The concept of Neural Network can be illustrated by the MNIST number recognizing data set. Each letter can be broken up into 28X28 pixels, with each cell given a value indicating its brightness, then the whole 28X28 pixel elements are flatten to one line/row = 28*28 = 784 cells. For each cell, a random value … Continue reading Back Propagation Calculus in Neural Network

# Author: Naixian Zhang

# Fourier Transform

Fourier Transform is highly regarded for its impact in advancing modern technology, while its format nevertheless looks quite similar to Laplace Transform. Let's take a look: Recall from the last blog, I re-pasted the Laplace Transform below: Laplace transform, from my view, seems to help solve abstract DE as a powerful tool, while Fourier Transform … Continue reading Fourier Transform

# From Taylor Series to Euler Equation

When I was in middle school, I was always puzzled why those mathematicians would invent complex, irrational number, negative number on top of positive number. Why bother and make the route to grasp math more difficult? I took some research on Euler equation, the famously most elegant one I am amazed by and greatly interested … Continue reading From Taylor Series to Euler Equation

# Laplace Transform

According to the wiki, "Laplace Transform" transforms a function of a real variable t (often time) to a function of a complex variables (complex frequency). The transform has many applications in science and engineering. It's similar to Fourier transform with a difference that Fourier transform of a function is a complex function of a real variable, while the Laplace … Continue reading Laplace Transform

# Eigenvector And Differential Equations

I stumbled upon an article by Changwei Jun - if my memory serves me correctly - that he recalled many years later when he was in college that all the differential equations he was forced by his middle/high school teachers to memorize mechanically suddenly make full sense for him when he saw the connection between … Continue reading Eigenvector And Differential Equations

# Solve Twin Paradox by Applying Minkowski Space Geometry

According to Einstein's special relativity, time inflates while length contracts in a moving object with high speed comparable to light speed. So people accepts that if one goes out to outer space he/she will be younger relative to the people on earth. However, speed or velocity is a relative concept - we constantly fall back … Continue reading Solve Twin Paradox by Applying Minkowski Space Geometry

# Journey To Learn The Glorious Maxwell Equations_4_Integral and Differentiation Forms

The Maxwell Equations contains four components. The first is about the electric field, the second is about the magnetic field, the third is about how the change electric field generates electricity, the last one is conversely, how the change of electric field or/and electric current generates magnetism. Previously, classical Newtonian physicists have deduced the following … Continue reading Journey To Learn The Glorious Maxwell Equations_4_Integral and Differentiation Forms

# Journey To Learn The Glorious Maxwell Equations_3_Math Tools

I encountered a big obstacle in the journey of grasping Maxwell Equations because there are some math tools I did't understand yet. Concepts such as scalar, vector, vector's multiplication, gradient, partial differentiation etc. So I need to get into these math first. Chanwei Jun provided the best lectures I ever have read to explain these … Continue reading Journey To Learn The Glorious Maxwell Equations_3_Math Tools

# Journey To Learn The Glorious Maxwell Equations_2

Going over Newtonian theory, energy and momentum are conservative, for one particle object, we have the below equation, in which, K represents kinetic energy, approximated as m c square plus the famous 1/2mv^2. The gravitational force is similar to electric force The U here denotes the concept of potential energy, then we deduce and understand … Continue reading Journey To Learn The Glorious Maxwell Equations_2

# Journey To Learn The Glorious Maxwell Equations_1

Prof. E. Carson's lectures are quite clear, supplemental to Prof. Shankar's Yale courses: https://www.youtube.com/watch?v=v57B_1ZBAho&list=PLZ6kagz8q0bvxaUKCe2RRvU_h7wtNNxxi . Coulomb's law What's if it's not a point charge in the center, but a sphere shell? This is when the superposition principle comes into play: Placing a point charge in a place, and then place an atom to a distance … Continue reading Journey To Learn The Glorious Maxwell Equations_1