Open content is a neologism coined by David Wiley in 1998 which describes a creative work that others can copy or modify without asking for permission. The term evokes the related concept of open-source software; such content is said to be under an open licence. The concept of applying free software licenses to content was introduced by Michael Stutz, who in 1994 wrote the paper "Applying Copyleft to Non-Software Information" for the GNU Project; the term "open content" was coined by David A. Wiley in 1998 and evangelized via the Open Content Project, describing works licensed under the Open Content License and other works licensed under similar terms, it has since come to describe a broader class of content without conventional copyright restrictions. The openness of content can be assessed under the'5Rs Framework' based on the extent to which it can be reused, revised and redistributed by members of the public without violating copyright law. Unlike free content and content under open-source licenses, there is no clear threshold that a work must reach to qualify as'open content'.
Although open content has been described as a counterbalance to copyright, open content licenses rely on a copyright holder's power to license their work as copyleft which utilizes copyright for such a purpose. In 2003 Wiley announced that the Open Content Project has been succeeded by Creative Commons and their licenses, where he joined as "Director of Educational Licenses". In 2006 the Creative Commons' successor project was the Definition of Free Cultural Works for free content, put forth by Erik Möller, Richard Stallman, Lawrence Lessig, Benjamin Mako Hill, Angela Beesley, others; the Definition of Free Cultural Works is used by the Wikimedia Foundation. In 2008, the Attribution and Attribution-ShareAlike Creative Commons licenses were marked as "Approved for Free Cultural Works" among other licenses. Another successor project is the Open Knowledge Foundation, founded by Rufus Pollock in Cambridge, UK in 2004 as a global non-profit network to promote and share open content and data. In 2007 the Open Knowledge Foundation gave an Open Knowledge Definition for "Content such as music, books.
In October 2014 with version 2.0 Open Works and Open Licenses were defined and "open" is described as synonymous to the definitions of open/free in the Open Source Definition, the Free Software Definition and the Definition of Free Cultural Works. A distinct difference is the focus given to the public domain and that it focuses on the accessibility and the readability. Among several conformant licenses, six are recommended, three own and the CC BY, CC BY-SA, CC0 creative commons licenses; the OpenContent website once defined OpenContent as'freely available for modification and redistribution under a license similar to those used by the open-source / free software community'. However, such a definition would exclude the Open Content License because that license forbade charging'a fee for the itself', a right required by free and open-source software licenses; the term since shifted in meaning. OpenContent "is licensed in a manner that provides users with free and perpetual permission to engage in the 5R activities."The 5Rs are put forward on the OpenContent website as a framework for assessing the extent to which content is open: Retain – the right to make and control copies of the content Reuse – the right to use the content in a wide range of ways Revise – the right to adapt, modify, or alter the content itself Remix – the right to combine the original or revised content with other open content to create something new Redistribute – the right to share copies of the original content, your revisions, or your remixes with others This broader definition distinguishes open content from open-source software, since the latter must be available for commercial use by the public.
However, it is similar to several definitions for open educational resources, which include resources under noncommercial and verbatim licenses. The Open Definition by the Open Knowledge Foundation define open knowledge with open content and open data as sub-elements and draws on the Open Source Definition. "Open access" refers to toll-free or gratis access to content published peer-reviewed scholarly journals. Some open access works are licensed for reuse and redistribution, which would qualify them as open content. Over the past decade, open content has been used to develop alternative routes towards higher education. Traditional universities are expensive, their tuition rates are increasing. Open content allows a free way of obtaining higher education, "focused on collective knowledge and the sharing and reuse of learning and scholarly content." There are multiple projects and organizations that promote learning through open content, including OpenCourseWare Initiative, The Saylor Foundation and Khan Academy.
