Jean-Pierre Eckmann is a mathematical physicist in the department of theoretical physics at the University of Geneva and a pioneer of chaos theory and social network analysis. Eckmann is the son of mathematician Beno Eckmann, he completed his Ph. D. in 1970 under the supervision of Marcel Guenin at the University of Geneva. He has been a member of the Academia Europaea since 2001. In 2012 he became a fellow of the American Mathematical Society, he is a member of the Göttingen Academy of Sciences and Humanities. With Pierre Collet and Oscar Lanford, Eckmann was the first to find a rigorous mathematical argument for the universality of period-doubling bifurcations in dynamical systems, with scaling ratio given by the Feigenbaum constants. In a cited 1985 review paper with David Ruelle, he bridged the contributions of mathematicians and physicists to dynamical systems theory and ergodic theory, put the varied work on dimension-like notions in these fields on a firm mathematical footing, formulated the Eckmann–Ruelle conjecture on the dimension of hyperbolic ergodic measures, "one of the main problems in the interface of dimension theory and dynamical systems".
A proof of the conjecture was published 14 years in 1999. Eckmann has done additional mathematical work in diverse fields such as statistical mechanics, partial differential equations, graph theory, his PhD students have included Viviane Baladi, Pierre Collet, Martin Hairer. Website of Jean-Pierre Eckmann
Mitchell Jay Feigenbaum is a mathematical physicist whose pioneering studies in chaos theory led to the discovery of the Feigenbaum constants. Feigenbaum was born to Polish and Ukrainian Jewish immigrants, he attended Samuel J. Tilden High School, in Brooklyn, New York, the City College of New York. In 1964 he began his graduate studies at the Massachusetts Institute of Technology. Enrolling for graduate study in electrical engineering, he changed his area to physics, he completed his doctorate in 1970 for a thesis on dispersion relations, under the supervision of Professor Francis E. Low. After short positions at Cornell University and the Virginia Polytechnic Institute and State University, he was offered a longer-term post at the Los Alamos National Laboratory in New Mexico to study turbulence in fluids. Although that group of researchers was unable to unravel the intractable theory of turbulent fluids, his research led him to study chaotic maps. In 1983, he was awarded a MacArthur Fellowship, in 1986, he was awarded the Wolf Prize in Physics "for his pioneering theoretical studies demonstrating the universal character of non-linear systems, which has made possible the systematic study of chaos".
He is a member of the Board of Scientific Governors at The Scripps Research Institute. He has been Toyota Professor at Rockefeller University since 1986; some mathematical mappings involving a single linear parameter exhibit the random behavior known as chaos when the parameter lies within certain ranges. As the parameter is increased towards this region, the mapping undergoes bifurcations at precise values of the parameter. At first there is one stable point bifurcating to an oscillation between two values bifurcating again to oscillate between four values and so on. In 1975, Dr. Feigenbaum, using the small HP-65 calculator he had been issued, discovered that the ratio of the difference between the values at which such successive period-doubling bifurcations occur tends to a constant of around 4.6692... He was able to provide a mathematical proof of that fact, he showed that the same behavior, with the same mathematical constant, would occur within a wide class of mathematical functions, prior to the onset of chaos.
For the first time, this universal result enabled mathematicians to take their first steps to unraveling the intractable "random" behavior of chaotic systems. This "ratio of convergence" is now known as the first Feigenbaum constant; the logistic map is a prominent example of the mappings that Feigenbaum studied in his noted 1978 article: Quantitative Universality for a Class of Nonlinear Transformations. Feigenbaum's other contributions include important new fractal methods in cartography, starting when he was hired by Hammond to develop techniques to allow computers to assist in drawing maps; the introduction to the Hammond Atlas states: Using fractal geometry to describe natural forms such as coastlines, mathematical physicist Mitchell Feigenbaum developed software capable of reconfiguring coastlines and mountain ranges to fit a multitude of map scales and projections. Dr. Feigenbaum created a new computerized type placement program which places thousands of map labels in minutes, a task that required days of tedious labor.
In another practical application of his work, he founded Numerix with Michael Goodkin in 1996. The company’s initial product was a software algorithm that reduced the time required for Monte Carlo pricing of exotic financial derivatives and structured products. Numerix remains one of the leading software providers to financial market participants; the press release made on the occasion of his receiving the Wolf Prize summed up his works: The impact of Feigenbaum's discoveries has been phenomenal. It has spanned new fields of theoretical and experimental mathematics... It is hard to think of any other development in recent theoretical science that has had so broad an impact over so wide a range of fields, spanning both the pure and the applied. Feigenbaum was referenced on the Season 5, Episode 15 of the TV series Angel. While suffering a deadly illness, the character Winnifred "Fred" Burkle—a major protagonist, a physicist—exchanges the following dialogue: FRED: Feigenbaum. WESLEY: What? FRED: I—I have to find him.
