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Translated and Revised by C. Cecilia Krieger. Critically acclaimed text presents detailed theory of Fréchet (V) spaces and a comprehensive examination of their relevance to topological spaces, plus in-depth discussions of metric and complete spaces. Numerous exercises reinforce teachings of each chapter. "...an elegant piece of work suitable for the beginning student and the mature mathematician."—Scripta Mathematica. 2nd ed.
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A Compendium of Facts about Pythagorean Triangles
A pythagorean triangle (PT) is a right triangle (i. e. a triangle which has a right angle) the lengths of the sides of which are all 'natural numbers' (i.e. positive integers)The smallest and best known PT is (4, 3, 5).

Chapter 1, section 1.2 begins: In a pythagorean triangle (as in any right triangle) the biggest side is obviously the hypoteneuse; the other two sides, called arms, contain the right angle.If these (i. e. their lengths) are x and y and the hypoteneuse is z, then by the theorem of Pythagoras,

. . x^2 + y^2 = z^2

[Sierpinski used superscripts, but Amazon's text box doesn't provide for them]

Many (I think most) Americans are used to calling the shorter sides of a right triangle legs, not arms, and we might be more comfortable with 'called' instead of 'obviously.'Also, we are used to naming the legs a and b and the hypoteneuse c, rather than x, y, and z.While differences such as these make this book require a bit more effort to read, the effort is worth it for the many interesting facts you will find here.

For example, on page 16 we learn that if the lengths of the two legs of a PT are consecutive numbers, b = a + 1, then (3a + 2c + 1, 3a + 2c + 2, 4a + 3c + 2) is another PT, On page 17 he lists the first six such triangles: (4, 3, 5), (20, 21, 29), (120, 119, 169), (696, 697, 985), (4060, 4059, 5741), and (23660, 23661, 33461).Perhaps because it is so obvious, Sierpinski doesn't mention that therefore PTs exist with acute angles arbitrarily close (but never =) to 45 degrees.

While extensive, the information about PTs in this slim volume (107 pages) is not exhaustive.In addition to the omission cited above, Sierpinski says nothing about infinite matrices of PTs, of which there are two that I know of.One is based on x=2r-1, y=2k, where r is the row number and k is the column number, and a(r,k)=2xy, b(r,k)=y^2-x^2, c(r,k)=y^2+x^2.This has the advantage of formulaic simplicity, compared to a(r,k)=4rk+2k(k-1), b(r,k)=4r(r+k-1)-2k+1, c(r,k)=4r(r+k-1)+2k(k-1)+1.However, the latter gives a matrix in which every row is an infinite family of PTs in each of which c exceeds a by the square of the rth odd number (1, 9, 25, 49, . . .) and every column is an infinite family in each of which c exceeds b by twice the square of k (2, 8, 18, 32, . . .).

watziznaym@gmail.com



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Since the publication of the first edition of this work, considerable progress has been made in many of the questions examined. This edition has been updated and enlarged, and the bibliography has been revised.

The variety of topics covered here includes divisibility, diophantine equations, prime numbers (especially Mersenne and Fermat primes), the basic arithmetic functions, congruences, the quadratic reciprocity law, expansion of real numbers into decimal fractions, decomposition of integers into sums of powers, some other problems of the additive theory of numbers and the theory of Gaussian integers.

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Amazing
This book is amazing, its excelent for the ones who love math. The only problem is the price (it is too high)! ... Read more

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This digital document, covering the life and work of Waclaw Sierpinski, is an entry from Contemporary Authors, a reference volume published by Thompson Gale. The length of the entry is 1540 words. The page length listed above is based on a typical 300-word page. Although the exact content of each entry from this volume can vary, typical entries include the following information:

• Place and date of birth and death (if deceased)
• Family members
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• Professional associations and honors
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• Writings, including books and periodicals
• A description of the author's work
• References to further readings about the author

