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In [[mathematics]], a [[metric space]] ''X'' with metric ''d'' is said to be '''doubling''' if there is some constant ''M''&nbsp;>&nbsp;0 such that for any ''x'' in ''X'' and ''r''&nbsp;>&nbsp;0, the ball ''B''(''x'',&nbsp;''r'')&nbsp;=&nbsp;{''y'':|''d(x, y)'' <&nbsp;''r''} may be contained in a union of no more than ''M'' many balls with radius ''r''/2.<ref>{{cite book| last = Heinonen| first = Juha | title = Lectures on Analysis on Metric Spaces | series = Universitext | publisher = Springer-Verlag | location = New York | year = 2001 | pages = x+140 | ISBN = 0-387-95104-0}}</ref> The base-2 logarithm of ''M'' is often referred to as the '''doubling dimension''' of ''X''. Euclidean spaces '''ℝ'''<sup>''d''</sup> equipped with the usual Euclidean metric are examples of doubling spaces where the doubling constant ''M'' depends on the dimension&nbsp;''d''.
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==Assouad's embedding theorem==
 
An important question in metric space geometry is to characterize those metric spaces that can be embedded in some Euclidean space by a [[Lipschitz continuity|bi-Lipschitz]] function. This means that one can essentially think of the metric space as a subset of Euclidean space. Not all metric spaces may be embedded in Euclidean space. Doubling metric spaces, on the other hand, would seem like they have more of a chance, since the doubling condition says, in a way, that the metric space is not infinite dimensional. However, this is still not the case in general. The [[Heisenberg group]] with its [[Carnot metric]] is an example of a doubling metric space which cannot be embedded in any Euclidean space.<ref>{{cite journal| last = Pansu| first= Pierre | title = Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un| journal = Ann. of Math. (2)| volume = 129 | year = 1989 | number = 1 | pages = 1–60}}</ref>
 
'''Assouad's Theorem''' states that, for a ''M''-doubling metric space ''X'', if we give it the metric ''d''(''x'',&nbsp;''y'')<sup>''ε''</sup> for some 0&nbsp;<&nbsp;''ε''&nbsp;<&nbsp;1, then there is a ''L''-bi-Lipschitz map ''f'':''X''&nbsp;→&nbsp;'''ℝ'''<sup>''d''</sup>, where ''d'' and ''L'' depend on ''M'' and&nbsp;''ε''.
 
==Doubling Measures==
 
===Definition===
A [[Measure (mathematics)|measure]] on a metric space ''X'' is said to be '''doubling''' if the measure of any ball is approximately the measure of its double, or more precisely, if there is a constant ''C''&nbsp;>&nbsp;0 such that
 
:<math> \mu(B(x,2r))\leq C\mu(B(x,r)) \, </math>
 
for all ''x'' in ''X'' and ''r''&nbsp;>&nbsp;0. In this case, we say ''μ'' is '''C-doubling'''.
 
A measure space that supports a doubling measure is necessarily a doubling metric space, where the doubling constant depends on the constant&nbsp;''C''.
 
Conversely, any [[complete metric space|complete]] doubling metric space supports a doubling measure.<ref>{{cite journal| last1 = Luukainen | last2 = Saksman | first1 = Jouni | first2 = Eero | title = Every complete doubling metric space carries a doubling measure | journal = Proc. Amer. Math. Soc. | volume = 126 | year = 1998 | number = 2 | pages = 531–534}}</ref>
 
===Examples===
 
A simple example of a doubling measure is [[Lebesgue measure]] on a Euclidean space. One can, however, have doubling measures on Euclidean space that are  [[Singular measures|singular]] with respect to Lebesgue measure. One example on the real line is the [[weak limit]] of the following sequence of measures:<ref>{{cite book| last=Zygmund | first = A. | title= Trigonometric Series. Vol. I,II | series = Cambridge Mathematical Library | edition = Third | publisher = Cambridge University Press | year = 2002 | pages = xii; Vol. I: xiv+383 pp.; Vol. II: viii+364| ISBN = 0-521-89053-5}}</ref>
 
