Glossary of invariant theory: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
← Older edit
en>Gilo1969
fix citation with wikilinks embedded in URL
en>R.e.b.
m →‎top: Copyedit (minor)
 
(One intermediate revision by one other user not shown)
Line 1: Line 1:
In [[mathematics]], an '''ambit field''' is a ''d''-dimensional [[random field]] describing the stochastic properties of a given system. The input is in general a ''d''-dimensional [[vector (mathematics and physics)|vector]] (e.g. d-dimensional space or (1-dimensional) time and (''d''&nbsp;&minus;&nbsp;1)-dimensional space) assigning a real value to each of the points in the field. In its most general form, the ambit field, <math>Y</math>, is defined by a constant plus a [[stochastic integral]], where the [[integral|integration]] is done with respect to a ''[[Lévy basis]]'', plus a smooth term given by an ordinary [[Lebesgue integration|Lebesgue integral]]. The integrations are done over so-called ''ambit sets'', which is used to model the [[sphere of influence]] (hence the name, ambit, [[latin]] for "sphere of influence" or "boundary") which affect a given point.
Including storm shutters in a home is most popular exterior exterior remodeling made by homeowners, and it also could add elegance and fineness or rural charm with respect to the kind of flat and it doesn't matter if you may have preferred the most appropriate theme to further improve your home or apartment. You'll also find certain residence styles that just don't look right around with them. In case you aren't certain and don't need a number of counsel, possess a drive in the [http://tinyurl.com/pch83be http://tinyurl.com/pch83be] areas around your country and take quick notice of which residences have window protection that are proportional and appropriate and which ones request that they definitely caught on as an afterthought. Hard plastic workers who exactly routinely install them is also able to provide you valuable suggestions about trend, proportion and overall attraction. Keep these pointers from neighborhood vinyl professionals as the primary goal once your all set to layer them showing  [http://tinyurl.com/pch83be ugg boots usa] on flat: Some constructions look superior without storm shutters. Indeed, the lines of the property can appear muddled and improper in case you  [http://tinyurl.com/pch83be http://tinyurl.com/pch83be] make sure to increase kind of pointless decrease. [http://tinyurl.com/pch83be ugg boots sale] The hot button is the idea of "additional." When the construction theme doesn't need these kind of people and these people would possibly not have been included in the very first plan, you should probably do without hurricane shutters. A number of examples include are Craftsman bungalows, many ranch residences, and mid-century moderns. Then again, many building patterns protest to the confident people, including colonials, Shawl Cods, Greek revivals, saltboxes and Tudors. All of these apartment patterns have needed shutters with their earliest forms in order to keep to have end, cool temperature or spring hot temperature and that they remain to look best at their side in these days. After you have decided that by the way shutters would boost the looks of your property, taking in facts right becomes principal. Understand that in the event that they always nearly provide the appearance of being truly functioning, even when you might not making use of them as anything beyond enhancing sleek. Siding painting contractors as you gain experience aren't going to only place level, fake [http://Search.about.com/?q=versions+strait versions strait] onto have to deal with of your property side the windows. This looks so one-dimensional and doesn't permit the appearance of being purposeful. Instead, plastic contractors on ocasions when position these items on the period slim with hinges making sure that they usually have some intensity and appear that they could have been swung shut down each time needed. The S-shaped braces that by the way hold one down towards the surface environs, it doesn't matter if toward the longer sides or towards the underside just about every  [http://tinyurl.com/pch83be http://tinyurl.com/pch83be] one of them one, undoubtedly are a another simple method apply authenticity and extent.<br><br><br>With countless styles to select from, it's best to speak with synthetic house painters about the ones which will be best sitting on your house. Regal and Headland Cod operators look impeccable along with old style, louvered storm shutters although a Tudor might look best that has a team and batten type. Lifted Created website content could possibly be the variety of do the job that most of one's "authorities" will show you to target on. The stuff that is focused on top secret approaches, loopholes, fast solutions and other stupidity that will be presupposed to take advantage of for yourself an thought imbalance in the whole on the web enterprise ecology (i.e., each one of everyone "underhanded codes" the search web sites are uninformed as to about ... and they do!). Customarily, contrived stuff pozycjonowanie very infrequently function ... and if they actually, it's just extremely for the count of 15. Creating content for Website positioning is basically a type of inspirations, that is actually by virtue of on account that you could make use of anchor message links as part of your resource box, you will likely plainly "black jack" the system to produce it appear like your blog, or business site or landing web page was really more applicable than its or was also. And naturally, the idea that thus constructing hundreds of well written articles and syndicating these items about the web to really very small listings  [http://tinyurl.com/pch83be ugg boots] which have exclusive attach wording should Good your site or blog, has probably done more to really murder otherwise truly creative material entrepreneurs versus the rest (because in the event that it didn't be effective ... or more painful, they had been mistreated because it by having even LOWER search results ... they gave up!). Resourceful stuff, on the reverse side of one's path, was birthed by having Want and purposes and a experience of non-public energy concerning your specific niche market, your old field, your business and just what you like to talk about. It has no give attention to keyword phrases or algorithms or a stupid tactic at all. It's no stun, which typically imaginative content Smashes it within the long-term ... and the interesting things naturally, is, it's suitable for Search engine optimizing too (e.g. - people naturally the way it, have in common it and apply it along the online within a all natural method that truly DOES benefit your web page).<br><br>In the end, within my enterprise, halting an emphasis on key-word attach content, and instead ... ONLY being [http://Search.huffingtonpost.com/search?q=bothered&s_it=header_form_v1 bothered] about crafting the very best meaning to have the merely click, directly stubbed my visitor traffic by perfectly north of big. The key reason why? Because instead of getting concerned about making use of right "words" inside resource box for Web optimization, I instead could specialize in discovering a intriguing concept that got friends to stay probing and take good the next phase to go to my place.
 
