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Kaliningrad
Kaliningrad (Russian: Калинингра́д; Polish: Królewiec; Lithuanian: Karaliaučius; German: Königsberg; Czech: Královec) is a seaport city and the administrative center of Kaliningrad Oblast, the Russian exclave between Poland and Lithuania on the Baltic Sea. The territory, the northern part of the former East Prussia, borders on NATO and EU members Poland and Lithuania, and is geographically separated from the rest of Russia.
Originally named Königsberg in German, the town was founded in 1255 by the order of the Teutonic Knights in honour of the Czech king Přemysl Otakar II and was part of Prussia and then of Germany until 1945, but was largely destroyed during World War II. Its ruins were occupied by the Red Army in 1945 and its German population forced out. It was renamed Kaliningrad in 1946 in honor of Mikhail Kalinin.
According to the preliminary results of the 2010 Census, its population was 431,491—an increase from 430,003 recorded in the 2002 Census. Its ethnic composition is 77.9% Russians, 8.0% Belarusians, 7.3% Ukrainians, 1.9% Lithuanians, 0.6% Germans, and 0.5% Poles.
wn.com/Kaliningrad -
The Geometry Center
The Geometry Center was a mathematics research and education center at the University of Minnesota. It was established by the National Science Foundation in the late 1980s and closed in 1998. The focus of the Center's work was the use of computer graphics and visualization for research and education in pure mathematics and geometry.
The Center's founding director was Al Marden. Richard McGehee directed the Center during its final years. The Center's governing board was chaired by David P. Dobkin.
Much of the work at the Center was of the development of Geomview, a three-dimensional interactive geometry program. This focused on mathematical visualization with options to allow hyperbolic space to be visualised. This originally ran on Silicon Graphics machines, and has been ported to run under X11 on unix machines. Geomview can run under Windows using Cygwin and under Mac OS X. Geomview is still being supported today.[1]
Geomview is built around the Object Oriented Graphics Library (OOGL). The displayed scene and the attributes of the objects in it may be manipulated by the graphical command language (GCL) of Geomview. Geomview may be set as a default 3-D viewer for Mathematica.
wn.com/The_Geometry_Center
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Cesare Arzelà
Cesare Arzelà (6 March 1847 – 15 March 1912) was an Italian mathematician who taught at the University of Bologna and is recognized for his contributions in the theory of functions, particularly for his characterization of sequences of continuous functions, generalizing the one given earlier by Giulio Ascoli in the famous Arzelà-Ascoli theorem.
He was a pupil of the Scuola Normale Superiore of Pisa where he graduated in 1869. Arzela came from a poor household; therefore he could not start his study until 1871, when he studied in Pisa under Enrico Betti and Ulisse Dini.
He was working in Florence (from 1875) and in 1878 obtained the Chair of Algebra at the University of Palermo.
After that he became a professor in 1880 at the University of Bologna at the department of analysis. He conducted research in the field of theory of functions. His most famous student was Leonida Tonelli.
In 1889 he generalized the Ascoli theorem to Arzelà–Ascoli theorem, an important theorem in theory of functions.
He was a member of the Accademia Nazionale dei Lincei, and of several other academies.
wn.com/Cesare_Arzelà -
Felix Hausdorff
Felix Hausdorff (November 8, 1868 – January 26, 1942) was a German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory, descriptive set theory, measure theory, function theory, and functional analysis.
Hausdorff studied at the University of Leipzig, obtaining his Ph.D. in 1891. He taught mathematics in Leipzig until 1910, when he became professor of mathematics at the University of Bonn. He was professor at the University of Greifswald from 1913 to 1921. He then returned to Bonn. When the Nazis came to power, Hausdorff, who was Jewish, felt that as a respected university professor he would be spared from persecution. However, his abstract mathematics was denounced as "Jewish", useless, and "un-German"[citation needed] and he lost his position in 1935. Though he could no longer publish in Germany, Hausdorff continued to be an active research mathematician, publishing in the Polish journal Fundamenta Mathematicae. After Kristallnacht in 1938 as persecution of Jews escalated, Hausdorff became more and more isolated. He wrote to George Pólya requesting a research fellowship in the United States, but these efforts came to nothing. Finally, in 1942 when he could no longer avoid being sent to a concentration camp, Hausdorff committed suicide together with his wife, Charlotte Goldschmidt Hausdorff, and sister-in-law, Edith Goldschmidt Pappenheim, on 26 January. They are buried in Bonn, Germany.
wn.com/Felix_Hausdorff -
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor (
/ˈkæntɔr/ KAN-tor; German: [ɡeˈɔʁk ˈfɛʁdinant ˈluːtvɪç ˈfiːlɪp ˈkantɔʁ]; March 3 [O.S. February 19] 1845 – January 6, 1918) was a German mathematician, best known as the inventor of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are "more numerous" than the natural numbers. In fact, Cantor's method of proof of this theorem implies the existence of an "infinity of infinities". He defined the cardinal and ordinal numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact of which he was well aware.
wn.com/Georg_Cantor -
Giulio Ascoli
Giulio Ascoli (20 January 1843, Trieste – 12 July 1896, Milan) was an Italian mathematician. He was a student of the Scuola Normale di Pisa, where he graduated in 1868.
In 1872 he became Professor of Algebra and Calculus of the Politecnico di Milano University. From 1879 he was professor of mathematics at the Reale Istituto Tecnico Superiore, where, in 1901, was affixed a plaque that remembers him.
He was also corresponding member of Istituto Lombardo.
He made contributions to the theory of functions of a real variable and to Fourier series. For example, Ascoli introduced equicontinuity in 1884, a topic regarded as one of the fundamental concepts in the theory of real functions. In 1889, Italian mathematician Cesare Arzelà generalized Ascoli's Theorem into the Arzelà–Ascoli theorem, a practical sequential compactness criterion of functions.
wn.com/Giulio_Ascoli -
Hassler Whitney
Hassler Whitney (March 23, 1907 – May 10, 1989) was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersions, and characteristic classes.
Hassler Whitney was the son of First District New York Supreme Court judge Edward Baldwin Whitney and Josepha (Newcomb) Whitney, the grandson of Yale University Professor of Ancient Languages William Dwight Whitney, the great-grandson of Connecticut Governor and US Senator Roger Sherman Baldwin, and the great-great-great-grandson of American founding father Roger Sherman.
Hassler Whitney's maternal grandparents were astronomer and celestial mechanician Simon Newcomb and Mary Hassler Newcomb (the granddaughter of the first superintendent of the Coast Survey Ferdinand Hassler).
He married Margaret R. Howell, May 30, 1930; children: James Newcomb, Carol, Marian; married Mary Barnett Garfield, January 16, 1955; children: Sarah Newcomb, Emily Baldwin; and married Barbara Floyd Osterman, February 8, 1986.
wn.com/Hassler_Whitney -
Henri Poincaré
Jules Henri Poincaré (29 April 1854 – 17 July 1912) (French pronunciation: [ˈʒyl ɑ̃ˈʁi pwɛ̃nkaˈʁe]) was a French mathematician, theoretical physicist, engineer, and a philosopher of science. He is often described as a polymath, and in mathematics as The Last Universalist, since he excelled in all fields of the discipline as it existed during his lifetime.
As a mathematician and physicist, he made many original fundamental contributions to pure and applied mathematics, mathematical physics, and celestial mechanics. He was responsible for formulating the Poincaré conjecture, which was one of the most famous unsolved problems in mathematics until it was solved in 2002–2003. In his research on the three-body problem, Poincaré became the first person to discover a chaotic deterministic system which laid the foundations of modern chaos theory. He is also considered to be one of the founders of the field of topology.
Poincaré made clear importance of paying attention to the invariance of laws of physics under different transformations, and was the first to present the Lorentz transformations in their modern symmetrical form. Poincaré discovered the remaining relativistic velocity transformations and recorded them in a letter to Dutch physicist Hendrik Lorentz (1853–1928) in 1905. Thus he obtained perfect invariance of all of Maxwell's equations, an important step in the formulation of the theory of special relativity.
wn.com/Henri_Poincaré -
Jacques Hadamard
Jacques Salomon Hadamard FRS (8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex function theory, differential geometry and partial differential equations.
The son of a teacher, Amédée Hadamard, of Jewish descent, and Claire Marie Jeanne Picard, Hadamard was born in Versailles, France and attended the Lycée Charlemagne and Lycée Louis-le-Grand, where his father taught. In 1884 Hadamard entered the École Normale Supérieure, having been placed first in the entrance examinations both there and at the École Polytechnique. His teachers included Tannery, Hermite, Darboux, Appell, Goursat and Picard. He obtained his doctorate in 1892 and in the same year was awarded the Grand Prix des Sciences Mathématiques for his essay on the Riemann zeta function.
