Stephen Wolfram

Stephen Wolfram is a British-American computer scientist and businessman. He is known for his work in computer science, in theoretical physics. In 2012, he was named an inaugural fellow of the American Mathematical Society; as a businessman, he is the founder and CEO of the software company Wolfram Research where he worked as chief designer of Mathematica and the Wolfram Alpha answer engine. His recent work has been on knowledge-based programming and refining the programming language of Mathematica into what is now called the Wolfram Language. Stephen Wolfram was born in London in 1959 to Hugo and Sybil Wolfram, both German Jewish refugees to the United Kingdom. Wolfram's father, Hugo Wolfram, a textile manufacturer born in Bochum, served as managing director of the Lurex Company, makers of the fabric Lurex, he was the author of three novels. He emigrated to England in 1933; when World War II broke out, he left school at 15 and subsequently found it hard to get a job since he was regarded as an "enemy alien".

As an adult, he took correspondence courses in psychology. Wolfram's mother, Sybil Wolfram from Berlin, was a Fellow and Tutor in Philosophy at Lady Margaret Hall at University of Oxford from 1964 to 1993, she published two books, Philosophical Logic: An Introduction and In-laws and Outlaws: Kinship and Marriage in England. She was the translator of Claude Lévi-Strauss's La pensée sauvage, but disavowed the translation, she was the daughter of criminologist and psychoanalyst Kate Friedlander, an expert on the subject of juvenile delinquency, the physician Walter Misch who, wrote Die vegetative Genese der neurotischen Angst und ihre medikamentöse Beseitigung. After the Reichstag fire in 1933, she emigrated from Berlin, Germany to England with her parents and Jewish psychoanalyst Paula Heimann. Stephen Wolfram is married to a mathematician, they have four children together. Stephen's son, began a degree course in mathematics and computer science in 2018, he is the co-inventor of a patented method and computing device for optically recognizing mathematical expressions.

Christopher has presented and led workshops at several highly-regarded conferences and events including South by Southwest Interactive, University of Oxford Summer School, MIT Independent Activities Period. In 2016, he was awarded as Best Technical Advisor at the Raw Science Film Festival for his work on the movie, Arrival. Among many other personal achievements, he serves as a programmer for Wolfram Research. Wolfram was educated at Eton College, but left prematurely in 1976, he entered St. John's College, Oxford at age 17 but found lectures "awful", left in 1978 without graduating to attend the California Institute of Technology, the following year, where he received a PhD in particle physics on 19 November 1979 at age 20. Wolfram's thesis committee was composed of Richard Feynman, Peter Goldreich, Frank J. Sciulli and Steven Frautschi, chaired by Richard D. Field; as a young child, Wolfram had difficulties learning arithmetic. At the age of 12, he wrote a dictionary on physics. By 13 or 14, he had written three books on particle physics.

Wolfram, at the age of 15, began research in applied quantum field theory and particle physics and published scientific papers. Topics included matter creation and annihilation, the fundamental interactions, elementary particles and their currents and leptonic physics, the parton model, published in professional peer-reviewed scientific journals including Nuclear Physics B, Australian Journal of Physics, Nuovo Cimento, Physical Review D. Working independently, Wolfram published a cited paper on heavy quark production at age 18 and nine other papers, continued research and to publish on particle physics into his early twenties. Wolfram's work with Geoffrey C. Fox on the theory of the strong interaction is still used in experimental particle physics. A 1981 letter from Feynman to Gerald Freund giving reference for Wolfram for the MacArthur grant appears in Feynman's collected letters, Perfectly Reasonable Deviations from the Beaten Track. Following his PhD, Wolfram joined the faculty at Caltech and became the youngest recipient of the MacArthur Fellowships in 1981, at age 21.

In 1983, Wolfram left for the School of Natural Sciences of the Institute for Advanced Study in Princeton, where he conducted research into cellular automata with computer simulations. He produced a series of papers systematically investigating the class of elementary cellular automata, conceiving the Wolfram code, a naming system for one-dimensional cellular automata, a classification scheme for the complexity of their behaviour, he conjectured that the Rule 110 cellular automaton might be Turing complete, proved correct. A 1985 letter from Feynman to Wolfram appears in Feynman's letters. In it, in response to Wolfram writing to him that he was thinking about creating some kind of institute where he might study complex systems, Feynman tells Wolfram, "You do not understand ordinary people," and advises him "find a way to do your research with as little contact with non-technical people as possible."In the mid-1980s, Wolfram worked on simulations of physical processes with cellular automata on the Connection Machine alongside Feynman and helped initiate the field of complex systems.

