Evolution of SNR

http://www.astro.uu.se/~hoefner/astro/teach/apd_files/apd_SNR_dyn.pdf
Size distribution of supernova remnants and the interstellarmedium: the case of M 33
https://www.astro.umd.edu/~richard/ASTR680/supernova%20remnants.pdf
https://heasarc.gsfc.nasa.gov/docs/objects/snrs/snrstext.html

TERMUX INSTALL MATPLOTLIB error: ft2build.h / fthread.h

error: can not find ft2build.h

ft2build.h in the below directory: /data/data/com.termux/files/usr/include/freetype2

solution:

Reference:

https://www.reddit.com/r/termux/comments/bhiinx/problem_installing_matplotlib/

https://github.com/matplotlib/matplotlib/issues/3029/

From pitch angle diffusion to spatial diffusion coefficient

$$dp/dt = qv/c \times (B_0 + \delta B)$$
$$dp_{\parallel}/dt  = q/c (v_{\perp} \times \delta B)$$
$$p_{\perp} = \mu p$$
$$d\mu/dt = q/(pc)v(1-\mu^2)^{1/2} \delta B \cos(\Omega t -kx +\psi)$$
$$\delta B(x,t) = \delta B \cos(\Omega t -kx +\psi)$$

Averaging over random phase of the waves $\psi$,
$$\int_0^{2\pi} \cos^2\psi d\psi = \pi$$
$$\cos(a+b) = \cos(a)\cos(b) - \sin(a)\sin(b)$$
in the frame in which the wave is at rest, we can write $x=\mu v t$,
$$\langle \delta \mu (t')\delta \mu (t'') \rangle_{\psi} = \frac{q^2v^2(1-\mu^2)(\delta B)^2}{2c^2p^2} \cos[(\Omega -kv\mu)(t-t'')]$$ .

Integrating over time,
$$\langle \delta \mu (t')\delta \mu (t'') \rangle_{t} = \frac{q^2v^2(1-\mu^2)(\delta B)^2}{2c^2p^2}  \int dt' \int dt'' \cos[(\Omega -kv\mu)(t-t'')]  \\ = \frac{q^2 v(1-\mu^2)(\delta B)^2}{c^2 p^2 \mu} \delta(k-\Omega/(v\mu)) \delta t$$ .

Chromebook linux beta

Open Chrome OS developer shell  Ctrl + Alt + T
vmc list
vmc start termina 

check for default Linux container 
lxc list

If the penguin container is listed, you can try manually starting it 
logout
vmc container termina penguin

destroy a virtual machine

vmc destroy vmc_name

Reference:
https://support.google.com/chromebook/thread/6127686?hl=en

Integration to sum

The definite integral of a continuous function $f$ over the interval $[a,b]$, denoted by $\int_a^b f(x) dx$, is the limit of a Riemann sum as the number of subdivisions approach infinity.
$$\int_a^b f(x)dx = \lim_{n \to \infty} \sum_{i=1}^n \delta x \cdot f(x_i)$$
where $\delta x = (b-a)/n$, and $x_i = a+\delta x \cdot i$.

Example
$$\int_{\pi}^{2\pi} cos(x)dx$$
$$\delta x = (b-a)/n = (2\pi-\pi)/n = pi/n$$
$$x_i = a + \delta x \cdot i = \pi + \pi/n \cdot i$$
Therefore
$$\int_{\pi}^{2\pi} \cos(x)dx = \lim_{n\to \infty}\sum_{i=1}^{n} \frac{\pi}{n}\cdot \cos(\pi+\frac{\pi i}{n})$$

Reference:
https://www.khanacademy.org/math/ap-calculus-ab/ab-integration-new/ab-6-3/a/definite-integral-as-the-limit-of-a-riemann-sum

Where to register your kindle with vitrification code

www.amazon.com/code

Reference:
https://uk.amazonforum.com/s/question/0D54P00006zSmkrSAC/new-paperwhite-where-do-i-enter-the-verification-code

Math resources

residue theorem
https://math.mit.edu/~jorloff/18.04/notes/topic8.pdf

A table of integrals of exponential integral - NIST Page
https://nvlpubs.nist.gov/nistpubs/jres/73B/jresv73Bn3p191_A1b.pdf

Ten lessons I should have been taught. Lecture delivered by Gian-Carlo Rota at MIT on April 20, 1996, on occasion of the Rotafest.
Gian-Carlo Rota

USEFUL VECTOR AND TENSOR OPERATIONS
https://onlinelibrary.wiley.com/doi/pdf/10.1002/9781118127575.app1

Ordinary Differential Equations

// with very good examples

Riccati equation

$$dy/dx = a(x) + b(x)y + c(x)y^2$$

quasi-linear newton Broyden’s Method

https://faculty.math.illinois.edu/~mlavrov/docs/484-spring-2019/ch3lec6.pdf

Table of Approximations for Derivatives // important for finite difference

https://www.geometrictools.com/Documentation/FiniteDifferences.pdf

SDE 

Computational solution of stochastic differential equations

A fast stimulation method for particle acceleration

A more accurate numerical scheme for diffusive shock acceleration

A Numerical Method for the Simulation of Skew Brownian Motion and its Application to

Diffusive Shock Acceleration of Charged Particles

Finite difference methods for diffusionprocesses (python)

Coordinate Transformation to get non uniform grid for finite difference method  pde

https://www.mathematik.uni-dortmund.de/~kuzmin/cfdintro/lecture4.pdf  pde

Two point bounary value problem solver: bvp solver

pde

https://py-pde.readthedocs.io/en/latest/index.html

https://www.ctcms.nist.gov/fipy/index.html

https://github.com/vladimir-ch/odeity

github.com/maroba/findiff

Scikit-fdiff // finite difference + method of line to solve PDE

course for FIPY

http://basilisk.fr/src/README 

laptops

HP Laptop - 15t   $579         min 480$
10th Gen Intel® Core™ i7 processor

2019 Dell Inspiron 14" Laptop Computer| 10th Gen Intel Quad-Core i5 1035G4  $398

HP 17.3" Laptop - 10th Gen Intel Core i5-10210U  (1600 x 900) Touchscreen  12G RAM 499

HP 17.3" HD+ Laptop, Intel Core i7-8565U 1600*900 8GB Memory, 256GB SSD  569

HP 17t-by300  i7-1065G7 17.3" 1600*900 8G RAM 256 GB SSD 559.99  12G RAM 599.99

HP 17t-by400 i7-11 1600*900 8G RAM 256 SSD  569.99 // 12G RAM 609.99

Hp 15t-by100 i7-1065G7 12G RAM 15.6'' 1920*1080 256 SSD 529

HP Pavilion Laptop - 15z-eh000  Ryzen™ 7 4700U 12G RAM  15.6'' 1920*1080 256 SSD  579.99 

HP 15-ef1073od Laptop, 15.6" Screen, AMD Ryzen 7, 16GB Memory, 256GB Solid State Drive, Wi-Fi 6

559

Inspiron 15 5000 Laptop 630

AMD 4700, 8G, 256 NvME 15.6'', 1920*1080

5505 16G, 512ssd, 779