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Jan
9
answered An inequality using convex functions: If $\frac{a^2}{1+a^2}+\frac{b^2}{1+b^2}+\frac{c^2}{1+c^2} = 1$ then $abc \le \frac1{2\sqrt 2}$
Jan
4
answered Irreducibility of $f(x)=x^4+3x^3-9x^2+7x+27$
Nov
10
revised Real guessing puzzle
added 237 characters in body
Nov
10
answered Real guessing puzzle
Nov
5
awarded  Nice Answer
Nov
1
comment Why can't $p^p-(p-1)^{p-1}=n^2$ be a square?
@user10676 First congruence: $m$ is even. Second congruence: all binomial coefficients ${p\choose k}$ except the two extremal cases $k=0,k=p$ are divisible by $p$ and $(iy)^p$ is purely imaginary.
Oct
31
comment Proving matrix $A$ is never similar to $A + I$
Similar matrices have the same eigenvalues.
Oct
31
comment Why can't $p^p-(p-1)^{p-1}=n^2$ be a square?
@user236182 Because having both $x+iy$ and $x-iy$ as a factor in any of them results in divisibility by $p$, which is not there (the real part is not divisible by $p$). It is not "relatively prime implies $p$-th powers" but "the two conjugate primes in the factorization of $p$ have to be completely separated, hence..."
Oct
31
comment Why can't $p^p-(p-1)^{p-1}=n^2$ be a square?
@user236182 $\pm i$ and $\pm 1$ are exact $p$-th powers, so why should I?
Oct
31
answered Why can't $p^p-(p-1)^{p-1}=n^2$ be a square?
Oct
31
comment upper bound of a double sequence
If we replace inequalities by identities, we trivially have by induction that for every fixed $k$, $a_{km}$ is decreasing in $m$ for the corresponding majorant. Thus $a_{km}\le\frac{1+L}{m+1}a_{k-1,m+1}$, whence $a_{n0}\le \frac{(1+L)^{n-1}}{n!}$. Of course, this is very rough too.
Oct
30
comment Interchange of summation and analytic continuation with absolute convergence and an analytic sum?
Just take for partial sums a sequence of functions that are very small in a fixed strip around the real axis but blow up otherwise. You do not need to construct anything explicitly, just apply Runge to the sequence of rectangles $[-n,n]\times[-1,1]$ united with some compact set above the line where you want a bad behaviour.
Oct
23
comment Surjective function ?
@M.LTA Does not the question read "Does it remain true if $f\in C^1$..."? Also the structure of the sentence is "Since .., {for almost... there are...}", not {Since... ,for almost...}, there are...
Oct
14
answered Proving that $e^{\pi}-{\pi}^e\lt 1$ without using a calculator
Oct
13
answered Surjective function ?
Jul
5
awarded  Yearling
May
17
answered why does exist infinite postive $k$, such $\lfloor \frac{n^k}{k}\rfloor$ is odd numbers
Feb
23
comment Can we construct Sturm Liouville problems from an orthogonal basis of functions?
To mention just one strong restriction, if $f_{1,2}$ are solutions of $f''+pf'+q_{1,2}f=0$, then their Wronskian $W=f_1'f_2-f_1f_2'$ satisfies $W'+pW+(q_1-q_2)f_1f_2=0$. If $q_1$ and $q_2$ differ by just a constant, this puts $p$ into a one-parametric family for each pair of eigenfunction candidates, so you have to work hard to get just 3 functions simultaneously as eigenfunctions, forget countably many...
Dec
30
comment Volume of the intersection of two lp balls.
No chance for an explicit formula except for trivial cases, but if you really need something more realistic, just restate the question accordingly and someone will answer.
Dec
27
comment Why is sample variance divided by $n-1$ and not $n$
It is a mathematical fact that the deviations around the sample mean tend to be a bit smaller than the deviations around the population mean and that dividing by n−1 rather than n provides exactly the right correction, the proof of which is shorter in length than the sentence you have just finished reading. If $EX_i=0$ and $X_i$ are iid, then $E[X_iX_j]=\sigma^2$ when $i=j$ and $0$ otherwise, so $E[\sum_i(X_i-\frac{\sum_j X_j}n)^2]=(n-1)\sigma^2$.