2015 Definition and 219 Threads

Here is this week's problem! ----- Let $$\cdots \rightarrow V_{i-1} \rightarrow V_i \rightarrow V_{i+1} \rightarrow \cdots$$ be an exact sequence of finite dimensional vector spaces. Show that that alternating sum of their dimensions is zero, i.e., show that $\sum\limits_i...

Let $a,\,b$ and $c$ be integers, and given $X=16(a^2+b^2+c^2)-5(a+b+c)^2$, prove that $X\ge 0$ and find the smallest positive number that $X$ can be. 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...

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. With one exception, the papers are listed chronologically: latest first. The exception was a...

Here is this week's POTW: ----- A student forgot the Product Rule for differentiation and made the mistake of thinking that $(fg)'=f'g'$. However, he was lucky and got the correct answer. The function $f$ that he used was $f(x)=e^{x^2}$ and the domain of his problem was the interval...

Here's this week's problem! Problem: Let $X\subseteq \Bbb N$ with the property that $2j\in X$ for all $j\in X$. Let $Y = \{j\in X : j/2\notin X\}$. Show that the number of partitions of a positive integer $n$ into parts from $Y$ is equinumerous with the number of partitions of $n$ into distinct...

Let $P(x)=\dfrac{99^x+19^x}{x!}$ for $x=1,\,2,\,3,\cdots$. Find $x$ such that $P(x)$ is greatest. 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!

Here's this week's problem!Problem. Prove that if $A$ is a left Noetherian ring, then $xy = 1$ implies $yx = 1$, for every pair $x,y\in A$. ___________________________________ Remember to read the...

Here is this week's POTW: ----- In freshman calculus-based physics, you studied the simple pendulum, and possibly the physical pendulum. Typically, in such problems, you used the small-angle approximation to simplify the resulting differential equation for $\theta$, the angle the pendulum...

In a school we have $a$ girl and $a$ boy students with $a>2013$. We know that the number of ways we can choose a club consisting of 6 girls and 5 boys is a square number. What is the least possible value of $a$? Remember to read the...

I am trying to decide between SUMSRI(geometric group theory) and Fairfield(Markov chains). Thoughts on either of these programs? I am a sophomore who will have completed calculus, linear algebra, intro to proofs, and the first half of real analysis by the end of this year. Thanks in advance!

Just to let everyone know here in the UK we are waiting expectantly for the Solar Eclipse tomorrow morning - max at around 9.30am. 98% coverage in the north of Scotland and the Orkneys, Hebrides and Shetland islands, 85% down here in the south. The totality shadow crosses from the North...

Hey, I am from UC Berkeley. I wonder if anyone here has applied to perimeter institute's 2015 summer program. I've applied but haven't heard back from them yet. I wonder if anybody here has got a reply.

Here is this week's POTW: ----- Evaluate the integral $$\int_0^1 \int_0^1 e^{\max(x^2, \, y^2)} \, dy \, dx.$$ ----- 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 problem! Show that the closed unit interval $[0,1]$ cannot be expressed into a disjoint union of closed intervals of length less than one. ---------------------------- Remember to read the...

Find the number of digits to the right of the decimal point needed to express the fraction $\dfrac{12345678910}{2^{36}\cdot 5^6}$ as a decimal. 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...

I created this animation to celebrate Pi Day yesterday and thought Id share it with you guys :) BENBOBBY.

Here's this week's problem! ________________ Problem. Let $A_1 = \{z \in \Bbb C : |z| < 1\}$ and $A_2 = \{z \in A_1 : \operatorname{Im}(z) > 0\}$. Prove $A_1$ is conformally equivalent to $A_2$. ________________ Remember to read the...

Here is this week's POTW: ----- In freshman physics, or high school physics, you analyzed free-fall motion by assuming that the gravitational force was a constant, $mg$, from which you can use the constant-acceleration kinematic equations. Let's improve on this approximation one notch by...