In 1892 Hadamard married Louise-Anna Trénel, also of Jewish descent, with whom he had three sons and two daughters. The following year he took up a lectureship in the University of Bordeaux, where he proved his celebrated inequality on determinants, which led to the discovery of Hadamard matrices when equality holds. In 1896 he made two important contributions: he proved the prime number theorem, using complex function theory (also proved independently by Charles Jean de la Vallée-Poussin); and he was awarded the Bordin Prize of the French Academy of Sciences for his work on geodesics in the differential geometry of surfaces and dynamical systems. In the same year he was appointed Professor of Astronomy and Rational Mechanics in Bordeaux. His foundational work on geometry and symbolic dynamics continued in 1898 with the study of geodesics on surfaces of negative curvature. For his cumulative work, he was awarded the Prix Poncelet in 1898.
wn.com/Jacques_Hadamard -
James Munkres
James Raymond Munkres (b. August 18, 1930) is a Professor Emeritus of mathematics at MIT and the author of several texts in the area of topology, including Topology (a well-known undergraduate-level text), Analysis on Manifolds, Elements of Algebraic Topology, and Elementary Differential Topology. He is also the author of Elementary Linear Algebra.
He completed his undergraduate education at Nebraska Wesleyan University and received his Ph.D. from the University of Michigan in 1956; his advisor was Edwin E. Moise. Earlier in his career he taught at the University of Michigan and at Princeton University.
Among his contributions to mathematics is the development of what is sometimes called the Munkres assignment algorithm.
wn.com/James_Munkres -
Kazimierz Kuratowski
Kazimierz Kuratowski (February 2, 1896 – June 18, 1980) was a Polish mathematician and logician. He was one of the leading representatives of the Warsaw School of Mathematics.
Kazimierz Kuratowski was born in Warsaw, then part of the Russian Empire, on February 2, 1896. He was a son of Marek Kuratow, a barrister, and Róża Karzewski. He completed a Warsaw secondary school, which was named after general Paweł Chrzanowski. In 1913, he enrolled in an engineering course at the University of Glasgow in Scotland, in part because he did not wish to study in Russian; instruction in Polish was prohibited. He completed only one year of study when the outbreak of World War I precluded any further enrollment. In 1915, Russian forces withdrew from Warsaw and Warsaw University was reopened with Polish as the language of instruction. Kuratowski restarted his university education there the same year, this time in mathematics. He obtained his Ph.D. in 1921, in newly independent Poland.
In autumn 1921 Kuratowski was awarded the Ph.D. degree for his groundbreaking work. His thesis statement consisted of two parts. One was devoted to an axiomatic construction of topology via the closure axioms (Sur la notion de l'ensemble fini, "Fundamenta Mathematicae", 1/1920). This first part (republished in a slightly modified form in 1922) has been cited in hundreds of scientific articles.
wn.com/Kazimierz_Kuratowski -
Leonhard Euler
Leonhard Euler (German pronunciation: [ˈɔʏlɐ], ( Swiss German pronunciation (help·info)) ( Standard German pronunciation (help·info)) English approximation, "Oiler"; 15 April 1707 – 18 September 1783) was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. He is also renowned for his work in mechanics, fluid dynamics, optics, and astronomy.
Euler spent most of his adult life in St. Petersburg, Russia, and in Berlin, Prussia. He is considered to be the preeminent mathematician of the 18th century, and one of the greatest of all time. He is also one of the most prolific mathematicians ever; his collected works fill 60–80 quarto volumes. A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master of us all."
wn.com/Leonhard_Euler -
Vito Volterra
Vito Volterra (3 May 1860 – 11 October 1940) was an Italian mathematician and physicist, known for his contributions to mathematical biology and integral equations.
Born in Ancona, then part of the Papal States, into a very poor Jewish family, Volterra showed early promise in mathematics before attending the University of Pisa, where he fell under the influence of Enrico Betti, and where he became professor of rational mechanics in 1883. He immediately started work developing his theory of functionals which led to his interest and later contributions in integral and integro-differential equations. His work is summarised in his book Theory of functionals and of Integral and Integro-Differential Equations (1930).
In 1892, he became professor of mechanics at the University of Turin and then, in 1900, professor of mathematical physics at the University of Rome La Sapienza. Volterra had grown up during the final stages of the Risorgimento when the Papal States were finally annexed by Italy and, like his mentor Betti, he was an enthusiastic patriot, being named by the king Victor Emmanuel III as a senator of the Kingdom of Italy in 1905. In the same year, he began to develop the theory of dislocations in crystals that was later to become important in the understanding of the behaviour of ductile materials. On the outbreak of World War I, already well into his 50s, he joined the Italian Army and worked on the development of airships under Giulio Douhet. He originated the idea of using inert helium rather than flammable hydrogen and made use of his leadership abilities in organising its manufacture.
wn.com/Vito_Volterra
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AlgTop0: Introduction to Algebraic Topology
- Order: Reorder
- Duration: 30:05
- Updated: 16 Feb 2013
This is the Introductory lecture to a beginner's course in Algebraic Topology given by NJ Wildberger of the School of Mathematics and Statistics at UNSW in 2010. This first lecture introduces some of the topics of the course and three problems. His YouTube site "Insights into Mathematics" at www.youtube.com under user: njwildberger also contains series on MathFoundations, History of Mathematics, LinearAlgebra, Rational Trigonometry and even one called Elementary Mathematics (K-6) Explained.
http://wn.com/AlgTop0_Introduction_to_Algebraic_Topology
This is the Introductory lecture to a beginner's course in Algebraic Topology given by NJ Wildberger of the School of Mathematics and Statistics at UNSW in 2010. This first lecture introduces some of the topics of the course and three problems. His YouTube site "Insights into Mathematics" at www.youtube.com under user: njwildberger also contains series on MathFoundations, History of Mathematics, LinearAlgebra, Rational Trigonometry and even one called Elementary Mathematics (K-6) Explained.
Bob Franzosa - Introduction to Topology
- Order: Reorder
- Duration: 54:04
- Updated: 07 Feb 2013
www.coa.edu 2010.02.09 Introduction to Topology From the Konigsberg Bridges to Geographic Information Systems. Topology is the study of the spatial properties of objects that are unchanged by stretching. It is considered one of the essential fields of modern pure mathematics, and has found many applications outside of mathematics as well.
http://wn.com/Bob_Franzosa_Introduction_to_Topology
www.coa.edu 2010.02.09 Introduction to Topology From the Konigsberg Bridges to Geographic Information Systems. Topology is the study of the spatial properties of objects that are unchanged by stretching. It is considered one of the essential fields of modern pure mathematics, and has found many applications outside of mathematics as well.
- published: 12 Feb 2010
- views: 23306
- author: collegeoftheatlantic
Hitler Learns Topology
- Order: Reorder
- Duration: 3:50
- Updated: 16 Feb 2013
Hitler gets confused about the topological definitions of open and closed sets. Then he totally freaks out.
http://wn.com/Hitler_Learns_Topology
Hitler gets confused about the topological definitions of open and closed sets. Then he totally freaks out.
- published: 01 Feb 2010
- views: 96809
- author: Benjatastic
Introduction to Topology
- Order: Reorder
- Duration: 8:30
- Updated: 09 Feb 2013
This first lecture of the series Comically Small Lectures On Point Set Topology. We will give: the definition of a topology, open and closed sets, and the basis for a topology. Examples of topologies, including the trivial topology, the discrete topology, particular point / excluded point topologies.
http://wn.com/Introduction_to_Topology
This first lecture of the series Comically Small Lectures On Point Set Topology. We will give: the definition of a topology, open and closed sets, and the basis for a topology. Examples of topologies, including the trivial topology, the discrete topology, particular point / excluded point topologies.
- published: 13 May 2009
- views: 45709
- author: jadenbane
General Topology Lecture 1 Part 1
- Order: Reorder
- Duration: 13:40
- Updated: 29 Jan 2013
Lecture written by Victor Victorov and edited and presented by James Dilts Definition and Examples
http://wn.com/General_Topology_Lecture_1_Part_1
Lecture written by Victor Victorov and edited and presented by James Dilts Definition and Examples
- published: 16 Mar 2011
- views: 22021
- author: vivictorov
AlgTop0a: Introduction to Algebraic Topology
- Order: Reorder
- Duration: 9:50
- Updated: 20 Jan 2013
NOTE: This entire series is now available in the full (hour long lectures), also at this channel under the playlist AlgTop (full lectures): www.youtube.com This is the first video of the introductory lecture of a beginner's course in Algebraic Topology. The subject is one of the most dynamic and exciting areas of 20th century mathematics, with its roots in the work of Riemann, Klein and Poincare in the latter half of the 19th century. This first lecture will outline the main topics, and will present three well-known but perhaps challenging problems for you to try. The course is for 3rd or 4th year undergraduate math students, but anyone with some mathematical maturity and a little background or willingness to learn group theory can benefit. The subject is particularly important for modern physics. Our treatment will have many standard features, but also some novelties. The lecturer is Assoc Prof NJ Wildberger of the School of Mathematics and Statistics at UNSW, Sydney, Australia.
http://wn.com/AlgTop0a_Introduction_to_Algebraic_Topology
NOTE: This entire series is now available in the full (hour long lectures), also at this channel under the playlist AlgTop (full lectures): www.youtube.com This is the first video of the introductory lecture of a beginner's course in Algebraic Topology. The subject is one of the most dynamic and exciting areas of 20th century mathematics, with its roots in the work of Riemann, Klein and Poincare in the latter half of the 19th century. This first lecture will outline the main topics, and will present three well-known but perhaps challenging problems for you to try. The course is for 3rd or 4th year undergraduate math students, but anyone with some mathematical maturity and a little background or willingness to learn group theory can benefit. The subject is particularly important for modern physics. Our treatment will have many standard features, but also some novelties. The lecturer is Assoc Prof NJ Wildberger of the School of Mathematics and Statistics at UNSW, Sydney, Australia.