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This digital document is an article from Science and Its Times, brought to you by Gale®, a part of Cengage Learning, a world leader in e-research and educational publishing for libraries, schools and businesses.The length of the article is 100 words.The article is delivered in HTML format and is available in your Amazon.com Digital Locker immediately after purchase.You can view it with any web browser.The histories of science, technology, and mathematics merge with the study of humanities and social science in this interdisciplinary reference work. Essays on people, theories, discoveries, and concepts are combined with overviews, bibliographies of primary documents, and chronological elements to offer students a fascinating way to understand the impact of science on the course of human history and how science affects everyday life. Entries represent people and developments throughout the world, from about 2000 B.C. through the end of the twentieth century. ... Read more

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Chapters: Michel Foucault, Wacław Sierpiński, Kazimierz Kuratowski, Zygmunt Bauman, Leszek Kołakowski, Jerzy Szacki, Edward Lipiński, Jan Niecisław Baudouin de Courtenay, Jan Łukasiewicz, Stanisław Grabski, Grigol Peradze, Leon Petrazycki, Krzysztof Kopczyński, Joachim Lelewel, Maria Ossowska, Elliott Leyton, Marek Siemek, Henryk Jabłoński, D. D. Morduhai-Boltovskoi, Magdalena Środa, Georgy Voronoy, Karol Borsuk, Andrzej Trautman, Tadeusz Stefan Zieliński, Benedykt Dybowski, Pilars de Pilar, Szymon Askenazy, Stanisław Ossowski, Zygmunt Vogel, Stefan Mazurkiewicz, Stanisław Srokowski, Wacław Maciejowski, Andrzej Mostowski, Juliusz Bardach, Agnieszka Graff, Zbigniew Szafrański, Tomasz Nałęcz, Nikolay Yakovlevich Sonin, Roman Rybarski, Vladimir Prokhorovich Amalitskii, Feliks Paweł Jarocki, Alfred Sokołowski, Paweł Śpiewak, Andrzej Białynicki-Birula. Source: Wikipedia. Pages: 158. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: Michel Foucault (French pronunciation: ), born Paul-Michel Foucault (15 October 1926 25 June 1984), was a French philosopher, sociologist, and historian. He held a chair at the prestigious Collège de France with the title "History of Systems of Thought," and also taught at the University at Buffalo and the University of California, Berkeley. Foucault is best known for his critical studies of social institutions, most notably psychiatry, medicine, the human sciences, and the prison system, as well as for his work on the history of human sexuality. His work on power, and the relationships among power, knowledge, and discourse has been widely discussed and taken up by others. In the 1960s Foucault was associated with Structuralism, a movement from which he distanced himself. Foucault also rejected the post-structuralist and postmodernist labels later attributed to him, pre...More: http://booksllc.net/?id=47643 ... Read more

