This article explains Reed-Solomon erasure codes and the problems they solve in gory detail, with the aim of providing enough background to understand how the PAR1 and PAR2 file formats work, the details of which will be covered in future articles.
I’m assuming that the reader is familiar with programming, but has not had much exposure to coding theory or linear algebra. Thus, I’ll review the basics and treat the results we need as a “black box”, stating them and moving on. However, I’ll give self-contained proofs of those results in a companion article.
So let’s start with the problem we’re trying to solve! Let’s say you have n files of roughly the same size, and you want to guard against m of them being lost or corrupted. To do so, you generate m parity files ahead of time, and if in the future you lose up to m of the data files, you can use an equal number of parity files to recover the lost data files.
www.deepideas.net/godels-incompleteness-theorem-and-its-implications-for-artificial-intelligence/, posted Dec '17 by peter in ai math toread
This text gives an overview of Gödel’s Incompleteness Theorem and its implications for artificial intelligence. Specifically, we deal with the question whether Gödel’s Incompleteness Theorem shows that human intelligence could not be recreated by a traditional computer.
https://www.sciencenews.org/article/odds-are-its-wrong, posted 2015 by peter in math science statistics toread
It’s science’s dirtiest secret: The “scientific method” of testing hypotheses by statistical analysis stands on a flimsy foundation. Statistical tests are supposed to guide scientists in judging whether an experimental result reflects some real effect or is merely a random fluke, but the standard methods mix mutually inconsistent philosophies and offer no meaningful basis for making such decisions. Even when performed correctly, statistical tests are widely misunderstood and frequently misinterpreted. As a result, countless conclusions in the scientific literature are erroneous, and tests of medical dangers or treatments are often contradictory and confusing.
Although there are a few different public-key encryption algorithms, the most popular — and fortunately, the easiest to understand — is the RSA algorithm, named after its three inventors Rivest, Shamir and Adelman. To apply the RSA algorithm, you must find three numbers e, d and n related such that ((m^e)^d) % n = m. Here, e and n comprise the public key and d is the private key. When one party wishes to send a message in confidence to the holder of the private key, he computes and transmits c = (m^e) % n. The recipient then recovers the original message m using m = (c^d) % n.
blogs.scientificamerican.com/roots-of-unity/2014/11/30/the-saddest-thing-i-know-about-the-integers/, posted 2014 by peter in math music
The integers are a unique factorization domain, so we can’t tune pianos. That is the saddest thing I know about the integers.
I talked to a Girl Scout troop about math earlier this month, and one of our topics was the intersection of math and music. I chose the way we perceive ratios of sound wave frequencies as intervals. We interpret frequencies that have the ratio 2:1 as octaves. (Larger frequencies sound higher.) We interpret frequencies that have the ratio 3:2 as perfect fifths. And sadly, I had to break it to the girls that these two facts mean that no piano is in tune. In other words, you can tuna fish, but you can’t tune a piano.
As you can see, the tiles overlap and interact to generate new patterns and colors. And as we’re using magical prime numbers, this pattern will not repeat for a long, long time. § Exactly how long? 29px × 37px × 53px… or 56,869px! § Now this was something of a revelation to me. I actually had to triple-check my calculations, but the math is rock solid. Remember these are tiny graphics — less than 7kb in total — yet they are generating an area of original texture of almost 57,000 pixels wide.
www.michaelnielsen.org/ddi/if-correlation-doesnt-imply-causation-then-what-does/, posted 2013 by peter in math science statistics toread
In this post I will describe one small but important part of the theory of causal inference, a causal calculus developed by Pearl. This causal calculus is a set of three simple but powerful algebraic rules which can be used to make inferences about causal relationships. In particular, I’ll explain how the causal calculus can sometimes (but not always!) be used to infer causation from a set of data, even when a randomized controlled experiment is not possible. Also in the post, I’ll describe some of the limits of the causal calculus, and some of my own speculations and questions.
We study fifteen months of human mobility data for one and a half million individuals and find that human mobility traces are highly unique. In fact, in a dataset where the location of an individual is specified hourly, and with a spatial resolution equal to that given by the carrier's antennas, four spatio-temporal points are enough to uniquely identify 95% of the individuals. We coarsen the data spatially and temporally to find a formula for the uniqueness of human mobility traces given their resolution and the available outside information. This formula shows that the uniqueness of mobility traces decays approximately as the 1/10 power of their resolution. Hence, even coarse datasets provide little anonymity. These findings represent fundamental constraints to an individual's privacy and have important implications for the design of frameworks and institutions dedicated to protect the privacy of individuals.
successfulsoftware.net/2013/03/11/the-1-percent-fallacy/, posted 2013 by peter in business math startup
If you are a small software company, you have got a much better chance of getting a decent sized chunk of a small market, than 1% of a huge market. As a general rule of thumb, I would say pick a market for which you have got a decent chance of getting in the top ten Google results for important search terms (power laws again). You can even do this by going after a small segment of a big market. e.g. a CRM solution aimed at companies that trade on EBay. Or perhaps a CRM solution aimed at companies that trade on EBay in the Spanish-speaking market. You can always broaden your focus if you are successful in a small market. Whatever you do, don’t stand in front of investors and pitch them the 1% fallacy. It makes you look an idiot. I should know, because I’ve done it.