$P,\,Q,\,R$ is a triangle. $A,\,B,\,C$ lie on the sides $QR,\,RP,\,PQ$ respectively so that $PBC$ and $ABC$ are equilateral. $QB$ and $RC$ meet at $K$. Prove that $BC^2=BK\cdot BQ$. Remember to read the...

Background: Applied to PhD physics programs (7) , Master EE, MQF, MSMF I took a year off after undergrad, unwillingly, because I was rejected from most of the physics PhD programs I applied to. One offered an unfunded master's. I spent a lot of time figuring things out, including re-taking the...

Here's this week's problem! _______________ Problem. Suppose $f : (a,b) \to \Bbb R$ is a function that is strictly increasing at every point $c \in (a,b)$, i.e., for every $c\in (a,b)$, there exists a $\delta > 0$ such that $f(x) < f(c)$ whenever $c - \delta < x < c$, and $f(c) < f(x)$...

Here is this week's POTW: ----- Find the positively oriented (counterclockwise) simple closed curve $C$ in the $xy$ plane for which the value of the line integral $\displaystyle\int_{C}(y^3-y) \, dx-2x^3 \, dy$ is a maximum. ----- Remember to read the...

For integers $n\ge 1$, determine the sum of $n$ terms of the series $\dfrac{2n}{2n-1}+\dfrac{2n(2n-2)}{(2n-1)(2n-3)}+\dfrac{2n(2n-2)(2n-4)}{(2n-1)(2n-3)(2n-5)}+\cdots$ Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines...

I am an aspiring Astrophysicist/Astronomer from Omaha, NE and I am competing in the National Science Olympiad in May. I would like to say I have relatively basic-mid level knowledge of the field in for a junior in high school, however I am unaware of the type of questions that will be asked (I...

Here is this week's POTW (sorry about its being late - it's crazy at home!): ----- One morning Mr. Brown left his tent and walked one mile south. Then he turned east and continued for the second mile. At this point he turned north, walked for another mile, and arrived right back at his tent...

Here's this week's problem! ____________ Problem. Let $X$ be a smooth vector field on an oriented Riemannian manifold $(M,g)$. Show that if $\nu$ is the volume form on $M$, then $d(i_X\nu) = (\text{div} X) \nu$. ____________Remember to read the...

Hi all! I have been accepted by a university and have been given the second round interview question. However, I am pretty stumped as to how to answer it. Here's the question: Please answer the following question. Your answer should be handwritten on no more than three (3) sides of A4 paper...

Solve the equation below: $(2\cos x -1)(2\cos 2x -1)(2\cos 4x -1)(2\cos 8x -1)=1$ 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!

When is Easter Day in 2015? Here is a method to calculate the day of Easter. Choose a year that we name A. R is the remainder of the division of A by 4. S is the remainder of the division of A by 7. T is the remainder of the division of A by 19. B = 19 x T + 24 M is the remainder of the...

What is the 2015th decimal number in the division 2015 by 7? This is how I did it but is there an simpler way of showing this... I did 2015/7 = 287.857142 857142 857142 857142 ... As we have a recurring decimal for the numbers 857142, I divided 2015 by 6 which gives me a non finite answer so...

Here is this week's POTW: ----- Two men meet on the street. They haven't seen each other for many years. They talk about various things, and then after some time one of them says: "Since you're a professor in mathematics, I'd like to give you a problem to solve. You know, today's a very...

Here's this week's problem! _____________ Problem. Let $L : \mathcal{C} \to \mathcal{C}$ be a left-adjoint functor from category $\mathcal{C}$ to category $\mathcal{C'}$. Show that if $F : \mathcal{D} \to \mathcal{C}$ is a functor such that $\operatorname{colim} F$ is an object of...

Has anyone heard back from math REU Programs for 2015 yet?

