Mutual fund separation theorem: Difference between revisions

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In the [[mathematics|mathematical]] field of [[representation theory]], a '''highest-weight category''' is a [[linear category|''k''-linear category]] '''C''' (here ''k'' is a [[field (mathematics)|field]]) that
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*is ''locally artinian''<ref>In the sense that it admits arbitrary [[direct limit]]s of [[subobject]]s and every object is a union of its subobjects of [[finite length object|finite length]].</ref>
*has [[enough injectives]]
*satisfies
::<math>B\cap\left(\bigcup_\alpha A_\alpha\right)=\bigcup_\alpha\left(B\cap A_\alpha\right)</math>
:for all subobjects ''B'' and each family of subobjects {''A''<sub>&alpha;</sub>} of each object ''X''
and such that there is a [[locally finite poset]] Λ (whose elements are called the '''weights''' of '''C''') that satisfies the following conditions:<ref name="dimensional-algebras">{{harvnb|Cline|Scott|1988|loc=§3}}</ref>
 
* The poset Λ indexes an exhaustive set of non-isomorphic [[simple object]]s {''S''(''λ'')} in '''C'''.
* Λ also indexes a collection of objects {''A''(''λ'')} of objects of '''C''' such that there exist embeddings ''S''(''λ'')&nbsp;→&nbsp;''A''(''λ'') such that all [[composition factor]]s ''S''(''μ'') of ''A''(''λ'')/''S''(''λ'') satisfy ''μ''&nbsp;<&nbsp;''λ''.<ref>Here, a composition factor of an object ''A'' in '''C''' is, by definition, a composition factor of one of its finite length subobjects.</ref>
* For all ''μ'', ''λ'' in Λ,
::<math>\dim_k\operatorname{Hom}_k(A(\lambda),A(\mu))</math>
:is finite, and the [[composition factor|multiplicity]]<ref>Here, if ''A'' is an object in '''C''' and ''S'' is a simple object in '''C''', the multiplicity [A:S] is, by definition, the supremum of the multiplicity of ''S'' in all finite length subobjects of ''A''.</ref>
::<math>[A(\lambda):S(\mu)]</math>
:is also finite.
*Each ''S''(''λ'') has an [[injective envelope]] ''I''(''λ'') in '''C''' equipped with an increasing [[filtration (mathematics)|filtration]]
::<math>0=F_0(\lambda)\subseteq F_1(\lambda)\subseteq\dots\subseteq I(\lambda)</math>
:such that
:# <math>F_1(\lambda)=A(\lambda)</math>
:# for ''n'' > 1, <math>F_n(\lambda)/F_{n-1}(\lambda)\cong A(\mu)</math> for some ''μ'' = ''μ''(''n'') > ''λ''
:# for each ''μ'' in Λ, ''μ''(''n'') = ''μ'' for only finitely many ''n''
:# <math>\bigcup_iF_i(\lambda)=I(\lambda).</math>
 
== Examples ==
 
* The module category of the <math>k</math>-algebra of upper triangular <math>n\times n</math> matrices over <math>k</math>.
* This concept is named after the category of [[highest-weight module]]s of Lie-algebras.
* A finite-dimensional <math>k</math>-algebra <math>A</math> is [[quasi-hereditary]] iff its module category is a highest-weight category. In particular all module-categories over [[semisimple ring|semisimple]] and [[hereditary ring|hereditary]] algebras are highest-weight categories.
* A [[cellular algebra]] over a field is quasi-hereditary (and hence its module category a highest-weight category) [[iff]] its Cartan-determinant is&nbsp;1.
 