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Chapters: Polish Mathematicians, Wacław Sierpiński, Stefan Banach, Alfred Korzybski, Yulian Vasilievich Sokhotski, Albert Brudzewski, Edward Kofler, Kazimierz Żorawski, Józef Maria Hoene-Wroński, Bruno Abakanowicz, Jan Łukasiewicz, Jerzy Neyman, Teofil Żebrawski, Jan Mikołaj Smogulecki, Emil Leon Post, Stanisław Leśniewski, Henryk Iwaniec, Zygmunt Janiszewski, Greg Kuperberg, Jerzy Różycki, List of Polish Mathematicians, Witelo, Jan Brożek, Witold Hurewicz, Kazimierz Bartel, Helena Rasiowa, Jacob Theodor Klein, Włodzimierz Kuperberg, Antoni Zygmund, Salomon Bochner, Stanisław Trybuła, Stanisław Saks, Stanisław Zaremba, Cecilia Krieger, Ignacy Zaborowski, Krystyna Kuperberg, Józef Marcinkiewicz, Leon Chwistek, Ryszard Syski, Tadeusz Banachiewicz, Karol Borsuk, Tomek Bartoszyński, Henryk Zygalski, Jan Graliński, Stanisław Jaśkowski, Stanisław Mazur, Przemysław Prusinkiewicz, Kazimierz Zarankiewicz, Jan Śniadecki, Grzegorz Rozenberg, Stefan Mazurkiewicz, Robert Schatten, Józef H. Przytycki, Stefan Bergman, Polish School of Mathematics, Marcin Odlanicki Poczobutt, Andrzej Mostowski, Jakob Rosanes, Jan Mycielski, Marek Karpinski, Franciszek Leja, Władysław Ślebodziński, Adam Adamandy Kochański, Stanisław Radziszowski, Tadeusz Iwaniec, Warsaw School of Mathematics, Siemion Fajtlowicz, Lucjan Zarzecki, Otto M. Nikodym, Kraków School of Mathematics, Stanisław Gołąb, Adam Danielewicz, Gabriel Judah Lichtenfeld, Jan Mikusinski, Zdzisław Krygowski, Stefan Kaczmarz, Andrzej Schinzel, Władysław Orlicz, Bronisław Knaster, Ryszard Engelking, Kraków School of Mathematics and Astrology, Roman Sikorski, Czesław Ryll-Nardzewski, Władysław Zajączkowski, Jerzy Browkin, Arnold Walfisz, Edward Jan Habich, Andrzej Białynicki-Birula, Włodzimierz Stożek, Jerzy Łoś, Antoni Łomnicki, Herman Auerbach, Andrzej Trybulec, Piotr Smoleński, Stanisław Ruziewicz, Feliks Barański, Studia Mathematica, Andrzej Grzegorczyk, S. L. Woronowicz, Tadeusz Ważewski, Stani...More: http://booksllc.net/?id=29646 ... Read more

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Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Chapters: Wacław Sierpiński, René Thom, Henri Poincaré, Kazimierz Kuratowski, Felix Hausdorff, John Milnor, Vladimir Arnold, William Thurston, Michael Atiyah, Stephen Smale, Vladimir Voevodsky, Pavel Samuilovich Urysohn, Leopold Vietoris, Elmer Rees, Luitzen Egbertus Jan Brouwer, Grigori Perelman, Heinz Hopf, John R. Stallings, James W. Cannon, Brian Bowditch, Karen Vogtmann, Mladen Bestvina, Anatoly Fomenko, Erik Christopher Zeeman, Jean-Pierre Serre, Raymond Louis Wilder, Leonard Gillman, Robert Lee Moore, Hassler Whitney, Raoul Bott, Shaun Wylie, Edwin E. Moise, Johannes de Groot, George David Birkhoff, J. H. C. Whitehead, Andreas Floer, Peter Hilton, R. H. Bing, Henry Wallman, Solomon Lefschetz, Sergei Novikov, Louis Kauffman, Albert W. Tucker, Judith Roitman, Hing Tong, Cahit Arf, George Mackey, Samuel Eilenberg, Frank Adams, Clifford Taubes, Eberhard Hopf, Andrey Nikolayevich Tychonoff, Zygmunt Janiszewski, Greg Kuperberg, Colin Adams, John Morgan, Daniel Quillen, Víctor Neumann-Lara, Oswald Veblen, Max Dehn, Morwen Thistlethwaite, James Waddell Alexander Ii, Wu Wenjun, Armand Borel, Oleg Viro, C. T. C. Wall, Włodzimierz Kuperberg, Dennis Sullivan, Ruth Lawrence, Shmuel Weinberger, Klaus Wagner, Allen Hatcher, Alice T. Schafer, Jean Leray, Shlomo Sternberg, Curtis T. Mcmullen, Michel Kervaire, Christos Papakyriakopoulos, John L. Kelley, Victor Anatolyevich Vassiliev, Hans Freudenthal, Augustin Banyaga, Krystyna Kuperberg, Abram Ilyich Fet, Sylvain Cappell, Dušan Repovš, Karol Borsuk, Richard Davis Anderson, J. Hyam Rubinstein, David Epstein, Valentin Poénaru, Lev Schnirelmann, Peter Shalen, William Floyd, Arthur Milgram, Robion Kirby, W. B. R. Lickorish, Jack Morava, Mary Ellen Rudin, Georges de Rham, Ralph Fox, Viktor Buchstaber, Daina Taimina, F. Burton Jones, Tudor Ganea, Clifford ...More: http://booksllc.net/?id=20698 ... Read more

