codethrasher

Childhood Flashback

Wed, 02 Sep 2026 23:17:46 -0700

I have very fuzzy memories of playing games like Tank Wars when I was a kid. It’s cool to be able to play them, but it’s also really cool to be able to spin them up quickly with just a few prompts in stacks I’ve never had the time to test out.

Here’s a wasm-based version of Tank Wars to waste time to.

Galactic-level Management

Sat, 15 Aug 2026 15:37:12 -0700

I’ve always been interested in simulations. In college I would write very simple simulations, graph the output and marvel at the patterns that would emerge from things like the periodic oscillations of a predator/prey population set given a stable environment. It was so much fun to do that even if I knew the outcome. It was sort of my way of playing a game without actually playing a game.

I wasn’t ever really a massive gamer. I had my various love affairs with a few games when I was in middle/high school (Age of Empires, Command & Conquer, and Zelda come to mind), but they were frequently put to the side in favor of being out skating with my friends. In college the only game I had any fascination with was Dwarf Fortress. It absolutely blew my mind when I found it (which was before its Steam release). The ASCII art felt indecipherable the first few times I eyed the screen and simply trying to understand what each little symbol represented was like learning a new language, to say nothing of the insane complexity of it.

The Feynman Technique

Tue, 17 Feb 2026 21:30:13 -0700

I’m reposting this.

I was looking through my old notes and stumbled onto this oldie. It really is such a superb way to dive into a subject.

A four-step process to test your understanding of a given subject matter.

Get a piece of paper and write the name of the technique/concept at the top
Explain the concept in the most basic language possible. Avoid all technical jargon and pretend you’re explaining this concept to someone who has zero prior knowledge.
Identify any areas that seemed shaky, or gave you pause. Go back to the source material and iron-out those wrinkles.
Comb through your explanation and remove all technical terms.

Org-mode cheatsheet

Tue, 08 Jun 2021 08:53:45 -0700

TODOs

See todo list (all entries): SPC o a t or SPC n t
Insert todo within current section (subheading): SPC m i t
Change state of todo: SPC m t (a menu will popup, d to change to done)

Blocks

insert code block SPC m i c
insert quote block SPC m i q

Zero-width character

To escape the normal org-mode syntax (to show something like the file links above) C-x 8 RET (search for ZERO WIDTH CHARACTER) RET

Motion

To fuzzy search/goto section C-c C-j (start typing)
Up to previous heading C-c C-p
Down to next heading C-c C-n

Timestamps

Inactive (doesn’t show up on the Agenda) SPC m d T
- If you simply do the above and just type in a time, e.g. 3pm it will render the below
- Or type it out manually, format: [2021-03-23 Tue 15:30]
Active (does show up on the Agenda) SPC m d t
- If you simply do the above and just type in a time, e.g. 3pm it will render the below
- Or type it out manually, format: <2021-03-23 Tue 15:30>
Or use custom command SPC m i d which will put an inactive timestamp (with time) on the line below your cursor and enter insert mode to insert a prefix string before the stamp, e.g. Started: [timestamp]

Time Management

To start a clock for the current headline (section): SPC m c i
To stop a clock for the current headline (section): SPC m c o
To create a time-management overview of the file: SPC m c R

Curl a GraphQL API

Tue, 11 May 2021 09:29:37 -0700

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curl -0 -v -X POST https://some.api.com/graphql \
-H 'Content-Type: application/json' \
-d @- << EOF
{
 "query": "query { someQuery (someArg: false) { name } }"
}
EOF

curl will probably complain about the -X POST being inferred, but it shouldn’t harm anything. It can be left out.

Function Overloading

Tue, 02 Feb 2021 09:17:33 -0800

Take a function like below…

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type Combinable = string | number

function add(a: Combinable, b: Combinable): Combinable {
 if (typeof a === 'string' || typeof b === 'string') {
 return a.toString() + b.toString()
 }
 return a + b
}

If we were to use this function like this:

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const x = add('black cat', 'white dog')
x.split(' ') // TS compiler will complain here

TypeScript cannot determine what the return type of add is at this point. All it knows is that it might be a string | number. We could cast it as a known type (i.e. as string), but this is unnecessary and more code than we should really have to write.

Nullish Coalescing

Sun, 17 Jan 2021 21:21:00 -0800

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// say you have some var hangin' around called `name`
let x: string = name ?? '(no name)'

?? is the nullish coalescing operator. It differs from || in that if you had (in the above example)

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let x: string = name || '(no name)'

name equal to an empty string (which is falsey) x would then be assigned the value (no name). The nullish coalescing operator will only use the right-hand assignment for “nullish” values (null or undefined).

Recursive Type Aliases

Sat, 16 Jan 2021 21:09:40 -0800

In TS@4 a type can reference itself, e.g.

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type JSONValue =
 | string
 | number
 | boolean
 | null
 | JSONValue[]
 | {
 [k: string]: JSONValue
 }

Previously, this would not be possible without some messy hacks.

Labeled Tuple Types

Sat, 16 Jan 2021 21:02:12 -0800

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type Address = [number, string, string, number]

Say you now have a function printAddress which takes an Address type as its arg.

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function printAddress(...address: Address) {
 // ...stuff
}

Your editor (before TS@4) would hint that the arguments were something like:

address_0: number, address_1: string, ...etc

which is not really helpful because address_0: number doesn’t really explain what that parameter corresponds to. With Labled Tuple Types we can do

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type Address = [streetNumber: number, city: string, state: string, zip: number]

NOTE: once you begin to label an element in a type definition, you must label all of the other elements.

Variadic Tuple Types

Sat, 16 Jan 2021 20:44:40 -0800

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type Foo<T extends any[]> = [boolean, ...T, boolean]

Before TS@4.0 ...T would need to be the last element, but now we can spread the T type nested between known types; e.g. “This array will have a boolean, with some stuff (strings, numbers, etc.), and another boolean”.

