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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 as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Network Synthesis
Network synthesis is a design technique for linear electrical circuits. Synthesis starts from a prescribed impedance function of frequency or frequency response and then determines the possible networks that will produce the required response. The technique is to be compared to network analysis in which the response (or other behaviour) of a given circuit is calculated. Prior to network synthesis, only network analysis was available, but this requires that one already knows what form of circuit is to be analysed. There is no guarantee that the chosen circuit will be the closest possible match to the desired response, nor that the circuit is the simplest possible. Network synthesis directly addresses both these issues. Network synthesis has historically been concerned with synthesising passive networks, but is not limited to such circuits. The field was founded by Wilhelm Cauer after reading Ronald M. Foster's 1924 paper '' A reactance theorem''. Foster's theorem provided a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Residue (complex Analysis)
In mathematics, more specifically complex analysis, the residue is a complex number proportional to the contour integral of a meromorphic function along a path enclosing one of its singularities. (More generally, residues can be calculated for any function f\colon \mathbb \setminus \_k \rightarrow \mathbb that is holomorphic except at the discrete points ''k'', even if some of them are essential singularities.) Residues can be computed quite easily and, once known, allow the determination of general contour integrals via the residue theorem. Definition The residue of a meromorphic function f at an isolated singularity a, often denoted \operatorname(f,a), \operatorname_a(f), \mathop_f(z) or \mathop_f(z), is the unique value R such that f(z)- R/(z-a) has an analytic antiderivative in a punctured disk 0<\vert z-a\vert<\delta. Alternatively, residues can be calculated by finding |
Mathematische Annalen
''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann. Subsequent managing editors were Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguignon, Wolfgang Lück, and Nigel Hitchin. Currently, the managing editor of Mathematische Annalen is Thomas Schick. Volumes 1–80 (1869–1919) were published by Teubner. Since 1920 (vol. 81), the journal has been published by Springer. In the late 1920s, under the editorship of Hilbert, the journal became embroiled in controversy over the participation of L. E. J. Brouwer on its editorial board, a spillover from the foundational Brouwer–Hilbert controversy. Between 1945 and 1947 the journal briefly ceased publication. References External links''Mathematische Annalen''homepage at Springer''Mathematische Annalen''archive (1869†... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Project Euclid
Project Euclid is a collaborative partnership between Cornell University Library and Duke University Press which seeks to advance scholarly communication in theoretical and applied mathematics and statistics through partnerships with independent and society publishers. It was created to provide a platform for small publishers of scholarly journals to move from print to electronic in a cost-effective way. Through a combination of support by subscribing libraries and participating publishers, Project Euclid has made 70% of its journal articles available as open access. As of 2010, Project Euclid provided access to over one million pages of open-access content. Mission and goals Project Euclid's stated mission is to advance scholarly communication in the field of theoretical and applied mathematics and statistics. Through a "mixture of open access, subscription, and hosted subscription content it provides a way for small publishers (especially societies) to host their math or statistic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bulletin Of The American Mathematical Society
The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society. Scope It publishes surveys on contemporary research topics, written at a level accessible to non-experts. It also publishes, by invitation only, book reviews and short ''Mathematical Perspectives'' articles. History It began as the ''Bulletin of the New York Mathematical Society'' and underwent a name change when the society became national. The Bulletin's function has changed over the years; its original function was to serve as a research journal for its members. Indexing The Bulletin is indexed in Mathematical Reviews, Science Citation Index, ISI Alerting Services, CompuMath Citation Index, and Current Contents/Physical, Chemical & Earth Sciences. See also *'' Journal of the American Mathematical Society'' *''Memoirs of the American Mathematical Society'' *''Notices of the American Mathematical Society'' *'' Proceedings of the American M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positive-definite Matrix
In mathematics, a symmetric matrix M with real entries is positive-definite if the real number z^\textsfMz is positive for every nonzero real column vector z, where z^\textsf is the transpose of More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number z^* Mz is positive for every nonzero complex column vector z, where z^* denotes the conjugate transpose of z. Positive semi-definite matrices are defined similarly, except that the scalars z^\textsfMz and z^* Mz are required to be positive ''or zero'' (that is, nonnegative). Negative-definite and negative semi-definite matrices are defined analogously. A matrix that is not positive semi-definite and not negative semi-definite is sometimes called indefinite. A matrix is thus positive-definite if and only if it is the matrix of a positive-definite quadratic form or Hermitian form. In other words, a matrix is positive-definite if and only if it defines a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermitian Matrix
