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Skewray Research

@skewray.mathstodon.xyz.ap.brid.gy
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#Math interests • #Mathematics Foundations [dilettante] • #Probability • #Bayesian #Statistics • #RealAnalysis and #NumericalComputing • Stable distributions • […] 🌉 bridged from ⁂ mathstodon.xyz/@skewray, follow @ap.brid.gy to interact

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Skewray Research @skewray.mathstodon.xyz.ap.brid.gy · 12/10/2026
The usual definition of conditional probability, P(A∩B) = P(A|B) P(B), is necessary but not sufficient. Someone forgot to point out that P(A|B) is a probability measure on B. Let's fix that and see what new we learn […]
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Skewray Research @skewray.mathstodon.xyz.ap.brid.gy · 09/10/2026
I spent today thinking about conditional probability, which is defined as P(A|B) = P(A∩B)/P(B) iff P(B) > 0, else dunno. This is rather unsatisfying, and using limits as P(B) → 0 is known to not work (Borel-Kolmogorov paradox). However, I can say with absolute certainty that B⊆A ⇒ P(A|B) = 1 […]
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Skewray Research @skewray.mathstodon.xyz.ap.brid.gy · 12/09/2026
Although those interested in measure theory here are a set of measure zero, I have nonetheless written an article on the subject. Along with a bit of inappropriate humor, I have careened off of continuity, equivalence classes, orderings, lattices and semilattices, order topologies, and metrics […]
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Skewray Research @skewray.mathstodon.xyz.ap.brid.gy · 17/08/2026
The power spectrum of imaging detector noise is relatively featureless. But while I work it out, I call the APS a bunch of sleazebags and discuss a bucket of yuck. Good times! www.skewray.com/articles/spatial-di… #math #mathematics #physics #optics […]
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Skewray Research @skewray.mathstodon.xyz.ap.brid.gy · 02/08/2026
SNAPPY neutrino-detection technology demonstration satellite achieved "first particle" today. www.skewray.com/articles/snappy-neu… #neutrinos #heliophysics #space #physics
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SNAPPY Neutrino-Detecting Satellite | Skewray Research
SNAPPY is a satellite to demonstrate neutrino detection in space.
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Skewray Research @skewray.mathstodon.xyz.ap.brid.gy · 01/06/2026
Honestly, I wrote this today just so I could submit something to Carnival of Math. www.skewray.com/articles/sigma-alge… #mathematics #math
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Sigma Algebra vs Topology - Epic Smackdown | Skewray Research
Measure spaces and topological spaces seem similar to newbies at first glance.
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Skewray Research @skewray.mathstodon.xyz.ap.brid.gy · 29/03/2026
Posted a blog entry on my current work, which is looking at how similar the stable distributions are to each other. The upshot is that translating a stable distribution along the axis has the minimal effect for α=1 (e.g.,the Cauchy distribution), and the maximal effect for very small α. #math […]
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Skewray Research @skewray.mathstodon.xyz.ap.brid.gy · 07/03/2026
I've been tinkering with SVG animation, ancient Chinese instruments, and harmonics. What are those weird dots? Why not mash these three subjects all up together into a pile of blog? www.skewray.com/articles/animated-g… #mathematics #music #guqin #animation #svg
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Animated GuQin Harmonics | Skewray Research
The guqin, a seemingly simple Chinese music instrument, has some interesting techniques involving harmonics.
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Skewray Research @skewray.mathstodon.xyz.ap.brid.gy · 31/01/2026
The Axiom of Choice can lead to some very weird wackiness in measure theory. The Banach-Tarski paradox is the famous example of cutting up a pea to make a ball the size of the sun. However, it is possible to break measure theory in only one dimension using the Axiom of Choice, and the example […]
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Skewray Research @skewray.mathstodon.xyz.ap.brid.gy · 02/01/2026
I added some tasty bits to my Notes on Probability article, about probability theory not contaminated by topology. • Convolution and Characteristic Functions • Probability Metric Spaces • Directly Fitting Data to Probability Measures The number of real people who looked at this article before […]
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Skewray Research @skewray.mathstodon.xyz.ap.brid.gy · 15/12/2025
I wrote up some notes on probability theory, mainly for my own amusement. Covers: Introduction Metaphysics Measure Spaces Probability Spaces Random Variables Product Measures Integration Change of Variable Probability Density Functions Probability Metrics #mathematics #probability […]
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Skewray Research @skewray.mathstodon.xyz.ap.brid.gy · 26/11/2025
What is randomness? That's a rather philosophical question. Randomness is some mysterious gremlin of chaos. A demonic entity that defies certainly in the future. What is it, mathematically? If you read this article, I hope to leave you without a satisfying answer. #mathematics #probability […]
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Skewray Research @skewray.mathstodon.xyz.ap.brid.gy · 02/09/2025
I recently dumped by notes on modulus of convergence for hypergeometric functions on my website. I also had some thoughts on numerical accuracy. Not very valuable thoughts, but thoughts nonetheless. For those who are wondering why anyone would care, many math libraries such as GSL, boost, and […]
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Skewray Research @skewray.mathstodon.xyz.ap.brid.gy · 27/08/2025
A few years ago (mid 2023), I wrote up some notes regarding the modulus of convergence of the generalized hypergeometric functions. This month I wrote up and posted a series of articles that are those notes, cleaned up a bit, and this article is the wrap-up of the series. #Math #RealAnalysis […]
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Skewray Research @skewray.mathstodon.xyz.ap.brid.gy · 25/08/2025
We all get the feeling that, day by day, the world is converging towards disaster. But what tells us how fast? The Modulus of Convergence does! #Math #RealAnalysis #NumericalComputing www.skewray.com/articles/modulus-of…
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Modulus of Convergence | Skewray Research
If you want to know how fast a sequence converges, then the Modulus of Convergence will hit the spot.
