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In combinatorial mathematics, a block design is a set together with a family of subsets (repeated subsets are allowed at times) whose members are chosen to satisfy some set of properties that are deemed useful for a particular application. These applications come from many areas, including experimental design, finite geometry, software testing, cryptography, and algebraic geometry. Many variations have been examined, but the most intensely studied are the balanced incomplete block designs (BIBDs or 2-designs) which historically were related to statistical issues in the design of experiments.^[1]^[2]

A block design in which all the blocks have the same size is called uniform. The designs discussed in this article are all uniform. Pairwise balanced designs (PBDs) are examples of block designs that are not necessarily uniform.

Definition of a BIBD (or 2-design)

Given a finite set X (of elements called points) and integers k, r, λ ≥ 1, we define a 2-design (or BIBD, standing for balanced incomplete block design) B to be a family of k-element subsets of X, called blocks, such that the number r of blocks containing x in X is not dependent on which x is chosen, and the number λ of blocks containing given distinct points x and y in X is also independent of the choices.

"Family" in the above definition can be replaced by "set" if repeated blocks are not allowed. Designs in which repeated blocks are not allowed are called simple.

Here v (the number of elements of X, called points), b (the number of blocks), k, r, and λ are the parameters of the design. (To avoid degenerate examples, it is also assumed that v > k, so that no block contains all the elements of the set. This is the meaning of "incomplete" in the name of these designs.) In a table:

v	points, number of elements of X
b	blocks
r	number of blocks containing a given point
k	number of points in a block
λ	number of blocks containing 2 (or more generally t) points

The design is called a (v, k, λ)-design or a (v, b, r, k, λ)-design. The parameters are not all independent; v, k, and λ determine b and r, and not all combinations of v, k, and λ are possible. The two basic equations connecting these parameters are

b k = v r,

λ (v - 1) = r (k - 1) .

These conditions are not sufficient as, for example, a (43,7,1)-design does not exist.^[3]

The order of a 2-design is defined to be n = r − λ. The complement of a 2-design is obtained by replacing each block with its complement in the point set X. It is also a 2-design and has parameters v′ = v, b′ = b, r′ = b−r, k′ = v−k, λ′ = λ+b−2r. A 2-design and its complement have the same order.

A fundamental theorem, Fisher's inequality, named after the statistician Ronald Fisher, is that b ≥ v in any 2-design.

Symmetric BIBDs

The case of equality in Fisher's inequality, that is, a 2-design with an equal number of points and blocks, is called a symmetric design.^[4] Symmetric designs have the smallest number of blocks amongst all the 2-designs with the same number of points.

In a symmetric design r = k holds as well as b = v, and, while it is generally not true in arbitrary 2-designs, in a symmetric design every two distinct blocks meet in λ points.^[5] A theorem of Ryser provides the converse. If X is a v-element set, and B is a v-element set of k-element subsets (the "blocks"), such that any two distinct blocks have exactly λ points in common, then (X, B) is a symmetric block design.^[6]

The parameters of a symmetric design satisfy

λ (v - 1) = k (k - 1) .

This imposes strong restrictions on v, so the number of points is far from arbitrary. The Bruck–Ryser–Chowla theorem gives necessary, but not sufficient, conditions for the existence of a symmetric design in terms of these parameters.

The following are important examples of symmetric 2-designs:

Projective planes

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Finite projective planes are symmetric 2-designs with λ = 1 and order n > 1. For these designs the symmetric design equation becomes:

v - 1 = k (k - 1) .

Since k = r we can write the order of a projective plane as n = k − 1 and, from the displayed equation above, we obtain v = (n + 1)n + 1 = n² + n + 1 points in a projective plane of order n.

As a projective plane is a symmetric design, we have b = v, meaning that b = n² + n + 1 also. The number b is the number of lines of the projective plane. There can be no repeated lines since λ = 1, so a projective plane is a simple 2-design in which the number of lines and the number of points are always the same. For a projective plane, k is the number of points on each line and it is equal to n + 1. Similarly, r = n + 1 is the number of lines with which a given point is incident.

For n = 2 we get a projective plane of order 2, also called the Fano plane, with v = 4 + 2 + 1 = 7 points and 7 lines. In the Fano plane, each line has n + 1 = 3 points and each point belongs to n + 1 = 3 lines.

Projective planes are known to exist for all orders which are prime numbers or powers of primes. They form the only known infinite family (with respect to having a constant λ value) of symmetric block designs.^[7]

Biplanes

A biplane or biplane geometry is a symmetric 2-design with λ = 2; that is, every set of two points is contained in two blocks ("lines"), while any two lines intersect in two points.^[7] They are similar to finite projective planes, except that rather than two points determining one line (and two lines determining one point), two points determine two lines (respectively, points). A biplane of order n is one whose blocks have k = n + 2 points; it has v = 1 + (n + 2)(n + 1)/2 points (since r = k).

The 18 known examples^[8] are listed below.

(Trivial) The order 0 biplane has 2 points (and lines of size 2; a 2-(2,2,2) design); it is two points, with two blocks, each consisting of both points. Geometrically, it is the digon.
The order 1 biplane has 4 points (and lines of size 3; a 2-(4,3,2) design); it is the complete design with v = 4 and k = 3. Geometrically, the points are the vertices and the blocks are the faces of a tetrahedron.
The order 2 biplane is the complement of the Fano plane: it has 7 points (and lines of size 4; a 2-(7,4,2)), where the lines are given as the complements of the (3-point) lines in the Fano plane.^[9]
The order 3 biplane has 11 points (and lines of size 5; a 2-(11,5,2)), and is also known as the Template:Visible anchor after Raymond Paley; it is associated to the Paley digraph of order 11, which is constructed using the field with 11 elements, and is the Hadamard 2-design associated to the size 12 Hadamard matrix; see Paley construction I.

Algebraically this corresponds to the exceptional embedding of the projective special linear group PSL(2,5) in PSL(2,11) – see projective linear group: action on p points for details.^[10]

There are three biplanes of order 4 (and 16 points, lines of size 6; a 2-(16,6,2)). These three designs are also Menon designs.
There are four biplanes of order 7 (and 37 points, lines of size 9; a 2-(37,9,2)).^[11]
There are five biplanes of order 9 (and 56 points, lines of size 11; a 2-(56,11,2)).^[12]
Two biplanes are known of order 11 (and 79 points, lines of size 13; a 2-(79,13,2)).^[13]

Hadamard 2-designs

An Hadamard matrix of size m is an m × m matrix H whose entries are ±1 such that HH^⊤ = mI_m, where H^⊤ is the transpose of H and I_m is the m × m identity matrix. An Hadamard matrix can be put into standardized form (that is, converted to an equivalent Hadamard matrix) where the first row and first column entries are all +1. If the size m > 2 then m must be a multiple of 4.

