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How do you call this system?
Is there a specific name for a dynamical system that depends on the relative indexation ${\displaystyle i\pm k}$ for some k? For example, consider the following dynamical system defined on a ring of cells by
$\begin{array}{rl}{\dot{u}}_i& =u_i+2(v_{i-1}+v_{i+1})\\ {\dot{v}}_i& =u_i-v_i\end{array}$
for each cell i, where the derivative is with respect to time t.
The main reason I ask this is because I won't to compare this kind of systems with systems involving spatial coordinates, u(x,t), as in reaction-diffusion equations.

I'm trying to determine the convergence of this series:
$\sum _{n=1}^{\infty }(\frac{{\displaystyle 1}}{{\displaystyle 2}}·\frac{{\displaystyle 3}}{{\displaystyle 4}}·\frac{{\displaystyle 5}}{{\displaystyle 6}}·...\frac{{\displaystyle 2n-3}}{{\displaystyle 2n-2}}·\frac{{\displaystyle 2n-1}}{{\displaystyle 2n}}{)}^a$

Find a polynomial f(x) of degree 3 that has the following zeros. 8 ( multiplicity 2), -4.

Find the values of b such that the function has the given maximum value.

Maximum value: 62

f(x) = −x2 + bx − 19 

b =            (smaller value)

b =           (larger value)

Find a matrix such that the systems are equivalent
Take the system
$$\{\begin{array}{l}w=v^{\prime }\\ v^{″}-\mu v^{\prime }+v=0\end{array}$$
for ${\displaystyle \mu \in \mathbb{R}}$
Find a constant matrix ${\displaystyle A^{\mu }}$ such that the above system is equivalent to
$\left[\begin{array}{c}v\text{'}\\ w\text{'}\end{array}\right]=A^{\mu }\left[\begin{array}{c}v\text{'}\\ w\text{'}\end{array}\right]$
My first thought was to find a solution to ${\displaystyle v{ }^{″}-\mu v^{\prime }+v=0}$. Taking the characteristic polynomial as ${\displaystyle r^2-\mu r+1=0}$, I get ${\displaystyle r=\frac{1}{2}(\mu \pm \sqrt{{\mu }^2-4})}$. Not really sure what to do at this point, however. Another way to solve this might be first to replace v' in the second equation with w to get ${\displaystyle v{ }^{″}-\mu w+v=0}$ but I'm not sure where to go from there either.

Consider the IVP
{ y' = 2y
{y(0) = 1
a) Find the exact solution of the IVP
b) Using step ℎ = 0.2, find an approximate solution of the PVI at point = 1.0
b1) Euler's method
b2) Improved Euler Method
b3) Fourth-order Runge-Kutta method
c) Compare the approximate solutions obtained in the previous items with the exact solution.

a) Consider the following functional relation, f, defined as:
${\displaystyle f:R\to R,f\left(x\right)=x^2}$
Determine whether or not fis a bijection. If it is, prove it. If it is not, show why it is not 
b) Consider the set 
${\displaystyle F=\{y\mid y=a{x}^3+b\}}$
a, b being constants such that $a\ne 0$ and ${\displaystyle x\in R}$. 
Is ${\displaystyle F=R}$? If so, prove it. If not, explain in details why it is not the case.