Find the largest possible real number $M$ such that for all pairs $(a,\,b)$ of real numbers with $a\ne b$, and $ab=2$, $\dfrac{((a+b)^2-6)((a-b)^2+8)}{(a-b)^2}\ge M$. Also, determine for which pairs $(a,\,b)$ equality holds. Remember to read the...

http://www.aapt.org/Conferences/sm2015/ "The 2015 AAPT Summer meeting will take place on the campus of the University of Maryland, College Park. College Park, Maryland is also the home of the American Association of Physics Teachers." Abstract submissions are now open... until Feb 25...

Here's this week's problem! _______________ Problem. Let $X$ be a surface imbedded in $\Bbb R^3$. Show that if $K$ is the Gaussian curvature of $X$, then $$K(x) = \lim_{V\downarrow x} \frac{\text{vol}_{\Bbb S^2}(N(V))}{\text{vol}_X(V)}$$ where $N : X\to \Bbb S^2$ is the Gauss map...

Here is this week's problem: ----- You are given a finite number of points in space with the property that any line that contains two of these points contains three of them. What must be true of all the points? Prove it. ----- Remember to read the...

\int_{0}^{\dfrac{\pi}{2}} \dfrac{\cos^4 x+\sin x\cos^3x+\sin^2 x\cos^2 x+\sin^3x\cos x}{\sin^4 x+\cos^4 x+2\sin x\cos^3 x+2\sin^2 x\cos^2 x+2\sin^3 x\cos x}\,dx Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find...

An interesting dimension of the Planck results that can use a thread on its own. http://www.cosmos.esa.int/documents/387566/522789/Planck_2015_Results_XX_Constraints_Inflation.pdf/ Planck 2015 results. XX. Constraints on inflation Preprint online version: February 7, 2015 ABSTRACT We present...

The Planck 2015 (formerly 2014) Data Release will finally be released to the public tomorrow February 5. Details at- http://www.cosmos.esa.int/web/planck

Here's this week's problem! __________________ Problem. Suppose $S$ is a partially ordered set such that for some positive integer $n$, every finite subset of $S$ is a union of $n$ chains. Prove that $S$ is itself the union of $n$ chains. __________________ Remember to read the...

Here is this week's Problem of the Week; you might notice a similarity with the University POTW # 147, two weeks ago, and there are. But there are also some subtle differences which completely change the solution. ----- Suppose you and your husband attended a party with three other married...

Find $a$ and $b$ such that the equation $x^6+ax^4+bx^2-225=0$ has six real roots in arithmetic progression. 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...

I loathe the Patriots so I'm going with Seattle 31-28 If anyone hits their prediction on the dot, they will get gold membership! Don't forget to watch the Puppy Bowl.

My apologies for not getting to this on time. I will try harder next week! Here is this week's POTW: ----- Imagine a man running from his parked car to a building. He runs a distance $d \, \text{m}$ in the rain. The rain is falling at a terminal velocity of $v \, \text{m/s}$. What is the best...

Here's this week's problem! _________________ Problem. A projectile is shot from the edge of a building of height $h$ with initial speed $v$ at an angle $\alpha$ that gives the greatest range $d$. Show that $$\alpha = \cos^{-1}\left(\sqrt{\frac{2gh + v^2}{2gh + 2v^2}}\right) \quad \text{and}...

If $a_1,\,a_2,\,a_3$ are the altitudes of a triangle and $r$ is the radius of its inscribed circle, show that $\dfrac{1}{a_1}+\dfrac{1}{a_2}+\dfrac{1}{a_3}=\dfrac{1}{r}$. Remember to read the...

Here's this week's problem! ________ Problem. Prove that $\sqrt[3]{2 + \sqrt{5}} + \sqrt[3]{2 - \sqrt{5}}$ is rational. ________Note. The cube roots involved are principal cube roots. Remember to read the...

Here is this week's problem: ----- Eight total people (four couples), including the host and hostess, arrive at a party, and a number of handshakes occur. No one shakes hands with their own spouse or with themselves, and everyone shakes at least one hand. The host asks the others how many...

A set of 8 problems was prepared for an examination. Each student was given 3 of them. No two students received more than one common problem. What is the largest possible number of students? Remember to read the...