All Questions

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Does $O(\log^2(x))$ imply $O(x)$

Does $O(\log^2(x))$ imply $O(x)$ I have to prove the following: $$\sum\limits_{\substack{n\in\mathbb N\\n\le x}}\Lambda(n)\log(n)=\psi(x)\log(x)+O(x)$$ Applying partial sum I get; ...
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How to get this very simplified demographic forecast?

I'm working on the simulation of a population growth. The variables and hypothesizes are the following: Lifetime: X years (X constant for everybody, yeah !) Initial population: Y people (with always ...
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Movement of birds - Acceleration, Velocity, Time and Displacement. Needed for an assignment

Hi so there are a quandary of birds sitting on a tree.There are 3 teams observing the movement of the birds. Team 1 observes that on their first flight the birds move a short distance across a branch ...
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a question on Frechet derivative

Suppose the derivative of a functional is given by $\int_{\Omega}(\vec{v}.\nabla u)|\nabla u|^{p-2} \phi$ for $\phi\in W_0^{1,p}(\Omega)$, then what is the functional?.
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Is it 3-D Catalan numbers?

I am studying Catalan numbers recently but I think that how about 3-D Catalan? So that I imagine following situation ; A man travel through the path-way parallel to $x, y, z$ axis from O ...
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How to find kth smallest value of a linear equation

Here's a question that was asked in IOITC 2009 India. Even though it should have a solution related to algorithms, yet I post it here as it is pretty "number-theoretic". Indraneel loves posing ...
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IMO 2015 Problem 3 [on hold]

Let $n$ and $k$ be positive integers. Prove that if $n$ is relatively prime with $30$, then there exist integers $a$ and $b$, each relatively prime with $n$, such that $\frac{a^2-b^2+k}{n}$ is an ...
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Finding surface of revolution isometric to helicoid

I'm trying to find a function $f(x)$ such that the two surfaces given below are isometric: $$f_1(x,y) = (ax \cos(y), ax \sin(y), y)$$ $$f_2(x,y) = (f(x)\cos(y), f(x)\sin(y), x)$$ Now I understand ...
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An⇀̸A in L1[−π;π] ( An is partial fourier sum )

Let \begin{equation*} (A_n x)(t) = \frac{a_0}{2} + \sum\limits_{k=1}^n a_k cos(kt) + b_k sin(kt), \\ a_k = \frac{1}{\sqrt{\pi}} \int_{-\pi}^{\pi} x(t) cos(kt) dt, \\ b_k = \frac{1}{\sqrt{\pi}} ...
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How many $10$ letter anagrams of KOLMOGOROV don't contain the subword GROOV?

How many $10$ letter anagrams of KOLMOGOROV don't contain the subword GROOV? Not sure how to do this one. Obviously there are $\frac{10!}{4!}$ anagrams of KOLMOGOROV but I'm not sure how to account ...
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Show that a given sigma field is the smallest one containing the given class of sets

I've been trying to solve the following question from Leo Breiman, Probability but getting stuck in how to proceed and have few doubts as well. Define $\mathcal{B}^{(\infty)}$ as the smallest ...