In 1984, he was a participant in the Founding Workshops of the Santa Fe Institute, along with Nobel laureates Murray Gell-Mann, Manfred Eigen, Philip Warren Anderson, future laureate Frank Wilczek. In 1986, he founded the Center for Complex Systems Research at the Uni

Wolfram Demonstrations Project

The Wolfram Demonstrations Project is an organized, open-source collection of small interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hosted by Wolfram Research, whose stated goal is to bring computational exploration to the widest possible audience. At its launch, it contained 1300 demonstrations but has grown to over 10,000; the site won a Parents' Choice Award in 2008. The Demonstrations run in Mathematica 6 or above and in Wolfram CDF Player, a free modified version of Wolfram's Mathematica and available for Windows and Macintosh and can operate as a web browser plugin, they consist of a direct user interface to a graphic or visualization, which dynamically recomputes in response to user actions such as moving a slider, clicking a button, or dragging a piece of graphics. Each Demonstration has a brief description of the concept. Demonstrations are now embeddable into any website or blog; each Demonstration page includes a snippet of JavaScript code in the Share section of the sidebar.

The website is organized by topic: for example, mathematics, computer science, art and finance. They cover a variety of levels, from elementary school mathematics to much more advanced topics such as quantum mechanics and models of biological organisms; the site is aimed at both educators and students, as well as researchers who wish to present their ideas to the broadest possible audience. Wolfram Research's staff organizes and edits the Demonstrations, which may be created by any user of Mathematica freely published and downloaded; the Demonstrations are open-source, which means that they not only demonstrate the concept itself but show how to implement it. The use of the web to transmit small interactive programs is reminiscent of Sun's Java applets, Adobe's Flash, the open-source Processing. However, those creating Demonstrations have access to the algorithmic and visualization capabilities of Mathematica making it more suitable for technical demonstrations; the Demonstrations Project has similarities to user-generated content websites like Wikipedia and Flickr.

Its business model is similar to Adobe's Acrobat and Flash strategy of charging for development tools but providing a free reader. Official site

Casorati–Weierstrass theorem

In complex analysis, a branch of mathematics, the Casorati–Weierstrass theorem describes the behaviour of holomorphic functions near their essential singularities. It is named for Karl Theodor Wilhelm Weierstrass and Felice Casorati. In Russian literature it is called Sokhotski's theorem. Start with some open subset U in the complex plane containing the number z 0, a function f, holomorphic on U ∖, but has an essential singularity at z 0; the Casorati–Weierstrass theorem states that if V is any neighbourhood of z 0 contained in U f is dense in C. This can be stated as follows: for any ε > 0, δ > 0, complex number w, there exists a complex number z in U with 0 < | z − z 0 | < δ and | f − w | < ε. Or in still more descriptive terms: f comes arbitrarily close to any complex value in every neighbourhood of z 0; the theorem is strengthened by Picard's great theorem, which states, in the notation above, that f assumes every complex value, with one possible exception, infinitely on V. In the case that f is an entire function and a = ∞, the theorem says that the values f approach every complex number and ∞, as z tends to infinity.

It is remarkable that this does not hold for holomorphic maps in higher dimensions, as the famous example of Pierre Fatou shows. The function f = exp has an essential singularity at 0. Consider the function f = e 1 / z; this function has the following Taylor series about the essential singular point at 0: f = ∑ n = 0 ∞ 1 n! Z − n; because f ′ = − e 1 z z 2 exists for all points z ≠ 0 we know that ƒ is analytic in a punctured neighborhood of z = 0. Hence it is an isolated singularity, as well as being an essential singularity. Using a change of variable to polar coordinates z = r e i θ our function, ƒ = e1/z becomes: f = e 1 r e − i θ = e 1 r cos e − 1 r i sin . Taking the absolute value of both sides: | f | = | e 1 r cos θ | | e − 1 r i sin | = e 1 r cos θ. Thus, for values of θ such that cos θ > 0, we have f → ∞ as r → 0, for cos θ < 0, f → 0 as r → 0. Consider what happens, for example when z takes values on a circle of diameter

Complex analysis

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics. By extension, use of complex analysis has applications in engineering fields such as nuclear, aerospace and electrical engineering; as a differentiable function of a complex variable is equal to the sum of its Taylor series, complex analysis is concerned with analytic functions of a complex variable. Complex analysis is one of the classical branches in mathematics, with roots in the 18th century and just prior. Important mathematicians associated with complex numbers include Euler, Riemann, Cauchy and many more in the 20th century. Complex analysis, in particular the theory of conformal mappings, has many physical applications and is used throughout analytic number theory. In modern times, it has become popular through a new boost from complex dynamics and the pictures of fractals produced by iterating holomorphic functions.