The use and development of ambit fields is motivated by the need of flexible stochastic models to describe [[turbulence]]<ref name="turb">Barndorff-Nielsen, O. E., Schmiegel, J. [http://data.imf.au.dk/publications/thiele/2005/imf-thiele-2005-14.pdf "Ambit processes; with applications to turbulence and tumour growth"], ''Research report, Thiele Centre'', December 2005</ref> and the evolution of [[electricity]] prices<ref name="forw">Barndorff-Nielsen, O. E., Benth, F. E., and Veraart, A., [http://pure.au.dk/portal/files/21655743/rp10_41.pdf "Modelling electricity forward markets by ambit fields"], ''CREATES research center'', 2010</ref> for use in e.g. [[risk management]] and [[derivative pricing]]. It was pioneered by [[Ole E. Barndorff-Nielsen]] and [[Jürgen Schmiegel]] to model turbulence and tumour growth.<ref name="turb" />
 
Note, that this article will use notation that includes time as a dimension, i.e. we consider (''d''&nbsp;&minus;&nbsp;1)-dimensional space together with 1-dimensional time. The theory and notation easily carries over to ''d''-dimensional space (either including time herin or in a setting involving no time at all).
 
==Intuition and motivation==
In stochastic analysis, the usual way to model a random process, or field, is done by specifying the ''dynamics'' of the process through a [[Stochastic partial differential equation|stochastic (partial) differential equation]] (SPDE). It is known, that solutions of (partial) differential equations can in some cases be given as an integral of a [[Green's function]] [[convolution|convolved]] with another function – if the differential equation is stochastic, i.e. contaminated by random noise (e.g. [[white noise]]) the corresponding solution would be a stochastic integral of the Green's function. This fact motivates the reason for modelling the field of interest ''directly'' through a stochastic integral, taking a similar form as a solution through a Green's Function, instead of first specifying a SPDE and then trying to find a solution to this. This provides a very flexible and general framework for modelling a variety of phenomena.<ref name="forw" />
 
