Minimax Condorcet: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
en>Arno Nymus
m disamb
 
Made short description at the top much clearer. The explanation below still explains in great detail.
Line 1: Line 1:
I am Sherman from Katowice. I love to play Pedal Steel Guitar. Other hobbies are Water sports.<br><br>Here is my blog [http://szhytdz.com/plus/guestbook.php how to get free fifa 15 coins]
{{Inadequate lead|date=March 2013}}
{{Use dmy dates|date=September 2011}}
The theory of '''association schemes''' arose in statistics, in the theory of [[design of experiments|experimental design]] for the [[analysis of variance]].<ref>{{harvnb|Bailey|2004|loc=pg. 387}}</ref><ref>{{harvnb|Bose|Mesner|1959}}</ref><ref>{{harvnb|Bose|Nair|1939}}</ref> In [[mathematics]], association schemes belong to both [[algebra]] and [[combinatorics]]. Indeed, in [[algebraic combinatorics]], association schemes provide a unified approach to many topics, for example [[combinatorial design]]s and [[coding theory]].<ref>{{harvnb|Bannai|Ito|1984}}</ref><ref>{{harvnb|Godsil|1993}}</ref> In algebra, association schemes generalize [[group (mathematics)|group]]s, and the theory of association schemes generalizes the [[group character|character theory]] of [[group representation|linear representations]] of groups.<ref>{{harvnb|Bailey|2004|loc=pg. 387}}</ref><ref>{{harvnb|Zieschang|2005b}}</ref><ref>{{harvnb|Zieschang|2005a}}</ref>
 
==Definition==
 
An n-class association scheme consists of a [[Set (mathematics)|set]] ''X''  together with a [[partition of a set|partition]] ''S'' of ''X'' &times; ''X'' into n + 1 [[binary relation]]s, R<sub>0</sub>, R<sub>1</sub>, ..., R<sub>n</sub>  which satisfy:
 
*<math>R_{0}=\{(x,x):x\in X\}</math> and is called the [[Identity relation]].
*Defining <math> R^* :=\{(x,y) | (y,x)\in R\}</math>, if ''R'' in ''S'', then ''R*'' in ''S''
*If <math>(x,y)\in R_{k}</math>, the number of <math>z\in X</math> such that <math>(x,z)\in R_{i}</math> and <math>(z,y)\in R_{j}</math> is a constant <math>p^k_{ij}</math> depending on <math>i</math>, <math>j</math>, <math>k</math> but not on the particular choice of <math>x</math> and <math>y</math>.
 
An association scheme is ''commutative'' if <math>p_{ij}^k=p_{ji}^k</math> for all <math>i</math>, <math>j</math> and <math>k</math>. Most authors assume this property.
 
A ''symmetric'' association scheme is one in which each relation <math>R_i</math> is a [[symmetric relation]]. That is:
 
* if (''x'',''y'')  ∈ ''R''<sub>''i''</sub>, then (''y'',''x'') ∈ ''R''<sub>''i''</sub> . (Or equivalently, ''R''* = ''R''.)
 
Every symmetric association scheme is commutative.
 
Note, however, that while the notion of an association scheme generalizes the notion of a group, the notion of a commutative association scheme only generalizes the notion of a commutative group.
 
Two points ''x'' and ''y'' are called ''i'' <sup>th</sup> associates if <math>(x,y)\in R_{i}</math>. The definition states that if ''x'' and ''y'' are ''i'' <sup>th</sup> associates so are ''y'' and ''x''. Every pair of points are ''i'' <sup>th</sup> associates for exactly one <math>i</math>. Each point is its own zeroth associate while distinct points are never zeroth associates. If ''x'' and ''y'' are ''k'' <sup>th</sup> associates then the number of points <math>z</math> which are both ''i'' <sup>th</sup> associates of <math>x</math> and ''j'' <sup>th</sup> associates of <math>y</math> is a constant <math>p^k_{ij}</math>.
 
