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81-520: Norbert Wiener (November 26, 1894 – March 18, 1964) was an American computer scientist , mathematician and philosopher . He became 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

162-612: A 2 , b 2 ) {\displaystyle (a_{2},b_{2})} be ordered pairs. Then the characteristic (or defining ) property of the ordered pair is: ( a 1 , b 1 ) = ( a 2 , b 2 )  if and only if  a 1 = a 2  and  b 1 = b 2 . {\displaystyle (a_{1},b_{1})=(a_{2},b_{2}){\text{ if and only if }}a_{1}=a_{2}{\text{ and }}b_{1}=b_{2}.} The set of all ordered pairs whose first entry

243-430: A ∉ { x } } = ⋃ { y } = y . {\displaystyle \pi _{2}(p)=\bigcup \left\{\left.a\in \bigcup p\,\right|\,\bigcup p\neq \bigcap p\rightarrow a\notin \bigcap p\right\}=\bigcup \left\{\left.a\in \{x,y\}\,\right|\,\{x,y\}\neq \{x\}\rightarrow a\notin \{x\}\right\}=\bigcup \{y\}=y.} (if x ≠ y {\displaystyle x\neq y} , then

324-463: A , b ) = ( b , a ) {\displaystyle (a,b)=(b,a)} is false unless b = a {\displaystyle b=a} . There are other definitions, of similar or lesser complexity, that are equally adequate: The reverse definition is merely a trivial variant of the Kuratowski definition, and as such is of no independent interest. The definition short

405-460: A , b , c , d , e , f ∉ N {\displaystyle a,b,c,d,e,f\notin \mathbb {N} } . In type theory and in outgrowths thereof such as the axiomatic set theory NF , the Quine–Rosser pair has the same type as its projections and hence is termed a "type-level" ordered pair. Hence this definition has the advantage of enabling a function , defined as

486-581: A PhD in June 1913, when he was only 19 years old, for a dissertation on mathematical logic (a comparison of the work of Ernst Schröder with that of Alfred North Whitehead and Bertrand Russell ), supervised by Karl Schmidt, the essential results of which were published as Wiener (1914). He was one of the youngest to achieve such a feat. 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

567-652: A PhD , M.S. , Bachelor's degree in computer science, or other similar fields like Information and Computer Science (CIS), or a closely related discipline such as mathematics or physics . Computer scientists are often hired by software publishing firms, scientific research and development organizations where they develop the theories and computer model that allow new technologies to be developed. Computer scientists are also employed by educational institutions such as universities . Computer scientists can follow more practical applications of their knowledge, doing things such as software engineering. They can also be found in

648-407: A vector space .) The entries of an ordered pair can be other ordered pairs, enabling the recursive definition of ordered n -tuples (ordered lists of n objects). For example, the ordered triple ( a , b , c ) can be defined as ( a , ( b , c )), i.e., as one pair nested in another. In the ordered pair ( a , b ), the object a is called the first entry , and the object b the second entry of

729-417: A ) K = ( d, c ) K . Therefore, b = d and a = c . Only if . If a = c and b = d , then {{ b }, { a, b }} = {{ d }, { c, d }}. Thus ( a, b ) reverse = ( c, d ) reverse . Short: If : If a = c and b = d , then { a , { a, b }} = { c , { c, d }}. Thus ( a, b ) short = ( c, d ) short . Only if : Suppose { a , { a, b }} = { c , { c, d }}. Then a is in the left hand side, and thus in

810-651: A , b ): ( a ,   b ) K :=   { { a } ,   { a ,   b } } . {\displaystyle (a,\ b)_{K}\;:=\ \{\{a\},\ \{a,\ b\}\}.} When the first and the second coordinates are identical, the definition obtains: ( x ,   x ) K = { { x } , { x ,   x } } = { { x } ,   { x } } = { { x } } {\displaystyle (x,\ x)_{K}=\{\{x\},\{x,\ x\}\}=\{\{x\},\ \{x\}\}=\{\{x\}\}} Given some ordered pair p ,

891-593: A great interest in the mathematical theory of Brownian motion (named after Robert Brown ) proving many results now widely known, such as the non-differentiability of the paths. Consequently, the one-dimensional version of Brownian motion was named the Wiener process . It is the best known of the Lévy processes , càdlàg stochastic processes with stationary statistically independent increments , and occurs frequently in pure and applied mathematics, physics and economics (e.g. on

