Activity for Peter Taylor‭

Type	On...	Excerpt	Status	Date
Answer	—	A: Concrete examples of set theorists thinking independence proofs only determine provability rather than that a statement is neither true nor false? >> …the independence of a set-theoretic assertion from ZFC tells us little about whether it holds or not in the universe. > > How can this be? If it is independent, then it cannot be proved from the axioms. Thus, one has the freedom to assume it or assume the negation, as an axiom. Why would someon... (more)	—	2 months ago
Answer	—	A: Is there a $(n_3)$ configuration which is not self-dual? OEIS has A001403 Number of combinatorial configurations of type (n3). A100001 Number of self-dual combinatorial configurations of type (n3). They first differ at $n=11$. An example of a non-self-dual configuration of type $(113)$ has points $0$ to $10$ and lines $[0, 1, 2]$, $[0, 3, 4]$, $[0... (more)	—	4 months ago
Answer	—	A: How can I improve contrast of red and green, to prove Reverse Triangle Inequality? The obvious answer to how to amplify the the difference in length between $|\vec{b}| - |\vec{r}|$ and $|\vec{b} - \vec{r}|$ is to set $r = -b$ so that one difference is zero and the other is arbitrarily large. (more)	—	7 months ago
Question	—	Is mathematical pedagogy in scope? Do we want to consider questions about how to teach mathematics in scope, or do we want to restrict questions to actually doing mathematics? (more)	—	7 months ago
Answer	—	A: Picture proof for expansion of $x^n−y^n$ $$\begin{array}{c} \times & \mid & x^{n−1} & +x^{n−2}y & +\ldots & +xy^{n−2} & +y^{n−1} \\ \hline x & \mid & x^n & {\color{blue} +x^{n−1}y} & +\ldots & +x^2 y^{n−2} & {\color{green} +x y^{n−1}} \\ -y & \mid & {\color{blue} -x^{n−1}y} & -x^{n−2}y^2 & -\ldots & {\color{green} -xy^{n−1}} & -y^n \end... (more)	—	8 months ago
Answer	—	A: Classification for involutory real infinite series If we look at formal power series and ignore questions of convergence for now, we can take $f(x) = \sum{i \ge 0} ai x^i$. Then the question is which sequences of $ai$ satisfy $$\sum{j \ge 0} aj \left(\sum{i \ge 0} ai x^i\right)^j = x$$ The case $a0 \neq 0$ is awkward, because we immediately get th... (more)	—	8 months ago
Answer	—	A: Should posting on Meta affect reputation? It seems like a good thing to reward useful contributions to Meta, but probably not as much as useful contributions to Q&A. I have mixed feelings about negative scores for downvotes, but bearing in mind the use of downvotes to express disagreement with good-faith proposals I'm inclined to remove them... (more)	—	10 months ago
Answer	—	A: How to intuit P(win the same lottery twice) $= p^{2}$ vs. P(win the same lottery twice | you won the lottery once) $= p$? > It feels contradictory for P(you win the same lottery twice) $\neq$ P(you win the same lottery twice|you won the lottery once). Would you expect P(you win the lottery exactly zero times) = P(you win the lottery exactly zero times | you won the lottery once)? > Intuitively, why aren't these ... (more)	—	about 1 year ago
Question	—	Prove that 49 is the only prime square to be followed by twice a prime square and then a semiprime Let $\tau(n)$ denote the number of divisors of $n$. OEIS sequence A309981 gives the smallest $k$ such that the tuple $(\tau(n), \tau(n+1), \ldots, \tau(n+k))$ uniquely determines $n$. For small $n$ the value can often be verified by case analysis in residues to a suitable modulus, but $n=49$ is mo... (more)	—	about 1 year ago
Answer	—	A: Dividing a cuboid in four Firstly, I assume that your usage of "quadrilateral" implies planarity, and I will largely assume that the hexahedron is convex. Then the hexahedron can be represented as six vectors, being the (unnormalised) normals for the planes under the convention that the plane is given by $v \cdot N = 1$. This... (more)	—	over 1 year ago
Answer	—	A: Posts with math are very, very tall in the feed Apart from my other answer about titles, which is posted separately so that people can vote on it separately, I don't see a problem which needs fixing. (more)	—	over 1 year ago