==Definition==
 
A [[Space-time|tempo-spatial]] ambit field, <math>Y</math>, is a random field in [[space-time]] <math>\chi \times \mathbb{R}</math> taking values in <math>\mathbb{R}</math>. Let <math>\mu \in \mathbb{R}, A_t (x), B_t (x)</math> be [[Ambit Field#Ambit Sets|ambit sets]] in <math>\chi \times \mathbb{R}_{+}, g, q</math> [[deterministic]] [[Integral kernel|kernel]] functions, <math>a</math> a [[stochastic]] [[function (mathematics)|function]], <math>\sigma \geq 0</math> a [[Random field|stochastic field]] (called the ''energy dissipation field'' in [[turbulence]] and ''[[volatility (finance)|volatility]]'' in [[finance]]) and <math>L</math> a Lévy basis. Now, the ambit field <math>Y</math> is
 
: <math>Y_t = \mu + \int_{A_{t}(x)} g(\eta,s,x,t)\sigma_{s}(\eta) L(d\eta,ds) + \int_{B_t(x)} q(\eta,s,x,t)a_{s}(\eta) \, d\eta \, ds</math>
 
===Ambit sets===
In the above, the ''ambit sets'' <math>A_t(x)</math> and <math>B_t(x)</math> describe the sphere of influence for a given point in space-time. I.e. at a given point, <math>(t,x) \in \chi \times \mathbb{R}</math> the sets <math>A_t(x)</math> and <math>B_t(x)</math> are the points in space-time which affect the value of the ambit field at <math>(t,x), Y_t(x)</math>. When time is considered as one of the dimensions, the sets are often taken to only include time-coordinates which are at or prior to the current time, t, so as to preserve [[causality]] of the field (i.e. a given point in space-time can only be affected by events that happened prior to time <math>t</math> and can thus not be affected by the future).
 
The ambit sets can be of a variety of forms and when using ambit fields for modelling purposes, the choice of ambit sets should be made in a way that captures the desired properties (e.g. [[stylized facts]]) of the system considered in the best possible way. In this sense, the sets can be used to make a particular model fit the data as closely as possible and thus provides a very flexible – yet general – way of specifying the model.
 
===Ambit process===
Often, the object of interest is not the ambit field itself, but instead a process taking a particular path through the field. Such a process is called an ''ambit process''. As an example such a process can represent the price of a particular financial object – e.g. the price of a [[forward contract]] for a certain time and point in space, space representing things such as time to delivery, [[spot price]], period of delivery etc.<ref name="forw" /> This motivates the following definition:
 
Let the ambit field, ''Y'', be given as above and consider a curve in space-time <math>\tau(\theta) = (x(\theta), t(\theta)) \in \chi \times \mathbb{R}</math>. An ambit process is defined as the value of the field along the curve, i.e.
 
: <math>X_\theta = Y_{t(\theta)}(x(\theta))</math>
 
===Stochastic intermittency/volatility===
The energy dissipation field/volatility, <math>\sigma</math>, is, in general, stochastic (called ''[[intermittency]]'' in the context of ''[[turbulence]]''), and can be modelled as a stochastic variable or field. Particularly, it may itself be modelled by another ambit field, i.e.
 
: <math>\sigma^2_t(x) = \int_{C_t(x)} h(\eta,s,x,t) \tilde{L}(d\eta,ds)</math>
 
where <math>\tilde{L}</math> is a non-negative Lévy basis.
 