Some universities, like MIT, Tufts are making their courses available on the internet. The textbook industry is one of the educational in
In mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exists a real number such that, for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is not greater than this real number. For instance, every function that has bounded first derivatives is Lipschitz. In the theory of differential equations, Lipschitz continuity is the central condition of the Picard–Lindelöf theorem which guarantees the existence and uniqueness of the solution to an initial value problem. A special type of Lipschitz continuity, called contraction, is used in the Banach fixed point theorem. We have the following chain of strict inclusions for functions over a closed and bounded non-trivial interval of the real line Continuously differentiable ⊂ Lipschitz continuous ⊂ α-Hölder continuous ⊂ uniformly continuous = continuouswhere 0 < α ≤ 1.
We have Lipschitz continuous ⊂ continuous ⊂ bounded variation ⊂ differentiable everywhere Given two metric spaces and, where dX denotes the metric on the set X and dY is the metric on set Y, a function f: X → Y is called Lipschitz continuous if there exists a real constant K ≥ 0 such that, for all x1 and x2 in X, d Y ≤ K d X. Any such K is referred to as a Lipschitz constant for the function f; the smallest constant is sometimes called the Lipschitz constant. If K = 1 the function is called a short map, if 0 ≤ K < 1 and f maps a metric space to itself, the function is called a contraction. In particular, a real-valued function f: R → R is called Lipschitz continuous if there exists a positive real constant K such that, for all real x1 and x2, | f − f | ≤ K | x 1 − x 2 |. In this case, Y is the set of real numbers R with the standard metric dY = |y1 − y2|, X is a subset of R. In general, the inequality is satisfied if x1 = x2. Otherwise, one can equivalently define a function to be Lipschitz continuous if and only if there exists a constant K ≥ 0 such that, for all x1 ≠ x2, d Y d X ≤ K.
For real-valued functions of several real variables, this holds if and only if the absolute value of the slopes of all secant lines are bounded by K. The set of lines of slope K passing through a point on the graph of the function forms a circular cone, a function is Lipschitz if and only if the graph of the function everywhere lies outside of this cone. A function is called locally Lipschitz continuous if for every x in X there exists a neighborhood U of x such that f restricted to U is Lipschitz continuous. Equivalently, if X is a locally compact metric space f is locally Lipschitz if and only if it is Lipschitz continuous on every compact subset of X. In spaces that are not locally compact, this is a necessary but not a sufficient condition. More a function f defined on X is said to be Hölder continuous or to satisfy a Hölder condition of order α > 0 on X if there exists a constant M > 0 such that d Y ≤ M d X α for all x and y in X. Sometimes a Hölder condition of order α is called a uniform Lipschitz condition of order α > 0.
If there exists a K ≥ 1 with 1 K d X ≤ d Y ≤ K d X f is called bilipschitz. A bilipschitz mapping is injective, is in fact a homeomorphism onto its image. A bilipschitz function is the same thing as an injective Lipschitz function whose inverse function is Lipschitz. Lipschitz continuous functions Lipschitz continuous functions that are not everywhere differentiable Lipschitz continuous functions that are everywhere differentia
Brouwer fixed-point theorem
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. Brouwer, it states that for any continuous function f mapping a compact convex set to itself there is a point x 0 such that f = x 0. The simplest forms of Brouwer's theorem are for continuous functions f from a closed interval I in the real numbers to itself or from a closed disk D to itself. A more general form than the latter is for continuous functions from a convex compact subset K of Euclidean space to itself. Among hundreds of fixed-point theorems, Brouwer's is well known, due in part to its use across numerous fields of mathematics. In its original field, this result is one of the key theorems characterizing the topology of Euclidean spaces, along with the Jordan curve theorem, the hairy ball theorem and the Borsuk–Ulam theorem; this gives it a place among the fundamental theorems of topology. The theorem is used for proving deep results about differential equations and is covered in most introductory courses on differential geometry.
