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Apr
30
comment If $au + bv + cw = 0$ with $a+b + c = 0$ then $u,v,w$ are collinear
Ok, however $a+b=0$ with $a\neq 0$ and $b\neq 0$, and $c=0$ won't help much: in that case, $u,v$ are collinear, and you know nothing about $w$. However, you know for sure $u,v,w$ are coplanar.
Apr
22
comment Complexity of Gaussian Process algorithms is $\mathcal{O}(n^3)$
EHHH, Gaussian elimination is an $O(n^3)$ algorithm, however it does not make use of matrix product. To solve a linear system $Au=v$, you may first inverse the matrix (also with Gaussian elimination, but in a slightly different way as you solve $n$ systems instead of a single one, still in $O(n^3)$) and compute afterwards $u=A^{-1}v$, but then this last matrix-vector product is $O(n^2)$.
Apr
22
comment Write $π = (3, 2, 5)(2, 5, 4)$ in “table” notation?
Chris, you really have to tell which convention you assume for the product of permutations, as both are used (left-to-right and right-to-left). JMoravitz and probablyme answers are both correct.
Apr
15
comment Sum of integers
With the usual definition of convergence of a series (that is, convergence of partial sums), then this seris is not convergent. There is absolutely no doubt about this. However, there are "transformations of series" (such as Cesàro or Borel summation) that can turn some divergent series into convergent ones. See also Divergent series on Wikipedia.
Apr
13
comment If $e^{i\pi}=-1$, then what does $e^{2i\pi}$ equal?
@labbhattacharjee You probably mean en.wikipedia.org/wiki/Exponentiation#Powers_of_complex_numbers since your second link is also about $a^b$ with real (and even positive) $a$. When $a$ is negative or complex, it's much more difficult to define $a^b$ consistently, but you already knew that I guess.
Apr
13
comment If $e^{i\pi}=-1$, then what does $e^{2i\pi}$ equal?
@labbhattacharjee This page is only about $a^b$ for real $a,b$.
Apr
9
comment “Subtracting” sets question
Of course, the answer below is correct, but a remark on notation: your "set difference" is often written $A\backslash B$. And, sometimes $A-B$ designs the set of $x-y$ for any $x\in A$ and $y\in B$.
Apr
8
comment $\int_{a}^b f(x)dx=\int_{a}^b x f(x)dx=0$, how to prove there are more than two zeros in $[a,b]$
@Nick I was a bit sloppy in my comment: you multiply by $(x-\alpha)$ only if $f(x)$ changes sign at $\alpha$. If it's a zero without sign change, you don't multiply, but it's similar to the case when there are no zeros at all. The same applies to the general case where there are severals (distinct) zeros $\alpha_1,\dots,\alpha_n$: only multiply by the factors $(x-\alpha_i$)$ where there is a sign change at $\alpha_i$.
Apr
8
comment $\int_{a}^b f(x)dx=\int_{a}^b x f(x)dx=0$, how to prove there are more than two zeros in $[a,b]$
@coffeemath The function is continuous. If it has one zero $\alpha$, $f(x)$ is positive on one "side" of $\alpha$, negative on the other. What happens if you integrate $(x-\alpha)f(x)$? And there is at least one zero because the integral of $f(x)$ vanishes. Interestingly, you can generalize to $(x-\alpha_1)(\dots)(x-\alpha_n)f(x)$: you get a function that is positive (or negative, but not both) on every subinterval, if $f(x)$ has $n$ zeros.
Apr
8
comment Show that $H_n$ is almost always a non-terminating decimal.
For the record: the theorem and proof in the question is an excerpt from Havil & Dyson, "Gamma: Exploring Euler's Constant", section 2.3.3, pp. 24-25.
Apr
8
comment For $0 \le x \le \frac{π}{4}$,let $f(x) = \sin(x) + \cos(x)$. Find the infimum of $f(x)$ over the interval $x \in [0,\frac{π}{4}]$
No infimum for a bounded continuous function on a compact interval? Really? Just an hint: an infimum is not necessarily a critical point, even if the function is differentiable everywhere, as here. There is another possibility that should be cristal clear if you draw the curve.
Apr
8
comment calculate the internally studentized residual
@whoisit Also have a look at stats.stackexchange.com/questions/115011/…
Apr
8
comment calculate the internally studentized residual
@whoisit Yes, but the estimate of the variance of the residuals is not what you think. It's obtained from the hat matrix. Really, have a look at this Wikipedia article, as it has everything you need.
Apr
8
comment Proving a binomial sum identity $\sum _{k=0}^n \binom nk \frac{(-1)^k}{2k+1} = \frac{(2n)!!}{(2n+1)!!}$
This is also a binomial transform.
Apr
8
comment Proving a binomial sum identity $\sum _{k=0}^n \binom nk \frac{(-1)^k}{2k+1} = \frac{(2n)!!}{(2n+1)!!}$
@yjj Then have a look at en.wikipedia.org/wiki/Wallis'_integrals
Apr
8
comment Proving Chi-squared Distribution
en.wikipedia.org/wiki/…
Apr
8
comment Show $\sqrt{3}+\sqrt[3]{5}$ is algebraic over $\mathbb{Q}$
mathoverflow.net/questions/26832/…
Apr
8
comment EllipticF problem with Maple
@СимонТыран Yes, it's what Robert Israel wrote in the other answer. I didn't check the details, since I don't have Maple nor Mathematica :-)
Apr
7
comment Scientific calculator
If you just want to compute a value of $(x-3)(x-5)(x-7)$ for a given $x$, then do the computation (it's only using subtraction and multiplication). If you want your calculator to expand this product and return $x^3-15x^2+71x-105$ as a formula, you need a calculator that has a Computer Algebra System (CAS), such as a TI-89 or HP-50. See also Karl's answer, there are infinitely many polynomials with roots $3,5,7$, just multiply one of them by any nonzero number.
Apr
6
comment combinatorics - why my reasoning is wrong?
@Daveddd What do you mean by replacement? Initially, your problem states that all $8$ balls are taken at once. Do you mean you want now to take them one at a time and replace each before taking the next?