Composite Builds

Fri, 15 Jan 2021 22:20:07 -0800

TypeScript has a way of describing a build process as multiple subpieces of a project. This saves from having to build every piece, and instead build independent parts and stitch them together as needed.

In a monorepo environment, multiple packages will have multiple builds and, possibly, refer to those builds amongst each other.

The root tsconfig.json file can be used, but a problem of independent sources of truth arises. For instance, one project may have a different level of type strictness, or use a different version of Node. Rather than forcing every package to abide by a given rule, it is advantageous to have thin configurations for independent packages, but have them all rely on the more general rules in the root-level tsconfig.json file.

Jest Setup for a Monorepo

Fri, 15 Jan 2021 22:06:47 -0800

Out of the box, Jest mostly works in a Monorepo environment, with the exception of a few Babel plugins so that (as an example) TypeScript works.

Needs:

@babel/preset-env (Babel will transform whatever it needs to transform based on the build target, e.g. TypeScript, IE11, ES6, Node10, etc.)
@babel/preset-typescript (Babel will strip out the symbols that are specific to TypeScript)

The root-level .babelrc file would look like:

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{
 "presets": [
 ["@babel/preset-env", { "targets": { "node": 10 } }],
 "@babel/preset-typescript"
 ]
}

The target above is arbitrary, and can be anything (or nothing).

Monorepos

Fri, 15 Jan 2021 21:42:29 -0800

Colocation

Thu, 14 Jan 2021 10:49:46 -0800

“colocation” is a pattern wherein you keep the query/mutation as close to the consuming component as possible. In many instances, it’s in the exact file. This is in contrast with the DRY approach, which would beg the author to push queries/mutations higher up and imported into whichever file needs it.

Yarn-NPM

Wed, 16 Dec 2020 10:38:33 -0800

package.json

resolutions field

yarn specific

Allows you to force the use of a particular version for a nested dependency. e.g.:

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"devDependencies": {
 "@angular/cli": "1.0.3",
 "typescript": "2.3.2"
}

yarn.lock will contain:

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"typescript@>=2.0.0 <2.3.0":
 version "2.2.2"
 resolved "https://registry.yarnpkg.com/typescript/-/typescript-2.2.2.tgz#606022508479b55ffa368b58fee963a03dfd7b0c"

typescript@2.3.2:
 version "2.3.2"
 resolved "https://registry.yarnpkg.com/typescript/-/typescript-2.3.2.tgz#f0f045e196f69a72f06b25fd3bd39d01c3ce9984"

and within node_modules you’d see:

typescript@2.3.2 in node_modules/typescript
typescript@2.2.2 in node_modules/@angular/cli/node_modules.

Because TS@2.2.2 is nested within @angular/cli it is impossilbe to force the use of TS@2.3.2 across the entire project (see flattening, for a counter to this).

Cipher

Tue, 01 Dec 2020 16:08:32 -0800

A cipher is defined over the spaces of:

All Keys, \(\mathscr{K}\)

All Messages, \(\mathscr{M}\)

All Cipher texts, \(\mathscr{C}\)

Cipher (defined as a triple, \((\mathscr{K}, \mathscr{M}, \mathscr{C})\)) as a pair of algorithms \((\mathbf{E}, \mathbf{D})\) where \(\mathbf{E}\) represents the encryption algorithm and \(\mathbf{D}\) represents the decryption algorithm.

\begin{equation} \mathbf{E}: \mathscr{K} \times \mathscr{M} \rightarrow \mathscr{C} \end{equation}

and

\begin{equation} \mathbf{D}: \mathscr{K} \times \mathscr{C} \rightarrow \mathscr{M} \end{equation}

Such that:

\begin{equation} \forall m \in \mathscr{M}, k \in \mathscr{K}: \mathbf{D}(k, \mathbf{E}(k, m)) = m \end{equation}

Distribution Vector

Tue, 01 Dec 2020 15:06:23 -0800

A vector with non-negative components (representing specific probabilities) which add up to one

e.g.

\begin{equation} (P(x_{0}), P(x_{1}),…,P(x_{n})) \end{equation}

Also known as:

Stochastic Vector

Uniform Distribution

Tue, 01 Dec 2020 15:03:05 -0800

\begin{equation} x \in U:P(x) = \frac{1}{|U|} \end{equation}

Where \(|U|\) is the size of the universe (set).

In Probability Theory, a uniform distribution assigns an equal probability to each element of a given set.

Point Distribution

Tue, 01 Dec 2020 14:58:59 -0800

x₀: P(x) = 1,∀ x ≠ x₀: P(x) = 0

In Probability Theory, a point distribution is a distribution which assigns all the probability to a given point (set element).

Cryptography

Tue, 01 Dec 2020 14:46:08 -0800

Digital Signature

Digital Signature

Tue, 01 Dec 2020 13:43:16 -0800

A function of the content being “signed”

Cost Function

Tue, 01 Dec 2020 11:27:21 -0800

The measurement of accuracy of a hypothesis function. The accuracy is given as an average difference of all the results of the hypothesis from the inputs (\(x\)’s) to the outputs (\(y\)’s).

\begin{equation} J(\Theta_{0},\Theta_{1})=\frac{1}{2m}\sum_{i=1}^{m}(h_{\Theta}(x_{i}) - y_{i})^{2} \end{equation}

where \(m\) is the number of inputs (e.g. training examples)

This function is also known as the squared error function or mean squared error. The \(\frac{1}{2}\) is a convenience for the cancellation of the 2 which will be present due to the squared term being derived (see gradient descent).

Machine Learning

Tue, 01 Dec 2020 11:21:27 -0800

Gradient Descent

Tue, 01 Dec 2020 11:13:06 -0800

An optimization algorithm for finding the local minimum of a differentiable function.