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Skewray Research @skewray.mathstodon.xyz.ap.brid.gy · 24/08/2025
We all get the feeling that, day by day, the world is converging towards disaster. But what tells us how fast? The Modulus of Convergence does! #Math #RealAnalysis #NumericalComputing www.skewray.com/articles/modulus-of…
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Modulus of Convergence | Skewray Research
If you want to know how fast a sequence converges, then the Modulus of Convergence will hit the spot.
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Skewray Research @skewray.mathstodon.xyz.ap.brid.gy · 22/08/2025
In mathematics, we say that a function is bounded if can restrict its image. Oddly, we never seem to 'bind' a function, though. I can find bounds on the the remainder of generalized hypergeometric functions, and I never used the word 'bind' either. Maybe 'bound' refers to bunnies and deer? […]
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Skewray Research @skewray.mathstodon.xyz.ap.brid.gy · 21/08/2025
The generalized hypergeometric series are ubiquitous in the world of computing special functions, for certain amounts of ubiquity. Turns out the speed of convergence is related to the obscure Conway-Maxwell-Poisson distribution, which no one has ever heard of - pretty much the opposite of […]
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Skewray Research @skewray.mathstodon.xyz.ap.brid.gy · 19/08/2025
Accurately computing generalized hypergeometric functions is hard. How many terms do we need? Guess we need a general expression for the size of the terms in the series. Oh, wait, I've got one right here! #math #RealAnalysis #NumericalComputing […]
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Skewray Research @skewray.mathstodon.xyz.ap.brid.gy · 17/08/2025
I have a recent project to stop (LLM training) crawlers from copyright-thefting my website. I find these bots with either a hidden-link tarpit or by looking for single access events (no css), which I then ban if they come from a cloud server. So far I have learned: • Amazon AWS, Google Cloud […]
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Skewray Research @skewray.mathstodon.xyz.ap.brid.gy · 16/08/2025
"Volkswagen Wants You To Pay Monthly To Unlock More Horsepower" But the seat heater is free, so it's a good deal.
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Skewray Research @skewray.mathstodon.xyz.ap.brid.gy · 16/08/2025
Leonhard Euler came up with a cool limit formula that he needed in order to derive an expression for the gamma function. Turns out I need bounds on the same expression. Thanks, Lenny! #math #RealAnalysis www.skewray.com/articles/a-limit-fo…
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A Limit Formula from Lenny Euler | Skewray Research
Euler discovered a handy ratio involving the gamma function that goes in the limit to 1, but how good is it before the limit is reached?
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Skewray Research @skewray.mathstodon.xyz.ap.brid.gy · 14/08/2025
"Be not afraid of math. Some are born to math, some achieve math, and some have math thrust upon ’em." - dude who wrote English in pentameter
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Skewray Research @skewray.mathstodon.xyz.ap.brid.gy · 14/08/2025
I use generalized hypergeometric functions a lot. That means I can type "generalized hypergeometric function" and not make a typo. Is it possible to compute these exactly? Sometimes! #math #NumericalComputing […]
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Skewray Research @skewray.mathstodon.xyz.ap.brid.gy · 09/08/2025
A bunch of apps and utilities on my Ubuntu didn't work right; the worst was Firefox from PPA (the not-snap version). Turns out that AppArmor, a giant software-defined condom to keep me safe, has to have a few holes gently punched through its membranes. Coincidentally, I created a Codeberg […]
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Skewray Research @skewray.mathstodon.xyz.ap.brid.gy · 28/07/2025
Most optimization algorithms assume nice conditions, such as sufficient smoothness. Sadly, the L-infinity norm of a smooth function is continuous but not everywhere smooth. We can fix this! www.skewray.com/articles/optimizing…
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Optimizing with the L-infinity Norm | Skewray Research
Most optimization algorithms assume nice conditions, such as sufficient smoothness. Sadly, the _L_ ∞ norm of a smooth function is continuous but not everywhere smooth. We can fix this!