==Integration with respect to a Lévy basis==
The stochastic integral, <math>\int_{A_{t}(x)} g(\eta,s,x,t)\sigma_{s}(\eta) L(d\eta,ds)</math>, in the definition of the ambit process is an integral of a stochastic field (the [[integrand]]) over Lévy basis (the [[integrator]]), and is thus more complicated than the usual stochastic [[Itô-integral]]. A new theory of integration was provided by Walsh (1987)<ref>Walsh, J., "An introduction to stochastic partial differential equations", ''Lecture Notes in Mathematics'', 1986</ref> where integration is done with respect to random fields and this theory can be extended to integration with respect to so-called Lévy bases,<ref name="spde">Barndorff–Nielsen, O. E., Benth, F. E.,
and Veraart, A., [ftp://ftp.econ.au.dk/creates/rp/10/rp10_17.pdf "Ambit processes and stochastic partial differential equations"], ''CREATES research center'', 2010</ref> which is the main building block of the ambit field.
 
===Definition of Lévy basis===
A family <math>(\Lambda(A) : A \in \mathbb{B}_b(S))</math> of random vectors in <math>\mathbb{R}^d</math> is called a ''Lévy basis'' on <math>S</math> if:
: 1. The law of <math>\Lambda(A)</math> is [[infinitely divisible]] for all <math>A \in \mathbb{B}_{b}(S)</math>.
: 2. If <math>A_1 , A_2, \ldots, A_n \in \mathbb{B}_b(S)</math> are disjoint, then <math>\Lambda(A_1), \Lambda(A_2),\ldots, \Lambda(A_n)</math> are independent.
: 3. If <math>A_1 , A_2, \ldots \in \mathbb{B}_b(S)</math> are disjoint with <math>\bigcup_{i=1}^\infty A_i \in \mathbb{B}_b(S)</math>, then
::: <math>\Lambda(\bigcup_{i=1}^\infty A_i) = \sum_{i=1}^\infty \Lambda(A_i)</math>,   a.s.
 
where the [[convergence (mathematics)|convergence]] on the right hand side of 3. is a.s.
 
Note that proporties 2. and 3. define an independently scattered [[random measure]].
 
==A stationary example==
In some data (e.g. commodity prices) there is often found a [[Time-invariant|stationary]] component, which a good model should be able to capture. The ambit field can be made stationary in a straight forward way. Consider the ambit field <math>Y</math>, defined as
 
: <math>Y_t = \mu + \int_{A_{t}(x)} g(\eta,t-s,x)\sigma_{s}(\eta) L(d\eta,ds) + \int_{B_{t}(x)} q(\eta,t-s,x)a_{s}(\eta) \, d\eta \, ds</math>
 
where the ambit sets, <math>A_{t}(x), B_{t}(x)</math> are of the form <math>A_{t}(x) = A + (x,t)</math> where the time-coordinates of <math>A</math> are negative (same for <math>B</math>). Furthermore, we take <math>g(\eta,t,x) = q(\eta,t,x) = 0 </math> for <math> t \leq 0</math> and that <math>\sigma</math> and <math>a</math> are also stationary random variables/fields. In particular, we can take <math>\sigma</math> to be a stationary ambit field itself:
 
: <math>\sigma^2_{t}(x) = \int_{C_{t}(x)} h(\eta,t-s,x) \tilde{L}(d\eta,ds)</math>
 
where <math>\tilde{L}</math> is a non-negative Lévy basis and <math>h</math> is a positive function.
 
==References==
<references/>
 
==External links==
* [http://www.ambitprocesses.au.dk/ Ambit Processes at University of Aarhus]
 
[[Category:Probability theory]]

