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Diffstat (limited to 'mindmap')
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diff --git a/mindmap/Fourier Transform.org b/mindmap/Fourier Transform.org index 9996abd..c34e6e9 100644 --- a/mindmap/Fourier Transform.org +++ b/mindmap/Fourier Transform.org @@ -1,5 +1,6 @@ :PROPERTIES: :ID: 262ca511-432f-404f-8320-09a2afe1dfb7 +:ROAM_ALIASES: "Heat Equation" "Fourier Series" :END: #+title: Fourier Transform #+author: Preston Pan @@ -9,4 +10,109 @@ #+options: broken-links:t * Introduction -The Fourier Transform is a generalization of the Fourier Series. +The Fourier Transform is a generalization of the Fourier Series. It has applications in solving [[id:4be41e2e-52b9-4cd1-ac4c-7ecb57106692][differential equations]] and +has applications in many different fields, including [[id:136e79df-106f-4989-ab19-89705929cf91][quantum mechanics]], radio, planetary motion, and even the study +of the heat equation. In this article we will study the heat equation, the Fourier Series, and the Fourier transform. + +** The Heat Equation +The heat equation is the study of how heat travels in a conductor with the unknown function in question being $f(\vec{r}, t)$, +giving the temperature at position $\vec{r}$ at time $t$. Now we want to describe the time rate of change of this function, so we use a +[[id:3993a45d-699b-4512-93f9-ba61f498f77f][partial derivative]]: +\begin{align} +\label{Heat equation 1} +\partial_{t}f = ?f +\end{align} +the left hand side should intuitively be some operator on $f$ dealing with how heat changes in space. A single spacial +derivative operator $\partial_{x}f$ might give you the rate of change of the temperature over space. The [[id:a871e62c-b4a0-4674-9dea-d377de2f780b][continuity equation]] for +heat might look like this: +\begin{align} +\label{} +\partial_{t}f + \vec{\nabla} \cdot \vec{j} = 0 +\end{align} +where $\vec{j}$ is the heat current. It makes sense that the heat equation ought to satisfy some continuity equation, because +heat can't just teleport without leaving some enclosed area (in order for this theory to be physical, it must respect +local conservation of energy). The heat current vector field and a scalar heat field might have a similar relation to +that of the potential-vector field relation in [[id:6f2aba40-5c9f-406b-a1fa-13018de55648][conservative force]] fields: +\begin{align} +\label{} +\vec{j} = -\vec{\nabla}u +\end{align} +substituting: +\begin{align} +\label{} +\partial_{t}f = \nabla^{2}u +\end{align} +and heat $u$ is proportional to temperature $f$, so we have: +\begin{align} +\label{} +\partial_{t}f = \alpha\nabla^{2}f +\end{align} +The result is a [[id:365190d8-0f3a-4728-9b09-83a216292256][PDE]] that we can solve analytically in one dimension. +*** Solving the Heat Equation in One Dimension +In one dimension, the heat equation looks like this: +\begin{align} +\label{} +\partial_{t}f = \alpha\partial_{xx}f +\end{align} +we assume this is a [[id:8e9c975a-cd75-447e-b094-16258147d83c][separable differential equation]] i.e. the solution can be written: +\begin{align} +\label{} +f(x, t) = X(x)T(t) +\end{align} +Plugging this in: +\begin{align} +\label{} +X(x)\frac{dT}{dt} = \alpha T(t)\frac{d^{2}X}{dx^{2}} \\ +\frac{1}{T}\frac{dT}{dt} = \alpha \frac{1}{X}\frac{d^{2}X}{dx^{2}} +\end{align} +we see that each side doesn't depend on any variables from the other side, so we can separate this into two [[id:5ef63bef-2d8f-4e00-b292-8206cf69469a][ODEs]], +because each side looks like a constant to the other side: +\begin{align} +\label{} +\frac{1}{T}\frac{dT}{dt} = \alpha \frac{1}{X}\frac{d^{2}X}{dx^{2}} = -k +\end{align} +Now we solve for $T(t)$, by reducing this problem to being a homogeneous differential equation: +\begin{align} +\label{} +\frac{dT}{dt} + kT = 0 \\ +\lambda + k = 0 \\ +\lambda = k \\ +T(t) = Ae^{-kt} +\end{align} +Solving for $X(x)$ is also a homogeneous case: +\begin{align} +\label{} +\frac{d^{2}X}{dx^{2}} + \frac{k}{\alpha}X = 0 \\ +\lambda^{2} + \frac{k}{\alpha} = 0 \\ +\lambda = \pm \sqrt{-\frac{k}{\alpha}} \\ +X(x) = Be^{x\sqrt{-\frac{k}{\alpha}}} + Ce^{-x\sqrt{-\frac{k}{\alpha}}} +\end{align} +re-using $\lambda$ to mean something else and casting $\sqrt{\frac{k}{\alpha}} = \lambda$: +\begin{align} +\label{} +X(x) = Be^{i\lambda x} + Ce^{-i\lambda x} \\ +T(t) = Ae^{-\lambda^{2}\alpha t} \\ +f(x, t) = ABe^{i\lambda x - \lambda^{2}\alpha t} + ACe^{-i\lambda x - \lambda^{2}\alpha t} \\ +f(x, t) = A_{1}e^{i\lambda x - \lambda^{2}\alpha t} + A_{2}e^{-i\lambda x - \lambda^{2}\alpha t} +\end{align} +we know that by the [[id:422653e2-daa4-422a-9cb7-3983a5a72554][superposition principle]]: +\begin{align} +\label{} +f(x, t) = \sum_{i=0}^{N}f_{i}(x, t) +\end{align} +Now, in order to proceed, we need to formulate this as an [[id:bc7e9e01-9721-4b3e-a886-74a2fd27daf3][initial value problem]] with boundary conditions. A classic +example would be a wire of dimension 1 and length $L$. Thus we set a couple of boundary conditions: + +* The Fourier Transform +Taking the Fourier series and letting $T \rightarrow \infty$: +\begin{align} +\label{} +f(x) = \sum_{n=-\infty}^{\infty}c_{n}e^{\frac{inx}{T}} \\ +Tc_{n} = \lim_{T\rightarrow\infty} \int_{-T}^{T}f(x)e^{\frac{inx}{T}}dx \\ +F(\omega) := Tc_{n} \\ +\omega := \frac{n}{T} \\ +F(\omega) = \int_{-\infty}^{\infty}f(x)e^{i\omega x}dx +\end{align} +$F(\omega)$ is the Fourier Transform of $f(x)$. Putting different functions as arguments gives you a Fourier Transform +table, and the inverse transform is same as the forward transform, scaled by a constant. This makes Fourier Transforms +useful tools for solving differential equations. diff --git a/mindmap/Heaviside Step Function.org b/mindmap/Heaviside Step Function.org new file mode 100644 index 0000000..c92a264 --- /dev/null +++ b/mindmap/Heaviside Step Function.org @@ -0,0 +1,28 @@ +:PROPERTIES: +:ID: 53dade38-21e1-4fa9-a552-6ceab8a75f82 +:END: +#+title: Heaviside Step Function +#+author: Preston Pan +#+html_head: <link rel="stylesheet" type="text/css" href="../style.css" /> +#+html_head: <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script> +#+html_head: <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> +#+options: broken-links:t + +* Introduction +the Heaviside Step Function $H(t)$ is an important function in signal analysis. It is defined as follows: +\begin{align} +\label{} +H(t) = +\[ \left\{ +\begin{array}{ll} +0 & t \leq 0 \\ +1 & t > 0 +\end{array} +\right. \] +\end{align} +and it is related to the [[id:90574fea-88f4-4b80-9cda-32cff0bcb76d][dirac delta]] distribution by taking a [[id:31d3944a-cddc-496c-89a3-67a56e821de3][derivative]]: +\begin{align} +\label{} +\frac{dH}{dt} = \delta(t) +\end{align} +Note that this definition of the derivative may be different than that of the regular derivative definition. diff --git a/mindmap/mass-spring system.org b/mindmap/Hilbert Space.org index 7b780f5..281c107 100644 --- a/mindmap/mass-spring system.org +++ b/mindmap/Hilbert Space.org @@ -1,11 +1,12 @@ :PROPERTIES: -:ID: 5004f2b4-f8f7-46e6-87b5-e547184f82ef +:ID: eac079b7-1144-4506-b8c5-4717ccac8a7b :END: -#+title: mass-spring system +#+title: Hilbert Space #+author: Preston Pan #+html_head: <link rel="stylesheet" type="text/css" href="../style.css" /> #+html_head: <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script> #+html_head: <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> #+options: broken-links:t + * Introduction -A mass-spring model uses Hooke's law $F = kx$ in order to model a mass attached to a spring with a differential equation. +A Hilbert Space $\mathcal{H}$ is an [[id:36a2715c-a8db-4b75-b799-61ce43be2d2d][inner product space]] whose inner product defines a complete metric space. diff --git a/mindmap/IAK.org b/mindmap/IAK.org new file mode 100644 index 0000000..f9404b8 --- /dev/null +++ b/mindmap/IAK.org @@ -0,0 +1,21 @@ +:PROPERTIES: +:ID: 700073f4-04d5-4d20-8bcd-ee9ba0a739c8 +:END: +#+title: IAK +#+author: Preston Pan +#+html_head: <link rel="stylesheet" type="text/css" href="../style.css" /> +#+html_head: <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script> +#+html_head: <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> +#+options: broken-links:t +* Introduction +Short for "I am king". Refers to the modern propensity for people to talk about political issues +/as if they personally ran policy/, without any care to think about the fact that they /don't/. + +For example, the statement "we should raise taxes" is an IAK statement because "we" doesn't mean anything here. +The person talking has no control over the "we", or the group of people making public policy. To say that they +do via democracy is very likely to be a case of [[id:7456da20-684d-4de6-9235-714eaafb2440][IEEDI]] syndrome. Talking about IAK without being aware of the lack +of direct input can put people in an unhealthy mindset. + +* Conclusion +It can be fun to make IAK statements, in which case do so, but many do not realize how useless they are, because +it is a direct consequence of [[id:7456da20-684d-4de6-9235-714eaafb2440][IEEDI]], which people also don't get in general. diff --git a/mindmap/IEEDI.org b/mindmap/IEEDI.org new file mode 100644 index 0000000..893c122 --- /dev/null +++ b/mindmap/IEEDI.org @@ -0,0 +1,40 @@ +:PROPERTIES: +:ID: 7456da20-684d-4de6-9235-714eaafb2440 +:END: +#+title: IEEDI +#+author: Preston Pan +#+html_head: <link rel="stylesheet" type="text/css" href="../style.css" /> +#+html_head: <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script> +#+html_head: <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> +#+options: broken-links:t + +* Introduction +Short for "if everyone else did it...". This particular /syndrome/ involves arguments about group mentality +that would be completely insolvent in game theory, yet are often brought up in defense of modern practices. +** The Democracy Game +Democracy is a good example of IEEDI syndrome. The gain of voting in an $n$ player game involving two candidates +and popular vote drops off at $\frac{1}{n}$, but the time it takes to be informed and vote has a constant value. The decisions +of individuals in this game most likely, in real life, at most influence the decision making of one or two other people +(in total, from the whole chain reaction), +so the effect of influence is not very significant (so you can't argue that you have an influence over the crowd to vote, +because you don't). Given all these conditions, for a large $n$, voting should not be +worth it for most people, because the choice of you voting is independent of everyone else voting. You voting or not voting +has no bearing over the crowd. Yet, the common retort is, "if everyone thought like you...". This logic is dead on arrival, +because /not/ everyone's going to think like you. The character of the system is that other people irrationally vote +regardless of if you do, and your decision to vote or not vote has no bearing over the crowd voting or not voting. + +This simple fact is IEEDI syndrome; people are quick to conform rather than think about the personal cost-benefit analysis, +even if the logic stops working for large societies. +** The Activism Game +The activism game is similar; your activism doesn't matter much, and the crowd would continue existing even if you weren't +a part of it. Thus, any attempt to engage in activism at almost any cost is useless. + +* Symptoms +One may diagnose people with IEEDI syndrome if they: +1. cannot affect a situation meaningfully, yet they try to anyways. +2. make [[id:700073f4-04d5-4d20-8bcd-ee9ba0a739c8][IAK]] statements regularly. +3. engage in activism. + +* Conclusion +Almost any decision to try to influence large societies where your influence is a fraction of everyone elses' +is unjustified from a cost benefit analysis standpoint. diff --git a/mindmap/Kirchhoff's Laws.org b/mindmap/Kirchhoff's Laws.org index be20a41..fc88086 100644 --- a/mindmap/Kirchhoff's Laws.org +++ b/mindmap/Kirchhoff's Laws.org @@ -3,13 +3,14 @@ :END: #+title: Kirchhoff's Laws #+author: Preston Pan +#+description: basic laws of circuit analysis #+html_head: <link rel="stylesheet" type="text/css" href="../style.css" /> #+html_head: <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script> #+html_head: <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> #+options: broken-links:t * Introduction -Kirchhoff's Laws, along with Ohm's law, create the axioms of circuit analysis. The two laws are the Kirchhoff Voltage Law +Kirchhoff's Laws, along with [[id:3cdce475-7644-4529-a447-6e790ad4055f][Ohm's Law]], create the axioms of [[id:a7d6d6e9-9f7a-446f-b6af-255c802f86b1][circuit analysis]]. The two laws are the Kirchhoff Voltage Law (KVL) and Kirchhoff's Current Law (KCL). They can be derived from an approximation of [[id:fde2f257-fa2e-469a-bc20-4d11714a515e][Maxwell's Equations]]. ** KCL :PROPERTIES: @@ -29,7 +30,33 @@ If the total amount of charge in the wires are conserved: Therefore: \begin{align} \label{} -\sum_{n}I_{n} = 0 +\sum_{n}^{N}I_{n} = 0 \end{align} where the total current $\vec{I}$ can be decomposed into many currents of each branched path $I_{n}$. ** KVL +:PROPERTIES: +:ID: 92c952ee-f1f3-4782-b9e2-6fecb56caac6 +:END: +The Kirchhoff voltage law can be derived also from [[id:fde2f257-fa2e-469a-bc20-4d11714a515e][Maxwell's Equations]], specifically the [[id:63713308-0ff7-433f-8103-8b64ba9bdbe1][electrostatics]] equations +that formulate the electric field as an [[id:951db9ac-3e8b-49a1-b609-2bbb795be834][electrostatic potential]]: +\begin{align} +\label{} +\vec{E} = -\vec{\nabla}V +\end{align} +more specifically, the [[id:951db9ac-3e8b-49a1-b609-2bbb795be834][potential difference]] across a circuit element can be defined by +$\int \vec{E} \cdot d\vec{l} = V(b) - V(a)$, where $a$ and $b$ correspond to the positions before and after the circuit element. +We know from electrostatics that: +\begin{align} +\label{} +\oint \vec{E} \cdot d\vec{l} = 0 +\end{align} +and from the superposition principle we know: +\begin{align} +\label{} +V_{tot} = \sum V_{i} +\end{align} +so the total voltage drop, or potential difference around the entire circuit must be zero: +\begin{align} +\label{} +\sum_{n=0}^{N}V_{n} = 0 +\end{align} diff --git a/mindmap/LRC circuit.org b/mindmap/LRC circuit.org index b3e8751..bd2f078 100644 --- a/mindmap/LRC circuit.org +++ b/mindmap/LRC circuit.org @@ -1,5 +1,6 @@ :PROPERTIES: :ID: 6dbe2931-cc18-48fc-8cc1-6c71935a6be3 +:ROAM_ALIASES: "mass-spring system" :END: #+title: LRC circuit #+author: Preston Pan @@ -90,7 +91,8 @@ note that this is actually a decaying solution because $m$ must be negative. * AC [[id:951db9ac-3e8b-49a1-b609-2bbb795be834][Voltage]] Source Here is the circuit diagram for the LRC circuit with a voltage source: #+name: LRC Circuit -#+header: :exports both :file lrc_circuit_source.png +#+header: :export +#+header: :exports both :file lrc_circuit.png #+header: :imagemagick yes :iminoptions -density 600 :imoutoptions -geometry 400 #+header: :fit yes :noweb yes :headers '("\\usepackage{circuitikz}") #+begin_src latex :exports both :file @@ -118,4 +120,140 @@ This new [[id:4be41e2e-52b9-4cd1-ac4c-7ecb57106692][differential equation]] look \begin{align*} [LD^{2} + RD + \frac{1}{C}]i(t) = V_{0}sin(\phi + 2\pi\omega t) \end{align*} -where the right hand side of the equation includes the term created by the AC [[id:951db9ac-3e8b-49a1-b609-2bbb795be834][voltage]] source. +where the right hand side of the equation includes the term created by the AC [[id:951db9ac-3e8b-49a1-b609-2bbb795be834][voltage]] source. Now we take the +[[id:e73baa24-1a29-4f35-9d3d-0fad4a3a8e59][Laplace Transform]] of both sides (using euler notation to keep track of the phase shift $\phi$): +\begin{align} +\label{} +\mathcal{L}\{LD^{2} + RD + \frac{1}{C}\}i(t) = V_{0}e^{i\phi}\mathcal{L}\{e^{i 2\pi\omega t}\} \\ +L(s^{2}I(s) - s i(0) - i'(0)) + R(sI(s) - i(0)) + \frac{1}{C}I(s) = V_{0}e^{i\phi}\mathcal{L}\{e^{i 2\pi\omega t}\} \\ +(Ls^{2} + (R - i(0))s + \frac{1}{C})I(s) = V_{0}e^{i\phi}\mathcal{L}\{e^{i 2\pi\omega t}\} + i'(0) + i(0) \\ +I(s) = \frac{V_{0}e^{i\phi}\frac{1}{s - 2\pi i \omega} + i'(0) + i(0)}{Ls^{2} + (R - i(0))s + \frac{1}{C}} \\ +i(t) = \mathcal{L}^{-1}\{\frac{V_{0}e^{i\phi}}{(s - 2\pi i \omega)(Ls^{2} + (R - i(0))s + \frac{1}{C})}\} + +\mathcal{L}^{-1}\{\frac{i'(0) + i(0)}{Ls^{2} + (R - i(0))s + \frac{1}{C}}\} +\end{align} +We want to use partial fraction decomposition in order to break these denominators apart, so that doing the inverse +transform is easier (the Laplace transform is [[id:ab024db7-6903-48ee-98fc-b2a228709c04][linear]]). Now we already know the roots of the polynomial, and we can +set the root $2\pi i\omega = z_{1}$. Let's also ignore the rightmost term for now: +\begin{align} +\label{Pain} +i(t) = V_{0}e^{i\phi}\mathcal{L}^{-1}\{\frac{1}{(s - z_{1})(s - z_{2})(s - z_{3})}\} +\end{align} +where $z_{2}$ and $z_{3}$ are the two roots we found for the homogeneous case. We then use partial fraction decomposition: +\begin{align} +\label{} +\frac{1}{(s - z_{1})(s - z_{2})(s - z_{3})} = \frac{A}{s - z_{1}} + \frac{B}{s - z_{2}} + \frac{C}{s - z_{3}} \\ +A(s - z_{2})(s - z_{3}) + B(s - z_{1})(s - z_{3}) + C(s - z_{1})(s - z_{2}) = 1 +\end{align} +from this we know: +\begin{align} +\label{} +A + B + C = 0 \\ +z_{2}z_{3}A + z_{1}z_{3}B + z_{1}z_{2}C = 1 \\ +(z_{2} + z_{3})A + (z_{1} + z_{3})B + (z_{1} + z_{2})C = 0 +\end{align} +Which is a linear system of equations. Eliminating C: +\begin{align} +\label{} +(z_{2}z_{3} - z_{1}z_{2})A + (z_{1}z_{3} - z_{1}z_{2})B = 1 \\ +(z_{2} + z_{3})A + (z_{1} + z_{3})B - (z_{1} + z_{2})(A + B) = 0 +\end{align} +Now we want to eliminate B: +\begin{align} +\label{} +[(z_{2} + z_{3}) - (z_{1} + z_{2})]A + [(z_{1} + z_{3}) - (z_{1} + z_{2})]B = 0 \\ +(z_{3} - z_{1})A + (z_{3} - z_{2})B = 0 \\ +B = -\frac{z_{3} - z_{1}}{z_{3} - z_{2}}A \\ +\end{align} +finally, we have one equation in terms of A: +\begin{align} +\label{} +[(z_{2}z_{3} - z_{1}z_{2}) - (z_{1}z_{3} - z_{1}z_{2})\frac{z_{3} - z_{1}}{z_{3} - z_{2}}]A = 1 \\ +[(z_{2}z_{3} - z_{1}z_{2})\frac{z_{3} - z_{2}}{z_{3} - z_{2}} +- (z_{1}z_{3} - z_{1}z_{2})\frac{z_{3} - z_{1}}{z_{3} - z_{2}}]A = 1 \\ +\frac{(z_{2}z_{3} - z_{1}z_{2})(z_{3} - z_{2}) - (z_{1}z_{3} - z_{1}z_{2})(z_{3} - z_{1})}{z_{3} - z_{2}}A = 1 \\ +\frac{z_{2}z_{3}^{2} - z_{1}z_{2}z_{3} - z_{2}^{2}z_{3} + z_{1}z_{2}^{2} - +z_{1}z_{3}^{2} + z_{1}z_{2}z_{3} + z_{1}^{2}z_{3} - z_{1}^{2}z_{2}}{z_{3} - z_{2}}A = 1 \\ +\frac{z_{2}z_{3}^{2} - z_{2}^{2}z_{3} + z_{1}z_{2}^{2} - +z_{1}z_{3}^{2} + z_{1}^{2}z_{3} - z_{1}^{2}z_{2}}{z_{3} - z_{2}}A = 1 \\ +A = \frac{z_{3} - z_{2}}{z_{2}z_{3}^{2} + z_{1}z_{2}^{2} + z_{1}^{2}z_{3} - z_{1}z_{3}^{2} - z_{1}^{2}z_{2} - z_{2}^{2}z_{3}} \\ +B = -\frac{z_{3} - z_{1}}{z_{3} - z_{2}}A \\ +B = -\frac{z_{3} - z_{1}}{z_{2}z_{3}^{2} + z_{1}z_{2}^{2} + z_{1}^{2}z_{3} - z_{1}z_{3}^{2} - z_{1}^{2}z_{2} - z_{2}^{2}z_{3}} \\ +C = -(A + B) \\ +C = \frac{z_{2} - z_{1}}{z_{2}z_{3}^{2} + z_{1}z_{2}^{2} + z_{1}^{2}z_{3} - z_{1}z_{3}^{2} - z_{1}^{2}z_{2} - z_{2}^{2}z_{3}} +\end{align} +So we have the three coefficients: +\begin{align} +\label{} +A = \frac{z_{3} - z_{2}}{z_{2}z_{3}^{2} + z_{1}z_{2}^{2} + z_{1}^{2}z_{3} - z_{1}z_{3}^{2} - z_{1}^{2}z_{2} - z_{2}^{2}z_{3}} \\ +B = \frac{z_{1} - z_{3}}{z_{2}z_{3}^{2} + z_{1}z_{2}^{2} + z_{1}^{2}z_{3} - z_{1}z_{3}^{2} - z_{1}^{2}z_{2} - z_{2}^{2}z_{3}} \\ +C = \frac{z_{2} - z_{1}}{z_{2}z_{3}^{2} + z_{1}z_{2}^{2} + z_{1}^{2}z_{3} - z_{1}z_{3}^{2} - z_{1}^{2}z_{2} - z_{2}^{2}z_{3}} +\end{align} +The resulting solution looks like this: +\begin{align} +\label{} +i(t) = V_{0}e^{i\phi}(Ae^{z_{1}t} + Be^{z_{2}t} + Ce^{z_{3}t}) +\end{align} +where: +\begin{align} +\label{} +z_{1} = 2\pi i \omega \\ +z_{2} = \frac{-(R - i(0)) + \sqrt{(R - i(0))^{2} - \frac{4L}{C}}}{2L} \\ +z_{3} = \frac{-(R - i(0)) - \sqrt{(R - i(0))^{2} - \frac{4L}{C}}}{2L} +\end{align} +by taking the inverse Laplace Transform. The other terms can be either ignored if $i'(0) = 0$ and $i(0) = 0$ or +one can solve for them in the same way. Solving for the two other terms: +\begin{align} +\label{} +\frac{1}{(s - z_{2})(s - z_{3})} = \frac{D}{s - z_{2}} + \frac{E}{s - z_{3}} \\ +D(s - z_{3}) + E(s - z_{2}) = 1 \\ +D + E = 0 \\ +D z_{3} + E z_{2} = -1 +\end{align} +We have a much easier linear system: +\begin{align} +\label{} +E = -D \\ +D(z_{3} - z_{2}) = -1 \\ +D = -\frac{1}{z_{3} - z_{2}} \\ +E = \frac{1}{z_{3} - z_{2}} +\end{align} +so the full solution including the terms used for the [[id:bc7e9e01-9721-4b3e-a886-74a2fd27daf3][initial value problem]] looks like this: +\begin{align} +\label{} +i(t) = V_{0}e^{i\phi}(Ae^{z_{1}t} + Be^{z_{2}t} + Ce^{z_{3}t}) + (i'(0) + i(0))(De^{z_{2}t} + Ee^{z_{3}t}) +\end{align} +the sinusoidal part of the solution looks like this: +\begin{align} +\label{hello world} +\frac{(z_{3} - z_{2})V_{0}e^{i\phi}e^{2\pi i\omega t}}{z_{2}z_{3}^{2} + z_{1}z_{2}^{2} + z_{1}^{2}z_{3} - z_{1}z_{3}^{2} - z_{1}^{2}z_{2} - z_{2}^{2}z_{3}} +\end{align} + +* Mass-Spring System +Starting from [[id:6e2a9d7b-7010-41da-bd41-f5b2dba576d3][Newtonian mechanics]] in a single dimension: +\begin{align} +\label{} +F_{net} = \sum_{i} m\frac{d^{2}x}{dt} +\end{align} +With Hooke's law: +\begin{align} +\label{} +F = -kx +\end{align} +then: +\begin{align} +\label{} +m\ddot{x} = -kx \\ +m\ddot{x} + kx = 0 +\end{align} +We can define some damping force to be: +\begin{align} +\label{} +F_{damp} = -a\dot{x} +\end{align} +which will always resist a change in the direction of motion. Then, the new equation is: +\begin{align} +\label{} +m\ddot{x} + a\dot{x} + kx = 0 +\end{align} +which has the same form as the above LRC circuit equation. Now, any external driving force will appear +on the right hand side. diff --git a/mindmap/L_{2} space.org b/mindmap/L_{2} space.org new file mode 100644 index 0000000..50bb154 --- /dev/null +++ b/mindmap/L_{2} space.org @@ -0,0 +1,19 @@ +:PROPERTIES: +:ID: 0ecee9ae-7786-4db0-940c-c3d64f6240f6 +:END: +#+title: L_{2} space +#+author: Preston Pan +#+html_head: <link rel="stylesheet" type="text/css" href="../style.css" /> +#+html_head: <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script> +#+html_head: <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> +#+options: broken-links:t + +* Introduction +The $L_{2}$ space is a [[id:9a1cc2d9-ef99-436c-8c21-9e68fd7df192][normed vector space]] with a particular norm: +\begin{align} +\label{} +(\sum_{n=0}^{N}x_{n}^{2})^{\frac{1}{2}} +\end{align} +It is easy to prove that this norm satisfies the triangle inequality, and it is even more trivial to prove +that this norm satisfies the other postulates. This norm is a part of a larger set of norms that all satisfy the +norm definition. diff --git a/mindmap/Laplace Transform.org b/mindmap/Laplace Transform.org new file mode 100644 index 0000000..942d54b --- /dev/null +++ b/mindmap/Laplace Transform.org @@ -0,0 +1,30 @@ +:PROPERTIES: +:ID: e73baa24-1a29-4f35-9d3d-0fad4a3a8e59 +:END: +#+title: Laplace Transform +#+author: Preston Pan +#+html_head: <link rel="stylesheet" type="text/css" href="../style.css" /> +#+html_head: <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script> +#+html_head: <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> +#+options: broken-links:t + +* Introduction +The dual-edge Laplace Transform is defined as: +\begin{align} +\label{Laplace Transform} +F(s) = \int_{-\infty}^{\infty}f(t)e^{-st}dt +\end{align} +when $s$ is complex (which it usually is), the [[id:262ca511-432f-404f-8320-09a2afe1dfb7][Fourier Transform]] can be taken to be a special case of the +dual-edge Laplace Transform. One can think of this as analyzing the complex exponential domain, rather than just +the frequency domain (imaginary exponential domain). Now, multiplying the signal by the [[id:53dade38-21e1-4fa9-a552-6ceab8a75f82][Heaviside Step Function]]: +\begin{align} +\label{Step Function} +F(s) = \int_{-\infty}^{\infty}H(t)f(t)e^{-st}dt = \int_{0}^{\infty}f(t)e^{-st}dt +\end{align} +gives you the conventional Laplace Transform. +** Usage +The Laplace Transform is primarily used for analyzing [[id:32a116d9-b813-4b5a-a2e8-6dd7b767ec16][linear differential equations]] as it converts these equations into +algebraic equations. The inverse Laplace Transform is complicated, and is therefore not used often. Instead, Laplace +Transform tables are used in order to convert back into the time-domain. Taking the Laplace transform of all terms in +a linear differential equation will yield this result. One of the simplest differential equations that the Laplace +Transform can solve is the [[id:6dbe2931-cc18-48fc-8cc1-6c71935a6be3][mass-spring system]], and it also generally has applications in [[id:a7d6d6e9-9f7a-446f-b6af-255c802f86b1][circuit analysis]]. diff --git a/mindmap/Ohm's Law.org b/mindmap/Ohm's Law.org new file mode 100644 index 0000000..16f48ee --- /dev/null +++ b/mindmap/Ohm's Law.org @@ -0,0 +1,19 @@ +:PROPERTIES: +:ID: 3cdce475-7644-4529-a447-6e790ad4055f +:END: +#+title: Ohm's Law +#+author: Preston Pan +#+html_head: <link rel="stylesheet" type="text/css" href="../style.css" /> +#+html_head: <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script> +#+html_head: <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> +#+options: broken-links:t +* Introduction +The name is a misnomer; Ohm's law simply refers to the tendency for real world materials to obey this relation: +\begin{align} +\label{} +V = IR +\end{align} +where $V$ is the [[id:951db9ac-3e8b-49a1-b609-2bbb795be834][voltage]], $I$ is the current, and $R$ is a proportionality