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -th row and -th column, for all indices and : or in matrix form: A \text \quad \iff \quad A = \overline . Hermitian matrices can be understood as the complex extension of real symmetric matrices. If the conjugate transpose of a matrix A is denoted by A^\mathsf, then the Hermitian property can be written concisely as Hermitian matrices are named after Charles Hermite, who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of always having real eigenvalues. Other, equivalent notations in common use are A^\mathsf = A^\dagger = A^\ast, although note that in quantum mechanics, A^\ast typically means the complex conjugate only, and not the conjugate transpose. Alternative characterizations Hermit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Admittance Parameters
Admittance parameters or Y-parameters (the elements of an admittance matrix or Y-matrix) are properties used in many areas of electrical engineering, such as power, electronics, and telecommunications. These parameters are used to describe the electrical behavior of linear electrical networks. They are also used to describe the small-signal ( linearized) response of non-linear networks. Y parameters are also known as short circuited admittance parameters. They are members of a family of similar parameters used in electronic engineering, other examples being: S-parameters, Z-parameters, H-parameters, T-parameters or ABCD-parameters. The Y-parameter matrix A Y-parameter matrix describes the behaviour of any linear electrical network that can be regarded as a black box with a number of ports. A ''port'' in this context is a pair of electrical terminals carrying equal and opposite currents into and out of the network, and having a particular voltage between them. The Y-matrix gives ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Impedance Parameters
Impedance parameters or Z-parameters (the elements of an impedance matrix or Z-matrix) are properties used in electrical engineering, electronic engineering, and communication systems engineering to describe the electrical behavior of linear electrical networks. They are also used to describe the small-signal ( linearized) response of non-linear networks. They are members of a family of similar parameters used in electronic engineering, other examples being: S-parameters, Y-parameters, H-parameters, T-parameters or ABCD-parameters. Z-parameters are also known as ''open-circuit impedance parameters'' as they are calculated under open circuit conditions. i.e., Ix=0, where x=1,2 refer to input and output currents flowing through the ports (of a two-port network in this case) respectively. The Z-parameter matrix A Z-parameter matrix describes the behaviour of any linear electrical network that can be regarded as a black box with a number of ports. A ''port'' in this context is a pair ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Port (circuit Theory)
In electrical circuit theory, a port is a pair of terminals connecting an electrical network or circuit to an external circuit, as a point of entry or exit for electrical energy. A port consists of two nodes (terminals) connected to an outside circuit which meets the ''port condition'' - the currents flowing into the two nodes must be equal and opposite. The use of ports helps to reduce the complexity of circuit analysis. Many common electronic devices and circuit blocks, such as transistors, transformers, electronic filters, and amplifiers, are analyzed in terms of ports. In multiport network analysis, the circuit is regarded as a "black box" connected to the outside world through its ports. The ports are points where input signals are applied or output signals taken. Its behavior is completely specified by a matrix of parameters relating the voltage and current at its ports, so the internal makeup or design of the circuit need not be considered, or even known, in determi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Branch Points
In the mathematical field of complex analysis, a branch point of a multi-valued function (usually referred to as a "multifunction" in the context of complex analysis) is a point such that if the function is n-valued (has n values) at that point, all of its neighborhoods contain a point that has more than n values. Multi-valued functions are rigorously studied using Riemann surfaces, and the formal definition of branch points employs this concept. Branch points fall into three broad categories: algebraic branch points, transcendental branch points, and logarithmic branch points. Algebraic branch points most commonly arise from functions in which there is an ambiguity in the extraction of a root, such as solving the equation ''w''2 = ''z'' for ''w'' as a function of ''z''. Here the branch point is the origin, because the analytic continuation of any solution around a closed loop containing the origin will result in a different function: there is non-trivial monodromy. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ladder Network
Electronic filter topology defines electronic filter circuits without taking note of the values of the components used but only the manner in which those components are connected. Filter design characterises filter circuits primarily by their transfer function rather than their topology. Transfer functions may be linear or nonlinear. Common types of linear filter transfer function are; high-pass, low-pass, bandpass, band-reject or notch and all-pass. Once the transfer function for a filter is chosen, the particular topology to implement such a prototype filter can be selected so that, for example, one might choose to design a Butterworth filter using the Sallen–Key topology. Filter topologies may be divided into passive and active types. Passive topologies are composed exclusively of passive components: resistors, capacitors, and inductors. Active topologies also include active components (such as transistors, op amps, and other integrated circuits) that require po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
