2016 Definition and 207 Threads

2016 (MMXVI) was a leap year starting on Friday of the Gregorian calendar, the 2016th year of the Common Era (CE) and Anno Domini (AD) designations, the 16th year of the 3rd millennium, the 16th year of the 21st century, and the 7th year of the 2010s decade.
2016 was designated as:

International Year of Pulses by the sixty-eighth session of the United Nations General Assembly.
International Year of Global Understanding (IYGU) by the International Council for Science (ICSU), the International Social Science Council (ISSC), and the International Council for Philosophy and Human Sciences (CIPSH).

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  1. anemone

    MHB What is the solution to evaluating this fraction series?

    Here is this week's POTW: ----- Evaluate $\dfrac{7}{1^2\cdot 6^2}+\dfrac{17}{6^2\cdot 11^2}+\dfrac{27}{11^2\cdot 16^2}+\cdots$----- 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...
  2. Euge

    MHB Is the Quotient Sheaf Isomorphic to the Image Sheaf?

    Here is this week's POTW: ----- Let $\mathscr{F} \overset{\eta}{\to} \mathscr{G}$ be a morphism of sheaves over a topological space $X$. Prove that quotient sheaf $\mathscr{F}/\operatorname{ker}(\eta)$ is isomorphic to the image sheaf $\operatorname{im}(\eta)$.----- Remember to read the...
  3. Orodruin

    News How Is Thouless Pronounced in the Context of the Nobel Prize Announcement?

    The Nobel prize in physics is being announced in 4 minutes. Apparently they will be on time this year according to the webcast at: http://www.nobelprize.org This probably means that they have already spoken to the laureate(s).
  4. Ackbach

    MHB What is the maximum sum of coefficients in a binomial expansion?

    Here is this week's POTW: ----- Let $a_{m,n}$ denote the coefficient of $x^n$ in the expansion of $(1+x+x^2)^m$. Prove that for all [integers] $k\geq 0$, \[0\leq \sum_{i=0}^{\lfloor \frac{2k}{3}\rfloor} (-1)^i a_{k-i,i}\leq 1.\] ----- Remember to read the...
  5. Euge

    MHB How to solve Problem of the Week #226?

    Here is this week's POTW: ----- Let $(X,\mu)$ be a measure space, $f\in \mathcal{L}^1(\mu)$, and $\phi_n\in \mathcal{L}^1(\mu)$ such that $\sup_{n,t}\lvert \phi_n(t)\rvert \le 1$ and $\|\phi_n\|_1 \to 0$ as $n\to \infty$. Show that $\|f\phi_n\|_1 \to 0$ as $n\to \infty$.----- Remember to read...
  6. anemone

    MHB How to Minimize the Sum of Squares with Given Distinct Elements?

    Here is this week's POTW: ----- Let $x_1,\,x_2,\,x_3,\,x_4,\,x_5,\,x_6,\,x_7$ and $x_8$ be distinct elements in the set $\{-7,\,-5,\,-3,\,-2,\,2,\,4,\,6,\,13\}$. What is the minimum possible value of $(x_1+\,x_2+\,x_3+\,x_4)^2+(x_5+\,x_6+\,x_7+\,x_8)^2$? ----- Remember to read the...
  7. collinsmark

    News 1st Presidential Debate (2016) thread

    Tonight (as of the time of this post) is the first televised presidential debate between Hillary Clinton and Donald Trump. Moderator: Lester Holt, Anchor, NBC Nightly News Location: Hofstra University, Hempstead, NY The first debate will be divided into six time segments of approximately 15...
  8. jtbell

    Who Won the 2016 Ig Nobel Prizes for Physics and Chemistry?

    ...have been announced. (BBC.com)
  9. Ackbach

    MHB What is the sum of a positive integer in fraction form?

    Here is this week's POTW: ----- For each positive integer $n$, write the sum $\displaystyle\sum_{m=1}^n \frac1m$ in the form $\dfrac{p_n}{q_n}$, where $p_n$ and $q_n$ are relatively prime positive integers. Determine all $n$ such that 5 does not divide $q_n$. ----- Remember to read the...
  10. Euge

    MHB Problem of the Week #225 - Sep 20, 2016

    Here is this week's POTW: ----- Suppose $X$ is a compact Hausdorff space. Show that there is homeomorphism between $X$ and the collection $Y$ of maximal ideals in $C(X,\Bbb R)$, the space of continuous real-valued functions on $X$ (here, $Y$ is topologized with the Zariski topology). -----...
  11. anemone

    MHB Can You Solve the Trigonometric Equation for \(\tan^2 9^\circ\)?