It appears in unlikely fields such as game theory. In economics, Brouwer's fixed-point theorem and its extension, the Kakutani fixed-point theorem, play a central role in the proof of existence of general equilibrium in market economies as developed in the 1950s by economics Nobel prize winners Kenneth Arrow and Gérard Debreu; the theorem was first studied in view of work on differential equations by the French mathematicians around Henri Poincaré and Charles Émile Picard. Proving results such as the Poincaré–Bendixson theorem requires the use of topological methods; this work at the end of the 19th century opened into several successive versions of the theorem. The general case was first proved in 1910 by Luitzen Egbertus Jan Brouwer; the theorem has several formulations, depending on the context in which it is used and its degree of generalization. The simplest is sometimes given as follows: In the plane Every continuous function from a closed disk to itself has at least one fixed point; this can be generalized to an arbitrary finite dimension: In Euclidean space Every continuous function from a closed ball of a Euclidean space into itself has a fixed point.
A more general version is as follows: Convex compact set Every continuous function from a convex compact subset K of a Euclidean space to K itself has a fixed point. An more general form is better known under a different name: Schauder fixed point theorem Every continuous function from a convex compact subset K of a Banach space to K itself has a fixed point; the theorem holds only for sets that are convex. The following examples show. Consider the function f = x + 1, a continuous function from R to itself; as it shifts every point to the right, it cannot have a fixed point. Note that R is convex and closed, but not bounded. Consider the function f = x + 1 2, a continuous function from the open interval to itself. In this interval, it shifts every point to the right, so it cannot have a fixed point. Note, convex and bounded, but not closed; the function f does have a fixed point for the closed interval, namely f = 1. Note that convexity is not necessary for BFPT; because the properties involved are invariant under homeomorphisms, BFPT is equivalent to forms in which the domain is required to be a closed unit ball D n.
For the same reason it holds for every set, homeomorphic to a closed ball. The following example shows. Consider the following function, defined in polar coordinates: f =, a continuous function from the unit circle to itself, it rotates every point on the unit circle 45 degrees counterclockwise, so it cannot have a fixed point. Note that the unit circle is closed and bounded, but it has a hole; the function f does have a fixed point for the unit disc. A formal generalization of BFPT for "hole-free" domains can be derived from the Lefschetz fixed-point theorem; the continuous function in this theorem is not required to be bijective or surjective. The theorem has several "real world" illustrations. Here are some examples. 1. Take two sheets of graph paper of equal size with coordinate systems on them, lay one flat on the table and crumple up the other one and place it, in any fashion, on top of the first so that the crumpled paper does not reach outside the flat one. There will be at least one point of the crumpled sheet that lies directly above its corresponding point of the flat sheet.
This is a consequence of the n = 2 case o
Infinite compositions of analytic functions
In mathematics, infinite compositions of analytic functions offer alternative formulations of analytic continued fractions, series and other infinite expansions, the theory evolving from such compositions may shed light on the convergence/divergence of these expansions. Some functions can be expanded directly as infinite compositions. In addition, it is possible to use ICAF to evaluate solutions of fixed point equations involving infinite expansions. Complex dynamics offers another venue for iteration of systems of functions rather than a single function. For infinite compositions of a single function see. For compositions of a finite number of functions, useful in fractal theory, see Iterated function system. Although the title of this article specifies analytic functions, there are results for more general functions of a complex variable as well. There are several notations describing infinite compositions, including the following: Forward compositions: Fk,n = fk ∘ fk+1 ∘... ∘ fn−1 ∘ fn. Backward compositions: Gk,n = fn ∘ fn−1 ∘... ∘ fk+1 ∘ fk In each case convergence is interpreted as the existence of the following limits: lim n → ∞ F 1, n, lim n → ∞ G 1, n.
For convenience, set Fn = F1,n and Gn = G1,n. One may write F n = R n k = 1 f k = f 1 ∘ f 2 ∘ ⋯ ∘ f n and G n = L n k = 1 g k = g n ∘ g n − 1 ∘ ⋯ ∘ g 1 Many results can be considered extensions of the following result: Contraction Theorem for Analytic Functions. Let f be analytic in a simply-connected region S and continuous on the closure S of S. Suppose f is a bounded set contained in S. For all z in S there exists an attractive fixed point α of f in S such that: F n = → α, Let be a sequence of functions analytic on a simply-connected domain S. Suppose there exists a compact set Ω ⊂ S such that for each n, fn ⊂ Ω. Forward Compositions Theorem. Converges uniformly on compact subsets of S to a constant function F = λ. Backward Compositions Theorem. Converges uniformly on compact subsets of S to γ ∈ Ω if and only if the sequence of fixed points of the converges to γ. Additional theory resulting from investigations based on these two theorems Forward Compositions Theorem, include location analysis for the limits obtained here.
