"Norbert Wiener Norbert Wiener (November 26, 1894 – March 18, 1964) was an American mathematician and philosopher. He was a professor of mathematics at the Massachusetts Institute of Technology (MIT). A child prodigy, Wiener later became an early researcher in stochastic and mathematical noise processes, contributing work relevant to electronic engineering, electronic communication, and control systems. Wiener is considered the originator of cybernetics, a formalization of the notion of feedback, with implications for engineering, systems control, computer science, biology, neuroscience, philosophy, and the organization of society. Wiener was born in Columbia, Missouri, the first child of Leo Wiener and Bertha"
"series is a modernized version of the theory of analytic functionals due to the Italian mathematician Vito Volterra in work dating from 1887. Norbert Wiener became interested in this theory in the 1920s from contact with Volterra's student Paul Lévy. He applied his theory of Brownian motion to the integration of Volterra analytic functionals. The use of Volterra series for system analysis originated from a restricted 1942 wartime report of Wiener, then professor of mathematics at MIT. It used the series to make an approximate analysis of the effect of radar noise in a nonlinear receiver circuit. The report became"
"""imagine the effects of natural selection on those organisms that fortuitously evolved the flagella ... without the concommitant control mechanisms"". An early concept of irreducibly complex systems comes from Ludwig von Bertalanffy (1901-1972), an Austrian biologist. He believed that complex systems must be examined as complete, irreducible systems in order to fully understand how they work. He extended his work on biological complexity into a general theory of systems in a book titled ""General Systems Theory"". After James Watson and Francis Crick published the structure of DNA in the early 1950s, General Systems Theory lost many of its adherents in"
"relations whose arguments were not necessarily all of the same type. In 1914, Norbert Wiener showed how to code the ordered pair as a set of sets, making it possible to eliminate relation types in favor of the linear hierarchy of sets described here. The well-formed formulas of New Foundations (NF) are the same as the well-formed formulas of TST, but with the type annotations erased. The axioms of NF are: By convention, NF's ""Comprehension"" schema is stated using the concept of stratified formula and making no direct reference to types. A formula formula_12 is said to be stratified if"
"Complex-base system In arithmetic, a complex-base system is a positional numeral system whose radix is an imaginary (proposed by Donald Knuth in 1955) or complex number (proposed by S. Khmelnik in 1964 and Walter F. Penney in 1965). Let formula_1 be an integral domain formula_2, and formula_3 the (Archimedean) absolute value on it. A number formula_4 in a positional number system is represented as an expansion where The cardinality formula_6 is called the ""level of decomposition"". A positional number system or coding system is a pair with radix formula_8 and set of digits formula_9, and we write the standard set"
"of set theory. In 1921, Kazimierz Kuratowski proposed a simplification of Wiener's definition of ordered pairs, and that simplification has been in common use ever since. It is (x, y) = . In 1914, Wiener traveled to Europe, to be taught by Bertrand Russell and G. H. Hardy at Cambridge University, and by David Hilbert and Edmund Landau at the University of Göttingen. During 1915–16, he taught philosophy at Harvard, then was an engineer for General Electric and wrote for the ""Encyclopedia Americana"". Wiener was briefly a journalist for the ""Boston Herald"", where he wrote a feature story on the"
"Systematics – study of multi-term systems Systematics is the name given by John Godolphin Bennett (1897–1974) to a branch of systems science that he developed in the mid-twentieth century. Also referred to as the theory of Multi-Term Systems or Bennettian Systematics, it focuses on types, levels, and degrees of complexity in systems, the qualities emergent at these levels, and the ability to represent and practically deal with (""understand"") complexity using abstract models. Thus to understand the notions of sameness and difference requires a system or universe of discourse with a minimum of two terms or elements. To understand the concept"
"by the ""Lee-Schetzen method"". The Wiener series is important in nonlinear system identification. In this context, the series approximates the functional relation of the output to the entire history of system input at any time. The Wiener series has been applied mostly to the identification of biological systems, especially in neuroscience. The name Wiener series is almost exclusively used in system theory. In the mathematical literature it occurs as the Itô expansion (1951) which has a different form but is entirely equivalent to it. The Wiener series should not be confused with the Wiener filter, which is another algorithm developed"
