en>Srleffler: Rv. No, I believe it is actually power per unit wavelength. Imagine the total power being divided up into "bins" by wavelength (eg. 1 nm wide bins).

2014-10-30T12:29:14Z

Rv. No, I believe it is actually power per unit wavelength. Imagine the total power being divided up into "bins" by wavelength (eg. 1 nm wide bins).

Show changes

en>Addbot: Bot: Migrating 9 interwiki links, now provided by Wikidata on d:q899219 (Report Errors)

2013-02-26T10:36:53Z

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← Older revision		Revision as of 12:36, 26 February 2013
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	~~This~~ is ~~important~~. ~~If you make half~~-~~hearted efforts~~ to ~~lose fat~~, ~~you~~'~~re not actually going to succeed~~. ~~So develop a strong determination~~, which this is ~~what you want plus you will do anything~~ it ~~takes to achieve~~ a ~~goal~~.<br><br>~~Often we think you are hungry whenever you are really thirsty~~. ~~This signifies we eat more than we require~~. ~~So we gain fat~~. ~~If we like to lose weight fast~~, ~~you require to train ourselves to drink more fluids~~, ~~throughout~~ the ~~day~~, ~~especially water~~. ~~We could even teach ourselves to carry water with us as you are running~~ the ~~errands or keeping it close in front~~ of ~~you at~~ the ~~desks, whilst you function~~. This can ~~keep you sipping throughout~~ the ~~day~~.~~<br><br>Losing weight~~ is ~~only possible when~~ a ~~stamina consumption~~ is ~~smaller than the stamina usage~~. ~~This technique you~~'~~ll usually~~ have a ~~bad stamina balance plus will lose weight~~.<br><br>~~Drink Plenty~~ of ~~Water: Drink minimal 10 glasses~~ of ~~water daily~~. ~~Water benefits~~ the ~~body~~ in ~~several techniques~~. ~~It boosts metabolism~~, ~~detoxifies~~ the ~~body~~, ~~reduces bloating plus most importantly, makes we feel full thereby cutting down your consumption of food~~.<br><br>~~And when I am seized with all~~ the ~~temptation~~ of the ~~evil doughnut, I could resist~~-~~at least~~ for a ~~time~~ [~~http:~~/~~/safedietplansforwomen~~.~~com/how-to-lose-weight-fast lose weight fast~~]. ~~And then another moment~~, ~~till I~~'~~ve put enough time between~~ the ~~impulse plus~~ the ~~action~~, and the ~~wanting has subsided~~.<br><br>~~Cardio~~ is ~~an aerobic exercise which you do for hours at slow - moderate pace with an intensity level where you~~ can ~~nonetheless talk~~. ~~It does burn fat but just during~~ the ~~exercise bout. Once you stop~~, ~~the fat-burning effect stops~~.<br><br>~~You do not have~~ to ~~remain away from all your favorite foods just discover to count~~ a ~~calories~~ and ~~confirm you are actually eating healthy foods~~. ~~It will likely~~ not ~~do you any good to consume 1200 calories if~~ the ~~calories are nothing over chocolate cake~~. ~~Your body actually requires~~ a ~~range~~ of ~~foods~~ to ~~stay healthy~~.		In [[mathematics]], '''monodromy''' is the study of how objects from [[mathematical analysis]], [[algebraic topology]] and [[algebraic geometry\|algebraic]] and [[differential geometry]] behave as they 'run round' a [[Mathematical singularity\|singularity]]. As the name implies, the fundamental meaning of ''monodromy'' comes from 'running round singly'. It is closely associated with [[covering map]]s and their degeneration into [[ramification]]; the aspect giving rise to monodromy phenomena is that certain [[function (mathematics)\|function]]s we may wish to define fail to be ''single-valued'' as we 'run round' a path encircling a singularity. The failure of monodromy is best measured by defining a '''monodromy group''': a [[group (mathematics)\|group]] of transformations acting on the data that encodes what does happen as we 'run round'.

