2015 Definition and 215 Threads

Here is this week's POTW: ----- Let $k$ be a field, and $x_1,\ldots, x_n$ indeterminates. Show that there is an isomorphism $$k[x_1,\ldots, x_n] \approx k[x_1]\otimes_k \cdots \otimes_k k[x_n].$$ -----Remember to read the...

Here is this week's POTW: ----- In a group of 17 nations, any two nations are either mutual friends, mutual enemies, or neutral to each other. Show that there is a subgroup of 3 or more nations such that any two nations in the subgroup share the same kind of relationship. ----- Remember to...

Here is this week's POTW: ----- Given $x=p+\sqrt{q}+\sqrt{r}+\sqrt{s}$ (where $q,\,r,\,s$ are square-free integers) is one root of the equation $x^4-4x^3-16x^2-8x+4=0$, find the values for $p,\,q,\,r$ and $s$. ----- Remember to read the...

Here is this week's POTW: ----- Let $f : \Bbb R\to \Bbb R$ be the function $f(x) = x^2$. Let $\mu$ be the pushforward of the Lebesgue measure $m$ with respect to $f$. Evaluate the Radon-Nikodym derivative of $\mu$ with respect to $m$. ----- Remember to read the...

Here is this week's POTW: ----- Solve the Diophantine equation $4x+51y=9$. ----- Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!

Today we will learn who wins the 2015 Nobel Prize in physics. The announcement will be available online at 11.45 CET at the earliest (depending on whether or not they have taken a decision and been able to inform the laureates). Link to announcement...

Here is this week's POTW: ----- Prove that if $\dfrac{a+b}{3x-y}=\dfrac{b+c}{3y-z}=\dfrac{c+a}{3z-x}$, then $\dfrac{a+b+c}{x+y+z}=\dfrac{ax+by+cz}{x^2+y^2+z^2}$. ----- Remember to read the...

Indicate the papers you think will prove most significant for future Loop-and-allied QG research. The poll is multiple choice, so it's possible to vote for several. Abstracts follow in the next post. http://arxiv.org/abs/1509.08899 Generalized effective description of loop quantum cosmology...

Hi I downloaded visual studio 15 and as i saw some of its components need to be installed I tried to install them but it seems that its not compatible with my internet download (20kb/s download breaks after few minutes) So is there any way i can download them from microsoft and install them...

Here is this week's POTW: ----- Evaluate $\cot x \cdot \cot(x+y)$ , if $\cos y = 17\cos(2x+y)$. ----- Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to...

Here is this week's POTW: ----- Let $M$ be a smooth manifold, $X, Y$ smooth vector fields on $M$, and $\phi_t$ the flow of $X$. The Lie derivative of $Y$ along $X$, $\mathcal{L}_XY$, is given by $$\mathcal{L}_XY:= \frac{d}{dt}\bigg|_{t=0} \phi_{-t*}Y.$$ Show that $\mathcal{L}_XY$ is equal to...

Here is this week's POTW: ----- Suppose $G$ is a group with no proper subgroups. What can be said about $G?$ Prove your statements. ----- Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to...

Next week (Oct 5-12), the Royal Swedish Academy of Sciences will announce the annual winners of science's most prestigious award, the Nobel Prize. In this thread, let's throw out some guesses as to who might win. Here are Thomson-Reuter's annual predictions. My predictions: Physiology and...

Below is the link to the Lunar super eclipse Sept. 27th. It won't happen again until 2033. http://finance.yahoo.com/news/theres-rare-supermoon-total-lunar-205421688.html?amp;soc_trk=ma&soc_src=mail&soc_trk=ma

Here is this week's POTW: ----- Let $x$ be a variable and $K = F(x)$ the field of rational functions in $x$ over a field $F$. Let $L = F\left(\frac{f(x)}{g(x)}\right)$, where $f(x),\, g(x)\in F[x]$ are relatively prime and $\frac{f}{g}\in K\setminus F$. Show that $K/L$ is a finite extension and...

Here is this week's POTW: ----- Solve the simply supported beam equations: \begin{align*} u_{tt}&=-u_{xxxx}, \qquad 0<x<1, \qquad 0<t<\infty \\ u(0,t)&=0 \\ u_{xx}(0,t)&=0 \\ u(1,t)&=0 \\ u_{xx}(1,t)&=0 \\ u(x,0)&=f(x) \\ u_{t}(x,0)&=g(x) \end{align*} ----- Remember to read the...

