https://codethrasher.com/Recent content on codethrasherHugoen-usTue, 17 Feb 2026 21:24:29 -0800https://codethrasher.com/posts/the_feynman_technique/Tue, 17 Feb 2026 21:30:13 -0700https://codethrasher.com/posts/the_feynman_technique/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.https://codethrasher.com/posts/org_mode_cheatsheet/Tue, 08 Jun 2021 08:53:45 -0700https://codethrasher.com/posts/org_mode_cheatsheet/Tags Search tags: SPC n m or SPC o a m Add tag: SPC m q Links Insert link: SPC m l l File links have the structure [[file:relative_link_to_file][some_title]​] To link to a specific heading in a file ​[[file:relative_link::header title][some_title]] header title can have spaces in it, e.g. relative_link.org::A Heading in a file External (web) links have the structure [​[full_url][some_title]​] To see the link (so you can more easily edit it): SPC m l t To store a link to the current file/headline you’re in: SPC n l and to “paste” that stored link: SPC m l l (Return) 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.https://codethrasher.com/posts/curl_a_graphql_api/Tue, 11 May 2021 09:29:37 -0700https://codethrasher.com/posts/curl_a_graphql_api/1 2 3 4 5 6 7 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.https://codethrasher.com/posts/function_overloading/Tue, 02 Feb 2021 09:17:33 -0800https://codethrasher.com/posts/function_overloading/Take a function like below… 1 2 3 4 5 6 7 8 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: 1 2 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.https://codethrasher.com/posts/nullish_coalescing/Sun, 17 Jan 2021 21:21:00 -0800https://codethrasher.com/posts/nullish_coalescing/1 2 // 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) 1 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).https://codethrasher.com/posts/recursive_type_aliases/Sat, 16 Jan 2021 21:09:40 -0800https://codethrasher.com/posts/recursive_type_aliases/In TS@4 a type can reference itself, e.g. 1 2 3 4 5 6 7 8 9 type JSONValue = | string | number | boolean | null | JSONValue[] | { [k: string]: JSONValue } Previously, this would not be possible without some messy hacks.https://codethrasher.com/posts/labeled_tuple_type/Sat, 16 Jan 2021 21:02:12 -0800https://codethrasher.com/posts/labeled_tuple_type/1 type Address = [number, string, string, number] Say you now have a function printAddress which takes an Address type as its arg. 1 2 3 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 dohttps://codethrasher.com/posts/variadic_tuple_types/Sat, 16 Jan 2021 20:44:40 -0800https://codethrasher.com/posts/variadic_tuple_types/1 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”.https://codethrasher.com/posts/composite_builds/Fri, 15 Jan 2021 22:20:07 -0800https://codethrasher.com/posts/composite_builds/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.https://codethrasher.com/posts/jest_setup_for_a_monorepo/Fri, 15 Jan 2021 22:06:47 -0800https://codethrasher.com/posts/jest_setup_for_a_monorepo/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: 1 2 3 4 5 6 { "presets": [ ["@babel/preset-env", { "targets": { "node": 10 } }], "@babel/preset-typescript" ] } The target above is arbitrary, and can be anything (or nothing).https://codethrasher.com/posts/monorepos/Fri, 15 Jan 2021 21:42:29 -0800https://codethrasher.com/posts/monorepos/https://codethrasher.com/posts/colocation/Thu, 14 Jan 2021 10:49:46 -0800https://codethrasher.com/posts/colocation/“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.https://codethrasher.com/posts/yarn_npm/Wed, 16 Dec 2020 10:38:33 -0800https://codethrasher.com/posts/yarn_npm/package.json resolutions field yarn specific Allows you to force the use of a particular version for a nested dependency. e.g.: 1 2 3 4 "devDependencies": { "@angular/cli": "1.0.3", "typescript": "2.3.2" } yarn.lock will contain: 