===Graph interpretation and adjacency matrices===
 
An association scheme can be visualized as a [[complete graph]] with labeled edges. The graph has <math>v</math> vertices, one for each point of <math>X</math>, and the edge joining vertices <math>x</math> and <math>y</math> is labeled <math>i</math> if <math>x</math> and <math>y</math> are <math>i</math> <sup>th</sup> associates. Each edge has a unique label, and the number of triangles with a fixed base labeled <math>k</math> having the other edges labeled <math>i</math> and <math>j</math> is a constant <math>p^k_{ij}</math>, depending on <math>i,j,k</math> but not on the choice of the base. In particular, each vertex is incident with exactly <math>p^0_{ii}=v_{i}</math> edges labeled <math>i</math>; <math>v_{i}</math> is the [[Adjacency relation|valency]] of the [[Relation (mathematics)|relation]] <math>R_{i}</math>. There are also loops labeled <math>0</math> at each vertex <math>x</math>, corresponding to <math>R_{0}</math>.
 
The [[Relation (mathematics)|relations]] are described by their [[Adjacency matrix|adjacency matrices]]. <math>A_{i}</math> is the [[adjacency matrix]] of <math>R_{i}</math> for <math>i=0,\ldots,n</math> and is a ''v'' &times; ''v'' [[Matrix (mathematics)|matrix]] with rows and columns labeled by the points of <math>X</math>.
:<math>\left(A_{i}\right)_{x,y}=\left\{\begin{matrix}
1, & \mbox{if } \left(x,y\right)\in R_{i},\\
0, & \mbox{otherwise.}  \end{matrix}\right. \qquad(1)</math>
 
The definition of an association scheme is equivalent to saying that the <math>A_{i}</math> are ''v'' &times; ''v'' (0,1)-[[Matrix (mathematics)|matrices]] which satisfy
:I. <math>A_{i} \,</math> is symmetric,
:II. <math>\sum_{i=0}^{n}A_{i}=J</math> (the all-ones matrix),
:III. <math>A_{0}=I \,</math>,
:IV. <math>A_{i}A_{j}=\sum_{k=0}^{n} p^k_{ij}A_{k}=A_{j}A_{i}, i,j=0,\ldots,n</math>.
 
The (''x'', ''y'')-th entry of the left side of (IV) is the number of paths of length two between ''x'' and ''y'' with labels i and j in the graph. Note that the rows and columns of <math>A_{i}</math> contain <math>v_{i}</math> <math>1</math>'s:
 
<math>A_{i} J=J A_{i}=v_{i} J. \qquad(2)</math>
 
===Terminology===
*The numbers <math>p_{ij}^k</math> are called the ''parameters'' of the scheme. They are also referred to as the ''structural constants''.
 
==History==
The term ''association scheme'' is due to {{harv|Bose|Shimamoto|1952}} but the concept is already inherent in {{harv|Bose|Nair|1939}}.<ref>{{harvnb|Dembowski|1968|loc=pg. 281, footnote 1}}</ref> These authors were studying what statisticians have called ''partially balanced incomplete block designs'' (PBIBDs). The subject became an object of algebraic interest with the publication of {{harv|Bose|Mesner|1959}} and the introduction of the Bose–Mesner algebra. The most important contribution to the theory was the thesis of P. Delsarte {{harv|Delsarte|1973}} who recognized and fully used the connections with coding theory and design theory.<ref>{{harvnb|Bannai|Ito|1984|loc=pg. vii}}</ref> Generalizations have been studied by D. G. Higman (coherent configurations) and B. Weisfeiler (distance regular graphs).
 
==Basic facts==
 
*<math>p_{00}^0 = 1</math>, i.e. if <math>(x,y) \in R_0</math> then <math>x = y</math> and the only <math>z</math> such that <math>(x,z) \in R_0</math> is <math>z=x</math>
*<math>\sum_{i=0}^{k} p_{ii}^0 = |X|</math>, this is because the <math>R_i</math> partition <math>X</math>.
 