- published: 22 Jul 2010
- views: 11198
- author: njwildberger
Computer Networking Tutorial - 3 - Network Topology
- Order: Reorder
- Duration: 6:59
- Updated: 13 Feb 2013
Introduction to computer networking. Any questions about the video? I will answer all questions at thenewboston.com
http://wn.com/Computer_Networking_Tutorial_3_Network_Topology
Introduction to computer networking. Any questions about the video? I will answer all questions at thenewboston.com
- published: 31 Jan 2009
- views: 221200
- author: thenewboston
Through one hole or two?
- Order: Reorder
- Duration: 0:45
- Updated: 16 Feb 2013
A topological magic trick
http://wn.com/Through_one_hole_or_two?
Topology #1 Metrics
- Order: Reorder
- Duration: 8:07
- Updated: 07 Feb 2013
Introduction to Metrics
http://wn.com/Topology_1_Metrics
GSM Network Topology
- Order: Reorder
- Duration: 5:47
- Updated: 17 Nov 2012
GSM Network Topology description using RF optimization tool OptPCS.
http://wn.com/GSM_Network_Topology
GSM Network Topology description using RF optimization tool OptPCS.
- published: 08 Feb 2011
- views: 17048
- author: optpcs
Spanning-Tree Protocol Topology Change Process
- Order: Reorder
- Duration: 9:25
- Updated: 16 Nov 2012
A general explanation of the Spanning-Tree Protocol, followed by an in-depth look at STP's topology change process.
http://wn.com/Spanning-Tree_Protocol_Topology_Change_Process
A general explanation of the Spanning-Tree Protocol, followed by an in-depth look at STP's topology change process.
- published: 23 Jan 2009
- views: 34243
- author: Trevor Landau
Robotics;Notes ED2 - Topology (Piano Transcription)
- Order: Reorder
- Duration: 1:47
- Updated: 17 Feb 2013
The book of my face: www.facebook.com Want to support my transcriptions? Donate here~ tinyurl.com Anime that I'm currently watching: myanimelist.net Sheet music + MIDI + mp3: sheethost.com ========== This transcription is a bit simple, mainly because I've been practicing Mirai Nikki's ED theme - Blood Teller. I'm quite liking Robotics;Notes, especially the latest episodes (Frau best girl ftw). The plot is finally picking up. Along with that, I'm also tuning in to AKB0048 Next Stage, Kotoura-san, and Psycho-Pass. Recording-wise: The audio of this video is from professional microphones: the Rode NT5 matched pair condenser mics. I'm testing it out for a week or two, and it's certainly an improvement from my Zoom H1. BUT, I think the quality of the mics is waaay overkill for my Yamaha grand, and thus I'm thinking of buying a cheaper microphone, but with similar good audio quality (like the Zoom H4n as suggested by pianohero87). If you have suggestions for cheaper microphones, please do tell~ And I will take this time to say: THANK YOU for the donators!!! I will not post names for privacy's sake, but you know who you are, and I truly appreciate your support. ;_; ========== Piano solo transcription by tehishter Original song: -Composer and Lyricist: Chiyomaru Shikura -Arranger: Toshimichi Isoe -Performer: Kanako Ito -Publisher: Aniplex ==========
http://wn.com/Robotics;Notes_ED2_Topology_Piano_Transcription
The book of my face: www.facebook.com Want to support my transcriptions? Donate here~ tinyurl.com Anime that I'm currently watching: myanimelist.net Sheet music + MIDI + mp3: sheethost.com ========== This transcription is a bit simple, mainly because I've been practicing Mirai Nikki's ED theme - Blood Teller. I'm quite liking Robotics;Notes, especially the latest episodes (Frau best girl ftw). The plot is finally picking up. Along with that, I'm also tuning in to AKB0048 Next Stage, Kotoura-san, and Psycho-Pass. Recording-wise: The audio of this video is from professional microphones: the Rode NT5 matched pair condenser mics. I'm testing it out for a week or two, and it's certainly an improvement from my Zoom H1. BUT, I think the quality of the mics is waaay overkill for my Yamaha grand, and thus I'm thinking of buying a cheaper microphone, but with similar good audio quality (like the Zoom H4n as suggested by pianohero87). If you have suggestions for cheaper microphones, please do tell~ And I will take this time to say: THANK YOU for the donators!!! I will not post names for privacy's sake, but you know who you are, and I truly appreciate your support. ;_; ========== Piano solo transcription by tehishter Original song: -Composer and Lyricist: Chiyomaru Shikura -Arranger: Toshimichi Isoe -Performer: Kanako Ito -Publisher: Aniplex ==========
- published: 04 Feb 2013
- views: 8602
- author: TehIshter
1/7 - Facial Intro Topology - Sintel
- Order: Reorder
- Duration: 6:45
- Updated: 11 Feb 2013
Part 1 of 7, created by Angela Guenette for "Sintel" DVD making-of extras, released September 2010. About this project: www.sintel.org Watch "Sintel" on YouTube youtu.be 4-disk DVD box : www.blender3d.org
http://wn.com/1_7_Facial_Intro_Topology_Sintel
Part 1 of 7, created by Angela Guenette for "Sintel" DVD making-of extras, released September 2010. About this project: www.sintel.org Watch "Sintel" on YouTube youtu.be 4-disk DVD box : www.blender3d.org
- published: 17 Jun 2011
- views: 36018
- author: PonderStudios
Working with Geodatabase Topology
- Order: Reorder
- Duration: 60:19
- Updated: 06 Feb 2013
العمل علي طبولوجي قواعد البيانات الجغرافية Geodatabase topology is used to maintain the spatial integrity of data. A geodatabase topology establishes and monitors topological relationships between features. By adding pre-defined topology rules, you can customize the topology to meet your needs and find errors. You can then use ArcMap topology tools to manage and edit these errors.
http://wn.com/Working_with_Geodatabase_Topology
العمل علي طبولوجي قواعد البيانات الجغرافية Geodatabase topology is used to maintain the spatial integrity of data. A geodatabase topology establishes and monitors topological relationships between features. By adding pre-defined topology rules, you can customize the topology to meet your needs and find errors. You can then use ArcMap topology tools to manage and edit these errors.
- published: 02 May 2011
- views: 6352
- author: Mahmoud Abdelrahman
Quanta, Symmetry, and Topology | Frank Wilczek
- Order: Reorder
- Duration: 69:25
- Updated: 29 Jan 2013
Frank Wilczek, Herman Feshbach Professor of Physics, Massachusetts Institute of Technology web.mit.edu Quantum theory radically transforms our fundamental understanding of physical reality. It reveals that the world contains a hidden richness of structure that we have barely begun to control and exploit. In this lecture, Frank Wilczek indicates the extraordinary potential of quantum engineering (the size and nature of Hilbert space); reviews one important ongoing effort to harness it (topological quantum computing); and speculates on its ultimate prospects (quantum minds). This lecture was part of the Institute for Advanced Study's celebration of its eightieth anniversary, and took place during the events related to the Schools of Mathematics and Natural Sciences. More videos at video.ias.edu
http://wn.com/Quanta,_Symmetry,_and_Topology_|_Frank_Wilczek
Frank Wilczek, Herman Feshbach Professor of Physics, Massachusetts Institute of Technology web.mit.edu Quantum theory radically transforms our fundamental understanding of physical reality. It reveals that the world contains a hidden richness of structure that we have barely begun to control and exploit. In this lecture, Frank Wilczek indicates the extraordinary potential of quantum engineering (the size and nature of Hilbert space); reviews one important ongoing effort to harness it (topological quantum computing); and speculates on its ultimate prospects (quantum minds). This lecture was part of the Institute for Advanced Study's celebration of its eightieth anniversary, and took place during the events related to the Schools of Mathematics and Natural Sciences. More videos at video.ias.edu
- published: 26 Apr 2012
- views: 3888
- author: videosfromIAS
Advanced Mobius Strip Topology (19087512 -210)
- Order: Reorder
- Duration: 9:09
- Updated: 01 Feb 2013
I've cut a 3 flip mobius counter clockwise strip down the middle using special mobius paper that I created. I then added half cuts to the plain so they can intersect for the purpose of getting rid of all twists. As you know, the resulting loop has 8 twists, a trefoil knot and two boundaries (surfaces), but it takes 6 lines to kill the 8 twists. The resulting paper soultion is then put on a page and turned into a knot diagram and knot notation. The knot diagram is then turned into a pseudograph consisting of vertices, lines (I meant to make them straight but I do this in my head) and loops. The findings are very very interesting! All the solutions have a knot crossing number of six (except for one so far - it had 8?), all pseudographs have a network Betti number of 7 and almost all the pseudographs have a Euler characteristic of 0. The original uncut mobius strip has a Euler number of 0! WOW. I think all of these findings are reproduceable and original because I cannot find any related papers so far. Can anyone help?
