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In [[mathematics]], '''directed-complete partial orders''' (abbreviated to '''dcpo''' or sometimes just '''cpo''') are special classes of [[partially ordered set]]s, characterized by particular [[completeness (order theory)|completeness properties]]. Complete partial orders play a central role in [[theoretical computer science]], in [[denotational semantics]] and [[domain theory]].
In [[combinatorics|combinatorial]] [[mathematics]], the '''Bell polynomials''', named in honor of [[Eric Temple Bell]], are a [[triangular array]] of polynomials given by


== Definitions ==
:<math>B_{n,k}(x_1,x_2,\dots,x_{n-k+1})</math>


A partially ordered set is a '''directed-complete partial order''' ('''dcpo''') if each of its [[directed set|directed subsets]] has a [[supremum]]. Recall that a subset of a partial order is directed if it is non-empty and every pair of elements has an upper bound in the set. In the literature, dcpos sometimes also appear under the label '''up-complete poset''' or simply '''cpo'''.
:<math>=\sum{n! \over j_1!j_2!\cdots j_{n-k+1}!}
\left({x_1\over 1!}\right)^{j_1}\left({x_2\over 2!}\right)^{j_2}\cdots\left({x_{n-k+1} \over (n-k+1)!}\right)^{j_{n-k+1}},</math>


In some contexts, the phrase [[chain complete|ω-cpo]] (or just '''cpo''') is used to describe a poset in which every ω-chain (''x''<sub>1</sub>≤''x''<sub>2</sub>≤''x''<sub>3</sub>≤''x''<sub>4</sub>≤...) has a supremum. Every dcpo is an ω-cpo.
where the sum is taken over all sequences ''j''<sub>1</sub>, ''j''<sub>2</sub>, ''j''<sub>3</sub>, ..., ''j''<sub>''n''−''k''+1</sub> of non-negative integers such that


An important role is played by dcpo's with a least element. They are sometimes called '''pointed dcpo'''s, or '''cppo'''s, or just '''cpo'''s.
:<math>j_1+j_2+\cdots = k\quad\mbox{and}\quad j_1+2j_2+3j_3+\cdots=n.</math>


Requiring the existence of directed suprema can be motivated by viewing directed sets as generalized approximation sequences and suprema as ''limits'' of the respective (approximative) computations. This intuition, in the context of denotational semantics, was the motivation behind the development of [[domain theory]].
==Complete Bell polynomials==


The [[duality (order theory)|dual]] notion of a directed complete poset is called a '''filtered complete partial order'''. However, this concept occurs far less frequently in practice, since one usually can work on the dual order explicitly.
The sum


== Examples ==
:<math>B_n(x_1,\dots,x_n)=\sum_{k=1}^n B_{n,k}(x_1,x_2,\dots,x_{n-k+1})</math>


* Every finite poset is directed complete.
is sometimes called the ''n''th '''complete Bell polynomial'''.  In order to contrast them with complete Bell polynomials, the polynomials ''B''<sub>''n'',&nbsp;''k''</sub> defined above are sometimes called "partial" Bell polynomials.
* All [[complete lattice]]s are also directed complete.
* For any poset, the set of all [[non-empty]] [[filter (mathematics)|filters]], ordered by subset inclusion, is a dcpo. Together with the empty filter it is also pointed. If the order has binary [[join and meet|meets]], then this construction (including the empty filter) actually yields a [[complete lattice]].
* The set of all [[partial function]]s on some given set ''S'' can be ordered by defining ''f''&nbsp;≤&nbsp;''g'' for functions ''f'' and ''g'' if and only if ''g'' extends ''f'', i.e. if the domain of ''f'' is a subset of the domain of ''g'' and the values of ''f'' and ''g'' agree on all inputs for which both functions are defined. (Equivalently, ''f''&nbsp;≤&nbsp;''g'' if and only if ''f''&nbsp;⊆&nbsp;''g'' where ''f'' and ''g'' are identified with their respective [[graph of a function|graphs]].) This order is a pointed dcpo, where the least element is the nowhere defined function (with empty domain). In fact, ≤ is also [[bounded complete]]. This example also demonstrates why it is not always natural to have a greatest element.
* The [[specialization order]] of any [[sober space]] is a dcpo.
* Let us use the term “[[deductive system]]” as a set of sentences closed under consequence (for defining notion of consequence, let us use e.g. Tarski's algebraic approach<ref name=Tar-BizIg>Tarski, Alfred: Bizonyítás és igazság / Válogatott tanulmányok. Gondolat, Budapest, 1990. (Title means: Proof and truth / Selected papers.)</ref><ref name=BurSan-UnivAlg>[http://www.math.uwaterloo.ca/~snburris/index.html Stanley N. Burris] and H.P. Sankappanavar: [http://www.math.uwaterloo.ca/~snburris/htdocs/ualg.html A Course in Universal Algebra]</ref>). There are interesting theorems which concern a set of deductive systems being a directed complete partial ordering.<ref name=seqdcpo>See online in p. 24 exercises 5–6 of §5 in [[#_note-BurSan-UnivAlg|BurSan:UnivAlg]]. Or, on paper, see [[#_note-Tar-BizIg|Tar:BizIg]].</ref> Also, a set of deductive systems can be chosen to have a least element in a natural way (so that it can be also a complete partial ordering), because the set of all consequences of the empty set (i.e. “the set of the logically provable / logically valid sentences”) is (1) a deductive system (2) contained by all deductive systems.