(The red arrows show the minimums of \(J(\Theta_{0},\Theta_{1})\), i.e. the cost function)

To find the minimum of the cost function, we take its derivative and “move along” the tangential line of steepest (negative) descent. Each “step” is determined by the coefficient \(\alpha\), which is called the Learning Rate.

\begin{equation} \Theta_{j_{new}} := \Theta_{j_{old}} - \alpha\frac{\partial}{\partial\Theta_{j}}J(\Theta_{0},\Theta_{1}) \end{equation}

\(\Theta_{j}\), initially, will be a randomly-chosen value. At each iteration of the algorithm, we want to update the “old” parameter \(\Theta_{j_{old}}\) with the newly-calculated value. This process should occur until the slope is zero (i.e. the derivative of the cost function is zero), indicating a minimum.

Hypothesis Function

Tue, 01 Dec 2020 10:47:16 -0800

A function which maps values \(x\) to an output value \(y\). Historically, in ML, hypothesis functions are denoted \(h(x^{(i)})\).

Artificial Neural Network

Thu, 26 Nov 2020 20:23:30 -0800

Artificial Neural Network (ANN)

Layers All learning occurs in the layers. In the image, below, there are three layers, but there could be only one, or many more. In the example image the first layer is known as the Input Layer, the second the Hidden Layer, and the third the Output Layer. In a 3+ layered ANN, any layer that is not the input/output layer is a Hidden Layer.
Input The data being fed into the ANN

Covectors

Tue, 17 Nov 2020 21:59:53 -0800

A linear mapping from a vector space to a field of scalars. In other words, a linear function which acts upon a vector resulting in a real number (scalar)

\begin{equation} \alpha\,:\,\mathbf{V} \longrightarrow \mathbb{R} \end{equation}

Simplistically, covectors can be thought of as “row vectors”, or:

\begin{equation} \begin{bmatrix} 1 & 2 \end{bmatrix} \end{equation}

This might look like a standard vector, which would be true in an orthonormal basis, but it is not true generally. Instead, a covector acts as a function on a standard “column vector”, e.g.

Differential Geometry

Tue, 17 Nov 2020 09:30:10 -0800

Linear Algebra

Tue, 17 Nov 2020 09:30:00 -0800

Linear Mapping

Mon, 16 Nov 2020 22:13:48 -0800

A mapping from \(\mathbf{V} \rightarrow \mathbf{W}\) that preserves the operations of addition and scalar multiplication.

Also known as

Linear Map
Linear Transformation
Linear Function

Multilinear Map

Mon, 16 Nov 2020 22:11:24 -0800

A function of several variables that is linear, separately, in each variable.

A multilinear map of one variable is a standard linear mapping.

Tensor Product

Mon, 16 Nov 2020 21:41:03 -0800

Dual Space

Thu, 12 Nov 2020 09:58:51 -0800

The space of all linear functionals \(f:V\rightarrow \mathbb{R}\), noted as \(V^{*}\)

The dual space has the same dimension as the corresponding vector space or, given a space \(V\), with bases \((v_{1},…,v_{n})\), there exists a dual space \(V^{*}\) with a dual basis \((v^{*}_{1},…,v^{*}_{n})\).

Dual Vector Space

Thu, 12 Nov 2020 09:58:21 -0800

The space of all linear functionals \(f:V\rightarrow \mathbb{R}\), noted as \(V^{*}\)

Heliosphere

Thu, 29 Oct 2020 16:07:43 -0700

The bubble, created by the Sun’s plasma (see Solar Wind), which encompasses the Sun itself. Outside of this bubble, the Sun’s plasma is overwhelmed by the Interstellar plasma.

Solar Wind

Thu, 29 Oct 2020 16:03:32 -0700

TODO

Tensors

Thu, 29 Oct 2020 09:46:58 -0700

As a linear representation

A tensor can be represented as a vector of x-number of dimensions. Basically, a generalization on top of scalars, vectors, and matrices. The specific “flavor” of the tensor (i.e. is it a scalar, vector, or matrix) is clarified by referring to the tensor’s “rank”. For instance; a rank 0 tensor is a scalr, rank 1 tensor is a one-dimensional vector, a rank 2 tensor is a two-dimensional vector (\(2x2\) matrix), etc.

Kronecker Delta

Thu, 29 Oct 2020 09:41:12 -0700

\begin{equation} \delta_{ij}= \begin{cases} 0 & \text{if i \(\neq\) j}\\\ 1 & \text{if i \(=\) j} \end{cases} \end{equation}

Bases

Thu, 29 Oct 2020 09:39:54 -0700

a basis for an n-dimensional vector space \(V\) is any ordered set of linearly independent vectors \((\mathbf{e}_{1}, \mathbf{e}_{2},…,\mathbf{e}_{n})\)

An arbitrary vector \(\mathbf{x}\) in \(V\) can be expressed as a linear combination of the basis vectors:

\begin{equation} \mathbf{x}\,=\,\sum\limits_{i = 1}^{n} \mathbf{e}_{i}x^{i} \end{equation}

See Bases Transformation, Coordinate Transformation

Orthonormal Basis

Tue, 27 Oct 2020 20:58:35 -0700

An orthonormal basis is a basis where all the vectors are one unit long and all perpendicular to each other (e.g. the Cartesian plane)

Bases Transformation

Tue, 27 Oct 2020 11:05:44 -0700

Consider two bases \((\mathbf{e}_{1},\mathbf{e}_{2})\) and \((\mathbf{\tilde{e}}_{1},\mathbf{\tilde{e}}_{2})\), where we consider the former the old basis and the latter the new basis.