http://wn.com/Advanced_Mobius_Strip_Topology_19087512_-210
I've cut a 3 flip mobius counter clockwise strip down the middle using special mobius paper that I created. I then added half cuts to the plain so they can intersect for the purpose of getting rid of all twists. As you know, the resulting loop has 8 twists, a trefoil knot and two boundaries (surfaces), but it takes 6 lines to kill the 8 twists. The resulting paper soultion is then put on a page and turned into a knot diagram and knot notation. The knot diagram is then turned into a pseudograph consisting of vertices, lines (I meant to make them straight but I do this in my head) and loops. The findings are very very interesting! All the solutions have a knot crossing number of six (except for one so far - it had 8?), all pseudographs have a network Betti number of 7 and almost all the pseudographs have a Euler characteristic of 0. The original uncut mobius strip has a Euler number of 0! WOW. I think all of these findings are reproduceable and original because I cannot find any related papers so far. Can anyone help?
- published: 04 Jun 2008
- views: 9461
- author: David Sparks
Topology #4 Open Balls
- Order: Reorder
- Duration: 8:33
- Updated: 25 Jan 2013
Open Balls in a Metric Space
http://wn.com/Topology_4_Open_Balls
Topology #5 Open Sets
- Order: Reorder
- Duration: 5:25
- Updated: 07 Feb 2013
Open Sets in a Metric Space
http://wn.com/Topology_5_Open_Sets
Creating Topology in Topogun 1.06
- Order: Reorder
- Duration: 20:03
- Updated: 09 Feb 2013
somuchmonsters.com Short(is it?) walk-through of the tools in topogun. Thought about clipping out the end, but as enough people have asked me to slow down a little, I decided to leave the 1 time span. Sidenote: bourbons+microphone= rambling. YA! This is from a previous version of topogun but I thought I'd upload it now that the length cap is changed.
http://wn.com/Creating_Topology_in_Topogun_106
somuchmonsters.com Short(is it?) walk-through of the tools in topogun. Thought about clipping out the end, but as enough people have asked me to slow down a little, I decided to leave the 1 time span. Sidenote: bourbons+microphone= rambling. YA! This is from a previous version of topogun but I thought I'd upload it now that the length cap is changed.
- published: 13 Aug 2011
- views: 9869
- author: officerbeardontcare
Computer Network Training Course 1.1- Topologies
- Order: Reorder
- Duration: 7:06
- Updated: 06 Feb 2013
Computer Networking Training is an introduction to computer networking and the first part of a new training series. This introduction discusses basic network types, topologies, protocol, LAN, MAN, and WAN networks.
http://wn.com/Computer_Network_Training_Course_11_Topologies
Computer Networking Training is an introduction to computer networking and the first part of a new training series. This introduction discusses basic network types, topologies, protocol, LAN, MAN, and WAN networks.
- published: 06 Dec 2009
- views: 21225
- author: Greg French
Computer Networking Tutorial - 13 - Ring Topology
- Order: Reorder
- Duration: 6:41
- Updated: 16 Feb 2013
Visit my website at thenewboston.com for all of my videos! My Google+ - https Facebook - www.facebook.com My Twitter - twitter.com My Other YouTube Channel - www.youtube.com Donate - www.paypal.com
http://wn.com/Computer_Networking_Tutorial_13_Ring_Topology
Visit my website at thenewboston.com for all of my videos! My Google+ - https Facebook - www.facebook.com My Twitter - twitter.com My Other YouTube Channel - www.youtube.com Donate - www.paypal.com
- published: 13 Dec 2012
- views: 5454
- author: thenewboston
Using the Topology Freeform Function - Part 1
- Order: Reorder
- Duration: 4:29
- Updated: 05 Feb 2013
Level: Advanced Recorded in: 3ds Max 2011 Files used: areadownloads.autodesk.com Description: This movie showcases the Topology function found in the Graphite Freeform Tools that enables you to build a low-poly model from a hight-poly counterpart. This can be useful when you purchase or receive a model, and then discover that its geometry is imperfect. You can rebuild this geometry using the Topology function. This first part shows how to set up the topology tool and use it to block out the geometry of your new model.
http://wn.com/Using_the_Topology_Freeform_Function_Part_1
Level: Advanced Recorded in: 3ds Max 2011 Files used: areadownloads.autodesk.com Description: This movie showcases the Topology function found in the Graphite Freeform Tools that enables you to build a low-poly model from a hight-poly counterpart. This can be useful when you purchase or receive a model, and then discover that its geometry is imperfect. You can rebuild this geometry using the Topology function. This first part shows how to set up the topology tool and use it to block out the geometry of your new model.
- published: 13 Aug 2010
- views: 30296
- author: 3dsMaxHowTos
Problems in Topology, Post-Perelman - Stephen Smale
- Order: Reorder
- Duration: 46:34
- Updated: 16 Feb 2013
2010 Clay Research Conference Problems in Topology, Post-Perelman Stephen Smale Clay Mathematics Institute
http://wn.com/Problems_in_Topology,_Post-Perelman_Stephen_Smale
2010 Clay Research Conference Problems in Topology, Post-Perelman Stephen Smale Clay Mathematics Institute
- published: 24 Nov 2011
- views: 14170
- author: PoincareDuality
photo: AP
16 Feb 2013
February 16, 2013 -- Updated 0115 GMT (0915 HKT). Did you see the meteor? Share your images with CNN iReport. Moscow (CNN) -- A meteor streaked through the skies above Russia's Urals region Friday morning before exploding with a flash and boom that shattered glass in buildings and left about 1,000 people hurt, authorities said ... The injured included more than 200 children ... The meteor leaves a white streak through the sky ... HIDE CAPTION ... 1. 2....(size: 14.9Kb)
photo: AP / Nasha gazeta, www.ng.kz
15 Feb 2013
A meteor streaked across the sky above Russia's Ural Mountains today morning, causing sharp explosions and injuring more than 400 people, many of them hurt by broken glass. "There was panic. People had no idea what was happening ... Fragments of the meteor fell in a thinly populated area of the Chelyabinsk region, the Emergency Ministry said in a statement ... ....(size: 1.7Kb)
photo: UN / Mark Garten
16 Feb 2013
The United States has called on North Korea not to take any further provocative actions after report of plans for more nuclear tests. Friday's statement came after the North conducted its third nuclear test on Tuesday, drawing global condemnation ... Meanwhile, North Korea's leader has made his first public appearance since Tuesday's controversial nuclear test. ... Al Jazeera's Harry Fawcett, reporting from Seoul, said ... 479 Source.....(size: 7.5Kb)
photo: AP / Office of the Supreme Leader
17 Feb 2013
TEHRAN — Iran’s supreme leader said Saturday that his country was not seeking nuclear weapons but added that if Iran ever decided to build them, no “global power” could stop it. Related. Nuclear Watchdog Says No Deal Reached With Iran (February 15, 2013). Connect With Us on Twitter. Follow @nytimesworld for international breaking news and headlines. Twitter List. Reporters and Editors ... Mr ... During the same session, Mr....(size: 5.9Kb)
photo: AP
14 Feb 2013
In their hurry to flee last month, al Qaeda fighters left behind a crucial document. Tucked under a pile of papers and trash is a confidential letter, spelling out the terror network's strategy for conquering northern Mali and reflecting internal discord over how to rule the region ... Comparing the relationship of al Qaeda to Mali as that of an adult to an infant, he urges them to be more gentle, like a parent ... Ad Feedback ... - AP. ....(size: 12.4Kb)
noodls
15 Feb 2013
(Source. University of Illinois System) Article». Three U ... Hughes is interested in condensed matter systems, working with materials such as superconductors, topological insulators and graphene ... For example, recent research explored topological insulators - materials that conduct electricity only on their surface, and with very little energy dissipation - using quantum entanglement to describe the quantum properties of these materials....(size: 4.2Kb)
Yahoo Daily News
14 Feb 2013
Over the years YESpay has been responding to the Pan-European and North American credit/ debit, Mobile and Internet payments landscape, helping people ‘pay the new way’ ... For those seeking innovative topologies in payment systems, this book will be a great starting point with interesting insights on the mechanisms, future trends and opportunities ... Ron Kalifa, CEO, WorldPay ... Click here to register now. 13.00 – 13.30. Registration ... ....(size: 5.0Kb)
PhysOrg
13 Feb 2013
"This is the first demonstration of the existence of topological insulators based on organic materials," says Liu ... While other researchers still must synthesize the new organic topological insulators, Liu says his team's previous work "shows we can engineer an interface between two different thin films to create topological insulators," in which electrons known as Dirac fermions move along the interface between two films, Liu adds....(size: 3.2Kb)
IIFL
13 Feb 2013
The world’s first programmable digital controller specifically optimized for lighting and power-supply applications, the new STLUX385 (Masterlux platform) simplifies the design of traditional power-conversion topologies, enabling faster development of innovative lighting systems....(size: 4.5Kb)
noodls
13 Feb 2013
Called organic topological insulators, these materials are made from a thin molecular sheet (left) that resembles chicken wire and conducts electricity on its right edge (blue line) -- with the electrons carrying ......(size: 1.1Kb)
