C++ is a general-purpose programming language, developed by Bjarne Stroustrup as an extension of the C language, or "C with Classes". It has imperative, object-oriented and generic programming features, while providing facilities for low-level memory manipulation, it is always implemented as a compiled language, many vendors provide C++ compilers, including the Free Software Foundation, Intel, IBM, so it is available on many platforms. C++ was designed with a bias toward system programming and embedded, resource-constrained software and large systems, with performance and flexibility of use as its design highlights. C++ has been found useful in many other contexts, with key strengths being software infrastructure and resource-constrained applications, including desktop applications and performance-critical applications. C++ is standardized by the International Organization for Standardization, with the latest standard version ratified and published by ISO in December 2017 as ISO/IEC 14882:2017.
The C++ programming language was standardized in 1998 as ISO/IEC 14882:1998, amended by the C++03, C++11 and C++14 standards. The current C++ 17 standard supersedes these with an enlarged standard library. Before the initial standardization in 1998, C++ was developed by Danish computer scientist Bjarne Stroustrup at Bell Labs since 1979 as an extension of the C language. C++20 is the next planned standard, keeping with the current trend of a new version every three years. In 1979, Bjarne Stroustrup, a Danish computer scientist, began work on "C with Classes", the predecessor to C++; the motivation for creating a new language originated from Stroustrup's experience in programming for his Ph. D. thesis. Stroustrup found that Simula had features that were helpful for large software development, but the language was too slow for practical use, while BCPL was fast but too low-level to be suitable for large software development; when Stroustrup started working in AT&T Bell Labs, he had the problem of analyzing the UNIX kernel with respect to distributed computing.
Remembering his Ph. D. experience, Stroustrup set out to enhance the C language with Simula-like features. C was chosen because it was general-purpose, fast and used; as well as C and Simula's influences, other languages influenced C++, including ALGOL 68, Ada, CLU and ML. Stroustrup's "C with Classes" added features to the C compiler, including classes, derived classes, strong typing and default arguments. In 1983, "C with Classes" was renamed to "C++", adding new features that included virtual functions, function name and operator overloading, constants, type-safe free-store memory allocation, improved type checking, BCPL style single-line comments with two forward slashes. Furthermore, it included the development of a standalone compiler for Cfront. In 1985, the first edition of The C++ Programming Language was released, which became the definitive reference for the language, as there was not yet an official standard; the first commercial implementation of C++ was released in October of the same year.
In 1989, C++ 2.0 was released, followed by the updated second edition of The C++ Programming Language in 1991. New features in 2.0 included multiple inheritance, abstract classes, static member functions, const member functions, protected members. In 1990, The Annotated C++ Reference Manual was published; this work became the basis for the future standard. Feature additions included templates, namespaces, new casts, a boolean type. After the 2.0 update, C++ evolved slowly until, in 2011, the C++11 standard was released, adding numerous new features, enlarging the standard library further, providing more facilities to C++ programmers. After a minor C++14 update released in December 2014, various new additions were introduced in C++17, further changes planned for 2020; as of 2017, C++ remains the third most popular programming language, behind Java and C. On January 3, 2018, Stroustrup was announced as the 2018 winner of the Charles Stark Draper Prize for Engineering, "for conceptualizing and developing the C++ programming language".
According to Stroustrup: "the name signifies the evolutionary nature of the changes from C". This name is credited to Rick Mascitti and was first used in December 1983; when Mascitti was questioned informally in 1992 about the naming, he indicated that it was given in a tongue-in-cheek spirit. The name comes from C's ++ operator and a common naming convention of using "+" to indicate an enhanced computer program. During C++'s development period, the language had been referred to as "new C" and "C with Classes" before acquiring its final name. Throughout C++'s life, its development and evolution has been guided by a set of principles: It must be driven by actual problems and its features should be useful in real world programs; every feature should be implementable. Programmers should be free to pick their own programming style, that style should be supported by C++. Allowing a useful feature is more important than preventing every possible misuse of C++, it should provide facilities for organising programs into separate, well-defined parts, provide facilities for combining separately developed parts.
No implicit violations of the type system (but allow explicit violations.
