Magnetic energy: Difference between revisions
italian version |
Undid revision 583727784 by 186.45.33.139 (talk) removing unsourced silliness |
||
Line 1: | Line 1: | ||
'''Difference algebra''' is a branch of [[mathematics]] concerned with the study of [[difference equation|difference]] (or [[functional equation|functional]]) equations from the algebraic point of view. Difference algebra is analogous to [[differential algebra]] but concerned with difference equations rather than differential equations. As an independent subject it was initiated by [[Joseph Ritt]] and his student Richard Cohn. | |||
== Difference rings, difference fields and difference algebras == | |||
A ''difference ring'' is a [[commutative ring]] <math> R </math> together with a ring endomorphism <math>\sigma\colon R\to R</math>. Often it is assumed that <math>\sigma</math> is injective. When <math>R</math> is a field one speaks of a ''difference field''. A classical example of a difference field is the field <math>K=\mathbb{C}(x)</math> of rational functions with the difference operator <math>\sigma</math> given by <math>\sigma(f(x))=f(x+1)</math>. The role of difference rings in difference algebra is similar to the role of commutative rings in [[commutative algebra]] and [[algebraic geometry]]. A morphism of difference rings is a morphism of rings that commutes with <math>\sigma</math>. A ''difference algebra'' over a difference field <math>K</math> is a difference ring <math>R</math> with a <math>K</math>-algebra structure such that <math>K\to R</math> is a morphism of difference rings, i.e. <math>\sigma\colon R\to R</math> extends <math>\sigma\colon K\to K</math>. A difference algebra which is a field is called a ''difference field extension''. | |||
== Algebraic difference equations == | |||
The difference polynomial ring <math>K\{y\}=K\{y_1,\ldots,y_n\}</math> over a difference field <math>K</math> in the (difference) variables <math>y_1,\ldots,y_n</math> is the polynomial ring over <math>K</math> in the infinitely many variables <math>\sigma^i(y_j),\ (i\in\mathbb{N}, 1\leq j\leq n)</math>. It becomes a difference algebra over <math>K</math> by extending <math>\sigma</math> from <math>K</math> to <math>K\{y\}</math> as suggested by the naming of the variables. | |||
By a ''system of algebraic difference'' equations over <math>K</math> one means any subset <math>F</math> of <math>K\{y\}</math>. If <math>R</math> is a difference algebra over <math>K</math> the solutions of <math>F</math> in <math>R</math> are | |||
:<math>\mathbb{V}_R(F)=\{a\in R^n|\ f(a)=0 \text{ for all } f\in F\}.</math> | |||
Classically one is mainly interested in solutions in difference field extensions of <math>K</math>. For example, if <math>K=\mathbb{C}(x)</math> and <math>R</math> is the field of meromorphic functions on <math>\mathbb{C}</math> with difference operator <math>\sigma</math> given by <math>\sigma(f(x))=f(x+1)</math>, then the fact that the [[gamma function]] <math>\Gamma</math> satisfies the functional equation <math>\Gamma(x+1)=x\Gamma(x)</math> can be restated abstractly as <math>\Gamma\in\mathbb{V}_R(x\sigma(y_1)-y_1)</math>. | |||
== Difference varieties == | |||
Intuitively, a ''difference variety'' over a difference field <math>K</math> is the set of solutions of a system of algebraic difference equations over <math>K</math>. This definition has to be made more precise by specifying where one is looking for the solutions. Usually one is looking for solutions in the so-called universal family of difference field extensions of <math>K</math>.<ref name=Cohn>{{cite book|last=Cohn|title=Difference algebra}} Chapter 4</ref><ref name=Levin>{{cite book|last=Levin|title=Difference algebra}} Section 2.6</ref> Alternatively, one may define a difference variety as a [[functor]] from the [[category]] of difference field extensions of <math>K</math> to the category of sets, which is of the form <math>R\rightsquigarrow \mathbb{V}_R(F)</math> for some <math>F\subset K\{y\}.</math>. | |||
There is a one-to-one correspondence between the difference varieties defined by algebraic difference equations in the variables <math>y_1,\ldots,y_n</math> and certain ideals in <math>K\{y\}</math>, namely the perfect difference ideals of <math>K\{y\}</math>.<ref>{{cite book|last=Levin|title=Difference algebra}} Theorem 2.6.4</ref> One of the basic theorems in difference algebra asserts that every ascending chain of perfect difference ideals in <math>K\{y\}</math> is finite. This result can be seen as a difference analog of [[Hilbert's basis theorem]]. | |||
== Applications == | |||
Difference algebra is related to many other mathematical areas, such as discrete dynamical systems, combinatorics, number theory or model theory. While some real life problems, such as population dynamics, can be modeled by algebraic difference equations, difference algebra also has applications in pure mathematics. For example, there is a proof of the [[Manin-Mumford conjecture]] using methods of difference algebra.<ref>{{cite journal|last=Hrushovski|first=Ehud|title=The Manin–Mumford conjecture and the model theory of difference fields|journal=Annals of Pure and Applied Logic|year=2001|volume=112|issue=1|pages=43–115|doi=10.1016/S0168-0072(01)00096-3}}</ref> The [[model theory]] of difference fields has been studied. | |||
