LL grammar

In mathematics, the infinity Laplace (or $L^{\infty }$-Laplace) operator is a 2nd-order partial differential operator, commonly abbreviated $\Delta _{\infty }$. It is alternately defined by

$\Delta _{\infty }u(x)=\langle Du,D^{2}u\,Du\rangle =\sum _{i,j}{\frac {\partial ^{2}u}{\partial x_{i}\,\partial x_{j}}}{\frac {\partial u}{\partial x_{i}}}{\frac {\partial u}{\partial x_{j}}}$

or

$\Delta _{\infty }u(x)={\frac {\langle Du,D^{2}u\,Du\rangle }{|Du|^{2}}}={\frac {1}{|Du|^{2}}}\sum _{i,j}{\frac {\partial ^{2}u}{\partial x_{i}\,\partial x_{j}}}{\frac {\partial u}{\partial x_{i}}}{\frac {\partial u}{\partial x_{j}}}.$

The first version avoids the singularity which occurs when the gradient vanishes, while the second version is homogeneous of order zero in the gradient. Verbally, the second version is the second derivative in the direction of the gradient. In the case of the infinity Laplace equation $\Delta _{\infty }u=0$, the two definitions are equivalent.

While the equation involves second derivatives, usually solutions are not twice differentiable, as evidenced by the well-known Aronsson solution $u(x,y)=|x|^{4/3}-|y|^{4/3}$. For this reason the correct notion of solutions is that given by the viscosity solutions.

Viscosity solutions to the equation $\Delta _{\infty }u=0$ are also known as infinity harmonic functions. This terminology arises from the fact that the infinity Laplace operator first arose in the study of absolute minimizers for $\|Du\|_{L^{\infty }}$, and it can be viewed in a certain sense as the limit of the p-Laplacian as $p\rightarrow \infty$. More recently, viscosity solutions to the infinity Laplace equation have been identified with the payoff functions from randomized tug-of-war games. The game theory point of view has significantly improved the understanding of the partial differential equation itself.

Discrete version and game theory

A defining property of the usual $L^{2}$-harmonic functions is the mean value property. That has a natural and important discrete version: a real-valued function $f$ on a finite or infinite graph $G(V,E)$ is discrete harmonic on a subset $U\subseteq V$ if

$f(x)=\sum _{y\sim x}f(y)$

for all $x\in U$. Similarly, the vanishing second derivative in the direction of the gradient has a natural discrete version:

$f(x)={\frac {\sup\{f(y):y\sim x\}+\inf\{f(y):y\sim x\}}{2}}$.

In this equation, we used sup and inf instead of max and min because the graph $G(V,E)$ does not have to be locally finite (i.e., to have finite degrees): a key example is when $V(G)$ is the set of points in a domain in $\mathbb {R} ^{d}$, and $(x,y)\in E(G)$ if their Euclidean distance is at most $\epsilon$. The importance of this example lies in the following.

Consider a bounded open set $D\subseteq \mathbb {R} ^{d}$ with smooth boundary $\partial D$, and a continuous function $f:\partial D\longrightarrow \mathbb {R}$. In the $L^{2}$-case, an approximation of the harmonic extension of f to D is given by taking a lattice $\mathbb {Z} _{\epsilon }^{d}$ with small mesh size $\epsilon$, letting $G_{\epsilon }(V,E)=D\cap \mathbb {Z} _{\epsilon }^{d}$ and $\partial V\subset V$ be the set of vertices with degree less than 2d, taking a natural approximation $f_{\epsilon }:\partial V\longrightarrow \mathbb {R}$, and then taking the unique discrete harmonic extension of $f_{\epsilon }$ to V. However, it is easy to see by examples that this does not work for the $L^{\infty }$-case. Instead, as it turns out, one should take the continuum graph with all edges of length at most $\epsilon$, mentioned above.

Now, a probabilistic way of looking at the $L^{2}$-harmonic extension of $f_{\epsilon }$ from $\partial V$ to $V$ is that

$f_{\epsilon }(x)=\mathbb {E} {\big [}f_{\epsilon }(X_{\tau })\,|\,X_{0}=x{\big ]}$,

where $X_{0},X_{1},\dots$ is the simple random walk on $G_{\epsilon }(V,E)$ started at $X_{0}=x$, and $\tau$ is the hitting time of $\partial V$.

For the $L^{\infty }$-case, we need game theory. A token is started at location $x\in V(G)$, and $f:\partial V\longrightarrow \mathbb {R}$ is given. There are two players, in each turn they flip a fair coin, and the winner can move the token to any neighbour of the current location. The game ends when the token reaches $\partial V$ at some time $\tau$ and location $X_{\tau }$, at which point the first player gets the amount $f(X_{\tau })$ from the second player. Therefore, the first player wants to maximize $f(X_{\tau })$, while the second player wants to minimize it. If both players play optimally (which has a well-defined meaning in game theory), the expected payoff $\mathbb {E} [f(X_{\tau })\,|\,X_{0}=x]$ to the first player is a discrete infinity harmonic function, as defined above.

There is a game theory approach to the p-Laplacian, too, interpolating between simple random walk and the above random tug-of-war game.

Sources

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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
  
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