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Modular Arithmetic Library

Modular Arithmetic Library in C++

This library provides the basics of modular arithmetic operations:

Great Common Divisor (GCD)
Least Common Multiple (LCM)
Bezout solver
Decomposition number algorithm
Euler’s totient function
$a^{b}(mod\:m)$
Primality Test
Find next Prime
Inverse function
Solver for 1 equation
Solver for multiple equations

-GitHub repository:

Modular Arithmetic Source Code

-Article

Modular Arithmetic Article

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Fluid Dynamics

Navier-Stokes Equation

$\rho$ [ $kg/m^3$ ] : fluid density

$\mu$ [ $Pa\cdot s$ ] : dynamic viscosity

$\nu$ [ $N/\rho=m^2/s$ ] : kinematic viscosity

$\vec{V}=(u,v,w)$ [ $m/s$ ] : velocity

$\forall$ : volume

Conservation of mass

For a system:
$\frac{d}{dt}M_{SYS}=0$
For a C.V. (Reynolds transport theorem):\]

$\frac{\partial}{\partial t}\int_{CV}{\rho d\forall}+\int_{CS}{\rho \vec{V}\cdot \hat{n} dA}=0$

Differential form (\delta x\delta y\delta z):

$\frac{\partial}{\partial t}\int_{CV}{\rho d\forall}=\frac{\rho}{\partial t} \delta x\delta y\delta z$

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RNN, LSTM & GRU

Recurrent Neural Network (RNN), Long-Short Term Memory (LSTM) & Gated Recurrent Unit (GRU)

Is a type of artificial neural network where connections between nodes form a sequence. This allows temporal dynamic behavior for time sequence.
There are 3 types of vanilla recurrent neural network: the simple (RNN), gated recurrent unit (GRU) and long short term memory unit (LSTM).

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Convolutional Neural Network (CNN)

Convolutional neural network introduction and tutorial

Introduction

Convolutional neural network (CNN or ConvNet) is a type of neural network used in artificial intelligence that is commonly applied to analyzing images.

They can be considered a pre-processing compared to image classification algorithms. They have applications in image and video recognition, recommender systems, image classification, natural language processing, etc.

Notation

$f^l$ : filter size.
$p^l$ : padding.
$s^l$ : stride.
$n^{l-1}_H \times n^{l-1}_W \times n^{l-1}_C$ : input.
$n^{l}_H \times n^{l}_W \times n^{l}_C$ : output.
$n^l = \floor{\frac{n^{l-1}+2p^l-f^l}{s^l}+1}$ .
$f^l \times f^l \times n_c^l$ : filter.
$a^l$ : activation function.
$n_c^l$ : bias.