2017 Definition and 169 Threads

Since I don't see a thread for this already, I'm making one. This thread is to post updates in your application process, notably acceptances or waitlists. I applied to SULI at Argonne; they first opened applications up for staff/group review Jan. 26th, and I got an offer in the HEP division...

Here is this week's POTW: ----- Consider the power series expansion \[\frac{1}{1-2x-x^2} = \sum_{n=0}^\infty a_n x^n.\] Prove that, for each integer $n\geq 0$, there is an integer $m$ such that \[a_n^2 + a_{n+1}^2 = a_m .\] ----- Remember to read the...

Here is this week's POTW: ----- Suppose $\Gamma$ is a finite group of homeomorphisms of a Hausdorff space $M$ such that every non-identity element of $\Gamma$ is fixed point free. Show that $\Gamma$ acts on $M$ properly discontinuously. ----- Remember to read the...

Here is this week's POTW: ----- Suppose that for two real numbers $a$ and $b$ the following equality is true: $(a+\sqrt{a^2+1})(b+\sqrt{b^2+1})=1$. Find the value of $a+b$. ----- Remember to read the...

Here is this week's POTW: ----- Let $P$ be a polynomial such that $P(x)=a_0+a_1x+\cdots+a_nx^n$ where $a_0,\,a_1,\cdots$ are non-negative integer. Given that $P(1)=4$ and $P(5)=152$, find $P(6)$. ----- Remember to read the...

Here is this week's POTW: ----- Let $p(x)$ be a polynomial that is nonnegative for all real $x$. Prove that for some $k$, there are polynomials $f_1(x),\dots,f_k(x$) such that \[p(x) = \sum_{j=1}^k (f_j(x))^2.\] ----- Remember to read the...

Here is this week's POTW: ----- Find polynomials $f(x), \: g(x),$ and $h(x),$ if they exist, such that for all $x,$ \[ |f(x)|-|g(x)|+h(x) = \begin{cases} -1 & \mbox{if $x<-1$} \\ 3x+2 & \mbox{if $-1 \leq x \leq 0$} \\ -2x+2 & \mbox{if $x>0$.}...

Here is this week's POTW: ----- Consider an analytic map $f : \Bbb D \to \Bbb C$ such that $f(z) = \sum\limits_{n = 0}^\infty a_n z^n$ for all $z\in \Bbb D$. Prove that $f$ is injective, provided $$\sum_{n = 2}^\infty n\lvert a_n\rvert < \lvert a_1\rvert.$$ ----- Remember to read the...

Here is this week's POTW: ----- Two people take turns rolling an N-sided die. The object of the game is to roll a higher number than the previous roll. Failure to roll a higher number results in losing the game. Assuming player one rolls first, find the probability that player one wins the...

Happy New Year, MHB! Since the year has just started I figured I'd start with a light problem which I'm sure several of you can solve. ----- Prove that a homeomorphism of the closed unit disk onto itself must map $S^1$ onto $S^1$. -----Remember to read the...

Here is this week's POTW, the first of the (prime) new year! ----- Prove that, for any integers $a, b, c$, there exists a positive integer $n$ such that $\sqrt{n^3+an^2+bn+c}$ is not an integer. ----- Remember to read the...

Here is this week New Year's POTW::) ----- Let $g(x)=a_1\sin x+a_2\sin 2x+\cdots+a_n\sin nx$, where $a_1,\,a_2,\,\cdots,\,a_n$ are real numbers. Suppose that $|g(x)|<|\sin x|$ for all real $x$. Prove $|a_1+2a_2+\cdots+na_n|<1$. ----- Remember to read the...

Hi everyone! I thought I would pose this question to all of you on PF. Which STEM field do you think could be the most employable in the US and Canada in 2017, based on the following: (1) Past demand in the US and Canada circa 2015 and 2016. (2) Current demand in the US and Canada as of today...

Homework Statement find 2017 times A Homework EquationsThe Attempt at a Solution the sum of last members denominator.is 2017 multiplied by 1008 the member before it is 2015 multiplied by 1008 before it 2015 times 1007 2013 times 1007 2013 times 2016 how can i move futher

Hi, I want some suggestion about christmas decorations, Because I`ll guest a party with my family & friends, So, I need some help. Cheers, Rebeca

In 2012 actor Alan Alda started a competition in which scientists are asked to explain by whatever means a designated phenomenon or concept to 11 year olds. The explanations are judged by students whose schools are participating in the competition worldwide some 26,000 students so far. This...

I've been waiting for this for a long time and it's just a little more than a year away now. This will be the opportunity of a lifetime for people in the U.S. The 2017 solar eclipse will be visible across the width of the entire U.S! The points of Greatest Eclipse and Greatest Duration are...

And goes directly through Oregon. Woohoo! http://eclipse.gsfc.nasa.gov/SEgoogle/SEgoogle2001/SE2017Aug21Tgoogle.html

Steve Mohr of Australia's Newcastle University has modeled the Earth's fossil fuel reserves and come up with this massive study (warning: 13mb). PROJECTION OF WORLD FOSSIL FUEL PRODUCTION WITH SUPPLY AND DEMAND INTERACTIONS...