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Skewray Research @skewray.mathstodon.xyz.ap.brid.gy · 20/07/2025
Say you want to fit a probability distribution to some data, but the distribution doesn't have a known PDF or CDF (e.g., stable distributions, for example). You can still do a non-parametric fit! #probability #statistics […]
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Skewray Research @skewray.mathstodon.xyz.ap.brid.gy · 10/07/2025
My previous attempt to calculate how fast sums of the Pareto and Student's-t for parameter = 2 converge to a normal distribution failed due to both a bad fitting formula and accuracy issues. Now I am simulating the summation of up to 1e4930 random variables before something goes awry […]
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Skewray Research @skewray.mathstodon.xyz.ap.brid.gy · 10/06/2025
I spent two weeks trying to numerically demonstrate convergence of the generalized central limit theorem for the critical case where the variance just becomes infinite. I failed! I suspect my choice of norm was a poor one. Modeling the sum of 10^30 random variables is challenging. #probability […]
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Skewray Research @skewray.mathstodon.xyz.ap.brid.gy · 30/05/2025
Stable distributions have two shape parameters that both affect the “chubbiness” of a distribution, α (tail parameter) and γ (core width). The γ parameter corresponds to σ for a normal distribution, while α gives the slope of the tails at large distances from the core. If we are comparing two […]
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Skewray Research @skewray.mathstodon.xyz.ap.brid.gy · 12/05/2025
Averaging a bunch of Pareto distributions together approaches a normal distribution, because of the central limit theorem. But how does the speed depend on the slope of the power-law tail? If nothing else, at least there are a lot of wild plots. #probability #statistics […]
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Skewray Research @skewray.mathstodon.xyz.ap.brid.gy · 08/05/2025
I have better things to be doing right now. And I am certainly not the first person to try this. And yes, it compiles and runs in C. #programmer_humor #define 中 main #define 数 int #define 回 return #define 零 (0) #define 印 printf #define 左 ( #define 右 ) #define 开 { #define 关 } #define 分 […]
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Skewray Research @skewray.mathstodon.xyz.ap.brid.gy · 07/05/2025
The cackling villain gesticulates wildly and pronounces, "You must flip this coin and take a step to the right for heads and to the left for tails. Somewhere is a pit that leads to the caldera of the FIERY VOLCANO OF INFINITE DEATH!" With much trepidation, you begin flipping the coin and […]
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Skewray Research @skewray.mathstodon.xyz.ap.brid.gy · 07/05/2025
If we multiply two irrational numbers together, like root-2 times root-2, we can get an integer. So, in the same way, we should be able to toss half a coin twice to get a whole coin toss, right? Right? www.skewray.com/articles/what-happe…
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What happens when someone tosses half a coin? Mayhem! | Skewray Research
Is that one toss of half a coin or a half toss of one coin?
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Skewray Research @skewray.mathstodon.xyz.ap.brid.gy · 21/04/2025
Averaging a bunch of symmetric beta distributions together approaches a normal distribution as 1/N. But how does this depend on the distribution parameters? #probability #statistics www.skewray.com/articles/central-li…
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Central Limit Theorem Speed of Convergence of the Beta Distribution | Skewray Research
How fast does a sum of symmetric beta distributions, as a function of the parameters, approach a normal distribution?
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Skewray Research @skewray.mathstodon.xyz.ap.brid.gy · 19/04/2025
The central limit theorem states that a sum of independent identically-distributed random variables with finite variance will ultimately converge to a normal distribution. But how fast? Is the convergence so slow that theorem is useless for real data? #probability #statistics […]
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Skewray Research @skewray.mathstodon.xyz.ap.brid.gy · 13/03/2025
Anyone know if this shape has a name? It is four outward-facing semi-circles. I've seen it on commercial mission revival buildings as an accent. I came across this shape when computing the Nyquist limit in 2 dimensions. My intuition before I started was that I would get a four-leaf clover, so […]
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Skewray Research @skewray.mathstodon.xyz.ap.brid.gy · 12/03/2025
The SVG web graphics standard has quadratic Bézier curves available. Not sure why - they cannot deal with a change in curvature sign. But I needed a short, simple, blindingly fast, bulletproof curve-drawing routine. ~30 lines of C get the job done. #C #graphics #SVG #plotting #BézierCurve […]
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