The default calculator on Android since Marshmallow apparently uses arbitrary precision arithmetic: https://m-cacm.acm.org/magazines/2017/8/219594-small-data-co...
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You can reach me at jwmerrill@gmail.com
“arch minute” is a term used in the article, but “arc minute” also sounds more standard to me.
This study claims that variations of the earth’s magnetic field direction that are more than 100 times slower than this have measurable effects on the behavior of dogs.
According to the authors’ own interpretation, variation of 2 arch minutes of heading per minute of time is enough to destroy the claimed effect. In other words, they’re talking about a 0.03 degree change in direction in the amount of time it takes a dog to poop.
Perhaps you believe dogs are sensitive to magnetic fields, but can you really believe that they are also sensitive to such tiny relative variations in magnetic fields? Much more sensitive than a handheld compass?
I also agree with the general criticisms about p-hacking, but I think it’s worth having some sense of the actual size of the thing the authors are talking about, and how implausible it all is, apart from any details of the statistics.
[0] See figure 4 in https://frontiersinzoology.biomedcentral.com/track/pdf/10.11...
Author here. The paraphrase makes this sound wrong, but what I actually meant is that I don’t know of any calculator that uses rationals while the numerator and denominator can be represented as integer floating point doubles, and then falls back to a single floating point double if the numerator or denominator overflow.
I am aware of many systems that use arbitrary precision rationals, and I mentioned this in the post:
> Rational numbers where the numerator and denominator are allowed to grow arbitrarily large are used by computer algebra systems (CAS) like Mathematica and Maple (and also some high-end handheld graphing calculators).
Most recent post, which links to other parts of the series: https://www.shapeoperator.com/2019/02/10/proving-theorems-ab...
An older post applying GA to the geometry of sunsets: https://www.shapeoperator.com/2016/12/12/sunset-geometry/
log2(355)+log2(113)≈15.3, so it seems like you could squeeze this into 16 bits, but what about a sign bit? And since 355>2^8, I think you also need to be reserving some bits to say how the remaining bits are partitioned between the numerator and denominator (see [1] for an interesting writeup of some experiments with "floating bar").
A fundamental fact of any fixed width number format is that you can only represent 2^n values with n bits, so designing a number representation is really all about deciding how you will distribute those values. Some properties you might want:
1. Enough dynamic range for your application 2. A smooth distribution of values within that range (clustering some values too close together implies gaps somewhere else). 3. Efficient arithmetic operations 4. Exact results in some important cases
Floating point optimizes for 1-3, somewhat at the expense of 4 although it is exact for (small enough) integers and fractions with a (small enough) power of 2 in the denominator.
Fixed-width rationals are less good at 1-3, but are better for 4. The distribution of rationals on the number line is pretty funky, and there's even redundancy because 2/4 and 1/2 represent the same value.
I don't think optimizing for exact results in more special cases is really the right move in most applications, so I don't think we should be surprised by floating point's dominance. On the other hand, I have worked on a project where we decided exact results in special cases really was important [2].
[1] https://iquilezles.org/www/articles/floatingbar/floatingbar.... [2] https://engineering.desmos.com/articles/intuitive-calculator...
Nitpick: the idea that Einstein struggled with school-level mathematics is a myth[1]. I would be very surprised if he ever had to look up the procedure for long division.
[1] http://content.time.com/time/specials/packages/article/0,288...
There’s a “strict” flag that you can use to turn on most of the more specific strictness flags all at once.
I think the writing in this article is not so good, but I've seen this usage by plenty of respectable writers. Stephen Hawking's "A Brief History of Time" is not comically bad writing, and it's full of even more outlandish usages of "million million million ..."
https://www.google.com/search?q=brief+history+of+time+"milli...
Maybe Rebol inspired some part of Mathematica that came later, like the manipulate/dynamic system?
Looks like f(x)=x+x^3/5 gets you two decimal digits everywhere [1], and you could improve that with more terms in the polynomial.
Edit: you can derive f(x) by doing a series expansion of tanh(x)/√(1-tanh(x)²) about x=0, and Mathematica tells me the first few terms are
f(x) = x + x^3/6 + x^5/120 + x^7/5040 + O(x^9)
The usual efficiency calculation for a Carnot cycle tells you what fraction of heat energy that's removed from the hot reservoir can be converted to mechanical work. This is always less than 100%.
However, when running the cycle in reverse as a heat pump, it is possible to transfer more than one Joule of energy from the cold reservoir to the hot reservoir per Joule of mechanical energy used. That's why the round trip efficiency from mechanical energy back to mechanical energy can be 100%.
I have no idea what the practical efficiency of this system can be. But it's wrong to immediately reject it on theoretical grounds.
So if you were ever in this situation, you would probably expect price to rise until the slope of the demand curve changes.
Convolution with a top-hat kernel is just a moving average. Each point of the output is the average of the input signal over a given radius about that point.
Convolution with other kernels is a weighted moving average. Each point of the output is the average of the input signal over a region with weights depending on the displacement from that point.
The Fourier Transform allows implementing convolution as multiplication in frequency space (because the Fourier transform turns translations into multiplications), which is sometimes formally useful, and sometimes more efficient to evaluate.
I'd be curious to hear what you think about the equation editor in Desmos[1] (full disclosure, my employer).
We use an open source WYSIWYG typesetting library called MathQuill[2]. The keystrokes you use are very similar to AsciiMath, but the result is direct editing of the typeset math. E.g. you type '_' to move into a subscript, '^' to move into a superscript, '/' to create a fraction, 'sqrt' to create a radical, etc.
I believe that I can edit much faster with this interface than with Microsoft Equation Editor, and that it's approximately similar difficulty to learn. But I'm biased.
The nice thing about Markdown is that it's quite legible in its source form, which makes it less distracting to edit. Same deal with AsciiMath and LaTeX: AsciiMath is more legible in its source form which means that it has lower overhead during editing.
One of these is more legible than the other:
(f'(x^2+y^2)^2)/(g'(x^2+y^2)^2)
\frac{f'\left(x^2+y^2\right)}{g'\left(x^2+y^2\right)^2}
It's a mistake to assume that anything that can be analyzed in terms of smaller pieces is less real than the pieces. Neither is real, or both are.
Hypocrisy isn't great, but practically everyone has been guilty of it in one way or another. So it's an easy charge to make that gets everyone worked up.
I don't know if we should tolerate hypocrisy, but I'd be in favor of talking about it less in favor of more direct and productive arguments.
http://djafari.free.fr/MaxEnt2014/papers/Tutorial4_paper.pdf
The main point is that KL-divergence is not a metric, so imagining it as a distance in a space may give you some wrong intuitions. Its matrix of 2nd derivatives, the Fischer Information, works as a local metric, but then many people want to draw global pictures that try to extend this back to a global metric, which doesn't actually work.
Is there a reason that those are worse options than Chrome apps?
One of the main things you "do" with expressions like these is to transform them into equivalent expressions that have some desirable form, or examine special cases that allow simplifications, or examine the result of applying some additional operation.
Compact symbols and single letter variables reduce the amount of writing you need to do for each step of such transformations, but they also make the "structure" of expressions more visually obvious.
Programmers also spend time transforming programs into equivalent programs (aka refactoring), but refactoring isn't quite as important to programmers as algebra is to mathematicians and physicists, so (popular) programming languages don't optimize for it to quite the same degree.
The other writings on that site are worth studying if you want to learn more about how v8 optimizes code [2].
[1] http://mrale.ph/irhydra/2/
[2] http://mrale.ph