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In [[mathematics]], a bounded sequence {''s''<sub>1</sub>, ''s''<sub>2</sub>, ''s''<sub>3</sub>, …} of [[real number]]s is said to be '''equidistributed''', or '''uniformly distributed''', if the proportion of terms falling in a subinterval is proportional to the length of that interval.  Such sequences are studied in [[Diophantine approximation]] theory and have applications to [[Monte Carlo integration]].
MinLaw formulates policies and techniques in help of Singapore's drive towards an innovation-driven financial system. Particular capabilities include ensuring a robust domestic IP protection and enforcement regime that's conscious of our economic needs; fostering the growth of a vibrant IP services sector, and creating Singapore as an Asian IP hub.<br><br>You might beto consult your lawyer instantly after paying the booking feeand obtaining the prescribed Option to Purchase the [http://www.primaboinca.com/view_profile.php?userid=5754278 industrial property developers in singapore]. The lawyer will theninvestigate the title and advise you on the phrases of the prescribed Settlement beforeyou settle for the identical by exercising the Option to Buy. The Law Society's Circumstances of Sale Back to Top 5. File Property Tax Claims Back to High 6. 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Make sure that you qualify for a mortgage, and focus only on a small variety of neighborhoods. Then yow will discover an skilled and knowledgeable New York real property attorney, and take his consultations before you go for any contract.<br><br>There are three-room, four-room, five-room and executive flats. A 3-room flat has two bedrooms , which is about 1,000 sq ft. A four-room flat has three bedrooms which is about 1,200 sq ft of house. A 5-room flat is about 1,400 sq ft. An Executive mansionette has three and they are relatively larger than some other HDB units- some even have double storey, with the rooms on the second degree. Any foreigners who attained ownership of residential properties in Plan out a finances and your annual revenue, to determine in case you can afford the worth. Get your very own Digital Property Film (VPM) once you listing your property COMPLETELY with me! And let's begin advertising and marketing your Property to everyone EFFECTIVELY & VIRTUALLY! 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Hire Property District 12 Rent Property District 13<br><br>Whether they're downsizing with empty nests , or they're realizing their retirement dream of moving to the beach or mountains, this principle states that there is going to be a massive sales glut when Boomers start to move. Because the core of this group is already getting into retirement and we're in a shaky restoration from the real property crash, this is not a happy prediction of what is coming. In any given market around the country, one among these market outcomes may very well be in play already. Real property is local, and people high dollar homes alongside the coast in California may lock up their homeowners with golden handcuffs. Nonetheless, go to a small city within the South or Midwest, and you possibly can see the exodus of retirees creating a surge in housing stock.<br><br>It is the sellers obligation handy over the property upon authorized completion as agreed in the contract. In the case of a HDB flat, vacant possession is a requirement. Plan your move early to stop a last minute rush. Take away all your belongings and discard any undesirable furniture and rubbish. Be certain that all fixtures and objects in the inventory list are left behind. Invite the client (as required by HDB) for an inspection before completion. Within the case of personal properties, they can be sold with existing tenancy or vacant possession. The sales contract will specify your authorized obligations. Examine with your agent if you are in doubt.
 
==Definition==
A bounded sequence {''s''<sub>1</sub>, ''s''<sub>2</sub>, ''s''<sub>3</sub>, …} of [[real number]]s is said to be ''equidistributed'' on an interval [''a'',&nbsp;''b''] if for any subinterval [''c'',&nbsp;''d''] of [''a'',&nbsp;''b''] we have
:<math>\lim_{n\to\infty}{ \left|\{\,s_1,\dots,s_n \,\} \cap [c,d] \right| \over n}={d-c \over b-a} . \,</math>
(Here, the notation |{''s''<sub>1</sub>,&hellip;,''s''<sub>''n''</sub>&nbsp;}∩[''c'',''d'']| denotes the number of elements, out of the first ''n'' elements of the sequence, that are between ''c'' and ''d''.)
 
For example, if a sequence is equidistributed in [0,&nbsp;2], since the interval [0.5,&nbsp;0.9] occupies 1/5 of the length of the interval [0,&nbsp;2], as ''n'' becomes large, the proportion of the first ''n'' members of the sequence which fall between 0.5 and 0.9 must approach 1/5. Loosely speaking, one could say that each member of the sequence is equally likely to fall anywhere in its range. However, this is not to say that {''s''<sub>''n''</sub>} is a sequence of random variables; rather, it is a determinate sequence of real numbers.
 