    Here is this week's POTW: ----- Prove that \tan^2 9^\circ=\sqrt{201+88\sqrt{5}}-\sqrt{200+88\sqrt{5}}. ----- 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...
  12. Ackbach

    MHB Problem of the Week # 233 - Sep 15, 2016

    Here is this week's POTW: ----- Let $f$ be a twice-differentiable real-valued function satisfying \[f(x)+f''(x)=-xg(x)f'(x),\] where $g(x)\geq 0$ for all real $x$. Prove that $|f(x)|$ is bounded. ----- Remember to read the...
  13. anemone

    MHB How Many Ways to Fit Six People in Four Rooms?

    Here is this week's POTW: ----- In how many different ways can six people be accommodated in the four rooms, if each room can accommodate a maximum of four people? ----- Remember to read the...
  14. Euge

    MHB Is the mapping from $\mathcal{L}^p(X,\mu)$ to $\mathcal{L}^1(X,\mu)$ continuous?

    Here is this week's POTW: ----- Let $(X,\mu)$ be a positive measure space. For $0 < p < \infty$, why is the mapping $\mathcal{L}^p(X,\mu) \to \mathcal{L}^1(X,\mu)$ sending $f$ to $\lvert f\rvert^p$, continuous?-----Remember to read the...
  15. liometopum

    SpaceX Investigating the SpaceX Rocket Explosion of September 1, 2016

    As already posted on PF, and you have likely seen in the news, a SpaceX rocket exploded, September 1, 2016. Elon Musk is reaching out for help in finding out how it happened. http://www.csmonitor.com/Science/2016/0911/SpaceX-needs-you-Musk-calls-on-public-government-in-explosion-probe I took...
  16. Ackbach

    MHB What is the solution to Problem A-6 in the 1997 Putnam Mathematical Competition?

    Here is this week's POTW: ----- For a positive integer $n$ and any real number $c$, define $x_k$ recursively by $x_0=0$, $x_1=1$, and for $k\geq 0$, \[x_{k+2}=\frac{cx_{k+1}-(n-k)x_k}{k+1}.\] Fix $n$ and then take $c$ to be the largest value for which $x_{n+1}=0$. Find $x_k$ in terms of $n$...
  17. Euge

    MHB Why must $G$ be the internal direct product of $H$ and $K$?

    Here is this week's POTW: ----- Let $G$ be a group with finite index subgroups $H$ and $K$. Suppose $H$ and $K$ have relatively prime indices in $G$. Why must $G$ be the internal direct product of $H$ and $K$? ----- Remember to read the...
  18. anemone

    MHB Find the Ratio of BG to FG in a Figure with Given Side Lengths

    Here is this week's POTW: ----- In the figure below, $AB=AF=15,\,FD=12,\,BD=18,\,BE=24$ and $CF=17$. Find \frac{BG}{FG}. https://www.physicsforums.com/attachments/5959._xfImport ----- Remember to read the...
  19. Safinaz

    A Status of cp - even neutral heavy Higgs after ICHEP 2016

    Hi all, It is well known that on ICHEP 2016 CMS and ATLAS have announced that the bump has been found on December 2015 around 750 GeV diphoton invariant mass was no more than a statistical fluctuation. So now what is the statues of the searchs for a cp - even neutral heavy Higgs state ? I...
  20. Ackbach

    MHB What is the area of region $S$?

    Here is this week's POTW: ----- Sorry for the shamefully late POTW this week. To compensate, I'll make it a bit easier. Find the area of the region $S=\{(x,y):x\ge 0, y\le 1, x^2+y^2\le 4y\}.$ ----- Remember to read the...
  21. DennisN

    Who Was the Hilarious Actor Behind These Unforgettable Characters?

    Gene Wilder (1933 – 2016) He was one of my favorite comic actors because of his timing, performances and the number of quite unusual and hilarious characters he played. He gave me many, many good laughs in movies, particularly Start The Revolution Without Me, The Adventure of Sherlock Holmes'...
  22. anemone

    MHB How Do You Solve the Integral Challenge from POTW #231?

    Here is this week's POTW: ----- Evaluate \int_{0}^{1}\left(\sqrt[4]{1-x^7}-\sqrt[7]{1-x^4}\right) 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...
  23. Euge

    MHB How Do You Solve the Integral in POTW #222?

    Here is this week's POTW: ----- Evaluate the integral $$\int_{-\infty}^\infty \frac{\ln^2\lvert x\rvert}{x^2+1}\, dx$$----- 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...
  24. lfdahl

    MHB Calculating the Sum of f(x) from 0 to 2016

    Let \[f(x) = \frac{a^{2x}}{a^{2x}+a}, \;\;\; a \in \Bbb{N}.\]Find the sum:\[ \sum_{j=0}^{2016}f \left ( \frac{j}{2016} \right )\]
  25. Ackbach

    MHB Is $N_{10}$ even or odd?