Here is this week's POTW: ----- Find $$\int_{}^{} \dfrac{c\cos x+d\sin x}{a\cos x+b\sin x}\,dx$$ given $a,\,b,\,c,\,d$ are constants such that $a^2+b^2=c^2+d^2=1$. ----- Remember to read the...

Here is this week's POTW: ----- Find a sequence of functions $\{f_n\}$ on $[0,1]$ such that $$\lim_{n\to \infty}\int_0^1 f_n(x) \, dx=\int_0^1 \lim_{n\to \infty} \, f_n(x) \, dx,$$ but $\{f_n\}$ does not converge uniformly to any function $f(x)$ on $[0,1]$. Thus, uniform convergence is not a...

Here is this week's POTW: ----- Find the pointwise limit of the sequence of integral functions $$ f_n(x) = \int_0^{2\pi} \frac{\cos n\phi}{1 - 2x\cos \phi + x^2}\, d\phi, \quad 0 < x < 1$$-----Remember to read the...

Here is this week's POTW: ----- The equation $x^5=ax^2+b$ where $b\ne 0$ has a double root. Show that $108a^5=-3125b^3$. ----- Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to...

Here is this week's POTW: ----- Let $G$ be a Lie group, and let $i : G \to G$, $i(g) = g^{-1}$, be the inversion mapping. Compute the pushforward of $i$ at the identity of $G$. ----- Remember to read the...

Here is this week's POTW: ----- A spider at corner $S$ of a cube of side length $1$ inch wishes to capture a fly at the opposite corner $F$. The spider, who must walk on the surface of the solid cube, wishes to find the shortest path. Find a shortest path with the aid of calculus. Find a...

Here is this week's POTW: ----- Two points are picked at random on the unit circle $x^2+y^2=1$. What is the probability that the chord joining the two points has length at least 1? ----- Remember to read the...

Since many schools have just started the school year, I'd like to begin September's POTW with an important, yet simple problem. ----- Suppose $X$ and $Y$ are $n \times n$ complex matrices such that $X$ is $Y$-invariant, i.e., $e^{zY}Xe^{-zY} = X$ for all $z\in \Bbb C$. Prove that $X$ and $Y$...

Here is this week's POTW: ----- Let $f:\mathbb{R}\to\mathbb{R}$ be the function given by $f(x)=x^3-3x$. Calculate $f^{-1}([-2,2]), f^{-1}((2,18)), f^{-1}([2,18)),$ and $f^{-1}([0,2])$. ----- Remember to read the...

Here is this week's POTW: ----- Find $$\left\lfloor{2\sqrt{x_n}}\right\rfloor$$ given $$x_n=10^{2n}-10^n+1$$ for all $$n\in N$$. ----- Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to...

http://news.yahoo.com/puerto-rico-braces-ts-erika-4-dead-dominica-040658141.html Maybe headed to the east coast of Florida. http://www.nhc.noaa.gov/archive/2015/refresh/ERIKA+shtml/234401.shtml?

Here is this week's POTW: ----- Without the help of calculator, evaluate $\sqrt[6]{1061520150601}$. ----- Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to...

Here is this week's POTW: ----- A pendulum consists of a mass $m$ suspended by a spring with negligible mass with unextended length $b$ and spring constant $k$. Find Lagrange's equations of motion. ----- Remember to read the...

Here is this week's POTW: ----- Prove that $$\int_0^\infty t^{-1/2}e^{-t}\cos(t\sqrt{3})\, dt = \frac{\sqrt{6\pi}}{4}.$$ ----- Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to...

Here is this week's POTW: ----- Let $X$ be a domain in $\Bbb R^2$. Suppose $u,v \in C^2(X\to\Bbb R)$ such that $$\oint_c uv\frac{\partial v}{\partial \mathbf{n}}\, ds = -\frac{1}{2}\oint_c v^2\frac{\partial u}{\partial \mathbf{n}}\, ds$$ for every simple closed curve $c$ in $X$. Prove that to...

Here is this week's POTW: ----- I am indebted to Philip Exeter's Math Problems for the following problem. Prove that the mapping $(x,y)\mapsto (u^2-v^2,2uv)$ is conformal everywhere except the origin. ----- Remember to read the...

Here is this week's POTW: ----- Compute $ab+bc+ca$ if $a=\tan 15^{\circ},\,b=\tan 25^{\circ},\,c=\tan 50^{\circ}$. ----- Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to...