== Properties ==
The complete Bell polynomials satisfy the following identity


An ordered set ''P'' is a pointed dcpo if and only if every [[chain (order theory)|chain]] has a supremum in ''P''. Alternatively, an ordered set ''P'' is a pointed dcpo if and only if every [[order-preserving]] self-map of ''P'' has a least [[fixpoint]]. Every set ''S'' can be turned into a pointed dcpo by adding a least element ⊥ and introducing a flat order with ⊥&nbsp;≤&nbsp;''s'' and s&nbsp;≤&nbsp;''s'' for every ''s''&nbsp;∈&nbsp;''S'' and no other order relations.
:<math>B_n(x_1,\dots,x_n) = \det\begin{bmatrix}x_1 & {n-1 \choose 1} x_2 & {n-1 \choose 2}x_3 & {n-1 \choose 3} x_4 & {n-1 \choose 4} x_5 & \cdots & \cdots & x_n \\  \\
-1 & x_1 & {n-2 \choose 1} x_2 & {n-2 \choose 2} x_3 & {n-2 \choose 3} x_4 & \cdots & \cdots & x_{n-1} \\  \\
0 & -1 & x_1 & {n-3 \choose 1} x_2 & {n-3 \choose 2} x_3 & \cdots & \cdots & x_{n-2} \\  \\
0 & 0 & -1 & x_1 & {n-4 \choose 1} x_2 & \cdots  & \cdots & x_{n-3} \\  \\
0 & 0 & 0 & -1 & x_1 & \cdots & \cdots & x_{n-4} \\  \\
0 & 0 & 0 & 0 & -1 & \cdots & \cdots & x_{n-5} \\  \\
\vdots & \vdots & \vdots &  \vdots & \vdots & \ddots & \ddots & \vdots  \\  \\
0 & 0 & 0 & 0 & 0 & \cdots & -1 & x_1  \end{bmatrix}.</math>


== Continuous functions and fixpoints ==
==Combinatorial meaning==


A function ''f'' between two dcpos ''P'' and ''Q'' is called [[Scott continuity|'''(Scott) continuous''']] if it maps directed sets to directed sets while preserving their suprema:
If the integer ''n'' is [[integer partition|partitioned]] into a sum in which "1" appears ''j''<sub>1</sub> times, "2" appears ''j''<sub>2</sub> times, and so on, then the number of [[partition of a set|partitions of a set]] of size ''n'' that collapse to that partition of the integer ''n'' when the members of the set become indistinguishable is the corresponding coefficient in the polynomial.
* <math>f(D) \subseteq Q</math> is directed for every directed <math>D \subseteq P</math>.
* <math>f(\sup D) = \sup f(D)</math> for every directed <math>D \subseteq P</math>.
Note that every continuous function between dcpos is a [[monotone|monotone function]].
This notion of continuity is equivalent to the topological continuity induced by the [[Scott topology]].