constant arguably defined by this equation. +Essentially this law posits that resistors scale the current in a [[id:ab024db7-6903-48ee-98fc-b2a228709c04][linear]] fashion. This law is used extensively in +the analysis of resistors in [[id:a7d6d6e9-9f7a-446f-b6af-255c802f86b1][circuit analysis]]. + diff --git a/mindmap/circuit analysis.org b/mindmap/circuit analysis.org new file mode 100644 index 0000000..849e56a --- /dev/null +++ b/mindmap/circuit analysis.org @@ -0,0 +1,15 @@ +:PROPERTIES: +:ID: a7d6d6e9-9f7a-446f-b6af-255c802f86b1 +:END: +#+title: circuit analysis +#+author: Preston Pan +#+html_head: <link rel="stylesheet" type="text/css" href="../style.css" /> +#+html_head: <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script> +#+html_head: <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> +#+options: broken-links:t + +* Introduction +Circuit analysis is the study of real world circuits using circuit elements and circuit components with idealized +wires. There are three basic axioms of circuit analysis, the two [[id:1d586d6b-bd97-4c59-ad57-8894ae4ac8ba][Kirchhoff's Laws]], and [[id:3cdce475-7644-4529-a447-6e790ad4055f][Ohm's Law]]. Circuit analysis +is generally visualized via circuit diagrams, but one can represent circuits using [[id:4be41e2e-52b9-4cd1-ac4c-7ecb57106692][differential equations]] as well. +Simple examples of circuits analyzed in this field include the [[id:6dbe2931-cc18-48fc-8cc1-6c71935a6be3][LRC circuit]] and various filters. diff --git a/mindmap/consequentialism.org b/mindmap/consequentialism.org new file mode 100644 index 0000000..d277173 --- /dev/null +++ b/mindmap/consequentialism.org @@ -0,0 +1,18 @@ +:PROPERTIES: +:ID: 4b4d4071-2ef4-4a6d-ada3-adc8cac425cc +:END: +#+title: consequentialism +#+author: Preston Pan +#+html_head: <link rel="stylesheet" type="text/css" href="../style.css" /> +#+html_head: <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script> +#+html_head: <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> +#+options: broken-links:t +* Introduction +Consequentialism is a set of metaethical views that give importance to the consequence of actions rather than some +inherent quality in the action. Examples of consequentialism include [[id:326eb3f8-680a-432c-bf69-42ba4d366116][egoism]] and utilitarianism. This is in contrast +to [[id:6d8c8bcc-58b0-4267-8035-81b3bf753505][frameworks]] that posit that there is some inherent moral quality to certain things or actions, such as virtue ethics +or deontologism. I posit that these non-consequentialist frameworks are special cases of consequentialist calculation, +less abstracted versions of the same phenomenon. Of course, to say anything is more or less abstract is itself +suspect as anything can be explained in terms of anything else, but it is a useful fiction to assume that one comes from +the other, because of heuristics that follow downstream from consequentialism, such as noticing [[id:7456da20-684d-4de6-9235-714eaafb2440][IEEDI]] syndrome. + diff --git a/mindmap/conservative force.org b/mindmap/conservative force.org index 9b01117..ca50664 100644 --- a/mindmap/conservative force.org +++ b/mindmap/conservative force.org @@ -26,11 +26,11 @@ integrate over \(\vec{f}\), we define \(V(\vec{r})\) such that: \int_{\vec{a}}^{\vec{b}}\vec{f} \cdot d\vec{l} = V(\vec{a}) - V(\vec{b}) \end{align*} because it is path independent, we do not need to consider the infinite paths between \(\vec{a}\) and \(\vec{b}\), which -allows us to define this function \(V(\vec{r})\). Then by the fundamental theorem of calculus: +allows us to define this function \(V(\vec{r})\). Then by the fundamental theorem of calculus, using the [[id:3587c3b4-c3d8-4ff1-b0ba-8eecb1ef0e4c][Gradient]]: \begin{align*} \vec{f} = -\vec{\nabla}V \end{align*} -Therefore, conservative forces can be represented by a scalar field. Now taking the curl of both sides we get: +Therefore, conservative forces can be represented by a scalar field. Now taking the [[id:b25e0e44-c764-4f0a-a5ad-7f9d79c7660d][Curl]] of both sides we get: \begin{align*} \vec{\nabla} \times \vec{f} = 0 \end{align*} diff --git a/mindmap/del operator.org b/mindmap/del operator.org index 3d9e2f5..657ff69 100644 --- a/mindmap/del operator.org +++ b/mindmap/del operator.org @@ -107,6 +107,6 @@ It returns a scalar field and is the multivariable analogue to the second deriva and gradient have been described, I feel it is trivial to understand the Laplacian. ** Product Rules -The product rules pertaining to the del operator are consistent with that of linear algebra and single variable derivative rules. +The [[id:d1e245f4-0b04-450e-8465-a9c85fe57f7e][product rules]] pertaining to the del operator are consistent with that of linear algebra and single variable derivative rules. For example, \( \vec{\nabla} \times \vec{\nabla}f = 0\). You can show this yourself quite easily, so I find no need to go over it here. When in doubt, just assume the del works the same way as any old vector except you apply the [[id:d1e245f4-0b04-450e-8465-a9c85fe57f7e][product rule]], and you will usually be correct. diff --git a/mindmap/differential equation.org b/mindmap/differential equation.org index 8229587..c509da0 100644 --- a/mindmap/differential equation.org +++ b/mindmap/differential equation.org @@ -17,10 +17,301 @@ initial value problems. Differential equations are often used to model real world systems, and are the main tool in numerical simulations of said systems. ** ODE +:PROPERTIES: +:ID: 5ef63bef-2d8f-4e00-b292-8206cf69469a +:END: An ODE is a differential equation involving single variable function solution classes and their derivatives. ** PDE +:PROPERTIES: +:ID: 365190d8-0f3a-4728-9b09-83a216292256 +:END: A PDE is a differential equation involving multivariable function solution classes and generally involve partial derivatives of the unknown function. ** initial value problem +:PROPERTIES: +:ID: bc7e9e01-9721-4b3e-a886-74a2fd27daf3 +:END: An initial value problem is a problem where one is given a differential equation and particular values of the unknown function and particular values of its derivatives, and the result is a particular solution. +** separable differential equation +:PROPERTIES: +:ID: 8e9c975a-cd75-447e-b094-16258147d83c +:END: +For [[id:5ef63bef-2d8f-4e00-b292-8206cf69469a][ODEs]], separable differential equations are differential equations of the form: +\begin{align} +\label{} +\frac{dy}{dx} = f(y)g(x) +\end{align} +which can be solved by taking an integral on both sides: +\begin{align} +\label{} +\frac{dy}{f(y)} = g(x)dx \\ +\int\frac{dy}{f(y)} = \int g(x)dx +\end{align} +evaluating the integrals and solving for $y$, you obtain solutions for $y$ in terms of $x$. +** [[id:ab024db7-6903-48ee-98fc-b2a228709c04][linear]] differential equation +:PROPERTIES: +:ID: 32a116d9-b813-4b5a-a2e8-6dd7b767ec16 +:END: +Linear differential equations are differential equations of the form: +\begin{align} +\label{} +[\sum_{i}f_{i}(x)D^{i}]y(x) = g(x) +\end{align} +where $D$ is the [[id:31d3944a-cddc-496c-89a3-67a56e821de3][derivative]] operator. They are linear because the unknown function $y(x)$ is being operated on by some +linear operator, and common methods in linear algebra can be used in order to analyze equations of this kind. For +example, a first-order linear differential equation would look like this: +\begin{align} +\label{} +[f(x)D + g(x)]y(x) = h(x) +\end{align} +which can be easily solved in the following way where $G(x) = \frac{g(x)}{f(x)}$ and $H(x) = \frac{h(x)}{f(x)}$: +\begin{align} +\label{} +[D + G(x)]y(x) = H(x) \\ +\mu(x)[D + G(x)]y(x) = \mu(x)H(x) \\ +\mu'(x) := G(x)\mu(x) \\ +D(\mu(x)y(x)) = \mu(x)H(x) \\ +y(x) = \frac{\int\mu(x)H(x)dx}{\mu(x)} +\end{align} +Now to solve for $\mu(x)$, using [[id:8e9c975a-cd75-447e-b094-16258147d83c][separable differential equation]] methods: +\begin{align} +\label{} +\frac{d\mu}{dx} = G(x)\mu(x) \\ +\frac{1}{\mu}d\mu = G(x)dx \\ +\int\frac{1}{\mu}d\mu = \int G(x)dx \\ +ln(\mu) = \int G(x)dx \\ +e^{\int G(x)dx} = \mu +\end{align} +Therefore: +\begin{align} +\label{} +y(x) = \frac{\int e^{\int G(x)dx}H(x)dx}{e^{\int G(x)dx}} +\end{align} +Then, to model any particular first order system, plug in functions for $G(x)$ and $H(x)$. +*** superposition principle +:PROPERTIES: +:ID: 422653e2-daa4-422a-9cb7-3983a5a72554 +:END: +The principle of superposition states that any solutions $f_i(x)$ add to a new solution: +\begin{align} +\label{} +\sum_{i=0}^{N}f_{i}(x) = f_{new}(x) +\end{align} +that also satisfies