    Here is this week's POTW: ----- Let $N_n$ denote the number of ordered $n$-tuples of positive integers $(a_1,a_2,\ldots,a_n)$ such that $1/a_1 + 1/a_2 +\ldots +1/a_n=1$. Determine whether $N_{10}$ is even or odd. ----- Remember to read the...
  26. Euge

    MHB Can Every Smooth Map from a Manifold to the Complex Numbers be Lifted to the Exponential Map?

    Here is this week's POTW: ----- Suppose $M$ is a smooth path-connected manifold. Consider the differential form $$\nu = \Re\left\{\frac{1}{2\pi i} \frac{dz}{z}\right\}$$ which generates $H^1_{dR}(\Bbb C^\times)$, the first de Rham cohomology of $\Bbb C^\times$. Show that every smooth map $f...
  27. anemone

    MHB Solution for POTW #230: Calculate g^(3)(0) for f(x)

    Here is this week's POTW: ----- Let f(x)=x+\frac{x^2}{2}+\frac{x^3}{3}+\frac{x^4}{4}+\frac{x^5}{5}. Set g(x)=f^{-1}(x), compute g^{(3)}(0). ----- 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...
  28. Ackbach

    MHB Can a Group Have Multiple Homomorphisms With the Same Identity?

    Here is this week's POTW: ----- Let $G$ be a group with identity $e$ and $\phi:G\rightarrow G$ a function such that \[\phi(g_1)\phi(g_2)\phi(g_3)=\phi(h_1)\phi(h_2)\phi(h_3)\] whenever $g_1g_2g_3=e=h_1h_2h_3$. Prove that there exists an element $a\in G$ such that $\psi(x)=a\phi(x)$ is a...
  29. anemone

    MHB Can You Solve the Cube of Cosines for Specific Angles?

    Here is this week's POTW: ----- Evaluate \cos^3 \left(\frac{2\pi}{7}\right)+\cos^3 \left(\frac{4\pi}{7}\right)+\cos^3 \left(\frac{8\pi}{7}\right). ----- Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out...
  30. Euge

    MHB Can a Function Belong to L^1 and have a Maximal Function in L^1?

    Here is this week's POTW: ----- Let $f : \Bbb R^n \to \Bbb R$ be a function such that $f$ and its maximal function $\mathcal{M}f$ belong to $\mathcal{L}^1(\Bbb R^n)$. Show that $f(x) = 0$ for almost every $x\in \Bbb R^n$.----- Remember to read the...
  31. G

    Studying Fall 2016 Schedule: Organic Chem 1, Gen Bio 1, Calc 1 & Effective Speech

    I have a question. Does this schedule look heavy or easy to handle? Organic Chemistry 1 Tuesday and Thursday 11:00-12:15 Tuesday lab: 1:00-4:50 General Biology 1 ( not sure, because I won't take it unless it will be useful for General biochemistry) Monday and Wednesday 3:00- 4:15 and Monday...
  32. Ackbach

    MHB Can You Solve the Origin-Inclusion Convex Polygon Problem?

    Here is this week's POTW: ----- Let $(a_1, b_1), (a_2, b_2), \ldots, (a_n, b_n)$ be the vertices of a convex polygon which contains the origin in its interior. Prove that there exist positive real numbers $x$ and $y$ such that $$ (a_1, b_1)x^{a_1} y^{b_1} + (a_2, b_2)x^{a_2}y^{b_2} + \cdots +...
  33. Euge

    MHB Can Nonsingular Matrices Be Generated by Multiplying and Adding?

    Here is this week's POTW: ----- Let $A$ and $B$ be nonsingular $n\times n$-matrices over a field $\Bbb k$. Show that for all but finitely many $x\in \Bbb k$, $xA + B$ is nonsingular. ----- Remember to read the...
  34. anemone

    MHB Can You Solve This Challenging Inequality Involving Real Numbers?

    Here is this week's POTW: ----- The reals $x$ and $y$ are such that $0 < x< 1$ , and $y> 0$, prove that (x+ y)\left(\frac{1}{x}+\frac{1}{y} -\frac{4}{(x+1)^2}\right) ≥ \frac{4}{(x+1)^2}. ----- Remember to read the...
  35. W

    B Strings 2016 Videos: Unwatchable & Unresponsive

    Is it just me or are the videos at Strings 2016 unwatchable? They start and stop and continually seem to buffer, as if the bandwidth is not sufficient. The organizers did not respond to my email. I don't see a way to download and view offline, and I can't tell anything about the streams, e.g...
  36. Ackbach

    MHB Is there a 2x2 matrix with real entries whose sine is [1, 2016; 0, 1]?

    Here is this week's POTW: ----- For any square matrix $A$, we can define $\sin(A)$ by the usual power series: \[ \sin(A) = \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!} A^{2n+1}. \] Prove or disprove: there exists a $2 \times 2$ matrix $A$ with real entries such that \[ \sin(A) = \left(...
  37. Euge

    MHB Can an entire function with exponential growth have bounded derivatives?