Each vector \((\mathbf{\tilde{e}}_{1},\mathbf{\tilde{e}}_{2})\) can be expressed as a linear combination of \((\mathbf{e}_{1},\mathbf{e}_{2})\):

\begin{equation} \mathbf{\tilde{e}}_{1}\,=\,\mathbf{e}_{1}S^{1}_{1}\,+\,\mathbf{e}_{2}S^{2}_{1}\\\ \tag{1.0}\\\ \mathbf{\tilde{e}}_{2}\,=\,\mathbf{e}_{1}S^{1}_{2}\,+\,\mathbf{e}_{2}S^{2}_{2} \end{equation}

(1.0) is the basis transformation formula, and the object \(S\) is the direct transformation \(\{S^{j}_{i},\,1\,\leq\,i,\,j\,\leq\,2\}\), (assuming a \(2x2\) matrix) which can also be written in matrix form:

\begin{equation} \begin{bmatrix} \mathbf{\tilde{e}}_{1} & \mathbf{\tilde{e}}_{2} \end{bmatrix}\,=\, \begin{bmatrix} \mathbf{e}_{1} & \mathbf{e}_{2}\, \end{bmatrix} \begin{bmatrix} S^{1}_{1} & S^{1}_{2}\\\ S^{2}_{1} & S^{2}_{2} \end{bmatrix}\\\ =\, \begin{bmatrix} \mathbf{e}_{1} & \mathbf{e}_{2} \end{bmatrix}\,S\tag{1.1} \end{equation}

Identity Matrix

Tue, 27 Oct 2020 10:32:05 -0700

\begin{equation} \mathbf{I}(\mathbf{X})=\mathbf{X} \end{equation}

Where any \(nxn\) matrix is established via the Kronecker Delta, e.g.

\begin{equation} \mathbf{I}_{ij}\,=\,\delta_{ij} \end{equation}

Invertible Matrix

Tue, 27 Oct 2020 10:14:42 -0700

A matrix, which when multiplied by another matrix, results in the identity matrix.

\begin{equation} \mathbf{A}\mathbf{A}^{-1} = I \end{equation}

e.g.

\begin{equation} \begin{bmatrix} a & b\\\ c & d \end{bmatrix} \begin{bmatrix} d & -b\\\ -c & a \end{bmatrix}= \begin{bmatrix} 1 & 0\\\ 0 & 1 \end{bmatrix} \end{equation}

Vector Space

Mon, 26 Oct 2020 22:29:06 -0700

also known as a linear space

A collection of objects known as “vectors”. In the Euclidean space these can be visualized as simple arrows with a direction and a length, but this analogy will not necessarily translate to all spaces.

Addition and multiplication of these objects (vectors) must adhere to a set of axioms for the set to be considered a “vector space”.

Addition (+)

\begin{equation} +\,:\,V\,\times\,V\,\longrightarrow\,V \end{equation}

Multiplication (\(\cdot\))

\begin{equation} \cdot\,:\,F\,\times\,V\,\longrightarrow\,V \end{equation}

Coordinate Transformation

Mon, 26 Oct 2020 22:23:41 -0700

Axioms (Vector Space)

Mon, 26 Oct 2020 20:54:32 -0700

To qualify as a vector space, a set \(V\) and its associated operations of addition (\(+\)) and multiplication/scaling (\(\cdot\)) must adhere to the below:

Associativity

\begin{equation} \mathbf{u}+(\mathbf{v}+\mathbf{w}) = (\mathbf{u} + \mathbf{v}) + \mathbf{w} \end{equation}

Commutivity

\begin{equation} \mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u} \end{equation}

Identity of Addition

There exists and element \(\mathbf{0}\,\in\,V\), called the zero vector, such that \(\mathbf{v} + \mathbf{0} = \mathbf{v}\) for all \(\mathbf{v}\,\in\,V\).

Inverse of Addition

For every \(\mathbf{v}\,\in\,V\), there exists an element \(\mathbf{-v}\,\in\,V\), such that \(\mathbf{v}\,+\,(\mathbf{-v})\,=\,\mathbf{0}\).

Invariance (Mathematics)

Mon, 26 Oct 2020 10:30:36 -0700

An property (of a mathematical object) is invariant if, after some operation(s) are applied, that property remains unchanged.

For instance, in a geometrical space where the concept of “length” is defined (by some metric); a physical object, say a pencil, will maintain its characteristics (length) despite a change of coordinates (e.g. polar to cartesian).

In short, vectors are invariant, but their components are not (under a transformation).

Cosmology

Fri, 23 Oct 2020 09:54:46 -0700

The Inflationary Universe

Fri, 23 Oct 2020 09:50:13 -0700

source The Inflationary Universe: A Possible Solution To The Horizon And Flatness Problems (Guth, 1980)

Questions I still have

DONE WHY#1: why is that approximation unstable?

See Flatness Problem

Abstract

The initial conditions defined in the Standard Model present two problems:

The early universe is defined to be homogeneous despite the massive distances between regions (causal disconnect)
The Hubble constant must be set very finely to produce a flat universe (like ours)

These problems could disappear if the universe (in its early stages) cooled to temperatures twenty-eight or more orders of magnitude below the critical temperature for “some phase transition”.

Standard Model (Cosmology)

Thu, 22 Oct 2020 22:03:48 -0700

This refers to the Cosmological “Standard Model”, i.e. the \(\Lambda\)CDM. This is not the same as the Standard Model of Particle Physics

Lambda-CDM

The lambda-cdm or, lambda cold-dark-matter, is a three-parameter description of the Big Bang Cosmological model parametrized by:

the Cosmological Constant \(\Lambda\)
dark matter
normal matter

Of the cosmological models, this presents the simplest one in terms of describing the universe as we see it observationally. Specifically, it succintly explains:

Cosmological Principle

Thu, 22 Oct 2020 22:01:59 -0700

The Cosmological Principle states that at large enough scales (>100Mpc) the universe’s matter distribution is isotropic and homogeneous and should not produce irregularities in the large-scale structure of the universe over the course of its evolution.