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972-559: A heart attack. Wiener and his wife are buried at the Vittum Hill Cemetery in Sandwich, New Hampshire . Information is information, not matter or energy. Wiener was an early studier of stochastic and mathematical noise processes, contributing work relevant to electronic engineering , electronic communication , and control systems . It was Wiener's idea to model a signal as if it were an exotic type of noise, giving it

1053-609: A journalist for the Boston Herald , where he wrote a feature story on the poor labor conditions for mill workers in Lawrence, Massachusetts , 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 eventually became a staunch pacifist, he eagerly contributed to the war effort in World War I. In 1916, with America's entry into

1134-622: A key piece of which is the Hermite-Laguerre expansion. This was developed in detail in Nonlinear Problems in Random Theory . Wiener applied Hermite-Laguerre expansion to nonlinear system identification and control. Specifically, a nonlinear system can be identified by inputting a white noise process and computing the Hermite-Laguerre expansion of its output. The identified system can then be controlled. Wiener took

1215-567: A participant of the Macy conferences . In 1926 Wiener married Margaret Engemann, an assistant professor of modern languages at Juniata College . They had two daughters. Wiener admitted in his autobiography I Am a Mathematician: The Later Life of a Prodigy to abusing benzadrine throughout his life without being fully aware of its dangers. Wiener died in March 1964, aged 69, in Stockholm , from

1296-721: A personal library from which the young Norbert benefited greatly. Leo also 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 (Norbert) failed to give a correct answer, at which his father would lose his temper. In "The Theory of Ignorance", a paper he wrote at the age of 10, he disputed "man’s presumption in declaring that his knowledge has no limits", arguing that all human knowledge "is based on an approximation", and acknowledging "the impossibility of being certain of anything." He graduated from Ayer High School in 1906 at 11 years of age, and Wiener then entered Tufts College . He

1377-540: A set a and b must be different, but in an ordered pair they may be equal and that while the order of listing the elements of a set doesn't matter, in an ordered pair changing the order of distinct entries changes the ordered pair. This "definition" is unsatisfactory because it is only descriptive and is based on an intuitive understanding of order . However, as is sometimes pointed out, no harm will come from relying on this description and almost everyone thinks of ordered pairs in this manner. A more satisfactory approach

1458-502: A set of ordered pairs, to have a type only 1 higher than the type of its arguments. This definition works only if the set of natural numbers is infinite. This is the case in NF , but not in type theory or in NFU . J. Barkley Rosser showed that the existence of such a type-level ordered pair (or even a "type-raising by 1" ordered pair) implies the axiom of infinity . For an extensive discussion of

1539-402: A sound mathematical basis. The example often given to students is that English text could be modeled as a random string of letters and spaces, where each letter of the alphabet (and the space) has an assigned probability. But Wiener dealt with analog signals, where such a simple example doesn't exist. Wiener's early work on information theory and signal processing was limited to analog signals, and

1620-400: A }, { a , b }} = {{ c }, { c , d }}. Thus ( a, b ) K = ( c , d ) K . Only if . Two cases: a = b , and a ≠ b . If a = b : If a ≠ b , then ( a , b ) K = ( c , d ) K implies {{ a }, { a , b }} = {{ c }, { c , d }}. Reverse : ( a, b ) reverse = {{ b }, { a, b }} = {{ b }, { b, a }} = ( b, a ) K . If . If ( a, b ) reverse = ( c, d ) reverse , ( b,

1701-416: Is φ ″ A ∪ ψ ″ B {\displaystyle \varphi ''A\cup \psi ''B} in alternate notation). Extracting all the elements of the pair that do not contain 0 and undoing φ {\displaystyle \varphi } yields A . Likewise, B can be recovered from the elements of the pair that do contain 0. For example,

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1782-585: Is a scientist who specializes in the academic study of computer science . Computer scientists typically work on the theoretical side of computation. Although computer scientists can also focus their work and research on specific areas (such as algorithm and data structure development and design, software engineering , information theory , database theory , theoretical computer science , numerical analysis , programming language theory , compiler , computer graphics , computer vision , robotics , computer architecture , operating system ), their foundation

