Glass transition

In mathematics, Weingarten functions are rational functions indexed by partitions of integers that can be used to calculate integrals of products of matrix coefficients over classical groups. They were first studied by Template:Harvtxt who found their asymptotic behavior, and named by Template:Harvtxt, who evaluated them explicitly for the unitary group.

Unitary groups

Weingarten functions are used for evaluating integrals over the unitary group U_d of products of matrix coefficients of the form

$\underset{U_d}{\int }{U}_{i_1{j}_1}\cdots U_{i_q{j}_q}{U}_{j_1^{\prime }{i}_1^{\prime }}^{*}\cdots U_{j_q^{\prime }{i}_q^{\prime }}^{*}dU.$

(Here $U^{*}$ denotes the conjugate transpose of $U$, alternatively denoted as $U^{†}$.)

This integral is equal to

$\sum _{\sigma ,\tau \in S_q}{\delta }_{i_1{i}_{\sigma 1}^{\prime }}\cdots {\delta }_{i_q{i}_{\sigma q}^{\prime }}{\delta }_{j_1{j}_{\tau 1}^{\prime }}\cdots {\delta }_{j_q{j}_{\tau q}^{\prime }}Wg(d,\sigma {\tau }^{-1})$

where Wg is the Weingarten function, given by

$Wg(d,\sigma )=\frac{1}{q{!}^2}\sum _{\lambda }\frac{{\chi }^{\lambda }(1{)}^2{\chi }^{\lambda }(\sigma )}{s_{\lambda ,d}(1)}$

where the sum is over all partitions λ of q Template:Harv. Here χ^λ is the character if S_q corresponding to the partition λ and s is the Schur polynomial of λ, so that s_λd(1) is the dimension of the representation of U_d corresponding to λ.

The Weingarten functions are rational functions in d. They can have poles for small values of d, which cancel out in the formula above. There is an alternative inequivalent definition of Weingarten functions, where one only sums over partitions with at most d parts. This is no longer a rational function of d, but is finite for all positive integers d. The two sorts of Weingarten functions coincide for d larger than q, and either can be used in the formula for the integral.

Examples

The first few Weingarten functions Wg(σd) are

$Wg(,d)=1$ (The trivial case where q = 0)

$Wg(1,d)=\frac{1}{d}$

$Wg(2,d)=\frac{-1}{d(d^2-1)}$

$Wg(1^2,d)=\frac{1}{d^2-1}$

$Wg(3,d)=\frac{2}{d(d^2-1)(d^2-4)}$

$Wg(21,d)=\frac{-1}{(d^2-1)(d^2-4)}$

$Wg(1^3,d)=\frac{d^2-2}{d(d^2-1)(d^2-4)}$

where permutations σ are denoted by their cycle shapes.

There exists a computer algebra program to produce these expressions.

Asymptotic behavior

For large d, the Weingarten function Wg has the asymptotic behavior

$Wg(\sigma ,d)=d^{-n-\left|\sigma \right|}\prod _i(-1{)}^{\left|C_i\right|-1}{c}_{\left|C_i\right|-1}+O(d^{-n-\left|\sigma \right|-2})$

where the permutation σ is a product of cycles of lengths C_i, and c_n = (2n)!/n!(n + 1)! is a Catalan number, and |σ| is the smallest number of transpositions that σ is a product of. There exists a diagrammatic method to systematically calculate the integrals over the unitary group as a power series in 1/d.

Orthogonal and symplectic groups

For orthogonal and symplectic groups the Weingarten functions were evaluated by Template:Harvtxt. Their theory is similar to the case of the unitary group. They are parameterized by partitions such that all parts have even size.

External links

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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