Cosmological Constant

Thu, 22 Oct 2020 21:57:03 -0700

TODO

Cosmic Microwave Background

Thu, 22 Oct 2020 21:52:44 -0700

TODO

The Structure and Interpretation of Computer Programs

Thu, 22 Oct 2020 10:15:40 -0700

Table of Contents

Chapter 1
- Predicates/Expressions
- Functions v. Procedures or Imperative v. Declarative

Chapter 1

Predicates/Expressions

Taking the following Lisp code:

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(defun abs (x)
 (cond ((> x 0) x)
 ((= x 0) 0)
 ((< x 0) (- x))))

which simply calculates the absolute value of a number, we see within the cond (conditional) three “clauses”. These parenthesized pairs (clauses) are formed of “predicates” and “expressions”, respectively.

Predicate an expression whose value is interpreted as either true or false.

Programming

Thu, 22 Oct 2020 10:11:19 -0700

The Structure and Interpretation of Computer Programs (SICP)

Manifold

Thu, 22 Oct 2020 10:07:50 -0700

A manifold is a topological space that locally resembles Euclidean space near each point.

As defined in Physics

An N-dimensional manifold of points is one for which N independent real coordinates \((x^{1}, x^{2},…,x^{N})\) are required to specify a point completely. These N coordinates are denoted collectively by \(x^{a}\), where it is understood that \(a\,=\,1,2,…,N\).

As an example, in \(\mathbb{R}^{2}\) we have a 2-dimensional manifold of points descrived by the real coordinates \((x^{1}, x^{2})\).

Symmetric Relation

Thu, 22 Oct 2020 10:07:00 -0700

This definition is more clearly described sybollically:

\begin{equation} \forall a,b \in X(aRb\,\iff\,bRa) \end{equation}

Tao’s definition:

Given any two objects \(x\) and \(y\) of the same type, if \(x = y\), then \(y = x\).

See also:

Transitive Relation

Thu, 22 Oct 2020 10:06:11 -0700

A relation \(R\) over a set \(X\) is transitive if for all elements x, y, z in \(X\), whenever \(R\) relates \(x\) to \(y\) and \(y\) to \(z\), then \(R\) also relates \(x\) to \(z\).

\begin{equation} \forall x,y,z \in X,if\,xRy\,and\,yRz,\,then\,xRz \end{equation}

Tao’s definition:

Given any three objects x, y, and z of the same type, if \(x = y\) and \(y = z\), then \(x = z\).

See also:

Symmetric Relation

Reflexive Relation

Thu, 22 Oct 2020 10:05:38 -0700

A binary relation \(R\) over a set \(X\) is reflexive if it relates every \(x \in X\) to itself.

\begin{equation} \forall x \in X \,|\, xRx \end{equation}

An example of a reflexive relation is “is equal to” since any number within \(\mathbb{R}\) would be mapped back to itself.

Tao’s definition:

Given any object x, we have \(x = x\).

See also:

Learning

Wed, 21 Oct 2020 22:31:25 -0700

The Feynman Technique

Physics

Wed, 21 Oct 2020 22:31:13 -0700

Articles

Black Holes as Primordial Dark Matter

Papers

The Inflationary Universe (Guth, 1980)

Concepts

Problems

Science

Wed, 21 Oct 2020 22:31:06 -0700

Branches

Physics

Articles

Papers

The Inflationary Universe: Guth, 1980 (Physics)

Concepts

Set Theory

Wed, 21 Oct 2020 22:31:03 -0700

Concepts

Teaching

Wed, 21 Oct 2020 22:30:58 -0700

The Feynman Technique

Math

Wed, 21 Oct 2020 22:28:30 -0700

General Concepts

Bijective Function

Injective Function

Surjective Function

Transitive Relation

Symmetric Relation

Locality

Peano Axioms

Chemistry

Wed, 21 Oct 2020 22:28:09 -0700

Articles

Phosphine, Life, and Venus

Biology

Wed, 21 Oct 2020 22:27:58 -0700

Articles

Phosphine, Life, and Venus

Triangle Inequality

Wed, 21 Oct 2020 21:30:28 -0700

For any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.

\(z \leq x + y\)

Topology

Wed, 21 Oct 2020 21:30:22 -0700

Concepts

Topological Space

Wed, 21 Oct 2020 21:30:19 -0700

An ordered pair \((X, \tau)\), where \(X\) is a set and \(\tau\) is a collection of subsets of \(X\) satisfying:

The empty set (\(\emptyset\)) and \(X\) belong to \(\tau\)
Any arbitrary (in)finite union of members of \(\tau\) still belongs to \(\tau\)
The intersection of any finite number of members of \(\tau\) still belongs to \(\tau\)

Surjective Function

Wed, 21 Oct 2020 21:30:06 -0700

A function is surjective or “onto” if

For every \(y \in Y\), there exists \(x \in X\) such that \(f(x) = y\).

Statistics

Wed, 21 Oct 2020 21:30:04 -0700

Concepts

Null Hypothesis

Shape of the Universe

Wed, 21 Oct 2020 21:29:57 -0700

Open/Closed/Flat

TODO

Relations (Sets)

Wed, 21 Oct 2020 21:29:49 -0700

Definition

A relation \(R\) from the elements of set \(A\) to the elements of set \(B\) is a subset of \(A \times B\).

…alternatively…

Let \(A\) and \(B\) be two non-empty sets, then every subset of \(A \times B\) defines a relation from \(A\) to \(B\) and ever relation from \(A\) to \(B\) is a subset of \(A \times B\).

Let \(R \subseteq A \times B\) and \((a, b) \in R\). Then we say that \(a\) is related to \(b\) by the relation \(R\) and write it as \(a R b\). If \((a, b) \in R\), we write it as \(a R b\)

Real Analysis

Wed, 21 Oct 2020 21:29:45 -0700

The theoretical foundation which underlies calculus

Principle of Relativity

Wed, 21 Oct 2020 21:29:37 -0700

The Principle of Relativity states that:

The laws of physics take the same form in every intertial frame.

No exception has been found to this principle, and it holds equally well in both Newtonian theory and Special Relativity.