In probability theory, the normal distribution is a common continuous probability distribution. Normal distributions are important in statistics and are used in the natural and social sciences to represent real-valued random variables whose distributions are not known. A random variable with a Gaussian distribution is said to be distributed and is called a normal deviate; the normal distribution is useful because of the central limit theorem. In its most general form, under some conditions, it states that averages of samples of observations of random variables independently drawn from independent distributions converge in distribution to the normal, that is, they become distributed when the number of observations is sufficiently large. Physical quantities that are expected to be the sum of many independent processes have distributions that are nearly normal. Moreover, many results and methods can be derived analytically in explicit form when the relevant variables are distributed; the normal distribution is sometimes informally called the bell curve.
However, many other distributions are bell-shaped. The probability density of the normal distribution is f = 1 2 π σ 2 e − 2 2 σ 2 where μ is the mean or expectation of the distribution, σ is the standard deviation, σ 2 is the variance; the simplest case of a normal distribution is known as the standard normal distribution. This is a special case when μ = 0 and σ = 1, it is described by this probability density function: φ = 1 2 π e − 1 2 x 2 The factor 1 / 2 π in this expression ensures that the total area under the curve φ is equal to one; the factor 1 / 2 in the exponent ensures that the distribution has unit variance, therefore unit standard deviation. This function is symmetric around x = 0, where it attains its maximum value 1 / 2 π and has inflection points at x = + 1 and x = − 1. Authors may differ on which normal distribution should be called the "standard" one. Gauss defined the standard normal as having variance σ 2 = 1 / 2, φ = e − x 2 π Stigler goes further, defining the standard normal with variance σ 2 = 1 /: φ = e − π x 2 Every normal distribution is a version of the standard normal distribution whose domain has been stretched by a factor σ and translated by μ: f = 1 σ φ.
The probability density must be scaled by 1 / σ so that the integral is still 1. If Z is a standard normal deviate X = σ Z + μ will have a normal distribution with expected value μ and standard deviation σ. Conversely, if X is a normal deviate with parameters μ and σ 2 Z = / σ
Numerical Recipes is the generic title of a series of books on algorithms and numerical analysis by William H. Press, Saul A. Teukolsky, William T. Vetterling and Brian P. Flannery. In various editions, the books have been in print since 1986; the most recent edition was published in 2007. In 2015 Numerical Recipes sold its historic two-letter domain name nr.com and became numerical.recipes instead. The Numerical Recipes books cover a range of topics that include both classical numerical analysis, signal processing, statistical treatment of data, a few topics in machine learning; the writing style has an informal tone. The emphasis is on understanding the underlying basics of techniques, not on the refinements that may, in practice, be needed to achieve optimal performance and reliability. Few results are proved with any degree of rigor, although the ideas behind proofs are sketched, references are given. All methods that are discussed are implemented in a programming language, with the code printed in the book.
Each version is keyed to a specific language. According to the publisher, Cambridge University Press, the Numerical Recipes books are the all-time best-selling books on scientific programming methods. In recent years, Numerical Recipes books have been cited in the scientific literature more than 3000 times per year according to ISI Web of Knowledge, and as of the end of 2017, the book had over 44000 citations on Google Scholar. The first publication was in 1986 with the title,”Numerical Recipes, The Art of Scientific Computing,” containing code in both Fortran and Pascal. Supplemental editions followed with code in Pascal, BASIC, C. Numerical Recipes took, from the start, an opinionated editorial position at odds with the conventional wisdom of the numerical analysis community: However, as it turned out, the 1980s were fertile years for the "black box" side, yielding important libraries such as BLAS and LAPACK, integrated environments like MATLAB and Mathematica. By the early 1990s, when Second Edition versions of Numerical Recipes were published, it was clear that the constituency for Numerical Recipes was by no means the majority of scientists doing computation, but only that slice that lived between the more mathematical numerical analysts and the larger community using integrated environments.