Answer	—	A: Posts with math are very, very tall in the feed I think that forcing inline display of a given MathJax segment in some contexts but not others is not a great solution because there are a few things which get treated differently in the two contexts, to the extent that I change some markup if I decide to change a block to an inline block. However... (more)	—	over 1 year ago
Answer	—	A: Is posting multiple answers encouraged, and under what circumstances? I'm not aware of any specific guidance, so for now it probably comes down to "Use your common sense". Certainly if the answers are long and independent then it makes sense to post them separately; if they're all one-liners then it probably makes more sense to post them in one answer. (more)	—	over 1 year ago
Answer	—	A: Find all integer solutions for $a_1(a_1+k)(a_1+2k)...(a_1+(2k+1)d)+1 = n^2$ The initial observation is explained by the identity $$(x^2 + 3x + 1)^2 = x(x+1)(x+2)(x+3) + 1$$ The generalisation is messed up in the question as it currently stands, but the only smallish similar identity which I can find is the uninteresting $$(x+1)^2 = x(x+2) + 1$$ (more)	—	over 1 year ago
Answer	—	A: What to do with the algebra-precalculus tag? After a week, I've taken action in line with the feedback received by eliminating the tag, retagging some of the questions which had it, and deleting some which had no redeeming features at all (negative score, no upvotes, no answers except for a self-answer by a deleted user). (more)	—	over 1 year ago
Question	—	What to do with the algebra-precalculus tag? This question is prompted by a recent question with the tag algebra-precalculus, although I see that it's by no means the first. This tag apparently means something to some people. I believe that specifically it means something to people who went through the US education system (whether in the US ... (more)	—	over 1 year ago
Answer	—	A: Do the Faber partition polynomials have integer coefficients? Let $B(x) = b1 x + b2 x^2 + \cdots$. Then $$\begin{eqnarray} Fn(b1, \ldots, bn) &=& - [x^n] n \log(1 + B(x)) \\\\ &=& [x^n] n \sum{i \ge 1} \frac{(-B(x))^i}{i} \\\\ &=& \sum{\lambda \\, \vdash \\, n} \frac{n}{\operatorname{len}(\lambda)} \binom{\operatorname{len}(\lambda)}{f1, \ldots, fn} \prodj ... (more)	—	almost 2 years ago
Question	—	Do the Faber partition polynomials have integer coefficients? The Online Encyclopedia of Integer Sequences includes A263916: Coefficients of the Faber partition polynomials. Perhaps the clearest definition given is > -log(1 + b(1) x + b(2) x^2 + ...) = Sum{n>=1} F(n,b(1),...,b(n)) x^n/n which in better notation is $$\sum{n \ge 1} Fn(b1, \ldots, bn) \f... (more)	—	almost 2 years ago
Answer	—	A: Optimising a 3 value problem (well 2 really) > This is the list of combinations I have found, the result size sequence changes depending on the three values so only the first and last four are always in the same place. > > ``A||B||C, A||B, A||C, A||(B+C), A, B||C, (A+C)||B, (A+B)||C, B, A+B||C, A||C+B, A+B, C, A||B+C, A+C, B+C, A+B+C`` I t... (more)	—	almost 2 years ago
Answer	—	A: Matrices with rotational symmetry Notation: $A^{\leftarrow}$ denotes $A$ with the columns reversed; $A^{\uparrow}$ denotes $A$ with the rows reversed; $A^{\leftarrow \uparrow} = A^{\uparrow \leftarrow}$ is denoted $A^{\circ}$ and is the rotation of $A$ by $180^{\circ}$. Consider first a $(2n+1)\times(2n+1)$ block matrix $\begin{pm... (more)	—	almost 2 years ago
Question	—	Matrices with rotational symmetry I've seen a claim without proof that the characteristic polynomials of matrices with rotational symmetry (i.e. $n \times n$ matrices $A$ with $A{i,j} = A{n+1-i,n+1-j}$) always factor into the product of the characteristic polynomials of smaller matrices which can be derived from blocks of the origina... (more)	—	almost 2 years ago
Answer	—	A: How to write the big Xi notation in MathJax? $\Xi$ may not be what you want, but (a) it's the correct answer to the question as stated; (b) I can't understand why you'd complain that the font used is more legible than a handwritten example deliberately chosen to be difficult to parse. $$\frac{\Xi}{\overline{\Xi}}$$ isn't marvellous, but it's ce... (more)	—	about 2 years ago
