My back ground is math with PhD, and have been publishing papers in graph theory and topology.
I find it awkward that u ask such question ignoring all cryptography, game Theory, Transaction Graph Analysis, AMM Automated Market Makers & their calculations to minimize the expected slippage (difference bet true & expected price) ,Merkle storing data structures,....
There's a lot of math & Theoretical CS here to learn & search in...
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1-If u want something to approach new up-to-date protocols & still math, read about the Geometric Brownian Motion (Stochastic random walk) how it's used to represent the TX arrival (how to adapt the AMM EQs to the dynamic input of TXs, with TX arrival rate distribution expected mea, still keep u inside some price margin,...etc)
https://uniswap.org/blog/uniswap-v3/https://github.com/Uniswap/uniswap-v3-core/(The blog will lead u to the white paper where u can find the math stuff about BM and the rest of details)
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2-The same random walk is used by this paper in EIP-1559 Simulations, where the BM is also used in simulating TXs arrival,
https://arxiv.org/abs/2102.10567I forgot I had also this
https://mobile.twitter.com/barnabemonnot/status/1364096212517855236Didn't re-read it after reading the paper, Oh yes there's an update of this she reached
https://notes.ethereum.org/@barnabe/rk5ue1WF_(My not answered, I think math related, Qs
https://mobile.twitter.com/ArafatShymaa/status/1397430382878994432 )
3-Here's a punch of papers addressing the strength of a blockchain, not just1559, against some attacks using Game Theory models
https://youtu.be/ndNyx-Oj9Wkhttp://timroughgarden.org/papers/eip1559.pdfInside the report u'll find a lot of references, one of them
"Double-Spend Counterattacks: Threat of Retaliation in Proof-of-Work Systems", Daniel J.Moroz, Daniel J. Aronoff, Neha Narula, David C. Parkes, Feb2020
&Has just passed by this
https://ethresear.ch/t/bayesian-network-model-of-witness-creation-feedback-request/9115.
4-The main public key cryptography used in most Blockchains is Elliptic Curve cryptography, that's gives a lot of room to math work in analysis, attacks,...etc
http://www.jbonneau.com/doc/B16a_BITCOIN_why_buy_when_you_can_rent.pdf(I'm not sure, can't remember now, this paper addresses the hash from cryptographic or game thoeritic prospective)
5-you can read like forever the continuous changes/enhancements in the Ethereum Merkleization data structures ( they have 4 tries, they removed RLP recursive Length Prefix, they switched to binary Tries, they now suggesting 2-level tries Verkle from Vitalik-Merkle)
https://blog.ethereum.org/2020/11/30/the-1x-files-code-merkleization/https://eips.ethereum.org/EIPS/eip-2926https://eips.ethereum.org/EIPS/eip-25846-you can compare Merkles in each cryptocurrency (just said Eth uses Tries, Cardano uses AVL, Monero uses BST balanced periodically, Algorand don't use Merkle, in Bitcoin a lot of research Utreexo forest is the famous example)
https://www.programmersought.com/article/94887721041/https://medium.com/coinmonks/merkle-trees-concepts-and-use-cases-5da873702318I can find Cardano & Monero white papers online, tell me if u didn't the post became too crowded
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There's a lot more, but reaching Utreexo,
7-u can help me in this dispute/contradiction/lack of mathematical understanding from my side, howcome what is described in this Stanford report as Chaotic (lifespan of UTXOs)
https://user-images.githubusercontent.com/83260375/119986617-5d047300-bfc4-11eb-942c-87c632113a04.jpghttps://user-images.githubusercontent.com/83260375/119986701-786f7e00-bfc4-11eb-87e8-82afea8106bf.jpghttps://user-images.githubusercontent.com/83260375/119986654-68579e80-bfc4-11eb-9725-fc2e1edc452c.jpghttps://user-images.githubusercontent.com/83260375/119986683-70afd980-bfc4-11eb-9876-d904cbcdd0e5.jpg is described by the Utreexo team (and another paper
https://eprint.iacr.org/2021/340; someone already provided u the link to Utreexo paper) as Pareto Distribution following the power law with alpha?
https://mobile.twitter.com/ArafatShymaa/status/13999639234120499218-Or in trying/ analyzing the performance at least of non famous although I believe could be more suitable data structures for the Merkle Tree problem; Buffer-Trees, cache oblivious Van Emde Boas layout(in Utreexo they do store it physically in memory as a sequential array, only with different ordering), priority queues, finger tree,...
(The objective is to minimize the proof size, ie siblings along the path from the requested leaf to the root, this is achieved by minimizing the tree height or/and by making the requested group of leaves adjacent to have a lot of siblings in common. That's why I find resemblance with cache oblivious, aside from that Utreexo project is doing real caching and I think researching now on predicting what to cache
https://mobile.twitter.com/ArafatShymaa/status/1402326526952095747.....
There's a lot more, but that what I can remember at the moment.
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9- Here's layer-2 u were asking about, I didn't get into it too much but maybe interests u the math in the sampling & the rest of stuff
https://web.stanford.edu/class/ee374/https://ethereum.org/en/developers/docs/layer-2-scaling/https://vitalik.ca/general/2021/01/05/rollup.htmlhttps://m.youtube.com/watch?v=BhMoJM1kuO8&feature=youtu.be.
10-Manipulation attacks r really interesting with a lot of research, but not exactly mathematical
https://m.facebook.com/story.php?story_fbid=1386287771725667&id=100010333725264...
& There is really much more areas here more than u could imagine
The idea of POB proof of Burn
https://youtu.be/Q5L8-GJVmZwI used/needed to know about it in analyzing UTXOs, & Transactions in general, since the resulting UTXOs live for infinity; I think it also involves cryptography since 2 of the authors main research area is security & cryptography
https://mobile.twitter.com/sol3ggahttps://mobile.twitter.com/dionyzizhttps://mobile.twitter.com/gtklocker