81. My Favorite Books By Shyam Sunder Gupta 55 sierpinski, waclaw and Schinzel, Andrzej. Elementary Theory ofNumbers, 2nd ed. Amsterdam, Netherlands NorthHolland,1988. 513 p. http://www.shyamsundergupta.com/referencebooks.htm

My Favorite Books [1] Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 59, 1987. [2] Beck, Anatole; Bleicher, Michael N.; and Crowe, Donald W. Excursions into Mathematics, the Millennium Edition. Natick, MA: AK Peters, 2000. 499 p. [3] Beiler, Albert H. Recreations in the Theory of Numbers. New York: Dover, 1966. [4] Bressoud, D. Factorization and Primality Testing. New York: Springer-Verlag, 1989. [5] Carroll, Lewis. Pillow Problems and A Tangles Tale. New York: Dover, 1958. [6] Cohen, Henri. Advanced Topics in Computational Number Theory. New York: Springer-Verlag, 2000. 578 p. [7] Cohen, Henri. A Course in Computational Algebraic Number Theory, 3rd. corr. ed. New York: Springer-Verlag, 1996. 534 p. [8] Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 33-38, 1996. [9] Crandall, Richard and Pomerance, Carl. Prime Numbers. New York: Springer-Verlag, 2001. 352 p. [10] Davenport, Harold. The Higher Arithmetic: An Introduction to the Theory of Numbers, 6th ed. Cambridge, England: Cambridge University Press, 1992. 217 p. [11] Bressoud, David M. and Wagon, Stan.

82. The Mathematics Genealogy Project - Index Of SI Translate this page 1975. Sierksma, Gerard, sierpinski, waclaw, Jagiellonian University,1906. Sierra-Cavazos, Jorge, North Carolina State University, 1992. http://genealogy.math.ndsu.nodak.edu/html/letter.phtml?letter=SI&fShow=1

83. Go2net | Internet | Deep Magic | The Natural Geometry Of Fractals Figure 1. waclaw sierpinski described a procedure for generating an interestingfigure known as sierpinski's gasket. Start with a filledin triangle. http://www.go2net.com/internet/deep/1996/12/11/body.html

11 December 1996 Fractals have become an extremely popular topic in math and computer science in recent years. Little else in these fields quite captivates like the beauty of regions of the Mandelbrot set or the striking realism of a fractally generated mountain range. It turns out that concepts very similar to what we today associate with the science of fractals were discovered quite a long time ago. It was in 1890 that Giuseppe Peano first postulated his Peano curve, and a year later, David Hilbert the Hilbert curve. However, not until the advent of the computer have we truly been able to explore the mysterious world of fractals. Today with the web and Java, fractals have virtually come knocking on your door. So sit back and enjoy the show. Order, Chaos and Gaskets The core fundaments of fractal geometry are the ideas of feedback and iteration. The creation of most fractals involve applying some simple rule to a set of geometric shapes or numbers and then repeating the process on the results. This feedback loop can result in very unexpected results, given the apparent simplicity of the rules followed for each iteration. Basic step in constructing Sierpinski's gasket Figure 1.

84. Automata And Ordinals, Bibliography 8, pp. 190194, 1965. waclaw sierpinski, Leçons sur les nombres transfinis. ParisGauthier-Villars, 1950. waclaw sierpinski, Cardinal and ordinal numbers. http://www-igm.univ-mlv.fr/~bedon/Recherche/bibliographie_en.shtml

[Up] Automata and ordinals, bibliography Topics Authors Ordinals Automata on finite words on infinite words on transfinite words Semigroups on finite words on infinite words on transfinite words Logics on finite words on infinite words on transfinite words Authors Conference Record of the Seventh Annual ACM Symposium on Principles of Programming Languages , (Las Vegas, Nevada), january 1980. Jorge Almeida, Finite semigroups and universal algebra , vol. 3 of Series in algebra . World Scientific, 1994. Theoretical Computer Science , vol. 39, pp. 333-335, 1985. Nicolas Bedon and Olivier Carton , "An Eilenberg theorem for words on countable ordinals", in Latin'98: Theoretical Informatics (Cláudio L. Lucchesi and Arnaldo V. Moura, eds.), vol. 1380 of Lecture Notes in Computer Science , (Campinas, Brazil), pp. 53-64, Springer , Apr. 1998. Nicolas Bedon "Finite automata and ordinals," Theoretical Computer Science , vol. 156, pp. 119-144, Mar. 1996. Nicolas Bedon "Autom% ata, semigroups and recognizability of words on ordinals," International Journal of Algebra and Computation , vol. 8, pp. 1-21, Feb. 1998.

