# The last part of Tate'sThesis

Is there a comprehensive reference dealing with the last part(theory in the large!) of Tate's thesis?

# Algorithmic Analysis Simplified under Big O

Just to let you know here is the example I am following:

f(0) = 0
f(n) = 3f(n-1)+1, n>0

f(n) = 3^n.We actually have

$$g(n) - g(0) = \sum _{j=1}^{n} \frac{-5}{6^j}$$

You missed the $g(0)$.

# What's the opposite of a cross product?

So we have:

$$\langle x_1, y_1, z_1\rangle\times\langle x_2, y_2, z_2\rangle = \langle y_1z_2 - z_1y_2, z_1x_2 - x_1z_2, x_1y_2 - y_1x_2\rangle$$

Then we have:

\begin{align}
\langle x_3, y_3, z_3\rangle\times\left(\langle x_1, y_1, z_1\rangle\times\langle x_2, y_2, z_2\rangle\right) = & \langle y_3(x_1y_2 - y_1x_2) - z_3(z_1x_2 - x_1z_2),\\
&z_3(y_1z_2 - z_1y_2) - x_3(x_1y_2 - y_1x_2),\\
& x_3(z_1x_2 - x_1z_2) - y_3(y_1z_2 - z_1y_2)\rangle
\end{align}

Now this looks very complicated (and it is--if you need to solve it "as is").Then this becomes:

\begin{align}
\langle x_3, y_3, z_3\rangle\times\vec{n}' = & \langle y_3n'_z - z_3n'_y, z_3n'_x - x_3n'_z, x_3n'_y - y_3n'_x\rangle
\end{align}

Now we are trying to find $\langle x_3, y_3, z_3\rangle$ which is the inverse.

# Matlab: how to create a function $f(x)$ and get the value of $f(x)$ given $x$?

I'm a newbie to matlab and wonder how to create a function (i.In the above example, say $x=2$, I'd expect the result to be $5$ (yep, trivial example) :-)

Thanks

The typical way to do this is to create a MATLAB.

# Question regarding Goursat's Theorem

I read through and pretty much understand most of Goursat's Theorem in $\textit{Complex Analysis}$ by Gamelin.The theorem states that if $f(z)$ is a complex-valued function on a domain $D$ such that $f'(z_0)$ exists at each point $z_0 \in D$, then $f(z)$ is analytic on $D$.

# Identifying maximal, greatest elements on a Hasse/lattice diagram?

So, while I understand the difference between maximal elements and greatest elements, I'm having trouble understanding how to identify them on a lattice diagram (just as an example, the lattice diagram of the power set of A = {1,2,3}).From what I understand (and this could be wrong), the element {1,2,3} of the power set would be both the greatest element (since {1,2,3} is drawn at least as high as every other element in the diagram) and maximal element (since nothing is drawn higher than {1,2,3}).

# Complex Numbers with the prey-predator equilibrium?

One possibility is $x=\frac{-h}{\alpha}, y=0$ and another one is $x=\frac{\gamma}{\rho}, y = \frac{\alpha}{\beta}+\frac{\rho h}{\beta\gamma}$ (plug these values in to see it for yourself).) neighborhood of any point in the state space, the nonlinear system $\dot x = f(x)$ behaves like a linear system $\dot x = Ax$.

# Must a solution defined for $(t_0,+\infty)$ with bounded limit in $+\infty$ tend to a constant solution?

Let $x'=f(t,x)$ be a differential equation with $f$ in the hypothesis of Picard's theorem.Let $\varphi$ be a solution such that its interval of definition contains $(t_0,+\infty)$ for some fixed $t_0\in \mathbb{R}$.