1 2 3 4 5 6 7 "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.https://codethrasher.com/posts/cipher/Tue, 01 Dec 2020 16:08:32 -0800https://codethrasher.com/posts/cipher/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}https://codethrasher.com/posts/distribution_vector/Tue, 01 Dec 2020 15:06:23 -0800https://codethrasher.com/posts/distribution_vector/ 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 Vectorhttps://codethrasher.com/posts/uniform_distribution/Tue, 01 Dec 2020 15:03:05 -0800https://codethrasher.com/posts/uniform_distribution/\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.https://codethrasher.com/posts/point_distribution/Tue, 01 Dec 2020 14:58:59 -0800https://codethrasher.com/posts/point_distribution/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).https://codethrasher.com/posts/cryptography/Tue, 01 Dec 2020 14:46:08 -0800https://codethrasher.com/posts/cryptography/ Digital Signaturehttps://codethrasher.com/posts/digital_signature/Tue, 01 Dec 2020 13:43:16 -0800https://codethrasher.com/posts/digital_signature/A function of the content being “signed”https://codethrasher.com/posts/cost_function/Tue, 01 Dec 2020 11:27:21 -0800https://codethrasher.com/posts/cost_function/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).https://codethrasher.com/posts/machine_learning/Tue, 01 Dec 2020 11:21:27 -0800https://codethrasher.com/posts/machine_learning/ Gradient Descent Cost Function Hypothesis Function Artificial Neural Networkhttps://codethrasher.com/posts/gradient_descent/Tue, 01 Dec 2020 11:13:06 -0800https://codethrasher.com/posts/gradient_descent/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.https://codethrasher.com/posts/hypothesis_function/Tue, 01 Dec 2020 10:47:16 -0800https://codethrasher.com/posts/hypothesis_function/A function which maps values \(x\) to an output value \(y\). Historically, in ML, hypothesis functions are denoted \(h(x^{(i)})\).https://codethrasher.com/posts/artificial_neural_network/Thu, 26 Nov 2020 20:23:30 -0800https://codethrasher.com/posts/artificial_neural_network/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 ANNhttps://codethrasher.com/posts/covectors/Tue, 17 Nov 2020 21:59:53 -0800https://codethrasher.com/posts/covectors/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.https://codethrasher.com/posts/differential_geometry/Tue, 17 Nov 2020 09:30:10 -0800https://codethrasher.com/posts/differential_geometry/ Tensors Tensor Producthttps://codethrasher.com/posts/linear_algebra/Tue, 17 Nov 2020 09:30:00 -0800https://codethrasher.com/posts/linear_algebra/ Bases Bases Transformation Coordinate Transformation Covectors Dual Space Identity Matrix Invertible Matrix Invertible Matrix Orthonormal Basis Tensor Product Tensors Vector Space Axioms (Vector Space)https://codethrasher.com/posts/linear_mapping/Mon, 16 Nov 2020 22:13:48 -0800https://codethrasher.com/posts/linear_mapping/ 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 Functionhttps://codethrasher.com/posts/multilinear_map/Mon, 16 Nov 2020 22:11:24 -0800https://codethrasher.com/posts/multilinear_map/A function of several variables that is linear, separately, in each variable. A multilinear map of one variable is a standard linear mapping.https://codethrasher.com/posts/tensor_product/Mon, 16 Nov 2020 21:41:03 -0800https://codethrasher.com/posts/tensor_product/https://codethrasher.com/posts/dual_space/Thu, 12 Nov 2020 09:58:51 -0800https://codethrasher.com/posts/dual_space/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 \((v^{*}_{1},…,v^{*}_{n})\).https://codethrasher.com/posts/dual_vector_space/Thu, 12 Nov 2020 09:58:21 -0800https://codethrasher.com/posts/dual_vector_space/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 \((v^{*}_{1},…,v^{*}_{n})\).https://codethrasher.com/posts/heliosphere/Thu, 29 Oct 2020 16:07:43 -0700https://codethrasher.com/posts/heliosphere/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.https://codethrasher.com/posts/solar_wind/Thu, 29 Oct 2020 16:03:32 -0700https://codethrasher.com/posts/solar_wind/TODOhttps://codethrasher.com/posts/tensors/Thu, 29 Oct 2020 09:46:58 -0700https://codethrasher.com/posts/tensors/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.https://codethrasher.com/posts/kronecker_delta/Thu, 29 Oct 2020 09:41:12 -0700https://codethrasher.com/posts/kronecker_delta/\begin{equation} \delta_{ij}= \begin{cases} 0 & \text{if i \(\neq\) j}\\\ 1 & \text{if i \(=\) j} \end{cases} \end{equation}https://codethrasher.com/posts/bases/Thu, 29 Oct 2020 09:39:54 -0700https://codethrasher.com/posts/bases/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 Transformationhttps://codethrasher.com/posts/orthonormal_basis/Tue, 27 Oct 2020 20:58:35 -0700https://codethrasher.com/posts/orthonormal_basis/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)https://codethrasher.com/posts/bases_transformation/Tue, 27 Oct 2020 11:05:44 -0700https://codethrasher.com/posts/bases_transformation/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.https://codethrasher.com/posts/identity_matrix/Tue, 27 Oct 2020 10:32:05 -0700https://codethrasher.com/posts/identity_matrix/\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}https://codethrasher.com/posts/invertible_matrix/Tue, 27 Oct 2020 10:14:42 -0700https://codethrasher.com/posts/invertible_matrix/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}https://codethrasher.com/posts/vector_space/Mon, 26 Oct 2020 22:29:06 -0700https://codethrasher.com/posts/vector_space/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}https://codethrasher.com/posts/coordinate_transformation/Mon, 26 Oct 2020 22:23:41 -0700https://codethrasher.com/posts/coordinate_transformation/https://codethrasher.com/posts/vector_space_axioms/Mon, 26 Oct 2020 20:54:32 -0700https://codethrasher.com/posts/vector_space_axioms/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 \(\mathbf{v}\,+\,(\mathbf{-v})\,=\,\mathbf{0}\).https://codethrasher.com/posts/invariance/Mon, 26 Oct 2020 10:30:36 -0700https://codethrasher.com/posts/invariance/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).https://codethrasher.com/posts/cosmology/Fri, 23 Oct 2020 09:54:46 -0700https://codethrasher.com/posts/cosmology/ The Inflationary Universe Standard Model Cosmic Inflation Cosmological Constant Cosmic Microwave Backgroundhttps://codethrasher.com/posts/the_inflationary_universe/Fri, 23 Oct 2020 09:50:13 -0700https://codethrasher.com/posts/the_inflationary_universe/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”.https://codethrasher.com/posts/standard_model/Thu, 22 Oct 2020 22:03:48 -0700https://codethrasher.com/posts/standard_model/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:https://codethrasher.com/posts/cosmological_principle/Thu, 22 Oct 2020 22:01:59 -0700https://codethrasher.com/posts/cosmological_principle/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.https://codethrasher.com/posts/cosmological_constant/Thu, 22 Oct 2020 21:57:03 -0700https://codethrasher.com/posts/cosmological_constant/TODOhttps://codethrasher.com/posts/cosmic_microwave_background/Thu, 22 Oct 2020 21:52:44 -0700https://codethrasher.com/posts/cosmic_microwave_background/TODOhttps://codethrasher.com/posts/sicp/Thu, 22 Oct 2020 10:15:40 -0700https://codethrasher.com/posts/sicp/Table of Contents Chapter 1 Predicates/Expressions Functions v. Procedures or Imperative v. Declarative Chapter 1 Predicates/Expressions Taking the following Lisp code: 1 2 3 4 (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.https://codethrasher.com/posts/programming/Thu, 22 Oct 2020 10:11:19 -0700https://codethrasher.com/posts/programming/ The Structure and Interpretation of