===Discrepancy===
We define the '''discrepancy''' ''D''(''N'') for a sequence {''s''<sub>1</sub>, ''s''<sub>2</sub>, ''s''<sub>3</sub>, …} with respect to the interval [''a'',&nbsp;''b''] as
 
:<math> D(N) = \sup_{a\le c\le d\le b} \left\vert \frac{\left|\{\,s_1,\dots,s_N \,\} \cap [c,d] \right|}{N} - \frac{d-c}{b-a} \right\vert . \,</math>
 
A sequence is thus equidistributed if the discrepancy ''D''(''N'') tends to zero as ''N'' tends to infinity.
 
Equidistribution is a rather weak criterion to express the fact that a sequence fills the segment leaving no gaps. For example, the drawings of a random variable uniform over a segment will be equidistributed in the segment, but there will be large gaps compared to a sequence which first enumerates multiples of ε in the segment, for some small ε, in an appropriately chosen way, and then continues to do this for smaller and smaller values of ε.  See [[low-discrepancy sequence]] for stronger criteria and [[constructions of low-discrepancy sequences]] for constructions of sequences which are more evenly distributed.
 
===Equidistribution modulo 1===
The sequence {''a''<sub>1</sub>, ''a''<sub>2</sub>, ''a''<sub>3</sub>, …} is said to be '''equidistributed modulo 1''' or '''uniformly distributed modulo 1''' if the sequence of the [[fractional part]]s of the ''a''<sub>''n''</sub>'s, (denoted by ''a''<sub>''n''</sub>−&lfloor;''a''<sub>''n''</sub>&rfloor;)
 
:<math> \{ a_1-\lfloor a_1\rfloor, a_2-\lfloor a_2\rfloor, a_3-\lfloor a_3\rfloor, \dots \} </math>
 
is equidistributed in the interval [0,&nbsp;1].
 
==Examples==
* The sequence of all multiples of an irrational α,
::0, &alpha;, 2&alpha;, 3&alpha;, 4&alpha;, &hellip;
is uniformly distributed modulo 1:<ref name=KN8>Kuipers & Niederreiter (2006) p.&nbsp;8</ref> this is the [[equidistribution theorem]].
* More generally, if ''p'' is a polynomial with at least one irrational coefficient (other than the constant term) then the sequence ''p''(''n'') is uniformly distributed modulo 1: this was proven by Weyl and is an application of van der Corput's difference theorem.<ref name=KN27>Kuipers & Niederreiter (2006) p.&nbsp;27</ref>
* The sequence log(''n'') is ''not'' uniformly distributed modulo 1.<ref name=KN8/>
* The sequence of all multiples of an irrational α by successive [[prime number]]s,
::2&alpha;, 3&alpha;, 5&alpha;, 7&alpha;, 11&alpha;, &hellip;
is equidistributed modulo 1.  This is a famous [[theorem]] of [[analytic number theory]], proven by [[I. M. Vinogradov]] in 1935.
* The [[van der Corput sequence]] is equidistributed.<ref name=KN127>Kuipers & Niederreiter (2006) p.&nbsp;127</ref>
 
==van der Corput's difference theorem==
A theorem of [[Johannes van der Corput]]<ref>{{Citation | last=van der Corput | first=J. | authorlink=Johannes van der Corput | title=Diophantische Ungleichungen. I. Zur Gleichverteilung Modulo Eins | publisher=Springer Netherlands | year=1931 | journal=[[Acta Mathematica]] | issn=0001-5962 | volume=56 | pages=373–456 | doi=10.1007/BF02545780 | zbl=0001.20102 | jfm=57.0230.05 }}</ref> states that if for each ''h'' the sequence ''s''<sub>''n''+''h''</sub>−s<sub>''n''</sub> is uniformly distributed modulo 1, then so is s<sub>''n''</sub>.<ref>Kuipers & Niederreiter (2006) p.&nbsp;26</ref><ref name=Mon18>Montgomery (1994) p.18</ref>
 
A '''van der Corput set''' is a set ''H'' of integers such that if for each ''h'' in ''H'' the sequence ''s''<sub>''n''+''h''</sub>−s<sub>''n''</sub> is uniformly distributed modulo 1, then so is s<sub>''n''</sub>.<ref name=Mon18/>
 
==Properties==
The following three conditions are equivalent:
# {''a''<sub>''n''</sub>} is equidistributed modulo 1.
# For every Riemann integrable function ''f'' on [0,&nbsp;1],
::<math>\lim_{n\to\infty} \frac{1}{n} \sum_{j=1}^n f(a_j)=\int_0^1 f(x)\, dx.</math>
<ol start=3><li>For every nonzero integer ''k'',
::<math>\lim_{n\to\infty} \frac{1}{n} \sum_{j=1}^n e^{2\pi ik a_j}=0.</math>
</ol>
The third condition is known as [[Weyl's criterion]].  Together with the formula for the sum of a finite [[geometric series]], the equivalence of the first and third conditions furnishes an immediate proof of the equidistribution theorem.
 