Here is this week's POTW: ----- Tile an $8 \times 8$ chessboard, with one square missing, with L-shaped pieces that cover three squares each. The missing square is arbitrary. ----- Remember to read the...

In today's Physics ArXiv: New constraints on primordial gravitational waves from Planck 2015. Authors Luca Pagano, Laura Salvati, and Alessandro Melchiorri of the Physics Department and INFN, Universita di Roma. Primordial gravitational waves from the universe exiting Inflation get more and...

Here is this week's POTW: ----- Let $p$ be a prime integer. Prove $p$ is irreducible in $\Bbb Z[(-1 + i\sqrt{3})/2]$ if and only if $p \equiv 2\pmod{3}$. ----- Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find...

Summer vacation is quickly wrapping up :nb) Time to post your 2015 fall class schedule and share with us what you're taking. Also post when your first day back is :smile:

Here is this week's POTW: ----- Evaluate $$\sum_{k=0}^{n}\dfrac{1}{(n-k)!(n+k)!}$$. ----- Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to...

Mw 6.6 180km SW of Dadali, Solomon Islands 2015-08-10 04:12:14 UTC from my seismograph located in Sydney, Australia ... the event below it was a M 5.2 NE of Raoul Is, Kermadecs, NZ Dave

Here is this week's POTW: ----- Let $z_0$ be complex constant. Consider the quadratic map $T : \Bbb C \to \Bbb C$ given by $T(w) = w^2 + z_0$. Show that the sequence $w_n = T^n(0)$ tends to infinity if $|z_0| > 2$. ----- Remember to read the...

Here is this week's POTW: ----- Find a non-trivial solution to the following nonlinear partial differential equation: $$u_t=u^3 u_{xxx}.$$ Check your solution by plugging it into the DE. ----- Remember to read the...

Here is this week's POTW: ----- Let $a,\,b,\,c$ be real numbers all different from $-1$ and $1$ such that $a+b+c=abc$. Prove that $\dfrac{a}{1-a^2}+\dfrac{b}{1-b^2}+\dfrac{c}{1-c^2}=\dfrac{4abc}{(1-a^2)(1-b^2)(1-c^2)}$. ----- Remember to read the...

Here is this week's POTW: ----- Let $S$ be a surface of revolution parametrized by $x(u,v) = (f(u)\cos v, f(u)\sin v, g(u))$, where $(u,v)$ ranges over some open connected set $\Omega \subset \Bbb R^2$. Assume $f$ and $g$ are smooth, $f > 0$ on its domain, and $(f')^2 + (g')^2 = 1$. Evaluate...

Here is this week's POTW: ----- Write a computer program in Python or R to check, given a particular even integer, whether it can be written as the sum of two primes. (N.B., that this can be done for all even numbers greater than $2$ is the famous unproven Goldbach Conjecture.) The output...

Here is this week's POTW: ----- Three positive real numbers $x,\,y,\,z$ are such that $x^2+5y^2+4z^2-4xy-4yz=0$. Can $x,\,y,\,z$ form the sides of a triangle? Justify your answer.----- Remember to read the...

Here is this week's POTW: ----- ​ Merlin and Arthur are magically transported to a land of pink and green fairies. Merlin explained to Arthur, "Real pink fairies always tell the truth, and real green fairies always lie. However, it is within my power to change a pink fairy to a green fairy, or...

Here is this week's POTW: ----- Find, with proof, the exact value of $$\int_0^{\pi/2} \frac{\cos^3u\sin^7 u}{(p\cos^2u + q\sin^2u)^{6}}\, du$$ where $p$ and $q$ are positive constants.----- Remember to read the...

Here is this week's POTW: ----- Find all real solutions of the equation $a^4+b^4+c^4+1=4abc$. ----- Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to...

Here is this week's POTW: ----- A number of different objects has been distributed into $n$ boxes $B_{1}, B_{2}, \dots ,B_{n}.$ All the objects from these boxes are removed and redistributed into $n+1$ new boxes $B_{1}^{*}, B_{2}^{*}, \dots , B_{n+1}^{*},$ with no new box empty (so the total...

Here is this week's POTW: ----- Consider the following system of ODE on $\Bbb R^2$. \begin{align} \dot{x} &= y\\ \dot{y} &= \lambda y(1 - x^2) - x \end{align} Determine a condition(s) on $\lambda$ such that the fixed point $(0,0)$ is asymptotically stable. ----- Remember to read the...