			==Definition==
			Let ''X'' be a connected and [[locally connected]] based [[topological space]] with base point ''x'', and let <math>p:\tilde{X}\to X</math> be a [[covering map\|covering]] with [[Fiber_(mathematics)\|fiber]] <math>F = p^{-1}(x)</math>. For a loop γ: [0, 1] → ''X'' based at ''x'', denote a [[homotopy lifting property\|lift]] under the covering map (starting at a point <math>\tilde{x}\in F</math>) by <math>\tilde{\gamma}</math>. Finally, we denote by <math>\tilde{x}\cdot\gamma</math> the endpoint <math>\tilde{\gamma}(1)</math>, which is generally different from <math>\tilde{x}</math>. There are theorems which state that this construction gives a well-defined [[group action]] of the [[fundamental group]] π<sub>1</sub>(''X'', ''x'') on ''F'', and that the [[stabilizer (group theory)\|stabilizer]] of <math>\tilde{x}</math> is exactly <math>p_{*}(\pi_1(\tilde{X},\tilde{x}))</math>, that is, an element [γ] fixes a point in ''F'' if and only if it is represented by the image of a loop in <math>\tilde{X}</math> based at <math>\tilde{x}</math>. This action is called the '''monodromy action''' and the corresponding [[group homomorphism\|homomorphism]] π<sub>1</sub>(''X'', ''x'') → Aut(''F'') into the [[automorphism group]] on ''F'' is the '''monodromy'''. The image of this homomorphism is the '''monodromy group'''.

			==Example==
			These ideas were first made explicit in [[complex analysis]]. In the process of [[analytic continuation]], a function that is an [[analytic function]] ''F''(''z'') in some open subset ''E'' of the punctured complex plane '''C''' \ {0} may be continued back into ''E'', but with different values. For example take

			::''F''(''z'') = log ''z''
			::''E'' = {''z'' ∈ '''C''': Re(''z'') > 0}

			then analytic continuation anti-clockwise round the circle

			::\|''z''\| = 0.5

			will result in the return, not to ''F''(''z'') but

			::''F''(''z'')+2π''i''.

			In this case the monodromy group is [[infinite cyclic]] and the covering space is the universal cover of the punctured complex plane. This cover can be visualized as the [[helicoid]] (as defined in the helicoid article) restricted to ρ > 0. The covering map is a vertical projection, in a sense collapsing the spiral in the obvious way to get a punctured plane.

			==Differential equations in the complex domain==
			One important application is to [[differential equation]]s, where a single solution may give further linearly independent solutions by [[analytic continuation]]. Linear differential equations defined in an open, connected set ''S'' in the complex plane have a monodromy group, which (more precisely) is a [[linear representation]] of the [[fundamental group]] of ''S'', summarising all the analytic continuations round loops within ''S''. The inverse problem, of constructing the equation (with [[regular singularity\|regular singularities]]), given a representation, is called the [[Riemann–Hilbert problem]].

			For a regular (and in particular Fuchsian) linear system one usually chooses as generators of the monodromy group the operators ''M<sub>j</sub>'' corresponding to loops each of which circumvents just one of the poles of the system counterclockwise. If the indices ''j'' are chosen in such a way that they increase from 1 to ''p'' + 1 when one circumvents the base point clockwise, then the only relation between the generators is the equality <math>M_1...M_{p+1}=id</math>. The [[Deligne–Simpson problem]] is the following realisation problem: For which tuples of conjugacy classes in GL(''n'', '''C''') do there exist irreducible tuples of matrices ''M<sub>j</sub>'' from these classes satisfying the above relation? The problem has been formulated by [[Pierre Deligne]] and [[Carlos Simpson]] was the first to obtain results towards its resolution. An additive version of the problem about residua of Fuchsian systems has been formulated and explored by [[Vladimir Kostov]]. The problem has been considered by other authors for matrix groups other than GL(''n'', '''C''') as well.<ref>{{Citation\|author=V.P. Kostov\|title=The Deligne–Simpson problem — a survey\|journal=J. Algebra\|volume=281\|year=2004\|issue=1\|pages=83–108\|mr=2091962}} and the references therein.</ref>

			==Topological and geometric aspects==
			In the case of a covering map, we look at it as a special case of a [[fibration]], and use the [[homotopy lifting property]] to 'follow' paths on the base space ''X'' (we assume it [[path-connected]] for simplicity) as they are lifted up into the cover ''C''. If we follow round a loop based at ''x'' in ''X'', which we lift to start at ''c'' above ''x'', we'll end at some ''c'' again above ''x''; it is quite possible that ''c'' ≠ ''c'', and to code this one considers the action of the [[fundamental group]] π<sub>1</sub>(''X'', ''x'') as a [[permutation group]] on the set of all ''c'', as a '''monodromy group''' in this context.

			In differential geometry, an analogous role is played by [[parallel transport]]. In a [[principal bundle]] ''B'' over a [[smooth manifold]] ''M'', a [[connection (mathematics)\|connection]] allows 'horizontal' movement from fibers above ''m'' in ''M'' to adjacent ones. The effect when applied to loops based at ''m'' is to define a '''[[holonomy]]''' group of translations of the fiber at ''m''; if the structure group of ''B'' is ''G'', it is a subgroup of ''G'' that measures the deviation of ''B'' from the product bundle ''M'' × ''G''.

