Rijndael S-box

In mathematical analysis, more precisely in microlocal analysis, the wave front (set) WF(f) characterizes the singularities of a generalized function f, not only in space, but also with respect to its Fourier transform at each point. The term "wave front" was coined by Lars Hörmander around 1970.

Introduction

In more familiar terms, WF(f) tells not only where the function f is singular (which is already described by its singular support), but also how or why it is singular, by being more exact about the direction in which the singularity occurs. This concept is mostly useful in dimension at least two, since in one dimension there are only two possible directions. The complementary notion of a function being non-singular in a direction is microlocal smoothness.

Intuitively, as an example, consider a function ƒ whose singular support is concentrated on a smooth curve in the plane at which the function has a jump discontinuity. In the direction tangent to the curve, the function remains smooth. By contrast, in the direction normal to the curve, the function has a singularity. To decide on whether the function is smooth in another direction v, one can try to smooth the function out by averaging in directions perpendicular to v. If the resulting function is smooth, then we regard ƒ to be smooth in the direction of v. Otherwise, v is in the wavefront set.

Formally, in Euclidean space, the wave front set of ƒ is defined as the complement of the set of all pairs (x₀,v) such that there exists a test function $\phi \in C_{0}^{\infty }$ with φ(x₀) ≠ 0 and an open cone Γ containing v such that the estimate

$|(\phi f)^{\wedge }(\xi )|\leq C_{N}(1+|\xi |)^{-N}\quad {\mbox{for all }}\ \xi \in \Gamma$

holds for all positive integers N. Here $(\phi f)^{\wedge }$ denotes the Fourier transform. Observe that the wavefront set is a conical set, since if (x,v) ∈ Wf(ƒ), then (x,λv) ∈ Wf(ƒ) for all λ > 0. In the example discussed in the previous paragraph, the wavefront set is the set-theoretic complement of the image of the tangent bundle of the curve inside the tangent bundle of the plane.

Because the definition involves cutoff by a compactly supported function, the notion of a wave front set can be transported to any differentiable manifold X. In this more general situation, the wave front set is a closed conical subset of the cotangent bundle T^*(X), since the ξ variable naturally localizes to a covector rather than a vector. The wave front set is defined such that its projection on X is equal to the singular support of the function.

Definition

In Euclidean space, the wave front set of a distribution ƒ is defined as

${\rm {WF}}(f)=\{(x,\xi )\in \mathbb {R} ^{n}\times \mathbb {R} ^{n}\mid \xi \in \Sigma _{x}(f)\}$

where $\Sigma _{x}(f)$ is the singular fibre of ƒ at x. The singular fibre is defined to be the complement of all directions $\xi$ such that the Fourier transform of f, localized at x, is sufficiently regular when restricted to an open cone containing $\xi$. More precisely, a direction v is in the complement of $\Sigma _{x}(f)$ if there is a compactly supported smooth function φ with φ(x) ≠ 0 and an open cone Γ containing v such that the following estimate holds for each positive integer N:

$(\phi f)^{\wedge }(\xi )<c_{N}(1+|\xi |)^{-N}\quad {\rm {forall}}\ \xi \in \Gamma .$

Once such an estimate holds for a particular cutoff function φ at x, it also holds for all cutoff functions with smaller support, possibly for a different open cone containing v.

On a differentiable manifold M, using local coordinates $x,\xi$ on the cotangent bundle, the wave front set WF(f) of a distribution ƒ can be defined in the following general way:

${\rm {WF}}(f)=\{(x,\xi )\in T^{*}(X)\mid \xi \in \Sigma _{x}(f)\}$

where the singular fibre $\Sigma _{x}(f)$ is again the complement of all directions $\xi$ such that the Fourier transform of f, localized at x, is sufficiently regular when restricted to a conical neighbourhood of $\xi$. The problem of regularity is local, and so it can be checked in the local coordinate system, using the Fourier transform on the x variables. The required regularity estimate transforms well under diffeomorphism, and so the notion of regularity is independent of the choice of local coordinates.

Generalizations

The notion of a wave front set can be adapted to accommodate other notions of regularity of a function. Localized can here be expressed by saying that f is truncated by some smooth cutoff function not vanishing at x. (The localization process could be done in a more elegant fashion, using germs.)

More concretely, this can be expressed as

$\xi \notin \Sigma _{x}(f)\iff \exists \phi \in {\mathcal {D}}_{x},\ \exists V\in {\mathcal {V}}_{\xi }:{\widehat {\phi f}}|_{V}\in O(V)$ (or $\xi =o$, never in $\Sigma _{x}(f)$)

where

• ${\mathcal {D}}_{x}$ are compactly supported smooth functions not vanishing at x,
• ${\mathcal {V}}_{\xi }$ are conical neighbourhoods of $\xi$, i.e. neighbourhoods V such that $c\cdot V\subset V$ for all $c>0$,
• ${\widehat {u}}|_{V}$ denotes the Fourier transform of the (compactly supported generalized) function u, restricted to V,
• and finally $O:\Omega \to O(\Omega )$ is the presheaf characterizing the regularity of the Fourier transform.

Typically, sections of O are characterized by some growth (or decrease) condition at infinity, e.g. such that $(1+|\xi |)^{s}v(\xi )$ belong to some L^p space. This definition makes sense, because the Fourier transform becomes more regular (in terms of growth at infinity) when f is truncated with the smooth cutoff $\phi$.

The most difficult "problem", from a theoretical point of view, is finding the adequate sheaf O characterizing functions belonging to a given subsheaf E of the space G of generalized functions.

Example

If we take G = D′ the space of Schwartz distributions and want to characterize distributions which are locally $C^{\infty }$ functions, we must take for O(Ω) the classical function spaces called O′_M(Ω) in the literature.

Then the projection on the first component of a distribution's wave front set is nothing else than its classical singular support, i.e. the complement of the set on which its restriction would be a smooth function.

Applications

The wave front set is useful, among others, when studying propagation of singularities by pseudodifferential operators.

See also

• FBI transform
• Singular spectrum
• Essential support

References

• Lars Hörmander, Fourier integral operators I, Acta Math. 127 (1971), pp. 79-183.
• 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 Chapter VIII, Spectral Analysis of Singularities