Answer	—	A: How can a 15 year old construe the LHS of Generalized Vandermonde's Identity, when it lacks summation limits and a summation index? You can rewrite it in different notation with sum limits if you want. You just need to use a different way to express the constraint on which terms to include in the sum. E.g. with the Iverson bracket notation the LHS becomes $$\sum{k1 = 0}^m \sum{k2 = 0}^m \cdots \sum{kp = 0}^m [k1+\cdots +kp = m... (more)	—	about 2 years ago
Answer	—	A: Why 1. multiply the number of independent options? 2. add the number of exclusive options? You might find it more helpful to instead consider only two choices at a time and draw a grid: | beef | chicken | fish | ------+---------+---------+---------+ | beef | chicken | fish | fries | + | + | + | | fries | fries |... (more)	—	over 2 years ago
Answer	—	A: How's it possible to arrange 0 objects? How can 0! = 1? Combinatorics is not necessarily tangible, so the question of what it means to physically rearrange objects is irrelevant. On the other hand, it's easy to write down the permutations of small finite numbers of objects: 0: [] 1: [1] 2: [1,2] [2,1] 3: [1,2,3] [1,3,2] [2,1,3] [2,3,... (more)	—	over 2 years ago
Answer	—	A: If C = Calvin wins the match, and $X \thicksim Bin(2, p) =$ how many of the first 2 games he wins — then why P(C|X = 1) = P(C)? The key is > the first player to win two games more than his opponent wins the match $P(C \mid X = 1)$ describes the situation when Calvin has won 1 of the first two matches: so the opponent won the other match, and they're currently even. Since a 1-1 score is equivalent to a 0-0 score for the ... (more)	—	over 2 years ago
Answer	—	A: How can we grow this community? Certainly low quality content is an issue. Being brutally honest, if I understood the rôle of moderator as being a ruthless dictator who pursues quality above all else I would delete 95% of the questions. But to put that in context, it's also true for most maths fora. I recently saw someone mention a... (more)	—	over 2 years ago
Answer	—	A: How can Cross Multiplication be intuited or pictured? average(average(a,b),c) vs. average(a,average(b,c)). 1. The two questions were closed as unclear because they're unclear. There's no dissimulation involved. 2. I don't see any evidence in r's comment that they understood the question. They say that you've ignored suggestions elsewhere for how to improve it, and they reference the introduction, but t... (more)	—	over 2 years ago
Answer	—	A: ratio of partial sums of the same geometric sequence I think your approach is the expected one, but a shortcut if rigour is not required would be to note that the absolute value of the ratio must be greater than 1, or the proportion couldn't exceed $\frac{12}8$; but then the largest term dominates, so $$q^4 \approx \frac{819}{51} \approx 16.05$$ and th... (more)	—	over 2 years ago
Answer	—	A: Consider the second of these integrals (What's the meaning of second right here?) > What did they mean by "second"? They've mentally expanded $$\frac{dJ}{d\alpha}=\int{x1}^{x2}\left(\frac{\partial f}{\partial y}\frac{\partial y}{\partial \alpha}+\frac{\partial f}{\partial \dot{y}}\frac{\partial \dot{y}}{\partial \alpha}\right)\mathrm dx$$ (which I've corrected to be what it say... (more)	—	over 2 years ago
Answer	—	A: Find limits of integration in polar coordinates You're going to need to differentiate the curves where they pass through the origin, and also find the angle of the intercept. $$\frac{\textrm{d}}{\textrm{d}x} \textrm{blue curve}\Big\vert{x=0} = 0$$ giving a limit $\theta0=0$. $$\frac{\textrm{d}}{\textrm{d}x} \textrm{red curve}\Big\vert{x=0} =... (more)	—	over 2 years ago
Question	—	Do these major triangle lines have names? Kimberling's Encyclopedia of Triangle Centers lists, among other things, lines on which each centre is found, but usually listing only two points on the line. As a little project I've assembled these triples to find lines which have many centres, which would seem to be a rough measure of how importan... (more)	—	over 2 years ago
Answer	—	A: Finding the smallest Mersenne-number multiple of an odd integer Add one to both sides and consider residues modulo $a$ to get $$2^n \equiv 1 \pmod a$$ So you want to find the multiplicative order of $2$ modulo $a$. As you note, $2^n \ge a + 1$ so $n \ge \lg(a+1)$; by Lagrange's theorem, $n \le \varphi(a)$, where $\varphi$ is Euler's totient function. More gene... (more)	—	over 2 years ago
