MA plot

In mathematics, the symbol of a linear differential operator associates to a differential operator a polynomial by, roughly speaking, replacing each partial derivative by a new variable. The symbol of a differential operator has broad applications to Fourier analysis. In particular, in this connection it leads to the notion of a pseudo-differential operator. The highest-order terms of the symbol, known as the principal symbol, almost completely controls the qualitative behavior of solutions of a partial differential equation. Linear elliptic partial differential equations can be characterized as those whose principal symbol is nowhere zero. In the study of hyperbolic and parabolic partial differential equations, zeros of the principal symbol correspond to the characteristics of the partial differential equation. Consequently, the symbol is often fundamental for the solution of such equations, and is one of the main computational devices used to study their singularities.

Definition

Operators on Euclidean space

Let P be a linear differential operator of order k on the Euclidean space R^d. Then P is a polynomial in the derivative D, which in multi-index notation can be written

$P=p(x,D)=\sum _{|\alpha |\leq k}a_{\alpha }(x)D^{\alpha }.$

The total symbol of P is the polynomial p:

$\sigma _{P}(\xi )=p(x,\xi )=\sum _{|\alpha |\leq k}a_{\alpha }(x)\xi ^{\alpha }.$

The leading symbol, also known as the principal symbol, is the highest degree component of σ_P :

$\sigma _{P}(\xi )=\sum _{|\alpha |=k}a_{\alpha }\xi ^{\alpha }$

and is of importance later because it is the only part of the symbol that transforms as a tensor under changes to the coordinate system.

The symbol of P appears naturally in connection with the Fourier transform as follows. Let ƒ be a Schwartz function. Then by the inverse Fourier transform,

$Pf(x)=\int _{\mathbf {R} ^{d}}e^{ix\cdot \xi }p(x,i\xi ){\hat {f}}(\xi )\,d\xi .$

This exhibits P as a Fourier multiplier. A more general class of functions p(x,ξ) which satisfy at most polynomial growth conditions in ξ under which this integral is well-behaved comprises the pseudo-differential operators.

Vector bundles

Let E and F be vector bundles over a closed manifold X, and suppose

$P:C^{\infty }(E)\to C^{\infty }(F)$

is a differential operator of order $k$. In local coordinates on X, we have

$Pu(x)=\sum _{|\alpha |=k}P^{\alpha }(x){\frac {\partial ^{\alpha }u}{\partial x^{\alpha }}}+{\text{lower order terms}}$

where, for each multi-index α, $P^{\alpha }(x):E\to F$ is a bundle map, symmetric on the indices α.

The k^th order coefficients of P transform as a symmetric tensor

$\sigma _{P}:S^{k}(T^{*}X)\otimes E\to F$

from the tensor product of the k^th symmetric power of the cotangent bundle of X with E to F. This symmetric tensor is known as the principal symbol (or just the symbol) of P.

The coordinate system xⁱ permits a local trivialization of the cotangent bundle by the coordinate differentials dxⁱ, which determine fiber coordinates ξ_i. In terms of a basis of frames e_μ, f_ν of E and F, respectively, the differential operator P decomposes into components

$(Pu)_{\nu }=\sum _{\mu }P_{\nu \mu }u_{\mu }$

on each section u of E. Here P_νμ is the scalar differential operator defined by

$P_{\nu \mu }=\sum _{\alpha }P_{\nu \mu }^{\alpha }{\frac {\partial }{\partial x^{\alpha }}}.$

With this trivialization, the principal symbol can now be written

$(\sigma _{P}(\xi )u)_{\nu }=\sum _{|\alpha |=k}\sum _{\mu }P_{\nu \mu }^{\alpha }(x)\xi _{\alpha }u^{\mu }.$

In the cotangent space over a fixed point x of X, the symbol $\sigma _{P}$ defines a homogeneous polynomial of degree k in $T_{x}^{*}X$ with values in $\operatorname {Hom} (E_{x},F_{x})$.

The differential operator $P$ is elliptic if its symbol is invertible; that is for each nonzero $\theta \in T^{*}X$ the bundle map $\sigma _{P}(\theta ,\dots ,\theta )$ is invertible. On a compact manifold, it follows from the elliptic theory that P is a Fredholm operator: it has finite dimensional kernel and cokernel.

See also

• Multiplier (Fourier analysis)
• Atiyah-Singer index theorem (section on symbol of operator)

References

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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.
  
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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.
• 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.