1863-408: Is a common notation for the i -th component of an n -tuple t . In some introductory mathematics textbooks an informal (or intuitive) definition of ordered pair is given, such as For any two objects a and b , the ordered pair ( a , b ) is a notation specifying the two objects a and b , in that order. This is usually followed by a comparison to a set of two elements; pointing out that in

1944-876: Is a natural number and leaves it as is otherwise; the number 0 does not appear in the range of σ {\displaystyle \sigma } . As x ∖ N {\displaystyle x\setminus \mathbb {N} } is the set of the elements of x {\displaystyle x} not in N {\displaystyle \mathbb {N} } go on with φ ( x ) := σ [ x ] = { σ ( α ) ∣ α ∈ x } = ( x ∖ N ) ∪ { n + 1 : n ∈ ( x ∩ N ) } . {\displaystyle \varphi (x):=\sigma [x]=\{\sigma (\alpha )\mid \alpha \in x\}=(x\setminus \mathbb {N} )\cup \{n+1:n\in (x\cap \mathbb {N} )\}.} This

2025-452: Is considered the originator of cybernetics , the science of communication as it relates to living things and machines, with implications for engineering , systems control , computer science , biology , neuroscience , philosophy , and the organization of society . His work heavily influenced computer pioneer John von Neumann , information theorist Claude Shannon , anthropologists Margaret Mead and Gregory Bateson , and others. Wiener

2106-492: Is credited as being one of the first to theorize that all intelligent behavior was the result of feedback mechanisms, that could possibly be simulated by machines and was an important early step towards the development of modern artificial intelligence . Wiener was born in Columbia, Missouri , the first child of Leo Wiener and Bertha Kahn, Jewish immigrants from Lithuania and Germany , respectively. Through his father, he

2187-446: Is due to Kuratowski (see below) and his definition was used in the second edition of Bourbaki's Theory of Sets , published in 1970. Even those mathematical textbooks that give an informal definition of ordered pairs will often mention the formal definition of Kuratowski in an exercise. If one agrees that set theory is an appealing foundation of mathematics , then all mathematical objects must be defined as sets of some sort. Hence if

2268-441: Is how the second coordinate can be extracted: π 2 ( p ) = ⋃ { a ∈ ⋃ p | ⋃ p ≠ ⋂ p → a ∉ ⋂ p } = ⋃ { a ∈ { x , y } | { x , y } ≠ { x } →

2349-464: Is in some set A and whose second entry is in some set B is called the Cartesian product of A and B , and written A × B . A binary relation between sets A and B is a subset of A × B . The ( a , b ) notation may be used for other purposes, most notably as denoting open intervals on the real number line . In such situations, the context will usually make it clear which meaning

2430-492: Is inadmissible in most modern formalized set theories and is methodologically similar to defining the cardinal of a set as the class of all sets equipotent with the given set. Morse–Kelley set theory makes free use of proper classes . Morse defined the ordered pair so that its projections could be proper classes as well as sets. (The Kuratowski definition does not allow this.) He first defined ordered pairs whose projections are sets in Kuratowski's manner. He then redefined

2511-474: Is intended. For additional clarification, the ordered pair may be denoted by the variant notation ⟨ a , b ⟩ {\textstyle \langle a,b\rangle } , but this notation also has other uses. The left and right projection of a pair p is usually denoted by π 1 ( p ) and π 2 ( p ), or by π ℓ ( p ) and π r ( p ), respectively. In contexts where arbitrary n -tuples are considered, π i ( t )

Norbert Wiener - Misplaced Pages Continue

2592-480: Is so-called because it requires two rather than three pairs of braces . Proving that short satisfies the characteristic property requires the Zermelo–Fraenkel set theory axiom of regularity . Moreover, if one uses von Neumann's set-theoretic construction of the natural numbers , then 2 is defined as the set {0, 1} = {0, {0}}, which is indistinguishable from the pair (0, 0) short . Yet another disadvantage of

2673-511: Is the set image of a set x {\displaystyle x} under σ {\displaystyle \sigma } , sometimes denoted by σ ″ x {\displaystyle \sigma ''x} as well. Applying function φ {\displaystyle \varphi } to a set x simply increments every natural number in it. In particular, φ ( x ) {\displaystyle \varphi (x)} never contains contain

2754-507: Is the theoretical study of computing from which these other fields derive. A primary goal of computer scientists is to develop or validate models, often mathematical, to describe the properties of computational systems ( processors , programs, computers interacting with people, computers interacting with other computers, etc.) with an overall objective of discovering designs that yield useful benefits (faster, smaller, cheaper, more precise, etc.). Most computer scientists are required to possess