==Metric theorems==
Metric theorems describe the behaviour of a parametrised sequence for [[almost all]] values of some parameter α: that is, for values of α not lying in some exceptional set of [[Lebesgue measure]] zero.
 
* For any sequence of distinct integers ''b''<sub>''n''</sub>, the sequence {''b''<sub>''n''</sub> α} is equidistributed mod 1 for almost all values of α.<ref>See {{citation | title=Über eine Anwendung der Mengenlehre auf ein aus der Theorie der säkularen Störungen herrührendes Problem | first=Felix | last=Bernstein | authorlink=Felix Bernstein | journal=[[Mathematische Annalen]] | volume=71 | number=3 | year=1911 | pages=417–439 | doi=10.1007/BF01456856 }}.</ref>
* The sequence {α<sup>''n''</sup>} is equidistributed mod 1 for almost all values of α > 1.<ref>{{citation | url=http://www.numdam.org/item?id=CM_1935__2__250_0 | title=Ein mengentheoretischer Satz über die Gleichverteilung modulo Eins | first=J. F. | last=Koksma | authorlink=Jurjen Ferdinand Koksma | journal=[[Compositio Mathematica]] | volume=2 | year=1935 | pages=250–258 | zbl=0012.01401  | jfm=61.0205.01 }}</ref>
It is not known whether the sequences {e<sup>''n''</sup>} or {&pi;<sup>''n''</sup>} are equidistributed mod 1.  However it is known that the sequence {α<sup>''n''</sup>} is ''not'' equidistributed mod 1 if α is a [[PV number]].
 
==Well-distributed sequence==
A bounded sequence {''s''<sub>1</sub>, ''s''<sub>2</sub>, ''s''<sub>3</sub>, …} of [[real number]]s is said to be '''well-distributed''' on [''a'',&nbsp;''b''] if for any subinterval [''c'',&nbsp;''d''] of [''a'',&nbsp;''b''] we have
:<math>\lim_{n\to\infty}{ \left|\{\,s_{k+1},\dots,s_{k+n} \,\} \cap [c,d] \right| \over n}={d-c \over b-a} \,</math>
 
uniformly in ''k''. Clearly every well-distributed sequence is uniformly distributed, but the converse does not hold. The definition of well-distributed modulo 1 is analogous.
 
==Sequences equidistributed with respect to an arbitrary measure==
For an arbitrary [[probability measure space]] <math>(X,\mu)</math>, a sequence of points <math>x_n</math> is said to be equidistributed with respect to <math>\mu</math> if the mean of [[Dirac delta function|point measures]] [[Convergence_of_measures#Weak_convergence_of_measures|converges weakly]] to <math>\mu</math>:
:<math>\frac{\sum_{k=1}^n \delta_{x_k}}{n}\Rightarrow \mu</math>
It is true, for example. that for any probabilistic [[borel measure]] on a [[separable space|separable]], [[metrizable]] space, there exists an equidistributed sequence (with respect to the measure).
 
== See also ==
*[[Equidistribution theorem]]
*[[Low-discrepancy sequence]]
 
== References ==
{{reflist}}
* {{cite book | first1=L. | last1=Kuipers | first2=H. | last2=Niederreiter | title=Uniform Distribution of Sequences | publisher=Dover Publishing | year=2006 | isbn=0-486-45019-8 }}
* {{cite book | first1=L. | last1=Kuipers | first2=H. | last2=Niederreiter | title=Uniform Distribution of Sequences | publisher=John Wiley & Sons Inc. | year=1974 | isbn=0-471-51045-9 }}
* {{cite book | last=Montgomery | first=Hugh L. | authorlink=Hugh Montgomery (mathematician) | title=Ten lectures on the interface between analytic number theory and harmonic analysis | series=Regional Conference Series in Mathematics | volume=84 | location=Providence, RI | publisher=[[American Mathematical Society]] | year=1994 | isbn=0-8218-0737-4 | zbl=0814.11001 }}
 
