Relationship between the quotient group and direct products Updated 2025-01-06 +Created 1970-01-01
Although quotients look a bit real number division, there are some important differences with the "group analog of multiplication" of direct product of groups.
If a group is isomorphic to the direct product of groups, we can take a quotient of the product to retrieve one of the groups, which is somewhat analogous to division: math.stackexchange.com/questions/723707/how-is-the-quotient-group-related-to-the-direct-product-group
The "converse" is not always true however: a group does not need to be isomorphic to the product of one of its normal subgroups and the associated quotient group. The wiki page provides an example:
Given G and a normal subgroup N, then G is a group extension of G/N by N. One could ask whether this extension is trivial or split; in other words, one could ask whether G is a direct product or semidirect product of N and G/N. This is a special case of the extension problem. An example where the extension is not split is as follows: Let , and which is isomorphic to Z2. Then G/N is also isomorphic to Z2. But Z2 has only the trivial automorphism, so the only semi-direct product of N and G/N is the direct product. Since Z4 is different from Z2 × Z2, we conclude that G is not a semi-direct product of N and G/N.
TODO find a less minimal but possibly more important example.
This is also semi mentioned at: math.stackexchange.com/questions/1596500/when-is-a-group-isomorphic-to-the-product-of-normal-subgroup-and-quotient-group
I think this might be equivalent to why the group extension problem is hard. If this relation were true, then taking the direct product would be the only way to make larger groups from normal subgroups/quotients. But it's not.
Theoretical framework on which quantum field theories are based, theories based on framework include:so basically the entire Standard Model
The basic idea is that there is a field for each particle particle type.
E.g. in QED, one for the electron and one for the photon: physics.stackexchange.com/questions/166709/are-electron-fields-and-photon-fields-part-of-the-same-field-in-qed.
And then those fields interact with some Lagrangian.
One way to look at QFT is to split it into two parts:Then interwined with those two is the part "OK, how to solve the equations, if they are solvable at all", which is an open problem: Yang-Mills existence and mass gap.
- deriving the Lagrangians of the Standard Model: why do symmetries such as SU(3), SU(2) and U(1) matter in particle physics?s. This is the easier part, since the lagrangians themselves can be understood with not very advanced mathematics, and derived beautifully from symmetry constraints
- the qantization of fields. This is the hard part Ciro Santilli is unable to understand, TODO mathematical formulation of quantum field theory.
There appear to be two main equivalent formulations of quantum field theory:
Quantum Field Theory visualized by ScienceClic English (2020)
Source. Gives one piece of possibly OK intuition: quantum theories kind of model all possible evolutions of the system at the same time, but with different probabilities. QFT is no different in that aspect.- youtu.be/MmG2ah5Df4g?t=209 describes how the spin number of a field is directly related to how much you have to rotate an element to reach the original position
- youtu.be/MmG2ah5Df4g?t=480 explains which particles are modelled by which spin number
Quantum Fields: The Real Building Blocks of the Universe by David Tong (2017)
Source. Boring, does not give anything except the usual blabla everyone knows from Googling:Quantum Field Theory: What is a particle? by Physics Explained (2021)
Source. Gives some high level analogies between high level principles of non-relativistic quantum mechanics and special relativity in to suggest that there is a minimum quanta of a relativistic quantum field.RSA vs Diffie-Hellman key exchange are the dominant public-key cryptography systems as of 2020, so it is natural to ask how they compare:
- security.stackexchange.com/questions/35471/is-there-any-particular-reason-to-use-diffie-hellman-over-rsa-for-key-exchange
- crypto.stackexchange.com/questions/2867/whats-the-fundamental-difference-between-diffie-hellman-and-rsa
- crypto.stackexchange.com/questions/797/is-diffie-hellman-mathematically-the-same-as-rsa
As its name indicates, Diffie-Hellman key exchange is a key exchange algorithm. TODO verify: this means that in order to transmit a message, both parties must first send data to one another to reach a shared secret key. For RSA on the other hand, you can just take the public key of the other party and send encrypted data to them, the receiver does not need to send you any data at any point.