The set of all continuous functions between two dcpos ''P'' and ''Q'' is denoted <nowiki>[</nowiki>''P''&nbsp;→&nbsp;''Q''<nowiki>]</nowiki>. Equipped with the pointwise order, this is again a dcpo, and a cpo whenever ''Q'' is a cpo.
===Examples===
Thus the complete partial orders with Scott continuous maps form a [[cartesian closed category]].<ref name="barendregt1984">[[Henk Barendregt|Barendregt, Henk]], [http://www.elsevier.com/wps/find/bookdescription.cws_home/501727/description#description ''The lambda calculus, its syntax and semantics''], [[North-Holland]] (1984)</ref>


Every order-preserving self-map ''f'' of a cpo (''P'', ⊥) has a least fixpoint.<ref>See [[Knaster–Tarski theorem]]; The foundations of program verification, 2nd edition, Jacques Loeckx and Kurt Sieber, John Wiley & Sons, ISBN 0-471-91282-4, Chapter 4; the Knaster-Tarski theorem, formulated over cpo's, is given to prove as exercise 4.3-5 on page 90.</ref> If ''f'' is continuous then this fixpoint is equal to the supremum of the [[Iterated function|iterates]] (⊥, ''f''(⊥), ''f''(''f''(⊥)), … ''f''<sup>''n''</sup>(⊥), …) of ⊥ (see also the [[Kleene fixpoint theorem]]).
For example, we have


== See also ==
:<math>B_{6,2}(x_1,x_2,x_3,x_4,x_5)=6x_5x_1+15x_4x_2+10x_3^2</math>


Directed completeness relates in various ways to other [[completeness (order theory)|completeness]] notions. Directed completeness alone is quite a basic property that occurs often in other order theoretic investigations, using for instance [[algebraic poset]]s and the [[Scott topology]].
because there are


== Notes ==
:6 ways to partition a set of 6 as 5&nbsp;+&nbsp;1,
:15 ways to partition a set of 6 as 4&nbsp;+&nbsp;2, and
:10 ways to partition a set of 6 as 3&nbsp;+&nbsp;3.


<references/>
Similarly,


== References ==
:<math>B_{6,3}(x_1,x_2,x_3,x_4)=15x_4x_1^2+60x_3x_2x_1+15x_2^3</math>
* {{cite book
 
| author = Davey, B.A., and Priestley, H. A.
because there are
  | year = 2002
 
  | title = '''Introduction to Lattices and Order'''
:15 ways to partition a set of 6 as 4&nbsp;+&nbsp;1&nbsp;+&nbsp;1,
| edition = Second
:60 ways to partition a set of 6 as 3&nbsp;+&nbsp;2&nbsp;+&nbsp;1, and
| publisher = Cambridge University Press
:15 ways to partition a set of 6 as 2&nbsp;+&nbsp;2&nbsp;+&nbsp;2.
| isbn = 0-521-78451-4
 
==Properties==
 
* <math>B_{n,k}(1!,2!,\dots,(n-k+1)!) = \binom{n}{k}\binom{n-1}{k-1} (n-k)!</math>
 
===Stirling numbers and Bell numbers===
 
The value of the Bell polynomial ''B''<sub>''n'',''k''</sub>(''x''<sub>1</sub>,''x''<sub>2</sub>,...) when all ''x''s are equal to 1 is a [[Stirling number of the second kind]]:
 
:<math>B_{n,k}(1,1,\dots)=S(n,k)=\left\{{n\atop k}\right\}.</math>
 
The sum
 
:<math>\sum_{k=1}^n B_{n,k}(1,1,1,\dots) = \sum_{k=1}^n \left\{{n\atop k}\right\}</math>
 
is the ''n''th [[Bell number]], which is the number of partitions of a set of size ''n''.
 
===Convolution identity===
 
For sequences ''x''<sub>''n''</sub>, ''y''<sub>''n''</sub>, ''n'' = 1, 2, ..., define a sort of [[convolution]] by:
 
:<math>(x \diamondsuit y)_n = \sum_{j=1}^{n-1} {n \choose j} x_j y_{n-j}</math>.
 
Note that the bounds of summation are 1 and ''n''&nbsp;−&nbsp;1, not 0 and ''n'' .
 