==The Bose–Mesner algebra==
 
The [[Adjacency matrix|adjacency matrices]] <math>A_i</math> of the [[Graph (mathematics)|graphs]] <math>\left(X,R_{i}\right)</math> generate a [[Commutativity|commutative]] and [[associative]] [[algebra]] <math>\mathcal{A}</math> (over the real or [[complex number]]s) both for the [[matrix product]] and the [[pointwise product]]. This [[Associative algebra|associative]], [[commutative algebra]] is called the [[Bose–Mesner algebra]] of the association scheme.
 
Since the [[Matrix (mathematics)|matrices]] in <math>\mathcal{A}</math> are [[Symmetric matrix|symmetric]] and [[Commutativity|commute]] with each other, they can be [[Diagonal matrix|diagonalized]] simultaneously. Therefore <math>\mathcal{A}</math> is [[Semisimple operator|semi-simple]] and has a unique basis of primitive [[idempotent]]s <math>J_{0},\ldots,J_{n}</math>.
 
There is another [[algebra]] of <math>\left(n+1\right)\times\left(n+1\right)</math> [[Matrix (mathematics)|matrices]] which is [[isomorphic]] to <math>\mathcal{A}</math>, and is often easier to work with.
 
==Examples==
 
*The [[Johnson scheme]], denoted ''J''(''v,k''), is defined as follows.  Let ''S'' be a set with ''v'' elements.  The points of the scheme ''J''(''v'',''k'') are the <math>{v \choose k}</math> subsets of S with ''k'' elements. Two ''k''-element subsets ''A'', ''B'' of ''S'' are ''i'' <sup>th</sup> associates when their intersection has size ''k''&nbsp;−&nbsp;''i''.
 
*The [[Hamming scheme]], denoted ''H''(''n'',''q''), is defined as follows.  The points of ''H''(''n'',''q'') are the ''q<sup>n</sup>'' ordered ''n''-[[tuple]]s over a set of size ''q''.  Two ''n''-tuples ''x'', ''y'' are said to be ''i'' <sup>th</sup> associates if they disagree in exactly ''i'' coordinates.  E.g., if ''x'' = (1,0,1,1), ''y'' = (1,1,1,1), ''z'' = (0,0,1,1), then ''x'' and ''y'' are 1st associates, ''x'' and ''z'' are 1st associates and ''y'' and ''z'' are 2nd associates in ''H(4,2)''.
 
*A [[distance-regular graph]], ''G'', forms an association scheme by defining two vertices to be ''i'' <sup>th</sup> associates if their distance is ''i''.
 
*A [[finite group]] ''G'' yields an association scheme on <math>X=G</math>, with a class ''R''<sub>''g''</sub> for each group element, as follows: for each <math>g \in G</math> let <math>R_g=\{(x,y)  |  x=g*y\}</math> where <math>*</math> is the group [[operation (mathematics)|operation]].  The class of the group identity is ''R''<sub>0</sub>.  This association scheme is commutative if and only if ''G'' is [[Abelian group|abelian]].
 