Latest revision as of 17:43, 15 October 2014

Including storm shutters in a home is most popular exterior exterior remodeling made by homeowners, and it also could add elegance and fineness or rural charm with respect to the kind of flat and it doesn't matter if you may have preferred the most appropriate theme to further improve your home or apartment. You'll also find certain residence styles that just don't look right around with them. In case you aren't certain and don't need a number of counsel, possess a drive in the http://tinyurl.com/pch83be areas around your country and take quick notice of which residences have window protection that are proportional and appropriate and which ones request that they definitely caught on as an afterthought. Hard plastic workers who exactly routinely install them is also able to provide you valuable suggestions about trend, proportion and overall attraction. Keep these pointers from neighborhood vinyl professionals as the primary goal once your all set to layer them showing ugg boots usa on flat: Some constructions look superior without storm shutters. Indeed, the lines of the property can appear muddled and improper in case you http://tinyurl.com/pch83be make sure to increase kind of pointless decrease. ugg boots sale The hot button is the idea of "additional." When the construction theme doesn't need these kind of people and these people would possibly not have been included in the very first plan, you should probably do without hurricane shutters. A number of examples include are Craftsman bungalows, many ranch residences, and mid-century moderns. Then again, many building patterns protest to the confident people, including colonials, Shawl Cods, Greek revivals, saltboxes and Tudors. All of these apartment patterns have needed shutters with their earliest forms in order to keep to have end, cool temperature or spring hot temperature and that they remain to look best at their side in these days. After you have decided that by the way shutters would boost the looks of your property, taking in facts right becomes principal. Understand that in the event that they always nearly provide the appearance of being truly functioning, even when you might not making use of them as anything beyond enhancing sleek. Siding painting contractors as you gain experience aren't going to only place level, fake versions strait onto have to deal with of your property side the windows. This looks so one-dimensional and doesn't permit the appearance of being purposeful. Instead, plastic contractors on ocasions when position these items on the period slim with hinges making sure that they usually have some intensity and appear that they could have been swung shut down each time needed. The S-shaped braces that by the way hold one down towards the surface environs, it doesn't matter if toward the longer sides or towards the underside just about every http://tinyurl.com/pch83be one of them one, undoubtedly are a another simple method apply authenticity and extent.


With countless styles to select from, it's best to speak with synthetic house painters about the ones which will be best sitting on your house. Regal and Headland Cod operators look impeccable along with old style, louvered storm shutters although a Tudor might look best that has a team and batten type. Lifted Created website content could possibly be the variety of do the job that most of one's "authorities" will show you to target on. The stuff that is focused on top secret approaches, loopholes, fast solutions and other stupidity that will be presupposed to take advantage of for yourself an thought imbalance in the whole on the web enterprise ecology (i.e., each one of everyone "underhanded codes" the search web sites are uninformed as to about ... and they do!). Customarily, contrived stuff pozycjonowanie very infrequently function ... and if they actually, it's just extremely for the count of 15. Creating content for Website positioning is basically a type of inspirations, that is actually by virtue of on account that you could make use of anchor message links as part of your resource box, you will likely plainly "black jack" the system to produce it appear like your blog, or business site or landing web page was really more applicable than its or was also. And naturally, the idea that thus constructing hundreds of well written articles and syndicating these items about the web to really very small listings ugg boots which have exclusive attach wording should Good your site or blog, has probably done more to really murder otherwise truly creative material entrepreneurs versus the rest (because in the event that it didn't be effective ... or more painful, they had been mistreated because it by having even LOWER search results ... they gave up!). Resourceful stuff, on the reverse side of one's path, was birthed by having Want and purposes and a experience of non-public energy concerning your specific niche market, your old field, your business and just what you like to talk about. It has no give attention to keyword phrases or algorithms or a stupid tactic at all. It's no stun, which typically imaginative content Smashes it within the long-term ... and the interesting things naturally, is, it's suitable for Search engine optimizing too (e.g. - people naturally the way it, have in common it and apply it along the online within a all natural method that truly DOES benefit your web page).

In the end, within my enterprise, halting an emphasis on key-word attach content, and instead ... ONLY being bothered about crafting the very best meaning to have the merely click, directly stubbed my visitor traffic by perfectly north of big. The key reason why? Because instead of getting concerned about making use of right "words" inside resource box for Web optimization, I instead could specialize in discovering a intriguing concept that got friends to stay probing and take good the next phase to go to my place.

Retrieved from "https://en.formulasearchengine.com/index.php?title=Glossary_of_invariant_theory&oldid=270192"