He's the master of— I have to have Feigenbaum here. WESLEY: Who is Feigenbaum? FRED: I don't remember. Logistic map Theory of Colours O'Connor, John J.. Feigenbaum's webpage at Rockefeller Feigenbaum, Mitchell J.. "The Theory of Relativity - Galileo's Child". ArXiv:0806.1234
ArXiv is a repository of electronic preprints approved for posting after moderation, but not full peer review. It consists of scientific papers in the fields of mathematics, astronomy, electrical engineering, computer science, quantitative biology, mathematical finance and economics, which can be accessed online. In many fields of mathematics and physics all scientific papers are self-archived on the arXiv repository. Begun on August 14, 1991, arXiv.org passed the half-million-article milestone on October 3, 2008, had hit a million by the end of 2014. By October 2016 the submission rate had grown to more than 10,000 per month. ArXiv was made possible by the compact TeX file format, which allowed scientific papers to be transmitted over the Internet and rendered client-side. Around 1990, Joanne Cohn began emailing physics preprints to colleagues as TeX files, but the number of papers being sent soon filled mailboxes to capacity. Paul Ginsparg recognized the need for central storage, in August 1991 he created a central repository mailbox stored at the Los Alamos National Laboratory which could be accessed from any computer.
Additional modes of access were soon added: FTP in 1991, Gopher in 1992, the World Wide Web in 1993. The term e-print was adopted to describe the articles, it began as a physics archive, called the LANL preprint archive, but soon expanded to include astronomy, computer science, quantitative biology and, most statistics. Its original domain name was xxx.lanl.gov. Due to LANL's lack of interest in the expanding technology, in 2001 Ginsparg changed institutions to Cornell University and changed the name of the repository to arXiv.org. It is now hosted principally with eight mirrors around the world, its existence was one of the precipitating factors that led to the current movement in scientific publishing known as open access. Mathematicians and scientists upload their papers to arXiv.org for worldwide access and sometimes for reviews before they are published in peer-reviewed journals. Ginsparg was awarded a MacArthur Fellowship in 2002 for his establishment of arXiv; the annual budget for arXiv is $826,000 for 2013 to 2017, funded jointly by Cornell University Library, the Simons Foundation and annual fee income from member institutions.
This model arose in 2010, when Cornell sought to broaden the financial funding of the project by asking institutions to make annual voluntary contributions based on the amount of download usage by each institution. Each member institution pledges a five-year funding commitment to support arXiv. Based on institutional usage ranking, the annual fees are set in four tiers from $1,000 to $4,400. Cornell's goal is to raise at least $504,000 per year through membership fees generated by 220 institutions. In September 2011, Cornell University Library took overall administrative and financial responsibility for arXiv's operation and development. Ginsparg was quoted in the Chronicle of Higher Education as saying it "was supposed to be a three-hour tour, not a life sentence". However, Ginsparg remains on the arXiv Scientific Advisory Board and on the arXiv Physics Advisory Committee. Although arXiv is not peer reviewed, a collection of moderators for each area review the submissions; the lists of moderators for many sections of arXiv are publicly available, but moderators for most of the physics sections remain unlisted.
Additionally, an "endorsement" system was introduced in 2004 as part of an effort to ensure content is relevant and of interest to current research in the specified disciplines. Under the system, for categories that use it, an author must be endorsed by an established arXiv author before being allowed to submit papers to those categories. Endorsers are not asked to review the paper for errors, but to check whether the paper is appropriate for the intended subject area. New authors from recognized academic institutions receive automatic endorsement, which in practice means that they do not need to deal with the endorsement system at all. However, the endorsement system has attracted criticism for restricting scientific inquiry. A majority of the e-prints are submitted to journals for publication, but some work, including some influential papers, remain purely as e-prints and are never published in a peer-reviewed journal. A well-known example of the latter is an outline of a proof of Thurston's geometrization conjecture, including the Poincaré conjecture as a particular case, uploaded by Grigori Perelman in November 2002.
Perelman appears content to forgo the traditional peer-reviewed journal process, stating: "If anybody is interested in my way of solving the problem, it's all there – let them go and read about it". Despite this non-traditional method of publication, other mathematicians recognized this work by offering the Fields Medal and Clay Mathematics Millennium Prizes to Perelman, both of which he refused. Papers can be submitted in any of several formats, including LaTeX, PDF printed from a word processor other than TeX or LaTeX; the submission is rejected by the arXiv software if generating the final PDF file fails, if any image file is too large, or if the total size of the submission is too large. ArXiv now allows one to store and modify an incomplete submission, only finalize the submission when ready; the time stamp on the article is set. The standard access route is through one of several mirrors. Sev
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883. Through consideration of this set and others helped lay the foundations of modern point-set topology. Although Cantor himself defined the set in a general, abstract way, the most common modern construction is the Cantor ternary set, built by removing the middle third of a line segment and repeating the process with the remaining shorter segments. Cantor himself mentioned the ternary construction only in passing, as an example of a more general idea, that of a perfect set, nowhere dense; the Cantor ternary set C is created by iteratively deleting the open middle third from a set of line segments. One starts by deleting the open middle third from the interval, leaving two line segments: ∪. Next, the open middle third of each of these remaining segments is deleted, leaving four line segments: ∪ ∪ ∪.
This process is continued ad infinitum, where the nth set is C n = C n − 1 3 ∪ for n ≥ 1, C 0 =. The Cantor ternary set contains all points in the interval that are not deleted at any step in this infinite process: C:= ⋂ n = 1 ∞ C n; the first six steps of this process are illustrated below. Using the idea of self-similar transformations, T L = x / 3, T R = / 3 and C n = T L ∪ T R, the explicit closed formulas for the Cantor set are C = ∖ ⋃ n = 1 ∞ ⋃ k = 0 3 n − 1, where every middle third is removed as the open interval from the closed interval = surrounding it, or C = ⋂ n = 1 ∞ ⋃ k = 0 3 n − 1 − 1 ( ∪ [ 3 k + 2