Phosphine, Life, and Venus

Wed, 21 Oct 2020 21:29:29 -0700

Source: Phosphine, Life, and Venus

Regarding the released report that Venus was found to “possibly harbor life” (see this Nature article)

What’s Phosphine?

A chemical compound found in small amounts in Earth’s atmosphere
Used in: electronics fabrication, rodenticide

On Earth it’s primarily found in its oxidized form, generally thought to be the result of lightning strikes, where the required amount of electrons would be present. But, it’s also been noticed in anaerobic environments (low/no oxygen). Some studies have suggested that glucose’s presence is very effective at increasing its (phosphine’s) production.

Peano Axioms

Wed, 21 Oct 2020 21:29:27 -0700

A natural number is any element of the set \(\mathbb{N} = \{0, 1, 2, 3…\}\)

I

0 is a natural number.

II

If \(n\) is a natural number, then n++ is also a natural number.

III

0 is not the successor of any natural number; i.e., we have n++ \(\ne\) 0 for every natural number n.

IV

Different natural numbers must have different successors; i.e., if \(n\) and \(m\) are natural numbers and \(n \ne m\), then n++ \(\ne\) m++. Equivalently, if n++ = m++, then we must have n = m.

Oxidation

Wed, 21 Oct 2020 21:29:23 -0700

A type of chemical reaction in which electrons are lost

Open Balls

Wed, 21 Oct 2020 21:29:21 -0700

Let \((X, d)\) be a metric space. Let \(a \in X\) and \(\delta > 0\). The subset of \(X\) consisting of those points \(x \in X\) such that \(d(a, x) < \delta\) is the called the open ball of radius \(\delta\) and denoted:

\begin{equation} B(a;\delta) \end{equation}

Olber's Paradox

Wed, 21 Oct 2020 21:29:18 -0700

The Problem

People used to believe that the Universe was infinite (in age and size) and static. If this were true

Why is the night sky not as bright as day?

That is to say, if the universe is infinite as are the stars, and the universe is not undergoing any sort of evolution, we should expect every point of the night sky to be filled with the light from any one of its inifnite stars. But we don’t observe this. Why not?

Null Hypothesis

Wed, 21 Oct 2020 21:29:12 -0700

Denoted H_0 (subscript 0)

The position that there is no relationship among two measured phenomenon or among groups. The central task of science is to prove that a relationship may exist between entities, implying that the null hypothesis is false.

Thus, centrally, science attempts to disprove the null hypothesis for a given field of questioning. This hypothesis is the default stance when approaching a statistical/scientific problem; i.e. the null hypothesis begins as true and it is up to the investigator to prove otherwise.

Newtons Laws of Motion

Wed, 21 Oct 2020 21:29:07 -0700

I

In an intertial frame of reference (i.e. a frame of reference undergoing zero acceleration), an object at rest stays at rest (or, similarly, keeps its initial velocity) unless acted upon by an outside force.

\begin{equation} \sum \mathbf{F} = 0 \Leftrightarrow \frac{d\mathbf{v}}{dt} = 0 \end{equation}

II

In the simplest of terms:

\begin{equation} \mathbf{F} = m\mathbf{a} \end{equation}

which states that

The vector sum of the forces on an object is equal to that objects mass times the acceleration

Newtonian Spacetime

Wed, 21 Oct 2020 21:28:57 -0700

Newtonian Spacetime differs from Relativistic spacetime in that time is considered an absolute (i.e. \(t^{\prime} = t\)). Transformations between reference frames can be achieved with Galilean transformations. A particle traveling along the x axis in \(S\) at a constant speed u, has a speed \(u^{\prime}\) found by:

\begin{equation} u^{\prime}_{x} = \frac{dx^{\prime}}{dt^{\prime}} = \frac{dx^{\prime}}{dt} = \frac{dx}{dt} - v = u_{x} - v\tag{1} \end{equation}

remembering that the first derivative of position is velocity, and the second derivative of motion is acceleration

Neighborhood

Wed, 21 Oct 2020 21:28:54 -0700

The set of points surrounding a point \(p\) in a set \(V\), such that \(p \in \mathbb{R}^{n}\), within a radius \(\epsilon > 0\)

See also:

Open Balls

Metric Space

Wed, 21 Oct 2020 21:28:52 -0700

A set and a function which defines the distance between two points on that set

An ordered pair \((X, d)\), where \(X\) is an arbitrary set and \(d\) is a metric (distance-defining function).

The metric defines the following properties:

\(d(a, b) = 0\) iff \(a = b\)
\(d(a, b) > 0\) (if \(a \neq b\))
\(d(a, b) = d(b, a)\)
\(d(a, c) \leq d(a, b) + d(b, c)\)

(Note: #4 is a restatement of the Triangle Inequality)

Mechanics

Wed, 21 Oct 2020 21:28:50 -0700

Locality (Math)

Wed, 21 Oct 2020 21:28:43 -0700

Locality refers to:

A property \(P\), of a point \(x\), which holds true near every point around \(x\).

As an example:

A sphere (and, more generally, a manifold) is locally Euclidean. For every point on the sphere there is a neighborhood which is identical to Euclidean space.

Isotropy (Physics)

Wed, 21 Oct 2020 21:28:24 -0700

Isotropy means that direction is not preferred, and that there is uniformity in any orientation.

Specifically, in Astrophysics and Cosmology, it means that there is no preferred direction in space. It looks the same no matter the direction you study (look).

Isotropy in space is only true at large scales (>100Mpc), the converse is true at smaller scales (anisotropy)

Injective Function

Wed, 21 Oct 2020 21:28:21 -0700

A function is injective (“one-to-one”) if

\(x \neq x^{\prime} \Longrightarrow f(x) \neq f(x^{\prime})\)

i.e. Given a set \(X\) and a set \(Y\), no two elements (say, \(x\) and \(x^{\prime}\)) from \(X\) map to the same element in \(Y\).