*A specific 3-class association scheme:<ref>{{harvnb|Street|Street|1987|loc=pg. 238}}</ref>
:Let ''A''(3) be the following association scheme with three associate classes on the set ''X'' = {1,2,3,4,5,6}. The (''i'',''j'') entry is ''s'' if elements ''i'' and ''j'' are in relation R<sub>s</sub>.
<center>
{| class="wikitable"
|-
! &nbsp;!! 1!! 2!! 3!! 4!! 5!! 6
|-
| '''1''' || <span style="color:white; background:blue"> &nbsp;0&nbsp;</span> || <span style="color:white; background:red"> &nbsp;1&nbsp; </span> || <span style="color:white; background:red">  &nbsp;1&nbsp; </span> || <span style="color:white; background:lime"> &nbsp;2&nbsp; </span> || <span style="color:white; background:fuchsia"> &nbsp;3&nbsp;</span> || <span style="color:white; background:fuchsia"> &nbsp;3&nbsp;</span>
|-
|'''2'''|| <span style="color:white; background:red">  &nbsp;1&nbsp; </span> || <span style="color:white; background:blue"> &nbsp;0&nbsp; </span> || <span style="color:white; background:red">  &nbsp;1&nbsp; </span> || <span style="color:white; background:fuchsia"> &nbsp;3&nbsp;</span> || <span style="color:white; background:lime"> &nbsp;2&nbsp;</span> || <span style="color:white; background:fuchsia"> &nbsp;3&nbsp;</span>
|-
| '''3''' || <span style="color:white; background:red">  &nbsp;1&nbsp; </span> || <span style="color:white; background:red">  &nbsp;1&nbsp; </span> || <span style="color:white; background:blue"> &nbsp;0&nbsp;</span> || <span style="color:white; background:fuchsia"> &nbsp;3&nbsp;</span> || <span style="color:white; background:fuchsia"> &nbsp;3&nbsp;</span> || <span style="color:white; background:lime"> &nbsp;2&nbsp;</span>
|-
| '''4''' || <span style="color:white; background:lime"> &nbsp;2&nbsp;</span> || <span style="color:white; background:fuchsia"> &nbsp;3&nbsp;</span> || <span style="color:white; background:fuchsia"> &nbsp;3&nbsp;</span> || <span style="color:white; background:blue"> &nbsp;0&nbsp;</span> || <span style="color:white; background:red">  &nbsp;1&nbsp; </span> || <span style="color:white; background:red">  &nbsp;1&nbsp; </span>
|-
| '''5''' || <span style="color:white; background:fuchsia"> &nbsp;3&nbsp;</span> || <span style="color:white; background:lime"> &nbsp;2&nbsp;</span> || <span style="color:white; background:fuchsia"> &nbsp;3&nbsp;</span> || <span style="color:white; background:red">  &nbsp;1&nbsp; </span> || <span style="color:white; background:blue">&nbsp;0&nbsp;</span> || <span style="color:white; background:red">  &nbsp;1&nbsp; </span>
|-
| '''6''' || <span style="color:white; background:fuchsia"> &nbsp;3&nbsp; </span> || <span style="color:white; background:fuchsia"> &nbsp;3&nbsp;</span> || <span style="color:white; background:lime"> &nbsp;2&nbsp;</span> || <span style="color:white; background:red"> &nbsp;1&nbsp; </span> || <span style="color:white; background:red"> &nbsp;1&nbsp; </span> || <span style="color:white; background:blue">&nbsp;0&nbsp;</span>
|}
</center>
 
==Coding theory==
 
The [[Hamming scheme]] and the [[Johnson scheme]] are of major significance in classical [[coding theory]].
 
In [[coding theory]], association scheme theory is mainly concerned with the [[Hamming distance|distance]] of a [[code]].  The [[linear programming]] method produces upper bounds for the size of a [[code]] with given minimum [[Hamming distance|distance]], and lower bounds for the size of a [[T-design|design]] with a given strength. The most specific results are obtained in the case where the underlying association scheme satisfies certain [[polynomial]] properties; this leads one into the realm of [[orthogonal polynomials]]. In particular, some universal bounds are derived for [[code]]s and [[T-design|designs]] in polynomial-type association schemes.
 
In classical [[coding theory]], dealing with [[code]]s in a [[Hamming scheme]], the MacWilliams transform involves a family of orthogonal polynomials known as the [[Krawtchouk polynomials]]. These polynomials give the [[eigenvalues]] of the distance relation [[Matrix (mathematics)|matrices]] of the [[Hamming scheme]].
 
==See also==
<!-- * [[Analysis of variance]] (Statistics) -->
* [[Block design]]
* [[Bose–Mesner algebra]]
* [[Combinatorial design]]
 
==Notes==
{{Reflist}}
 
==References==
 
* {{citation| first=Rosemary A.| last=Bailey | authorlink=Rosemary A. Bailey | url=http://www.maths.qmul.ac.uk/~rab/Asbook | title=Association Schemes: Designed Experiments, Algebra and Combinatorics|publisher=Cambridge University Press|year=2004|isbn=978-0-521-82446-0| mr=2047311}}. (Chapters from preliminary draft are [http://www.maths.qmw.ac.uk/~rab available on-line].)
 