			===Monodromy groupoid and foliations===
			Analogous to the [[Fundamental group#Fundamental groupoid\|fundamental groupoid]] it is possible to get rid of the choice of a base point and to define a monodromy groupoid. Here we consider (homotopy classes of) lifts of paths in the base space ''X'' of a fibration <math>p:\tilde X\to X</math>. The result has the structure of a [[groupoid]] over the base space ''X''. The advantage is that we can drop the condition of connectedness of ''X''.

			Moreover the construction can also be generalized to [[foliation]]s: Consider <math>(M,\mathcal{F})</math> a (possibly singular) foliation of ''M''. Then for every path in a leaf of <math>\mathcal{F}</math> we can consider its induced diffeomorphism on local [[foliation#transversal section\|transversal section]]s through the endpoints. Within a simply connected chart this diffeomorphism becomes unique and especially canonical between different transversal sections if we go over to the [[Germ (mathematics)\|germ]] of the diffeomorphism around the endpoints. In this way it also becomes independent of the path (between fixed endpoints) within a simply connected chart and is therefore invariant under homotopy.

			==Definition via Galois theory==
			Let '''F'''(''x'') denote the field of the [[rational function]]s in the variable ''x'' over the [[field (mathematics)\|field]] '''F''', which is the [[field of fractions]] of the [[polynomial ring]] '''F'''[''x'']. An element ''y'' = ''f''(''x'') of '''F'''(''x'') determines a finite [[field extension]] ['''F'''(''x'') : '''F'''(''y'')].

			This extension is generally not Galois but has [[Galois closure]] ''L''(''f''). The associated [[Galois group]] of the extension [''L''(''f'') : '''F'''(''y'')] is called the monodromy group of the ''f''.

			In the case of '''F''' = '''C''' [[Riemann surface]] theory enters and allows for the geometric interpretation given above. In the case that the extension ['''C'''(''x'') : '''C'''(''y'')] is already Galois, the associated monodromy group is sometimes called a [[Covering map\|group of deck transformations]].

			This has connections with the [[Grothendieck's Galois theory\|Galois theory of covering spaces]] leading to the [[Riemann existence theorem]].

			==See also==
			* [[Braid group]]
			* [[Monodromy theorem]]
			* [[Mapping class group]] (of a punctured disk)

			==References==
			{{Reflist}}
			*{{springer\|author=V.I. Danilov\|title=Monodromy\|id=M/m064700}}
			*{{planetmath reference\|id=4000\|title=Monodromy}}

			[[Category:Mathematical analysis]]
			[[Category:Complex analysis]]
			[[Category:Differential geometry]]
			[[Category:Algebraic topology]]
			[[Category:Homotopy theory]]

en>Dicklyon: en dash

2012-06-25T05:25:13Z

en dash

New page

This is important. If you make half-hearted efforts to lose fat, you're not actually going to succeed. So develop a strong determination, which this is what you want plus you will do anything it takes to achieve a goal. Often we think you are hungry whenever you are really thirsty. This signifies we eat more than we require. So we gain fat. If we like to lose weight fast, you require to train ourselves to drink more fluids, throughout the day, especially water. We could even teach ourselves to carry water with us as you are running the errands or keeping it close in front of you at the desks, whilst you function. This can keep you sipping throughout the day. Losing weight is only possible when a stamina consumption is smaller than the stamina usage. This technique you'll usually have a bad stamina balance plus will lose weight. Drink Plenty of Water: Drink minimal 10 glasses of water daily. Water benefits the body in several techniques. It boosts metabolism, detoxifies the body, reduces bloating plus most importantly, makes we feel full thereby cutting down your consumption of food. And when I am seized with all the temptation of the evil doughnut, I could resist-at least for a time [http://safedietplansforwomen.com/how-to-lose-weight-fast lose weight fast]. And then another moment, till I've put enough time between the impulse plus the action, and the wanting has subsided. Cardio is an aerobic exercise which you do for hours at slow - moderate pace with an intensity level where you can nonetheless talk. It does burn fat but just during the exercise bout. Once you stop, the fat-burning effect stops. You do not have to remain away from all your favorite foods just discover to count a calories and confirm you are actually eating healthy foods. It will likely not do you any good to consume 1200 calories if the calories are nothing over chocolate cake. Your body actually requires a range of foods to stay healthy.

Luminosity function - Revision history

en>Srleffler: Rv. No, I believe it is actually power per unit wavelength. Imagine the total power being divided up into "bins" by wavelength (eg. 1 nm wide bins).

en>Addbot: Bot: Migrating 9 interwiki links, now provided by Wikidata on d:q899219 (Report Errors)

en>Dicklyon: en dash