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Activity for Peter Taylor‭

Type On... Excerpt Status Date
Comment Post #281724 Would I be correct in guessing that the lack of any comment on my answer is because you haven't seen the final version?
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12 days ago
Comment Post #281987 This appears to be exactly the same as your earlier question https://math.codidact.com/posts/280741 , and certainly suffers the same flaw that I raised then in the comments which makes it unanswerable.
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19 days ago
Edit Post #281764 Post edited:
Complete argument
about 1 month ago
Edit Post #281764 Post edited:
Actually I was a bit too blithe in how easy it is to show congruence
about 1 month ago
Edit Post #281764 Post edited:
about 1 month ago
Edit Post #281764 Initial revision about 1 month ago
Answer A: proving relative lengths on a secant
This image matches the description in the question (note that, in violation of what I consider to be conventional, $O$ is not the centre of the circle but the midpoint of $AD$). I add a perpendicular to $AD$ from $O$ which intersects $AC$ at $G$, and lines $OF$ and $DG$ which intersect at $H$. As ...
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about 1 month ago
Edit Post #281319 Initial revision 2 months ago
Answer A: How can I deduce which operation removes redundacies?
> 1. How can I deduce which operation ought fill in the red blank beneath? You can't. It's a hideous phrasing. The issue at question is not "redundancies" (which would carry the implication that they're merely unnecessary) but multiple counting: that is, counting the same assignment more than onc...
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2 months ago
Comment Post #280851 Do you have a definition of $\lim_{a \to \infty} f(a) = \infty$ in first order logic (i.e. as a simple statement with $\exists$ and $\forall$)?
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4 months ago
Comment Post #280741 How are you quantifying "contrast the base rate fallacy"?
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4 months ago
Comment Post #280639 @TechnologicallyIlliterate, yes. The wording around arbitrary constants and families of solutions indicates that you need to be more careful than just eliminating the common $y$. The $c$ is (32) is not necessarily equal to the $c$ in (33).
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5 months ago
Edit Post #280639 Post edited:
5 months ago
Edit Post #280639 Initial revision 5 months ago
Answer A: Isn't it wrong to write that Indefinite Integral = Definite Integral with a variable in its Upper Limit?
> ${\int{f(t) \\; dt} = \int{t0}^t f(s) \\; ds \quad \text{ where $t0$ is some convenient lower limit of integration.}}$ isn't actually in the source text at all. Unpacking some of the surrounding text to more formal notation, it goes from equation (32) $$\exists c: \mu(t) y = \int \mu(t) g(t) \\...
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5 months ago
Comment Post #278332 For $n=3$ you want three rectangles of 1/3 by 1. Beyond there it gets more complicated; I suspect that the initial cuts will tend to leave a rough circle, but if so then IIRC some calculations I made a few months ago showed that the diameter of a sector sliced from the circle would eventually be grea...
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5 months ago
Comment Post #280068 @Derek Elkins, I would say that the key difference is that division by 2 isn't really division in binary floating point representations: it's subtraction applied to the exponent. (I'm sure you know this already, but I didn't think it came through clearly in the explanation).
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6 months ago
Comment Post #280118 To be clear: am I correct to understand that by "*the last two points*" you mean everything from "*and we define*" until the end?
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6 months ago
Comment Post #279400 What is the division ring in your "intuitive" instantiation?
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7 months ago
Comment Post #278431 Or the property $P(x) = x \not\in x$ cannot exist in such an axiomatic system, or such an axiomatic system can contain a set of all sets but at the cost of consistency, or possibly such an axiomatic system can contain a set of all sets as long as it doesn't have the law of the excluded middle.
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9 months ago
Comment Post #278270 Note to site admins: I haven't wrapped the multiline stuff in `$$` because it was rendering identically in the preview. IMO it would be a perfectly reasonable approach to look for/write a Markdown plugin to treat the MathJax delimiters `$` and `$$` as start and end delimiters of a section where escap...
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9 months ago
Comment Post #278270 @Derek, now that you mention it, I'd noticed that I had to escape the backslashes for backslash-curlybrace to get the multiset notation to work. I should have put 2 and 2 together myself. Thanks for the diagnosis.
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9 months ago
Comment Post #278269 @tommi, see https://math.codidact.com/q/278270
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9 months ago
Edit Post #278270 Initial revision 9 months ago
Question MathJax config: newlines in eqnarray contexts
In my answer to https://math.codidact.com/questions/278268 I have a couple of `eqnarray` contexts which are being rendered by MathJax but aren't being broken into lines as they should. This works fine in other sites with MathJax which I used to use, so I suspect that it's a problem with the configura...
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9 months ago
Edit Post #278269 Initial revision 9 months ago
Answer A: Asymptotics of counting integers by prime signature
Let $\alpha$ be the smallest exponent such that we know how to calculate $\pi(n)$ in time $\tilde O(n^\alpha)$. Courtesy of Deléglise and Rivat, who removed the $+ \epsilon$ from Lagarias, Miller and Odlyzko's bound, we know that $\alpha \le \tfrac23$, but I'm going to work in terms of $\alpha$ becau...
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9 months ago
Edit Post #278268 Initial revision 9 months ago
Question Asymptotics of counting integers by prime signature
The prime counting function $\pi(n)$ which counts the number of primes up to $n$ is well-known, and it's also fairly well-known that using a well-optimised implementation of the Meissel-Lehmer algorithm it can be calculated in $\tilde O(n^{2/3})$ time. What about numbers of other forms? To be spec...
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9 months ago
Comment Post #278141 The case $k=1$ is also easy: we take $X = \max_{i=1}^n(X_i)$ and observe that for $x \in [1, s]$, $P(X \le x) = \left(\frac{x}{s}\right)^n$ because each independent die must roll no more than $x$. From that we can get $P(X = x)$ in closed form and $E(X)$ in terms of Faulhaber's formulas.
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9 months ago