I read a Putnam question recently. I am having trouble remembering which year it was, but it was probably from before the 1950's     Let $f\in[0,\pi]$ be a continuous function. If $$\int_0^\pi{f(\theta)\cos\theta \;{\rm d}\theta}=\int_0^\pi{f(\theta)\sin\theta\; {\rm d}\theta}$$ prove that $f$ has two zeroes in $(0,\pi)$   Consider the function $f(\theta)=e^{-\theta}$. It satisfies the condition given above, which can be seen through integration by parts, and does not have any zeroes. Where am I going wrong?
Let $ f $ be a continuous function defined on $ [0,\pi] $. Suppose that      $$ \int_{0}^{\pi}f(x)\sin {x} dx=0,   \int_{0}^{\pi}f(x)\cos {x} dx=0 $$   Prove that $ f(x) $ has at least two real roots in $ (0,\pi) $
WPF Get Element(s) under mouse I'm struggling to understand the core reason to invest in stocks as a minority shareholder. Majority shareholding makes more sense as you effectively control the company, so you can pay yourself dividends or even sell the entire company at will. You can guarantee your own payback.  However, as a minority shareholder, you don't get to make any of these decisions. You're basically just hoping to eventually, some day get paid, if the majority holders decide so.  Why do so many investors agree to pay now huge sums of money (trillions in total) for something that may eventually pay dividends, and even in the best case, these dividends are only a fraction of the price you paid to acquire these stocks?  The average annual dividend rate for the US stock market is 2%. That means if you bought 100 million worth of shares, you'd have to wait 50 years to recover your investment.  Of course, the most probable way you'd eventually get liquidity for these shares is by selling them to some other starry-eyed investor - which makes the whole thing seem like a giant Ponzi scheme.  Given all that, it's unclear to me why the public is willing to spend so much money to acquire minority shares in companies. Function whose inverse is also its derivative? What are some good examples of a function  $f : \mathbb{R} \to \mathbb{R}$ where its derivative is equal to its inverse? I attempted to find a monomial that satisfied it by starting with $f(x) = ax^b$ and showing that $f^{-1}(x) = f'(x) \implies b-1=\frac{1}{b} \implies b=\phi$ and got $$f(x) = \frac{x^\phi}{\sqrt[\phi]{\phi}}$$ Which seems to work according to WolframAlpha, but I'm having trouble double-checking it. Any other ideas? Why didn't Thanos come to Earth to retrieve Tesseract by himself? In the Avengers movie, why didn't Thanos personally come to Earth instead of Loki? Or, if Loki's skills were required, why didn't he come with Chitauri? He knew that humans were primitive; there wouldn't be any threat and he could personally secure Tesseract.  AFAIK, the movie didn't display much. Is there any other source answering this? So, this sounds like a stupid question I know but in the bleachverse 'names' are very important and you can only unleash the power of your zanpakuto if you can call out its name. Therefore, if a shinigami is mute, will they ever be able to achieve shikai/bankai? If so then how? Only one case that I know of is for shinigamis who have already achieved bankai to be able to release their shikais without a command.  Answers based on LN are welcome, but if you are citing from a dubious source or a source whose canon status can be question, please specifically mention it. I am mostly interested in an answer based on a  canon source. Finding an uncountable chain of subsets the integers How to find an uncountable subset of $P(\mathbb{N})$ such that every two elements of it can be compared. In fact, give an uncountable subset of $P(\mathbb{N})$ such that has totality property. We mean by $P(\mathbb{N})$, the powerset of natural numbers set $\mathbb{N}$. Divergence of $\frac{ \hat {\bf r}}{r^2} \equiv \frac{{\bf r}}{r^3}$, what is the 'paradox'? I just started in Griffith's Introduction to electrodynamics and I stumbled upon the divergence of $\frac{ \hat r}{r^2} \equiv \frac{{\bf r}}{r^3}$, now from the book, Griffiths says:  Now what is the paradox, exactly? Ignoring any physical intuition behind this (point charge at the origin) how are we supposed to believe that the source of $\vec v$ is concentrated at the origin mathematically? Or are we forced to believe that because there was a contradiction with the divergence theorem? Also how would the situation differ if $\vec v$ was the same vector function but not for a point charge? Or is it impossible? Maximum power of $2$ which divides $3^{1024}-1$ What is the maximum power of $2$ which completely divides $3^{1024}-1$?  I proceeded thus: $\phi(2^n)=2^{n-1}$ for all $n\ge1$ $$3^{1024}=3^{2^{10}}\equiv1\pmod {2^{11}}$$ $$3^{1024}-1\equiv0\pmod {2^{11}}$$ Since $\phi(2^{11})=2^{10}$.  So, maximum power of $2$ must be $11$. But the answer says it is $12$. Where am I wrong and how to solve it correctly? Prove by induction that $n! > 2^n$ Suppose that when $n=k$  $(k\geq4)$, we have that $k!