2835-424: Is to observe that the characteristic property of ordered pairs given above is all that is required to understand the role of ordered pairs in mathematics. Hence the ordered pair can be taken as a primitive notion , whose associated axiom is the characteristic property. This was the approach taken by the N. Bourbaki group in its Theory of Sets , published in 1954. However, this approach also has its drawbacks as both

2916-607: Is trivially true, since Y 1 ≠ Y 2 is never the case. If p = ( x , y ) = { { x } , { x , y } } {\displaystyle p=(x,y)=\{\{x\},\{x,y\}\}} then: This is how we can extract the first coordinate of a pair (using the iterated-operation notation for arbitrary intersection and arbitrary union ): π 1 ( p ) = ⋃ ⋂ p = ⋃ { x } = x . {\displaystyle \pi _{1}(p)=\bigcup \bigcap p=\bigcup \{x\}=x.} This

2997-522: The Cold War . He was a strong advocate of automation to improve the standard of living, and to end economic underdevelopment. His ideas became influential in India , whose government he advised during the 1950s. After the war, Wiener became increasingly concerned with what he believed was political interference with scientific research, and the militarization of science. His article "A Scientist Rebels" from

3078-538: The Khinchin – Kolmogorov theorem ), states that the power spectral density of a wide-sense-stationary random process is the Fourier transform of the corresponding autocorrelation function. 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. Wiener's original construction only applied to

3159-563: The Tauberian theorems . In 1926, Wiener's parents arranged his marriage to a German immigrant, Margaret Engemann; they had two daughters. His sister, Constance (1898–1973), married mathematician Philip Franklin . Their daughter, Janet, Wiener's niece, married mathematician Václav E. Beneš . Norbert Wiener's sister, Bertha (1902–1995), married the botanist Carroll William Dodge . Many tales, perhaps apocryphal, were told of Norbert Wiener at MIT, especially concerning his absent-mindedness. It

3240-531: The University of Göttingen . At Göttingen he also attended three courses with Edmund Husserl "one on Kant's ethical writings, one on the principles of Ethics, and the seminary on Phenomenology." (Letter to Russell, c. June or July, 1914). During 1915–16, he taught philosophy at Harvard, then was an engineer for General Electric and wrote for the Encyclopedia Americana . Wiener was briefly

3321-407: The Wiener equation , named after Wiener, assumes the current velocity of a fluid particle fluctuates randomly. For signal processing, the Wiener filter is a filter proposed by Wiener during the 1940s and published in 1942 as a classified document. Its purpose is to reduce the amount of noise present in a signal by comparison with an estimate of the desired noiseless signal. Wiener developed

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3402-493: The Wiener filter . (The now-standard practice of modeling an information source as a random process—in other words, as a variety of noise—is due to Wiener.) Initially his anti-aircraft work led him to write, with Arturo Rosenblueth and Julian Bigelow , the 1943 article 'Behavior, Purpose and Teleology', which was published in Philosophy of Science . Subsequently his anti-aircraft work led him to formulate cybernetics . After

3483-421: The short pair is the fact that, even if a and b are of the same type, the elements of the short pair are not. (However, if a  =  b then the short version keeps having cardinality 2, which is something one might expect of any "pair", including any "ordered pair".) Prove: ( a , b ) = ( c , d ) if and only if a = c and b = d . Kuratowski : If . If a = c and b = d , then {{

3564-471: The types of Principia Mathematica as sets. Principia Mathematica had taken types, and hence relations of all arities, as primitive . Wiener used {{ b }} instead of { b } to make the definition compatible with type theory where all elements in a class must be of the same "type". With b nested within an additional set, its type is equal to { { a } , ∅ } {\displaystyle \{\{a\},\emptyset \}} 's. About

3645-640: The Galaxy by Robert Heinlein . The song Dedicated to Norbert Wiener appears as the second track on the 1980 album Why? by G.G. Tonet (Luigi Tonet), released on the Italian It Why label. Wiener wrote many books and hundreds of articles: Wiener's papers are collected in the following works: Fiction: Autobiography: Under the name "W. Norbert": Wiener's life and work has been examined in many works: Books and theses: Articles: Archives: Computer scientist A computer scientist