* {{citation| last1=Bannai | first1=Eiichi | last2=Ito | first2=Tatsuro | title=Algebraic combinatorics I: Association schemes |  publisher=The Benjamin/Cummings Publishing Co., Inc. | location=Menlo Park, CA | year=1984 | pages=xxiv+425 | isbn=0-8053-0490-8 | mr=0882540 | unused_data=<!-- authorlink1=Eiichi Bannai | authorlink2= Tatsuro Ito --> }}
 
* {{citation| last1=Bose|first1=R. C.| authorlink1=R. C. Bose| last2=Mesner|first2=D. M.|year=1959|title=On linear associative algebras corresponding to association schemes of partially balanced designs|journal=[[Annals of Mathematical Statistics]]|volume=30|issue=1|pages=21–38| url=http://projecteuclid.org/euclid.aoms/1177706356 | doi=10.1214/aoms/1177706356 | mr = 102157 | jstor = 2237117}}
 
* {{citation| last1=Bose|first1=R.&nbsp;C.|authorlink1=R. C. Bose|first2= K.&nbsp;R.|last2= Nair|title= Partially balanced incomplete block designs|journal= [[Sankhya (journal)|Sankhyā]]|volume= 4|year=1939|pages= 337–372}}
 
* {{citation|last=Bose|first=R.&nbsp;C.|last2=Shimamoto|first2=T.|title=Classification and analysis of partially balanced incomplete block designs with two associate classes|journal=Journal of the American Statistical Association|year=1952|volume=47|pages=151–184|authorlink=R. C. Bose}}
 
* P. Camion (1998), Codes and Association Schemes: Basic Properties of Association Schemes Relevant to Coding, in ''Handbook of Coding Theory'', V. S. Pless and W. C. Huffman, Eds., Elsevier, The Netherlands.
 
* {{citation|last=Delsarte |first= P.|year=1973 |title= An Algebraic Approach to the Association Schemes of Coding Theory|journal=  Philips Research Reports, Supplement No. 10}}
 
* {{cite journal | last1 = Delsarte | first1 = P. | last2 = Levenshtein | first2 = V. I. | year = 1998 | title = Association schemes and coding theory | url = | journal = IEEE Transactions on Information Theory | volume = 44 | issue = 6| pages = 2477–2504 }}
 
*{{citation|last=Dembowski|first=P.|title=Finite Geometry|publisher=Springer-Verlag|location=Berlin|year=1968}}
 
* {{citation|first=C. D.| last=Godsil|authorlink = Chris Godsil|title=Algebraic Combinatorics|publisher=Chapman and Hall|year=1993|location=New York|isbn=0-412-04131-6 | mr=1220704 }}
 
* F. J. MacWilliams and N. J. A. Sloane, ''The Theory of Error-Correcting Codes'', Elsevier, New York, 1978.
 
*{{cite book
|author=Street, Anne Penfold and Street, Deborah J.
|title=Combinatorics of Experimental Design
|publisher=Oxford U. P. [Clarendon]
|year=1987
|pages=400+xiv
|isbn=0-19-853256-3
}}
 
* van Lint, J.H., and Wilson, R.M. (1992), ''A Course in Combinatorics''.  Cambridge, Eng.: Cambridge University Press.  ISBN 0-521-00601-5
 
* {{citation|doi=10.1090/S0273-0979-05-01077-3|url=http://www.ams.org/bull/2006-43-02/S0273-0979-05-01077-3/S0273-0979-05-01077-3.pdf|title=''Association Schemes: Designed Experiments, Algebra and Combinatorics'' by Rosemary A. Bailey, Review|first= Paul-Hermann|last=Zieschang|journal=Bulletin of the American Mathematical Society|volume=43|year=2005a|issue=02|pages=249–253}}
 