(Note: a function can be both injective and surjective, this image is injective only)

Inertial Frames

Wed, 21 Oct 2020 21:28:18 -0700

Above are two frames in Cartesian coordinates, \(S\) and \(S^{\prime}\). We have coordinates \((x, y, z)\), which define our dimensions.

Inertial Frame

An inertial frame is a frame of reference for which acceleration is zero. In other words:

\begin{equation} \frac{d^{2}x}{dt^{2}} = \frac{d^{2}y}{dt^{2}} = \frac{d^{2}z}{dt^{2}} = 0 \end{equation}

In the absence of gravity if \(S\) and \(S^{\prime}\) are two inertial frames they can only differ from each other by (and/or):

Homogeneous (Physics)

Wed, 21 Oct 2020 21:28:15 -0700

Homogeneous means that location is not preferred, that the physics are the same at any point.

Specifically, in Astrophysics and Cosmology, it means that there is no preferred direction in space, it behaves the same no matter where you (as an observer) stand.

This is subtly different from isotropy, which is only concerned with the physical characteristics of the object (universe) rather than the object’s behavior (wrt physical law).

Generalized Coordinates

Wed, 21 Oct 2020 21:28:12 -0700

Generalized coordinates are useful when calculating Lagrangians or Hamiltonians.

The term itself, refers to the parameters that describe the configuration of a system.

For instance, to describe the position of a point on a circle, one could use its rectangular (Cartesian) coordinates, spherical coordinates, or cylindrical coordinates. Buth with generalized coordinates a reference doesn’t need to be made to the specific coordinate system, rather it could be abstracted into something more general as coordinates \((q_{1}, q_{2},…q_{n})\). Describing the position of a point particle in Euclidean space could be described in terms of its Cartesian coordinates, and translating that to generalized coordinates would also require 3 constituents, i.e. \((q_{1}, q_{2}, q_{3})\).

General Relativity

Wed, 21 Oct 2020 21:28:09 -0700

Concepts

Galilean Transformations

Wed, 21 Oct 2020 21:28:07 -0700

In Newton’s formulation of spacetime, time is an absolute; meaning that every observer experiences the same flow of time. Assuming an inertial frame with only translation across the x axis, an event \(P\), can be translated between \(S\) and \(S^{\prime}\) by the following linear transformations:

\begin{equation} t^{\prime} = t\tag{1} \end{equation}

\begin{equation} x^{\prime} = x - vt\tag{2} \end{equation}

\begin{equation} y^{\prime} = y\tag{3} \end{equation}

\begin{equation} z^{\prime} = z\tag{4} \end{equation}

Switching between coordinates (\(S\) v. \(S^{\prime}\)) requires only switching the sign on \(v\).

Flatness Problem

Wed, 21 Oct 2020 21:28:05 -0700

Overview

In the Big Bang model there are multiple parameters which appear to be “fine tuned” in that, small changes to a given parameter have massive effects on the evolution of the universe as a whole.

Wrt the flatness problem itself, the questionable parameter is the Density Parameter (\(\Omega\)), which is defined as

\begin{equation} \Omega \equiv \frac{\rho}{\rho_{c}}\tag{1} \end{equation}

where the subscript \(c\) is in reference to the “critical value” of density (\(\rho\)). Which, today, is measured to be–roughly–five atoms of Hyrodgen per cubic meter. \(\rho_{c}\) is defined:

Equations of Motion

Wed, 21 Oct 2020 21:28:03 -0700

Linear Motion

(under constant \(\hat{\mathbf{a}}\))

\begin{equation} \mathbf{v} = \mathbf{a}t + \mathbf{v_{0}}\tag{1} \end{equation}

\begin{equation} \mathbf{r} = \mathbf{r_{0}} + \mathbf{v_{0}}t + \frac{1}{2}\mathbf{a}t^{2}\tag{2} \end{equation}

\begin{equation} \mathbf{r} = \mathbf{r_{0}} + \frac{1}{2}(\mathbf{v} + \mathbf{v_0})t\tag{3} \end{equation}

\begin{equation} \mathbf{v^{2}} = \mathbf{v^{2}_0} + 2\mathbf{a}(\mathbf{r} - \mathbf{r_{0}})\tag{4} \end{equation}

\begin{equation} \mathbf{r} = \mathbf{r_{0}} + \mathbf{v}t - \frac{1}{2}\mathbf{a}t^{2}\tag{5} \end{equation}

where:

\(\mathbf{r_{0}}\) is the initial position
\(\mathbf{r}\) is the final position
\(\mathbf{v_{0}}\) is the initial velocity
\(\mathbf{v}\) is the final velocity
\(\mathbf{a}\) is the acceleration
\(t\) is the time interval

Emacs Notes

Wed, 21 Oct 2020 21:28:00 -0700

Emacs notes (doom emacs)

Org

handy cheatsheet: https://orgmode.org/orgcard.pdf

To get a TODO item created on the same level as the one in which you’re working (org-insert-todo-heading)
Add tag: To add a tag to a heading (* line) SPC m q (or C-c C-q) then type the tag name, should appear in gray
Tag search: To search all the tags: SPC o a m
Schedule/Deadline: To set something as scheduled in org-agenda (so it shows up when you SPC o A n) SPC m d s, similarly for setting a deadline for something SPC m d d

Degenerate Coordinate Systems

Wed, 21 Oct 2020 21:27:55 -0700

Any coordinate system which cannot describe an entirety of a space

As an example, polar coordinates \(r,\psi)\) are degenerate at \((0, \psi)\)

Curvature

Wed, 21 Oct 2020 21:27:53 -0700

TODO

Cosmic Inflation

Wed, 21 Oct 2020 21:27:51 -0700

Inflation

Inflation (Cosmic) is a theory put forth (originally, by Alan Guth in his 1980 paper) to explain why, if the universe is full of mass, is it not curved in some way? Instead, it’s flat (see: Curvature).

