Here is a very direct description of the system:Code 1. "Sample Bitcoin transaction graph" illustrates these concepts:
- each transaction (transaction is often abbreviated "tx") has a list of inputs, and a list of outputs
- each input is the output of a previous transaction. You verify your identity as the indented receiver by producing a digital signature for the public key specified on the output
- each output specifies the public key of the receiver and the value being sent
- the sum of output values cannot obvious exceed the sum of input values. If it is any less, the leftover is sent to the miner of the transaction as a transaction fee, which is an incentive for mining.
- once an output is used from an input, it becomes marked as spent, and cannot be reused again. Every input uses the selected output fully. Therefore, if you want to use an input of 1 BTC to pay 0.1 BTC, what you do is to send 0.1 BTC to the receiver, and 0.9 BTC back to yourself as change. This is why the vast majority of transactions has two outputs: one "real", and the other change back to self.
tx0: magic transaction without any inputs, i.e. either Genesis block or a coinbase mining reward. Since it is a magic transaction, it produces 3 Bitcoins from scratch: 1 inout0and 2 inout1. The initial value was actually 50 BTC and reduced with time: Section "Bitcoin halving"tx1: regular transaction that takes:Since this is a regular transaction, no new coins are produced.tx2: regular transaction with a single input and a single output. It uses up the entire input, leading to 0 miner fees, so this greedy one might (will?) never get mined.tx3: regular transaction with two inputs and one output. The total input is 2.3, and the output is 1.8, so the miner fee will be 0.5
tx1 tx3
tx0 +---------------+ +---------------+
+----------+ | in0 | | in0 |
| out0 |<------out: tx0 out0 | +------out: tx1 out1 |
| value: 1 | +---------------+ | +---------------+
+----------+ | out0 | | | in1 |
| out1 |<-+ | value: 0.5 | | +----out: tx2 out0 |
| value: 2 | | +---------------+ | | +---------------+
+----------+ | | out1 |<-+ | | out1 |
| | value: 0.3 | | | value: 1.8 |
| +---------------+ | +---------------+
| |
| |
| |
| tx2 |
| +---------------+ |
| | in0 | |
+----out: tx0 out1 | |
+---------------+ |
| out0 |<---+
| value: 2 |
+---------------+Code 1.
Sample Bitcoin transaction graph
. Since every input must come from a previous output, there must be some magic way of generating new coins from scratch to bootstrap the system. This mechanism is that when the miner mines successfully, they get a mining fee, which is a magic transaction without any valid inputs and a pre-agreed value, and an incentive to use their power/compute resources to mine. This magic transaction is called a "coinbase transaction".
The key innovation of Bitcoin is how to prevent double spending, i.e. use a single output as the input of two different transactions, via mining.
For example, what prevents me from very quickly using a single output to pay two different people in quick succession?
The solution are the blocks. Blocks discretize transactions into chunks in a way that prevents double spending.
A block contains:
- a list of transactions that are valid amongst themselves. Notably, there can't be double spending within a block.People making transactions send them to the network, and miners select which ones they want to add to their block. Miners prefer to pick transactions that are:
- the ID of its parent block. Blocks therefore form a linear linked list of blocks, except for temporary ties that are soon resolved. The longest known list block is considered to be the valid one.
- a nonce, which is an integer chosen "arbitrarily by the miner"
For a block to be valid, besides not containing easy to check stuff like double spending, the miner must also select a nonce such that the hash of the block starts with N zeroes.
For example, considering the transactions from Code 1. "Sample Bitcoin transaction graph", the block structure shown at Code 2. "Sample Bitcoin blockchain" would be valid. In it
block0 contains two transactions: tx0 and tx1, and block1 also contains two transactions: tx2 and tx3. block0 block1 block2
+------------+ +--------------+ +--------------+
| prev: |<----prev: block0 |<----prev: block1 |
+------------+ +--------------+ +--------------+
| txs: | | txs: | | txs: |
| - tx0 | | - tx2 | | - tx4 |
| - tx1 | | - tx3 | | - tx5 |
+------------+ +--------------+ +--------------+
| nonce: 944 | | nonce: 832 | | nonce: 734 |
+------------+ +--------------+ +--------------+
Code 2.