Given an Hadamard matrix of size 4a in standardized form, remove the first row and first column and convert every -1 to a 0. The resulting 0-1 matrix M is the incidence matrix of a symmetric 2-(4a-1, 2a-1, a-1) design called an Hadamard 2-design.^[14] This construction is reversible, and the incidence matrix of a symmetric 2-design with these parameters can be used to form an Hadamard matrix of size 4a.

Resolvable 2-designs

A resolvable 2-design is a BIBD whose blocks can be partitioned into sets (called parallel classes), each of which forms a partition of the point set of the BIBD. The set of parallel classes is called a resolution of the design.

If a 2-(v,k,λ) resolvable design has c parallel classes, then b ≥ v + c - 1.^[15]

Consequently, a symmetric design can not have a non-trivial (more than one parallel class) resolution.^[16]

Archetypical resolvable 2-designs are the finite affine planes. A solution of the famous 15 schoolgirl problem is a resolution of a 2-(15,3,1) design.^[17]

Generalization: t-designs

Given any positive integer t, a t-design B is a class of k-element subsets of X, called blocks, such that every point x in X appears in exactly r blocks, and every t-element subset T appears in exactly λ blocks. The numbers v (the number of elements of X), b (the number of blocks), k, r, λ, and t are the parameters of the design. The design may be called a t-(v,k,λ)-design. Again, these four numbers determine b and r and the four numbers themselves cannot be chosen arbitrarily. The equations are

λ_{i} = λ (\binom{v - i}{t - i}) / (\binom{k - i}{t - i}) for i = 0, 1, \dots, t,

where λ_i is the number of blocks that contain any i-element set of points.

Theorem:^[18] Any t-(v,k,λ)-design is also an s-(v,k,λ_s)-design for any s with 1 ≤ s ≤ t. (Note that the "lambda value" changes as above and depends on s.)

A consequence of this theorem is that every t-design with t ≥ 2 is also a 2-design.

There are no known examples of non-trivial t-(v,k,1)-designs with $t > 5$ .

The term block design by itself usually means a 2-design.

Derived and extendable t-designs

Let D = (X, B) be a t-(v,k,λ) design and p a point of X. The derived design D_p has point set X − {p} and as block set all the blocks of D which contain p with p removed. It is a (t − 1)-(v − 1, k − 1, λ) design. Note that derived designs with respect to different points may not be isomorphic. A design E is called an extension of D if E has a point p such that E_p is isomorphic to D; we call D extendable if it has an extension.

Theorem:^[19] If a t-(v,k,λ) design has an extension, then k + 1 divides b(v + 1).

The only extendable projective planes (symmetric 2-(n² + n + 1, n + 1, 1) designs) are those of orders 2 and 4.^[20]

Every Hadamard 2-design is extendable (to an Hadamard 3-design).^[21]

Theorem:.^[22] If D, a symmetric 2-(v,k,λ) design, is extendable, then one of the following holds:

D is an Hadamard 2-design,
v = (λ + 2)(λ² + 4λ + 2), k = λ² + 3λ + 1,
v = 495, k = 39, λ = 3.

Note that the projective plane of order two is an Hadamard 2-design; the projective plane of order four has parameters which fall in case 2; the only other known symmetric 2-designs with parameters in case 2 are the order 9 biplanes, but none of them are extendable; and there is no known symmetric 2-design with the parameters of case 3.^[23]

Inversive planes

A design with the parameters of the extension of an affine plane, i.e., a 3-(n² + 1, n + 1, 1) design, is called a finite inversive plane, or Möbius plane, of order n.

It is possible to give a geometric description of some inversive planes, indeed, of all known inversive planes. An ovoid in PG(3,q) is a set of q² + 1 points, no three collinear. It can be shown that every plane (which is a hyperplane since the geometric dimension is 3) of PG(3,q) meets an ovoid O in either 1 or q + 1 points. The plane sections of size q + 1 of O are the blocks of an inversive plane of order q. Any inversive plane arising this way is called egglike. All known inversive planes are egglike.

An example of an ovoid is the elliptic quadric, the set of zeros of the quadratic form

x₁x₂ + f(x₃, x₄),

where f is an irreducible quadratic form in two variables over GF(q). [f(x,y) = x² + xy + y² for example].

If q is an odd power of 2, another type of ovoid is known – the Suzuki-Tits ovoid.

Theorem. Let q be a positive integer, at least 2. (a) If q is odd, then any ovoid is projectively equivalent to the elliptic quadric in a projective geometry PG(3,q); so q is a prime power and there is a unique egglike inversive plane of order q. (But it is unknown if non-egglike ones exist.) (b) if q is even, then q is a power of 2 and any inversive plane of order q is egglike (but there may be some unknown ovoids).

Steiner systems

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A Steiner system (named after Jakob Steiner) is a t-design with λ = 1 and t ≥ 2.

A Steiner system with parameters t, k, n, written S(t,k,n), is an n-element set S together with a set of k-element subsets of S (called blocks) with the property that each t-element subset of S is contained in exactly one block. In the general notation for block designs, an S(t,k,n) would be a t-(n,k,1) design.

This definition is relatively modern, generalizing the classical definition of Steiner systems which in addition required that k = t + 1. An S(2,3,n) was (and still is) called a Steiner triple system, while an S(3,4,n) was called a Steiner quadruple system, and so on. With the generalization of the definition, this naming system is no longer strictly adhered to.

Projective planes and affine planes are examples of Steiner systems under the current definition while only the Fano plane (projective plane of order 2) would have been a Steiner system under the older definition.

Partially balanced designs (PBIBDs)

An n-class association scheme consists of a set X of size v together with a partition S of X × X into n + 1 binary relations, R₀, R₁, ..., R_n. A pair of elements in relation R_i are said to be i^th–associates. Each element of X has n_i i^th associates. Furthermore:

$R_{0} = {(x, x) : x \in X}$ and is called the Identity relation.
Defining $R^{*} : = {(x, y) | (y, x) \in R}$ , if R in S, then R* in S
If $(x, y) \in R_{k}$ , the number of $z \in X$ such that $(x, z) \in R_{i}$ and $(z, y) \in R_{j}$ is a constant $p_{i j}^{k}$ depending on i, j, k but not on the particular choice of x and y.