For a different approach to Backward Compositions Theorem, see. Regarding Backward Compositions Theorem, the example f2n = 1/2 and f2n−1 = −1/2 for S = demonstrates the inadequacy of requiring contraction into a compact subset, like Forward Compositions Theorem. For functions not analytic the Lipschitz condition suffices: Theorem. Suppose S is a connected compact subset of C and let t n: S → S be a family of functions that satisfies ∀ n, ∀ z 1, z 2 ∈ S, ∃ ρ: | t n − t n | ≤ ρ | z 1 − z 2 |, ρ < 1. Define: G n = F n = ( t 1
Guido Stampacchia was a 20th-century Italian mathematician, known for his work on the theory of variational inequalities, the calculus of variation and the theory of elliptic partial differential equations. Stampacchia was born in Italy from Emanuele Stampacchia and Giulia Campagnano, he obtained his high school certification from the Liceo-Ginnasio Giambattista Vico in Naples in classical subjects, although he showed stronger aptitude for mathematics and physics. In 1940 he was admitted to the Scuola Normale Superiore of Pisa for undergraduate studies in pure mathematics, he was drafted in March 1943 but managed to take examinations during the summer before joining the resistance movement against the Germans in the defense of Rome in September. He was discharged in June 1945. In 1944 he won a scholarship to the University of Naples. In the 1945-1946 academic year he declined a specialization at the Scuola Normale in the Faculty of Sciences in favour of an assistant position at the Istituto Universitario Navale.
In 1949 he was appointed as assistant with tenure to the chair of mathematical analysis, in 1951 he obtained his "Libera docenza". In 1952 won a national competition for the chair at the University of Palermo, he was nominated Professor on Probation at the University of Genoa the same year and was promoted to full Professor in 1955. He married fellow student Sara Naldini in October 1948. Children Mauro, Renata and Franca were born in 1949, 1951, 1955 and 1956 respectively. Stampacchia was active in teaching throughout his career, he made key contributions to a number of fields, including calculus of variation, variational inequalities and differential equations. In 1967 Stampacchia was elected President of the Unione Matematica Italiana, it was about this time that his research efforts shifted toward the emerging field of variational inequalities, which he modeled after boundary value problems for partial differential equations. He was director of the Istituto per le Applicazioni del Calcolo of Consiglio Nazionale delle Ricerche from December 1968 to 1974.
Stampacchia accepted the position of Professor Mathematical Analysis at the University of Rome in 1968 and returned to Pisa in 1970. He died of heart arrest on 27 April that year; the Stampacchia Medal, an international prize awarded every three years for contributions to the Calculus of Variations, has been established in 2003. Stampacchia, Guido, "Second order elliptic equations and boundary value problems", Proceedings of the International Congress of Mathematicians, 15–22 August 1962, Stockholm, ICM Proceedings, 1962, Vol. 1, Stockholm: Almqvist & Wiksells, pp. 405–413, MR 0176198, Zbl 0137.06803. With Sergio Campanato, Sulle maggiorazioni in Lp nella teoria delle equazioni ellittiche, Bollettino dell’Unione Matematica Italiana, Zanichelli, 1965. With Jaurès Cecconi, Lezioni di analisi matematica, I: Funzioni di una variabile, Napoli: Liguori editore, 1974, ISBN 88-207-0127-8 with Jaurès Cecconi, Lezioni di analisi matematica, II: Funzioni di più variabili, Napoli: Liguori, 1980, ISBN 88-207-1022-6 with Jaurès Cecconi and Livio Clemente Piccinini, Esercizi e problemi di analisi matematica, Napoli: Liguori, 1996, ISBN 88-207-0744-6, ISBN 978-88-207-0744-6 Piccinini, Livio C..