Computer Programs (SICP)https://codethrasher.com/posts/manifold/Thu, 22 Oct 2020 10:07:50 -0700https://codethrasher.com/posts/manifold/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 \((x^{1}, x^{2})\).https://codethrasher.com/posts/symmetric_relation/Thu, 22 Oct 2020 10:07:00 -0700https://codethrasher.com/posts/symmetric_relation/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 Reflexive Relationhttps://codethrasher.com/posts/transitive_relation/Thu, 22 Oct 2020 10:06:11 -0700https://codethrasher.com/posts/transitive_relation/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 Relationhttps://codethrasher.com/posts/reflexive_relation/Thu, 22 Oct 2020 10:05:38 -0700https://codethrasher.com/posts/reflexive_relation/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: Symmetric Relation Transitive Relationhttps://codethrasher.com/posts/learning/Wed, 21 Oct 2020 22:31:25 -0700https://codethrasher.com/posts/learning/The Feynman Techniquehttps://codethrasher.com/posts/physics/Wed, 21 Oct 2020 22:31:13 -0700https://codethrasher.com/posts/physics/Articles Black Holes as Primordial Dark Matter Papers The Inflationary Universe (Guth, 1980) Concepts Configuration Space Cosmic Inflation Equations of Motion Galilean Transformation Generalized Coordinates Homogeneous Inertial Frames Isotropy Newtonian Spacetime Newtons Laws of Motion Principle of Relativity Problems Flatness Problem Olber’s Paradox Black Hole Information Paradoxhttps://codethrasher.com/posts/science/Wed, 21 Oct 2020 22:31:06 -0700https://codethrasher.com/posts/science/Branches Physics Articles Phosphine, Life, and Venus (Biology, Chemistry) Black Holes as Primordial Dark Matter (Physics) Papers The Inflationary Universe: Guth, 1980 (Physics) Concepts Chemical Oxidation Abiotic Processes Cosmic Inflation Newtons Laws of Motionhttps://codethrasher.com/posts/set_theory/Wed, 21 Oct 2020 22:31:03 -0700https://codethrasher.com/posts/set_theory/Concepts Relations Cardinality Binary Relation Symmetric Relation Transitive Relation Cartesian Product Reflexive Relation Surjective Functionhttps://codethrasher.com/posts/teaching/Wed, 21 Oct 2020 22:30:58 -0700https://codethrasher.com/posts/teaching/The Feynman Techniquehttps://codethrasher.com/posts/math/Wed, 21 Oct 2020 22:28:30 -0700https://codethrasher.com/posts/math/ Real Analysis Differential Geometry Linear Algebra Topology Set Theory General Concepts Bijective Function Injective Function Surjective Function Transitive Relation Symmetric Relation Locality Peano Axiomshttps://codethrasher.com/posts/chemistry/Wed, 21 Oct 2020 22:28:09 -0700https://codethrasher.com/posts/chemistry/Articles Phosphine, Life, and Venushttps://codethrasher.com/posts/biology/Wed, 21 Oct 2020 22:27:58 -0700https://codethrasher.com/posts/biology/Articles Phosphine, Life, and Venushttps://codethrasher.com/posts/triangle_inequality/Wed, 21 Oct 2020 21:30:28 -0700https://codethrasher.com/posts/triangle_inequality/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. \(z \leq x + y\)https://codethrasher.com/posts/topology/Wed, 21 Oct 2020 21:30:22 -0700https://codethrasher.com/posts/topology/Concepts Metric Space Open Balls Neighborhood Manifold Topological Spacehttps://codethrasher.com/posts/topological_space/Wed, 21 Oct 2020 21:30:19 -0700https://codethrasher.com/posts/topological_space/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 \(\tau\)https://codethrasher.com/posts/surjective_function/Wed, 21 Oct 2020 21:30:06 -0700https://codethrasher.com/posts/surjective_function/A function is surjective or “onto” if For every \(y \in Y\), there exists \(x \in X\) such that \(f(x) = y\).https://codethrasher.com/posts/statistics/Wed, 21 Oct 2020 21:30:04 -0700https://codethrasher.com/posts/statistics/Concepts Null Hypothesishttps://codethrasher.com/posts/shape_of_the_universe/Wed, 21 