An association scheme is commutative if $p_{i j}^{k} = p_{j i}^{k}$ for all i, j and k. Most authors assume this property.

A partially balanced incomplete block design with n associate classes (PBIBD(n)) is a block design based on a v-set X with b blocks each of size k and with each element appearing in r blocks, such that there is an association scheme with n classes defined on X where, if elements x and y are i^th associates, 1 ≤ i ≤ n, then they are together in precisely λ_i blocks.

A PBIBD(n) determines an association scheme but the converse is false.^[24]

Example

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_s.

	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

The blocks of a PBIBD(3) based on A(3) are:

124	134	235	456
125	136	236	456

The parameters of this PBIBD(3) are: v = 6, b = 8, k = 3, r = 4 and λ₁ = λ₂ = 2 and λ₃ = 1. Also, for the association scheme we have n₀ = n₂ = 1 and n₁ = n₃ = 2.^[25]

Properties

The parameters of a PBIBD(m) satisfy:^[26]

A PBIBD(1) is a BIBD and a PBIBD(2) in which λ₁ = λ₂ is a BIBD.^[27]

Two associate class PBIBDs

PBIBD(2)s have been the studied the most since they are the simplest and most useful of the PBIBDs.^[28] They fall into six types^[29] based on a classification of the then known PBIBD(2)s by Template:Harvtxt:^[30]

group divisible;
triangular;
Latin square type;
cyclic;
partial geometry type;
miscellaneous.

Applications

The mathematical subject of block designs originated in the statistical framework of design of experiments. These designs were especially useful in applications of the technique of analysis of variance (ANOVA). This remains a significant area for the use of block designs.

While the origins of the subject are grounded in biological applications (as is some of the existing terminology), the designs are used in many applications where systematic comparisons are being made, such as in software testing.

The incidence matrix of block designs provide a natural source of interesting block codes that are used as error correcting codes.

Notes

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References

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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.

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R. C. Bose, "A Note on Fisher's Inequality for Balanced Incomplete Block Designs", Annals of Mathematical Statistics, 1949, pages 619–620.

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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

R. A. Fisher, "An examination of the different possible solutions of a problem in incomplete blocks", Annals of Eugenics, volume 10, 1940, pages 52–75.

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

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

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

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

S. S. Shrikhande, and Vasanti N. Bhat-Nayak, Non-isomorphic solutions of some balanced incomplete block designs I – Journal of Combinatorial Theory, 1970

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

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

External links

DesignTheory.Org: Databases of combinatorial, statistical, and experimental block designs. Software and other resources hosted by the School of Mathematical Sciences at Queen Mary College, University of London.
Design Theory Resources: Peter Cameron's page of web based design theory resources.

22 year-old Systems Analyst Rave from Merrickville-Wolford, has lots of hobbies and interests including quick cars, property developers in singapore and baking. Always loves visiting spots like Historic Monuments Zone of Querétaro.

Here is my web site - cottagehillchurch.com

Template:Experimental design

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↑ Proved by Tarry in 1900 who showed that there was no pair of orthogonal Latin squares of order six. The 2-design with the indicate parameters is equivalent to the existence of five mutually orthogonal Latin squares of order six.
↑ They have also been referred to as projective designs or square designs. These alternatives have been used in an attempt to replace the term "symmetric", since there is nothing symmetric (in the usual meaning of the term) about these designs. The use of projective is due to P.Dembowski (Finite Geometries, Springer, 1968), in analogy with the most common example, projective planes, while square is due to P. Cameron (Designs, Graphs, Codes and their Links, Cambridge, 1991) and captures the implication of v = b on the incidence matrix. Neither term has caught on as a replacement and these designs are still universally referred to as symmetric.
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↑ 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:Harvnb
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[3] Proved by Tarry in 1900 who showed that there was no pair of orthogonal Latin squares of order six. The 2-design with the indicate parameters is equivalent to the existence of five mutually orthogonal Latin squares of order six.

[4] They have also been referred to as projective designs or square designs. These alternatives have been used in an attempt to replace the term "symmetric", since there is nothing symmetric (in the usual meaning of the term) about these designs. The use of projective is due to P.Dembowski (Finite Geometries, Springer, 1968), in analogy with the most common example, projective planes, while square is due to P. Cameron (Designs, Graphs, Codes and their Links, Cambridge, 1991) and captures the implication of v = b on the incidence matrix. Neither term has caught on as a replacement and these designs are still universally referred to as symmetric.

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[martinsingerman-10] 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

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[29] Not a mathematical classification since one of the types is a catch-all "and everything else".

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Jensen's alpha: Difference between revisions

Revision as of 18:52, 26 December 2013

Contents

Definition of a BIBD (or 2-design)

Symmetric BIBDs

Projective planes

Biplanes

Hadamard 2-designs

Resolvable 2-designs

Generalization: t-designs

Derived and extendable t-designs

Inversive planes

Steiner systems

Partially balanced designs (PBIBDs)

Example

Properties

Two associate class PBIBDs

Applications

See also

Notes

References

External links

Navigation menu

@@ Line 1: / Line 1: @@
-Hello and welcome. My title is Numbers Wunder. To do aerobics is a thing that I'm completely addicted to. Puerto Rico is exactly where he's been residing for many years and he will by no means move. Bookkeeping is what I do.<br><br>Feel free to visit my weblog home std test kit [[http://drive.ilovetheory.com/content/find-out-about-infections-and-preventing-them click here for more info]]
+{{about|block designs with fixed block size (uniform)|block designs with variable block sizes|Combinatorial design|[[design of experiments|experimental design]]s in [[statistics]]|randomized block design}}
+In [[combinatorics|combinatorial]] [[mathematics]],  a '''block design''' is a set together with a [[Family of sets|family of subsets]] (repeated subsets are allowed at times) whose members are chosen to satisfy some set of properties that are deemed useful for a particular application. These applications come from many areas, including [[experimental design]], [[finite geometry]], [[software testing]], [[cryptography]], and [[algebraic geometry]].  Many variations have been examined, but the most intensely studied are the '''balanced incomplete block designs''' (BIBDs or 2-designs) which historically were related to statistical issues in the [[design of experiments]].<ref>{{harvnb|Colbourn|Dinitz|2007}}</ref><ref>{{harvnb|Stinson|2003|loc=pg.1}}</ref>
+A block design in which all the blocks have the same size is called ''uniform''. The designs discussed in this article are all uniform.  [[Combinatorial design#pairwise balanced design|Pairwise balanced designs (PBDs)]] are examples of block designs that are not necessarily uniform.
+==Definition of a BIBD (or 2-design)==
+Given a finite set ''X'' (of elements called points) and integers ''k'', ''r'', ''λ'' ≥ 1, we define a ''2-design'' (or ''BIBD'', standing for balanced incomplete block design) ''B'' to be a family of ''k''-element subsets of ''X'', called ''blocks'', such that the number ''r'' of blocks containing ''x'' in ''X'' is not dependent on which ''x'' is chosen, and the number ''λ'' of blocks containing given distinct points ''x'' and ''y'' in ''X'' is also independent of the choices.
+"Family" in the above definition can be replaced by "set" if repeated blocks are not allowed. Designs in which repeated blocks are not allowed are called ''simple''.
+Here ''v'' (the number of elements of ''X'', called points), ''b'' (the number of blocks), ''k'', ''r'', and λ are the ''parameters'' of the design.  (To avoid degenerate examples, it is also assumed that ''v'' > ''k'', so that no block contains all the elements of the set. This is the meaning of "incomplete" in the name of these designs.) In a table:
+:{| class="wikitable"
+| ''v'' || points, number of elements of ''X''
+|-
+| ''b'' || blocks
+|-
+| ''r'' || number of blocks containing a given point
+|-
+| ''k'' || number of points in a block
+|-
+| ''λ'' || number of blocks containing 2 (or more generally ''t'') points
+|-
+|}
+The design is called a (''v'', ''k'', ''λ'')-design or a (''v'', ''b'', ''r'', ''k'', ''λ'')-design.  The parameters are not all independent; ''v'', ''k'', and λ determine ''b'' and ''r'', and not all combinations of ''v'', ''k'', and ''λ'' are possible.  The two basic equations connecting these parameters are
+:<math> bk = vr, \,</math>
+:<math> \lambda(v-1) = r(k-1). \,</math>
+These conditions are not sufficient as, for example, a (43,7,1)-design  does not exist.<ref>Proved by Tarry in 1900 who showed that there was no pair of orthogonal [[Latin square]]s of order six. The 2-design with the indicate parameters is equivalent to the existence of five mutually orthogonal Latin squares of order six.</ref>
+The ''order'' of a 2-design is defined to be ''n'' = ''r''&nbsp;&minus;&nbsp;''λ''.  The '''complement''' of a 2-design is obtained by replacing each block with its complement in the point set ''X''.  It is also a 2-design and has parameters ''v''′ = ''v'', ''b''′ = ''b'', ''r''′ = ''b''−''r'', ''k''′ = ''v''−''k'', ''λ''′ = ''λ''+''b''−2''r''.  A 2-design and its complement have the same order.
+A fundamental theorem, [[Fisher's inequality]], named after the statistician [[Ronald Fisher]], is that ''b''&nbsp;≥&nbsp;''v'' in any 2-design.
+==Symmetric BIBDs==
+The case of equality in Fisher's inequality, that is, a 2-design with an equal number of points and blocks, is called a '''symmetric design'''.<ref>They have also been referred to as ''projective designs'' or ''square designs''. These alternatives have been used in an attempt to replace the term "symmetric", since there is nothing symmetric (in the usual meaning of the term) about these designs. The use of ''projective'' is due to P.Dembowski (''Finite Geometries'', Springer, 1968), in analogy with the most common example, projective planes, while ''square'' is due to P. Cameron (''Designs, Graphs, Codes and their Links'', Cambridge, 1991) and captures the implication of v = b on the incidence matrix. Neither term has caught on as a replacement and these designs are still universally referred to as ''symmetric''.</ref> Symmetric designs have the smallest number of blocks amongst all the 2-designs with the same number of points.
+In a symmetric design ''r'' = ''k'' holds as well as ''b'' = ''v'', and, while it is generally not true in arbitrary 2-designs, in a symmetric design every two distinct blocks meet in ''λ'' points.<ref>{{harvnb|Stinson|2003|loc=pg.23, Theorem 2.2}}</ref> A theorem of [[Herbert Ryser|Ryser]] provides the converse.  If ''X'' is a ''v''-element set, and ''B'' is a ''v''-element set of ''k''-element subsets (the "blocks"), such that any two distinct blocks have exactly λ points in common, then (''X, B'') is a symmetric block design.<ref>{{harvnb|Ryser|1963|loc = pp. 102-104}}</ref>
+The parameters of a symmetric design satisfy
+::<math> \lambda (v-1) = k(k-1). </math>
+This imposes strong restrictions on ''v'', so the number of points is far from arbitrary.  The [[Bruck–Ryser–Chowla theorem]] gives necessary, but not sufficient, conditions for the existence of a symmetric design in terms of these parameters.
+The following are important examples of symmetric 2-designs:
+===Projective planes===
+{{main|Projective plane}}
+[[Projective plane#Finite projective planes|Finite projective planes]] are symmetric 2-designs with ''λ'' = 1 and order ''n'' > 1. For these designs the symmetric design equation becomes:
+::<math>v-1 = k(k-1).\,</math>
+Since  ''k'' = ''r'' we can write the ''order of a projective plane'' as ''n'' = ''k''&nbsp;&minus;&nbsp;1 and, from the displayed equation above, we obtain ''v'' = (''n''&nbsp;+&nbsp;1)''n'' &nbsp;+&nbsp; 1 = ''n''<sup>2</sup> &nbsp;+&nbsp; ''n'' &nbsp;+&nbsp; 1 points in a projective plane of order ''n''.
+As a projective plane is a symmetric design, we have ''b'' = ''v'', meaning that ''b'' = ''n''<sup>2</sup> &nbsp;+&nbsp; ''n'' &nbsp;+&nbsp; 1 also. The number ''b'' is the number of ''lines'' of the projective plane. There can be no repeated lines since λ&nbsp;=&nbsp;1, so a projective plane is a simple 2-design in which the number of lines and the number of points are always the same. For a projective plane, ''k'' is the number of points on each line and it is equal to ''n''&nbsp;+&nbsp;1.  Similarly, ''r'' = ''n''&nbsp;+&nbsp;1 is the number of lines with which a given point is incident.
+For ''n'' = 2 we get a projective plane of order 2, also called the [[Fano plane]], with ''v'' = 4 + 2 + 1 = 7 points and 7 lines. In the Fano plane, each line has ''n''&nbsp;+&nbsp;1 = 3 points and each point belongs to ''n''&nbsp;+&nbsp;1&nbsp;=&nbsp;3 lines.
+Projective planes are known to exist for all orders which are prime numbers or powers of primes. They form the only known infinite family (with respect to having a constant λ value) of symmetric block designs.<ref name="Hughes 1985 loc=pg.109">{{harvnb|Hughes|Piper|1985|loc=pg.109}}</ref>
+===Biplanes===
+A '''biplane''' or '''biplane geometry''' is a symmetric 2-design with ''λ'' = 2; that is, every set of two points is contained in two blocks ("lines"), while any two lines intersect in two points.<ref name="Hughes 1985 loc=pg.109">{{harvnb|Hughes|Piper|1985|loc=pg.109}}</ref> They are similar to  finite projective planes, except that rather than two points determining one line (and two lines determining one point), two points determine two lines (respectively, points). A biplane of order ''n'' is one whose blocks have ''k''&nbsp;=&nbsp;''n''&nbsp;+&nbsp;2 points; it has ''v''&nbsp;=&nbsp;1&nbsp;+&nbsp;(''n''&nbsp;+&nbsp;2)(''n''&nbsp;+&nbsp;1)/2 points (since ''r''&nbsp;=&nbsp;''k'').
+The 18 known examples<ref>{{harvnb|Hall|1986|loc=pp.320-335}}</ref> are listed below.
+* (Trivial) The order 0 biplane has 2 points (and lines of size 2;  a 2-(2,2,2) design); it is two points, with two blocks, each consisting of both points. Geometrically, it is the [[digon]].
+* The order 1 biplane has 4 points (and lines of size 3; a 2-(4,3,2) design); it is the complete design with ''v'' = 4 and ''k'' = 3. Geometrically, the points are the vertices and the blocks are the faces of a tetrahedron.
+* The order 2 biplane is the complement of the [[Fano plane]]: it has 7 points (and lines of size 4; a 2-(7,4,2)), where the lines are given as the ''complements'' of the (3-point) lines in the Fano plane.<ref>{{harvnb|Assmus|Key|1992|loc=pg.55}}</ref>
+* The order 3 biplane has 11 points (and lines of size 5; a 2-(11,5,2)), and is also known as the '''{{visible anchor|Paley biplane}}''' after [[Raymond Paley]]; it is associated to the [[Paley digraph]] of order 11, which is constructed using the field with 11 elements, and is the [[#Hadamard 2-designs|Hadamard 2-design]] associated to the size 12 Hadamard matrix; see [[Paley construction#Paley construction I|Paley construction I]].
+:Algebraically this corresponds to the exceptional embedding of the [[projective special linear group]] ''PSL''(2,5) in ''PSL''(2,11) – see [[Projective linear group#Action on p points|projective linear group: action on ''p'' points]] for details.<ref name="martinsingerman">{{citation | title = From Biplanes to the Klein quartic and the Buckyball | first1 = Pablo | last1 = Martin | first2 = David | last2 = Singerman | date = April 17, 2008 | url = http://www.neverendingbooks.org/DATA/biplanesingerman.pdf | page = 4}}</ref>
+* There are three biplanes of order 4 (and 16 points, lines of size 6; a 2-(16,6,2)).  These three designs are also [[Regular Hadamard matrix|Menon designs]].
+* There are four biplanes of order 7 (and 37 points, lines of size 9; a 2-(37,9,2)).<ref>{{harvnb|Salwach|Mezzaroba|1978}}</ref>
+* There are five biplanes of order 9 (and 56 points, lines of size 11; a 2-(56,11,2)).<ref>{{harvnb|Kaski|Östergård|2008}}</ref>
+* Two biplanes are known of order 11 (and 79 points, lines of size 13; a 2-(79,13,2)).<ref>{{harvnb|Aschbacher|1971|loc=pp. 279–281}}</ref>
+===Hadamard 2-designs===
+An [[Hadamard matrix]] of size ''m'' is an ''m'' × ''m'' matrix '''H''' whose entries are ±1 such that '''HH'''<sup>⊤</sup> &nbsp;=&nbsp;m'''I'''<sub>m</sub>, where '''H'''<sup>⊤</sup> is the transpose of '''H''' and '''I'''<sub>m</sub> is the ''m'' × ''m'' identity matrix. An Hadamard matrix can be put into ''standardized form'' (that is, converted to an equivalent Hadamard matrix) where the first row and first column entries are all +1. If the size ''m'' > 2 then ''m'' must be a multiple of 4.