85. Eliana Argenti E Tommaso Bientinesi - Caos E Oggetti Frattali - Il Triangolo Di Translate this page IL TAPPETO DI sierpinski. http://www.webfract.it/FRATTALI/ntappeto.htm

Pagina iniziale Introduzione Che cosa sono i frattali? Come si realizzano i frattali? ... Area Download CARATTERISTICHE Autosimilarità Perimetro infinito e area finita Dimensione non intera Struttura complessa a tutte le scale di riproduzione ... Dinamica caotica PERSONAGGI Niels Fabian Helge von Koch Waclaw Sierpinski Gaston Maurice Julia Benoit Mandelbrot TIPI DI FRATTALI Curva di von Kock Triangolo di Sierpinski Tappeto di Sierpinski Insieme di Mandelbrot ... Nuvole frattali FRATTALI E REALTA' ...fisiologia umana ...arte ...musica ...altri campi ... Bibliografia e indirizzi utili IL TAPPETO DI SIERPINSKI Attenzione: le linee tratteggiate non fanno parte del frattale; sono state aggiunte soltanto per una migliore chiarezza espositiva. Prendiamo come figura di partenza un quadrato e dividiamolo in nove quadrati uguali. Eliminiamo dalla sua superficie il quadrato centrale. Ripetiamo il procedimento su ognuno degli otto quadrati restanti: quindi al centro di ognuno di essi resterà un quadrato vuoto.

86. TeleMath - ÌáèçìáôéêÜ êáé Öéëïôåëéóìüò The summary for this Greek page contains characters that cannot be correctly displayed in this language/character set. http://www.telemath.gr/mathematical_stamps/stamps_mathematicians/persons/sierpin

SIERPINSKI, Waclaw

87. The Magic Sierpinski Triangle This design is called sierpinski's Triangle (or gasket), after the Polish mathematicianWaclaw sierpinski who described some of its interesting properties in http://serendip.brynmawr.edu/playground/sierpinski.html

The Magic Sierpinski Triangle Order dependent on randomness This design is called Sierpinski's Triangle (or gasket), after the Polish mathematician Waclaw Sierpinski who described some of its interesting properties in 1916. Among these is its fractal or self-similar character. The large blue triangle consists of three smaller blue triangles, each of which itself consists of three smaller blue triangles, each of which ..., a process of subdivision which could, with adequate screen resolution, be seen to continue indefinitely. Fractals and self-similarity are of considerable interest in their own right, but our interest here is in how to construct Sierpinski's triangle. One way to do so is to inscribe a second triangle inside the original one, by joining the midpoints of the three sides, and then repeat the process for the resulting three outer triangles, for the three outer triangles that result from that, and so forth. But there is a more intriguing way to construct Sierpinski's triangle, sometimes called the Chaos Game Lots of interesting questions have probably occurred to you. Does the pattern depend on the particular triangle you start with? Find out by clicking on the Custom button and creating your own triangle. Does the choice of the initial point matter? Try that out too by clicking the Clear button and selecting a new point inside the triangle. How come this construction gives the same (itself rather remarkable) pattern as inscribing triangles? We'll leave that and some other questions to

88. Sierpinski Triangle The sierpinski Triangle , introduced in 1916 by the Polish mathematician Waclawsierpinski (18821969), is a rather simple example of order out of disorder http://www.i1776.com/cs/sierpinski_triangle.html

89. Perturbed Sierpinski Gasket Monopole Perturbation of the Fractal sierpinski gasket enables design flexibility in tuningfor the required operating bands within a single antenna structure. http://members.fortunecity.com/scienziatopazzo/sierpinski.htm

web hosting domain names email addresses related sites ... BACK TO MY RESERACH Perturbation of the Fractal Sierpinski gasket enables design flexibility in tuning for the required operating bands within a single antenna structure. Poor matching characteristics are obtained when the perturbed Sierpinski gasket antenna ((h1/h2)=0.75 ) is mounted over a large metallic ground plane. By using a planar microstrip feed, this impedance mismatch can be overcome. The microstrip feed technique also play an important role in tuning the operating frequency ratio between the first and second resonance band (f1/f2). C. Puente ( Fractus ) has been a wonderful source of inspiration. Many thanks for his encouragement and sharing of his excellent thesis A picture of us at AP2000 conference. Simulated E-Field strength Band 1 ......................... Band 2 ........................ Band 3 ......................... Band 4 Back to top S11 performance of microstrip feed over large ground plane feed Back to top Dual periodic ratio Dual periodic ratio of 0.5 and 0.75 is achieved. This ratio is very close to the operating band ratios of GSM, DCS/DECT and WLAN