Problems and methods, Applied Mathematical Sciences, 39, translated by LoBello, A. New York: Springer-Verlag, pp. xii+385, doi:10.1007/978-1-4612-5188-0, ISBN 0-387-90723-8, MR 0740539, Zbl 0535.34001. With David Kinderlehrer: An introduction to variational inequalities and their applications, NY, Academic Press, 1980Reprint: Society for Industrial and Applied Mathematics, 2000 ISBN 978-0-89871-466-1 De Angelis, P. L.. "Guido Stampacchia", Matematici all'Istituto Universitario Navale, Napoli: Istituto Universitario Navale/RCE Edizioni, pp. 37–38. The chapter on Stampacchia in the a book collecting brief biographical sketches and bibliographies of the scientific the work produced by the mathematicians who worked at the Parthenope University of Naples during their stay. Ridolfi, Roberto, ed. "Guido Stampacchia", Biografie e bibliografie degli Accademici Lincei, Roma: Accademia Nazionale dei Lincei, pp. 637–638. The biographical and bibliographical entry on Guido Stampacchia, published under the auspices of the Accademia dei Lincei in a book collecting many profiles of its living members up to 1976.
Mazzone, Silvia, "Guido Stampacchia", in Giannessi, Franco. O'Connor, John J.. Guido Stampacchia at the Mathematics Genealogy Project
David Samuel Kinderlehrer is an American mathematician, who works on partial differential equations and related mathematics applied to materials in biology and physics. Kinderlehrer received in 1963 his bachelor's degree from MIT and in 1968 his Ph. D. from the University of California, Berkeley under Hans Lewy with thesis Minimal surfaces whose boundaries contain spikes. He became in 1968 an instructor and in 1975 a full professor at the University of Minnesota in Minneapolis. For the academic year 1971–1972 he was a visiting professor at the Scuola Normale Superiore di Pisa. In 2003 he became a professor at Carnegie Mellon University, he works on partial differential equations, minimal surfaces, variational inequalities, with mathematical applications to the microstructure of biological materials, to solid state physics, to materials science, including crystalline microstructure, liquid crystals, molecular mechanisms of intracellular transport, models of ion transport. In 2012 Kinderlehrer was elected a Fellow of the American Mathematical Society.
In 1974 in Vancouver he was an invited speaker at the International Mathematical Congress. In 2002 he was the editor of the Hans Lewy Selecta published by Birkhäuser, his doctoral students include Irene Fonseca. "How a minimal surface leaves an obstacle". Bull. Amer. Math. Soc. 78: 969–970. 1972. Doi:10.1090/s0002-9904-1972-13071-4. MR 0306741. "Variational inequalities and free boundary problems". Bull. Amer. Math. Soc. 84: 7–26. 1978. Doi:10.1090/s0002-9904-1978-14397-3. MR 0466885. With Guido Stampacchia: An introduction to variational inequalities and their applications, Academic Press 1980, SIAM Press 2000 as editor with Jerald L. Ericksen and Constantine Dafermos: Amorphous Polymers and Non-Newtonian Fluids, Springer Verlag 1987 as editor with J. L. Ericksen: Theory and application of liquid crystals, Springer Verlag 1987 as editor with Richard James, Mitchell Luskin, J. L. Ericksen: Microstructure and phase transition, Springer Verlag 1993 as editor with Mark J. Bowick, Govind Menon, Charles Radin: Mathematics and Materials, American Mathematical Society 2017 Homepage at Carnegie Mellon University
Renato Caccioppoli was an Italian mathematician, known for his contributions to mathematical analysis, including the theory of functions of several complex variables, functional analysis, measure theory. Born in Naples, he was the son of Giuseppe Caccioppoli, a surgeon, his second wife Sofia Bakunin, daughter of the Russian revolutionary Mikhail Bakunin. After earning his diploma in 1921, he enrolled in the department of engineering, but in November, 1923 changed to mathematics. After earning his laurea, in 1925, he became the assistant of Mauro Picone, who in that year was called to the University of Naples, where he remained until 1932. Picone discovered Caccioppoli's gifts and pointed him towards research in mathematical analysis. In the course of the next five years, Caccioppoli published about thirty works on topics developed in the complete autonomy provided by a ministerial award for mathematics in 1931, a competition he won at the age of 27 and the chair of algebraic analysis at the University of Padova.