Other Bitcon analysis:
Analyses in other blockchains:
- Ethereum
- reidjs.medium.com/top-6-weird-innovative-and-hilarious-findings-in-the-ethereum-blockchain-83dbbca461ca Top 6 Weird, Innovative, and Hilarious findings in the Ethereum Blockchain by Reid Sherman (2018)
- "Annotated blockchain project"Does the same as this page, just that it is an uncomprehensible mess of broken links. But they have soe good ideas!
- etherpad.mit.edu/p/r.e33d2e7230fafc0612a0f2e7ebc87bae
- etherpad.mit.edu/p/r.19b7b3e2c5ea08a61cb0bef0aeb213fd image list (February 8, 2017) We tried going over it, but it is just too much work, the huge majority of the results are just AtomSea & EMBII so not that interesting.
- archive.ph/Zz7m5
- www.reddit.com/r/Bitcoin/comments/5wax5v/a_group_is_working_on_building_a_fully_annotated/
- archive.4plebs.org/pol/thread/111742853/
Their main techniques seem to be:and:mkdir binout for file in blk*dat; do echo "$file"; binwalk --dd='.*' "$file" -C binout/. --log=binout/"$file""res.txt"; donewhich seem promising.mkdir subfileout for file in blk*dat; do mkdir subfileout/"$file"; done for file in blk*dat; do echo "$file"; hachoir-subfile --category=image,video,audio,container,archive,misc "$file" subfileout/"$file" > subfileout/"$file""subfile.txt"; doneThese are installable on Ubuntu 23.10 with:sudo apt install binwalk hachoirTODO how to they automatically map back to transaction IDs? There is a line "Script to add the TX ID to each file." Our attempts: Section "Get transaction id from position in dat file"
.data is section 1:00000080 01 00 00 00 01 00 00 00 03 00 00 00 00 00 00 00 |................|
00000090 00 00 00 00 00 00 00 00 00 02 00 00 00 00 00 00 |................|
000000a0 0d 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 |................|
000000b0 04 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 |................|- 80 0:
sh_name=01 00 00 00: index 1 in the.shstrtabstring tableHere,1says the name of this section starts at the first character of that section, and ends at the first NUL character, making up the string.data..datais one of the section names which has a predefined meaning according to www.sco.com/developers/gabi/2003-12-17/ch4.strtab.html:These sections hold initialized data that contribute to the program's memory image.
- 80 4:
sh_type=01 00 00 00:SHT_PROGBITS: the section content is not specified by ELF, only by how the program interprets it. Normal since a.datasection. - 80 8:
sh_flags=037x00:SHF_WRITEandSHF_ALLOC: www.sco.com/developers/gabi/2003-12-17/ch4.sheader.html#sh_flags, as required from a.datasection - 90 0:
sh_addr= 8x00: TODO: standard says:but I don't understand it very well yet.If the section will appear in the memory image of a process, this member gives the address at which the section's first byte should reside. Otherwise, the member contains 0.
- 90 8:
sh_offset=00 02 00 00 00 00 00 00=0x200: number of bytes from the start of the program to the first byte in this section - a0 0:
sh_size=0d 00 00 00 00 00 00 00If we take 0xD bytes starting atsh_offset200, we see:00000200 48 65 6c 6c 6f 20 77 6f 72 6c 64 21 0a 00 |Hello world!.. |AHA! So our"Hello world!"string is in the data section like we told it to be on the NASM.Once we graduate fromhd, we will look this up like:readelf -x .data hello_world.owhich outputs:Hex dump of section '.data': 0x00000000 48656c6c 6f20776f 726c6421 0a Hello world!.NASM sets decent properties for that section because it treats.datamagically: www.nasm.us/doc/nasmdoc7.html#section-7.9.2Also note that this was a bad section choice: a good C compiler would put the string in.rodatainstead, because it is read-only and it would allow for further OS optimizations.- a0 8:
sh_linkandsh_info= 8x 0: do not apply to this section type. www.sco.com/developers/gabi/2003-12-17/ch4.sheader.html#special_sections - b0 0:
sh_addralign=04= TODO: why is this alignment necessary? Is it only forsh_addr, or also for symbols insidesh_addr? - b0 8:
sh_entsize=00= the section does not contain a table. If != 0, it means that the section contains a table of fixed size entries. In this file, we see from thereadelfoutput that this is the case for the.symtaband.rela.textsections.
- a0 8:
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