==Notes== | |||
{{Reflist}} | |||
==References== | |||
*Alexander Levin (2008), [http://books.google.co.uk/books?id=15pgjT5PeY0C Difference algebra], Springer, ISBN 978-1-4020-6946-8 | |||
*Richard M. Cohn (1979), [http://books.google.co.uk/books?id=Fs8oAAAACAAJ& Difference algebra], R.E. Krieger Pub. Co., ISBN 978-0-88275-651-6 | |||
==External links== | |||
*{{cite book|last=Wibmer|first=Michael|title=Lecture Notes - Algebraic difference equations|year=2013|pages=80 pages|url=http://www.algebra.rwth-aachen.de/de/Mitarbeiter/Wibmer/Algebraic%20difference%20equations.pdf}} | |||
* The [http://www.logique.jussieu.fr/~zoe/ home page of Zoé Chatzidakis] has several online surveys discussing (the model theory of) difference fields. | |||
== See also == | |||
* [[Finite difference]] | |||
* [[Recurrence relation]] | |||
* [[Functional equation]] | |||
* [[Differential algebra]] | |||
[[Category:Algebras]] |
Revision as of 22:50, 30 November 2013
Difference algebra is a branch of mathematics concerned with the study of difference (or functional) equations from the algebraic point of view. Difference algebra is analogous to differential algebra but concerned with difference equations rather than differential equations. As an independent subject it was initiated by Joseph Ritt and his student Richard Cohn.
Difference rings, difference fields and difference algebras
A difference ring is a commutative ring together with a ring endomorphism . Often it is assumed that is injective. When is a field one speaks of a difference field. A classical example of a difference field is the field of rational functions with the difference operator given by . The role of difference rings in difference algebra is similar to the role of commutative rings in commutative algebra and algebraic geometry. A morphism of difference rings is a morphism of rings that commutes with . A difference algebra over a difference field is a difference ring with a -algebra structure such that is a morphism of difference rings, i.e. extends . A difference algebra which is a field is called a difference field extension.
Algebraic difference equations
The difference polynomial ring over a difference field in the (difference) variables is the polynomial ring over in the infinitely many variables . It becomes a difference algebra over by extending from to as suggested by the naming of the variables.
By a system of algebraic difference equations over one means any subset of . If is a difference algebra over the solutions of in are
Classically one is mainly interested in solutions in difference field extensions of . For example, if and is the field of meromorphic functions on with difference operator given by , then the fact that the gamma function satisfies the functional equation can be restated abstractly as .
Difference varieties
Intuitively, a difference variety over a difference field is the set of solutions of a system of algebraic difference equations over . This definition has to be made more precise by specifying where one is looking for the solutions. Usually one is looking for solutions in the so-called universal family of difference field extensions of .[1][2] Alternatively, one may define a difference variety as a functor from the category of difference field extensions of to the category of sets, which is of the form for some .
There is a one-to-one correspondence between the difference varieties defined by algebraic difference equations in the variables and certain ideals in , namely the perfect difference ideals of .[3] One of the basic theorems in difference algebra asserts that every ascending chain of perfect difference ideals in is finite. This result can be seen as a difference analog of Hilbert's basis theorem.
Applications
Difference algebra is related to many other mathematical areas, such as discrete dynamical systems, combinatorics, number theory or model theory. While some real life problems, such as population dynamics, can be modeled by algebraic difference equations, difference algebra also has applications in pure mathematics. For example, there is a proof of the Manin-Mumford conjecture using methods of difference algebra.[4] The model theory of difference fields has been studied.
Notes
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
References
- Alexander Levin (2008), Difference algebra, Springer, ISBN 978-1-4020-6946-8
- Richard M. Cohn (1979), Difference algebra, R.E. Krieger Pub. Co., ISBN 978-0-88275-651-6
External links
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - The home page of Zoé Chatzidakis has several online surveys discussing (the model theory of) difference fields.
See also
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 Chapter 4 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 Section 2.6 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 Theorem 2.6.4 - ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang