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In multilinear algebra, a dyadic or dyadic tensor is a second order tensor written in a special notation, formed by juxtaposing pairs of vectors, along with a notation for manipulating such expressions analogous to the rules for matrix algebra. The notation and terminology is relatively obsolete today. Its uses in physics include stress analysis and electromagnetism.

Dyadic notation was first established by Josiah Willard Gibbs in 1884.

In this article, upper-case bold variables denote dyadics (including dyads) whereas lower-case bold variables denote vectors. An alternative notation uses respectively double and single over- or underbars.

Definitions and terminology

Dyadic, outer, and tensor products

A dyad is a tensor of order two and rank one, and is the result of the dyadic product of two vectors (complex vectors in general), whereas a dyadic is a general tensor of order two.

There are several equivalent terms and notations for this product:

the dyadic product of two vectors a and b is denoted by the juxtaposition ab,
the outer product of two column vectors a and b is denoted and defined as a ⊗ b or ab^T, where T means transpose,
the tensor product of two vectors a and b is denoted a ⊗ b,

In the dyadic context they all have the same definition and meaning, and are used synonymously, although the tensor product is an instance of the more general and abstract use of the term.

Three-dimensional Euclidean space

To illustrate the equivalent usage, consider three-dimensional Euclidean space, letting:

a = a_{1} i + a_{2} j + a_{3} k

b = b_{1} i + b_{2} j + b_{3} k

be two vectors where i, j, k (also denoted e₁, e₂, e₃) are the standard basis vectors in this vector space (see also Cartesian coordinates). Then the dyadic product of a and b can be represented as a sum:

\begin{array}{llll} a b = & a_{1} b_{1} i i & + a_{1} b_{2} i j & + a_{1} b_{3} i k \\ + a_{2} b_{1} j i & + a_{2} b_{2} j j & + a_{2} b_{3} j k \\ + a_{3} b_{1} k i & + a_{3} b_{2} k j & + a_{3} b_{3} k k \end{array}

or by extension from row and column vectors, a 3×3 matrix (also the result of the outer product or tensor product of a and b):

a b \equiv a \otimes b \equiv {a b}^{T} = (\begin{matrix} a_{1} \\ a_{2} \\ a_{3} \end{matrix}) (\begin{matrix} b_{1} & b_{2} & b_{3} \end{matrix}) = (\begin{matrix} a_{1} b_{1} & a_{1} b_{2} & a_{1} b_{3} \\ a_{2} b_{1} & a_{2} b_{2} & a_{2} b_{3} \\ a_{3} b_{1} & a_{3} b_{2} & a_{3} b_{3} \end{matrix}) .

A dyad is a component of the dyadic (a monomial of the sum or equivalently entry of the matrix) - the juxtaposition of a pair of basis vectors scalar multiplied by a number.

Just as the standard basis (and unit) vectors i, j, k, have the representations:

i = (\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}), j = (\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}), k = (\begin{matrix} 0 \\ 0 \\ 1 \end{matrix})

(which can be transposed), the standard basis (and unit) dyads have the representation:

i i = (\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}), \dots j i = (\begin{matrix} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}), \dots j k = (\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix}) \dots

For a simple numerical example in the standard basis:

\begin{aligned} A & = 2 i j + \frac{\sqrt{3}}{2} j i - 8 π j k + \frac{2 \sqrt{2}}{3} k k \\ = 2 (\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}) + \frac{\sqrt{3}}{2} (\begin{matrix} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}) - 8 π (\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix}) + \frac{2 \sqrt{2}}{3} (\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}) \\ = (\begin{matrix} 0 & 2 & 0 \\ \sqrt{3} / 2 & 0 & - 8 π \\ 0 & 0 & \frac{2 \sqrt{2}}{3} \end{matrix}) \end{aligned}

N-dimensional Euclidean space

If the Euclidean space is N-dimensional, and

a = \sum_{i = 1}^{N} a_{i} e_{i} = a_{1} e_{1} + a_{2} e_{2} + \dots a_{N} e_{N}

b = \sum_{j = 1}^{N} b_{j} e_{j} = b_{1} e_{1} + b_{2} e_{2} + \dots b_{N} e_{N}

where e_i and e_j are the standard basis vectors in N-dimensions (the index i on e_i selects a specific vector, not a component of the vector as in a_i), then in algebraic form their dyadic product is:

A = \sum_{j = 1}^{N} \sum_{i = 1}^{N} a_{i} b_{j} e_{i} e_{j} .

This is known as the nonion form of the dyadic. Their outer/tensor product in matrix form is:

a b = {a b}^{T} = (\begin{matrix} a_{1} \\ a_{2} \\ ⋮ \\ a_{N} \end{matrix}) (\begin{matrix} b_{1} & b_{2} & \dots & b_{N} \end{matrix}) = (\begin{matrix} a_{1} b_{1} & a_{1} b_{2} & \dots & a_{1} b_{N} \\ a_{2} b_{1} & a_{2} b_{2} & \dots & a_{2} b_{N} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{N} b_{1} & a_{N} b_{2} & \dots & a_{N} b_{N} \end{matrix}) .

