Mnemonic: the gradient shows the direction in which the function increases fastest.
Therefore, it has to:
- take a scalar field as input. Otherwise, how do you decide which vector is larger than the other?
- output a vector field that contains the direction in which the scalar increases fastest locally at each point. This has to give out vectors, since we are talking about directions
This is the standard model.
This important and common simple case has easy properties.
Techniques to get numerical approximations to numeric mathematical problems.
The entire field comes down to estimating the true values with a known error bound, and creating algorithms that make those error bounds asymptotically smaller.
Not the most beautiful field of pure mathematics, but fundamentally useful since we can't solve almost any useful equation without computers!
The solution visualizations can also provide valuable intuition however.
Important numerical analysis problems include solving:
This is the hardest one to do iteratively.
It is hard to decide what makes Ciro Santilli more sad: the usage of Greek letters, the invention of new symbols, or the fifty million alternative font styles used.
Only Chinese characters could be worse than that!
Repeat this mantra:
Only decentralize when inevitable.
Only decentralize when inevitable.
Only decentralize when inevitable.
GPIO generally only supports discrete outputs.
But for some types of hardware, like LEDs and some motors, the system has some inertia, and if you switch on and off fast enough, you get a result similar to having an intermediate voltage.
So with pulse width modulation we can fake analog output from digital output in a good enough manner.
They might have shut down, but they still have the cutest name! And they've made some cute inscriptions too, see: HHTT
77822fd6663c665104119cb7635352756dfc50da76a92d417ec1a12c518fad69 by 
Ciro Santilli 40 Updated 2025-07-16
Ciro Santilli 40 Updated 2025-07-16Ouptut 0 disassembles as:The large constant contains an ASCII Bitcoin Core patch entitled
OP_IF OP_INVALIDOPCODE 4effffffff <large constant> OP_ENDIFRemove (SINGLE|DOUBLE)BYTE so presumably this is a proof of concept:From a3a61fef43309b9fb23225df7910b03afc5465b9 Mon Sep 17 00:00:00 2001
From: Satoshi Nakamoto <satoshin@gmx.com>
Date: Mon, 12 Aug 2013 02:28:02 -0200
Subject: [PATCH] Remove (SINGLE|DOUBLE)BYTE
I removed this from Bitcoin in f1e1fb4bdef878c8fc1564fa418d44e7541a7e83
in Sept 7 2010, almost three years ago. Be warned that I have not
actually tested this patch.
---
backends/bitcoind/deserialize.py | 8 +-------
1 file changed, 1 insertion(+), 7 deletions(-)
diff --git a/backends/bitcoind/deserialize.py b/backends/bitcoind/deserialize.py
index 6620583..89b9b1b 100644
--- a/backends/bitcoind/deserialize.py
+++ b/backends/bitcoind/deserialize.py
@@ -280,10 +280,8 @@ opcodes = Enumeration("Opcodes", [
"OP_WITHIN", "OP_RIPEMD160", "OP_SHA1", "OP_SHA256", "OP_HASH160",
"OP_HASH256", "OP_CODESEPARATOR", "OP_CHECKSIG", "OP_CHECKSIGVERIFY", "OP_CHECKMULTISIG",
"OP_CHECKMULTISIGVERIFY",
- ("OP_SINGLEBYTE_END", 0xF0),
- ("OP_DOUBLEBYTE_BEGIN", 0xF000),
"OP_PUBKEY", "OP_PUBKEYHASH",
- ("OP_INVALIDOPCODE", 0xFFFF),
+ ("OP_INVALIDOPCODE", 0xFF),
])
@@ -293,10 +291,6 @@ def script_GetOp(bytes):
vch = None
opcode = ord(bytes[i])
i += 1
- if opcode >= opcodes.OP_SINGLEBYTE_END and i < len(bytes):
- opcode <<= 8
- opcode |= ord(bytes[i])
- i += 1
if opcode <= opcodes.OP_PUSHDATA4:
nSize = opcode
--
1.7.9.4bitcointalk.org/index.php?topic=5231222.0 discusses what happens if there is an invalid opcode in a branch that is not taken.
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