3726-511: The January 1947 issue of The Atlantic Monthly urged scientists to consider 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

3807-440: The breach. Patrick D. Wall speculated that after the publication of Cybernetics , Wiener asked McCulloch for some physiological facts about the brain that he could then theorize. McCulloch told him "a mixture of what was known to be true and what McCulloch thought should be". Wiener then theorized it, went to a physiology congress, and was shot down. Wiener was convinced that McCulloch had set him up. Wiener later helped develop

3888-551: The case that the left and right coordinates are identical, the right conjunct ( ∀ Y 1 , Y 2 ∈ p : Y 1 ≠ Y 2 → ( x ∉ Y 1 ∨ x ∉ Y 2 ) ) {\displaystyle (\forall Y_{1},Y_{2}\in p:Y_{1}\neq Y_{2}\rightarrow (x\notin Y_{1}\lor x\notin Y_{2}))}

3969-415: The existence of ordered pairs and their characteristic property must be axiomatically assumed. Another way to rigorously deal with ordered pairs is to define them formally in the context of set theory. This can be done in several ways and has the advantage that existence and the characteristic property can be proven from the axioms that define the set theory. One of the most cited versions of this definition

4050-463: The field of information technology consulting , and may be seen as a type of mathematician, given how much of the field depends on mathematics. Computer scientists employed in industry may eventually advance into managerial or project leadership positions. Employment prospects for computer scientists are said to be excellent. Such prospects seem to be attributed, in part, to very rapid growth in computer systems design and related services industry, and

4131-618: The filter at the Radiation Laboratory at MIT to predict the position of German bombers from radar reflections. What emerged was a mathematical theory of great generality—a theory for predicting the future as best one can on the basis of incomplete information about the past. It was a statistical theory that included applications that did not, strictly speaking, predict the future, but only tried to remove noise. It made use of Wiener's earlier work on integral equations and Fourier transforms . Wiener studied polynomial chaos ,

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4212-595: The number 0, so that for any sets x and y , φ ( x ) ≠ { 0 } ∪ φ ( y ) . {\displaystyle \varphi (x)\neq \{0\}\cup \varphi (y).} Further, define ψ ( x ) := σ [ x ] ∪ { 0 } = φ ( x ) ∪ { 0 } . {\displaystyle \psi (x):=\sigma [x]\cup \{0\}=\varphi (x)\cup \{0\}.} By this, ψ ( x ) {\displaystyle \psi (x)} does always contain

4293-604: The number 0. Finally, define the ordered pair ( A , B ) as the disjoint union ( A , B ) := φ [ A ] ∪ ψ [ B ] = { φ ( a ) : a ∈ A } ∪ { φ ( b ) ∪ { 0 } : b ∈ B } . {\displaystyle (A,B):=\varphi [A]\cup \psi [B]=\{\varphi (a):a\in A\}\cup \{\varphi (b)\cup \{0\}:b\in B\}.} (which

4374-474: The ordered pair in the context of Quinian set theories, see Holmes (1998). Early in the development of the set theory, before paradoxes were discovered, Cantor followed Frege by defining the ordered pair of two sets as the class of all relations that hold between these sets, assuming that the notion of relation is primitive: ( x , y ) = { R : x R y } . {\displaystyle (x,y)=\{R:xRy\}.} This definition

4455-622: 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( see also ). Norbert Wiener proposed the first set theoretical definition of the ordered pair in 1914: ( a , b ) := { { { a } , ∅ } , { { b } } } . {\displaystyle \left(a,b\right):=\left\{\left\{\left\{a\right\},\,\emptyset \right\},\,\left\{\left\{b\right\}\right\}\right\}.} He observed that this definition made it possible to define

4536-529: The pair ( { { a , 0 } , { b , c , 1 } } , { { d , 2 } , { e , f , 3 } } ) {\displaystyle (\{\{a,0\},\{b,c,1\}\},\{\{d,2\},\{e,f,3\}\})} is encoded as { { a , 1 } , { b , c , 2 } , { d , 3 , 0 } , { e , f , 4 , 0 } } {\displaystyle \{\{a,1\},\{b,c,2\},\{d,3,0\},\{e,f,4,0\}\}} provided

4617-694: The pair ( x , y ) = ( { 0 } × s ( x ) ) ∪ ( { 1 } × s ( y ) ) {\displaystyle (x,y)=(\{0\}\times s(x))\cup (\{1\}\times s(y))} where the component Cartesian products are Kuratowski pairs of sets and where s ( x ) = { ∅ } ∪ { { t } ∣ t ∈ x } {\displaystyle s(x)=\{\emptyset \}\cup \{\{t\}\mid t\in x\}} This renders possible pairs whose projections are proper classes. The Quine–Rosser definition above also admits proper classes as projections. Similarly

4698-424: The pair. Alternatively, the objects are called the first and second components , the first and second coordinates , or the left and right projections of the ordered pair. Cartesian products and binary relations (and hence functions ) are defined in terms of ordered pairs, cf. picture. Let ( a 1 , b 1 ) {\displaystyle (a_{1},b_{1})} and (

4779-875: The property " x is the first coordinate of p " can be formulated as: ∀ Y ∈ p : x ∈ Y . {\displaystyle \forall Y\in p:x\in Y.} The property " x is the second coordinate of p " can be formulated as: ( ∃ Y ∈ p : x ∈ Y ) ∧ ( ∀ Y 1 , Y 2 ∈ p : Y 1 ≠ Y 2 → ( x ∉ Y 1 ∨ x ∉ Y 2 ) ) . {\displaystyle (\exists Y\in p:x\in Y)\land (\forall Y_{1},Y_{2}\in p:Y_{1}\neq Y_{2}\rightarrow (x\notin Y_{1}\lor x\notin Y_{2})).} In

4860-534: The remainder of his career. For many years his photograph was prominently displayed in the Infinite Corridor and often used in giving directions, but by 2017 it had been removed. 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, and

4941-885: The right hand side. Because equal sets have equal elements, one of a = c or a = { c, d } must be the case. Again, we see that { a, b } = c or { a, b } = { c, d }. Rosser (1953) employed a definition of the ordered pair due to Quine which requires a prior definition of the natural numbers . Let N {\displaystyle \mathbb {N} } be the set of natural numbers and define first σ ( x ) := { x , if  x ∉ N , x + 1 , if  x ∈ N . {\displaystyle \sigma (x):={\begin{cases}x,&{\text{if }}x\notin \mathbb {N} ,\\x+1,&{\text{if }}x\in \mathbb {N} .\end{cases}}} The function σ {\displaystyle \sigma } increments its argument if it

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5022-643: The run-up to World War II (1939–45) Wiener became a member of the China Aid Society and the Emergency Committee in Aid of Displaced German Scholars. He was interested in placing scholars such as Yuk-Wing Lee and Antoni Zygmund who had lost their positions. During World War II , his work on the automatic aiming and firing of anti-aircraft guns caused Wiener to investigate information theory independently of Claude Shannon and to invent

5103-413: The same time as Wiener (1914), Felix Hausdorff proposed his definition: ( a , b ) := { { a , 1 } , { b , 2 } } {\displaystyle (a,b):=\left\{\{a,1\},\{b,2\}\right\}} "where 1 and 2 are two distinct objects different from a and b." In 1921 Kazimierz Kuratowski offered the now-accepted definition of the ordered pair (

5184-512: The sense that their domains and codomains are proper classes . The above Kuratowski definition of the ordered pair is "adequate" in that it satisfies the characteristic property that an ordered pair must satisfy, namely that ( a , b ) = ( x , y ) ↔ ( a = x ) ∧ ( b = y ) {\displaystyle (a,b)=(x,y)\leftrightarrow (a=x)\land (b=y)} . In particular, it adequately expresses 'order', in that (

5265-526: The set {y} could be obtained more simply: { y } = { a ∈ { x , y } | a ∉ { x } } {\displaystyle \{y\}=\{\left.a\in \{x,y\}\,\right|\,a\notin \{x\}\}} , but the previous formula also takes into account the case when x=y) Note that π 1 {\displaystyle \pi _{1}} and π 2 {\displaystyle \pi _{2}} are generalized functions , in

5346-460: The singleton set s ( x ) {\displaystyle s(x)} which has an inserted empty set allows tuples to have the uniqueness property that if a is an n -tuple and b is an m -tuple and a = b then n = m . Ordered triples which are defined as ordered pairs do not have this property with respect to ordered pairs. Ordered pairs can also be introduced in Zermelo–Fraenkel set theory (ZF) axiomatically by just adding to ZF