:<math> d\mu_n = \prod_{i=1}^n (1+a\cos (3^i 2\pi x))\,dx,\;\;\; |a|<1.</math>
 
One can construct another singular doubling measure ''μ'' on the interval [0,&nbsp;1] as follows: for each ''k''&nbsp;≥&nbsp;0, partition the unit interval [0,1] into 3<sup>''k''</sup> intervals of length 3<sup>&minus;''k''</sup>. Let Δ be the collection of all such intervals in [0,1] obtained for each ''k'' (these are the ''triadic intervals''), and for each such interval ''I'', let ''m''(''I'') denote its "middle third" interval. Fix 0&nbsp;<&nbsp;''δ''&nbsp;<&nbsp;1 and let ''μ'' be the measure such that ''μ''([0,&nbsp;1])&nbsp;=&nbsp;1 and for each triadic interval ''I'', ''μ''(''m''(''I''))&nbsp;=&nbsp;''δμ''(''I''). Then this gives a doubling measure on [0,&nbsp;1] singular to Lebesgue measure.<ref>{{cite journal| last = Kahane | first = J.-P. | title = Trois notes sur les ensembles parfaits linéaires| journal = Enseignement Math. (2) | volume = 15 | year = 1969 | pages = 185–192}}</ref>
 
==Applications==
 
The definition of a doubling measure may seem arbitrary, or purely of geometric interest. However, many results from classical harmonic analysis extend to the setting of metric spaces with doubling measures.  
 
==References==
<references/>
 
{{DEFAULTSORT:Doubling Measure}}
[[Category:Metric geometry]]

Revision as of 11:20, 1 March 2014

As much more and more growers are choosing LED grow lights for indoor developing, there have been lots of new solutions introduced on the market place. The study, which is published at HortTechnology , located that every single plant grown utilizing regular HPS lighting made use of 1224 kilowatt-hours of energy by means of the season, when these beneath LEDs consumed just 294 kilowatt-hours. LED develop lights are exceptionally economical due to the efficiency of transferring power to light power alternatively of heat. LED's need approximately 75% significantly less energy compared to MH or HPS lights. The developing final results in studies speak for themselves.



In the course of the vegetative phase (first two months based on strain) the plants should really be given 16 to 20 hours of light per 24 hours. The special LED fixtures were created by GE and emit light at wavelengths optimal for plant development. I knew how to grow excellent vegetables biologically and I wanted to integrate that understanding with hardware to make points occur,” Shimamura says. Herbs grow prolifically indoors under LED lights.

Seedlings normally demand a blue light when flowering plants are far more suiteda red light. LED lights are typically in the ultraviolet spectrum and invisible to naked eyes. Just use these and watch your plants grow more rapidly. This website includes copyrighted material the use of which has not generally been specifically authorized by the copyright owner. These assist in cultivating plants indoor and places where the sun is impossible.

Increasing GaN on silicon is fairly a complex process,” mentioned Humphreys. Humphreys believes that the day will arrive when GaN devices will light our residences, energy our mobile phones, laptops, automobiles and aircraft engines, and connect us wirelessly to details transmitted from website traffic lights and street lights. If you liked this write-up, please give it a fast review on reddit or StumbleUpon Thanks! These enhanced LED lights have been received with a lot of constructive response.

The unit comes with 82 individual high energy LED lighting chips that are complete replaceable - this is critical as you will discover the odd 1 fails over time. This model operates excellent for growing vegetables and seedlings, if you want to develop flowers then a slightly different panel is suggested. Even though the G8 LED is a tiny bit more pricey it features a full 8 band spectrum creating it one particular of the finest possibilities for indoor increasing. A single-to-one replacement for 1000W HPS.

LED develop lights have an benefit in that they run at a temperature that is basically warm to the touch, rather than approaching 400 or even 500 degrees. When you use the finest LED develop lights, you will have the possibility to grow for years without possessing to replace your lighting program. For comparison, many other develop lights tend to last for around 20,000 hours just before needing to be replaced If you have any inquiries relating to where and how you can make use of Cannabis Developing With Led Lights (click through the following document), you could call us at our web-site. .

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