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What is $\int^{\frac{\pi}{2}}_0 (x -[\sin x])dx$ equal to ( where [.] denotes the greatest integer function) 27 Aug 2013 | 08:17 pm
Problem : $\int^{\frac{\pi}{2}}_0(x-[\sin x])~dx$ is equal to (where [.] denotes the greatest integer function) I have solved it the following way by separating two functions: i.e. $x$ and $[\sin x]...
Power automorphism and abelian groups 27 Aug 2013 | 07:13 pm
First of all I'm not sure if "Power automorphism" is the correct term, so I apologize if it is not. "Let $G$ be an abelian group of order $n$, and $m$ an integer. $f:G\rightarrow G$ s.t. $f(a)=a^m$. ...
Find the farthest points in d-dimensional space 27 Aug 2013 | 07:05 pm
We have $n$ points with $d$ coordinates each and we want to find two of them for which distance between them is the biggest, in Manhattan metric. The obvious algorithm has complexity $O(n^2 \cdot d)$...
gaussian function 27 Aug 2013 | 05:36 pm
I want to scale the gaussian function $\exp(-x^2)$ to the unit disc. In particular, I wish to represent $\int_0^\infty \exp(-x^2) dx$ as $\int_0^1 g(x) dx$, where $g$ should be the rescaed gaussian fu...
Divisibility of sequence 27 Aug 2013 | 05:18 pm
Let the sequence $x_n$ be defined by $x_1=1,\,x_{n+1}=x_n+x_{[(n+1)/2]},$ where $[x]$ is the integer part of a real number $x$. This is A033485. How to prove or disprove that 4 is not a divisor of any...
About the possible forms of the function $f$ 27 Aug 2013 | 11:34 am
Let $f$ be a real analytic function that have infinitely many real zeros. However, the function $f(x)-c$ has only a finite number of zeros. My question is about the possible forms of the function $f$....
If $\|F(x,y)\|\geq 1$ for all $(x,y)$ such that $\|(x,y)\|=1$, is $F(U)$ closed where $U$ is the open disk? 27 Aug 2013 | 11:30 am
This is an old qual problem that showed up several times in 2002 and 2003. Suppose $F:\mathbb{R}^2\to\mathbb{R}^2$ is a $C^1$ function such that the Jacobian matrix is everywhere nonsingular. Suppose...
Title of my thesis that includes Generative bayesian and optimization techniques 27 Aug 2013 | 11:28 am
I am struggling to get a single word that would imply both Generative Bayesian and Optimization technique. I am looking for my thesis topic that goes currently like "Generative Bayesian and optimizati...
lebesgue measurable subset of $\mathbb{R}$ such that every subset of $S$ is measurable, Then measure of $S$ equal to 27 Aug 2013 | 11:23 am
$S$ be a non empty lebesgue measurable subset of $\mathbb{R}$ such that every subset of $S$ is measurable, Then measure of $S$ equal to $1.$ measure of any bounded subset of $S$. $2.$ measure of any...
Prove that every highly abundant or highly composite number $k$ is a prime distance from the nearest primes $\ne k \pm 1$ on either side 27 Aug 2013 | 11:14 am
Prove that if $k$ is highly abundant or highly composite and $q,p$ are the nearest primes with $q+1<k<p-1$, then $k-q,p-k$ are primes. This immediately implies that all highly abundant and highly comp...