+Given an Hadamard matrix of size 4''a'' in standardized form, remove the first row and first column and convert every -1 to a 0. The resulting 0-1 matrix '''M''' is the [[incidence matrix]] of a symmetric 2-(4''a''-1, 2''a''-1, ''a''-1) design called an '''Hadamard 2-design'''.<ref>{{harvnb|Stinson|2003|loc=pg. 74, Theorem 4.5}}</ref> This construction is reversible, and the incidence matrix of a symmetric 2-design with these parameters can be used to form an Hadamard matrix of size 4''a''.
+==Resolvable 2-designs==
+A '''resolvable 2-design''' is a BIBD whose blocks can be partitioned into sets (called ''parallel classes''), each of which forms a partition of the point set of the BIBD. The set of parallel classes is called a ''resolution'' of the design.
+If a 2-(''v'',''k'',λ) resolvable design has ''c'' parallel classes, then ''b'' &nbsp;≥&nbsp;''v''&nbsp;+&nbsp;''c''&nbsp;-&nbsp;1.<ref>{{harvnb|Hughes|Piper|1985|loc=pg. 156, Theorem 5.4}}</ref>
+Consequently, a symmetric design can not have a non-trivial (more than one parallel class) resolution.<ref>{{harvnb|Hughes|Piper|1985|loc=pg. 158, Corollary 5.5}}</ref>
+Archetypical resolvable 2-designs are the finite [[projective plane#Affine planes|affine planes]].  A solution of the famous [[15 schoolgirl problem]] is a resolution of a 2-(15,3,1) design.<ref>{{harvnb|Beth|Jungnickel|Lenz|1986|loc=pg. 40 Example 5.8}}</ref>
+==Generalization: ''t''-designs==
+Given any positive integer ''t'', a ''t''-design ''B'' is a class of ''k''-element subsets of ''X'', called ''blocks'', such that every point ''x'' in ''X'' appears in exactly ''r'' blocks, and every ''t''-element subset ''T'' appears in exactly λ blocks.  The numbers ''v'' (the number of elements of ''X''), ''b'' (the number of blocks), ''k'', ''r'', λ, and ''t'' are the ''parameters'' of the design.  The design may be called a ''t''-(''v'',''k'',λ)-design.  Again, these four numbers determine ''b'' and ''r'' and the four numbers themselves cannot be chosen arbitrarily.  The equations are
+:<math> \lambda_i = \lambda \left.\binom{v-i}{t-i} \right/ \binom{k-i}{t-i} \text{ for } i = 0,1,\ldots,t, </math>
+where ''λ<sub>i</sub>'' is the number of blocks that contain any ''i''-element set of points.
+'''Theorem''':<ref>{{harvnb|Stinson|2003|loc=pg.203, Corollary 9.6}}</ref> Any ''t''-(''v'',''k'',λ)-design is also an ''s''-(''v'',''k'',λ<sub>s</sub>)-design for any ''s'' with 1 ≤ ''s'' ≤ ''t''. (Note that the "lambda value" changes as above and depends on ''s''.)
+A consequence of this theorem is that every ''t''-design with ''t'' ≥ 2 is also a 2-design.
+There are no known examples of non-trivial ''t''-(''v'',''k'',1)-designs with <math>t > 5</math>.
+The term ''block design'' by itself usually means a 2-design.
+===Derived and extendable t-designs===
+Let '''D''' = (''X'', ''B'') be a t-(''v'',''k'',''λ'') design and ''p'' a point of ''X''. The ''derived design'' ''D''<sub>''p''</sub> has point set ''X''&nbsp;&minus;&nbsp;{''p''} and as block set all the blocks of '''D''' which contain p with p removed. It is a (''t''&nbsp;&minus;&nbsp;1)-(''v''&nbsp;&minus;&nbsp;1, ''k''&nbsp;&minus;&nbsp;1, ''λ'') design. Note that derived designs with respect to different points may not be isomorphic. A design '''E''' is called an ''extension'' of '''D''' if '''E''' has a point p such that '''E'''<sub>p</sub> is isomorphic to '''D'''; we call '''D''' ''extendable'' if it has an extension.
+'''Theorem''':<ref>{{harvnb|Hughes|Piper|1985|loc=pg.29}}</ref> If a  ''t''-(''v'',''k'',''λ'') design has an extension, then ''k''&nbsp;+&nbsp;1 divides ''b''(''v''&nbsp;+&nbsp;1).
+The only extendable [[projective plane]]s (symmetric 2-(''n''<sup>2</sup>&nbsp;+&nbsp;''n''&nbsp;+&nbsp;1, ''n''&nbsp;+&nbsp;1, 1) designs) are those of orders 2 and&nbsp;4.<ref>{{harvnb|Cameron|van Lint|1991|loc=pg. 11, Proposition 1.34}}</ref>
+Every Hadamard 2-design is extendable (to an '''Hadamard 3-design''').<ref>{{harvnb|Hughes|Piper|1985|loc=pg. 132, Theorem 4.5}}</ref>
+'''Theorem''':.<ref>{{harvnb|Cameron|van Lint|1991|loc=pg. 11, Theorem 1.35}}</ref>
+If '''D''', a symmetric 2-(''v'',''k'',λ) design, is extendable, then one of the following holds:
+# '''D''' is an Hadamard 2-design,
+# ''v'' &nbsp;=&nbsp; (λ&nbsp;+&nbsp;2)(λ<sup>2</sup>&nbsp;+&nbsp;4λ&nbsp;+&nbsp;2), ''k''&nbsp;=&nbsp;λ<sup>2</sup>&nbsp;+&nbsp;3λ&nbsp;+&nbsp;1,
+# ''v''&nbsp;=&nbsp;495, ''k''&nbsp;=&nbsp;39, λ&nbsp;=&nbsp;3.
+Note that the projective plane of order two is an Hadamard 2-design; the projective plane of order four has parameters which fall in case 2; the only other known symmetric 2-designs with parameters in case 2 are the order 9 biplanes, but none of them are extendable; and there is no known symmetric 2-design with the parameters of case 3.<ref>{{harvnb|Colbourn|Dinitz|2007|loc=pg. 114, Remarks 6.35}}</ref>
+====Inversive planes====
+A design with the parameters of the extension of an [[affine plane (incidence geometry)#Finite affine plane|affine plane]], i.e., a 3-(''n''<sup>2</sup>&nbsp;+&nbsp;1, ''n''&nbsp;+&nbsp;1, 1) design, is called a finite '''inversive plane''', or [[Möbius plane]], of order&nbsp;''n''.
+It is possible to give a geometric description of some inversive planes, indeed, of all known inversive planes. An ''[[Ovoid (projective geometry)|ovoid]]'' in PG(3,''q'') is a set of ''q''<sup>2</sup>&nbsp;+&nbsp;1 points, no three collinear. It can be shown that every plane (which is a hyperplane since the geometric dimension is 3) of PG(3,''q'') meets an ovoid ''O'' in either 1 or ''q''&nbsp;+&nbsp;1 points. The plane sections of size ''q''&nbsp;+&nbsp;1 of ''O'' are the blocks of an inversive plane of order&nbsp;''q''. Any inversive plane arising this way is called ''egglike''. All known inversive planes are egglike.
