Fuss–Catalan number: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
en>Nick Number
m spelling fixes WP:TYPO
 
en>BG19bot
m WP:CHECKWIKI error fix for #61. Punctuation goes before References. Do general fixes if a problem exists. - using AWB (10469)
 
Line 1: Line 1:
In [[mathematics]], in [[linear algebra]], the '''cyclic decomposition theorem''' is an assertion of a certain property of [[dimension|finite-dimenstional]] [[vector space]]s in relation to [[linear transformation]]s of the spaces. The theorem states that given a linear transformation of a finite-dimensional vector space over an [[algebraically closed field]], the vector space can be expressed as a [[direct sum]] of [[linear subspace|subspace]]s each of which is [[Invariant subspace|invariant]] under the transformation and is cyclically generated by the transformation. This result is considered to be "one of the deepest results in linear algebra".<ref name=LA>{{cite book|last=Kenneth Hoffmann, Ray Kunze|title=Linear Algebra (Second Edition)|year=1971|publisher=Prentice-Hall|page=231}}</ref>
BUNGALOWS, SEMI-DETACHED & TERRACE HOMES IN WOODLANDS, Privately Held; 1001-5000 employees;Actual Estate industry Seascape Launched Today! Guide Now for Non-public Preview! By Appointment Only! Direct Showflat Agent Call +65 9668 8021 to Take pleasure in VVIP Pricing! JLL acquires YY Property Solutions Sdn Bhd in Malaysia Landed terrace (+5.02% nominal) and non-landed residences (+5.30% nominal) enjoyed comparatively good development, whereas non-landed condominums have been weaker (+zero.ninety eight% nominal), based on the figures from URA.<br><br>SLA assists the Authorities in managing its land sales system. Our roles are to realize optimum returns from the gross sales of State lands and to nominate and handle the State's land sales agents. The place land is slated for launch via the Government tenders and auctions, we make sure that land is accessible and the land sale processes comply with statutory guidelines. Misc. Franchise Resales Enterprise For Sale, Advertising & Advertising and marketing Business For Sale ASSOCIATE WITH A HIGHLY MOTIVATED STAFF About our Enterprise We are a global advertising enterprise who partners with an Award Profitable Leadership Growth Firm and are committed to turning into the chosen partner for individuals in search of their very own enterprise alternative. Business Des Tenure Any Freehold 999 Years ninety nine Years<br><br>The Hexacube is providing new industrial property on the market Singapore traders want to think about. Residents of the area will seemingly respect that they will customize the commercial house that they need to purchase as effectively. This challenge represents probably the most expansive buildings provided in Singapore so far. Many buyers will probably want to check out how they will customise the property that they do determine to buy through right here. This location has provided individuals the possibility that they need to perceive more about how this process can work as well.<br><br>Actual property providers embody selling, buying, leasing and property management for residential and industrial properties. eight Temasek Boulevard, Penthouse Level Suntec Metropolis Tower 3, Singapore 038988. Actual estate agency promoting and letting property. 190 Lorong 6 Toa Payoh, #02-514, Singapore 310190. Real property providers to patrons and sellers on the lookout for residential and commercial properties for sale and hire. The flexibleness of proudly owning a private property such as making modifications to the home is a significant purpose as to why people buy non-public property in Singapore Apart from the pliability, the prestige and standing that derive from owning a personal property is the last word for many. Freehold Condominium near Orchard The Edge on Cairnhill – Condominium For Sale Terrace Home<br><br>Pretty house in the east, near MRT Casa Merah (D16) – Residence For Lease Very spacious condo close to international faculty Apartment For Lease – Leonie Towers (D09) The Residential Property Act defines a "overseas particular person" as any person who shouldn't be a citizen of Singapore, a everlasting resident and any overseas firm or transformed international company. Any individual or entity falling beneath this category is totally restricted from owning landed residential property except approval is obtained from the Controller of Residential Property. S$100,000 cash is enough for a down cost of a primary property in prime London , or personal prime properties in Iskandar, KL, Manila & Cambodia Semi-Detached( For Sale), 34 /36 Jalan Tupai, Singapore, How does property public sale work?<br><br>To write down this ebook, I went out and interviewed the [http://www.treehousejournal.com/members/yvonnbillingt/activity/18608/ top property developers in singapore] Singapore property consultants, bringing to you their (and my very own) invaluable insights on easy methods to make your fortune investing in real estate. It will value hundreds of dollars to get simply considered one of these consultants to seek the advice of with you on your property purchase – when you could even get entry to them. The intention of Chesney Actual Estate is to cement our #1 place because the premier property firm in Singapore. We are going to present real property providers' from a workforce of professional managers whom are committed to meeting our shoppers wants. For Hire Function Condominium for Sale Feature Condominium for Rent Luxurious rental close to Somerset MRT Condominium For Sale – St Thomas Suites (D09) Unfurnished condominium for sale # 500 sf
 
