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Richards' Theorem
Richards' theorem is a mathematical result due to Paul I. Richards in 1947. The theorem states that for, :R(s) = \frac if Z(s) is a positive-real function (PRF) then R(s) is a PRF for all real, positive values of k. The theorem has applications in electrical network synthesis. The PRF property of an impedance function determines whether or not a passive network can be realised having that impedance. Richards' theorem led to a new method of realising such networks in the 1940s. Proof : R(s) = \frac where Z(s) is a PRF, k is a positive real constant, and s= \sigma + i \omega is the complex frequency variable, can be written as, : R(s) = \dfrac where, : W(s) = \left ( \frac \right ) Since Z(s) is PRF then : 1 + \dfrac is also PRF. The zeroes of this function are the poles of W(s). Since a PRF can have no zeroes in the right-half ''s''-plane, then W(s) can have no poles in the right-half ''s''-plane and hence is analytic in the right-half ''s''-plane. L ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul I
Paul I may refer to: *Paul of Samosata (200–275), Bishop of Antioch *Paul I of Constantinople (died c. 350), Archbishop of Constantinople *Pope Paul I (700–767) *Paul I Šubić of Bribir (c. 1245–1312), Ban of Croatia and Lord of Bosnia *Paul I, Serbian Patriarch, Archbishop of Peć and Serbian Patriarch (c. 1530–1541) *Paul I of Russia (1754–1801), Emperor of Russia *Paul Peter Massad (1806–1890), Maronite Patriarch of Antioch *Paul of Greece (1901–1964), King of Greece * Pavle, Serbian Patriarch Pavle ( sr-cyr, Павле, ''Paul''; 11 September 1914 – 15 November 2009) was the patriarch of the Serbian Orthodox Church from 1990 to his death. His full title was ''His Holiness the Archbishop of Peć, Metropolitan of Belgrade and ... (1914–2009), Patriarch of the Serbian Orthodox Church See also * Patriarch Paul I (other) {{hndis, Paul 01 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Raoul Bott
Raoul Bott (September 24, 1923 – December 20, 2005) was a Hungarian- American mathematician known for numerous basic contributions to geometry in its broad sense. He is best known for his Bott periodicity theorem, the Morse–Bott functions which he used in this context, and the Borel–Bott–Weil theorem. Early life Bott was born in Budapest, Hungary, the son of Margit Kovács and Rudolph Bott. His father was of Austrian descent, and his mother was of Hungarian Jewish descent; Bott was raised a Catholic by his mother and stepfather. Bott grew up in Czechoslovakia and spent his working life in the United States. His family emigrated to Canada in 1938, and subsequently he served in the Canadian Army in Europe during World War II. Career Bott later went to college at McGill University in Montreal, where he studied electrical engineering. He then earned a PhD in mathematics from Carnegie Mellon University in Pittsburgh in 1949. His thesis, titled ''Electrical Network Theory'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electronic Engineering
Electronics engineering is a sub-discipline of electrical engineering which emerged in the early 20th century and is distinguished by the additional use of active components such as semiconductor devices to amplify and control electric current flow. Previously electrical engineering only used passive devices such as mechanical switches, resistors, inductors and capacitors. It covers fields such as: analog electronics, digital electronics, consumer electronics, embedded systems and power electronics. It is also involved in many related fields, for example solid-state physics, radio engineering, telecommunications, control systems, signal processing, systems engineering, computer engineering, instrumentation engineering, electric power control, robotics. The Institute of Electrical and Electronics Engineers (IEEE) is one of the most important professional bodies for electronics engineers in the US; the equivalent body in the UK is the Institution of Engineering and Te ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorems In Complex Analysis
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Malcolm C
Malcolm, Malcom, Máel Coluim, or Maol Choluim may refer to: People * Malcolm (given name), includes a list of people and fictional characters * Clan Malcolm * Maol Choluim de Innerpeffray, 14th-century bishop-elect of Dunkeld Nobility * Máel Coluim, Earl of Atholl, Mormaer of Atholl between 1153/9 and the 1190s * Máel Coluim, King of Strathclyde, 10th century * Máel Coluim of Moray, Mormaer of Moray 1020–1029 * Máel Coluim (son of the king of the Cumbrians), possible King of Strathclyde or King of Alba around 1054 * Malcolm I of Scotland (died 954), King of Scots * Malcolm II of Scotland, King of Scots from 1005 until his death * Malcolm III of Scotland, King of Scots * Malcolm IV of Scotland, King of Scots * Máel Coluim, Earl of Angus, the fifth attested post 10th-century Mormaer of Angus * Máel Coluim I, Earl of Fife, one of the more obscure Mormaers of Fife * Maol Choluim I, Earl of Lennox, Mormaer * Máel Coluim II, Earl of Fife, Mormaer * Maol Choluim II, Earl of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John H
John is a common English name and surname: * John (given name) * John (surname) John may also refer to: New Testament Works * Gospel of John, a title often shortened to John * First Epistle of John, often shortened to 1 John * Second Epistle of John, often shortened to 2 John * Third Epistle of John, often shortened to 3 John People * John the Baptist (died c. AD 30), regarded as a prophet and the forerunner of Jesus Christ * John the Apostle (lived c. AD 30), one of the twelve apostles of Jesus * John the Evangelist, assigned author of the Fourth Gospel, once identified with the Apostle * John of Patmos, also known as John the Divine or John the Revelator, the author of the Book of Revelation, once identified with the Apostle * John the Presbyter, a figure either identified with or distinguished from the Apostle, the Evangelist and John of Patmos Other people with the given name Religious figures * John, father of Andrew the Apostle and Saint Peter * P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electrical Element