"Abstract Wiener space An abstract Wiener space is a mathematical object in measure theory, used to construct a ""decent"" (strictly positive and locally finite) measure on an infinite-dimensional vector space. It is named after the American mathematician Norbert Wiener. Wiener's original construction only applied to the space of real-valued continuous paths on the unit interval, known as classical Wiener space; Leonard Gross provided the generalization to the case of a general separable Banach space. The structure theorem for Gaussian measures states that all Gaussian measures can be represented by the abstract Wiener space construction. Let ""H"" be a separable Hilbert"
"Polynomial chaos Polynomial chaos (PC), also called Wiener chaos expansion, is a non-sampling-based method to determine evolution of uncertainty in a dynamical system, when there is probabilistic uncertainty in the system parameters. PC was first introduced by Norbert Wiener where Hermite polynomials were used to model stochastic processes with Gaussian random variables. It can be thought of as an extension of Volterra's theory of nonlinear functionals for stochastic systems. According to Cameron and Martin such an expansion converges in the formula_1 sense for any arbitrary stochastic process with finite second moment. This applies to most physical systems. Xiu (in his"
"is a mathematical object in measure theory, used to construct a ""decent"", strictly positive and locally finite measure on an infinite-dimensional vector space. Wiener's original construction only applied to the space of real-valued continuous paths on the unit interval, known as classical Wiener space. Leonard Gross provided the generalization to the case of a general separable Banach space. The notion of a Banach space itself was discovered independently by both Wiener and Stefan Banach at around the same time. The Norbert Wiener Center for Harmonic Analysis and Applications (NWC) in the Department of Mathematics at the University of Maryland, College"
"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 merely 17 years old, for a dissertation on mathematical logic, supervised by Karl Schmidt, the essential results of which were published as Wiener (1914). 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"
"held a MacArthur Fellowship between 1987–1992. He also holds an Honorary Degree in Science from the University of Louvain (1997); He was awarded the Norbert Wiener Memorial Gold Medal for Cybernetics in 1973, the Gold Medal of the Accademia dei Lincei in Rome in 1990, the Trotter Prize for Information and Complexity in 2001, and the Herbert Simon award for Complex Systems in 2013. He became a Fellow of the Royal Society of Canada in 2009. Kauffman is best known for arguing that the complexity of biological systems and organisms might result as much from self-organization and far-from-equilibrium dynamics as"
"studied both theoretically and by simulation experiments. The study of complex systems in general has been grouped under the heading of ""general systems theory"", particularly by Ludwig von Bertalanffy, Anatol Rapoport, Ralph Gerard, and Kenneth Boulding. These sciences include psychology and cognitive science, cellular automata, generative linguistics, natural language processing, connectionism, self-organization, evolutionary biology, neural network, social network, neuromusicology, quantum cellular automata, information theory, systems theory, genetic algorithms, computational sociology, communication networks, artificial life, chaos theory, complexity theory, network science, epistemology, quantum dot cellular automaton, quantum computer, systems thinking, genetics, economy, philosophy of science, quantum mechanics, cybernetics, digital physics, digital"
"mathematical objects must be defined as sets of some sort. Hence if the ordered pair is not taken as primitive, it must be defined as a set. Several set-theoretic definitions of the ordered pair are given below. Norbert Wiener proposed the first set theoretical definition of the ordered pair in 1914: He observed that this definition made it possible to define the types of ""Principia Mathematica"" as sets. ""Principia Mathematica"" had taken types, and hence relations of all arities, as primitive. Wiener used <nowiki> instead of {""b""} to make the definition compatible with type theory where all elements in a"
"laureates Murray Gell-Mann and Philip Anderson, economics Nobel laureate Kenneth Arrow, and Manhattan Project scientists George Cowan and Herb Anderson. Today, there are over 50 institutes and research centers focusing on complex systems. A scientific society called Complex Systems Society organizes every year a general conference on these topics. The traditional approach to dealing with complexity is to reduce or constrain it. Typically, this involves compartmentalisation: dividing a large system into separate parts. Organizations, for instance, divide their work into departments that each deal with separate issues. Engineering systems are often designed using modular components. However, modular designs become susceptible"
"Stochastic calculus Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. It is used to model systems that behave randomly. The best-known stochastic process to which stochastic calculus is applied is the Wiener process (named in honor of Norbert Wiener), which is used for modeling Brownian motion as described by Louis Bachelier in 1900 and by Albert Einstein in 1905 and other physical diffusion processes in space of particles subject to random forces. Since the 1970s, the"