Oct 2020 21:29:57 -0700https://codethrasher.com/posts/shape_of_the_universe/Open/Closed/Flat TODOhttps://codethrasher.com/posts/relations_sets/Wed, 21 Oct 2020 21:29:49 -0700https://codethrasher.com/posts/relations_sets/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 \((a, b) \in R\).https://codethrasher.com/posts/real_analysis/Wed, 21 Oct 2020 21:29:45 -0700https://codethrasher.com/posts/real_analysis/ The theoretical foundation which underlies calculus Peano Axioms Surjective Function Bijective Function Injective Function Cartesian Producthttps://codethrasher.com/posts/principle_of_relativity/Wed, 21 Oct 2020 21:29:37 -0700https://codethrasher.com/posts/principle_of_relativity/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.https://codethrasher.com/posts/phosphine_life_and_venus/Wed, 21 Oct 2020 21:29:29 -0700https://codethrasher.com/posts/phosphine_life_and_venus/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).https://codethrasher.com/posts/peano_axioms/Wed, 21 Oct 2020 21:29:27 -0700https://codethrasher.com/posts/peano_axioms/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++.https://codethrasher.com/posts/oxidation/Wed, 21 Oct 2020 21:29:23 -0700https://codethrasher.com/posts/oxidation/A type of chemical reaction in which electrons are losthttps://codethrasher.com/posts/open_balls/Wed, 21 Oct 2020 21:29:21 -0700https://codethrasher.com/posts/open_balls/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}https://codethrasher.com/posts/olber_s_paradox/Wed, 21 Oct 2020 21:29:18 -0700https://codethrasher.com/posts/olber_s_paradox/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.https://codethrasher.com/posts/null_hypothesis/Wed, 21 Oct 2020 21:29:12 -0700https://codethrasher.com/posts/null_hypothesis/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.https://codethrasher.com/posts/newtons_laws_of_motion/Wed, 21 Oct 2020 21:29:07 -0700https://codethrasher.com/posts/newtons_laws_of_motion/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 accelerationhttps://codethrasher.com/posts/newtonian_spacetime/Wed, 21 Oct 2020 21:28:57 -0700https://codethrasher.com/posts/newtonian_spacetime/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 accelerationhttps://codethrasher.com/posts/neighborhood/Wed, 21 Oct 2020 21:28:54 -0700https://codethrasher.com/posts/neighborhood/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 Ballshttps://codethrasher.com/posts/metric_space/Wed, 21 Oct 2020 21:28:52 -0700https://codethrasher.com/posts/metric_space/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)https://codethrasher.com/posts/mechanics/Wed, 21 Oct 2020 21:28:50 -0700https://codethrasher.com/posts/mechanics/https://codethrasher.com/posts/locality/Wed, 21 Oct 2020 21:28:43 -0700https://codethrasher.com/posts/locality/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.https://codethrasher.com/posts/isotropy/Wed, 21 Oct 2020 21:28:24 -0700https://codethrasher.com/posts/isotropy/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)https://codethrasher.com/posts/injective_function/Wed, 21 Oct 2020 21:28:21 -0700https://codethrasher.com/posts/injective_function/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)https://codethrasher.com/posts/inertial_frames/Wed, 21 Oct 2020 21:28:18 -0700https://codethrasher.com/posts/inertial_frames/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):https://codethrasher.com/posts/homogeneous/Wed, 21 Oct 2020 21:28:15 -0700https://codethrasher.com/posts/homogeneous/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).https://codethrasher.com/posts/generalized_coordinates/Wed, 21 Oct 2020 21:28:12 -0700https://codethrasher.com/posts/generalized_coordinates/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 \((q_{1}, q_{2},…q_{n})\).https://codethrasher.com/posts/general_relativity/Wed, 21 Oct 2020 