New articles - OurBigBook.com

Dirac Lagrangian by

Ciro Santilli  40 Updated 2025-07-16

L = \overset{ˉ}{ψ} (i ℏ c \partial / - m c^{2}) ψ

where:

$\partial$ : Feynman slash notation
$\overset{ˉ}{ψ}$ : Dirac adjoint

Remember that

ψ

is a 4-vetor, gamma matrices are 4x4 matrices, so the whole thing comes down to a dot product of two 4-vectors, with a modified

ψ

by matrix multiplication/derivatives, and the result is a scalar, as expected for a Lagrangian.

Like any other Lagrangian, you can then recover the Dirac equation, which is the corresponding equations of motion, by applying the Euler-Lagrange equation to the Lagrangian.

Cryogen-free dilution refrigerator by

Ciro Santilli  40 Updated 2025-07-16

Dilution refrigerator manufacturer by

Ciro Santilli  40 Updated 2025-07-16

Frequency divider by

Ciro Santilli  40 Updated 2025-07-16

Huygens-Fresnel principle by

Ciro Santilli  40 Updated 2025-07-16

Developmental genetics by

Ciro Santilli  40 Updated 2025-07-16

How genes form bodies.

Video 1.

Developmental Genetics 1 by Joseph Ross (2020)

Source. Talks about homeobox genes.

Heavy water by

Ciro Santilli  40 Updated 2025-07-16

Cody'sLab had a nice 5 video series on making it at home! But the United States Government asked him to take it down as suggested at Video "What's Been Going On With Cody'sLab? by Cody'sLab (2019)" at youtu.be/x1mv0vwb08Y?t=84.

Here's a copy online as of 2020: www.youtube.com/watch?v=bCXB6BdMh9Y

Pre-order depth-first search by

Ciro Santilli  40 Updated 2025-07-16

This is the order in which you would want to transverse to read the chapters of a book.

Like breadth-first search, this also has the property of visiting parents before any children.

In-order depth-first search by

Ciro Santilli  40 Updated 2025-07-16

This is the order in which a binary search tree should be traversed for ordered output, i.e.:

everything to the left is smaller than parent
everything to the right is larger than parent

This ordering makes sense for binary trees and not k-ary trees in general because if there are more than two nodes it is not clear what the top node should go in the middle of.

This is unlike pre-order depth-first search and post-order depth-first search which generalize obviously to general trees.

Post-order depth-first search by

Ciro Santilli  40 Updated 2025-07-16

Has the property of visiting all descendants before the parent.

The orthogonal group is the group of all matrices that preserve the dot product by

Ciro Santilli  40 Updated 2025-07-16

When viewed as matrices, it is the group of all matrices that preserve the dot product, i.e.:

O (n) = O \in M (n) ∣\forall x, y, x^{T} y = (O x)^{T} (O y)

This implies that it also preserves important geometric notions such as norm (intuitively: distance between two points) and angles.

This is perhaps the best "default definition".

The orthogonal group is the group of all invertible matrices where the inverse is equal to the transpose by

Ciro Santilli  40 Updated 2025-07-16

Let's show that this definition is equivalent to the orthogonal group is the group of all matrices that preserve the dot product.

Note that:

x^{T} y = (O x)^{T} (O y) = x^{T} O^{T} O y

and for that to be true for all possible

x

and

y

then we must have:

O^{T} O = I

(2)

i.e. the matrix inverse is equal to the transpose.

Conversely, if:

O^{T} O = I

(3)

is true, then

(O x)^{T} (O y) = x^{T} (O^{T} O) y = x^{T} y

(4)

These matricese are called the orthogonal matrices.

TODO is there any more intuitive way to think about this?

The orthogonal group is the group of all matrices with orthonormal rows and orthonormal columns by

Ciro Santilli  40 Updated 2025-07-16

Or equivalently, the set of rows is orthonormal, and so is the set of columns. TODO proof that it is equivalent to the orthogonal group is the group of all matrices that preserve the dot product.

Taylor expansion definition of the exponential function by

Ciro Santilli  40 Updated 2025-07-16

The Taylor series expansion is the most direct definition of the expontial as it obviously satisfies the exponential function differential equation:

the first constant term dies
each other term gets converted to the one before
because we have infinite many terms, we get what we started with!

e^{x} = \sum_{n = 0}^{\infty} \frac{x ^{n}}{n !} = 1 + \frac{x}{1} + \frac{x ^{2}}{2} + \frac{x ^{3}}{2 \times 3} + \frac{x ^{4}}{2 \times 3 \times 4} + \dots