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 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 $E[\sum_i(X_i-\frac{\sum_j X_j}n)^2]=(n-1)\sigma^2$. Dec 27 comment Why is sample variance divided by $n-1$ and not $n$ In addition, it reminds you that it may be not a great idea to estimate the variance from a single observation, i.e., when $n=1$ (yeah, I have seen some students who happily tried doing exactly that). Dec 24 comment What operation would satisfy this identity? What's the point of discussing an isolated (and, most likely distorted) formula? Just read around it and in 20 lines it will become clear what should be there. Once it is clear what should be there, just put it there and erase what was there from both the notes and the memory. Dec 21 comment Using probability to detect exam cheating (identical wrong answers) What's interesting is that most multiple choice quizzes are just like that, except when the teacher takes special effort to hide the right answer or does a nasty thing of the kind $\sqrt 7$ is closest to a) 2.6456 b) 2.6457 c) 2.6458 d) 2.6459 (to which any normal person should just respond "and who would care remembering it with this precision or seeing one of them as any different from the other three?") :-) Dec 21 answered Using probability to detect exam cheating (identical wrong answers) Dec 21 comment “Advice to young mathematicians” If the question is to be understood literally, the answer is "No, I wouldn't mind." If you prefer me to answer the implied question rather than the formal one, can you, please, be more specific? @nayrb I spent 3 hours today in the classroom (preparing people for qualifiers) showing how a problem solving process is akin to the bargaining on the market (we did 4 reasonably hard problems in measure theory), but do you really expect me to post a transcript of my crazy spiel here? The most I can do (if I have a long evening with nothing else to do) is to choose one problem and to go over it... Dec 19 awarded Constituent Dec 17 comment A conjectural inequality of the form $\int_a^b g(B'_1(t)) dt \le \int_a^b g(B'_2(t)) dt$ with convex increasing $g$ It is a finite process when $B_1$ is piecewise linear, so it is completely rigorous in that case. The general case follows by approximation as soon as you show that the piecewise linear functions inscribed into the graph of $B_1$ converge to $B_1$ in a strong enough sense to justify passing to the limit in the integral. In general, "a proof" is not as much a formal argument as a clear mental picture translated into a comprehensible language, and the ability to translate just comes with training and experience. :-) Dec 17 comment A conjectural inequality of the form $\int_a^b g(B'_1(t)) dt \le \int_a^b g(B'_2(t)) dt$ with convex increasing $g$ Or, even simpler, make the area over the top graph of stone, the area over the bottom one of paper, take scissors, and start cutting pieces off the area over the bottom graph carefully avoiding the stone. Translate what you see into induction/limits/whatever other math. dialect you like. Dec 16 awarded Caucus Nov 28 comment Poincare type inequality on compact manifold What exactly do you mean? You can always add constants without changing the derivative, be it the Euclidean space or the compact manifold, so the offset by the value at some point (or the mean over the ball, or whatever else killing the constant terms) is inevitable on the left hand side. Nov 28 comment Does this type of function exist? It is not even true in general if we have just radial limits and that's exactly when Baire comes to the rescue. If it is bounded around zero, it is just a boring exercise about removable singularities and you seemed to have absolutely no problem with it from the start :-). Nov 28 comment Does this type of function exist? Baire Category Theorem is the full name (it is just a very smart way to use the nested interval lemma). The answer to your second question is that my ingenuity is insufficient for that, but you can never guarantee that the elementary approach does not exist. All I can guarantee is that it cannot be too simple because of the reason I mentioned in the end of my post. Nov 28 answered conformal mapping, regions of the complex plane marked +/-, find the function f, Nov 28 comment Integral/infinite sum related to Bessels which pop up in optical coherence theory An explicit formula is certainly out of question here but we can discuss the ways to compute it quickly with decent accuracy though in that case more information is needed (range of arguments, precision, etc.) Nov 28 answered Does this type of function exist?