Answer	—	A: Is replacing the entire question with a different one appropriate? No, this is not appropriate. If someone no longer wishes to keep their unanswered question on the site, they can delete it. If someone wishes to ask a new question, they can do so as a new question. The only reasons I can see for not behaving straightforwardly are (i) some idea that it's more ecol... (more)	—	over 2 years ago
Answer	—	A: Why was my question closed: If Alice must've have classes on at least 2 days, why do you need the intersection of 3 's? 1. It wasn't actually unilateral. 2. It's not so much that the question was unconstructive, as the wholesale replacement of the content which removed all context to the existing comment. The question was flagged as "seems like some weird attempt to game the system", which indeed it does. So I pos... (more)	—	over 2 years ago
Answer	—	A: Prove $(\cos^3\theta+\sin^3\theta)^2= \cos^6\theta(1+\tan^3\theta)^2$ $$(a+b)^2 = a^2 \left(1 + \frac ba\right)^2$$ (more)	—	over 2 years ago
Answer	—	A: Intuitively, why would organisms — that after one minute, will either die, split into two, or stay the same, with equal probability — all die ultimately? This can be recast as a random walk on a line. Let $nt$ be the number of amoebae after $t$ events, and process the events in any order which makes sense. (It may help to think of this as serialising a parallel process on a single-core CPU). For example, you could choose to number the the amoebae by t... (more)	—	over 2 years ago
Answer	—	A: Why would skyrocketing the numbers of doors help laypeople intuit the Monty Hall Problem? The only thing special about the door you chose is that you chose it, and you did so without any information, so objectively it isn't special at all. The door which the host leaves closed is special because it was chosen from the remaining doors by someone with information, so objectively it is actua... (more)	—	over 2 years ago
Answer	—	A: A formal-logic formula for decimal to binary conversion The formal formula for base conversion of a non-negative number is $$x = \left\lfloor \frac{x}{b} \right\rfloor b + (x \bmod b)$$ For binary, $b=2$. (more)	—	almost 3 years ago
Answer	—	A: What story and TWO-digit Natural Numbers best fit Bayes' Theorem chart? I flagged it as a duplicate. I don't recall seeing the subtle difference, and in any case it's your responsibility as the asker of both questions to put them in context with respect to each other: both of them link to an external site but don't mention the other question on this site. I still beli... (more)	—	almost 3 years ago
Answer	—	A: proving relative lengths on a secant This image matches the description in the question (note that, in violation of what I consider to be conventional, $O$ is not the centre of the circle but the midpoint of $AD$). I add a perpendicular to $AD$ from $O$ which intersects $AC$ at $G$, and lines $OF$ and $DG$ which intersect at $H$. As ... (more)	—	almost 3 years ago
Answer	—	A: How can I deduce which operation removes redundacies? > 1. How can I deduce which operation ought fill in the red blank beneath? You can't. It's a hideous phrasing. The issue at question is not "redundancies" (which would carry the implication that they're merely unnecessary) but multiple counting: that is, counting the same assignment more than onc... (more)	—	about 3 years ago
Answer	—	A: Isn't it wrong to write that Indefinite Integral = Definite Integral with a variable in its Upper Limit? > ${\int{f(t) \\; dt} = \int{t0}^t f(s) \\; ds \quad \text{ where $t0$ is some convenient lower limit of integration.}}$ isn't actually in the source text at all. Unpacking some of the surrounding text to more formal notation, it goes from equation (32) $$\exists c: \mu(t) y = \int \mu(t) g(t) \\... (more)	—	about 3 years 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... (more)	—	over 3 years 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... (more)	—	over 3 years 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... (more)	—	over 3 years ago