Let <math>x_n^{k\diamondsuit}\,</math> be the ''n''th term of the sequence
 
:<math>\displaystyle\underbrace{x\diamondsuit\cdots\diamondsuit x}_{k\ \mathrm{factors}}.\,</math>
 
Then
 
:<math>B_{n,k}(x_1,\dots,x_{n-k+1}) = {x_{n}^{k\diamondsuit} \over k!}.\,</math>
 
For example, let us compute <math> B_{4,3}(x_1,x_2) </math>. We have
 
:<math> x = ( x_1 \ , \ x_2 \ , \ x_3 \ , \ x_4 \ , \dots ) </math>
 
:<math> x \diamondsuit x = ( 0,\  2 x_1^2 \ ,\  6 x_1 x_2 \ , \  8 x_1 x_3 + 6 x_2^2 \ , \dots ) </math>
 
:<math> x \diamondsuit x \diamondsuit x = (  0 \ ,\ 0 \  , \ 6 x_1^3 \ , \ 36 x_1^2 x_2 \ , \dots ) </math>
 
and thus,
 
:<math> B_{4,3}(x_1,x_2) = \frac{ ( x \diamondsuit x \diamondsuit x)_4 }{3!} = 6 x_1^2 x_2. </math>
 
==Applications of Bell polynomials==
===Faà di Bruno's formula===
{{main|Faà di Bruno's formula}}
 
[[Faà di Bruno's formula]] may be stated in terms of Bell polynomials as follows:
 
:<math>{d^n \over dx^n} f(g(x)) = \sum_{k=0}^n f^{(k)}(g(x)) B_{n,k}\left(g'(x),g''(x),\dots,g^{(n-k+1)}(x)\right).</math>
 
Similarly, a power-series version of Faà di Bruno's formula may be stated using Bell polynomials as follows. Suppose
 
:<math>f(x)=\sum_{n=1}^\infty {a_n \over n!} x^n \qquad
\mathrm{and} \qquad g(x)=\sum_{n=1}^\infty {b_n \over n!} x^n.</math>
 
Then
 
:<math>g(f(x)) = \sum_{n=1}^\infty
{\sum_{k=1}^{n} b_k B_{n,k}(a_1,\dots,a_{n-k+1}) \over n!} x^n.</math>
 
In particular, the complete Bell polynomials appear in the exponential of a [[formal power series]]:
 
:<math>\exp\left(\sum_{n=1}^\infty {a_n \over n!} x^n \right)
= \sum_{n=0}^\infty {B_n(a_1,\dots,a_n) \over n!} x^n.</math>
 
===Moments and cumulants===
 
The sum
 
:<math>B_n(\kappa_1,\dots,\kappa_n)=\sum_{k=1}^n B_{n,k}(\kappa_1,\dots,\kappa_{n-k+1})</math>
 
is the ''n''th [[moment (mathematics)|moment]] of a [[probability distribution]] whose first ''n'' [[cumulant]]s are κ<sub>1</sub>, ..., κ<sub>''n''</sub>.  In other words, the ''n''th moment is the ''n''th complete Bell polynomial evaluated at the first ''n'' cumulants.
 
===Representation of polynomial sequences of binomial type===
 
For any sequence ''a''<sub>1</sub>, ''a''<sub>2</sub>, ''a''<sub>3</sub>, ... of scalars, let
 
:<math>p_n(x)=\sum_{k=1}^n B_{n,k}(a_1,\dots,a_{n-k+1}) x^k.</math>
 
Then this polynomial sequence is of [[binomial type]], i.e. it satisfies the binomial identity
 
:<math>p_n(x+y)=\sum_{k=0}^n {n \choose k} p_k(x) p_{n-k}(y)</math>
 
for ''n'' ≥ 0.  In fact we have this result:
 
:'''Theorem:''' All polynomial sequences of binomial type are of this form.
 
If we let
 
:<math>h(x)=\sum_{n=1}^\infty {a_n \over n!} x^n</math>
 
taking this power series to be purely formal, then for all ''n'',
 
:<math>h^{-1}\left( {d \over dx}\right) p_n(x) = n p_{n-1}(x).</math>
 
==Software==
 
* Bell polynomials, complete Bell polynomials and generalized Bell polynomials are implemented in [[Mathematica]] as [http://reference.wolfram.com/mathematica/ref/BellY.html BellY].
 