PhysOrg
13 Feb 2013
Ions are produced in Hall thrusters when electrons from an electric current collide with the propellant atoms to form a plasma in the discharge chamber ... Based on the numerical predictions, the effect of this magnetic field topology would be to accelerate ions away from walls while also significantly reducing their energy adjacent to the walls ... ....(size: 1.7Kb)
noodls
13 Feb 2013
(Source. GE - General Electric Company) ... GE) is working with SCE to put the smart grid to work by upgrading and modernizing the energy infrastructure ... About the Technologies ... Year 2020 distribution system including distribution automation with looped circuit topology, advanced voltage/VAR control, advanced distribution equipment, smart metering, utility-scale storage and renewable distributed generation ... About GE. GE (NYSE ... Doing. GE works....(size: 5.3Kb)
Independent online (SA)
13 Feb 2013
Cape Town - If there ever was a road to nowhere this is it. The infamous unfinished Eastern Boulevard freeway ends abruptly, challenging, mocking the function of the highway. It is not quite the idyllic Cape Town vista I hankered to see, though it is an unofficial landmark ... It’s not very photogenic ... It’s a nice fantasy ... It may have something to do with their differing topologies and the kind of lifestyles they engender ... ....(size: 18.7Kb)
noodls
12 Feb 2013
These materials are known as topological insulators ... There prove to be many more of these kinds of material than was thought possible.' The Leiden group identified 18 different topological insulators in two-dimensional crystals, and at least 70 in three-dimensional crystals. The Leiden reserachers identified 18 different topological insulators in two-dimensional crystals and at least 70 in three-dimensional crystals....(size: 4.0Kb)
Business Wire
12 Feb 2013
In order to meet these challenges Tarana has created a new wireless paradigm with their CMP topology”. The AbsoluteAir product line was designed and built as carrier grade from the ground up and is based upon the Tarana Topology™, a unique Concentrating Multipoint™ (CMP) architecture that features high capacity and availability....(size: 9.1Kb)
Yahoo Daily News
12 Feb 2013
This Small Form Factor module supports transfer rates to 1 Megabits per second and has jumper selectable termination resistors so it can be used in various topology systems ... The +5VDC power supply includes overvoltage, overcurrent, and short-circuit protection ... The PCM-CAN-2-ISO supports transfer rates to 1 Megabits per second and has jumper selectable termination resistors so it can be used in various topology systems ... PRODUCT PAGES ... ....(size: 3.3Kb)
noodls
12 Feb 2013
(Source. Infineon Technologies AG). Munich and Santa Clara, Calif. - February 12, 2013 - Today at Strategies in Light, Infineon Technologies AG (FSE. IFX / OTCQX ... The new drivers extend Infineon's portfolio of switched-mode, buck (step-down) topology LED drivers for low voltage driven lighting applications, including indoor and outdoor systems in residential, commercial and industrial environments ... Availability, Package, Pricing ... (noodl....(size: 4.0Kb)
Stockhouse
12 Feb 2013
Industry's smallest and easy-to-use drivers reduce component count and solution size. DALLAS, Feb. 12, 2013 /PRNewswire/ -- Texas Instruments Incorporated (TI) (NASDAQ ... Key features and benefits of TPS92075 and TPS92560 ... Use TI's online design calculator to determine external component values when configuring the TPS92560 and TPS92075 with boost or SEPIC topologies. www.ti.com/tps92560calc and www.ti.com/tps92075calc....(size: 4.2Kb)
Not to be confused with topography.
For other uses, see Topology (disambiguation).
Topology (from the Greek τόπος, “place”, and λόγος, “study”) is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing. It emerged through the development of concepts from geometry and set theory, such as space, dimension, and transformation.
Ideas that are now classified as topological were expressed as early as 1736. Toward the end of the 19th century, a distinct discipline developed, which was referred to in Latin as the geometria situs (“geometry of place”) or analysis situs (Greek-Latin for “picking apart of place”). This later acquired the modern name of topology. By the middle of the 20th century, topology had become an important area of study within mathematics.
The word topology is used both for the mathematical discipline and for a family of sets with certain properties that are used to define a topological space, a basic object of topology. Of particular importance are homeomorphisms, which can be defined as continuous functions with a continuous inverse.
Topology includes many subfields. The most basic and traditional division within topology is point-set topology, which establishes the foundational aspects of topology and investigates concepts inherent to topological spaces (basic examples include compactness and connectedness); algebraic topology, which generally tries to measure degrees of connectivity using algebraic constructs such as homotopy groups and homology; and geometric topology, which primarily studies manifolds and their embeddings (placements) in other manifolds. Some of the most active areas, such as low dimensional topology and graph theory, do not fit neatly in this division. Knot theory studies mathematical knots.
See also: topology glossary for definitions of some of the terms used in topology and topological space for a more technical treatment of the subject.
Contents |
History[link]
Topology began with the investigation of certain questions in geometry. Leonhard Euler's 1736 paper on the Seven Bridges of Königsberg[1] is regarded as one of the first academic treatises in modern topology.
The term "Topologie" was introduced in German in 1847 by Johann Benedict Listing in Vorstudien zur Topologie,[2] who had used the word for ten years in correspondence before its first appearance in print. "Topology," its English form, was first used in 1883 in Listing's obituary in the journal Nature[3] to distinguish "qualitative geometry from the ordinary geometry in which quantitative relations chiefly are treated". The term topologist in the sense of a specialist in topology was used in 1905 in the magazine Spectator.[citation needed] However, none of these uses corresponds exactly to the modern definition of topology.
Modern topology depends strongly on the ideas of set theory, developed by Georg Cantor in the later part of the 19th century. Cantor, in addition to establishing the basic ideas of set theory, considered point sets in Euclidean space as part of his study of Fourier series.
Henri Poincaré published Analysis Situs in 1895,[4] introducing the concepts of homotopy and homology, which are now considered part of algebraic topology.
Maurice Fréchet, unifying the work on function spaces of Cantor, Volterra, Arzelà, Hadamard, Ascoli, and others, introduced the metric space in 1906.[5] A metric space is now considered a special case of a general topological space. In 1914, Felix Hausdorff coined the term "topological space" and gave the definition for what is now called a Hausdorff space.[6] In current usage, a topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski.[citation needed]
For further developments, see point-set topology and algebraic topology.
Elementary introduction[link]
Topology, as a branch of mathematics, can be formally defined as "the study of qualitative properties of certain objects (called topological spaces) that are invariant under certain kind of transformations (called continuous maps), especially those properties that are invariant under a certain kind of equivalence (called homeomorphism)." To put it more simply, topology is the study of continuity and connectivity.
The term topology is also used to refer to a structure imposed upon a set X, a structure that essentially 'characterizes' the set X as a topological space by taking proper care of properties such as convergence, connectedness and continuity, upon transformation.
Topological spaces show up naturally in almost every branch of mathematics. This has made topology one of the great unifying ideas of mathematics.
The motivating insight behind topology is that some geometric problems depend not on the exact shape of the objects involved, but rather on the way they are put together. For example, the square and the circle have many properties in common: they are both one dimensional objects (from a topological point of view) and both separate the plane into two parts, the part inside and the part outside.
One of the first papers in topology was the demonstration, by Leonhard Euler, that it was impossible to find a route through the town of Königsberg (now Kaliningrad) that would cross each of its seven bridges exactly once. This result did not depend on the lengths of the bridges, nor on their distance from one another, but only on connectivity properties: which bridges are connected to which islands or riverbanks. This problem, the Seven Bridges of Königsberg, is now a famous problem in introductory mathematics, and led to the branch of mathematics known as graph theory.
Similarly, the hairy ball theorem of algebraic topology says that "one cannot comb the hair flat on a hairy ball without creating a cowlick." This fact is immediately convincing to most people, even though they might not recognize the more formal statement of the theorem, that there is no nonvanishing continuous tangent vector field on the sphere. As with the Bridges of Königsberg, the result does not depend on the exact shape of the sphere; it applies to pear shapes and in fact any kind of smooth blob, as long as it has no holes.