* {{citation| last=Zieschang | first=Paul-Hermann |  title=Theory of association schemes |  publisher=Springer |  year=2005b | pages=xii+283 | isbn=3-540-26136-2 }}
 
* {{citation | last1=Zieschang | first1=Paul-Hermann | title=The exchange condition for association schemes | doi=10.1007/BF02777367 | mr=2214129 | year=2006 | journal=Israel Journal of Mathematics | issn=0021-2172 | volume=151 | issue=3 | pages=357–380}}
 
{{Experimental design|state=collapsed}}
{{Statistics|collection|state=collapsed}}
 
[[Category:Articles with inconsistent citation formats]]
[[Category:Design of experiments]]
[[Category:Analysis of variance]]
[[Category:Algebraic combinatorics]]
[[Category:Representation theory]]
<!-- [[Category:Algebra]]
[[Category:Combinatorics]] -->

Revision as of 05:05, 6 November 2013

Template:Inadequate lead 30 year-old Entertainer or Range Artist Wesley from Drumheller, really loves vehicle, property developers properties for sale in singapore singapore and horse racing. Finds inspiration by traveling to Works of Antoni Gaudí. The theory of association schemes arose in statistics, in the theory of experimental design for the analysis of variance.[1][2][3] In mathematics, association schemes belong to both algebra and combinatorics. Indeed, in algebraic combinatorics, association schemes provide a unified approach to many topics, for example combinatorial designs and coding theory.[4][5] In algebra, association schemes generalize groups, and the theory of association schemes generalizes the character theory of linear representations of groups.[6][7][8]

Definition

An n-class association scheme consists of a set X together with a partition S of X × X into n + 1 binary relations, R0, R1, ..., Rn which satisfy:

An association scheme is commutative if pijk=pjik for all i, j and k. Most authors assume this property.

A symmetric association scheme is one in which each relation Ri is a symmetric relation. That is:

  • if (x,y) ∈ Ri, then (y,x) ∈ Ri . (Or equivalently, R* = R.)

Every symmetric association scheme is commutative.

Note, however, that while the notion of an association scheme generalizes the notion of a group, the notion of a commutative association scheme only generalizes the notion of a commutative group.

Two points x and y are called i th associates if (x,y)Ri. The definition states that if x and y are i th associates so are y and x. Every pair of points are i th associates for exactly one i. Each point is its own zeroth associate while distinct points are never zeroth associates. If x and y are k th associates then the number of points z which are both i th associates of x and j th associates of y is a constant pijk.

Graph interpretation and adjacency matrices

An association scheme can be visualized as a complete graph with labeled edges. The graph has v vertices, one for each point of X, and the edge joining vertices x and y is labeled i if x and y are i th associates. Each edge has a unique label, and the number of triangles with a fixed base labeled k having the other edges labeled i and j is a constant pijk, depending on i,j,k but not on the choice of the base. In particular, each vertex is incident with exactly pii0=vi edges labeled i; vi is the valency of the relation Ri. There are also loops labeled 0 at each vertex x, corresponding to R0.

The relations are described by their adjacency matrices. Ai is the adjacency matrix of Ri for i=0,,n and is a v × v matrix with rows and columns labeled by the points of X.

(Ai)x,y={1,if (x,y)Ri,0,otherwise.(1)

The definition of an association scheme is equivalent to saying that the Ai are v × v (0,1)-matrices which satisfy

I. Ai is symmetric,
II. i=0nAi=J (the all-ones matrix),
III. A0=I,
IV. AiAj=k=0npijkAk=AjAi,i,j=0,,n.

The (x, y)-th entry of the left side of (IV) is the number of paths of length two between x and y with labels i and j in the graph. Note that the rows and columns of Ai contain vi 1's:

AiJ=JAi=viJ.(2)

Terminology

  • The numbers pijk are called the parameters of the scheme. They are also referred to as the structural constants.

