p a nonzero polynomial and q a nonconstant polynomial, or f(z) = integral(z) p(z)e(q(z)) dz, where p, q are nonconstant polynomials, such that f(g) = g(f) for a nonconstant entire function g, then J(f) = J(g).

In this paper first we prove that if f and g are two permutable transcendental entire functions satisfying f = f(1) (h) and g = g(1) (h), for some transcendental entire function h, rational function f(1) and a function g(1), which is analytic in the range of h, then F(g) subset of F(f). Then as an application of this result, we show that if f(z) = p(z)e(q(z)) + c, where c is a constant, p a nonzero polynomial and q a nonconstant polynomial, or f(z) = integral(z) p(z)e(q(z)) dz, where p, q are nonconstant polynomials, such that f(g) = g(f) for a nonconstant entire function g, then J(f) = J(g).

In this paper first we prove that if f and g are two permutable transcendental entire functions satisfying f = f(1) (h) and g = g(1) (h), for some transcendental entire function h, rational function f(1) and a function g(1), which is analytic in the range of h, then F(g) subset of F(f). Then as an application of this result, we show that if f(z) = p(z)e(q(z)) + c, where c is a constant, p a nonzero polynomial and q a nonconstant polynomial, or f(z) = integral(z) p(z)e(q(z)) dz, where p, q are nonconstant polynomials, such that f(g) = g(f) for a nonconstant entire function g, then J(f) = J(g).

In this paper we show that the Julia set J(G) of a finitely generated rational semigroup G is connected if the union of the Julia sets of generators is contained in a subcontinuum of J(G). Under a nonseparating condition, we prove that the Julia set of a finitely generated polynomial semigroup is connected if its postcritical set is bounded.

In this paper we show that the Julia set J(G) of a finitely generated rational semigroup G is connected if the union of the Julia sets of generators is contained in a subcontinuum of J(G). Under a nonseparating condition, we prove that the Julia set of a finitely generated polynomial semigroup is connected if its postcritical set is bounded.

In this paper we show that the Julia set J(G) of a finitely generated rational semigroup G is connected if the union of the Julia sets of generators is contained in a subcontinuum of J(G). Under a nonseparating condition, we prove that the Julia set of a finitely generated polynomial semigroup is connected if its postcritical set is bounded.

Diplomová práce se zabývá historií fraktální geometrie a popisuje vývoj nauky o fraktálech. Po počátečním seznámení se základními pojmy jsou popsány jednotlivé druhy fraktálů a jejich typické příklady. Dále jsou uvedeny oblasti, ve kterých je možno se s fraktály setkat mimo obor počítačové grafiky. Práce seznamuje s praktickým využitím fraktální geometrie. V textu jsou uvedeny v současné době známé programy a softwarové balíky vhodné pro zobrazování fraktálů a jsou popsány jejich možnosti. Praktickou část diplomové práce tvoří slajdy, demonstrační program a plakát. Elektronické slajdy představují osnovou využitelnou pro přednášky o problematice fraktální geometrie. Program slouží k demonstraci vybraných druhů fraktálů. Plakát je grafickým shrnutím výsledků práce., This Master's thesis deals with history of Fractal geometry and describes the fractal science development. In the begining there are essential Fractal science terms explained. Then description of fractal types and typical or most known examples of them are mentioned. Fractal knowledge application besides computer graphics area is discussed. Thesis informs about fractal geometry practical usage. Few present software packages or more programs which can be used for making fractal pictures are described in this work. Some of theirs capabilities are described. Thesis' practical part consists of slides, demonstrational program and poster. Electronical slides represents brief scheme usable for fractal geometry realm lectures. Program generates selected fractal types. Thesis results are projected on poster.

Práce se zabývá návrhem a implementací jednoduché strategické hry pomocí OpenGL. Herní logiku realizuje grafická karta a zobrazování je realizované grafickou kartou v jednom shaderu. Práce dále obsahuje úvod do OpenGL, přehled metod tvorby procedurálních textur a přehled zobrazovacích technik založených na metodě Ray Casting., This thesis deals with design and implementation of a simple strategic game using OpenGL, whose game logic is realized by the graphics card and rendering is realized by the graphics card in one shader. It also contains introduction to OpenGL, overview of techniques used for generating procedural textures and overview of rendering methods based on Ray Casting.

Diplomová práce se zabývá historií fraktální geometrie a popisuje vývoj nauky o fraktálech. Po počátečním seznámení se základními pojmy jsou popsány jednotlivé druhy fraktálů a jejich typické příklady. Dále jsou uvedeny oblasti, ve kterých je možno se s fraktály setkat mimo obor počítačové grafiky. Práce seznamuje s praktickým využitím fraktální geometrie. V textu jsou uvedeny v současné době známé programy a softwarové balíky vhodné pro zobrazování fraktálů a jsou popsány jejich možnosti. Praktickou část diplomové práce tvoří slajdy, demonstrační program a plakát. Elektronické slajdy představují osnovou využitelnou pro přednášky o problematice fraktální geometrie. Program slouží k demonstraci vybraných druhů fraktálů. Plakát je grafickým shrnutím výsledků práce., This Master's thesis deals with history of Fractal geometry and describes the fractal science development. In the begining there are essential Fractal science terms explained. Then description of fractal types and typical or most known examples of them are mentioned. Fractal knowledge application besides computer graphics area is discussed. Thesis informs about fractal geometry practical usage. Few present software packages or more programs which can be used for making fractal pictures are described in this work. Some of theirs capabilities are described. Thesis' practical part consists of slides, demonstrational program and poster. Electronical slides represents brief scheme usable for fractal geometry realm lectures. Program generates selected fractal types. Thesis results are projected on poster.

Práce se zabývá návrhem a implementací jednoduché strategické hry pomocí OpenGL. Herní logiku realizuje grafická karta a zobrazování je realizované grafickou kartou v jednom shaderu. Práce dále obsahuje úvod do OpenGL, přehled metod tvorby procedurálních textur a přehled zobrazovacích technik založených na metodě Ray Casting., This thesis deals with design and implementation of a simple strategic game using OpenGL, whose game logic is realized by the graphics card and rendering is realized by the graphics card in one shader. It also contains introduction to OpenGL, overview of techniques used for generating procedural textures and overview of rendering methods based on Ray Casting.

Práce se zabývá návrhem a implementací jednoduché strategické hry pomocí OpenGL. Herní logiku realizuje grafická karta a zobrazování je realizované grafickou kartou v jednom shaderu. Práce dále obsahuje úvod do OpenGL, přehled metod tvorby procedurálních textur a přehled zobrazovacích technik založených na metodě Ray Casting., This thesis deals with design and implementation of a simple strategic game using OpenGL, whose game logic is realized by the graphics card and rendering is realized by the graphics card in one shader. It also contains introduction to OpenGL, overview of techniques used for generating procedural textures and overview of rendering methods based on Ray Casting.