&gt;2^k$.  Now, we have to prove that $(k+1)!\geq2^{k+1}$ when $n=(k+1)$ $(k\geq4)$.  $$(k+1)! = (k+1)k! &gt; (k+1)2^k  \text{ (since }k!&gt;2^k)$$  That implies $(k+1)!&gt;2^k2$, since $(k+1)&gt;2$ because of $k$ is greater than or equal to 4.  Therefore $(k+1)!&gt;2^{k+1}$  Finally  we may conclude that $n!&gt;2^n$ for all integers $n\geq4$    Solved, thank you! Blender 2.78 file size grows and does never shrink I made a huge project, wanted to reduce file size from 1.3GB to below 1GB. Compressing did not help. I removed objects from scene. Also textures. I then realized, the file size even was the same when I delete ALL objects. Why?  Download of sample:   Before deleting all: 1.2GB, after: 1.1GB. How to echo output of a top into a text file? I want to write batch or something, that will write output of top into a file on login.  I did top &gt;&gt; output-file, but it contains some strange character!  Can somebody give a simple tutorial about how to write batch file in Linux? Let $b_n$ decrease monotonically to zero, prove $\sum b_nz^n$ converges for $|z|\leq 1$ and $z\neq 1$ I am able to parse strings containing date/time with time.strptime  &gt;&gt;&gt; import time &gt;&gt;&gt; time.strptime('30/03/09 16:31:32', '%d/%m/%y %H:%M:%S') (2009, 3, 30, 16, 31, 32, 0, 89, -1)   How can I parse a time string that contains milliseconds?  &gt;&gt;&gt; time.strptime('30/03/09 16:31:32.123', '%d/%m/%y %H:%M:%S') Traceback (most recent call last):   File "&lt;stdin&gt;", line 1, in &lt;module&gt;   File "/usr/lib/python2.5/_strptime.py", line 333, in strptime     data_string[found.end():]) ValueError: unconverted data remains: .123 I just made a minor , and  the grammar correction was unnecessary.  The original text was     Another factor--sometimes life happens and you can't fly. In the old days you could simply sell your ticket to someone else, now you either have to eat a hefty change fee or lose it outright. That's money in their pockets that they didn't used to get.   It was corrected to:     Another factor--sometimes life happens and you can't fly. In the old days you could simply sell your ticket to someone else, now you either have to eat a hefty change fee or lose it outright. That's money in their pockets that they didn't use to get.    Was this grammatical correction an improvement? Was the original text grammatically correct? Is the edited text grammatically correct? With powershell get exactly the same application list as in Add/Remove programms There are various methods to find installed applications on the system but to my knowledge none returns exactly the same list as control panel item Add Remove Programs. Some of the ways I tried are:   WMI via Win32_Product and Win32Reg_AddRemovePrograms Registry, both x32 and x64 branches   For instance, I have 217 entries in AR and   gwmi  Win32_Product   returns 261 items.  gp HKLM:\Software\Microsoft\Windows\CurrentVersion\Uninstall\*, HKLM:\SOFTWARE\Wow6432Node\Microsoft\Windows\CurrentVersion\Uninstall\* | ? {![string]::IsNullOrWhiteSpace($_.DisplayName) } | select DisplayName   returns 119 items.   (gwmi Win32Reg_AddRemovePrograms) + (gwmi Win32Reg_AddRemovePrograms64) | sort DisplayName -Unique    returns 107 items.  I tried to nail it down with more complex filter that included InstallLocation, InstallSource, InstallDate etc without success. When should end punctuation go inside quotes? I have been/am being taught that end punctuation should always go inside quotes. For example, you are supposed to write:     Marvin thought it was "awful."   The problem is I do not see how does this make sense. Intuitively, I always wrote:     Marvin thought it was "awful".   as that makes more logical sense — you want a quote to be an exact replication of what somebody else said, so why should you add punctuation inside?  I always thought it made more sense to not touch the quote and add anything after or before if it must be added.  So, why should I put end punctuation inside quotes? What is the quickest way to send some text from desktop\laptop to iPhone? I usually send emails to myself, but I wonder if there is a better way to share something quickly from my PC\laptop to iPhone. Deriving the conditional distributions of a multivariate normal distribution Does the English language have an official Academy? For some languages, there are academies that decide topics such as grammar and spelling of things, for example, for the Spanish language, there are 22 academies in 22 different countries, all making decisions on spanish grammar in their country. I would like to know if there is something that serves the same purpose for the English language. Does $\mathbb R^2$ remain connected when countably many points are removed? Does it remain path connected?  This is not homework but is in response to working several problems where countable or uncountable many points are removed and the pattern always holds.  Thank you.