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Home Page - Metamath
A: Metamath is a tiny language that can express theorems in abstract mathematics, accompanied by proofs that can be verified by a computer program. This site has a collection of web pages generated from those proofs and lets you see mathematics developed in complete detail from first principles, with absolute rigor. Hopefully it will amuse you, amaze you, and possibly enlighten you in its own special way.
logic  proof  math 
2 days ago by euler
[1811.00547] Geometric Mean of Partial Positive Definite Matrices with Missing Entries
In this paper the geometric mean of partial positive definite matrices with missing entries is considered. The weighted geometric mean of two sets of positive matrices is defined, and we show whether such a geometric mean holds certain properties which the weighted geometric mean of two positive definite matrices satisfies. Additionally, counterexamples demonstrate that certain properties do not hold. A Loewner order on partial Hermitian matrices is also defined. The known results for the maximum determinant positive completion are developed with an integral representation, and the results are applied to the weighted geometric mean of two partial positive definite matrices with missing entries. Moreover, a relationship between two positive definite completions is established with respect to their determinants, showing relationship between their entropy for a zero-mean,multivariate Gaussian distribution. Computational results as well as one application are shown.
matrices  to-understand  inference  proof  looking-to-see  to-write-about  consider:algorithms  missing-data  data-cleaning 
3 days ago by Vaguery
[1902.04312] Irrationality and transcendence of continued fractions with algebraic integers
We extend a result of Hančl, Kolouch and Nair on the irrationality and transcendence of continued fractions. We show that for a sequence {αn} of algebraic integers of bounded degree, each attaining the maximum absolute value among their conjugates and satisfying certain growth conditions, the condition
limsupn→∞|αn|1Ddn−1∏n−2i=1(Ddi+1)=∞
implies that the continued fraction α=[0;α1,α2,…] is not an algebraic number of degree less than or equal to D.
number-theory  what-if  continued-fractions  complex-numbers  out-of-the-box  looking-to-see  proof  nudge-targets  consider:looking-to-see 
3 days ago by Vaguery
Scientific totalism
Argues that science asserts that it is the only authority for explaining the world; self-referentially it discredits anything that isn't "scientific"
science  totalism  authoritarianism  history  church  renaissance  pointofview  astrology  spirituality  proof 
3 days ago by csrollyson
Odd at Heart - Futility Closet
A problem proposed by Charles W. Trigg in the Spring 1970 issue of the Pi Mu Epsilon Journal (Volume 5, Number 4):
constraint-satisfaction  puzzles  mathematical-recreations  proof  consider:generalizing  to-write-about  consider:hypergraphs 
16 days ago by Vaguery
“Fake Stake” attacks on chain-based Proof-of-Stake cryptocurrencies
This article is the public disclosure of a series of resource exhaustion vulnerabilities investigated by a team of students consisting of Sanket Kanjalkar (sanket1729, smk7@illinois. via Pocket
IFTTT  Pocket  attack  disclosure  pos  proof  stake  vulnerability 
4 weeks ago by ChristopherA
GitHub - stevenroose/reserves: Proof-of-Reserves tool for Bitcoin
reservesFeatures two-step procedure to ease use with hardware wallet or HSMs: first collecting UTXOs to be bundled, then signing the proof proofs are made at a specific block number and can be verified even if the funds moved after the point of proving relying on existing standards: Final proofs ar via Pocket
IFTTT  Pocket  bitcoin  proof  reserve 
4 weeks ago by ChristopherA

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