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Chapters: Wacław Sierpiński, Marian Rejewski, Jan Kowalewski, Bomba, Zygalski Sheets, Stanisław Leśniewski, Cadix, Cyclometer, Jerzy Różycki, Pc Bruno, Kabaty, Polish Enigma Doubles, Gwido Langer, Maksymilian Ciężki, Lacida, Henryk Zygalski, Jan Graliński, Card Catalog, Grill, Antoni Palluth, Stefan Mazurkiewicz, Franciszek Pokorny, Wiktor Michałowski, Pyry, Clock, Piotr Smoleński. Source: Wikipedia. Pages: 124. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: Marian Adam Rejewski ( (help·info); 16 August 1905 13 February 1980) was a Polish mathematician and cryptologist who in 1932 solved the plugboard-equipped Enigma machine, the main cipher device used by Germany. The success of Rejewski and his colleagues Jerzy Róycki and Henryk Zygalski jump-started British reading of Enigma in World War II; the intelligence so gained, code-named "Ultra", contributed, perhaps decisively, to the defeat of Nazi Germany. While studying mathematics at Pozna University, Rejewski had attended a secret cryptology course conducted by the Polish General Staff's Biuro Szyfrów (Cipher Bureau), which he joined full-time in 1932. The Bureau had achieved little success reading Enigma and in late 1932 set Rejewski to work on the problem. After only a few weeks, he deduced the secret internal wiring of the Enigma. Rejewski and his two mathematician colleagues then developed an assortment of techniques for the regular decryption of Enigma messages. Rejewski's contributions included devising the cryptologic "card catalog," derived using his "cyclometer," and the "cryptologic bomb." Five weeks before the German invasion of Poland in 1939, Rejewski and his colleagues presented their results on Enigma decryption to French and British intelligence representatives. Shortly after the outbreak of war, the Polish cryptologists were ...http://booksllc.net/?id=4349420 ... Read more

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Chapters: Frédéric Chopin, Menachem Begin, Wacław Sierpiński, Kazimierz Kuratowski, Alfred Tarski, Witold Gombrowicz, Janusz Zajdel, Stanisław Wojciechowski, Mieczysław Karłowicz, Yitzhak Shamir, Bolesław Prus, Leonid Hurwicz, Roman Dmowski, Zygmunt Bauman, Lech Kaczyński, Joseph Rotblat, Ryszard Kapuściński, Adam Michnik, Jan Karski, Tadeusz Mazowiecki, Bronisław Geremek, Leszek Kołakowski, Jacek Kuroń, Maja Trochimczyk, Kazimierz Żorawski, Julian Tuwim, Jerzy Szacki, Alpha Oumar Konaré, Tadeusz Borowski, Jan T. Gross, Robert Kowalski, Kornel Morawiecki, Włodzimierz Cimoszewicz, Jan Niecisław Baudouin de Courtenay, Jan Ludwik Popławski, Bohdan Paczyński, Jan Olszewski, Tom R. Burns, Longin Pastusiak, Edward Kossoy, Jan Łukasiewicz, Krzysztof Kamil Baczyński, Wanda Wasilewska, Zygmunt Rumel, Đuro Kurepa, Rafał A. Ziemkiewicz, Samuel Eilenberg, Janusz Korwin-Mikke, Stanisław Leśniewski, Joseph Epstein, Jan Brzechwa, Gustaw Herling-Grudziński, Janusz Onyszkiewicz, Krzysztof Zanussi, Aleksander Szczygło, Włodzimierz Kuperberg, Adam Dziewonski, Antoni Zygmund, Stanisław Saks, Maciej Zembaty, Andrzej Czuma, Krzysztof Piesiewicz, Jan Wejchert, Krystyna Kuperberg, Stanisław Kuczborski, Jerzy Andrzejewski, Ludwik Dorn, Marek Kotański, Zdzisław Najder, Adam Przeworski, Aleksandra Ziółkowska Boehm, Krzysztof Zaleski, Karol Borsuk, Adolf Berman, Bolesław Piasecki, Kazimierz Michałowski, Tomek Bartoszyński, Andrzej Tadeusz Kijowski, Stanisław Jaśkowski, Tadeusz Iwiński, Jenia Taversky, Raman Skirmunt, Kazimierz Sikorski, Paweł Machcewicz, Małgorzata Bocheńska, Yosef Shofman, Andrzej Celiński, Ze'ev Herring, Eliyahu Meridor, Kazimierz Zarankiewicz, Daniel Passent, Jerzy Stefan Stawiński, Ryszard Czerniawski, Shalom Zisman, Roman Wapiński, Stefan Mazurkiewicz, Józef H. Przytycki, Walter Dana, Georg Wannagat, Dimitrie Botgros, Andrzej Mostowski, Aleksander Kamiński, Wojciech Wierzejski, Janusz Minkiewicz, Tomasz Nałęcz, Witold Zawadowski, A...More: http://booksllc.net/?id=10823 ... Read more