the linear differential equation. This works because the operator is [[id:ab024db7-6903-48ee-98fc-b2a228709c04][linear]], so additive properties +work over this space. +*** Higher Order Linear Differential Equations +Solving higher order linear differential equations requires a couple of tricks. For example, transforms such +as the [[id:e73baa24-1a29-4f35-9d3d-0fad4a3a8e59][Laplace Transform]] or the [[id:262ca511-432f-404f-8320-09a2afe1dfb7][Fourier Transform]] may be used. Such transforms reduce differential equation +problems to algebraic problems, thus simplifying their solution methods. Other methods include guessing (I'm not +pulling your leg here, this is real), formulation as an eigenvalue problem, and taylor polynomial solutions. We will +take a look at all of these in this section. +*** Homogeneous Case +Take the case $Ay'' + By' + Cy = 0$, the substitute the form $y = De^{kt}$. Then: +\begin{align} +\label{} +De^{kt}(Ak^{2} + Bk + C) = 0 \\ +Ak^{2} + Bk + C = 0 +\end{align} +Then, use the quadratic formula to solve for $k$ in terms of the other constants. Such a polynomial is called the +characteristic polynomial of this differential equation. +*** Eigenvalue Problems +Eigenvalue problems can be solved just like in the familiar linear algebra case. For instance, take some differential +equation in this form: +\begin{align} +\label{} +A(f) = \lambda f +\end{align} +where $A$ is a linear operator in [[id:b1f9aa55-5f1e-4865-8118-43e5e5dc7752][function]] space, and $\lambda$ is any constant. Traditionally, one would solve such an eigenvalue +problem like so: +\begin{align} +\label{} +\det{(A - \lambda I)} = 0 +\end{align} +In the simple example of a polynomial basis, this function $f$ can be represented as some linear combination of linearly +independent polynomials. A simple basis to choose could be the Taylor series basis i.e. $\vec{e_{n}} = x^{n}$ where $e_{n}$ is the nth +basis vector. Note that there are many polynomial bases that are an orthogonal basis and span this subset of function +space, but this is a simple example. In this case, the matrix $A$ would represent an operation on an infinite polynomial, +and the $\lambda I$ tells you to subtract $\lambda$ from all its diagonals. You can interpret this literally, using the following example. +**** Example +\begin{align} +\label{} +D(r^{2}D(f(r)) = \lambda f(r) +\end{align} +is such an example of an eigenvalue problem. Now, using the Taylor basis, we need to know two things: what $D$ is as an +infinite dimensional matrix in this basis, and what $t^{2}$ is as an infinite dimensional matrix. $f$ is some unknown vector +we are trying to solve for in this system. Note this observation: +\begin{align} +\label{} +\begin{pmatrix} +0 & 1 & 0 & 0 & 0 \\ +0 & 0 & 2 & 0 & 0 \\ +0 & 0 & 0 & 3 & 0 \\ +0 & 0 & 0 & 0 & 4 \\ +\end{pmatrix} +\begin{pmatrix} +a \\ +b \\ +c \\ +d \\ +e +\end{pmatrix} += +\begin{pmatrix} +b \\ +2c \\ +3d \\ +4e +\end{pmatrix} +\end{align} + +That this matrix encodes the power rule for the Taylor eigenbasis for 4 dimensions; each entry in the vectors encodes the nth +power monomial term, which means, for example: +\begin{align} +\label{} +\begin{pmatrix} +a \\ +b \\ +c \\ +d \\ +e +\end{pmatrix} := a + bx + cx^{2} + dx^{3} + ex^{4} +\end{align} +then the derivative of this vector would be: +\begin{align} +\label{derivative} +b + 2cx + 3dx^{2} + 4ex^{3} +\end{align} +which is exactly the coefficients in the resultant vector! Now, if we generalize this to an infinite amount of dimensions +(where the vector has an infinite length and the matrix has infinite entries), this corresponds to the same effect. + +Thus, the infinite matrix with the off-diagonal increasing entries is the $D$ matrix, or $D$ operator. But what is the +$r^{2}$ operator? We know it must be a matrix operation that shifts the entire vector up by two, and pads the first two +entries of the vector with two zeros. If we find this matrix, the matrix multiplication $DRD$ should yield a new +infinite matrix $M$, which we can use in order to solve the eigenvalue problem $\det{(M - \lambda I)} = 0$. Now this matrix is: +\begin{align} +\label{R matrix} +\begin{pmatrix} +0 & 0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 & 0 \\ +1 & 0 & 0 & 0 & 0 \\ +0 & 1 & 0 & 0 & 0 \\ +0 & 0 & 1 & 0 & 0 \\ +\end{pmatrix} +\end{align} +and so on. This matrix does the exact same thing to a polynomial vector as what multiplying $r^{2}$ does to a polynomial. +We then multiply the two matrices to get this new matrix: +\begin{align} +\label{new} +\begin{pmatrix} +0 & 0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 & 0 \\ +1 & 0 & 0 & 0 & 0 \\ +0 & 1 & 0 & 0 & 0 \\ +0 & 0 & 1 & 0 & 0 \\ +\end{pmatrix} +\begin{pmatrix} +0 & 1 & 0 & 0 & 0 \\ +0 & 0 & 2 & 0 & 0 \\ +0 & 0 & 0 & 3 & 0 \\ +0 & 0 & 0 & 0 & 4 \\ +0 & 0 & 0 & 0 & 0 +\end{pmatrix} = +\begin{pmatrix} +0 & 0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 & 0 \\ +0 & 1 & 0 & 0 & 0 \\ +0 & 0 & 2 & 0 & 0 \\ +0 & 0 & 0 & 3 & 0 \\ +\end{pmatrix} +\end{align} +and so on, as you can see the pattern. Now we multiply in another $D$: +\begin{align} +\label{DS} +\begin{pmatrix} +0 & 1 & 0 & 0 & 0 \\ +0 & 0 & 2 & 0 & 0 \\ +0 & 0 & 0 & 3 & 0 \\ +0 & 0 & 0 & 0 & 4 \\ +0 & 0 & 0 & 0 & 0 +\end{pmatrix} +\begin{pmatrix} +0 & 0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 & 0 \\ +0 & 1 & 0 & 0 & 0 \\ +0 & 0 & 2 & 0 & 0 \\ +0 & 0 & 0 & 3 & 0 \\ +\end{pmatrix} = +\begin{pmatrix} +0 & 0 & 0 & 0 & 0 \\ +0 & 2 \cdot 1 & 0 & 0 & 0 \\ +0 & 0 & 3 \cdot 2 & 0 & 0 \\ +0 & 0 & 0 & 4 \cdot 3 & 0 \\ +0 & 0 & 0 & 0 & 5 \cdot 4 +\end{pmatrix} +\end{align} +of course the $5 \cdot 4$ isn't actually resultant from the image above, but it is from the infinite version +of this process. Now, we can finally subtract $\lambda$ from this infinite matrix: +\begin{align} +\label{lambda} +\begin{pmatrix} +-\lambda & 0 & 0 & 0 & 0 \\ +0 & 2 \cdot 1 - \lambda & 0 & 0 & 0 \\ +0 & 0 & 3 \cdot 2 - \lambda & 0 & 0 \\ +0 & 0 & 0 & 4 \cdot 3 - \lambda & 0 \\ +0 & 0 & 0 & 0 & 5 \cdot 4 - \lambda +\end{pmatrix} +\end{align} +Now, you might be wondering how we're going to take the determinant of this infinite matrix. We can take +a [[id:122fd244-ffeb-47d0-89ce-bf9bc6f01b70][limit]] of finite matrices to find out what the generalization might be. For instance, the 3d case might look like this: +\begin{align} +\label{} +\lambda^{2}(2\cdot 1 - \lambda) +\end{align} +(as the very last diagonal entry in the finite case does not extend infinitely, there is no $3 \cdot 2$ term). Now taking some +higher dimensions: +\begin{align} +\label{higher dimensions} +\lambda^{2}(2 \cdot 1 - \lambda)(3 \cdot 2 - \lambda) \\ +\lambda^{2}(2 \cdot 1 - \lambda)(3 \cdot 2 - \lambda)(4 \cdot 3 - \lambda) \\ +\lambda^{2}(2 \cdot 1 - \lambda)(3 \cdot 2 - \lambda)(4 \cdot 3 - \lambda)(5 \cdot 4 - \lambda) +\end{align} +using inductive reasoning we should expect the infinite form to be: +\begin{align} +\label{} +det(M) = -\lambda(2 \cdot 1 - \lambda)(3 \cdot 2 - \lambda)(4 \cdot 3 - \lambda)(5 \cdot 4 - \lambda)(6 \cdot 5 - \lambda)\dots +\end{align} +(note that it isn't $\lambda^{2}$ because the very last $-\lambda$ never gets multiplied, and it's negative for that reason too). Note +that if we want to set $det(M) = 0$, $\lambda = n(n - 1)$ where $n$ is a natural number (including zero). Then we substitute +back in the $\lambda$ for some $n$, let's use $n = 2$ as an example: +\begin{align} +\label{n=2} +\begin{pmatrix} +-2 & 0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 & 0 \\ +0 & 0 & 4 & 0 & 0 \\ +0 & 0 & 0 & 10 & 0 \\ +0 & 0 & 0 & 0 & 18 +\end{pmatrix} +\begin{pmatrix} +a_{1} \\ +a_{2} \\ +a_{3} \\ +a_{4} \\ +a_{5} \\ +\end{pmatrix} = +\begin{pmatrix} +0 \\ +0 \\ +0 \\ +0 \\ +0 +\end{pmatrix} +\end{align} +clearly, when we choose $n = 2$, the second value $a_{2}$ is free, and the rest for the given eigenfunction +must be zero, meaning for a given $n$, the resulting eigenvector is $ax^{n}$ for any value $a$. This is one of the solutions +to this differential equation. + +It turns out there's another solution in a space that the Taylor space does not span, but I'll leave it as an exercise +to find the other solution using this method, by extending it to include other kinds of functions. Note that for your +eigenbasis one can use the [[id:262ca511-432f-404f-8320-09a2afe1dfb7][Fourier Transform]] to make a Fourier basis, but that's also easy to generalize. diff --git a/mindmap/egoism.org b/mindmap/egoism.org index fb16307..9afef56 100644 --- a/mindmap/egoism.org +++ b/mindmap/egoism.org @@ -11,6 +11,9 @@ Egoism, broadly defined, is the philosophical view that one should maximize for their own utility function, or well-being, or self-satisfaction, etc..., whatever you want to call it. The general statement is that there is some value one would want to maximize for themselves. This philosophical view is a metaethical view in general but can be applied descriptively. + +This view is a part of a broader view of [[id:4b4d4071-2ef4-4a6d-ada3-adc8cac425cc][consequentialism]], a view that I broadly subscribe to; sometimes utilitarian +analysis is useful for describing things inside the egoist framework, and sometimes it is not. * Descriptively In economics, homo-econimus is a kind of idealized economic agent that maximizes for their own economic value. Descriptive egoism is in a sense the statement that people tend to maximize for some utility function. In a sense, there are many games diff --git a/mindmap/electrostatics.org b/mindmap/electrostatics.org index d865f86..7072bf8 100644 --- a/mindmap/electrostatics.org +++ b/mindmap/electrostatics.org @@ -25,14 +25,14 @@ these charges have no mass or volume; we call these /point charges/. The equatio on \( P_{1} \) is as follows: \begin{align*} -\vec{F(\vec{r})} = \frac{1}{4\pi\epsilon_{0}}\frac{qQ}{r^{2}} \hat{r}. +\vec{F(\vec{r})} = \frac{1}{4\pi\epsilon_{0}}\frac{qQ}{r^{2}} \hat{r} \end{align*} Where \( \hat{r} \) is the unit vector pointing in the direction of \( P_{2} \). Note that there are a couple of interesting things about this force. First, it is an [[id:2a543b79-33a0-4bc8-bd1c-e4d693666aba][inverse square]] law, and the formula looks a lot like the one for gravitation, only charge can be negative and mass cannot. Second, it is symmetrical, in the sense that the force felt by \( P_{2} \) is going to be the same, only \( \hat{r} \) -is pointing in the other direction. Also, note that due to linearity, this force calculation follows the /superposition principle/. +is pointing in the other direction. Also, note that due to linearity, this force calculation follows the [[id:422653e2-daa4-422a-9cb7-3983a5a72554][superposition principle]]. That is, if we have different electrostatic forces acting on one particle: \begin{align*} @@ -83,7 +83,7 @@ The result is we find a way to express force in a /test charge independent way/. because we often want to find the force if an arbitrary object with an arbitrary charge is next to the particle in question, instead of focusing specifically on two charges. -Note that it is trivial to prove that \( \vec{E} \) also follows the superposition principle. +Note that it is trivial to prove that \( \vec{E} \) also follows the [[id:422653e2-daa4-422a-9cb7-3983a5a72554][superposition principle]]. ** Continuous Charge Distributions diff --git a/mindmap/framework.org b/mindmap/framework.org index 114913f..f311e19 100644 --- a/mindmap/framework.org +++ b/mindmap/framework.org @@ -1,5 +1,6 @@ :PROPERTIES: :ID: 6d8c8bcc-58b0-4267-8035-81b3bf753505 +:ROAM_ALIASES: system :END: #+title: framework #+author: Preston Pan diff --git a/mindmap/index.org b/mindmap/index.org index 1177694..28f80f4 100644 --- a/mindmap/index.org +++ b/mindmap/index.org @@ -1,3 +1,4 @@ +# -*- org-publish-use-timestamps-flag: nil; -*- #+title: Mindmap #+author: Preston Pan #+date: <2023-06-10 Sat> diff --git a/mindmap/infinity.org b/mindmap/infinity.org index 8d41105..0f01b17 100644 --- a/mindmap/infinity.org +++ b/mindmap/infinity.org @@ -19,3 +19,4 @@ extends the real numbers to include infinites. However, only order is well defin \begin{align*} \forall a \in \mathbb{R}, -\infty < a < \infty \end{align*} +This definition of infinity is generally used to define concepts such as [[id:122fd244-ffeb-47d0-89ce-bf9bc6f01b70][limits]] in [[id:a6bc601a-7910-44bb-afd5-dffa5bc869b1][mathematics]]. diff --git a/mindmap/inner product space.org b/mindmap/inner product space.org new file mode 100644 index 0000000..e9448a0 --- /dev/null +++ b/mindmap/inner product space.org @@ -0,0 +1,26 @@ +:PROPERTIES: +:ID: 36a2715c-a8db-4b75-b799-61ce43be2d2d +:END: +#+title: inner product space +#+author: Preston Pan +#+html_head: <link rel="stylesheet" type="text/css" href="../style.css" /> +#+html_head: <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script> +#+html_head: <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> +#+options: broken-links:t + +* Introduction +An inner product space is a [[id:9a1cc2d9-ef99-436c-8c21-9e68fd7df192][normed vector space]] with an inner product defined. This inner product obeys the +following properties: +\begin{align} +\label{} +\langle x,y \rangle = \overline{\langle y,x \rangle} \\ +\langle ax + by, z \rangle = a\langle x,z \rangle + b\langle y,z \rangle \\ +\langle x,x \rangle > 0, x > 0 \\ +\langle x,x \rangle = 0, x = 0 +\end{align} +where $\overline{\langle y,x \rangle}$ is the complex conjugate of $\langle x,y \rangle$. This gives rise to a normed vector space: +\begin{align} +\label{} +\lVert x \rVert = \langle x,x \rangle +\end{align} + diff --git a/mindmap/inverse square.org b/mindmap/inverse square.org index d55c24a..205c5f7 100644 --- a/mindmap/inverse square.org +++ b/mindmap/inverse square.org @@ -189,4 +189,10 @@ V(\vec{r}) := k\int_{space}\frac{\sigma(r')}{r}d\tau \end{align*} Note that because this field does not require keeping track of vector orientation, it is significantly easier to solve for \(V\) then convert to \(\vec{f}\). Additionally, setting a reference point to something that is not infinity would be valid as well -- we just choose infinity because it cancels off the constant term. However, -the /difference/ in potentials is absolute and does not require any constant adjustment. +the /difference/ in potentials is absolute and does not require any constant adjustment. Then, the divergence of inverse +square fields can be reformulated with the [[id:65004429-a6b7-41f2-8489-07605841da3d][Laplacian]] operator: +\begin{align} +\label{} +\nabla^{2}V(\vec{r''}) = k\sigma(\vec{r''}) +\end{align} + diff --git a/mindmap/limit.org b/mindmap/limit.org index 65ae15f..22b3280 100644 --- a/mindmap/limit.org +++ b/mindmap/limit.org @@ -49,10 +49,22 @@ We know: \begin{align*} \lim s_{n} = s \iff \forall \epsilon > 0, \exists N, n > N \implies | s_{n} - s | < \epsilon \\ \end{align*} - +which is equivalent to: +\begin{align*} +\lim s_{n} = s \iff \forall \epsilon > 0, \exists N, n > N \implies s - \epsilon < s_{n} < s + \epsilon +\end{align*} and our sequence $\{s_{n}\}$ is monotone. If $\{s_{n}\}$ is increasing, we have: \begin{align*} s_{n + 1} \ge s_{n} \end{align*} -for all n. +for all n. Without loss of generality we shall assume $\{s_{n}\}$ is increasing. Then we take two cases: +1. $\{s_{n}\}$ is bounded. +2. $\{s_{n}\}$ is unbounded. +In the case $\{s_{n}\}$ is bounded: +\begin{align} +\label{} +\exists M, \forall n, s_{n} \le M \\ +s_{0} \le ... \le s_{n} \le s_{n + 1} \le s_{n + 2} \le ... \le M +\end{align} + #+end_proof diff --git a/mindmap/logic.org b/mindmap/logic.org index b43a76b..1de226b 100644 --- a/mindmap/logic.org +++ b/mindmap/logic.org @@ -12,3 +12,4 @@ Logic is the foundation of everything mathematical, and can even study itself; this is known as metalogic. 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index dc51536..3a5e1ec 100644 --- a/mindmap/mathematics.org +++ b/mindmap/mathematics.org @@ -7,22 +7,17 @@ #+html_head: <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> #+html_head: <link rel="stylesheet" type="text/css" href="../style.css" /> -* Mathematics is Logic -With a couple of set theory axioms. One might -describe it as an extrapolation framework -without grounding (i.e. a set of implications; -if p then q, but never specifying if p is a property -of a real system or not). Therefore, mathematics -is suitable for modeling other things if we believe -those other things have some rules and are logically -consistent. +* Mathematics is [[id:29ebc4f9-0fd8-4203-8bfe-84f8558e09cf][Logic]] +With a couple of set theory axioms. One might describe it as an extrapolation framework +without grounding (i.e. a set of implications; if p then q, but never specifying if p is a property +of a real system or not). Therefore, mathematics is suitable for modeling other things if we believe +those other things have some rules and are logically consistent. -There are many subfields in math, ranging from group theory -to complex analysis. However, much of the time we are able to find -morphisms between these different fields