A dyadic polynomial A, otherwise known as a dyadic, is formed from multiple vectors a_i and b_j:

A = \sum_{i} a_{i} b_{i} = a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3} + \dots

A dyadic which cannot be reduced to a sum of less than N dyads is said to be complete. In this case, the forming vectors are non-coplanar,To succeed in selling a home, it is advisable be competent in real estate advertising and marketing, authorized, monetary, operational aspects, and other information and skills. This is essential as a result of you want to negotiate with more and more sophisticated buyers. You could outperform rivals, use latest technologies, and stay ahead of the fast altering market.

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Classification

The following table classifies dyadics:

	Determinant	Adjugate	Matrix and its rank
Zero	= 0	= 0	= 0; rank 0: all zeroes
Linear	= 0	= 0	≠ 0; rank 1: at least one non-zero element and all 2 × 2 subdeterminants zero (single dyadic)
Planar	= 0	≠ 0 (single dyadic)	≠ 0; rank 2: at least one non-zero 2 × 2 subdeterminant
Complete	≠ 0	≠ 0	≠ 0; rank 3: non-zero determinant

Identities

The following identities are a direct consequence of the definition of the tensor product:^[1]

Template:Ordered list

Dyadic algebra

Product of dyadic and vector

There are four operations defined on a vector and dyadic, constructed from the products defined on vectors.

	Left	Right
Dot product	$c \cdot a b = (c \cdot a) b$	$(a b) \cdot c = a (b \cdot c)$
Cross product	$c \times (a b) = (c \times a) b$	$(a b) \times c = a (b \times c)$

Product of dyadic and dyadic

There are five operations for a dyadic to another dyadic. Let a, b, c, d be vectors. Then:

		Dot	Cross
Dot	Dot product $(a b) \cdot (c d) = a (b \cdot c) d = (b \cdot c) a d$	Double dot product $a b : c d = (a \cdot d) (b \cdot c)$ or $(a b) : (c d) = c \cdot (a b) \cdot d = (a \cdot c) (b \cdot d)$	Dot–cross product $(a b) \begin{array}{c} \cdot \\ ^{\times} \end{array} (c d) = (a \cdot c) (b \times d)$
Cross		Cross–dot product $(a b) \begin{array}{c} \times \\ ^{\cdot} \end{array} (c d) = (a \times c) (b \cdot d)$	Double cross product $(a b) \begin{array}{c} \times \\ ^{\times} \end{array} (c d) = (a \times c) (b \times d)$

Letting

A = \sum_{i} a_{i} b_{i} B = \sum_{i} c_{i} d_{i}

be two general dyadics, we have:

		Dot	Cross
Dot	Dot product $A \cdot B = \sum_{j} \sum_{i} (b_{i} \cdot c_{j}) a_{i} d_{j}$	Double dot product $A : B = \sum_{j} \sum_{i} (a_{i} \cdot d_{j}) (b_{i} \cdot c_{j})$ or $A : B = \sum_{j} \sum_{i} = (a_{i} \cdot c_{j}) (b_{i} \cdot d_{j})$	Dot–cross product $A \begin{array}{c} \cdot \\ ^{\times} \end{array} B = \sum_{j} \sum_{i} (a_{i} \cdot c_{j}) (b_{i} \times d_{j})$
Cross		Cross–dot product $A \begin{array}{c} \times \\ ^{\cdot} \end{array} B = \sum_{j} \sum_{i} (a_{i} \times c_{j}) (b_{i} \cdot d_{j})$	Double cross product $A \begin{array}{c} \times \\ ^{\times} \end{array} B = \sum_{i, j} (a_{i} \times c_{j}) (b_{i} \times d_{j})$

Double-dot product

There are two ways to define the double dot product, one must be careful when deciding which convention to use. As there are no analogous matrix operations for the remaining dyadic products, no ambiguities in their definitions appear.

The double-dot product is commutative due to commutativity of the normal dot-product:

A : B = B : A

There is a special double dot product with a transpose

A : B^{T} = A^{T} : B

Another identity is:

A : B = (A \cdot B^{T}) : I = (B \cdot A^{T}) : I

Double-cross product

We can see that, for any dyad formed from two vectors a and b, its double cross product is zero.

