Parametric process (optics): Difference between revisions

Revision as of 04:59, 27 December 2013

In Morse theory, a mathematical discipline, Jacobi sets provide a method of studying the relationship between two or more Morse functions.

For two Morse functions, the Jacobi set is defined as the set of critical points of the restriction of one function to the level sets of the other function.^[1]

The Jacobi set can also be defined as the set of points where the gradients of the two functions are parallel.

If both the functions are generic, the Jacobi set is a smoothly embedded 1-manifold.

Definition

Consider two generic Morse functions $f, g : M \to ℝ$ defined on a smooth $d$ -manifold, then the Jacobi set $J$ of $f$ and $g$ is $J = c l {x \in M ∣ x is critical point of f_{t}}$ , where, for a regular value $t \in ℝ$ , let $g^{- 1} (t)$ be the level set of $g$ . The restriction of $f$ to the level set $f_{t} : g^{- 1} \to ℝ$ is a Morse function.

Alternatively, the Jacobi set is the collection of points where the gradients of the functions align with each other or one of the gradients vanish (cite?), for some $λ \in ℝ$ , $J = {x \in M ∣ \nabla f (x) + λ \nabla g (x) = 0 or λ \nabla f (x) + \nabla g (x) = 0} .$

Equivalently, the Jacobi set can be described as the collection of critical points of the family of functions $f + λ g$ , for some $λ \in ℝ$ , $J = {x \in M ∣ x is a critical point of f + λ g or λ f + g} .$

References

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The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

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Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

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+{{Multiple issues|
+{{orphan|date=September 2011}}
+{{one source|date=August 2011}}
+{{expert-subject|date=August 2011}}
+}}
+In [[Morse theory]], a mathematical discipline, '''Jacobi sets''' provide a method of studying the relationship between two or more [[Morse function]]s.
+For two Morse functions, the Jacobi set is defined as the set of critical points of the restriction of one function to the level sets of the other function.<ref name="edelsbrunner">{{cite journal|last=Edelsbrunner|first=Herbert|coauthors=John Harer|year=2002|title=Jacobi sets of multiple morse functions|journal=Foundations of Computational Mathematics|publisher=Cambridge University Press|pages=37–57}}</ref>
+The Jacobi set can also be defined as the set of points where the [[gradients]] of the two functions are parallel.
+If both the functions are generic, the Jacobi set is a smoothly embedded 1-manifold.
+== Definition ==
+Consider two generic Morse functions <math>f, g: M \to \R</math> defined on a smooth <math>d</math>-manifold, then the Jacobi set <math>J</math> of <math> f </math> and <math>g</math> is
+<math> J = cl{\{x \in M \mid x \mbox{ is critical point of } f_t \}} </math>,
+where, for a regular value <math>t \in \R</math>, let <math>g^{-1}(t)</math> be the level set of <math>g</math>.
+The restriction of <math>f</math> to the level set <math>f_t: g^{-1} \to \R</math> is a Morse function.
+Alternatively, the Jacobi set is the collection of points where the gradients of the functions align with each other or one of the gradients vanish (cite?), for some <math>\lambda \in \R</math>,
+<math>
+J = \{x \in M \mid \nabla{f(x)} + \lambda \nabla{g(x)} = 0 \mbox{ or } \lambda \nabla{f(x)} + \nabla{g(x)} = 0\}.
+</math>
+Equivalently, the Jacobi set can be described as the collection of critical points of the family of functions <math>f+ \lambda g</math>, for some <math>\lambda \in \R</math>,
+<math>
+J = \{x \in M \mid x \mbox{ is a critical point of } f + \lambda g \mbox{ or } \lambda f + g\}.
+</math>
+==References==
+{{reflist}}
+[[Category:Morse theory]]

Parametric process (optics): Difference between revisions

Revision as of 04:59, 27 December 2013

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