==See also==
 
* [[Bell matrix]]
* [[Exponential formula]]
 
==References==
* {{cite journal | author=[[Eric Temple Bell]] |title=Partition Polynomials | jstor=1967979 |journal=[[Annals of Mathematics]] |volume=29 |issue=1/4 |year=1927–1928 |pages=38–46 |doi=10.2307/1967979 | mr=1502817 }}
* {{cite book |author=Louis Comtet |title=Advanced Combinatorics: The Art of Finite and Infinite Expansions |publisher=Reidel Publishing Company |place=Dordrecht, Holland / Boston, U.S. |year=1974}}
* {{cite book |author=[[Steven Roman]] |title=''The Umbral Calculus'' |publisher=[[Dover Publications]]}}
* {{cite journal |author=Vassily G. Voinov, Mikhail S. Nikulin |title=On power series, Bell polynomials, Hardy-Ramanujan-Rademacher problem and its statistical applications |journal= Kybernetika | volume=30 |issue=3 |pages=343–358 |year=1994 |issn=00235954}}
* {{cite book | first=George E. | last=Andrews | authorlink=George Andrews (mathematician) | title=The Theory of Partitions | series=Cambridge Mathematical Library | publisher=[[Cambridge University Press]] | year=1998 | edition=1st pbk | isbn=0-521-63766-X | pages=204–211 }}
* {{cite journal |author=Silvia Noschese, Paolo E. Ricci |title=Differentiation of Multivariable Composite Functions and Bell Polynomials |journal=Journal of Computational Analysis and Applications | volume=5 |issue=3 |pages=333–340 |year=2003 |doi=10.1023/A:1023227705558}}
*{{ cite journal|first1=Moncef
|last1=Abbas
|first2=Sadek
|last2=Bouroubi
|title=On new identities for Bell's polynomial
|journal=Disc. Math
|year=2005
|number=293
|pages=5-10
|mr=2136048
|doi=10.1016/j.disc.2004.08.023
}}
* {{cite journal |author=Khristo N. Boyadzhiev |title=Exponential Polynomials, Stirling Numbers, and Evaluation of Some Gamma Integrals |journal=[[Abstract and Applied Analysis]] |volume=2009 |pages=Article ID 168672 |year=2009 |doi=10.1155/2009/168672}} (contains also elementary review of the concept Bell-polynomials)
* {{cite arxiv| author=V. V. Kruchinin
|year=2011|eprint=1104.5065
|title=Derivation of Bell Polynomials of the Second Kind}}
* {{cite journal
|first1=Martin
|last1=Griffiths
|title=Families of sequences from a class of multinomial sums
|journal=Journal of Integer Sequences
|url=http://www.cs.uwaterloo.ca/journals/JIS/VOL15/Griffiths/griffiths20.html
|year=2012
|mr=2872465
|volume=15
}}
}}


{{DEFAULTSORT:Complete Partial Order}}
{{DEFAULTSORT:Bell Polynomials}}
[[Category:Order theory]]
[[Category:Enumerative combinatorics]]
[[Category:Polynomials]]


[[fr:Ordre partiel complet (informatique)]]
[[fr:Polynômes de Bell]]
[[pl:Porządek zupełny]]
[[ru:Полиномы Белла]]
[[ru:Частично упорядоченное множество#Полное частично упорядоченное множество]]
[[zh:完全偏序]]

Revision as of 08:38, 12 August 2014

In combinatorial mathematics, the Bell polynomials, named in honor of Eric Temple Bell, are a triangular array of polynomials given by

Bn,k(x1,x2,…,xn−k+1)
=∑n!j1!j2!⋯jn−k+1!(x11!)j1(x22!)j2⋯(xn−k+1(n−k+1)!)jn−k+1,

where the sum is taken over all sequences j1, j2, j3, ..., jn−k+1 of non-negative integers such that

j1+j2+⋯=kandj1+2j2+3j3+⋯=n.

Complete Bell polynomials

The sum

Bn(x1,…,xn)=∑k=1nBn,k(x1,x2,…,xn−k+1)

is sometimes called the nth complete Bell polynomial. In order to contrast them with complete Bell polynomials, the polynomials Bn, k defined above are sometimes called "partial" Bell polynomials.

The complete Bell polynomials satisfy the following identity

Bn(x1,…,xn)=det⁡[x1(n−11)x2(n−12)x3(n−13)x4(n−14)x5⋯⋯xn−1x1(n−21)x2(n−22)x3(n−23)x4⋯⋯xn−10−1x1(n−31)x2(n−32)x3⋯⋯xn−200−1x1(n−41)x2⋯⋯xn−3000−1x1⋯⋯xn−40000−1⋯⋯xn−5⋮⋮⋮⋮⋮⋱⋱⋮00000⋯−1x1].