90. Math Forum - Ask Dr. Math Date 01/20/97 at 112636 From Doctor Toby Subject Re sierpinski Triangle Waclawsierpinski invented the triangle (or gasket) named after him in 1916. http://mathforum.org/library/drmath/view/54524.html

Associated Topics Dr. Math Home Search Dr. Math Sierpinski Triangle Date: 01/15/97 at 10:28:11 From: Anonymous Subject: Sierpinski Triangle Hi, My name is Ryan and I would like to ask you a question. What is a Sierpinski Triangle? http://mathforum.org/dr.math/ Associated Topics High School Fractals Search the Dr. Math Library: Find items containing (put spaces between keywords): Click only once for faster results: [ Choose "whole words" when searching for a word like age. all keywords, in any order at least one, that exact phrase parts of words whole words Submit your own question to Dr. Math Math Forum Home Math Library Quick Reference ... Math Forum Search Ask Dr. Math TM http://mathforum.org/dr.math/

91. NetLogo Models Library: Sierpinski Simple The fractal that this model produces was discovered by the great Polish mathematicianWaclaw sierpinski in 1916. sierpinski was a professor at Lvov and Warsaw. http://ccl.northwestern.edu/netlogo/models/SierpinskiSimple

Home Page Download Models Library NetLogo User ... Community Models User Manual: HTML Printable (PDF) FAQ Contact Us NetLogo Models Library: Sample Models/Mathematics/Fractals (back to the library) Sierpinski Simple Run Sierpinski Simple in your browser system requirements Note: If you download the NetLogo application, every model in the Models Library (besides the Community Models) is included. If you have trouble running this model in your browser, you may wish to download the application instead. WHAT IS IT? The fractal that this model produces was discovered by the great Polish mathematician Waclaw Sierpinski in 1916. Sierpinski was a professor at Lvov and Warsaw. He was one of the most influential mathematicians of his time in Poland and had a worldwide reputation. In fact, one of the moon's craters is named after him. The basic geometric construction of the Sierpinski tree goes as follows. We begin with a single point on the plane and then apply a repetitive scheme of operations to it. Grow a "spider" centered at this point by drawing three equal line segments directed to the vertices of an equilateral triangle. Then at each vertex of the triangle repeat the construction grow a similar "spider" only scale it down by the factor of two. . Step 0: Start with a point

92. Eliana Argenti E Tommaso Bientinesi - Caos E Oggetti Frattali - Il Triangolo Di Translate this page IL TRIANGOLO DI sierpinski. http://space.tin.it/computer/eargenti/FRATTALI/ntriangolo.htm

Pagina iniziale Introduzione Che cosa sono i frattali? Come si realizzano i frattali? Area Download CARATTERISTICHE Autosimilarità Perimetro infinito e area finita Dimensione non intera Struttura complessa a tutte le scale di riproduzione ... Dinamica caotica PERSONAGGI Niels Fabian Helge von Koch Waclaw Sierpinski Gaston Maurice Julia Benoit Mandelbrot TIPI DI FRATTALI Curva di von Kock Triangolo di Sierpinski Tappeto di Sierpinski Insieme di Mandelbrot ... Alberi frattali FRATTALI E REALTA' ...fisiologia umana ...arte ...musica ...altri campi ... Bibliografia e indirizzi utili IL TRIANGOLO DI SIERPINSKI Prendiamo come figura di partenza un triangolo equilatero. Eliminiamo dalla sua superficie il triangolo che ha come lati i segmenti che uniscono i punti medi dei lati del triangolo precedente. Ripetiamo il procedimento su ognuno dei tre triangoli che si sono così formati. Ripetiamo il procedimento su ognuno dei nove triangoli che si sono così formati.