(a b) \begin{array}{c} \times \\ ^{\times} \end{array} (a b) = (a \times a) (b \times b) = 0

However, by definition, a dyadic double-cross product on itself will generally be non-zero. For example, a dyadic A composed of six different vectors

A = \sum_{i = 1}^{3} a_{i} b_{i}

has a non-zero self-double-cross product of

A \begin{array}{c} \times \\ ^{\times} \end{array} A = 2 [(a_{1} \times a_{2}) (b_{1} \times b_{2}) + (a_{2} \times a_{3}) (b_{2} \times b_{3}) + (a_{3} \times a_{1}) (b_{3} \times b_{1})]

Tensor contraction

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The spur or expansion factor arises from the formal expansion of the dyadic in a coordinate basis by replacing each juxtaposition by a dot product of vectors:

\begin{array}{llll} | A | & = A_{11} i \cdot i + A_{12} i \cdot j + A_{31} i \cdot k \\ + A_{21} j \cdot i + A_{22} j \cdot j + A_{23} j \cdot k \\ + A_{31} k \cdot i + A_{32} k \cdot j + A_{33} k \cdot k \\ = A_{11} + A_{22} + A_{33} \end{array}

in index notation this is the contraction of indices on the dyadic:

| A | = \sum_{i} A_{i}^{i}

In three dimensions only, the rotation factor arises by replacing every juxtaposition by a cross product

\begin{array}{llll} ⟨ A ⟩ & = A_{11} i \times i + A_{12} i \times j + A_{31} i \times k \\ + A_{21} j \times i + A_{22} j \times j + A_{23} j \times k \\ + A_{31} k \times i + A_{32} k \times j + A_{33} k \times k \\ = A_{12} k - A_{31} j - A_{21} k \\ + A_{23} i + A_{31} j - A_{32} i \\ = (A_{23} - A_{32}) i + (A_{31} - A_{13}) j + (A_{12} - A_{21}) k \end{array}

In index notation this is the contraction of A with the Levi-Civita tensor

⟨ A ⟩ = \sum_{j k} {ϵ_{i}}^{j k} A_{j k} .

Special dyadics

Unit dyadic

For any vector a, there exist a unit dyadic I, such that

I \cdot a = a \cdot I = a

For any basis of 3 vectors a, b and c, with reciprocal basis $\hat{a}, \hat{b}, \hat{c}$ , the unit dyadic is defined by

I = a \hat{a} + b \hat{b} + c \hat{c}

In the standard basis,

I = i i + j j + k k

The corresponding matrix is

I = (\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix})

This can be put on more careful foundations (explaining what the logical content of "juxtaposing notation" could possibly mean) using the language of tensor products. If V is a finite-dimensional vector space, a dyadic tensor on V is an elementary tensor in the tensor product of V with its dual space.

The tensor product of V and its dual space is isomorphic to the space of linear maps from V to V: a dyadic tensor vf is simply the linear map sending any w in V to f(w)v. When V is Euclidean n-space, we can use the inner product to identify the dual space with V itself, making a dyadic tensor an elementary tensor product of two vectors in Euclidean space.

In this sense, the unit dyadic ij is the function from 3-space to itself sending a₁i + a₂j + a₃k to a₂i, and jj sends this sum to a‍₂j. Now it is revealed in what (precise) sense ii + jj + kk is the identity: it sends a₁i + a₂j + a₃k to itself because its effect is to sum each unit vector in the standard basis scaled by the coefficient of the vector in that basis.

Properties of unit dyadics

(a \times I) \cdot (b \times I) = a b - (a \cdot b) I

I \begin{array}{c} \times \\ ^{\cdot} \end{array} (a b) = b \times a

I \begin{array}{c} \times \\ ^{\times} \end{array} A = (A \begin{array}{c} \times \\ ^{\times} \end{array} I) I - A^{T}

I : (a b) = (I \cdot a) \cdot b = a \cdot b = t r (a b)

where "tr" denotes the trace.

Rotation dyadic

For any vector a in two dimensions, the left-cross product with the identity dyad I:

a \times I

is a 90 degree anticlockwise rotation dyadic around a. Alternatively the dyadic tensor

J = ji − ij =

(\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix})

is a 90° anticlockwise rotation operator in 2d. It can be left-dotted with a vector to produce the rotation:

(j i - i j) \cdot (x i + y j) = x j i \cdot i - x i j \cdot i + y j i \cdot j - y i j \cdot j = - y i + x j,

or in matrix notation

(\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix}) (\begin{matrix} x \\ y \end{matrix}) = (\begin{matrix} - y \\ x \end{matrix}) .

A general 2d rotation dyadic for θ angle anti-clockwise is

I \cos θ + J \sin θ = (\begin{matrix} \cos θ & - \sin θ \\ \sin θ & \cos θ \end{matrix})

where I and J are as above.

Related terms

Some authors generalize from the term dyadic to related terms triadic, tetradic and polyadic.^[2]

References

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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.
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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.

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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

Template:Tensor

↑ Spencer (1992), page 19.
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[1] Spencer (1992), page 19.

[2] For example, 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 [1]

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