21:28:09 -0700https://codethrasher.com/posts/general_relativity/Concepts Inertial Frames Newtons Laws of Motionhttps://codethrasher.com/posts/galilean_transformations/Wed, 21 Oct 2020 21:28:07 -0700https://codethrasher.com/posts/galilean_transformations/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.https://codethrasher.com/posts/flatness_problem/Wed, 21 Oct 2020 21:28:05 -0700https://codethrasher.com/posts/flatness_problem/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 (\(\rho\)).https://codethrasher.com/posts/equations_of_motion/Wed, 21 Oct 2020 21:28:03 -0700https://codethrasher.com/posts/equations_of_motion/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 intervalhttps://codethrasher.com/posts/emacs/Wed, 21 Oct 2020 21:28:00 -0700https://codethrasher.com/posts/emacs/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 dhttps://codethrasher.com/posts/degenerate_coordinate_systems/Wed, 21 Oct 2020 21:27:55 -0700https://codethrasher.com/posts/degenerate_coordinate_systems/Any coordinate system which cannot describe an entirety of a space As an example, polar coordinates \(r,\psi)\) are degenerate at \((0, \psi)\)https://codethrasher.com/posts/curvature/Wed, 21 Oct 2020 21:27:53 -0700https://codethrasher.com/posts/curvature/TODOhttps://codethrasher.com/posts/cosmic_inflation/Wed, 21 Oct 2020 21:27:51 -0700https://codethrasher.com/posts/cosmic_inflation/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.https://codethrasher.com/posts/configuration_space/Wed, 21 Oct 2020 21:27:48 -0700https://codethrasher.com/posts/configuration_space/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 \(Q = \mathbb{R}^{3}\)https://codethrasher.com/posts/cartesian_product/Wed, 21 Oct 2020 21:27:43 -0700https://codethrasher.com/posts/cartesian_product/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.https://codethrasher.com/posts/cardinality/Wed, 21 Oct 2020 21:27:40 -0700https://codethrasher.com/posts/cardinality/Equality in Cardinality Two sets \(X\) and \(Y\) are equal iff there exists a bijection \(f : X \longrightarrow Y\) from \(X\) to \(Y\).https://codethrasher.com/posts/black_holes_as_primordial_dark_matter/Wed, 21 Oct 2020 21:27:37 -0700https://codethrasher.com/posts/black_holes_as_primordial_dark_matter/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.https://codethrasher.com/posts/black_hole_information_paradox/Wed, 21 Oct 2020 21:27:34 -0700https://codethrasher.com/posts/black_hole_information_paradox/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 detailshttps://codethrasher.com/posts/binary_relation/Wed, 21 Oct 2020 21:27:27 -0700https://codethrasher.com/posts/binary_relation/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.https://codethrasher.com/posts/abiotic_process/Wed, 21 Oct 2020 21:27:20 -0700https://codethrasher.com/posts/abiotic_process/An abiotic process is a process which occurs via abiotic resources (non-living physical/chemical elements) in the environmenthttps://codethrasher.com/posts/content/abiotic_process/Wed, 21 Oct 2020 21:27:12 -0700https://codethrasher.com/posts/content/abiotic_process/An abiotic process is a process which occurs via abiotic resources (non-living physical/chemical elements) in the environmenthttps://codethrasher.com/posts/bijective_function/Wed, 21 Oct 2020 21:26:54 -0700https://codethrasher.com/posts/bijective_function/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 versahttps://codethrasher.com/about/Mon, 01 Jan 0001 00:00:00 +0000https://codethrasher.com/about/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.https://codethrasher.com/posts/content/set_theory/Mon, 01 Jan 0001 00:00:00 +0000https://codethrasher.com/posts/content/set_theory/Concepts Relations Cardinality Binary Relation Symmetric Relation Transitive Relation Cartesian Product Reflexive Relation Surjective Function