in mathematics, which we +There are many subfields in math, ranging from group theory to calculus. +However, much of the time we are able to find morphisms between these different fields in mathematics, which we model using category theory. - ** Mathematical Models +There are many fields where mathematics is the main method of analysis. *** [[id:ece8bf94-4e3c-4939-a77a-9949c1ec0dc6][Physics]] Physics is the study of the natural world and its laws, usually resorting to a mathematical framework. diff --git a/mindmap/metric space.org b/mindmap/metric space.org new file mode 100644 index 0000000..df750db --- /dev/null +++ b/mindmap/metric space.org @@ -0,0 +1,22 @@ +:PROPERTIES: +:ID: 6f24f731-60e5-4904-88d7-c63869505981 +:ROAM_ALIASES: metric +:END: +#+title: metric space +#+author: Preston Pan +#+html_head: <link rel="stylesheet" type="text/css" href="../style.css" /> +#+html_head: <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script> +#+html_head: <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> +#+options: broken-links:t + +* Introduction +A metric space $(G, d)$ is a set with a metric $d(x,y): G \times G \rightarrow \mathbb{R}$ defined on members of the set. +This metric is a generalization of distance, with the following properties: +\begin{align} +\label{} +d(x, x) = 0 \\ +x \ne y \implies d(x, y) > 0 \\ +d(x, y) = d(y, x) \\ +d(x, z) \le d(x, y) + d(x, z) +\end{align} +where property $(4)$ is the triangle inequality. diff --git a/mindmap/normed vector space.org b/mindmap/normed vector space.org new file mode 100644 index 0000000..82f1f6f --- /dev/null +++ b/mindmap/normed vector space.org @@ -0,0 +1,23 @@ +:PROPERTIES: +:ID: 9a1cc2d9-ef99-436c-8c21-9e68fd7df192 +:END: +#+title: normed vector space +#+author: Preston Pan +#+html_head: <link rel="stylesheet" type="text/css" href="../style.css" /> +#+html_head: <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script> +#+html_head: <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> +#+options: broken-links:t + +* Introduction +A normed vector space is a [[id:ab024db7-6903-48ee-98fc-b2a228709c04][vector space]] with a norm defined, which describes the "length" of the vector. +This norm obeys these properties: +\begin{align} +\label{} +\lVert ax \rVert = \lvert a \rvert \lVert x \rVert \\ +\lVert x + y \rVert \le \lVert x \rVert + \lVert y \rVert +\end{align} +this gives rise to a [[id:6f24f731-60e5-4904-88d7-c63869505981][metric]] $d(x, y)$: +\begin{align} +\label{} +d(x, y) = \lVert x - y \rVert +\end{align} diff --git a/mindmap/philosophy.org b/mindmap/philosophy.org index b1a26e5..188cd23 100644 --- a/mindmap/philosophy.org +++ b/mindmap/philosophy.org @@ -24,7 +24,45 @@ that they are two different things in many senses, but that the fundamental asse It is just a different kind of emotion, but there is no underlying fact of the matter that one can point to with regards to moral theory. -Note that there are several arguments that facts are treated on a separate footing to moral theory under such a framework. +Note that there are several arguments that facts are treated on a separate footing to moral theory under such a [[id:6d8c8bcc-58b0-4267-8035-81b3bf753505][framework]]. Indeed it is true that this mindmap will rest on some emperical facts, but this mindmap maintains that doing this is a perfectly internally consistent and descriptive standpoint. From here on, we will use ethical and moral statements as a description of people, rather than a description of some real moral fact. + +Generally speaking, one can use [[id:29ebc4f9-0fd8-4203-8bfe-84f8558e09cf][logical deduction]] in order to reach conclusions from initial epistemological or +metaphysical assertions in philosophy. People also apply the same reasoning to moral intuition, but as I explained above, +I do not hold moral philosophy to be important. +* Philosophy and Egoism +Egoism is a generally acceptable bootstrapping belief; it can get oneself into talking about moral facts (or the lack +thereof) without many buy-ins, and it can describe a wide variety of other beliefs from within its own framework. The logical +consequence of choosing egoism as an acceptable framework is that philosophy becomes the study of maximizing for the +goals created by oneself; in esscence, egoism is the weak assertion that there is something in life to be optimized. + +One can, in general, create an optimal life by doing two things: +1. When a value is easier to get rid of than to satisfy, get rid of it. +2. Derive as many current values from deeper, more fundamental values as possible, using [[id:29ebc4f9-0fd8-4203-8bfe-84f8558e09cf][logic]]. +Being attached to moral values is itself in contradiction to satisfying said moral values; you're creating more work +for yourself, much of which one can't do. One should view values themselves as tools to achieve some optimal end, +whatever that may mean to you, and deriving current values from deeper values using logic allows you to rule out +values that you hold for no good reason. This metavalue system is efficient because it gets rid of values that do harm +to the egoist goal. + +For instance, some may care about climate change, and wish to do something about climate change because of some moral +value that they hold. I hold that this is, in many cases, ill advised, because singular people cannot do anything about +climate change. However, many still hold onto the belief that they are somehow important in the cause, when they just +objectively aren't (it would be [[id:7456da20-684d-4de6-9235-714eaafb2440][IEEDI]] syndrome), and hold that they should still do it for "moral reasons". If these "moral reasons" are just tools +that you can bend given you convince yourself of something else, why would one subject themselves to doing something +suboptimal? + +The answer is that two things could be going on: either it is hard mentally for them to accept getting rid of their +values, in which case they should keep their values, or they haven't thought about the fact that it isn't a good idea +from an egoist standpoint. I think this is common. + +Very few modern ideas that people would consider "moral things to do" are built into the human condition. As long +as you can escape those ideas easily, you'd have more time and energy to allocate towards doing something that satisfies +goals that are more tangible (you can't fix climate change on your own, but you can fix your own life). Give up on +things that don't give you an advantage, or gives you a disadvantage (advantage and disadvantage being with respect +to values that are hard to give up on, such as having friends, eating food, drinking water, etc...). +* Isn't This Value Itself A Tool? +Yes, and I could've described this in many different ways using many different metaframeworks, as they all probably +have the same perscriptive power. However, I hold that this would "work better" for most people who try it. diff --git a/mindmap/physics.org b/mindmap/physics.org index f0fcf89..eeb3a0c 100644 --- a/mindmap/physics.org +++ b/mindmap/physics.org @@ -20,7 +20,7 @@ There is no currently known theory that can describe [[id:2b8515d8-9f3c-44a3-a40 many correspondences and [[id:1b1a8cff-1d20-4689-8466-ea88411007d7][dualities]]. ** Classical Mechanics Classical mechanics deals with fields of physics that do not deal with time dilation or quantum -weirdness. It is called classical mechanics because most of these theories wer invented before +weirdness. It is called classical mechanics because most of these theories were invented before their quantum or relativistic counterparts, and usually serve as a baseline understanding for the later topics. *** [[id:6e2a9d7b-7010-41da-bd41-f5b2dba576d3][Newtonian mechanics]] @@ -30,8 +30,10 @@ governed all things. This is a different strain of classical mechanics. Instead of looking at direction and vectors, it looks at a single fundamental principle, even more fundamental than inertia and conservation of momentum: optimization. We believe that nature always optimizes for some parameters in all physical phenomena. Almost -all physical theories including electromagnetism, general relativity, Newtonian mechanics, and even quantum mechanics can be explained -within the Lagrangian framework. +all physical theories including electromagnetism, general relativity, Newtonian mechanics, and even quantum +mechanics can be explained within the Lagrangian framework. +*** Fluid Mechanics +*** Thermodynamics *** Classical Electrodynamics Classical