Interspersions and fractal sequences associated with fractions c d. Kimberling, the first column of an interspersion, fibonacci quarterly 32 1994 3014. Nonlinear forecasting has recently been shown to distinguish between deterministic chaos and uncorrelated white noise added to periodic signals1, and. The first camera image shows the surface dotted with craters ranging from small to giant. Necessary and sucient conditions for the latter are given. Pdf fractal sequences and restricted nim semantic scholar. This research studies the properties of integer sequences related to a memorization technique. Proper selfcontaining sequences, fractal sequences, and. The precise definition of fractal sequence depends on a preliminary definition. The fractal property of the main sequence is investigated as well as the formula for the related.
Card sorting related to fibonacci numbers springerlink. The section on triangle geometry contains the worlds largest internet collection of triangle centers. Related fractal sequences and permutations of the set of nonnegative integers are also discussed. Clark kimberling, fractal sequences clark kimberling, interspersions clark kimberling and john e.
Kimberling proved that signature sequences are doubly fractal and conjectured the converse. One o these is that when we examine a fractal shape without a suitable frame of reference, it is often impossible to tell the scale o. Kimberling 10, 12 has given characterizations of fractal sequencesthe notions of interspersion and disp ersion which on the surface have nothing to do with selfsimilarity. Kimberling, numeration systems and fractal sequences, acta arithmetica 73 1995 103117 fractal sequences have in common with the more familiar geometric fractals the property of selfcontainment. Illustrating number sequences the bridges organization. Two operations that, when applied to kimberling fractal sequences, produce fractal sequences are upper trimming and lower trimming. Certain classes of restricted nim are found to produce sequences of grundy numbers with a selfsimilar fractal structure. The fractal dimension of a time series is important because it recognizes that process can be somewhere between deterministic a line with fractal dimension of 1 and random a fractal dimension of 1. Brown, partial complements and transposable dispersions, j. Interspersions and fractal sequences associated with. The gain of excitation was adjusted until seemingly chaotic motion was observed.
Nonlinear prediction as a way of distinguishing chaos from. The gain of excitation was adjusted until seemingly. Fractals and self similarity indiana university math. Suppose and are integers, and let be the set of integers, where and range over the nonnegative integers. Clark kimberling noticed that a sequence of integers can. A laser vibrometer was used to record the beam tip velocity and the analog signal streamed to a national instruments data acquisition board. Online encyclopedia of integer sequences neil sloanes online encyclopedia of integer sequences oeis is a treasury of mathematical patterns, including references, links, and comments. Upper trimming strikes out the first instance of every value, as illustrated by the first example in this article. It can be argued that the mathematics behind these images is even prettier than the pictures themselves.
A new fractal algorithm to model discrete sequences. Oeis sequence a125158 the fractal sequence associated with a125150 oeis sequence a125159 the fractal sequence associated with a125151 oeis sequence a108712 a fractal sequence, the almostnatural numbers references. Hutchinson this is a retyped texd version of the article from indiana university mathematics journal 30 1981, 7747 with some minor formatting changes, a few. Signature sequences are one category of kimberling fractal sequences 3. A sequence is a fractal sequence if it contains itself as a proper subsequence. Building predictive models from fractal representations of. Fractal sequences and restricted nim cornell university. The numbers in form interspersions in various ways. The fact that a fractal object is, in some sense, composed o smaller copies of itself, has interesting implications. Some reasons for the effectiveness of fractals in mathematics education benoit b. In fact, the fractal dimension of a line can range from 1 3 hurst, h.
More generally, it is an example of the fractal sequences studied by kimberling 11, 12. Mandelbrot and michael frame short is the distance between the elementary and the most sophisticated results, which brings rank beginners close to certain current concerns of the specialists. Simple online calculator which helps to generate kimberling sequence conjecture of numbers. Doubly interspersed sequences, double interspersions and fractal. An explicit formula for the grundy sequence 2 is given by truncating. Named by mandelbrot in 1975, from latin fractus meaning to break. The selfcontainment property resembles that of visual fractals a doubly fractal sequence of integers is defined by operations called upper trimming and lower trimming.