Electrical elements are conceptual abstractions representing idealized electrical components, such as resistors, capacitors, and inductors, used in the analysis of electrical networks. All electrical networks can be analyzed as multiple electrical elements interconnected by wires. Where the elements roughly correspond to real components, the representation can be in the form of a schematic diagram or circuit diagram. This is called a lumped-element circuit model. In other cases, infinitesimal elements are used to model the network, in a distributed-element model. These ideal electrical elements represent real, physical electrical or electronic components but they do not exist physically and they are assumed to have ideal properties, while actual electrical components have less than ideal properties, a degree of uncertainty in their values and some degree of nonlinearity. To model the nonideal behavior of a real circuit component may require a combination of multiple i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard Duffin
Richard James Duffin (1909 – October 29, 1996) was an American physicist, known for his contributions to electrical transmission theory and to the development of geometric programming and other areas within operations research. Education and career Duffin obtained a BSc in physics at the University of Illinois, where he was elected to Sigma Xi in 1932. He stayed at Illinois for his PhD, which was advised by Harold Mott-Smith and David Bourgin, producing a thesis entitled ''Galvanomagnetic and Thermomagnetic Phenomena'' (1935). Duffin lectured at Purdue University and Illinois before joining the Carnegie Institute in Washington, D.C. during World War II.Richard J. Duffin from the |
Necessary And Sufficient
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of is guaranteed by the truth of (equivalently, it is impossible to have without ). Similarly, is sufficient for , because being true always implies that is true, but not being true does not always imply that is not true. In general, a necessary condition is one that must be present in order for another condition to occur, while a sufficient condition is one that produces the said condition. The assertion that a statement is a "necessary ''and'' sufficient" condition of another means that the former statement is true if and only if the latter is true. That is, the two statements must be either simultaneously true, or simultaneously false. In ordinary English (also natural language) "necessary" and "sufficient" indicate relations bet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positive-real Function
Positive-real functions, often abbreviated to PR function or PRF, are a kind of mathematical function that first arose in electrical network synthesis. They are complex functions, ''Z''(''s''), of a complex variable, ''s''. A rational function is defined to have the PR property if it has a positive real part and is analytic in the right half of the complex plane and takes on real values on the real axis. In symbols the definition is, : \begin & \Re (s)0 \quad\text\quad \Re(s) > 0 \\ & \Im (s)0 \quad\text\quad \Im(s)=0 \end In electrical network analysis, ''Z''(''s'') represents an impedance expression and ''s'' is the complex frequency variable, often expressed as its real and imaginary parts; :s=\sigma+i\omega \,\! in which terms the PR condition can be stated; : \begin & \Re (s)0 \quad\text\quad \sigma > 0 \\ & \Im (s)0 \quad\text\quad \omega=0 \end The importance to network analysis of the PR condition lies in the realisability condition. ''Z''(''s'') is realisable a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Otto Brune
Otto Walter Heinrich Oscar Brune (10 January 1901 – 1982) undertook some key investigations into network synthesis at the Massachusetts Institute of Technology (MIT) where he graduated in 1929. His doctoral thesis was supervised by Wilhelm Cauer and Ernst Guillemin, who the latter ascribed to Brune the laying of "the mathematical foundation for modern realization theory". Biography Brune was born in Bloemfontein, Orange Free State 10 January 1901 and grew up in Kimberley, Cape Colony. He enrolled in the University of Stellenbosch in 1918, receiving a Bachelor of Science in 1920 and Master of Science in 1921. He taught German, mathematics, and science at the Potchefstroom Gymnasium, Transvaal in 1922, and lectured in mathematics at the Transvaal University College, Pretoria 1923–1925. In 1926 Brune moved to the US to attend the Massachusetts Institute of Technology (MIT) under the sponsorship of the General Electric Company, receiving Batchelor and Master's degrees in 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schwarz's Lemma
In mathematics, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex analysis about holomorphic functions from the open unit disk to itself. The lemma is less celebrated than deeper theorems, such as the Riemann mapping theorem, which it helps to prove. It is, however, one of the simplest results capturing the rigidity of holomorphic functions. Statement Let \mathbf = \ be the open unit disk in the complex plane \mathbb centered at the origin, and let f : \mathbf\rightarrow \mathbb be a holomorphic map such that f(0) = 0 and , f(z), \leq 1 on \mathbf. Then , f(z), \leq , z, for all z \in \mathbf, and , f'(0), \leq 1. Moreover, if , f(z), = , z, for some non-zero z or , f'(0), = 1, then f(z) = az for some a \in \mathbb with , a, = 1.Theorem 5.34 in Proof The proof is a straightforward application of the maximum modulus principle on the function :g(z) = \begin \frac\, & \mbox z \neq 0 \\ f'(0) & \mbox z = 0, \end which is holomorphic on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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