Sample Bitcoin blockchain
. nonces are on this example arbitrary chosen numbers that would lead to a desired hash for the block.block0 is the Genesis block, which is magic and does not have a previous block, because we have to start from somewhere. The network is hardcoded to accept that as a valid starting point.Now suppose that the person who created Clearly, this transaction would try to spend Notably, it is not possible that
tx2 had tried to double spend and also created another transaction tx2' at the same time that looks like this: tx2'
+---------------+
| in0 |
| out: tx0 out1 |
+---------------+
| out0 |
| value: 2 |
+---------------+tx0 out1 one more time in addition to tx2, and should not be allowed! If this were attempted, only the following outcomes are possible:block1containstx2. Then whenblock2gets made, it cannot containtx2', becausetx0 out1was already spent bytx2block1containstx2'.tx2cannot be spent anymore
block1 contains both tx2 and tx2', as that would make the block invalid, and the network would not accept that block even if a miner found a nonce.Since hashes are basically random, miners just have to try a bunch of nonces randomly until they find one that works.
The more zeroes, the harder it is to find the hash. For example, on the extreme case where N is all the bits of the hash output, we are trying to find a hash of exactly 0, which is statistically impossible. But if e.g. N=1, you will in average have to try only two nonces, N=2 four nonces, and so on.
The value N is updated every 2 weeks, and aims to make blocks to take 10 minutes to mine on average. N has to be increased with time, as more advanced hashing hardware has become available.
Once a miner finds a nonce that works, they send their block to the network. Other miners then verify the block, and once they do, they are highly incentivized to stop their hashing attempts, and make the new valid block be the new parent, and start over. This is because the length of the chain has already increased: they would need to mine two blocks instead of one if they didn't update to the newest block!
Therefore if you try to double spend, some random miner is going to select only one of your transactions and add it to the block.
They can't pick both, otherwise their block would be invalid, and other miners wouldn't accept is as the new longest one.
Then sooner or later, the transaction will be mined and added to the longest chain. At this point, the network will move to that newer header, and your second transaction will not be valid for any miner at all anymore, since it uses a spent output from the first one that went in. All miners will therefore drop that transaction, and it will never go in.
The goal of having this mandatory 10 minutes block interval is to make it very unlikely that two miners will mine at the exact same time, and therefore possibly each one mine one of the two double spending transactions. When ties to happen, miners randomly choose one of the valid blocks and work on top of it. The first one that does, now has a block of length L + 2 rather than L + 1, and therefore when that is propagated, everyone drops what they are doing and move to that new longest one.
Most of what follows is part of the Universal Chess Interface. Tested on Ubuntu 22.10, Stockfish 14.1.
After starting Sweet ASCII art. where:
stockfish on the command line, d (presumably display) contains: +---+---+---+---+---+---+---+---+
| r | n | b | q | k | b | n | r | 8
+---+---+---+---+---+---+---+---+
| p | p | p | p | p | p | p | p | 7
+---+---+---+---+---+---+---+---+
| | | | | | | | | 6
+---+---+---+---+---+---+---+---+
| | | | | | | | | 5
+---+---+---+---+---+---+---+---+
| | | | | | | | | 4
+---+---+---+---+---+---+---+---+
| | | | | | | | | 3
+---+---+---+---+---+---+---+---+
| P | P | P | P | P | P | P | P | 2
+---+---+---+---+---+---+---+---+
| R | N | B | Q | K | B | N | R | 1
+---+---+---+---+---+---+---+---+
a b c d e f g h
Fen: rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR w KQkq - 0 1
Key: 8F8F01D4562F59FBFen: FEN notationKey: TODO
Move white king's pawn from e2 to e4:Then display again:gives:so we see that the pawn moved.