Another important application of complex analysis is in string theory which studies conformal invariants in quantum field theory. A complex function is a function from complex numbers to complex numbers. In other words, it is a function that has a subset of the complex numbers as a domain and the complex numbers as a codomain. Complex functions are supposed to have a domain that contains a nonempty open subset of the complex plane. For any complex function, the values z from the domain and their images f in the range may be separated into real and imaginary parts: z = x + i y and f = f = u + i v, where x, y, u, v are all real-valued. In other words, a complex function f: C → C may be decomposed into u: R 2 → R and v: R 2 → R, i.e. into two real-valued functions of two real variables. Any complex-valued function f on an arbitrary set X can be considered as an ordered pair of two real-valued functions: or, alternatively, as a vector-valued function from X into R 2; some properties of complex-valued functions are nothing more than the corresponding properties of vector valued functions of two real variables.

Other concepts of complex analysis, such as differentiability are direct generalizations of the similar concepts for real functions, but may have different properties. In particular, every differentiable complex function is analytic, two differentiable functions that are equal in a neighborhood of a point are equal on the intersection of their domain; the latter property is the basis of the principle of analytic continuation which allows extending every real analytic function in a unique way for getting a complex analytic function whose domain is the whole complex plane with a finite number of curve arcs removed. Many basic and special complex functions are defined in this way, including exponential functions, logarithmic functions, trigonometric functions. Complex functions that are differentiable at every point of an open subset Ω of the complex plane are said to be holomorphic on Ω. In the context of complex analysis, the derivative of f at z 0 is defined to be f ′ = lim z → z 0 f − f z − z 0, z ∈ C.

Superficially, this definition is formally analogous to that of the derivative of a real function. However, complex derivatives and differentiable functions behave in different ways compared to their real counterparts. In particular, for this limit to exist, the value of the difference quotient must approach the same complex number, regardless of the manner in which we

Classification of discontinuities

Continuous functions are of utmost importance in mathematics and applications. However, not all functions are continuous. If a function is not continuous at a point in its domain, one says that it has a discontinuity there; the set of all points of discontinuity of a function may be a discrete set, a dense set, or the entire domain of the function. This article describes the classification of discontinuities in the simplest case of functions of a single real variable taking real values; the oscillation of a function at a point quantifies these discontinuities as follows: in a removable discontinuity, the distance that the value of the function is off by is the oscillation. For each of the following, consider a real valued function f of a real variable x, defined in a neighborhood of the point x0 at which f is discontinuous. Consider the function f = { x 2 for x < 1 0 for x = 1 2 − x for x > 1 The point x0 = 1 is a removable discontinuity. For this kind of discontinuity: The one-sided limit from the negative direction: L − = lim x → x 0 − f and the one-sided limit from the positive direction: L + = lim x → x 0 + f at x0 both exist, are finite, are equal to L = L− = L+.

In other words, since the two one-sided limits exist and are equal, the limit L of f as x approaches x0 exists and is equal to this same value. If the actual value of f is not equal to L x0 is called a removable discontinuity; this discontinuity can be removed to make f continuous at x0, or more the function g = { f x ≠ x 0 L x = x 0 is continuous at x = x0. The term removable discontinuity is sometimes an abuse of terminology for cases in which the limits in both directions exist and are equal, while the function is undefined at the point x0; this use is abusive because continuity and discontinuity of a function are concepts defined only for points in the function's domain. Such a point not in the domain is properly named a removable singularity. Consider the function f = { x 2 for x < 1 0 for x = 1 2 − 2 for x > 1 Then, the point x0 = 1 is a jump discontinuity. In this case, a single limit does not exist because the one-sided limits, L− and L+, exist and are finite, but are not equal: since, L− ≠ L+, the limit L does not exist.

X0 is called a jump discontinuity, step discontinuity, or discontinuity of the first kind. For this type of discontinuity, the function f may have any value at x0. For an essential discontinuity, only one of the two one-sided limits needs not be infinite. Consider the function f = { sin 5 x − 1 for x < 1 0 for x = 1 1 x − 1 for x > 1 Then, the point x 0 = 1 is an essential discontinuity. In this case, L − doesn't exist and L