The Second Edition versions occupied a stable role in this niche environment. By the mid-2000s, the practice of scientific computing had been radically altered by the mature Internet and Web. Recognizing that their Numerical Recipes books were valued more for their explanatory text than for their code examples, the authors expanded the scope of the book, rewrote a large part of the text, they continued to include code, still printed in the book, for every method discussed. The Third Edition was released as an electronic book made available on the Web for free or by paid or institutional subscription. Numerical Recipes as a single volume that covers broad range of algorithms; that format skewed the choice of algorithms towards simpler and shorter early algorithms which were not as acurate, efficient or stable as more complex algorithms. First edition had some minor bugs, which were fixed in the latter editions, they attributed this to people using outdated versions of the code, bugs in other parts of the code and misuse of routines which require some understanding to use correctly..
The code listings are commercially licensed by the Numerical Recipes authors. However, Numerical Recipes includes the following statement regarding copyrights on computer programs:Copyright does not protect ideas, but only the expression of those ideas in a particular form. In the case of a computer program, the ideas consist of the program's methodology and algorithm, including the necessary sequence of steps adopted by the programmer; the expression of those ideas is the program source code... If you analyze the ideas contained in a program, express those ideas in your own different implementation that new program implementation belongs to you. One early motivation for the GNU Scientific Library was that a free library was needed as a substitute for Numerical Recipes. Another line of criticism centers on the coding style of the books, which strike some modern readers as "Fortran-ish", though written in contemporary, object-oriented C++; the authors have defended their terse coding style as necessary to the format of the book because of space limitations and for readability.
The books differ by the computer language in which the code is given. Numerical Recipes; the Art of Scientific Computing, 1st Edition, 1986, ISBN 0-521-30811-9. Numerical Recipes; the Art of Scientific Computing, 3rd Edition, 2007, ISBN 0-521-88068-8. Numerical Recipes in C; the Art of Scientific Computing, 2nd Edition, 1992, ISBN 0-521-43108-5. Numerical Recipes in Fortran; the Art of Scientific Computing, 2nd Edition, 1992, ISBN 0-521-43064-X. Numerical Recipes in Fortran 90; the Art of Parallel Scien
George E. P. Box
George Edward Pelham Box FRS was a British statistician, who worked in the areas of quality control, time-series analysis, design of experiments, Bayesian inference. He has been called "one of the great statistical minds of the 20th century", he was born in Gravesend, England. Upon entering university he began to study chemistry, but was called up for service before finishing. During World War II, he performed experiments for the British Army exposing small animals to poison gas. To analyze the results of his experiments, he taught himself statistics from available texts. After the war, he enrolled at University College London and obtained a bachelor's degree in mathematics and statistics, he received a PhD from the University of London under the supervision of Egon Pearson. From 1948 to 1956, Box worked as a statistician for Imperial Chemical Industries. While at ICI, he took a leave of absence for a year and served as a visiting professor at North Carolina State University at Raleigh, he went to Princeton University where he served as Director of the Statistical Research Group.
In 1960, Box moved to the University of Wisconsin–Madison to create the Department of Statistics. He was appointed Vilas Research Professor of Statistics in 1980. Box and Bill Hunter co-founded the Center for Quality and Productivity Improvement at the University of Wisconsin–Madison in 1984. Box retired in 1992, becoming an Emeritus Professor. Box published books including Statistics for Experimenters, Time Series Analysis: Forecasting and Control and Bayesian Inference in Statistical Analysis.. Box co-founded anomaly.io, a company which provides real-time anomaly detection and alerting services. Box served as President of the American Statistical Association in 1978 and of the Institute of Mathematical Statistics in 1979, he received the Shewhart Medal from the American Society for Quality Control in 1968, the Wilks Memorial Award from the American Statistical Association in 1972, the R. A. Fisher Lectureship in 1974, the Guy Medal in Gold from the Royal Statistical Society in 1993. Box was elected a member of the American Academy of Arts and Sciences in 1974 and a Fellow of the Royal Society in 1985.
