en>David Eppstein: It's essential to the statement of the problem that this is the first discovery of an empty box. Otherwise it's not well defined; e.g. after he lights 2N cigarettes, both boxes are empty with probability 1 as he will discover when he tries to light 2N+1

2013-07-11T20:10:33Z

It's essential to the statement of the problem that this is the *first* discovery of an empty box. Otherwise it's not well defined; e.g. after he lights 2N cigarettes, both boxes are empty with probability 1 as he will discover when he tries to light 2N+1

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	~~Andera~~ is ~~what you can contact her but she~~ by ~~no means truly liked that name~~. ~~Mississippi~~ is the ~~only place I~~'~~ve been residing~~ in ~~but I will have to move~~ in a ~~yr or two. As~~ a ~~lady what she truly likes is fashion and she~~'~~s been performing it~~ for ~~fairly a while~~. ~~Office supervising~~ is ~~exactly where her primary earnings comes from but she~~'~~s currently utilized for an additional 1.~~<br><br>~~Feel~~ [~~http~~:~~//gcjcteam~~.~~org~~/~~index~~.~~php?mid~~=~~etc_video&document_srl~~=~~696611~~&~~sort_index~~=~~regdate&order_type~~=~~desc free online tarot card readings~~] ~~to surf to my clairvoyants~~ ([~~http~~://~~www.article~~-~~galaxy.com~~/~~profile.php?a=143251 simply click~~ the ~~up coming article~~]) ~~web page~~ - ~~live psychic reading~~ - [~~http~~://~~checkmates~~.co.za/~~index~~.~~php?do~~=~~/profile~~-~~56347/info/ please click checkmates.co.za~~],		In [[mathematics]], the '''Fraňková–Helly selection theorem''' is a generalisation of [[Helly's selection theorem]] for functions of [[bounded variation]] to the case of [[regulated function]]s. It was proved in 1991 by the [[Czech Republic\|Czech]] [[mathematician]] [[Dana Fraňková]].

			==Background==

			Let ''X'' be a [[separable space\|separable]] [[Hilbert space]], and let BV([0, ''T'']; ''X'') denote the [[normed vector space]] of all functions ''f'' : [0, ''T''] → ''X'' with finite total variation over the [[interval (mathematics)\|interval]] [0, ''T''], equipped with the total variation norm. It is well-known that BV([0, ''T'']; ''X'') satisfies the [[compact space\|compactness theorem]] known as '''Helly's selection theorem''': given any sequence of functions (''f''<sub>''n''</sub>)<sub>''n''∈'''N'''</sub> in BV([0, ''T'']; ''X'') that is uniformly bounded in the total variation norm, there exists a subsequence

			:<math>\left( f_{n(k)} \right) \subseteq (f_{n}) \subset \mathrm{BV}([0, T]; X)</math>

			and a limit function ''f'' ∈ BV([0, ''T'']; ''X'') such that ''f''<sub>''n''(''k'')</sub>(''t'') [[weak convergence\|converges weakly]] in ''X'' to ''f''(''t'') for every ''t'' ∈ [0, ''T'']. That is, for every [[continuous linear functional]] ''λ'' ∈ ''X''*,

			:<math>\lambda \left( f_{n(k)}(t) \right) \to \lambda(f(t)) \mbox{ in } \mathbb{R} \mbox{ as } k \to \infty.</math>

			Consider now the [[Banach space]] Reg([0, ''T'']; ''X'') of all regulated functions ''f'' : [0, ''T''] → ''X'', equipped with the [[supremum norm]]. Helly's theorem does not hold for the space Reg([0, ''T'']; ''X''): a [[counterexample]] is given by the sequence

			:<math>f_{n} (t) = \sin (n t).</math>

			One may ask, however, if a weaker selection theorem is true, and the '''Fraňková–Helly selection theorem''' is such a result.

			==Statement of the Fraňková–Helly selection theorem==

			As before, let ''X'' be a separable Hilbert space and let Reg([0, ''T'']; ''X'') denote the space of regulated functions ''f'' : [0, ''T''] → ''X'', equipped with the supremum norm. Let (''f''<sub>''n''</sub>)<sub>''n''∈'''N'''</sub> be a sequence in Reg([0, ''T'']; ''X'') satisfying the following condition: for every ''ε'' > 0, there exists some ''L''<sub>ε</sub> > 0 so that each ''f''<sub>''n''</sub> may be approximated by a ''u''<sub>''n''</sub> ∈ BV([0, ''T'']; ''X'') satisfying

			:<math>\\| f_{n} - u_{n} \\|_{\infty} < \varepsilon</math>

			and

			:<math>\| u_{n}(0) \| + \mathrm{Var}(u_{n}) \leq L_{\varepsilon},</math>

			where \|-\| denotes the [[norm (mathematics)\|norm]] in ''X'' and Var(''u'') denotes the variation of ''u'', which is defined to be the [[supremum]]

			:<math>\sup_{\Pi} \sum_{j=1}^{m} \| u(t_{j}) - u(t_{j-1}) \|</math>

			over all [[partition of an interval\|partitions]]

			:<math>\Pi = \{ 0 = t_{0} < t_{1} < \dots < t_{m} = T , m \in \mathbf{N} \}</math>

			of [0, ''T'']. Then there exists a subsequence

			:<math>\left( f_{n(k)} \right) \subseteq (f_{n}) \subset \mathrm{Reg}([0, T]; X)</math>

			and a limit function ''f'' ∈ Reg([0, ''T'']; ''X'') such that ''f''<sub>''n''(''k'')</sub>(''t'') converges weakly in ''X'' to ''f''(''t'') for every ''t'' ∈ [0, ''T'']. That is, for every continuous linear functional ''λ'' ∈ ''X''*,

			:<math>\lambda \left( f_{n(k)}(t) \right) \to \lambda(f(t)) \mbox{ in } \mathbb{R} \mbox{ as } k \to \infty.</math>

			==References==

			* {{cite journal \| last=Fraňková \| first=Dana \| title=Regulated functions \| journal=Math. Bohem. \| volume=116 \| year=1991 \| issue=1 \| pages=20–59 \| issn=0862-7959 }}

			{{DEFAULTSORT:Frankova-Helly selection theorem}}
			[[Category:Mathematical analysis]]
			[[Category:Compactness theorems]]

en>Artturimehto at 16:33, 2 October 2011

2011-10-02T16:33:20Z

New page

Andera is what you can contact her but she by no means truly liked that name. Mississippi is the only place I've been residing in but I will have to move in a yr or two. As a lady what she truly likes is fashion and she's been performing it for fairly a while. Office supervising is exactly where her primary earnings comes from but she's currently utilized for an additional 1.<br><br>Feel [http://gcjcteam.org/index.php?mid=etc_video&document_srl=696611&sort_index=regdate&order_type=desc free online tarot card readings] to surf to my clairvoyants ([http://www.article-galaxy.com/profile.php?a=143251 simply click the up coming article]) web page - live psychic reading - [http://checkmates.co.za/index.php?do=/profile-56347/info/ please click checkmates.co.za],

Banach's matchbox problem - Revision history

en>David Eppstein: It's essential to the statement of the problem that this is the *first* discovery of an empty box. Otherwise it's not well defined; e.g. after he lights 2N cigarettes, both boxes are empty with probability 1 as he will discover when he tries to light 2N+1

en>Artturimehto at 16:33, 2 October 2011

en>David Eppstein: It's essential to the statement of the problem that this is the first discovery of an empty box. Otherwise it's not well defined; e.g. after he lights 2N cigarettes, both boxes are empty with probability 1 as he will discover when he tries to light 2N+1