To deal with these problems that do not rely on the exact shape of the objects, one must be clear about just what properties these problems do rely on. From this need arises the notion of homeomorphism. The impossibility of crossing each bridge just once applies to any arrangement of bridges homeomorphic to those in Königsberg, and the hairy ball theorem applies to any space homeomorphic to a sphere.
Intuitively two spaces are homeomorphic if one can be deformed into the other without cutting or gluing. A traditional joke is that a topologist can't distinguish a coffee mug from a doughnut, since a sufficiently pliable doughnut could be reshaped to the form of a coffee cup by creating a dimple and progressively enlarging it, while shrinking the hole into a handle. A precise definition of homeomorphic, involving a continuous function with a continuous inverse, is necessarily more technical.
Homeomorphism can be considered the most basic topological equivalence. Another is homotopy equivalence. This is harder to describe without getting technical, but the essential notion is that two objects are homotopy equivalent if they both result from "squishing" some larger object.
| Equivalence classes of the English alphabet: | |
|---|---|
| Homeomorphism | Homotopy equivalence |
 |
 |
An introductory exercise is to classify the uppercase letters of the English alphabet according to homeomorphism and homotopy equivalence. The result depends partially on the font used. The figures use a sans-serif font named Myriad. Notice that homotopy equivalence is a rougher relationship than homeomorphism; a homotopy equivalence class can contain several of the homeomorphism classes. The simple case of homotopy equivalence described above can be used here to show two letters are homotopy equivalent, e.g. O fits inside P and the tail of the P can be squished to the "hole" part.
Thus, the homeomorphism classes are: one hole two tails, two holes no tail, no holes, one hole no tail, no holes three tails, a bar with four tails (the "bar" on the K is almost too short to see), one hole one tail, and no holes four tails.
The homotopy classes are larger, because the tails can be squished down to a point. The homotopy classes are: one hole, two holes, and no holes.
To be sure we have classified the letters correctly, we not only need to show that two letters in the same class are equivalent, but that two letters in different classes are not equivalent. In the case of homeomorphism, this can be done by suitably selecting points and showing their removal disconnects the letters differently. For example, X and Y are not homeomorphic because removing the center point of the X leaves four pieces; whatever point in Y corresponds to this point, its removal can leave at most three pieces. The case of homotopy equivalence is harder and requires a more elaborate argument showing an algebraic invariant, such as the fundamental group, is different on the supposedly differing classes.
Letter topology has some practical relevance in stencil typography. The font Braggadocio, for instance, has stencils that are made of one connected piece of material.
Mathematical definition[link]
Main article: Topological space
Let X be a set and let τ be a family of subsets of X. Then τ is called a topology on X if:
- Both the empty set and X are elements of τ
- Any union of elements of τ is an element of τ
- Any intersection of finitely many elements of τ is an element of τ
If τ is a topology on X, then the pair (X, τ) is called a topological space. The notation Xτ may be used to denote a set X endowed with the particular topology τ.
The members of τ are called open sets in X. A subset of X is said to be closed if its complement is in τ (i.e., its complement is open). A subset of X may be open, closed, both (clopen set), or neither. The empty set and X itself are always clopen.
A function or map from one topological space to another is called continuous if the inverse image of any open set is open. If the function maps the real numbers to the real numbers (both spaces with the Standard Topology), then this definition of continuous is equivalent to the definition of continuous in calculus. If a continuous function is one-to-one and onto, and if the inverse of the function is also continuous, then the function is called a homeomorphism and the domain of the function is said to be homeomorphic to the range. Another way of saying this is that the function has a natural extension to the topology. If two spaces are homeomorphic, they have identical topological properties, and are considered topologically the same. The cube and the sphere are homeomorphic, as are the coffee cup and the doughnut. But the circle is not homeomorphic to the doughnut.
Topology topics[link]
Some theorems in general topology[link]
- Every closed interval in R of finite length is compact. More is true: In Rn, a set is compact if and only if it is closed and bounded. (See Heine–Borel theorem).
- Every continuous image of a compact space is compact.
- Tychonoff's theorem: the (arbitrary) product of compact spaces is compact.
- A compact subspace of a Hausdorff space is closed.
- Every continuous bijection from a compact space to a Hausdorff space is necessarily a homeomorphism.
- Every sequence of points in a compact metric space has a convergent subsequence.
- Every interval in R is connected.
- Every compact finite-dimensional manifold can be embedded in some Euclidean space Rn.
- The continuous image of a connected space is connected.
- Every metric space is paracompact and Hausdorff, and thus normal.
- The metrization theorems provide necessary and sufficient conditions for a topology to come from a metric.
- The Tietze extension theorem: In a normal space, every continuous real-valued function defined on a closed subspace can be extended to a continuous map defined on the whole space.
- Any open subspace of a Baire space is itself a Baire space.
- The Baire category theorem: If X is a complete metric space or a locally compact Hausdorff space, then the interior of every union of countably many nowhere dense sets is empty.
- On a paracompact Hausdorff space every open cover admits a partition of unity subordinate to the cover.
- Every path-connected, locally path-connected and semi-locally simply connected space has a universal cover.
General topology also has some surprising connections to other areas of mathematics. For example:
- In number theory, Fürstenberg's proof of the infinitude of primes.
See also some counter-intuitive theorems, e.g. the Banach–Tarski one.
Some useful notions from algebraic topology[link]
See also list of algebraic topology topics.
- Homology and cohomology: Betti numbers, Euler characteristic, degree of a continuous mapping.
- Operations: cup product, Massey product
- Intuitively attractive applications: Brouwer fixed-point theorem, Hairy ball theorem, Borsuk–Ulam theorem, Ham sandwich theorem.
- Homotopy groups (including the fundamental group).
- Chern classes, Stiefel–Whitney classes, Pontryagin classes.
Generalizations[link]
Occasionally, one needs to use the tools of topology but a "set of points" is not available. In pointless topology one considers instead the lattice of open sets as the basic notion of the theory, while Grothendieck topologies are certain structures defined on arbitrary categories that allow the definition of sheaves on those categories, and with that the definition of quite general cohomology theories.
See also[link]
- List of algebraic topology topics
- List of general topology topics
- List of geometric topology topics
- List of topology topics
- Publications in topology
- Topology glossary
- General topology
- List of examples in general topology
References[link]
- ^ Euler, Leonhard, Solutio problematis ad geometriam situs pertinentis
- ^ Listing, Johann Benedict, "Vorstudien zur Topologie", Vandenhoeck und Ruprecht, Göttingen, p. 67, 1848
- ^ Tait, Peter Guthrie, "Johann Benedict Listing (obituary)", Nature *27*, 1 February 1883, pp. 316–317
- ^ Poincaré, Henri, "Analysis situs", Journal de l'École Polytechnique ser 2, 1 (1895) pp. 1–123
- ^ Fréchet, Maurice, "Sur quelques points du calcul fonctionnel", PhD dissertation, 1906
- ^ Hausdorff, Felix, "Grundzüge der Mengenlehre", Leipzig: Veit. In (Hausdorff Werke, II (2002), 91–576)
Further reading[link]
- Basener, William (2006). Topology and Its Applications (1st ed.). Wiley. ISBN 0-471-68755-3.
- Bourbaki; Elements of Mathematics: General Topology, Addison–Wesley (1966).
- Breitenberger, E. (2006). "Johann Benedict Listing". In I.M. James. History of Topology. North Holland. ISBN 978-0-444-82375-5.
- Kelley, John L. (1975). General Topology. Springer-Verlag. ISBN 0-387-90125-6.
- Munkres, James (1999). Topology (2nd ed.). Prentice Hall. ISBN 0-13-181629-2.
- Pickover, Clifford A. (2006). The Möbius Strip: Dr. August Möbius's Marvelous Band in Mathematics, Games, Literature, Art, Technology, and Cosmology. Thunder's Mouth Press (Provides a popular introduction to topology and geometry). ISBN 1-56025-826-8.
- Boto von Querenburg (2006). Mengentheoretische Topologie. Heidelberg: Springer-Lehrbuch. ISBN 3-540-67790-9 (German)
- Richeson, David S. (2008) Euler's Gem: The Polyhedron Formula and the Birth of Topology. Princeton University Press.
- Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.
External links[link]
 |
Wikimedia Commons has media related to: Topology |
 |
Wikibooks has more on the topic of |
- Elementary Topology: A First Course Viro, Ivanov, Netsvetaev, Kharlamov
- Topology at the Open Directory Project
- The Topological Zoo at The Geometry Center
- Topology Atlas
- Topology Course Lecture Notes Aisling McCluskey and Brian McMaster, Topology Atlas
- Topology Glossary
- Moscow 1935: Topology moving towards America, a historical essay by Hassler Whitney.