This isotropy in its temperature/geometry seems to contradict Thermodynamics in that we should see varying pockets of temperature around us since it would take time for the heat to dissipate throughout the universe.

The theory of Cosmic Inflation gets around this issue by stating that the baby universe expanded faster than light, which prevented anisotropy.

Configuration Space

Wed, 21 Oct 2020 21:27:48 -0700

The vector space defined by generalized coordinates.

Example:

The position of a particle in Euclidean 3-space is defined by the generalized coordinate \(q = (x, y, z)\), therefore its configuration space is \(Q = \mathbb{R}^{3}\)

Cartesian Product

Wed, 21 Oct 2020 21:27:43 -0700

If \(X\) and \(Y\) are sets, then we define the Cartesian Product \(X \times Y\) to be the collection of ordered pairs, e.g. \((x, y)\), whose first component lies in \(X\) and whose second component lies in \(Y\)

\(X \times Y = \{(x, y) | x \in X, y \in Y\}\)

An example would be \(\mathbb{R} \times \mathbb{R} = \mathbb{R}^{2}\) (i.e. the cartesian plane). \(\mathbb{R}^{2}\) is the cartesian product of “crossing” \(\mathbb{R}\) with itself.

Cardinality

Wed, 21 Oct 2020 21:27:40 -0700

Equality in Cardinality

Two sets \(X\) and \(Y\) are equal iff there exists a bijection \(f : X \longrightarrow Y\) from \(X\) to \(Y\).

Black Holes as Primordial Dark Matter

Wed, 21 Oct 2020 21:27:37 -0700

source: Physicists Argue That Black Holes From the Big Bang Could Be the Dark Matter

Primordial Black Holes

A “Primordial Black Hole” (PBH), is a black hole which has been around since nearly the time of the Big Bang.

PBH as Dark Matter

The hypothesis that PBHs could be the dark matter cosomolgists tangentially observe was put forth in many papers (like this one from Johns Hopkins U.) around 2016. This idea was dismantled by the calculation which showed that if black holes had been around as long as PBHs are/were thought to have been around for, and in high enough numbers, we would see more pairs forming. These pairs would, through gravity, begin to orbit each other and be detectable by LIGO, observationally. We don’t observe this, so the idea (then) was abandoned.

Black Hole Information Paradox

Wed, 21 Oct 2020 21:27:34 -0700

High-level

The result of the marrying of QM with GR. (Initially, Hawking’s) calculations suggested that information would permanently disappear in a black hole. This runs afoul with the postulates of the Copenhagen Interpretation, in that information about a system is encoded in its wave function \(\Psi\) and, further, that that information is conserved.

TODO more technical details

Binary Relation

Wed, 21 Oct 2020 21:27:27 -0700

A binary relation \(R\) over sets \(X\) and \(Y\) is a subset of the cartesian product of \(X \times Y\). Where \(X\) is the domain, or set of departure of \(R\), and \(Y\) is the codomain, or set of destination of \(R\).

An element \(x \in X\) is related to \(y \in Y\), iff the ordered pair \((x, y)\) is found within the (above) subset.

Abiotic Process

Wed, 21 Oct 2020 21:27:20 -0700

An abiotic process is a process which occurs via abiotic resources (non-living physical/chemical elements) in the environment

Abiotic Process

Wed, 21 Oct 2020 21:27:12 -0700

An abiotic process is a process which occurs via abiotic resources (non-living physical/chemical elements) in the environment

Bijective Function

Wed, 21 Oct 2020 21:26:54 -0700

A function is bijective or “invertible” if it is

Both one-to-one and onto (injective and surjective)

In other words: each element of one set is paired with exactly one element of the other set, and vice versa

About

Mon, 01 Jan 0001 00:00:00 +0000

I’m Matt.

Welcome to the blog. It’s at times incoherent and unfocused… a general dumping ground of thought.

I’m an Engineering leader with well over a decade of hands-on technical experience scaling and architecting systems. I’m motivated by long-term mastery and uncovering the shadows in my own thinking. I don’t like ambiguity.

Outside of my day job I prioritize the physical: Jiu Jitsu, skateboarding, and slowing down the steady march of time.

Set Theory

Mon, 01 Jan 0001 00:00:00 +0000

codethrasher

Childhood Flashback

Galactic-level Management

The Feynman Technique

Org-mode cheatsheet

Tags

Links

TODOs

Blocks

Zero-width character

Motion

Timestamps

Time Management

Curl a GraphQL API

Function Overloading

Nullish Coalescing

Recursive Type Aliases

Labeled Tuple Types

Variadic Tuple Types

Composite Builds

Jest Setup for a Monorepo

Monorepos

Colocation

Yarn-NPM

package.json

resolutions field

Cipher

Distribution Vector

Uniform Distribution

Point Distribution

Cryptography

Digital Signature

Cost Function

Machine Learning

Gradient Descent

Hypothesis Function

Artificial Neural Network

Covectors

Differential Geometry

Linear Algebra

Linear Mapping

Also known as

Multilinear Map

Tensor Product

Dual Space

Dual Vector Space

Heliosphere

Solar Wind

Tensors

As a linear representation

Kronecker Delta

Bases

Orthonormal Basis

Bases Transformation

Identity Matrix

Invertible Matrix

Vector Space

Coordinate Transformation

Axioms (Vector Space)

Associativity

Commutivity

Identity of Addition

Inverse of Addition

Invariance (Mathematics)

Cosmology

The Inflationary Universe

Questions I still have

DONE WHY#1: why is that approximation unstable?

Abstract

Standard Model (Cosmology)

Lambda-CDM

Cosmological Principle

Cosmological Constant

Cosmic Microwave Background

The Structure and Interpretation of Computer Programs

Chapter 1

Predicates/Expressions

Programming

Manifold

As defined in Physics