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Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Chapters: Wacław Sierpiński, Stefan Banach, Kazimierz Kuratowski, Hugo Steinhaus, Edward Flatau, George Zarnecki, Jerzy Kuryłowicz, Stanisław Kutrzeba, Emil Godlewski, Andrzej Gawroński, Jerzy Strzelczyk, Kazimierz Bartel, Antoni Zygmund, Zygmunt Wojciechowski, Gustaw Holoubek, Tadeusz Banachiewicz, Henryk Samsonowicz, Władysław Konopczyński, Benedykt Dybowski, Szymon Askenazy, Oskar Halecki, Karol Lanckoroński, Gerard Labuda, Stanisław Lorentz, Adam Chętnik, Maria Janion, Henryk Łowmiański, Kazimierz Piwarski, Wacław Sobieski, Stanisław Salmonowicz. Source: Wikipedia. Free updates online. Not illustrated. Excerpt: Stefan Banach ( listen) (18921945) was a Polish mathematician who worked in interwar Poland and in Soviet Ukraine. A self-taught mathematics prodigy, Banach was the founder of modern functional analysis and a founder of the Lwów School of Mathematics. Among his most prominent achievements was the 1932 book, Théorie des opérations linéaires (Theory of Linear Operations), the first monograph on the general theory of linear-metric space. Notable mathematical concepts named after Banach include the BanachTarski paradox, HahnBanach theorem, BanachSteinhaus theorem, Banach-Mazur game and Banach space. Stefan Banach was born on March 30, 1892, at St. Lazarus General Hospital in Kraków, then part of Austro-Hungarian Galicia. Banach's parents were Stefan Greczek and one Katarzyna Banach, both natives of the Podhale region. Stefan Greczek was born in Ostrowsko near the town of Nowy Targ and at one time was a soldier in the Austro-Hungarian Army stationed in Kraków. Stefan Greczek's father, Józef, was a farmer and a village mayor and Józef's wife, Antonina (née Borkowska) bore the Pomian coat of arms. Banach's mother left him after baptizing him when he was four days old. Her name on the bi...More: http://booksllc.net/?id=29646 ... Read more