Combinatorial meaning

If the integer n is partitioned into a sum in which "1" appears j1 times, "2" appears j2 times, and so on, then the number of partitions of a set of size n that collapse to that partition of the integer n when the members of the set become indistinguishable is the corresponding coefficient in the polynomial.

Examples

For example, we have

B6,2(x1,x2,x3,x4,x5)=6x5x1+15x4x2+10x32

because there are

6 ways to partition a set of 6 as 5 + 1,
15 ways to partition a set of 6 as 4 + 2, and
10 ways to partition a set of 6 as 3 + 3.

Similarly,

B6,3(x1,x2,x3,x4)=15x4x12+60x3x2x1+15x23

because there are

15 ways to partition a set of 6 as 4 + 1 + 1,
60 ways to partition a set of 6 as 3 + 2 + 1, and
15 ways to partition a set of 6 as 2 + 2 + 2.

Properties

Stirling numbers and Bell numbers

The value of the Bell polynomial Bn,k(x1,x2,...) when all xs are equal to 1 is a Stirling number of the second kind:

Bn,k(1,1,…)=S(n,k)={nk}.

The sum

∑k=1nBn,k(1,1,1,…)=∑k=1n{nk}

is the nth Bell number, which is the number of partitions of a set of size n.

Convolution identity

For sequences xn, yn, n = 1, 2, ..., define a sort of convolution by:

(x♢y)n=∑j=1n−1(nj)xjyn−j.

Note that the bounds of summation are 1 and n − 1, not 0 and n .

Let xnk♢ be the nth term of the sequence

x♢⋯♢x⏟kfactors.

Then

Bn,k(x1,…,xn−k+1)=xnk♢k!.

For example, let us compute B4,3(x1,x2). We have

x=(x1,x2,x3,x4,…)
x♢x=(0,2x12,6x1x2,8x1x3+6x22,…)
x♢x♢x=(0,0,6x13,36x12x2,…)

and thus,

B4,3(x1,x2)=(x♢x♢x)43!=6x12x2.

Applications of Bell polynomials

Faà di Bruno's formula

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Faà di Bruno's formula may be stated in terms of Bell polynomials as follows:

dndxnf(g(x))=∑k=0nf(k)(g(x))Bn,k(g′(x),g″(x),…,g(n−k+1)(x)).

Similarly, a power-series version of Faà di Bruno's formula may be stated using Bell polynomials as follows. Suppose

f(x)=∑n=1∞ann!xnandg(x)=∑n=1∞bnn!xn.

Then

g(f(x))=∑n=1∞∑k=1nbkBn,k(a1,…,an−k+1)n!xn.

In particular, the complete Bell polynomials appear in the exponential of a formal power series:

exp⁡(∑n=1∞ann!xn)=∑n=0∞Bn(a1,…,an)n!xn.

Moments and cumulants

The sum

Bn(κ1,…,κn)=∑k=1nBn,k(κ1,…,κn−k+1)

is the nth moment of a probability distribution whose first n cumulants are κ1, ..., κn. In other words, the nth moment is the nth complete Bell polynomial evaluated at the first n cumulants.

Representation of polynomial sequences of binomial type

For any sequence a1, a2, a3, ... of scalars, let

pn(x)=∑k=1nBn,k(a1,…,an−k+1)xk.

Then this polynomial sequence is of binomial type, i.e. it satisfies the binomial identity

pn(x+y)=∑k=0n(nk)pk(x)pn−k(y)

for n ≥ 0. In fact we have this result:

Theorem: All polynomial sequences of binomial type are of this form.

If we let

h(x)=∑n=1∞ann!xn

taking this power series to be purely formal, then for all n,

h−1(ddx)pn(x)=npn−1(x).

Software

  • Bell polynomials, complete Bell polynomials and generalized Bell polynomials are implemented in Mathematica as BellY.

See also

References

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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
  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  • One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    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
  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  • One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    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
  • One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    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
  • One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    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 (contains also elementary review of the concept Bell-polynomials)
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  • One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    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

fr:Polynômes de Bell ru:Полиномы Белла

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