==Preliminaries==
A knowledge of certain concepts and terminology related to linear transformations is an essential prerequisite for stating and understanding the cyclic decomposition theorem and its proof. To explain these, let <math>T</math> be a [[linear operator]] on a finite-dimensional vector space <math>V</math> over a [[Field (mathematics)|field]] <math>F</math>. Let <math>v</math> be a vector in <math>V</math>.  
 
===Cyclic subspace===
 
The subspace of <math>V</math> generated by the set <math>\{ v, Tv, T^2v, \ldots, T^kv, \ldots\}</math> is called the <math>T</math>-''[[cyclic subspace]] generated by'' <math>v</math>. This subspace is denoted by <math>Z(v;T)</math>.
 
===Annihilator of a vector===
 
Let <math>F[x]</math> be the [[Ring (mathematics)|ring]] of all [[polynomial]]s in <math>x</math> over the field <math>F</math>.
The set of all polynomials <math>g(x)</math> in <math>F[x]</math> such that <math>g(T)v=0</math> is called the <math>T</math>-''annihilator'' of <math>v</math>. It is denoted by <math>M(v;T)</math>. <math>M(v;T)</math> is an ideal in the ring <math>F[x]</math>. The ideal <math>M(v;T)</math> consists of all multiples by elements of <math>F[x]</math> of some fixed monic polynomial in <math>F[x]</math>. This fixed  monic polynomial is denoted by <math>p_v(x)</math> and it is also sometimes referred to as the <math>T</math>-annihilator of <math>v</math>.
 
===Conductor===
 
Let <math>W</math> be subspace of <math>V</math> which is invariant under <math>T</math>. Let <math>F[x]</math> be the [[Ring (mathematics)|ring]] of all [[polynomial]]s in <math>x</math> over the field <math>F</math>.
The set of all polynomials <math>g(x)</math> in <math>F[x]</math> such that <math>g(T)v\in W</math> is called the <math>T</math>-''conductor of <math>v</math> into <math>W</math>'' and is denoted by <math>S_T(v;W)</math>. <math>S_T(v;W)</math> is an ideal in the ring <math>F[x]</math>.  The unique monic polynomial <math>g(x)</math> of least degree such that <math>g(T)v\in W</math>, which is the generator of the ideal <math>S_T(v;W)</math>, is also called the <math>T</math>-''conductor of <math>v</math> into <math>W</math>''.
 
===Admissible subspace===
 
Let <math>W</math> be a [[linear subspace]] of <math>V</math>. <math>W</math> is called a <math>T</math>-admissible subspace of <math>V</math> if the following conditions are satisfied.
# <math>W</math> is invariant under <math>T</math>;  that is, for any <math>w</math> in <math>W</math>, the vector <math>Tw</math> is in <math>W</math>.  
# For any polynomial <math>f(x)</math> in <math>F[x]</math> and any vector <math>v</math> in <math>V</math>, if <math>f(T)v</math> is in <math>W</math> then there is a vector <math>w</math> in <math>W</math> such that <math>f(T)v=f(T)w</math>.
 
==Cyclic decomposition theorem==
Let <math>T</math>  be a linear operator on a finite-dimensional vector space <math>V</math> and let <math>W_0</math> be a proper <math>T</math>-admissible subspace of <math>V</math>. There exist non-zero vectors <math>v_1,v_2, \ldots, v_r</math> in <math>V</math> with respective <math>T</math>-annihilators <math>p_1(x),p_2(x), \ldots, p_r(x)</math> such that
 
# <math> V = W_0 \oplus Z(v_1;T) \oplus Z(v_2;T) \oplus \cdots \oplus Z(v_r;T)</math>.  
# <math>p_k(x)</math>  divides <math>p_{k-1}(x)</math> for <math> k = 2, \ldots , r</math>.
 