position startpos moves e2e4d +---+---+---+---+---+---+---+---+
| r | n | b | q | k | b | n | r | 8
+---+---+---+---+---+---+---+---+
| p | p | p | p | p | p | p | p | 7
+---+---+---+---+---+---+---+---+
| | | | | | | | | 6
+---+---+---+---+---+---+---+---+
| | | | | | | | | 5
+---+---+---+---+---+---+---+---+
| | | | | P | | | | 4
+---+---+---+---+---+---+---+---+
| | | | | | | | | 3
+---+---+---+---+---+---+---+---+
| P | P | P | P | | P | P | P | 2
+---+---+---+---+---+---+---+---+
| R | N | B | Q | K | B | N | R | 1
+---+---+---+---+---+---+---+---+
a b c d e f g h
Fen: rnbqkbnr/pppppppp/8/8/4P3/8/PPPP1PPP/RNBQKBNR b KQkq - 0 1
Key: B46022469E3DD31BNow let's make Stockfish think for one second what is the next best move for black:gives as the last line:TODO:
go movetime 1000bestmove c7c5 ponder g1f3- what is ponder? Something to do with thinking on the opponent's turn: permanent brain.
- understand the previous lines
To make the move it as suggested for black, we have to either repeat the entire sequence of movements:or alternatively we could also use the previous FEN notation as a starting point;Note how the Universal Chess Interface interface is very simple: we just load a state and then decide what to do next for that one state. The engine holds only one and exactly one state at a time, and you can't even modify it differentially without loading new one from scratch.
position startpos moves e2e4 c7c5d: +---+---+---+---+---+---+---+---+
| r | n | b | q | k | b | n | r | 8
+---+---+---+---+---+---+---+---+
| p | p | | p | p | p | p | p | 7
+---+---+---+---+---+---+---+---+
| | | | | | | | | 6
+---+---+---+---+---+---+---+---+
| | | p | | | | | | 5
+---+---+---+---+---+---+---+---+
| | | | | P | | | | 4
+---+---+---+---+---+---+---+---+
| | | | | | | | | 3
+---+---+---+---+---+---+---+---+
| P | P | P | P | | P | P | P | 2
+---+---+---+---+---+---+---+---+
| R | N | B | Q | K | B | N | R | 1
+---+---+---+---+---+---+---+---+
a b c d e f g h
Fen: rnbqkbnr/pp1ppppp/8/2p5/4P3/8/PPPP1PPP/RNBQKBNR w KQkq - 0 2
Key: 4CA78BCE9C2980B0position fen rnbqkbnr/pppppppp/8/8/4P3/8/PPPP1PPP/RNBQKBNR b KQkq - 0 1 moves c7c5
Computer science masters course of the University of Oxford Updated 2025-07-11 +Created 1970-01-01
Concurrent programming course of the University of Oxford Updated 2025-07-11 +Created 1970-01-01
Opens a virtual MIDI piano GUI. It just works on Ubuntu 20.04: askubuntu.com/questions/34391/virtual-midi-piano-keyboard-setup/1298026#1298026
VMPK is a virtual device that replicates what you would get by connecting a physical MIDI keyboard to your computer. It is not a software synthesizer on its own. But it does connect to a working synthesizer by default (Sonivox EAS) which makes it produce sounds out-of-the box.
TODO: then I messed with my sound settings, and then it stopped working by default on the default "MIDI Connection" > "MIDI Out Driver" > "Network". But it still works on "SonivoxEAS".
A hello world of actually connecting it to a specific software synthesizer manually on Advanced Linux Sound Architecture with
aconnect can be found at: askubuntu.com/questions/34391/virtual-midi-piano-keyboard-setup/1298026#1298026Save to a MIDI file: askubuntu.com/questions/709673/save-as-midi-when-playing-from-vmpk-qsynth/1298231#1298231
Reasonable default key mappings to keyboard covering 2 octaves.
3 multiple simultaneous keys did not work (tested "ZQI"). This might just be a limitation of my keyboard however.
TODO how to save to a
.mid file? askubuntu.com/questions/709673/save-as-midi-when-playing-from-vmpk-qsynth If a product of a big company has a catchy name it came from an acquisition Updated 2025-07-11 +Created 1970-01-01
Applications of power, we have to remember it is there to notice how awesome it is!
- lightning
- motors
- sending nad receiving communication signals
- computers, which in turn can do computations and improved communication
Not to be confused with algebra over a field, which is a particular algebraic structure studied within algebra.
