This is about the polarization of a string in 3D space. That is the first concept of polarization you must have in mind!
Far field approximation to Kirchhoff's diffraction formula, i.e. when the plane of observation is far from the object diffracting.
Near field approximation to Kirchhoff's diffraction formula, i.e. when the plane of observation is near the object diffracting.
Notably used for the pattern of the double-slit experiment.
Shows up when trying to solve 2D wave equation on a circular domain in polar coordinates with separation of variables, where we have to decompose the initial condition in termes of a fourier-Bessel series, exactly like the Fourier series appears when solving the wave equation in linear coordinates.
For the same fundamental reasons, also appears when calculating the Schrödinger equation solution for the hydrogen atom.
Completeness: math.stackexchange.com/questions/2192665/is-this-set-of-bessel-functions-a-basis-for-all-c10-a-functions TODO
This is the Bessel function analogue to Fourier basis is complete for .
eigenvalue problem of Laplace's equation.
Existence and uniqueness of solutions of partial differential equations by 
Ciro Santilli 37 Updated 2025-07-16
Ciro Santilli 37 Updated 2025-07-16Unlike for ordinary differential equations which have the Picard–Lindelöf theorem, the existence and uniqueness of solution is not well solved for PDEs.
For example, Navier-Stokes existence and smoothness was one of the Millennium Prize Problems.
This would allow us to index inscriptions in the .dat files directly with fast C tools, and then retrive the transaction ID to get cleaner data and metadata.
It should be possible if we managed to take the information from bitcoindev.network/understanding-the-data/ and dump into an indexed SQLite database.
I tried to start things off with LevelDBDumper:but that consumed all 64 GB of RAM on P51... github.com/mdawsonuk/LevelDBDumper/issues/15
LevelDBDumper -d ~/snap/bitcoin-core/common/.bitcoin/indexes/txindex -f btc.csv -q -o . -t csvStarted in 1987 and written in Pascal, by the French from Pierre and Marie Curie University, the French are really strong in numerical analysis.
The fact that French wrote it can be seen in the documentation, for example doc.freefem.org/tutorials/index.html uses file extension
mycode.edp instead of mycode.pde where dep stands for "Équation aux dérivées partielles".Besides the painful build, using FreeFem is relatively simple, as can be seen from the examples on the website.
They do use a domain-specific language on the examples, which appears to be the main/only interface, which is a bad thing, Ciro would rather have a Python API as the "main API", which is more the approach taken by the FEniCS Project, but so be it. This domain-specific language business means that you always stumble upon basic stuff you want to do but can't, and then you have to think about how to share data between the simulation and the plotting. The plotting notably is super complex and they can't implement all of what people want, upstream examples often offload that to gnuplot. This is potentially a big advantage of FEniCS Project.
It nice though that they do have some graphics out of the box, as that allows to quickly debug common problems.
Uses variational formulation of a partial differential equation, which is not immediately obvious to beginners? The introduction doc.freefem.org/tutorials/poisson.html gives an ultra quick example, but your are mostly on your own with that.
On Ubuntu 20.04, the
freefem is a bit out-of-date (3.5.8, there isn't even a tag for that in the GitHub repo, and refs/tags/release_3_10 is from 2010!) and fails to run the examples from the website. It did work with the example package though, but the output does not have color, which makes me sad :-)sudo apt install freefem freefem-examples
freefem /usr/share/doc/freefem-examples/heat.pdeSo let's just compile the latest v4.6 it from source, on Ubuntu 20.04:
sudo apt build-dep freefem
git clone https://github.com/FreeFem/FreeFem-sources
cd FreeFem-sources
# Post v4.6 with some fixes.
git checkout 3df0e2370d9752801ac744b11307b14e16743a44
# Won't apply automatically due to tab hell.
# https://superuser.com/questions/607410/how-to-copy-paste-tab-characters-via-the-clipboard-into-terminal-session-on-gnom
git apply <<'EOS'
diff --git a/3rdparty/ff-petsc/Makefile b/3rdparty/ff-petsc/Makefile
index dc62ab06..13cd3253 100644
--- a/3rdparty/ff-petsc/Makefile
+++ b/3rdparty/ff-petsc/Makefile
@@ -204,7 +204,7 @@ $(SRCDIR)/tag-make-real:$(SRCDIR)/tag-conf-real
$(SRCDIR)/tag-install-real :$(SRCDIR)/tag-make-real
cd $(SRCDIR) && $(MAKE) PETSC_DIR=$(PETSC_DIR) PETSC_ARCH=fr install
-test -x "`type -p otool`" && make changer
- cd $(SRCDIR) && $(MAKE) PETSC_DIR=$(PETSC_DIR) PETSC_ARCH=fr check
+ #cd $(SRCDIR) && $(MAKE) PETSC_DIR=$(PETSC_DIR) PETSC_ARCH=fr check
test -e $(DIR_INSTALL_REAL)/include/petsc.h
test -e $(DIR_INSTALL_REAL)/lib/petsc/conf/petscvariables
touch $@
@@ -293,7 +293,6 @@ $(SRCDIR)/tag-tar:$(PACKAGE)
-tar xzf $(PACKAGE)
patch -p1 < petsc-hpddm.patch
ifeq ($(WIN32DLLTARGET),)
- patch -p1 < petsc-metis.patch
endif
touch $@
$(PACKAGE):
EOS
autoreconf -i
./configure --enable-download --enable-optim --prefix="$(pwd)/../FreeFem-install"
./3rdparty/getall -a
cd 3rdparty/ff-petsc
make petsc-slepc
cd -
./reconfigure
make -j`nproc`
make install
cd ../FreeFem-install
PATH="${PATH}:$(pwd)/bin" ./bin/FreeFem++ ../FreeFem-sources/examples/tutorial/Ciro's initial build experience was a bit painful, possibly because it was done on a relatively new Ubuntu 20.04 as of June 2020, but in the end it worked: github.com/FreeFem/FreeFem-sources/issues/141
Pinned article: Introduction to the OurBigBook Project
Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
Intro to OurBigBook
. Source. We have two killer features:
- topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculusArticles of different users are sorted by upvote within each article page. This feature is a bit like:
- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1. Screenshot of the "Derivative" topic page. View it live at: ourbigbook.com/go/topic/derivativeVideo 2. OurBigBook Web topics demo. Source. - local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
Figure 3. Visual Studio Code extension installation.
Figure 4. Visual Studio Code extension tree navigation.
Figure 5. Web editor. You can also edit articles on the Web editor without installing anything locally.Video 3. Edit locally and publish demo. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension.Video 4. OurBigBook Visual Studio Code extension editing and navigation demo. Source. 
- Infinitely deep tables of contents:
All our software is open source and hosted at: github.com/ourbigbook/ourbigbook
Further documentation can be found at: docs.ourbigbook.com
Feel free to reach our to us for any help or suggestions: docs.ourbigbook.com/#contact