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This page contains text from Wikipedia, the Free Encyclopedia - http://en.wikipedia.org/wiki/Topology
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This article is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported License, which means that you can copy and modify it as long as the entire work (including additions) remains under this license.
| Frank Wilczek | |
|---|---|
 |
|
| Born | (1951-05-15) May 15, 1951 (age 61) Mineola, New York, U.S. |
| Nationality | United States |
| Fields | Physics |
| Institutions | MIT |
| Alma mater | University of Chicago (B.S.), Princeton University (M.A., Ph.D.) |
| Doctoral advisor | David Gross |
| Doctoral students | Mark Alford (*) Michael Forbes Martin Greiter Christoph Holzhey David Kessler Finn Larsen Richard MacKenzie John March-Russell (*) Chetan Nayak Maulik Parikh Krishna Rajagopal David Robertson Sean Robinson Alfred Shapere Stephen Wandzura (*): Jointly a Sidney Coleman student |
| Known for | Quantum chromodynamics |
| Notable awards | Lorentz Medal (2002) Nobel Prize in Physics (2004) |
| Spouse | Betsy Devine |
Frank Anthony Wilczek (born May 15, 1951) is a theoretical physicist from the United States and a Nobel laureate. He is currently the Herman Feshbach Professor of Physics at the Massachusetts Institute of Technology (MIT).
Wilczek, along with David Gross and H. David Politzer, was awarded the Nobel Prize in Physics in 2004 for their discovery of asymptotic freedom in the theory of the strong interaction.
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Biography[link]
Born in Mineola, New York, of Polish and Italian origin, Wilczek was educated in the public schools of Queens, attending Martin Van Buren High School. It was around this time Wilczek's parents realized that he was exceptional - in part as a result of Frank Wilczek having been administered an IQ test. [1]
He received his Bachelor of Science in Mathematics at the University of Chicago in 1970, a Master of Arts in Mathematics at Princeton University, 1972, and a Ph.D. in physics at Princeton University in 1974. Wilczek holds the Herman Feshbach Professorship of Physics at MIT Center for Theoretical Physics. He worked at the Institute for Advanced Study in Princeton and the Institute for Theoretical Physics at the University of California, Santa Barbara.
He was awarded the Lorentz Medal in 2002. Wilczek won the Lilienfeld Prize of the American Physical Society in 2003. In the same year he was awarded the Faculty of Mathematics and Physics Commemorative Medal from Charles University in Prague. He was the co-recipient of the 2003 High Energy and Particle Physics Prize of the European Physical Society. Wilczek was also the co-recipient of the 2005 King Faisal International Prize for Science.
He currently serves on the board for Society for Science & the Public.
Wilczek was married to Betsy Devine on July 3, 1973, and together have two daughters, Amity and Mira.
Wilczek has also appeared on an episode of Penn & Teller: Bullshit!, where Penn referred to him as "the smartest person [they have] ever had on the show."
Research[link]
In 1973 Wilczek, a graduate student working with David Gross at Princeton University, discovered asymptotic freedom, which holds that the closer quarks are to each other, the less the strong interaction (or color charge) between them; when quarks are in extreme proximity, the nuclear force between them is so weak that they behave almost as free particles. The theory, which was independently discovered by H. David Politzer, was important for the development of quantum chromodynamics.
Wilczek has helped to reveal and develop axions, anyons, asymptotic freedom, the color superconducting phases of quark matter, and other aspects of quantum field theory. He has worked on an unusually wide range of topics, ranging across condensed matter physics, astrophysics, and particle physics.
- Current research
- "Pure" particle physics: connections between theoretical ideas and observable phenomena;
- Behavior of matter: phase structure of quark matter at ultra-high temperature and density; color superconductivity;
- Application of particle physics to cosmology;
- Application of field theory techniques to condensed matter physics;
- Quantum theory of black holes.
Publications[link]
For lay readers[link]
- 2008. The Lightness of Being: Mass, Ether, and the Unification of Forces. Basic Books. ISBN 978-0-465-00321-1.
- 2007. La musica del vuoto. Roma: Di Renzo Editore.
- 2006. Fantastic Realities: 49 Mind Journeys And a Trip to Stockholm. World Scientific. ISBN 978-981-256-655-3.
- 2002, "On the world's numerical recipe (an ode to physics)," Daedalus 131(1): 142-47.
- 1989 (with Betsy Devine). Longing for the Harmonies: Themes and Variations from Modern Physics. W W Norton. ISBN 978-0-393-30596-8.
Technical[link]
- 1988. Geometric Phases in Physics.
- 1990. Fractional Statistics and Anyon Superconductivity.
- Wilczek, F.; Gross, D.J. (1973). "Asymptotically Free Gauge Theories. I". Physical Review D 8 (10): 3633. Bibcode 1973PhRvD...8.3633G. DOI:10.1103/PhysRevD.8.3633. OSTI 4312175.
- Wilczek, F.; Gross, D.J. (1973). "Ultraviolet Behavior of non-Abelian Gauge Theories". Physical Review Letters 30 (26): 1343. Bibcode 1973PhRvL..30.1343G. DOI:10.1103/PhysRevLett.30.1343.
- Wilczek, F., Zee, A. & S. B. Treiman (1974). Scaling Deviations for Neutrino Reactions in Aysmptotically Free Field Theories. Joseph Henry Laboratories. DOI:10.2172/4256152. OSTI 4256152
- Wilczek, F., Zee, A., Kingsley, R. L. & S. B. Treiman. (1975). "Weak Interaction Models with New Quarks and Right-handed Currents". Physical Review D 12 (9): 2768–2780. Bibcode 1975PhRvD..12.2768W. DOI:10.1103/PhysRevD.12.2768. OSTI 4082874.
- Wilczek, F. (1978). "Problem of Strong P and T Invariance in the Presence of Instantons". Physical Review Letters 40 (5): 279–282. Bibcode 1978PhRvL..40..279W. DOI:10.1103/PhysRevLett.40.279.
- Wilczek, F. (1982). "Quantum Mechanics of Fractional Spin Particles". Physical Review Letters 49 (14): 957. Bibcode 1982PhRvL..49..957W. DOI:10.1103/PhysRevLett.49.957.
- Wilczek, F. & M. S. Turner (1990). "Inflationary Axion Cosmology". Physical Review Letters 66 (1): 5–8. Bibcode 1991PhRvL..66....5T. DOI:10.1103/PhysRevLett.66.5. OSTI 6099352.
- Wilczek, F.; Alford, M.G.; Rajagopal, K. (1998). "QCD at finite baryon density: Nucleon droplets and color superconductivity". Physics Letters B 422 (422): 247–256. arXiv:hep-ph/9711395. Bibcode 1998PhLB..422..247A. DOI:10.1016/S0370-2693(98)00051-3.
- Wilczek, F. (1998). "Riemann-Einstein structure from volume and gauge symmetry". Physical Review Letters 80 (22): 4851–4854. arXiv:hep-th/9801184. Bibcode 1998PhRvL..80.4851W. DOI:10.1103/PhysRevLett.80.4851.
- Wilczek, F.; Fradkin, E.H.; Nayak, C,; Tsvelik, A (1998). "A Chern-Simons effective field theory for the Pfaffian quantum Hall state". Nuclear Physics B 516 (3): 704–718. arXiv:cond-mat/9711087. Bibcode 1998NuPhB.516..704F. DOI:10.1016/S0550-3213(98)00111-4.
- Wilczek, F.; Alford, M.G.; Rajagopal, K. (1999). "Color-flavor locking and chiral symmetry breaking in high density QCD". Nuclear Physics B 537: 443–458. arXiv:hep-ph/9804403. Bibcode 1999NuPhB.537..443A. DOI:10.1016/S0550-3213(98)00668-3.
- Wilczek, F. (1999). "Quantum field theory". Reviews of Modern Physics 71 (71): S85–S95. arXiv:hep-th/9803075. Bibcode 1999RvMPS..71...85W. DOI:10.1103/RevModPhys.71.S85.
- Wilczek, F.; T. Schafer and (1999). "Continuity of quark and hadron matter". Physical Review Letters 82 (82): 3956–3959. arXiv:hep-ph/9811473. Bibcode 1999PhRvL..82.3956S. DOI:10.1103/PhysRevLett.82.3956.
- Wilczek, F.; Babu, K.S.; Pati, J.C. (2000). "Fermion masses, neutrino oscillations, and proton decay in the light of SuperKamiokande". Nuclear Physics B 566: 33–91. arXiv:hep-ph/9812538. Bibcode 1998hep.ph...12538B. DOI:10.1016/S0550-3213(99)00589-1.