electrodynamics attempts to explain the electromagnetic force from a classical perspective. What is light? How does electricity actually work? All these questions you will find (half) answered in the study @@ -39,7 +41,13 @@ of electrodynamics, giving rise to the complete theory encoded in [[id:fde2f257- *** General Relativity General Relativity aims unify gravity with [[id:e38d94f2-8332-4811-b7bd-060f80fcfa9b][special relativity]], in a unified theory of macroscopic forces including classical electromagnetism. - -** Quantum Mechanics +** [[id:136e79df-106f-4989-ab19-89705929cf91][Quantum Mechanics]] Quantum mechanics renounces all hope of having a deterministic view of physics, and instead replaces direct causation with statistical causation, at least according to some interpretations. +** QFT +Quantum Field Theory is an approach that combines [[id:e38d94f2-8332-4811-b7bd-060f80fcfa9b][special relativity]] with [[id:136e79df-106f-4989-ab19-89705929cf91][quantum mechanics]], then utilizing a process +called canonical quantization in order to quantize a given field. This is the closest thing we have to a theory of [[id:2b8515d8-9f3c-44a3-a40d-147f6a2bbb25][everything]] +in physics. + +* Note +I did not exhaustively cover all fields in physics but this is most of it. diff --git a/mindmap/python.org b/mindmap/python.org deleted file mode 100644 index 6234beb..0000000 --- a/mindmap/python.org +++ /dev/null @@ -1,6 +0,0 @@ -:PROPERTIES: -:ID: 5d2e2f3b-96ac-4196-9baf-4c3d6d349c98 -:END: -#+title: python -#+author: Preston Pan -#+html_head: <link rel="stylesheet" type="text/css" href="../style.css" /> diff --git a/mindmap/quantum mechanics.org b/mindmap/quantum mechanics.org new file mode 100644 index 0000000..e0df273 --- /dev/null +++ b/mindmap/quantum mechanics.org @@ -0,0 +1,45 @@ +:PROPERTIES: +:ID: 136e79df-106f-4989-ab19-89705929cf91 +:END: +#+title: quantum mechanics +#+author: Preston Pan +#+html_head: <link rel="stylesheet" type="text/css" href="../style.css" /> +#+html_head: <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script> +#+html_head: <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> +#+options: broken-links:t +* Introduction +Quantum mechanics was discovered as a predictive [[id:6d8c8bcc-58b0-4267-8035-81b3bf753505][framework]] in the early 1900's after a set of experiments (i.e. the +photoelectric effect and the Stern Gerlach experiment) showed that particles were better described as waves, and +that the probability of measurement of particle states were random. Indeed, other experiments that tested Bell's +inequality have confirmed this assertion of quantum mechanics that the world is fundamentally random. Let's take +a look at some of the postulates of quantum mechanics (for which the [[id:a6bc601a-7910-44bb-afd5-dffa5bc869b1][mathematics]] borrow from linear algebra quite a bit). +* Postulates +Here, we describe all the postulates of quantum mechanics, with some motivation when needed. +** Postulate 1 +$| \psi \rangle$ is the state vector in a [[id:eac079b7-1144-4506-b8c5-4717ccac8a7b][Hilbert Space]] $\mathcal{H}$ which describes the entire system. +** Postulate 2 +The norm $\langle \psi | \psi \rangle$ of all state vectors are 1. +** Postulate 3 +If $| \psi \rangle$ and $| \phi\rangle$ represent two different quantum systems, the composite system can be described by +$| \psi \rangle \otimes | \phi \rangle$, where $\otimes$ is the tensor product. +** Postulate 4 +Observable quantities are represented by Hermitian operators $\hat{A}$ whose eigenvectors form a basis for $\mathcal{H}$. +Solutions to $\hat{A}|\psi\rangle = a|\psi\rangle$ for eigenvalues $a$ are the only possible observable values for a given eigenvector $|\psi\rangle$. +** Postulate 5 +The time-evolution of the state vector $|\psi\rangle$ can be given by the Schrodinger equation, a [[id:365190d8-0f3a-4728-9b09-83a216292256][PDE]]: +\begin{align} +\label{} +\hat{H}|\psi\rangle = i\hbar\partial_{t}|\psi\rangle +\end{align} +which is motivated by the De Broglie hypothesis: +\begin{align} +\label{} +p = hf +\end{align} +where $p$ is the momentum, and $f$ is the frequency. De Broglie hypothesized that all matter behaves like a wave, +after Einstein came up with the same relation for light: +\begin{align} +\label{} +p = \frac{E}{c} = hf +\end{align} +where $h$ is Planck's constant, which is specifically a relation diff --git a/mindmap/recursion.org b/mindmap/recursion.org index 5191203..80fb92d 100644 --- a/mindmap/recursion.org +++ b/mindmap/recursion.org @@ -58,7 +58,7 @@ factorial [ dup 0 <= [ 1 + ] [ dup 1 - factorial * ] if ] def #+RESULTS: : 120 -and in stem, we can print out the stack every step of the way with the builtin word ~?~: +and in stem, we can print out the [[id:52d255d2-114c-42f4-b362-f0b4a2f7b83d][stack]] every step of the way with the builtin word ~?~: #+begin_src stem :exports both factorial-debug [ dup 0 <= [ 1 + ] [ ? "\n" . dup 1 - factorial-debug ? "\n" . * ] if ] def 5 factorial-debug . @@ -118,3 +118,16 @@ that doesn't cause recursion in the if statement), in which case we "go back up" is present in all programming languages, although in stem the operations are transparent as the stack is accessible by regular program users. In short, we keep on going down and down until we hit the bottom, base case, in which case we have all the pieces we need in order to go back up again, where the stack stores the information from most recent tasks to be done and we work back up in order to do the less recent tasks. + +This concept is important in programming because it allows one to build definitions in an intuitive way, simply by +specifying the base case and specifying the case that is not the base case. Such an algorithm absolves oneself from having +to design complicated patterns, as instead the entire computation emerges out of simple rules. + +In general, we see recursive definitions and design patterns in nature in the form of fractals. +* Self Reference Problems +A big part of [[id:654280d8-82e8-4a0e-a914-bd32181c101b][infinite]] [[id:8f265f93-e5fd-4150-a845-a60ab7063164][recursion]] has to do with self reference problems. For instance, Russel's paradox with respect to +set theory: does a set that contains all sets that do not contain themselves contain itself? + +Such a set would contain itself if and only if it didn't contain itself. This apparent contradiction in set theory is an +example of using recursion to reach self reference paradoxes. There are more examples, such as Godel's theorems and +Turing's computability theorem. diff --git a/mindmap/stack.org b/mindmap/stack.org index 74d0138..9f05a5e 100644 --- a/mindmap/stack.org +++ b/mindmap/stack.org @@ -4,3 +4,10 @@ #+title: stack #+author: Preston Pan #+html_head: <link rel="stylesheet" type="text/css" href="../style.css" /> +* Introduction +A stack in programming is a data structure that satisfies the following API rules: +1. Push: puts something on the top of the stack +2. Pop: takes item off stack and gives it to the caller +3. Peek: hands copy of top item on the stack to caller +4. isEmpty: checks if the stack size is zero +where stacks generally store their data in a list, either an array or a singly linked list. diff --git a/mindmap/vector space.org b/mindmap/vector space.org new file mode 100644 index 0000000..55f37c8 --- /dev/null +++ b/mindmap/vector space.org @@ -0,0 +1,27 @@ +:PROPERTIES: +:ID: ab024db7-6903-48ee-98fc-b2a228709c04 +:ROAM_ALIASES: vector linear "linear space" +:END: +#+title: vector space +#+author: Preston Pan +#+html_head: <link rel="stylesheet" type="text/css" href="../style.css" /> +#+html_head: <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script> +#+html_head: <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> +#+options: broken-links:t + +* Introduction +A vector space $V$ is a set with addition and scalar multiplication defined. It obeys the following axioms: +\begin{align} +\label{} +\vec{a} + (\vec{b} + \vec{c}) = (\vec{a} + \vec{b}) + \vec{c} \\ +\vec{a} + \vec{b} = \vec{b} + \vec{a} \\ +\exists \vec{0},\forall \vec{a}, \vec{a} + \vec{0} = \vec{a} \\ +\forall \vec{a},\exists\vec{-a}, \vec{a} + \vec{-a} = \vec{0} \\ +(cd)\vec{a} = c(d\vec{a}) \\ +1\vec{a} = \vec{a} \\ +c(\vec{a} + \vec{b}) = c\vec{a} + c\vec{b} \\ +(c + d)\vec{a} = c\vec{a} + d\vec{a} +\end{align} + +vector spaces are an Abelian [[id:ba7b95b0-0ce6-4b33-9a79-5e5fddaea710][group]] under addition. $\vec{a}$, $\vec{b}$, and $\vec{c}$ are considered vectors so long as they fulfill these +properties. |
