Specht module

2013-12-02T00:32:43Z

68.44.67.172: A tabloid is an equivalence class of labelings of the Young diagram that are not necessarily tableaux; there is a unique tableau in each equivalence class.

In [[set theory]], a set is called '''hereditarily countable''' if it is a [[countable set]] of [[hereditary property|hereditarily]] countable sets. This [[inductive definition]] is in fact [[well-founded]] and can be expressed in the language of [[first-order logic|first-order]] set theory. A set is hereditarily countable if and only if it is countable, and every element of its [[transitive set|transitive closure]] is countable. If the [[axiom of countable choice]] holds, then a set is hereditarily countable if and only if its transitive closure is countable.

The [[class (set theory)|class]] of all hereditarily countable sets can be proven to be a set from the axioms of [[Zermelo–Fraenkel set theory]] (ZF) without any form of the [[axiom of choice]], and this set is designated <math>H_{\aleph_1}</math>. The hereditarily countable sets form a model of [[Kripke–Platek set theory]] with the [[axiom of infinity]] (KPI), if the axiom of countable choice is assumed in the [[metatheory]].

If <math>x \in H_{\aleph_1}</math>, then <math>L_{\omega_1}(x) \subset H_{\aleph_1}</math>.

More generally, a set is '''hereditarily of cardinality less than κ''' if and only it is of [[cardinality]] less than κ, and all its elements are hereditarily of cardinality less than κ; the class of all such sets can also be proven to be a set from the axioms of ZF, and is designated <math>H_\kappa \!</math>. If the axiom of choice holds and the cardinal κ is regular, then a set is hereditarily of cardinality less than κ if and only if its transitive closure is of cardinality less than κ.

==See also==
*[[Hereditarily finite set]]
*[[Constructible universe]]

==External links==
*[http://www.jstor.org/pss/2273380 "On Hereditarily Countable Sets"] by [[Thomas Jech]]

[[Category:Set theory]]
[[Category:Large cardinals]]

{{settheory-stub}}

Quantum depolarizing channel

2013-09-10T13:06:06Z

68.44.88.133: /* Outline of the proof of the additivity of Holevo information */

In [[mathematics]], specifically [[abstract algebra]], if (''G'', +) is an [[abelian group]] then <math>\scriptstyle \nu\colon G \to \mathbb{R}</math> is said to be a '''norm on the abelian group''' (''G'', +) if:
<ol>
<li><math>\scriptstyle \nu(g) > 0</math> if <math>\scriptstyle g\ne 0</math>,</li>
<li><math>\scriptstyle \nu(g+h) \le \nu(g) + \nu(h)</math>,</li>
<li><math>\scriptstyle \nu(mg) = |m| \nu(g)</math> if <math>\scriptstyle m \in \mathbb{Z}</math>.</li>
</ol>

The norm ''ν'' is '''discrete''' if there is some [[real number]] ''ρ'' > 0 such that ''ν''(''g'') > ''ρ'' whenever ''g'' ≠ 0.

== Free abelian groups ==
An abelian group is a [[free abelian group]] [[if and only if]] it has a discrete norm.<ref>{{citation
| last = Steprāns | first = Juris
| doi = 10.2307/2044776
| issue = 2
| journal = Proceedings of the American Mathematical Society
| mr = 770551
| pages = 347–349
| title = A characterization of free abelian groups
| volume = 93
| year = 1985}}</ref>

==References==
{{reflist}}

[[Category:Abelian group theory]]

{{Abstract-algebra-stub}}

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Specht module

Quantum depolarizing channel