"the ethical implications of their work. After the war, he refused to accept any government funding or to work on military projects. The way Wiener's beliefs concerning nuclear weapons and the Cold War contrasted with those of von Neumann is the major theme of the book ""John Von Neumann and Norbert Wiener"". Wiener was a participant of the Macy conferences. He died in March 1964, aged 69, in Stockholm, from a heart attack. Wiener and his wife are buried at the Vittum Hill Cemetery in Sandwich, New Hampshire. Wiener was an early studier of stochastic and mathematical noise processes, contributing"
"or communities. Complex systems may have the following features: Although it is arguable that humans have been studying complex systems for thousands of years, the modern scientific study of complex systems is relatively young in comparison to established fields of science such as physics and chemistry. The history of the scientific study of these systems follows several different research trends. In the area of mathematics, arguably the largest contribution to the study of complex systems was the discovery of chaos in deterministic systems, a feature of certain dynamical systems that is strongly related to nonlinearity. The study of neural networks"
"by Norbert Wiener used in signal processing. Given a system with an input/output pair formula_1 where the input is white noise with zero mean value and power A, we can write the output of the system as sum of a series of Wiener G-functionals formula_2 In the following the expressions of the G-functionals up to the fifth order will be given: formula_3 formula_4 formula_5 formula_6 formula_7 formula_8 formula_9 formula_10 formula_11 Wiener series In mathematics, the Wiener series (or ""Wiener G-functional expansion"") originates from the 1958 book of Norbert Wiener. It is an orthogonal expansion for nonlinear functionals closely related to"
"Hypercomplex number In mathematics, a hypercomplex number is a traditional term for an element of a unital algebra over the field of real numbers. The study of hypercomplex numbers in the late 19th century forms the basis of modern group representation theory. In the nineteenth century number systems called quaternions, tessarines, coquaternions, biquaternions, and octonions became established concepts in mathematical literature, added to the real and complex numbers. The concept of a hypercomplex number covered them all, and called for a discipline to explain and classify them. The cataloguing project began in 1872 when Benjamin Peirce first published his ""Linear"
"Wiener measure. Abstract Wiener space An abstract Wiener space is a mathematical object in measure theory, used to construct a ""decent"" (strictly positive and locally finite) measure on an infinite-dimensional vector space. It is named after the American mathematician Norbert Wiener. Wiener's original construction only applied to the space of real-valued continuous paths on the unit interval, known as classical Wiener space; Leonard Gross provided the generalization to the case of a general separable Banach space. The structure theorem for Gaussian measures states that all Gaussian measures can be represented by the abstract Wiener space construction. Let ""H"" be a"
"system and numerous forms of relationships among the elements. However, what one sees as complex and what one sees as simple is relative and changes with time. Warren Weaver posited in 1948 two forms of complexity: disorganized complexity, and organized complexity. Phenomena of 'disorganized complexity' are treated using probability theory and statistical mechanics, while 'organized complexity' deals with phenomena that escape such approaches and confront ""dealing simultaneously with a sizable number of factors which are interrelated into an organic whole"". Weaver's 1948 paper has influenced subsequent thinking about complexity. The approaches that embody concepts of systems, multiple elements, multiple relational"
"after Norbert Wiener, who proved its mathematical existence, but the process is also called the Brownian motion process or just Brownian motion due to its historical connection as a model for Brownian movement in liquids. Playing a central role in the theory of probability, the Wiener process is often considered the most important and studied stochastic process, with connections to other stochastic processes. Its index set and state space are the non-negative numbers and real numbers, respectively, so it has both continuous index set and states space. But the process can be defined more generally so its state space can"
"The most general and powerful theory was developed by Aulbach and Wanner. They addressed non-autonomous dynamical systems formula_25 in infinite dimensions, with potentially infinite dimensional stable, unstable and center manifolds. Further, they usefully generalised the definition of the manifolds so that the center manifold is associated with eigenvalues such that formula_26, the stable manifold with eigenvalues formula_27, and unstable manifold with eigenvalues formula_28. They proved existence of these manifolds, and the emergence of a center manifold, via nonlinear coordinate transforms. Potzsche and Rasmussen established a corresponding approximation theorem for such infinite dimensional, non-autonomous systems. As the stability of the equilibrium"