+An example of an ovoid is the [[Quadric (projective geometry)|elliptic quadric]], the set of zeros of the quadratic form
+::: ''x''<sub>1</sub>''x''<sub>2</sub> + ''f''(''x''<sub>3</sub>, ''x''<sub>4</sub>),
+where f is an irreducible [[quadratic form]] in two variables over GF(''q''). [''f''(''x'',''y'') = ''x''<sup>2</sup> + ''xy'' + ''y''<sup>2</sup> for example].
+If ''q'' is an odd power of 2, another type of ovoid is known – the [[Ovoid (projective geometry)|Suzuki-Tits ovoid]].
+'''Theorem'''.  Let ''q'' be a positive integer, at least 2.  (a) If ''q'' is odd, then any ovoid is projectively equivalent to the elliptic quadric in a projective geometry PG(3,''q''); so ''q'' is a prime power and there is a unique egglike inversive plane of order ''q''. (But it is unknown if non-egglike ones exist.)  (b) if ''q'' is even, then ''q'' is a power of 2 and any inversive plane of order ''q'' is egglike (but there may be some unknown ovoids).
+==Steiner systems==
+{{main|Steiner system}}
+A '''[[Steiner system]]''' (named after [[Jakob Steiner]]) is a t-design with λ = 1 and ''t'' ≥ 2.
+A Steiner system with parameters ''t'', ''k'', ''n'', written S(''t'',''k'',''n''), is an ''n''-element [[Set (mathematics)|set]] ''S'' together with a set of ''k''-element [[subset]]s of ''S'' (called '''blocks''') with the property that each ''t''-element subset of ''S'' is contained in exactly one block. In the general notation for block designs, an S(''t'',''k'',''n'') would be a ''t''-(''n'',''k'',1) design.
+This definition is relatively modern, generalizing the ''classical'' definition of Steiner systems which in addition required that ''k'' = ''t'' + 1. An S(2,3,''n'') was (and still is) called a ''Steiner triple system'', while an S(3,4,''n'') was called a ''Steiner quadruple system'', and so on. With the generalization of the definition, this naming system is no longer strictly adhered to.
+[[Projective plane]]s and [[Affine plane (incidence geometry)|affine planes]] are examples of Steiner systems under the current definition while only the [[Fano plane]] (projective plane of order 2) would have been a Steiner system under the older definition.
+==Partially balanced designs (PBIBDs)==
+An ''n''-class [[association scheme]] consists of a [[Set (mathematics)|set]] ''X'' of size ''v'' 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>. A pair of elements in relation R<sub>i</sub> are said to be ''i''<sup>th</sup>&ndash;''associates''. Each element of ''X'' has ''n''<sub>i</sub>&nbsp; ''i''<sup>th</sup> associates. Furthermore:
+*<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 ''i'', ''j'', ''k'' but not on the particular choice of ''x'' and ''y''.
+An association scheme is ''commutative'' if <math>p_{ij}^k=p_{ji}^k</math> for all ''i'', ''j'' and ''k''. Most authors assume this property.
+A '''partially balanced incomplete block design''' with ''n'' associate classes (PBIBD(''n'')) is a block design based on a ''v''-set X with ''b'' blocks each of size ''k'' and with each element appearing in ''r'' blocks, such that there is an association scheme with ''n'' classes defined on ''X'' where, if elements ''x'' and ''y'' are ''i''<sup>th</sup> associates, 1 ≤ ''i'' ≤ ''n'', then they are together in precisely λ<sub>i</sub> blocks.
+A PBIBD(''n'') determines an association scheme but the converse is false.<ref>{{harvnb|Street|Street|1987|loc=pg. 237}}</ref>
+===Example ===
+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>
+The blocks of a PBIBD(3) based on ''A''(3) are:
+<center>
+{|
+|-
+|&nbsp;124&nbsp;||&nbsp; 134 &nbsp;||&nbsp; 235&nbsp; ||&nbsp;456&nbsp;
+|-
+|&nbsp; 125&nbsp; || &nbsp;136&nbsp; ||&nbsp; 236&nbsp; ||&nbsp; 456&nbsp;
+|}
+</center>
+The parameters of this PBIBD(3) are: ''v'' &nbsp;=&nbsp; 6, ''b'' &nbsp;=&nbsp; 8, ''k'' &nbsp;=&nbsp; 3, ''r'' &nbsp;=&nbsp; 4 and λ<sub>1</sub>&nbsp;=&nbsp;λ<sub>2</sub>&nbsp;=&nbsp;2 and λ<sub>3</sub>&nbsp;=&nbsp;1. Also, for the association scheme we have ''n''<sub>0</sub> &nbsp;=&nbsp; ''n''<sub>2</sub> &nbsp;=&nbsp; 1 and ''n''<sub>1</sub> &nbsp;=&nbsp; ''n''<sub>3</sub> &nbsp;=&nbsp; 2.<ref>{{harvnb|Street|Street|1987|loc=pg. 238}}</ref>
+===Properties===
+The parameters of a PBIBD(''m'') satisfy:<ref>{{harvnb|Street|Street|1987|loc=pg. 240, Lemma 4}}</ref>
+# <math> vr = bk </math>
+# <math> \sum_{i=1}^{m} n_i = v-1 </math>
+# <math> \sum_{i=1}^{m} n_i \lambda_i = r(k-1) </math>
+# <math> \sum_{u=0}^{m} p_{ju}^h = n_j </math>
+# <math> n_i p_{jh}^{i} = n_j p_{ih}^{j} </math>
+A PBIBD(1) is a BIBD and a PBIBD(2) in which λ<sub>1</sub> &nbsp;=&nbsp; λ<sub>2</sub> is a BIBD.<ref>{{harvnb|Colburn|Dinitz|2007|loc=pg. 562, Remark 42.3 (4)}}</ref>
+===Two associate class PBIBDs===
+PBIBD(2)s have been the studied the most since they are the simplest and most useful of the PBIBDs.<ref>{{harvnb|Street|Street|1987|loc=pg. 242}}</ref> They fall into six types<ref>Not a mathematical classification since one of the types is a catch-all "and everything else".</ref> based on a classification of the ''then known'' PBIBD(2)s by {{harvtxt|Bose|Shimamoto|1952}}:<ref>{{harvnb|Raghavarao|1988|loc=pg. 127}}</ref>
+# group divisible;
+# triangular;
+# Latin square type;
+# cyclic;
+# partial geometry type;
+# miscellaneous.
+==Applications==
+The mathematical subject of block designs originated in the statistical framework of [[design of experiments]]. These designs were especially useful in applications of the technique of [[Analysis of variance|analysis of variance (ANOVA)]]. This remains a significant area for the use of block designs.