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Chapters: Members of the Polish Academy of Learning, Wacław Sierpiński, Stefan Banach, Kazimierz Kuratowski, Hugo Steinhaus, Edward Flatau, George Zarnecki, Jerzy Kuryłowicz, Stanisław Kutrzeba, Emil Godlewski, Andrzej Gawroński, Jerzy Strzelczyk, Kazimierz Bartel, Antoni Zygmund, Zygmunt Wojciechowski, Gustaw Holoubek, Tadeusz Banachiewicz, Henryk Samsonowicz, Władysław Konopczyński, Benedykt Dybowski, Szymon Askenazy, Oskar Halecki, Karol Lanckoroński, Gerard Labuda, Stanisław Lorentz, Adam Chętnik, Maria Janion, Henryk Łowmiański, Kazimierz Piwarski, Wacław Sobieski, Stanisław Salmonowicz. Source: Wikipedia. Pages: 101. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: Stefan Banach ( listen) (18921945) was a Polish mathematician who worked in interwar Poland and in Soviet Ukraine. A self-taught mathematics prodigy, Banach was the founder of modern functional analysis and a founder of the Lwów School of Mathematics. Among his most prominent achievements was the 1932 book, Théorie des opérations linéaires (Theory of Linear Operations), the first monograph on the general theory of linear-metric space. Notable mathematical concepts named after Banach include the BanachTarski paradox, HahnBanach theorem, BanachSteinhaus theorem, Banach-Mazur game and Banach space. Stefan Banach was born on March 30, 1892, at St. Lazarus General Hospital in Kraków, then part of Austro-Hungarian Galicia. Banach's parents were Stefan Greczek and one Katarzyna Banach, both natives of the Podhale region. Stefan Greczek was born in Ostrowsko near the town of Nowy Targ and at one time was a soldier in the Austro-Hungarian Army stationed in Kraków. Stefan Greczek's father, Józef, was a farmer and a village mayor and Józef's wife, Antonina (née Borkowska) bore the Pomian coat of arms. Banach's mother left him after bap...More: http://booksllc.net/?id=29646 ... Read more

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Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Chapters: Pope John Paul Ii, Wisława Szymborska, Wacław Sierpiński, Carl Menger, John Iii Sobieski, Ivo Andrić, Jan Długosz, Conrad Celtes, Antoni Kępiński, Nicolaus Copernicus, Stanisław Koniecpolski, Norman Davies, Bronisław Malinowski, Kazimierz Żorawski, George Zarnecki, Charniele Herring, Jan Kochanowski, Józef Cyrankiewicz, Andrzej Frycz Modrzewski, Zvi Zimmerman, Zbigniew Czajkowski, Jan Buzek, Marcin Kromer, Tadeusz Pankiewicz, Teofil Żebrawski, Józef Buzek, Klymentiy Sheptytsky, John Cantius, Paulo Szot, Sebastian Petrycy, Krzysztof Zanussi, Mieczysław Domaradzki, Abraomas Kulvietis, Emil Zegadłowicz, Francysk Skaryna, Andrzej Zoll, Leo Sternbach, Pelbartus Ladislaus of Temesvár, Krystyna Chojnicka, Tadeusz Estreicher, Jerzy Adamuszek, Władysław Pobóg-Malinowski, Stanisław Estreicher, Zbigniew Gołąb, Jan Bystroń, Jan Olbrycht, Edward Długajczyk, Jerzy Zathey, Janusz Spyra, Zdzisław Żygulski, Jr., Juliusz Bardach, Kazimierz Godłowski, Jerzy Hordyński, Zbigniew Ćwiąkalski, Marcin Baran, Kaz Dziamka, Stanisław Sołtys, Ryszard Legutko, Dominic of Prussia. Source: Wikipedia. Free updates online. Not illustrated. Excerpt: The Venerable Pope John Paul II (Latin: , Italian: , Polish: , French: , German: ), born Karol Józef Wojtya (pronounced ; 18 May 1920 2 April 2005) served as Supreme Pontiff of the Catholic Church and Sovereign of Vatican City from 16 October 1978 until his death over 26 years later. His was the second-longest documented pontificate; only Pope Pius IX served longer (St Peter the Apostle is reputed to have served for more than thirty years as the first pontiff; however documentation is too sparse to definitively support this). He has been the only Polish Pope to date, and was the first non-Italian Pope since Dutch Pope Adrian VI in the 1520s. John Paul II has been acclai...More: http://booksllc.net/?id=23805 ... Read more