See also[link]
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Wikimedia Commons has media related to: Frank Wilczek |
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Wikiquote has a collection of quotations related to: Frank Wilczek |
- coupling unification
- Quantum chromodynamics
- black holes
- axion
- dark matter
- WIMP
- quantum number
- soliton
- fractional statistics
- Hall effect
- MIT Physics Department
- List of Poles
References[link]
- ^ Dreifus, Claudia (December 28, 2009). "Discovering the Mathematical Laws of Nature". The New York Times. http://www.nytimes.com/2009/12/29/science/29conv.html?_r=1. Retrieved 22 May 2012.
- "Frank Wilczek, Herman Feshbach Professor of Physics". Department of Physics, MIT. 2011. http://web.mit.edu/physics/people/faculty/wilczek_frank.html. Retrieved 2011-06-14.
- Tore Frängsmyr, editor (2005). location=Stockholm "The Nobel Prizes 2004". Les Prix Nobel. Nobel Foundation. http://nobelprize.org/physics/laureates/2004/wilczek-autobio.html location=Stockholm. Retrieved 2008-04-30.
External links[link]
- Biography and Bibliographic Resources, from the Office of Scientific and Technical Information, United States Department of Energy
- Longer biography at Lifeboat Foundation website
- Frank Wilczek explains Einstein's massive contributions to science
- Papers in ArXiv
- Frank Wilczek discusses his book "The Lightness of Being" on the 7th Avenue Project Radio Show
- The World's Numerical Recipe
- Scientific articles by Wilczek in the SLAC database
- Wilczek on anyons and superconductivity
- Blog of the Wilczek family's Nobel adventures
- Freeman Dyson, "Leaping into the Grand Unknown: Review of The Lightness of Being," The New York Review of Books 56(6), April 9, 2009.
- ForaTV: The Large Hadron Collider and Unified Field Theory
- A radio interview with Frank Wilczeck Aired on the Lewis Burke Frumkes Radio Show the 10th of April 2011.
- A television interview with Frank Wilczek from February 2011 for Cambridge University Television
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| Persondata | |
|---|---|
| Name | Wilczek, Frank |
| Alternative names | |
| Short description | Physicist |
| Date of birth | May 15, 1951 |
| Place of birth | Mineola, New York, U.S. |
| Date of death | |
| Place of death | |
http://wn.com/Frank_Wilczek
Related pages:
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http://cs.wn.com/Frank Wilczek
http://id.wn.com/Frank Wilczek
http://es.wn.com/Frank Wilczek
http://ru.wn.com/Вильчек, Фрэнк
http://nl.wn.com/Frank Wilczek
http://pl.wn.com/Frank Wilczek
http://fr.wn.com/Frank Wilczek
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This page contains text from Wikipedia, the Free Encyclopedia - http://en.wikipedia.org/wiki/Frank_Wilczek
This article is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported License, which means that you can copy and modify it as long as the entire work (including additions) remains under this license.
This article is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported License, which means that you can copy and modify it as long as the entire work (including additions) remains under this license.
| Stephen Smale | |
|---|---|
 |
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| Born | (1930-07-15) July 15, 1930 (age 81) Flint, Michigan |
| Nationality | American |
| Fields | Mathematics |
| Institutions | City University of Hong Kong University of Chicago Columbia University University of California, Berkeley |
| Alma mater | University of Michigan |
| Doctoral advisor | Raoul Bott |
| Doctoral students | Rufus Bowen John Guckenheimer Morris Hirsch Jacob Palis Themistocles M. Rassias |
| Notable awards | Wolf Prize (2006/07) National Medal of Science (1996) Fields Medal (1966) |
Steven Smale a.k.a. Steve Smale, Stephen Smale (born July 15, 1930) is an American mathematician from Flint, Michigan. He was awarded the Fields Medal in 1966, and spent more than three decades on the mathematics faculty of the University of California, Berkeley (1960–61 and 1964–1995).
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Education and career[link]
He entered the University of Michigan in 1948. Initially, Smale was a good student, placing into an honors calculus sequence taught by Bob Thrall and earning himself A's. However, his sophomore and junior years were marred with mediocre grades, mostly Bs, Cs and even an F in nuclear physics. However, with some luck, Smale was accepted as a graduate student at the University of Michigan's mathematics department. Yet again, Smale performed poorly his first years, earning a C average as a graduate student. It was only when the department chair, Hildebrant, threatened to kick out Smale, that he began to work hard[citation needed]. Smale finally earned his Ph.D. in 1957, under Raoul Bott.
Smale began his career as an instructor at the college at the University of Chicago. In 1958, he astounded the mathematical world with a proof of a sphere eversion. He then cemented his reputation with a proof of the Poincaré conjecture for all dimensions greater than or equal to 5; he later generalized the ideas in a 107 page paper that established the h-cobordism theorem.
After having made great strides in topology, he then turned to the study of dynamical systems, where he made significant advances as well. His first contribution is the Smale horseshoe that jumpstarted significant research in dynamical systems. He also outlined a research program carried out by many others. Smale is also known for injecting Morse theory into mathematical economics, as well as recent explorations of various theories of computation.
In 1998 he compiled a list of 18 problems in mathematics to be solved in the 21st century, known as Smale's problems. This list was compiled in the spirit of Hilbert's famous list of problems produced in 1900. In fact, Smale's list contains some of the original Hilbert problems, including the Riemann hypothesis and the second half of Hilbert's sixteenth problem, both of which are still unsolved. Other famous problems on his list include the Poincaré conjecture, the P = NP problem, and the Navier-Stokes equations, all of which have been designated Millennium Prize Problems by the Clay Mathematics Institute.
Earlier in his career, Smale was involved in controversy over remarks he made regarding his work habits while proving the higher dimensional Poincaré conjecture. He said that his best work had been done "on the beaches of Rio".[1] This led to the withholding of his grant money from the NSF. He has been politically active in various movements in the past, such as the Free Speech movement. At one time he was subpoenaed by the House Un-American Activities Committee.
In 1960 Smale was appointed an associate professor of mathematics at the University of California, Berkeley, moving to a professorship at Columbia University the following year. In 1964 he returned to a professorship at UC Berkeley where he has spent the main part of his career. He retired from UC Berkeley in 1995 and took up a post as professor at the City University of Hong Kong. He also amassed over the years one of the finest private mineral collections in existence. Many of Smale's mineral specimens can be seen in the book—The Smale Collection: Beauty in Natural Crystals.[2]
Since 2002 Smale is a Professor at the Toyota Technological Institute at Chicago; starting August 1, 2009, he is also a Distinguished University Professor at the City University of Hong Kong.[3]
In 2007, Smale was awarded the Wolf Prize in mathematics.[4]
Important publications[link]
- S. Smale, Generalized Poincaré's conjecture in dimensions greater than four, Annals of Mathematics, 2nd Ser., 74 (1961), no. 2, 391 – 406. (via JSTOR)
- S. Smale, Differentiable dynamical systems, Bulletin of the American Mathematical Society, 73 (1967), 747 – 817. ([1])
- F. Cucker & R Wong, The Collected Papers of Stephen Smale, ISBN 978-981-02-4307-4
- L. Blum, F. Cucker, M. Shub and S.Smale, Complexity and Real Computation, ISBN 0-387-98281-7.
References[link]
- ^ He discovered the famous Smale horseshoe map on a beach in Leme, Rio de Janeiro. See: S. Smale (1996), Chaos: Finding a Horseshoe on the Beaches of Rio.
- ^ http://www.lithographie.org/bookshop/the_smale_collection.htm
- ^ Stephen Smale Vita. Accessed November 18, 2009.
- ^ Press release
External links[link]
- Stephen Smale at the Mathematics Genealogy Project
- O'Connor, John J.; Robertson, Edmund F., "Stephen Smale", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Smale.html .
- Weisstein, Eric W., "Smale's Problems" from MathWorld.
- Robion Kirby, Stephen Smale: The Mathematician Who Broke the Dimension Barrier, a book review of a biography in the Notices of the AMS.
Personal Website at Universities[link]
- Steven Smale at the City University of Hong Kong
- Stephen Smale at the City University of Hong Kong
- Stephen Smale at the University of Chicago
- Steve Smale at the University of California, Berkeley
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| Persondata | |
|---|---|
| Name | Smale, Stephen |
| Alternative names | |
| Short description | Mathematician |
| Date of birth | July 15, 1930 |
| Place of birth | Flint, Michigan |
| Date of death | |
| Place of death | |
http://wn.com/Stephen_Smale
Related pages:
http://it.wn.com/Stephen Smale
http://es.wn.com/Stephen Smale
http://ru.wn.com/Смэйл, Стивен
http://nl.wn.com/Stephen Smale
http://pt.wn.com/Stephen Smale
http://pl.wn.com/Stephen Smale
http://fr.wn.com/Stephen Smale
http://de.wn.com/Stephen Smale
This page contains text from Wikipedia, the Free Encyclopedia - http://en.wikipedia.org/wiki/Stephen_Smale
This article is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported License, which means that you can copy and modify it as long as the entire work (including additions) remains under this license.
This article is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported License, which means that you can copy and modify it as long as the entire work (including additions) remains under this license.