    Here is this week's POTW: ----- Let $F$ be an entire function for which there exists $t > 0$ such that $\lvert F(z)\rvert = O(\exp(\lvert z\rvert^t))$ as $\lvert z\rvert \to \infty$. Show that there is a constant $M > 0$ such that for all $n$ sufficiently large, $$\lvert F^{(n)}(0)\rvert \le...
  38. anemone

    MHB Is Sin 1 Less Than Log Base 3 of Root 7?

    Here is this week's POTW: ----- Prove $\sin 1 < \log_3 \sqrt{7}$. ----- 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!
  39. Ackbach

    MHB What is the maximum sum of products of consecutive integers from 1 to n?

    Here is this week's POTW: ----- Given that $\{x_1, x_2, \ldots, x_n\} = \{1, 2, \ldots, n\}$, find, with proof, the largest possible value, as a function of $n$ (with $n \geq 2$), of \[ x_1x_2 + x_2x_3 + \cdots + x_{n-1}x_n + x_nx_1. \] ----- Remember to read the...
  40. Euge

    MHB How Does the Limit of the Integral Relate to the Sinc Integral?

    Here is this week's POTW: ----- Prove $$\lim_{n\to \infty} \int_0^{\pi/(2n)} \frac{\sin 2nx}{\sin x}\, dx = \int_0^\pi \frac{\sin x}{x}\, dx.$$ ----- Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how...
  41. anemone

    MHB Proving the Non-Perfect Square Property of 4 Consecutive Positive Integers

    Here is this week's POTW: ----- Prove that the product of 4 consecutive positive integers is never a perfect square. ----- 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...
  42. B

    Exploring the Physics of Ghosts: A Creative Workshop at Perimeter Institute

    Just watched Ghostbusters yesterday. Quite entertaining. Let's say there would be a creative toy model making workshop for physicists to come up with the most brilliant theory to explain ghosts (in brane worlds where ghosts exist).. what would they be.. perhaps dark matter entities? higgs field...
  43. Ackbach

    MHB Can you prove the inequality for positive integers?

    My apologies for being late this week. Here is this week's POTW: ----- Show that for every positive integer $n$, \[ \left( \frac{2n-1}{e} \right)^{\frac{2n-1}{2}} < 1 \cdot 3 \cdot 5 \cdots (2n-1) < \left( \frac{2n+1}{e} \right)^{\frac{2n+1}{2}}. \] ----- Remember to read the...
  44. Mech_Engineer

    Happy Moon Day 2016! Celebrate Apollo 11 Anniversary!

    Happy Moon Day 2016! I was lax in my duties last year and [gasp] missed Moon Day 2015, but that makes Moon Day 2016 all that more important! For those that didn't realize, today marks 47 years since mankind first set foot on the moon. In celebration of this achievement, we are enjoying...
  45. anemone

    MHB How Can You Rewrite x²/(x⁴+x²+1) Using u?

    Here is this week's POTW: ----- Given that $u=\dfrac{x}{x^2+x+1}$. Express in terms of $u$ the value of the expression $\dfrac{x^2}{x^4+x^2+1}$. ----- Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how...
  46. Euge

    MHB What is the sum of quadratic residues in $\Bbb Z/p\Bbb Z$ for a prime $p > 3$?

    Here is this week's POTW: ----- Let $p$ be a prime greater than $3$. Compute the sum of the quadratic residues in $\Bbb Z/p\Bbb Z$. ----- 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...
  47. Ackbach

    MHB Why Is the Sum of Certain Binomial Coefficients Divisible by \( p^2 \)?

    Here is this week's POTW: ----- If $p$ is a prime number greater than 3 and $k = \lfloor 2p/3 \rfloor$, prove that the sum \[ \binom p1 + \binom p2 + \cdots + \binom pk \] of binomial coefficients is divisible by $p^2$. ----- Remember to read the...
  48. Euge

    MHB Can a Function Be Discontinuous Only at Irrationals?

    Here is this week's POTW: ----- Does there exist a real-valued function on $\Bbb R$ that is discontinuous only on the irrationals? ----- 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...
  49. anemone

    MHB Triangle Inequality Proof: Shortest Side in Relation to Sides a, b, and c

    Here is this week's POTW: ----- The sides $a,\,b$ and $c$ of a triangle satisfy $a^2+ b^2> 5c^2$. Prove that $c$ is the shortest side of this triangle. ----- Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find...
  50. Euge

    MHB What is a non-trivial unit in the integral group ring Z[S_3]?

    Here is this week's POTW: ----- Give an example of a unit of the integral group ring $\Bbb Z[S_3]$ that is not of the form $1x$ for some $x\in S_3$. ----- Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out...
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