In 1934 he returned to Naples to accept the chair in group theory. In 1931 he became a correspondent member of the Academy of Physical and Mathematical Sciences of Naples, becoming an ordinary member in 1938. In 1944 he became an ordinary member of the Accademia Pontaniana, in 1947 a correspondent member of the Accademia Nazionale dei Lincei, a national member in 1958, he was a correspondent member of the Paduan Academy of Sciences and Arts. In the years from 1947 to 1957 he directed, together with Carlo Miranda, the journal Giornale di Matematiche, founded by Giuseppe Battaglini. In 1948 he became a member of the editing committee of Annali di Matematica, starting in 1952 he was a member of the editing committee of Ricerche di Matematica. In 1953 the Academia dei Lincei bestowed on him the national prize of physical and natural sciences, he was an excellent pianist, noted as well for his nonconformist temperament. He tried out the vagrant life, was arrested for begging. In May 1938 he gave a speech against Adolf Hitler and Benito Mussolini, when the latter was visiting Naples.
Together with his companion Sara Mancuso, he had the French national anthem played by an orchestra, after which he began to speak against fascism and Nazism in the presence of OVRA agents. He was again arrested, but his aunt, Maria Bakunin, who at the time was a professor of chemistry at the University of Naples, succeeded in having him released by convincing the authorities that her nephew was non compos mentis, thus Caccioppoli was interned, but he continued his studies in mathematics, playing the piano. In his last years, the disappointments of politics and his wife's desertion, together with the weakening of his mathematical vein, pushed him into alcoholism, his growing instability had sharpened his "strangenesses", to the point that the news of his suicide on May 8, 1959 by a gunshot to the head did not surprise those who knew him. He died at his home in Palazzo Cellamare, his most important works, out of a total of around eighty publications, relate to functional analysis and the calculus of variations.
Beginning in 1930 he dedicated himself to the study of differential equations, the first to use a topological-functional approach. Proceeding in this way, in 1931 he extended the Brouwer fixed point theorem, applying the results obtained both from ordinary differential equations and partial differential equations. In 1932 he introduced the general concept of inversion of functional correspondence, showing that a transformation between two Banach spaces is invertible only if it is locally invertible and if the only convergent sequences are the compact ones. Between 1933 and 1938 he applied his results to elliptic equations, establishing the majorizing limits for their solutions, generalizing the two-dimensional case of Felix Bernstein. At the same time he studied analytic functions of several complex variables, i.e. analytic functions whose domain belongs to the vector space Cn, proving in 1933 the fundamental theorem on normal families of such functions: if a family is normal with respect to every complex variable, it is normal with respect to the set of the variables.
He proved a logarithmic residue formula for functions of two complex variables in 1949. In 1935 Caccioppoli proved the analyticity of class C2 solutions of elliptic equations with analytic coefficients; the year 1952 saw the publication of his masterwork on the area of a surface and measure theory, the article Measure and integration of dimensionally oriented sets. The article is concerned with the theory of dimensionally oriented sets. In this paper, the family of sets approximable by polygonal domains of finite perimeter, known today as Caccioppoli sets or sets of finite perimeter, was introduced and studied, his last works, produced between 1952 and 1953, deal with a class of pseudoanalytic functions, introduced by him to extend certain properties of analytic functions. In 1992 his tormented personality inspired the plot of a film directed by Mario Martone, The Death of a Neapolitan Mathematician, in which he was portrayed by Carlo Cecchi. An asteroid, 9934 Caccioppoli, has been named after him.
Caccioppoli, Opere scelte, Roma: Edizioni Cremonese, Zbl 0112.28201 ISBN 88-7083-505-7 AND ISBN 88-7083-506-5. His "Sele