"Wiener in a series of studies culminating in his articles of 1921 on Brownian motion. They developed a rigorous method (now known as the Wiener measure) for assigning a probability to a particle's random path. Richard Feynman developed another functional integral, the path integral, useful for computing the quantum properties of systems. In Feynman's path integral, the classical notion of a unique trajectory for a particle is replaced by an infinite sum of classical paths, each weighted differently according to its classical properties. Functional integration is central to quantization techniques in theoretical physics. The algebraic properties of functional integrals are"
"work by pushing on it). In 1850, the German physicist Rudolf Clausius generalized this picture to include the concept of the surroundings and began to use the term ""working body"" when referring to the system. The biologist Ludwig von Bertalanffy (1901–1972) became one of the pioneers of the general systems theory. In 1945 he introduced ""models, principles, and laws that apply to generalized systems or their subclasses, irrespective of their particular kind, the nature of their component elements, and the relation or 'forces' between them."" Norbert Wiener (1894–1964) and Ross Ashby (1903–1972), who pioneered the use of mathematics to study"
"Generalized Wiener filter The Wiener filter as originally proposed by Norbert Wiener is a signal processing filter which uses knowledge of the statistical properties of both the signal and the noise to reconstruct an optimal estimate of the signal from a noisy one-dimensional time-ordered data stream. The generalized Wiener filter generalizes the same idea beyond the domain of one-dimensional time-ordered signal processing, with two-dimensional image processing being the most common application. Consider a data vector formula_1 which is the sum of independent signal and noise vectors formula_2 with zero mean and covariances formula_3 and formula_4. The generalized Wiener Filter is"
"the specific abstract machines used to define them. Specifically, each logical system produces a set of queries expressible in it. The queries – when restricted to finite structures – correspond to the computational problems of traditional complexity theory. The first main result of descriptive complexity was Fagin's theorem, shown by Ronald Fagin in 1974. It established that NP is precisely the set of languages expressible by sentences of existential second-order logic; that is, second order logic excluding universal quantification over relations, functions, and subsets. Many other classes were later characterized in such a manner, most of them by Neil Immerman:"
"of these in probability theory. The dimension of the unit interval is not an obstacle, as was clear already to Norbert Wiener. He constructed the Wiener process (also called Brownian motion) in the form of a measurable map from the unit interval to the space of continuous functions. The theory of standard probability spaces was started by von Neumann in 1932 and shaped by Vladimir Rokhlin in 1940. For modernized presentations see , , and . Nowadays standard probability spaces may be (and often are) treated in the framework of descriptive set theory, via standard Borel spaces, see for example"
"Wiener series In mathematics, the Wiener series (or ""Wiener G-functional expansion"") originates from the 1958 book of Norbert Wiener. It is an orthogonal expansion for nonlinear functionals closely related to the Volterra series and having the same relation to it as an orthogonal Hermite polynomial expansion has to a power series. For this reason it is also known as the ""Wiener-Hermite expansion"". The analogue of the coefficients are referred to as Wiener kernels. The terms of the series are orthogonal (uncorrelated) with respect to a statistical input of white noise. This property allows the terms to be identified in applications"
"set of three coupled, nonlinear ordinary differential equations. A detailed derivation may be found, for example, in nonlinear dynamics texts. The Lorenz system is a reduced version of a larger system studied earlier by Barry Saltzman. Lorenz system The Lorenz system is a system of ordinary differential equations first studied by Edward Lorenz. It is notable for having chaotic solutions for certain parameter values and initial conditions. In particular, the Lorenz attractor is a set of chaotic solutions of the Lorenz system which, when plotted, resemble a butterfly or figure eight. In 1963, Edward Lorenz developed a simplified mathematical model"
"Wiener algebra In mathematics, the Wiener algebra, named after Norbert Wiener and usually denoted by , is the space of absolutely convergent Fourier series. Here denotes the circle group. The norm of a function is given by where is the th Fourier coefficient of . The Wiener algebra is closed under pointwise multiplication of functions. Indeed, therefore Thus the Wiener algebra is a commutative unitary Banach algebra. Also, is isomorphic to the Banach algebra , with the isomorphism given by the Fourier transform. The sum of an absolutely convergent Fourier series is continuous, so where is the ring of continuous"