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The Mandelbrot Set
"... The fact, that after a Gilbreath shuffle (=give the deck a random cut, deal some cards facedown into a pile on the table, riffle shuffle this pile with the rest of the deck) , a deck of cards with a constantly repeated pattern has certain properties, which the following example shows: A deck of 52 ..."
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The fact, that after a Gilbreath shuffle (=give the deck a random cut, deal some cards facedown into a pile on the table, riffle shuffle this pile with the rest of the deck) , a deck of cards with a constantly repeated pattern has certain properties, which the following example shows: A deck of 52 cards is arranged: Clubs – Hearts – Spades – Diamonds – Clubs – Hearts – etc… After a Gilbreath shuffle, the top 4 cards consist one of each suit, the next four cards have one of each suit, etc. The Ultimate Gilbreath Principle For a permutation π of {1, 2, …, N}, the following four properties are equivalent: π is a Gilbreath permutation A deck of N cards starting in order 1, 2, …, N is in order π(1), π(2), …, π(N) after a Gilbreath shuffle for each j, the top j cards {π(1), π(2), π(3),…, π(j)} are distinct modulo j The top j cards have distinct remainders, when they are divided by j For each j and k with kj ≤ N, the j cards {π((k1)j+1), π((k1)j+2),…, π(kj)} are distinct modulo j N doesn’t need to be divisible by j The last k cards still have distinct remainders The top j cards are distinct modulo j, the cards π(j+1), …, π(2j) are distinct modulo j, etc For each j, the top j cards are consecutive in 1, 2, 3,…, N The top j cards are now out of order, but they were consecutive in the original deck
SOME ESTIMATIONS OF THE MANDELBROT SET
"... The aim of this article is to give some estimations of the Mandelbrot set. We also compute how big is the radius of a ball contained entirely into the Mandelbrot set. ..."
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The aim of this article is to give some estimations of the Mandelbrot set. We also compute how big is the radius of a ball contained entirely into the Mandelbrot set.
Is the Mandelbrot set computable?
 MATH. LOGIC QUART
, 2005
"... We discuss the question whether the Mandelbrot set is computable. The computability notions which we consider are studied in computable analysis and will be introduced and discussed. We show that the exterior of the Mandelbrot set, the boundary of the Mandelbrot set, and the hyperbolic components sa ..."
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Cited by 13 (0 self)
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We discuss the question whether the Mandelbrot set is computable. The computability notions which we consider are studied in computable analysis and will be introduced and discussed. We show that the exterior of the Mandelbrot set, the boundary of the Mandelbrot set, and the hyperbolic components
The Mandelbrot Set and The Farey Tree
 Am. Math. Monthly
, 1999
"... this paper is to explain and to make precise several "folk theorems" involving the Mandelbrot set and the Farey tree [D]. Recall that the Mandelbrot set is the parameter plane for iteration of the complex quadratic function Q c (z) = z ..."
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this paper is to explain and to make precise several "folk theorems" involving the Mandelbrot set and the Farey tree [D]. Recall that the Mandelbrot set is the parameter plane for iteration of the complex quadratic function Q c (z) = z
The Mandelbrot set is universal
, 1997
"... We show small Mandelbrot sets are dense in the bifurcation locus for any holomorphic family of rational maps. 1 ..."
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Cited by 9 (1 self)
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We show small Mandelbrot sets are dense in the bifurcation locus for any holomorphic family of rational maps. 1
MUTATIONS OF THE MANDELBROT SET (Revised)
"... The Mandelbrot set is the most famous of all fractals. It is easy to generate on a home computer and full of fascinatingly beautiful detail. The Mandelbrot set (shown in Figure 1) is sometimes called “the bug ” for obvious reasons. In this paper, we show how to make simple modifications of the itera ..."
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The Mandelbrot set is the most famous of all fractals. It is easy to generate on a home computer and full of fascinatingly beautiful detail. The Mandelbrot set (shown in Figure 1) is sometimes called “the bug ” for obvious reasons. In this paper, we show how to make simple modifications
π in the Mandelbrot set
, 2001
"... The Mandelbrot set is arguably one of the most beautiful sets in mathematics. In 1991, Dave Boll discovered a surprising occurrence of the number π while exploring a seemingly unrelated property of the Mandelbrot set. Boll’s Þnding is easy to describe and understand, and yet it is not widely known — ..."
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The Mandelbrot set is arguably one of the most beautiful sets in mathematics. In 1991, Dave Boll discovered a surprising occurrence of the number π while exploring a seemingly unrelated property of the Mandelbrot set. Boll’s Þnding is easy to describe and understand, and yet it is not widely known
The Complex Geometry of the Mandelbrot Set
"... Abstract In this paper, we give a brief overview of the geometry of the Mandelbrot set. We show how to distinguish each of the principal bulbs hanging off the main cardioid of this set by counting the spokes of the antennas attached to each bulb. We also use these antennas to attach a fraction to ea ..."
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Abstract In this paper, we give a brief overview of the geometry of the Mandelbrot set. We show how to distinguish each of the principal bulbs hanging off the main cardioid of this set by counting the spokes of the antennas attached to each bulb. We also use these antennas to attach a fraction
Hypercomputing the Mandelbrot Set?
, 2008
"... Abstract: The Mandelbrot set is an extremely wellknown mathematical object that can be described in a quite simple way but has very interesting and nontrivial properties. This paper surveys some results that are known concerning the (non)computability of the set. It considers two models of decida ..."
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Abstract: The Mandelbrot set is an extremely wellknown mathematical object that can be described in a quite simple way but has very interesting and nontrivial properties. This paper surveys some results that are known concerning the (non)computability of the set. It considers two models
Geometry of the Antennas in the Mandelbrot Set
, 2000
"... In the Mandelbrot set, the bulbs attached directly to the main cardioid are called the p=qbulbs. The reason for this is that the largest component of the interior of these bulbs consists of cvalues for which the quadratic function Q c (z) = z 2 + c admits an attracting cycle with rotation numbe ..."
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In the Mandelbrot set, the bulbs attached directly to the main cardioid are called the p=qbulbs. The reason for this is that the largest component of the interior of these bulbs consists of cvalues for which the quadratic function Q c (z) = z 2 + c admits an attracting cycle with rotation
Results 1  10
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