+While the origins of the subject are grounded in biological applications (as is some of the existing terminology), the designs are used in many applications where systematic comparisons are being made, such as in [[software testing]].
+The [[incidence matrix]] of block designs provide a natural source of interesting [[block code]]s that are used as [[error correcting code]]s.
+==See also==
+* [[Randomized block design]]
+* [[Combinatorial design]]
+* [[Incidence geometry]]
+==Notes==
+{{reflist}}
+==References==
+* {{cite journal
+  | author = Aschbacher, Michael
+  | title = On collineation groups of symmetric block designs
+  | journal = Journal of Combinatorial Theory, Series A
+  | volume = 11
+  | issue = 3
+  | year = 1971
+  | pages = 272–281
+  | doi = 10.1016/0097-3165(71)90054-9}}
+* {{citation|last1=Assmus|first1=E.F.|last2=Key|first2=J.D.|title=Designs and Their Codes|publisher=Cambridge University Press|year=1992|place=Cambridge|isbn=0-521-41361-3}}
+*{{citation|first1=Thomas|last1=Beth|first2=Dieter|last2=Jungnickel|first3=Hanfried|last3=Lenz|title=Design Theory|publisher=[[Cambridge University Press]]|location=Cambridge|year=1986}}. 2nd ed. (1999) ISBN 978-0-521-44432-3.
+* [[R. C. Bose]], "A Note on Fisher's Inequality for Balanced Incomplete Block Designs", ''[[Annals of Mathematical Statistics]]'', 1949, pages 619&ndash;620.
+* {{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}}
+* {{citation|last1=Cameron|first1=P. J.|last2=van Lint|first2=J. H.|title=Designs, Graphs, Codes and their Links|year=1991|publisher=Cambridge University Press|place=Cambridge|isbn=0-521-42385-6}}
+* {{citation|last1=Colbourn|first1=Charles J.|last2=Dinitz|first2=Jeffrey H.|title=Handbook of Combinatorial Designs|year=2007|publisher=Chapman & Hall/ CRC|location=Boca Raton|isbn=1-58488-506-8|edition=2nd Edition}}
+* R. A. Fisher, "An examination of the different possible solutions of a problem in incomplete blocks", ''[[Annals of Eugenics]]'', volume 10, 1940, pages 52&ndash;75.
+* {{citation|last=Hall, Jr. |first=Marshall|title=Combinatorial Theory|edition=2nd|publisher= Wiley-Interscience|place=New York|year=1986|isbn=0-471-09138-3}}
+* {{citation|last1=Hughes|first1=D.R.|last2=Piper|first2=E.C.|title=Design theory|publisher=Cambridge University Press|place=Cambridge|year=1985|isbn=0-521-25754-9}}
+* {{cite journal
+  | author = Kaski, Petteri and Östergård, Patric
+  | title = There Are Exactly Five Biplanes with ''k'' = 11
+  | journal = Journal of Combinatorial Designs
+  | volume = 16
+  | year = 2008
+  | issue = 2
+  | pages = 117–127
+  | doi = 10.1002/jcd.20145
+  | mr = 2008m:05038 }}
+* {{citation|last=Lander|first=E. S.|title=Symmetric Designs: An Algebraic Approach|year=1983|publisher=Cambridge University Press|location=Cambridge}}
+* {{citation|last=Lindner|first=C.C.|last2=Rodger|first2=C.A.|title=Design Theory|year=1997|publisher=CRC Press|location=Boca Raton|isbn=0-8493-3986-3}}
+* {{cite book
+|title=Constructions and Combinatorial Problems in Design of Experiments
+|author=[[Damaraju Raghavarao|Raghavarao, Damaraju]]
+|location=New York
+|year=1988
+|edition=corrected reprint of the 1971 Wiley
+|publisher=Dover
+}}
+* {{cite book
+|title=Block Designs: Analysis, Combinatorics and Applications
+|author=[[Damaraju Raghavarao|Raghavarao, Damaraju]] and Padgett, L.V.
+|location=
+|year=2005
+|edition=
+|publisher=World Scientific
+}}
+* {{citation|last=Ryser|first=Herbert John|title=Combinatorial Mathematics (Carus Monograph #14)|year=1963|publisher=Mathematical Association of America|chapter=Chapter 8: Combinatorial Designs}}
+* {{cite journal
+  | first1 = Chester J.
+  | last1 = Salwach
+  | first2 = Joseph A.
+  | last2 = Mezzaroba
+  | title = The four biplanes with ''k'' = 9
+  | journal = Journal of Combinatorial Theory, Series A
+  | volume = 24
+  | issue = 2
+  | year = 1978
+  | pages = 141–145
+  | doi = 10.1016/0097-3165(78)90002-X}}
+* [[S. S. Shrikhande]], and [[Bhat-Nayak Vasanti N.|Vasanti N. Bhat-Nayak]], Non-isomorphic solutions of some balanced incomplete block designs I – [[Journal of Combinatorial Theory]], 1970
+* {{citation|last=Stinson|first=Douglas R.|title=Combinatorial Designs: Constructions and Analysis|year=2003|publisher=Springer|location=New York|isbn=0-387-95487-2}}
+*{{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
+}}
+* {{cite book | last1 = van Lint | first1 = J.H. | first2 = R.M. | last2 = Wilson | year = 1992 | title = A Course in Combinatorics | location = Cambridge | publisher = Cambridge University Press}}
+==External links==
+*[http://designtheory.org DesignTheory.Org]: Databases of combinatorial, statistical, and experimental block designs. Software and other resources hosted by the School of Mathematical Sciences at Queen Mary College, University of London.
+*[http://www.maths.qmul.ac.uk/~pjc/design/resources.html Design Theory Resources]: [[Peter Cameron (mathematician)|Peter Cameron]]'s page of web based design theory resources.
+* {{mathworld |urlname=BlockDesign|title=Block Designs}}
+{{Experimental design|state=expanded}}
+[[Category:Combinatorics]]
+[[Category:Design theory]]
+[[Category:Set families]]
+[[Category:Design of experiments]]

Jensen's alpha: Difference between revisions

Revision as of 18:52, 26 December 2013

Definition of a BIBD (or 2-design)

Symmetric BIBDs

Projective planes

Biplanes

Hadamard 2-designs

Resolvable 2-designs

Generalization: t-designs

Derived and extendable t-designs

Inversive planes

Steiner systems

Partially balanced designs (PBIBDs)

Example

Properties

Two associate class PBIBDs

Applications

See also

Notes

References

External links

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