The GARCH model is a model for the heteroskedastic variations where the changes in variance is assumed to be auto correlated: that, though the variance changes, it changes in a predictable manner.

It is especially useful to

GARCH 1,1

Conditional mean:

\begin{equation} y_{t} = x’_{t} \theta + \epsilon_{t} \end{equation}

Then, the epsilon parameter:

\begin{equation} \epsilon_{t} = \sigma_{t}z_{t} \end{equation}

where:

\begin{equation} z_{t} \sim \mathcal{N}(0,1) \end{equation}

and:

conditional variance

\begin{equation} {\sigma_{t}}^{2} = \omega + \lambda {\sigma_{t-1}}^{2} + \beta {\sigma_{t-1}}^{2} \end{equation}

The Gauss’ Law is a principle of electric flux of uniformly distributed electric field along a surface: that, the electric flux through a closed surface is the sum of the electric charge enclosed divided by the permittivity of free space.

That is:

\begin{equation} \oint E \cdot dA = \frac{\sum Q}{\epsilon_{0}} \end{equation}

somewhat motivating Gauss’ Law

Consider a sphere with uniformly distributed charge on its surface. It has surface area \(4 \pi r^{2}\). Given the expression of electric flux and the fact that the origin change is in the center, and the test change is evenly distributed (i.e. \(E\) is held constant):

The Gaussian, in general, gives:

\begin{equation} e^{-\frac{ax^{2}}{2}} \end{equation}

which is a Bell-Shaped curve. It’s pretty darn important

solving heat equation without boundary

for general expression:

\begin{equation} \pdv{U}{t} = \alpha \pdv[2]{U}{x} \end{equation}

\begin{equation} U(t,x) = \frac{1}{\sqrt{4\pi \alpha t}}\int_{\mathbb{R}} f(y) e^{-\frac{(x-y)^{2}}{4\alpha t}} \dd{y} \end{equation}

where,

\begin{equation} \hat{U}(t,\lambda) = \hat{f}(\lambda)e^{-\alpha t \lambda^{2}} \end{equation}

\begin{equation} \hat{U}(t,\lambda) = \hat{f}(\lambda)e^{-\lambda^{2}(t)} \end{equation}

Heat Equation and Gaussian

\begin{equation} H(t,x) = \frac{1}{\sqrt{2\pi} t}e^{-\frac{x^{2}}{2t}} \end{equation}

You will note that \(H\) does satisfy the heat equation:

The point of Gaussian elimination is to solve/identiy-ify a linear equation. Take, if you have a matrix expression:

\begin{equation} Ax = b \end{equation}

We can apply \(A^{-1}\) to both side, we then have:

\begin{equation} A^{-1}Ax = A^{-1} b \end{equation}

Applying the definition of the identity:

\begin{equation} Ix = A^{-1}b \end{equation}

Therefore, to solve for some \(A^{-1}\), which would yield \(x\).

GDB is gnu’s very own debugger

• b main or b 72 (set breakpoint on main function or line 72)
• r args (run with args)
• p thingname or p 3+5 (print a variable or return value)
  • p/t print as binary
  • p/x print as hex
• info (get args, locals)
• n s continue next, step, continue

int test;
short lsb = 0xff;
test |= lsb

printf("%d\n",lsb);

int test;