==Further reading==
* {{cite book | zbl=pre06110460 | last=Tao | first=Terence | authorlink=Terence Tao | title=Higher order Fourier analysis | series=Graduate Studies in Mathematics | volume=142 | location=Providence, RI | publisher=[[American Mathematical Society]] | year=2012 | isbn=978-0-8218-8986-2 | url=http://terrytao.wordpress.com/books/higher-order-fourier-analysis/ }}
 
==External links==
* {{MathWorld|title=Equidistributed Sequence|urlname=EquidistributedSequence}}
 
[[Category:Diophantine approximation]]
[[Category:Ergodic theory]]

Latest revision as of 05:46, 28 August 2014

MinLaw formulates policies and techniques in help of Singapore's drive towards an innovation-driven financial system. Particular capabilities include ensuring a robust domestic IP protection and enforcement regime that's conscious of our economic needs; fostering the growth of a vibrant IP services sector, and creating Singapore as an Asian IP hub.

You might beto consult your lawyer instantly after paying the booking feeand obtaining the prescribed Option to Purchase the industrial property developers in singapore. The lawyer will theninvestigate the title and advise you on the phrases of the prescribed Settlement beforeyou settle for the identical by exercising the Option to Buy. The Law Society's Circumstances of Sale Back to Top 5. File Property Tax Claims Back to High 6. Different Taxes on Property Whenever you purchase any property, you should pay Stamp Obligation inside 14 days of the date of Contract/Settlement. In most cases, your lawyer performing for you within the buy of the property will arrange for cost of Stamp Duty. For enquiries on Our Properties, please make reference to the property identify) Our Properties Rent Home District 03 real property brokers

If in any respect, dwelling in New York City, under the skyscrapers of Manhattan, has always been your dream, a condominium is the best choice. Going for a rental is a brilliant selection and terrific investment in NYC. The only factor that matters is that if you may get a mortgage to purchase one. The most important thing is to develop a technique that would come with your sensible and personal needs. Know what you can afford and clearly underline your property by consulting a monetary skilled. Make sure that you qualify for a mortgage, and focus only on a small variety of neighborhoods. Then yow will discover an skilled and knowledgeable New York real property attorney, and take his consultations before you go for any contract.

There are three-room, four-room, five-room and executive flats. A 3-room flat has two bedrooms , which is about 1,000 sq ft. A four-room flat has three bedrooms which is about 1,200 sq ft of house. A 5-room flat is about 1,400 sq ft. An Executive mansionette has three and they are relatively larger than some other HDB units- some even have double storey, with the rooms on the second degree. Any foreigners who attained ownership of residential properties in Plan out a finances and your annual revenue, to determine in case you can afford the worth. Get your very own Digital Property Film (VPM) once you listing your property COMPLETELY with me! And let's begin advertising and marketing your Property to everyone EFFECTIVELY & VIRTUALLY! Maintain abreast of the market prices

First, there are usually more rental transactions than gross sales transactions, to allow AV to be decided for each property based mostly on comparable properties. Second, movements in sale costs are extra risky than leases. Therefore, utilizing rental transactions to derive the AV helps to maintain property tax extra steady for property owners. In case you are buying or looking to hire a property. It's tiring to name up individual property agent, arrange appointments, coordinate timing and to go for particular person property viewing. What most individuals do is to have a property agent representing them who will prepare and coordinate the viewings for all the properties obtainable primarily based in your necessities & most well-liked timing. Hire Property District 12 Rent Property District 13

Whether they're downsizing with empty nests , or they're realizing their retirement dream of moving to the beach or mountains, this principle states that there is going to be a massive sales glut when Boomers start to move. Because the core of this group is already getting into retirement and we're in a shaky restoration from the real property crash, this is not a happy prediction of what is coming. In any given market around the country, one among these market outcomes may very well be in play already. Real property is local, and people high dollar homes alongside the coast in California may lock up their homeowners with golden handcuffs. Nonetheless, go to a small city within the South or Midwest, and you possibly can see the exodus of retirees creating a surge in housing stock.

It is the sellers obligation handy over the property upon authorized completion as agreed in the contract. In the case of a HDB flat, vacant possession is a requirement. Plan your move early to stop a last minute rush. Take away all your belongings and discard any undesirable furniture and rubbish. Be certain that all fixtures and objects in the inventory list are left behind. Invite the client (as required by HDB) for an inspection before completion. Within the case of personal properties, they can be sold with existing tenancy or vacant possession. The sales contract will specify your authorized obligations. Examine with your agent if you are in doubt.

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