Rectorbc aschool of ee engineering, north china electric power university, beijing 102206, china bdepartment of civil engineering, university of california, berkeley 94530, usa clawrence berkeley laboratory, berkeley 94530, usa. A fractal is a geometrical figure in which an identical motif repeats itself on an ever diminishing scale. Fractal analysis of time series and distribution properties of hurst exponent malhar kale ferry butar butar abstract fractal analysis is done by conducting rescaled range rs analysis of time series. Nonlinear forecasting has recently been shown to distinguish between deterministic chaos and uncorrelated white noise added to periodic signals1, and can be used to estimate the degree of chaos. Clark kimberling professor of mathematics, university of evansville, 1800 lincoln avenue, evansville, in 47722. Fractal dimensions of time sequences sciencedirect. A novel proof for kimberlings conjecture on doubly fractal.
Many contributors have developed prototypical sequences that introduce newly discovered relationships among numbers, some proved and others conjectural. S s implies that s contains a copy of itself, and hence contains in. Devaney we choose to do mathematics, not because it is easy, but because it is hard. It is proved that distinct positive fibonacci sequences eventually intersperse or else doubly intersperse.
A novel proof for kimberlings conjecture on doubly. In mathematics, a fractal sequence is one that contains itself as a proper subsequence. Clark kimberling, university of evansville professor of mathematics, presented his recent research on beatty sequences and fractal sequences, fractal trees, and linear recurrence sequences on march 8 and 9 to the undergraduate honours seminar and the faculty colloquium at dalhousie university, halifax, nova scotia. Kimberling, numeration systems and fractal sequences, acta arithmetica 73 1995 103117. Kimberling, we obtain new characterizations of these fractal sequences and give a bijection between these sequences and certain uppertriangular arrays. The sequence begun in 6, and also the paraphrases of trinary and the other ary number systems, are examples of fractal sequences. Brown, partial complements and transposable dispersions, journal of integer sequences 7 2004 147159. Imagine a camera on a space probe hurtling toward the moons surface.
The numbertheoretic topics deal mostly with integer sequences. A pdf of the shape and history of the ellipse in washington, d. The paper presents algorithms for measuring the fractal dimension of real and complex digital. An efficient algorithm for measuring fractal dimension of complex sequences paul cristea university politehnica of bucharest, biomedical eng ineering center spl. Watson research center yorktown heights, new york 10598. Interspersions and fractal sequences associated with fractions c j d k clark kimberling department of mathematics university of evansville 1800 lincoln avenue. Introduction introduction to fractals and the fractal. Fractal sequences have in common with the more familiar. Doubly interspersed sequences, double interspersions, and fractal sequences clark kimberling abstract. Fractal dimension was considered right from its invention to be a good parameter to characterize time sequences of values of natural variables.
Kimberling 10, 12 has given characterizations of fractal sequences the notions of interspersion and disp ersion which on the surface have nothing to do with selfsimilarity. S s, then s is a fractal sequence, so named, in 3, because the selfsimilarity property. In mathematics, the kolakoski sequence, sometimes also known as the oldenburgerkolakoski sequence, is an infinite sequence of symbols 1,2 that is the sequence of run lengths in its own runlength encoding, and the prototype for an infinite family of related sequences. The selfcontainment property resembles that of visual fractals a doubly fractal sequence of inte. Interspersions are closely related to fractal sequences. It is proved that distinct positive fibonacci sequences eventually intersperse or else doubly. It was initially named after the recreational mathematician william kolakoski 194497, who described it in. Lower trimming subtracts 1 from each value and discards 0s. Proper selfcontaining sequences, fractal sequences, and parasequences clark kimberling august 20, 2007 abstract various kinds of ordered sets are discussed. First, were going to figure out the fibonacci sequence. The math forums internet math library is a comprehensive catalog of web sites and web pages relating to the study of mathematics.
Concepts in fractal research such as the fractal dimension are not strictly defined and much of the. A fractal is an image that repeats itself over and over again within itself. The hurst exponent and the fractal fractional dimension of a time series can be estimated with the help of rs analysis. Fractals and self similarity indiana university math ematics.
In the 1990s, mathematician clark kimberling studied sequences of counting numbers satisfying properties 1 and 2 and found a simple method for generating uncountably many of them. One o these is that when we examine a fractal shape without a suitable frame of reference, it is often impossible to tell the scale o magnification at which it is being viewed. A new fractal algorithm to model discrete sequences zhai mingyueay, heidi kuzumab, and james w. There is a host of simple observations that everyone can appreciate. Classic sequences in the online encyclopedia of integer. Then, the ys are sorted in increasing order and the corresponding set of ps is the signature sequence of x. Probably, fractal sequences are first defined in the following article.
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