# Reciprocity theorem wikipedia

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# Reciprocity theorem wikipedia

The Artin reciprocity lawwhich was established by Emil Artin in a series of papers ; ;is a general theorem in number theory that forms a central part of global class field theory. Artin's result provided a partial solution to Hilbert's ninth problem. One of the statements of the Artin reciprocity law is that there is a canonical isomorphism called the global symbol map [2] [3]. This is the content of the local reciprocity lawa main theorem of local class field theory.

Artin's reciprocity law implies a description of the abelianization of the absolute Galois group of a global field K which is based on the Hasse localâ€”global principle and the use of the Frobenius elements. Together with the Takagi existence theoremit is used to describe the abelian extensions of K in terms of the arithmetic of K and to understand the behavior of the nonarchimedean places in them. Therefore, the Artin reciprocity law can be interpreted as one of the main theorems of global class field theory.

It can be used to prove that Artin L-functions are meromorphic and for the proof of the Chebotarev density theorem. Two years after the publication of his general reciprocity law inArtin rediscovered the transfer homomorphism of I.

Schur and used the reciprocity law to translate the principalization problem for ideal classes of algebraic number fields into the group theoretic task of determining the kernels of transfers of finite non-abelian groups. The Artin reciprocity law or global reciprocity law states that there is a modulus c of K such that the Artin map induces an isomorphism.

Then, quadratic reciprocity states that. Robert Langlands interpreted Hecke characters as automorphic forms on the reductive algebraic group GL 1 over the ring of adeles of K. The formulation of the Artin reciprocity law as an equality of L -functions allows formulation of a generalisation to n -dimensional representations, though a direct correspondence is still lacking.

In fact, a more precise version of the reciprocity law keeps track of the ramification. Categories : Class field theory. Namespaces Article Talk. Views Read Edit View history. Help Learn to edit Community portal Recent changes Upload file. Download as PDF Printable version.Reciprocity in electrical networks is a property of a circuit that relates voltages and currents at two points. The reciprocity theorem states that the current at one point in a circuit due to a voltage at a second point is the same as the current at the second point due to the same voltage at the first.

The reciprocity theorem is valid for almost all passive networks. The reciprocity theorem is a feature of a more general principle of reciprocity in electromagnetism.

Any network that consists entirely of ideal capacitancesinductances including mutual inductancesand resistancesthat is, elements that are linear and bilateralwill be reciprocal. Any component containing ferromagnetic material is likely to be non-reciprocal. Examples of passive components deliberately designed to be non-reciprocal include circulators and isolators. The transfer function of a reciprocal network has the property that it is symmetrical about the main diagonal if expressed in terms of a z-parametery-parameteror s-parameter matrix.

### Stanley's reciprocity theorem

A non-symmetrical matrix implies a non-reciprocal network. A symmetric matrix does not imply a symmetric network. In some parametisations of networks, the representative matrix is not symmetrical for reciprocal networks. Common examples are h-parameters and ABCD-parametersbut they all have some other condition for reciprocity that can be calculated from the parameters.

These representations mix voltages and currents in the same column vector and therefore do not even have matching units in transposed elements. An example of reciprocity can be demonstrated using an asymmetrical resistive attenuator. An asymmetrical network is chosen as the example because a symmetrical network is fairly self-evidently reciprocal.

Injecting six amps into port 1 of this network produces 24 volts at port 2. Hence, the network is reciprocal. In this example, the port that is not injecting current is left open circuit. This is because a current generator applying zero current is an open circuit. If, on the other hand, one wished to apply voltages and measure the resulting current, then the port to which the voltage is not applied would be made short circuit. This is because a voltage generator applying zero volts is a short circuit.

Reciprocity of electrical networks is a special case of Lorentz reciprocitybut it can also be proven more directly from network theorems. This proof shows reciprocity for a two-node network in terms of its admittance matrix, and then shows reciprocity for a network with an arbitrary number of nodes by an induction argument. A linear network can be represented as a set of linear equations through nodal analysis.

These equations can be expressed in the form of an admittance matrix, [6]. The matrix is therefore symmetrical. In words, the ratio of the current at one port to the voltage at another is the same ratio if the ports being driven and measured are interchanged. For the case of a matrix of arbitrary size, the order of the matrix can be reduced through node elimination. After eliminating the s th node, the new admittance matrix will have the form.

It can be seen that this new matrix is also symmetrical. Since this matrix is symmetrical it is proved that reciprocity applies to a matrix of arbitrary size when one node is driven by a voltage and current measured at another. A similar process using the impedance matrix from mesh analysis demonstrates reciprocity where one node is driven by a current and voltage is measured at another.

Reciprocity Theorem for DC networks

Namespaces Article Talk.The Etherington's distance-duality equation is the relationship between the luminosity distance of standard candles and the angular diameter distance. When Etherington introduced this equation inhe mentioned that this equation was proposed by Tolman as a way to test a cosmological model. Ellis proposed a proof of this equation in the context of Riemannian geometry. This statement is fundamental in the derivation of the reciprocity theorem. The Etherington's distance-duality equation has been validated from astronomical observations based on the X-ray surface brightness and the Sunyaevâ€”Zel'dovich effect of galaxy clusters.

Etherington Philosophical Magazine ser. General Relativity and Gravitation. Physical Review D. International Journal of Modern Physics D. The Astrophysical Journal. IOP Publishing. Monthly Notices of the Royal Astronomical Society. Categories : Physical quantities. Namespaces Article Talk. Views Read Edit View history. Help Learn to edit Community portal Recent changes Upload file.

Download as PDF Printable version.Stanleystates that a certain functional equation is satisfied by the generating function of any rational cone defined below and the generating function of the cone's interior.

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A d -tuple satisfying the corresponding strict inequalities, i. The generating function F int x 1It can be shown that these are rational functions. Matthias Beck and Mike Develin have shown how to prove this by using the calculus of residues. Develin has said that this amounts to proving the result "without doing any work". Stanley's reciprocity theorem generalizes Ehrhart-Macdonald reciprocity for Ehrhart polynomials of rational convex polytopes.

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## Etherington's reciprocity theorem

Namespaces Article Talk. Views Read Edit View history. Help Learn to edit Community portal Recent changes Upload file. Download as PDF Printable version. Add links.Create an accountCommunity DashboardRandom ArticleAbout UsCategoriesRecent ChangesWrite an ArticleRequest a New ArticleAnswer a RequestMore Ideas. Odds represent which team, horse, or athlete has the highest probability of winning. While there are different ways to write odds, they all indicate how likely one outcome is in comparison to another.

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