His name is associated with results in statistics such as Box–Jenkins models, Box–Cox transformations, Box–Behnken designs, others. Additionally, Box famously wrote, in various books and papers, that "all models are wrong, but some are useful". Box married the second of Ronald Fisher's five daughters. In 1978, Joan Fisher Box published a biography of Ronald Fisher, with substantial collaboration with Box. Box married Claire Quist in 1985. Box died on 28 March 2013, he was 93 years old. Box-Behnken designs from a handbook on engineering statistics at NIST ASQ: George E. P. Box Accomplishments in statistics Articles and Reports by George Box Statistics for Experimenters - Second Edition, 2005 by George Box, William G. Hunter and Stuart Hunter Biography of George E. P. Box from the Institute for Operations Research and the Management Sciences https://anomaly.io/about/
Norbert Wiener was an American mathematician and philosopher. He was a professor of mathematics at the Massachusetts Institute of Technology. A child prodigy, Wiener became an early researcher in stochastic and mathematical noise processes, contributing work relevant to electronic engineering, electronic communication, control systems. Wiener is considered the originator of cybernetics, a formalization of the notion of feedback, with implications for engineering, systems control, computer science, neuroscience and the organization of society. Norbert Wiener is credited as being one of the first to theorize that all intelligent behavior was the result of feedback mechanisms, that could be simulated by machines and was an important early step towards the development of modern AI. Wiener was born in Columbia, the first child of Leo Wiener and Bertha Kahn, Jews from Poland and Germany, respectively. Through his father, he was related to Maimonides, the famous rabbi and physician from Al Andalus, as well as to Akiva Eger, chief rabbi of Posen from 1815 to 1837.
Leo had educated Norbert at home until 1903, employing teaching methods of his own invention, except for a brief interlude when Norbert was seven years of age. Earning his living teaching German and Slavic languages, Leo read and accumulated a personal library from which the young Norbert benefited greatly. Leo had ample ability in mathematics and tutored his son in the subject until he left home. In his autobiography, Norbert described his father as calm and patient, unless he failed to give a correct answer, at which his father would lose his temper, he became an agnostic. After graduating from Ayer High School in 1906 at 11 years of age, Wiener entered Tufts College, he was awarded a BA in mathematics in 1909 at the age of 14, whereupon he began graduate studies of zoology at Harvard. In 1910 he transferred to Cornell to study philosophy; the next year he returned to Harvard. Back at Harvard, Wiener became influenced by Edward Vermilye Huntington, whose mathematical interests ranged from axiomatic foundations to engineering problems.
Harvard awarded Wiener a Ph. D. in 1912, when he was 17 years old, for a dissertation on mathematical logic, supervised by Karl Schmidt, the essential results of which were published as Wiener. In that dissertation, he was the first to state publicly that ordered pairs can be defined in terms of elementary set theory. Hence relations can be defined by set theory, thus the theory of relations does not require any axioms or primitive notions distinct from those of set theory. In 1921, Kazimierz Kuratowski proposed a simplification of Wiener's definition of ordered pairs, that simplification has been in common use since, it is =. In 1914, Wiener traveled to Europe, to be taught by Bertrand Russell and G. H. Hardy at Cambridge University, by David Hilbert and Edmund Landau at the University of Göttingen. During 1915–16, he taught philosophy at Harvard was an engineer for General Electric and wrote for the Encyclopedia Americana. Wiener was a journalist for the Boston Herald, where he wrote a feature story on the poor labor conditions for mill workers in Lawrence, but he was fired soon afterwards for his reluctance to write favorable articles about a politician the newspaper's owners sought to promote.
Although Wiener became a staunch pacifist, he eagerly contributed to the war effort in World War I. In 1916, with America's entry into the war drawing closer, Wiener attended a training camp for potential military officers, but failed to earn a commission. One year Wiener again tried to join the military, but the government again rejected him due to his poor eyesight. In the summer of 1918, Oswald Veblen invited Wiener to work on ballistics at the Aberdeen Proving Ground in Maryland. Living and working with other mathematicians strengthened his interest in mathematics. However, Wiener was still eager to serve in uniform, decided to make one more attempt to enlist, this time as a common soldier. Wiener wrote in a letter to his parents, "I should consider myself a pretty cheap kind of a swine if I were willing to be an officer but unwilling to be a soldier." This time the army accepted Wiener into its ranks and assigned him, by coincidence, to a unit stationed at Aberdeen, Maryland. World War I ended just days after Wiener's return to Aberdeen and Wiener was discharged from the military in February 1919.