5427-430: The software publishing industry, which are projected to be among the fastest growing industries in the U.S. economy. Ordered pair Ordered pairs are also called 2-tuples , or sequences (sometimes, lists in a computer science context) of length 2. Ordered pairs of scalars are sometimes called 2-dimensional vectors . (Technically, this is an abuse of terminology since an ordered pair need not be an element of

5508-578: 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. His work with Mary Brazier is referred to in Avis DeVoto 's As Always, Julia . A flagship named after him appears briefly in Citizen of

5589-438: The stock-market). Wiener's tauberian theorem , a 1932 result of Wiener, developed Tauberian theorems in summability theory , on the face of it a chapter of real analysis , by showing that most of the known results could be encapsulated in a principle taken from harmonic analysis . In its present formulation, the theorem of Wiener does not have any obvious association with Tauberian theorems, which deal with infinite series ;

5670-496: The theories of cybernetics, robotics , computer control, and automation . He discussed the modeling of neurons with John von Neumann , and in a letter from November 1946 von Neumann presented his thoughts in advance of a meeting with Wiener. Wiener always shared his theories and findings with other researchers, and credited the contributions of others. These included Soviet researchers and their findings. Wiener's acquaintance with them caused him to be regarded with suspicion during

5751-459: 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, and that simplification has been in common use ever since. It is (x, y) = {{x}, {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

5832-408: The translation from results formulated for integrals, or using the language of functional analysis and Banach algebras , is however a relatively routine process. The Paley–Wiener theorem relates growth properties of entire functions on C and Fourier transformation of Schwartz distributions of compact support. The Wiener–Khinchin theorem , (also known as the Wiener – Khintchine theorem and

5913-410: The triple is defined as a 3-tuple as follows: ( x , y , z ) = ( { 0 } × s ( x ) ) ∪ ( { 1 } × s ( y ) ) ∪ ( { 2 } × s ( z ) ) {\displaystyle (x,y,z)=(\{0\}\times s(x))\cup (\{1\}\times s(y))\cup (\{2\}\times s(z))} The use of

5994-599: The war drawing closer, Wiener attended a training camp for potential military officers but failed to earn a commission. One year later 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

6075-669: The war, his fame helped MIT to recruit a research team in cognitive science , composed of researchers in neuropsychology and the mathematics and biophysics of the nervous system, including Warren Sturgis McCulloch and Walter Pitts . These men later made pioneering contributions to computer science and artificial intelligence . Soon after the group was formed, Wiener suddenly ended all contact with its members, mystifying his colleagues. This emotionally traumatized Pitts, and led to his career decline. In their biography of Wiener, Conway and Siegelman suggest that Wiener's wife Margaret, who detested McCulloch's bohemian lifestyle, engineered

6156-497: 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. He graduated in 1911 at 17 years of age. The next year he returned to Harvard, while still continuing his philosophical studies. Back at Harvard, Wiener became influenced by Edward Vermilye Huntington , whose mathematical interests ranged from axiomatic foundations to engineering problems. Harvard awarded Wiener

6237-526: Was discharged from the military in February 1919. Wiener was unable to secure a permanent position at Harvard, a situation he attributed largely to anti-Semitism at the university and in particular the antipathy of Harvard mathematician G. D. Birkhoff . He was also rejected for a position at the University of Melbourne . At W. F. Osgood's suggestion, Wiener was hired as an instructor of mathematics at MIT , where, after his promotion to professor, he spent

6318-509: Was largely forgotten with the development of the digital theory. Wiener is one of the key originators of cybernetics , a formalization of the notion of feedback , with many implications for engineering , systems control , computer science , biology , philosophy , and the organization of society . His work with cybernetics influenced Gregory Bateson and Margaret Mead , and through them, anthropology , sociology , and education . A simple mathematical representation of Brownian motion ,

6399-416: Was related to Maimonides , the famous rabbi , philosopher 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 widely and accumulated

6480-444: Was said that he returned home once to find his house empty. He inquired of a neighborhood girl the reason, and she said that the family had moved elsewhere that day. He thanked her for the information and she replied, "It's ok, Daddy, Mommy sent me to get you". Asked about the story, Wiener's daughter reportedly asserted that "he never forgot who his children were! The rest of it, however, was pretty close to what actually happened…" In

6561-479: Was still eager to serve in uniform and 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

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