Furthermore, the integer <math>r</math> and the annihilators <math>p_1(x),p_2(x), \ldots, p_r(x)</math> are uniquely
determined by (1), (2), and the fact that no <math> v_k</math> is 0.
 
==References==
{{reflist}}
 
==External links==
 
*[http://planetmath.org/cyclicdecompositiontheorem PlanetMath: Cyclic decomposition theorem]
 
 
[[Category:Linear algebra]]
[[Category:Vector spaces]]

Latest revision as of 07:22, 6 October 2014

BUNGALOWS, SEMI-DETACHED & TERRACE HOMES IN WOODLANDS, Privately Held; 1001-5000 employees;Actual Estate industry Seascape Launched Today! Guide Now for Non-public Preview! By Appointment Only! Direct Showflat Agent Call +65 9668 8021 to Take pleasure in VVIP Pricing! JLL acquires YY Property Solutions Sdn Bhd in Malaysia Landed terrace (+5.02% nominal) and non-landed residences (+5.30% nominal) enjoyed comparatively good development, whereas non-landed condominums have been weaker (+zero.ninety eight% nominal), based on the figures from URA.

SLA assists the Authorities in managing its land sales system. Our roles are to realize optimum returns from the gross sales of State lands and to nominate and handle the State's land sales agents. The place land is slated for launch via the Government tenders and auctions, we make sure that land is accessible and the land sale processes comply with statutory guidelines. Misc. Franchise Resales Enterprise For Sale, Advertising & Advertising and marketing Business For Sale ASSOCIATE WITH A HIGHLY MOTIVATED STAFF About our Enterprise We are a global advertising enterprise who partners with an Award Profitable Leadership Growth Firm and are committed to turning into the chosen partner for individuals in search of their very own enterprise alternative. Business Des Tenure Any Freehold 999 Years ninety nine Years

The Hexacube is providing new industrial property on the market Singapore traders want to think about. Residents of the area will seemingly respect that they will customize the commercial house that they need to purchase as effectively. This challenge represents probably the most expansive buildings provided in Singapore so far. Many buyers will probably want to check out how they will customise the property that they do determine to buy through right here. This location has provided individuals the possibility that they need to perceive more about how this process can work as well.

Actual property providers embody selling, buying, leasing and property management for residential and industrial properties. eight Temasek Boulevard, Penthouse Level Suntec Metropolis Tower 3, Singapore 038988. Actual estate agency promoting and letting property. 190 Lorong 6 Toa Payoh, #02-514, Singapore 310190. Real property providers to patrons and sellers on the lookout for residential and commercial properties for sale and hire. The flexibleness of proudly owning a private property such as making modifications to the home is a significant purpose as to why people buy non-public property in Singapore Apart from the pliability, the prestige and standing that derive from owning a personal property is the last word for many. Freehold Condominium near Orchard The Edge on Cairnhill – Condominium For Sale Terrace Home

Pretty house in the east, near MRT Casa Merah (D16) – Residence For Lease Very spacious condo close to international faculty Apartment For Lease – Leonie Towers (D09) The Residential Property Act defines a "overseas particular person" as any person who shouldn't be a citizen of Singapore, a everlasting resident and any overseas firm or transformed international company. Any individual or entity falling beneath this category is totally restricted from owning landed residential property except approval is obtained from the Controller of Residential Property. S$100,000 cash is enough for a down cost of a primary property in prime London , or personal prime properties in Iskandar, KL, Manila & Cambodia Semi-Detached( For Sale), 34 /36 Jalan Tupai, Singapore, How does property public sale work?

To write down this ebook, I went out and interviewed the top property developers in singapore Singapore property consultants, bringing to you their (and my very own) invaluable insights on easy methods to make your fortune investing in real estate. It will value hundreds of dollars to get simply considered one of these consultants to seek the advice of with you on your property purchase – when you could even get entry to them. The intention of Chesney Actual Estate is to cement our #1 place because the premier property firm in Singapore. We are going to present real property providers' from a workforce of professional managers whom are committed to meeting our shoppers wants. For Hire Function Condominium for Sale Feature Condominium for Rent Luxurious rental close to Somerset MRT Condominium For Sale – St Thomas Suites (D09) Unfurnished condominium for sale # 500 sf

Retrieved from "https://en.formulasearchengine.com/index.php?title=Fuss–Catalan_number&oldid=318246"