Wiener was unable to secure a permanent position at Harvard, a situation he blamed on anti-semitism at the university and in particular on the antipathy of Harvard mathematician G. D. Birkhoff, he was rejected for a position at the University of Melbourne. At W. F. Osgood's suggestion, Wiener became an instructor of mathematics at MIT, where he spent the remainder of his career, becoming promoted to professor. There is a photograph of him prominently displayed in one of the hallways used in giving directions. In 1926, Wiener returned to Europe as a Guggenheim scholar, he spent most of his time at Göttingen and with Hardy at Cambridge, working on Brownian motion, the Fourier integral, Dirichlet's problem, harmonic analysis, the Tauberian theorems. In 1926, Wiener's parents arranged his marriage to Margaret Engemann, his sister, married Philip Franklin. Their daughter, Wiener's niece, married Václav E. Beneš. Many tales apocryphal, were told of him at MIT concerning his absent-mindedness, it was said.
He inquired of a neighborhood girl the reason, she said that the family
Raymond Edward Alan Christopher Paley was an English mathematician. Paley was born in England, he was educated at Eton. From there he entered Cambridge where he showed himself as a brilliant student, he was elected a fellow of Trinity College. His contributions include the Paley construction for Hadamard matrices and his collaboration with Norbert Wiener in the Paley–Wiener theorem, he collaborated with A. Zygmund on Fourier series and worked with J. E. Littlewood on what became known as Littlewood–Paley theory, an application of real-variable techniques in complex analysis. In 1932, he introduced the dyadic basis with regard to Walsh functions. On 7 April 1933, Paley was killed by an avalanche when skiing alone at an altitude of 9,600 ft at Fossil Mountain, Alberta in the Canadian Rockies, his death was witnessed by companions lower down the mountainside. Park wardens and a member of the Royal Canadian Mounted Police recovered the body: he is buried in the Banff Town Cemetery. O'Connor, John J..
Raymond Paley at the Mathematics Genealogy Project
Inverse transform sampling
Inverse transform sampling is a basic method for pseudo-random number sampling, i.e. for generating sample numbers at random from any probability distribution given its cumulative distribution function. Inverse transformation sampling takes uniform samples of a number u between 0 and 1, interpreted as a probability, returns the largest number x from the domain of the distribution P such that P ≤ u. For example, imagine that P is the standard normal distribution with mean zero and standard deviation one; the table below shows samples taken from the uniform distribution and their representation on the standard normal distribution. We are randomly choosing a proportion of the area under the curve and returning the number in the domain such that this proportion of the area occurs to the left of that number. Intuitively, we are unlikely to choose a number in the far end of tails because there is little area in them which would require choosing a number close to zero or one. Computationally, this method involves computing the quantile function of the distribution — in other words, computing the cumulative distribution function of the distribution and inverting that function.
This is the source of the term "inverse" or "inversion" in most of the names for this method. Note that for a discrete distribution, computing the CDF is not in general too difficult: we add up the individual probabilities for the various points of the distribution. For a continuous distribution, however, we need to integrate the probability density function of the distribution, impossible to do analytically for most distributions; as a result, this method may be computationally inefficient for many distributions and other methods are preferred. For the normal distribution, the lack of an analytical expression for the corresponding quantile function means that other methods may be preferred computationally, it is the case that for simple distributions, the inverse transform sampling method can be improved on: see, for example, the ziggurat algorithm and rejection sampling. On the other hand, it is possible to approximate the quantile function of the normal distribution accurately using moderate-degree polynomials, in fact the method of doing this is fast enough that inversion sampling is now the default method for sampling from a normal distribution in the statistical package R.
The probability integral transform states that if X is a continuous random variable with cumulative distribution function F X the random variable Y = F X has a uniform distribution on. The inverse probability integral transform is just the inverse of this: if Y has a uniform distribution on and if X has a cumulative distribution F X the random variable F X − 1 has the same distribution as X. From U ∼ U n i f, we want to generate X with CDF F X. We assume F X to be a increasing function, which provides good intuitions but is not true in general. We want to see if we can find some monotone transformation T: ↦ R, such that T = d X. Still, the condition of monotone may not be true for general situations. We will have F X = Pr = Pr = Pr = T − 1, for x ∈ R. So we got F X to be the inverse function of T, or, equivalently T = F