History

The term association scheme is due to Template:Harv but the concept is already inherent in Template:Harv.[9] These authors were studying what statisticians have called partially balanced incomplete block designs (PBIBDs). The subject became an object of algebraic interest with the publication of Template:Harv and the introduction of the Bose–Mesner algebra. The most important contribution to the theory was the thesis of P. Delsarte Template:Harv who recognized and fully used the connections with coding theory and design theory.[10] Generalizations have been studied by D. G. Higman (coherent configurations) and B. Weisfeiler (distance regular graphs).

Basic facts

The Bose–Mesner algebra

The adjacency matrices Ai of the graphs (X,Ri) generate a commutative and associative algebra 𝒜 (over the real or complex numbers) both for the matrix product and the pointwise product. This associative, commutative algebra is called the Bose–Mesner algebra of the association scheme.

Since the matrices in 𝒜 are symmetric and commute with each other, they can be diagonalized simultaneously. Therefore 𝒜 is semi-simple and has a unique basis of primitive idempotents J0,,Jn.

There is another algebra of (n+1)×(n+1) matrices which is isomorphic to 𝒜, and is often easier to work with.

Examples

  • The Johnson scheme, denoted J(v,k), is defined as follows. Let S be a set with v elements. The points of the scheme J(v,k) are the (vk) subsets of S with k elements. Two k-element subsets A, B of S are i th associates when their intersection has size k − i.
  • The Hamming scheme, denoted H(n,q), is defined as follows. The points of H(n,q) are the qn ordered n-tuples over a set of size q. Two n-tuples x, y are said to be i th associates if they disagree in exactly i coordinates. E.g., if x = (1,0,1,1), y = (1,1,1,1), z = (0,0,1,1), then x and y are 1st associates, x and z are 1st associates and y and z are 2nd associates in H(4,2).
  • A distance-regular graph, G, forms an association scheme by defining two vertices to be i th associates if their distance is i.
  • A finite group G yields an association scheme on X=G, with a class Rg for each group element, as follows: for each gG let Rg={(x,y)|x=g*y} where * is the group operation. The class of the group identity is R0. This association scheme is commutative if and only if G is abelian.
  • A specific 3-class association scheme:[11]
Let A(3) be the following association scheme with three associate classes on the set X = {1,2,3,4,5,6}. The (i,j) entry is s if elements i and j are in relation Rs.
  1 2 3 4 5 6
1  0   1   1   2   3   3 
2  1   0   1   3   2   3 
3  1   1   0   3   3   2 
4  2   3   3   0   1   1 
5  3   2   3   1   0   1 
6  3   3   2   1   1   0 

Coding theory

The Hamming scheme and the Johnson scheme are of major significance in classical coding theory.

In coding theory, association scheme theory is mainly concerned with the distance of a code. The linear programming method produces upper bounds for the size of a code with given minimum distance, and lower bounds for the size of a design with a given strength. The most specific results are obtained in the case where the underlying association scheme satisfies certain polynomial properties; this leads one into the realm of orthogonal polynomials. In particular, some universal bounds are derived for codes and designs in polynomial-type association schemes.

In classical coding theory, dealing with codes in a Hamming scheme, the MacWilliams transform involves a family of orthogonal polynomials known as the Krawtchouk polynomials. These polynomials give the eigenvalues of the distance relation matrices of the Hamming scheme.

See also

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010. (Chapters from preliminary draft are available on-line.)
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • P. Camion (1998), Codes and Association Schemes: Basic Properties of Association Schemes Relevant to Coding, in Handbook of Coding Theory, V. S. Pless and W. C. Huffman, Eds., Elsevier, The Netherlands.
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • 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
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier, New York, 1978.
  • 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
  • van Lint, J.H., and Wilson, R.M. (1992), A Course in Combinatorics. Cambridge, Eng.: Cambridge University Press. ISBN 0-521-00601-5
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

Template:Experimental design Template:Statistics