Connect two computers directly without a router with an Ethernet cable Updated 2025-07-11 +Created 1970-01-01
For IP-level communication, askubuntu.com/questions/22835/how-to-network-two-ubuntu-computers-using-ethernet-without-a-router/116680#116680 just worked between P51 and P14s both on Ubuntu 23.10 connected with a regular Cat 5e cable.
On both machines, first we found the Ethernet cable interface name with the which outputs on the P41s:so the interface was
ip CLI tool:ip a1: lo: <LOOPBACK,UP,LOWER_UP> mtu 65536 qdisc noqueue state UNKNOWN group default qlen 1000
link/loopback 00:00:00:00:00:00 brd 00:00:00:00:00:00
inet 127.0.0.1/8 scope host lo
valid_lft forever preferred_lft forever
inet6 ::1/128 scope host
valid_lft forever preferred_lft forever
2: enp1s0f0: <BROADCAST,MULTICAST,UP,LOWER_UP> mtu 1500 qdisc pfifo_fast state UP group default qlen 1000
link/ether fc:5c:ee:24:fb:b4 brd ff:ff:ff:ff:ff:ff
3: wlp2s0: <BROADCAST,MULTICAST,UP,LOWER_UP> mtu 1500 qdisc noqueue state UP group default qlen 1000
link/ether 04:7b:cb:cc:1b:10 brd ff:ff:ff:ff:ff:ff
inet 192.168.1.123/24 brd 192.168.1.255 scope global dynamic noprefixroute wlp2s0
valid_lft 61284sec preferred_lft 61284sec
inet6 fe80::3597:15d8:74ff:e112/64 scope link noprefixroute
valid_lft forever preferred_lft foreverenp1s0f0, because wlp is wireless and lo is localhost.So on the P14s we assign an IP of 10.0.0.10 to the P51:
sudo ip address add 10.0.0.10/24 dev enp1s0f0Then on the P51 analogously, giving IP of 10.0.0.20 to the P14s:
sudo ip address add 10.0.0.20/24 dev enp0s31f6And after that, P14s can:and P51 can:
ping 10.0.0.10ping 10.0.0.20Ciro Santilli intends to move his beauty list here little by little: github.com/cirosantilli/mathematics/blob/master/beauty.md
The most beautiful things in mathematics are results that are:
- simple to state but hard to prove:
- Fermat's Last Theorem
- number of unknown rationality, e.g. is rational?
- transcendental number conjectures, e.g. is transcendental?
- basically any conjecture involving prime numbers:
- many combinatorial game questions, e.g.:
- surprising results: we had intuitive reasons to believe something as possible or not, but a theorem shatters that conviction and brings us on our knees, sometimes via pathological counter-examples. General surprise themes include:Lists:
- classification of potentially infinite sets like: compact manifolds, etc.
- problems that are more complicated in low dimensions than high like:
- generalized Poincaré conjectures. It is also fun to see how in many cases complexity peaks out at 4 dimensions.
- classification of regular polytopes
- unpredictable magic constants:
- why is the lowest dimension for an exotic sphere 7?
- why is 4 the largest degree of an equation with explicit solution? Abel-Ruffini theorem
- undecidable problems, especially simple to state ones:
- mortal matrix problem
- sharp frontiers between solvable and unsolvable are also cool:
- attempts at determining specific values of the Busy beaver function for Turing machines with a given number of states and symbols
- related to Diophantine equations:
- applications: make life easier and/or modeling some phenomena well, e.g. in physics. See also: explain how to make money with the lesson
Good lists of such problems Lists of mathematical problems.
Whenever Ciro Santilli learns a bit of mathematics, he always wonders to himself:Unfortunately, due to how man books are written, it is not really possible to reach insight without first doing a bit of memorization. The better the book, the more insight is spread out, and less you have to learn before reaching each insight.
Am I achieving insight, or am I just memorizing definitions?
It is exactly what you'd expect from the name, Waring was watching Netflix with Goldbach, when they suddenly came up with this.
There are unlisted articles, also show them or only show them.