93. Fractal Antennas - Front Page The sierpinski Gasket is one of the oldest fractal shapes. It is named afterWaclaw sierpinski, the Polish mathematician that extensively studied it. http://www-tsc.upc.es/eef/research_lines/antennas/fractals/gallery/spk60.html

The Sierpinski Antenna Introduction The Sierpinski Gasket is one of the oldest fractal shapes. It is named after Waclaw Sierpinski, the Polish mathematician that extensively studied it. The fractal form is composed by 3 triangular sets, being each one itself a Sierpinksi Gasket. A classical procedure for constructing it is burning a triangular hole in the central part of a solid triangular shape, and keep iterating the same procedure over each new triangle formed this way. The object has a fractal dimension D and a characteristic scale factor d relating the several gaskets within the structure. Antenna Description The antenna was built by printing a iteration, 8.9 cm tall Sierpinski gasket over a Cuclad 250-GT microwave circuit dielectric substrate. The monopole configuration was chosen first for its simplicity and feeding scheme. The antenna was mounted orthogonally to a 80 cm x 80 cm ground plane and fed through an SMA coaxial connector. Owing to the antenna geometry, one expected current flowing from the feeding vertex to the antenna tips and power becoming radiated, i.e., driven out from the antenna, at those fractal iterations that matched the operating wavelength. Since the antenna contained 5 scale-levels with a characteristic scale-factor of relating all of them, it appeared reasonable to assume that the antenna would perform in a similar way at

94. Sierpinski-Dreieck Translate this page Das Sierpinksi-Dreieck ist nach dem polnischen Mathematiker WaclawSierpinski (1882-1969) benannt. Es entsteht folgendermaßen http://m.holzapfel.bei.t-online.de/themen/sierpinski/sierpinski.htm

Das Sierpinski-Dreieck 1. Entstehung des Sierpinski-Dreiecks Das Sierpinksi-Dreieck ist nach dem polnischen Mathematiker Waclaw Sierpinski (1882-1969) benannt. Es entsteht folgendermaßen: Das Dreieck (Initiator) als Ausgangsfigur wird in vier kongruente Dreiecke aufgeteilt und das mittlere Dreieck herausgenommen (Generator); A n A A l n l , l = Umfang des Ausgangsdreiecks. 3. Entstehung des Sierpinski-Dreiecks mit der Mehrfach-Verkleinerungs-Kopiermaschine (MVKM) Hier sind drei verschiedene Linsensysteme dargestellt: Entsprechend sind die erzeugten Sierpinski-Dreiecke, links dargestellt. 4. Dimension D eines Fraktals D = , a Anzahl der Teile, in die die Struktur zerlegt werden kann, s Verkleinerungsfaktor Beispiele: Strecke: D = Quadrat: D = Sierpinski-Dreieck: D MVKM mit Verkleinerungsfaktor > 0,5: D > 1,585 MVKM mit Verkleinerungsfaktor 0,5: D 5. Das Chaos-Spiel (Michael F. Barnsley) Nach etwa 500 Spielpunkten wird ein Muster sichtbar und nach etwa 10000 Punkten (s. nebenstehendes Bild) wird bereits deutlich die Struktur des Sierpinski-Dreiecks sichtbar. Die Menge aller s s s s mit s S Jedes Element G aus dem Adressenraum S kennzeichnet einen Punkt z im Sierpinski-Dreieck. Zwei verschiedene Elemente von

95. TeleMath - ÌáèçìáôéêÜ êáé Öéëïôåëéóìüò The summary for this Greek page contains characters that cannot be correctly displayed in this language/character set. http://www.telemath.gr/mathematical_stamps/stamps_mathematicians/

ABEL, Niels Henrik. AL-KHOWARIZMI. 'Añáâáò Ìáèçìáôéêüò( 790850 ) ARCHIMEDES. BABBAGE, Charles. BANACH, Stefan. BANACHIEWICZ, Tadeusz. ... KHAYYAM, Omar. 'Añáâáò Ìáèçìáôéêüò( 10481122 ) KIRCHHOFF, Gustav. KOVALEVSKAYA, Sofia. KRYLOV, Aleksei N. LAGRANGE, Joseph-Louis. ... ZAREMBA, Stanislaw.

96. Index For 1900-1909 Index for 19001909. This is the index into entries in the TCS Genealogyfor doctorates granted in the decade 1900-1909. Contents. http://sigact.acm.org/genealogy/index-190x.html

Index for 1900-1909 This is the index into entries in the TCS Genealogy for doctorates granted in the decade 1900-1909. Contents This is the index into entries in the TCS Genealogy for doctorates granted in the year 1903. Veblen, Oswald University of Chicago This is the index into entries in the TCS Genealogy for doctorates granted in the year 1904. Richardson, O.W. University of London This is the index into entries in the TCS Genealogy for doctorates granted in the year 1905. Schmidt, Erhard This is the index into entries in the TCS Genealogy for doctorates granted in the year 1906. Sierpinski, Waclaw Jagiellonian University This is the index into entries in the TCS Genealogy for doctorates granted in the year 1908. Weyl, Herman This is the index into entries in the TCS Genealogy for doctorates granted in the year 1909. Ham, W.R. University of Chicago

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