==Notes==
{{Reflist}}
 
==References==
*{{cite journal
  | last1 = Cline
| first1 = E.
  | last2 = Parshall
| first2 = B.
| last3 = Scott
| first3 = L.
| year = 1988
| month = January
| title = Finite-dimensional algebras and highest-weight categories
| journal = [[Journal für die reine und angewandte Mathematik]]
| volume = 1988
| issue = 391
| pages = 85–99
| location = [[Berlin, Germany]]
| publisher = [[Walter de Gruyter]]
| format = pdf
| issn = 0075-4102
| oclc = 1782270
| doi = 10.1515/crll.1988.391.85
| accessdate = 7 January 2011
| id = {{citeseerx|10.1.1.112.6181}}
| url = http://u.math.biu.ac.il/~margolis/Representation%20Theory%20Seminar/Highest%20Weight%20Categories.pdf
| accessdate=2012-07-17
}}
 
== See also ==
*[[Category O]]
 
[[Category:Representation theory]]

Latest revision as of 14:38, 20 February 2014



Prior to you start packing your soft sided cooler there are a quantity of issues that you must take into consideration very first. This cooler is ruggedized, claims 7-day ice retention at 90°, and has stainless steel hardware. Again, it characteristics stainless steel, and is more ruggedized, but it only promises to retain ice cold for 5 days. We still never consider it's worth paying that kind of dollars for a cooler when the $50 Coleman will meet most people's needs. If you have any queries concerning where by and how to use bestcoolerreviewshq.com, you can call us at our own site. With some very simple plumbing gear and a small bit of perform, the cooler can filter out the liquid from the solids.

Normally the fans in inferior laptop coolers will be messed up within a year but not the super durable Zalman. Targus Lap Chill Mat AWE55US This laptop cooler is fairly significant and can support a 17 inch laptop easily with no any troubles. The rubber stops at the four edges is a good touch and prevents the laptop from slipping off the laptop cooler. I started this internet site to share my experiences with laptop coolers.

Rubbermaid created this cooler with a split lid, so you could grab a drink with out letting too considerably cold air out. The Yeti Tundra Series 50-quart Cooler ( ) featured a freezer-type gasket to make certain a tight seal when the lid was closed, and a sturdy hinge system extended the complete length of the lid, so there was no concern about durability. These coolers float in the pool, lakes, oceans more.

They claimed the industry is so vast that new arrival will very easily settle as much more leads nonetheless untouched such as Spa, Gyms, Religion centres and so on. Yeti coolers are personal-use coolers with the insulation and cooling properties of industrial coolers. The Yeti cooler offers maximum ice retention as such, they give high high-quality and insulation properties. Yeti gives coolers in its Tundra and Roadie lines, all manufactured for rugged use. Discover great deals on Slide major beer coolers Refrigerators.

Pack into your cooler and it acts as an ice block that won&39t melt all over your food! Find coleman 12 volt cooler details and discount costs with our big selection. Applying the Coleman Power Chill Iceless 12-Volt Cooler can make the difference in between obtaining warm drinks and completely chilled treats. Our 12v coolers and warmers by Coleman are engineered to meet all of your outdoor dining requirements. Opt for from soft-sided or sturdy really hard-sided coolers.

The Antec USB Notebook Cooler and Thermaltake's MASSIVE23 NB COOLER are two good options to the aforementioned Zalman models. Yet another benefit of evaporative coolers is the decrease expense of installation, which is 50 % the expense of installing a central air unit. With a portable cooler, you can conveniently eradicate the swamp cooler when you move out of your dorm or apartment.

On trips, I carry a waterproof container like the ones sold in the accessories section here, and just fill it with ice either at the hotel, or at a gas station and it will retain the cooler wel inside temps all day. I havent utilized the coolers yet but by seeking at them they appear extremely sturdy and ought to hold up to what ever the races can throw at them. Cooler and accessories to me beneath such terrible climate conditions.

Kenji's takeaway from his experiments: for those of you who have believed of playing around with sous-vide cookery (and I hugely encourage that sort of behavior!) but have been thrown by the expenses, this is a cheap, trusted, and quite a lot foolproof way to do specifically what the extra costly machines do. Play away.” Per Se Chef Thomas Keller recommends a related cooler approach with vacuum packed food on the LATimes , employing a 28-quart cooler which is preheated with hot tap water.

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