Multiple boundary conditions for different parts of the boundary.
Variational formulation of a partial differential equation Updated 2025-04-24 +Created 1970-01-01
Output: another sequence of complex numbers such that:Intuitively, this means that we are braking up the complex signal into sinusoidal frequencies:and is the amplitude of each sine.
- : is kind of magic and ends up being a constant added to the signal because
- : sinusoidal that completes one cycle over the signal. The larger the , the larger the resolution of that sinusoidal. But it completes one cycle regardless.
- : sinusoidal that completes two cycles over the signal
- ...
- : sinusoidal that completes cycles over the signal
Motivation: similar to the Fourier transform:In particular, the discrete Fourier transform is used in signal processing after a analog-to-digital converter. Digital signal processing historically likely grew more and more over analog processing as digital processors got faster and faster as it gives more flexibility in algorithm design.
- compression: a sine would use N points in the time domain, but in the frequency domain just one, so we can throw the rest away. A sum of two sines, only two. So if your signal has periodicity, in general you can compress it with the transform
- noise removal: many systems add noise only at certain frequencies, which are hopefully different from the main frequencies of the actual signal. By doing the transform, we can remove those frequencies to attain a better signal-to-noise
Sample software implementations:
- numpy.fft, notably see the example: numpy/fft.py
DFT of with 25 points
. This is a simple example of a discrete Fourier transform for a real input signal. It illustrates how the DFT takes N complex numbers as input, and produces N complex numbers as output. It also illustrates how the discrete Fourier transform of a real signal is symmetric around the center point.Flow Cytometry Animation by StarCellBio (2015)
Source. Some possible/not possible sources that could be used to manually bootstrap content:
- LibreTexts. Good project. "Teacher-only-content" unfortunately as usual. But besides that fundamental flaw, they do exactly what we want to do in a sense.
- OpenStax: CC BY. This could be a great entry point, as they already have some university integration going on, and might be interested in this project.
- physics.stackexchange.com/questions/6157/list-of-freely-available-physics-books "List of freely available physics books" explicitly asks for:but the thread was locked, and basically none of the sources in the answers have free licenses, nor do they note it. It just seems that the physicists don't know what a free license is.
a list of physics books with open-source licenses, like Creative Commons, GPL
- MIT OpenCourseWare: CC BY-NC-SA, so not really usable
- github.com/certik/theoretical-physics: MIT License. Workable but wonky.
- subwiki.org/: wiki with some upper graduate math subjects presumably by this Indian dude: www.linkedin.com/in/vipul-naik-0ab1898/. Description on his homepage: vipulnaik.com/subwiki/. He's also got other interesting but not so relevant projects:He's also into Stack Overflow, Quora and Wikipedia editing. That's a cool dude. He's into in LessWrong it seems.
- pro freer immigration laws: vipulnaik.com/openborders/
- vipulnaik.com/cognito-mentoring/ free mentoring project for interested students
- massive mathematics books
- Infinite Napkin.CC BY-SA mathematics infinite book: github.com/vEnhance/napkin/issues/77. Very similar type of content to what we want in this project!
- Stacks Project
Existing lecture notes by students:
- github.com/mb2g17/NotesNetworkArchive Google Docs-based: docs.google.com/document/d/1OIcQ8dJ_FAhdkirU94M29-ZbNZ4oQs1LbWF3Nz-mq_U/edit#heading=h.vehxib58w1iw. An actual student uploading tons of lecture notes in one coherent system. CC BY-NC-SA unfortunately.
- academia.stackexchange.com/questions/148261/do-you-keep-your-study-notes-publicly-available mentions:Related: academia.stackexchange.com/questions/40381/how-common-is-it-that-professors-have-their-students-write-textbooks
- Cambridge Mathematics Lecture Notes by Dexter Chua (2014-2018)Comments:
Lecture note upload website:
- nexusnotes.com likely illegal reuploads of PDFs from teachers
- www.studocu.com/en-gb Paywall. PDF uploads. Unclear if simple teacher reuploads or actual novel notes.
- www.studydrive.net/
- Chinese GitHub repos. Some of these are very advanced in terms of content quantity and organizational quality! The Chinese are miles ahead in this area:
- github.com/PKUanonym/REKCARC-TSC-UHT Guidance for courses in Department of Computer Science and Technology, Tsinghua University. Chinese. Appears to try and store all past exams.
- github.com/lib-pku/libpku
- github.com/openwhu/OpenWHU: Wuhan University
- github.com/USTC-Resource/USTC-Course: USTC
- github.com/Zeal-L/UNSW: UNSW from Australia, but by a Chinese dude
- github.com/apachecn/mit-18.06-linalg-notes: translation of MIT course to Chinese
- github.com/chenyang1999/MyComputerCollegeCourses: TODO which univeresity
- github.com/elder-frog/OpenCourseCatalog: nothing to do with this project, but since I'm making a list, this dude is copying YouTube videos to Bilibili. And he's edgy anti-CCP on Twitter, what a legend.
- github.com/TheBloodthirster/BUAA_Course_Sharing: en.wikipedia.org/wiki/Beihang_University
- github.com/1051727403/SHU-CS-Source-Share: ShangHai University CS course source code
- github.com/Willie169/tw-gifted-k12-notes: Taiwanese high school notes
Exams uploads:
- questions.tripos.org/part-ib/all/ University of Cambridge Mathematics past examinations
Teachers have the incentive of making open source to get more students.
Students pay when they want help to learn something.
Metric space but where the distance between two distinct points can be zero.
Notable example: Minkowski space.
This is a term invented by Ciro Santilli, and refers to a loose set of uncommon Bitcoin inscription methods that involve inscribing one or a small number of payloads per Bitcoin transaction.
These methods are both inefficient and hard to detect and decode, partly because Bitcoin Core does not index spending transactions: bitcoin.stackexchange.com/questions/61794/bitcoin-rpc-how-to-find-the-transaction-that-spends-a-txo. This makes finding them all that more rewarding however.
On the other hand, they do have the advantage of not depending on any block size limits, as their individual transactions are very small.
There are unlisted articles, also show them or only show them.