Let $f:[0, 1] \to \mathbb{R}$ be a continuous function such that $\int^1_0 f(x)x^ndx=0$ $\forall n \in \mathbb{N}$. Show that $f \equiv 0$ in [0, 1].  I need help.
As the title says, I'm wondering if there is a continuous function such that $f$ is nonzero on $[0, 1]$, and for which $\int_0^1 f(x)x^n dx = 0$ for all $n \geq 1$. I am trying to solve a problem proving that if (on $C([0, 1])$) $\int_0^1 f(x)x^n dx = 0$ for all $n \geq 0$, then $f$ must be identically zero. I presume then we do require the $n=0$ case to hold too, otherwise it wouldn't be part of the statement. Is there ay function which is not identically zero which satisfies $\int_0^1 f(x)x^n dx = 0$ for all $n \geq 1$?  The statement I am attempting to prove is homework, but this is just idle curiosity (though I will tag it as homework anyway since it is related). Thank you!
I know the  processors have been around for a long time and are mostly used in servers, but I am curious, why do people not use the Xeons in a high performance desktop?    As far as I know about the best desktop processor out there now is the  line. The i7's and Xeons are both quad-core processors, what is the main difference in these? I just saw that the MacBook Pro's are using the quad core Xeons instead of the i7's. How do I see a C/C++ source file after preprocessing in Visual Studio? Checking integrity of my server data In $Sets$, the initial object is $0 = \emptyset$. We have $C \times \emptyset = \emptyset$ for any set $C$. As there are no maps to the empty set, we don't get much of the universal property of the product.  I am now wondering what is special about arbitrary categories $\mathscr{C}$ with $C \times 0 \cong 0$ for all objects $C$ of $\mathscr{C}$, where $0$ is the initial object of $\mathscr{C}$.   From Sheaves in Geometry and Logic by MacLane (p.194), in any topos $\mathscr{C}$, any arrow $k:C \to 0$ is an isomorphism. The universal property of the product would then also not yield very much.   Are there any categories of higher interest? \pause in tikzpicture breaks footline I've discovered that you can \pause within a tikzpicture, which works well until I switched the theme to one with a footline.  I've created a "test case" that illustrates the problem.  \documentclass{beamer} \providecommand\thispdfpagelabel[1]{} % Not sure what this does but our installation requires it. \usepackage{tikz} \usetheme{Madrid} % Has a footline.  \begin{document}   \begin{frame}     \frametitle{test}     \begin{tikzpicture}       \node at (0, 1) {Hello};       \pause       \node at (0, 0) {World};     \end{tikzpicture}   \end{frame} \end{document}   This creates two slides, but the footline is only drawn on the second (the headline/title appears on both). Is there anything I can do about this? How do you generate dynamic (parameterized) unit tests in Python? How can I move an installed game from one Steam library folder to another? I use (and have it much bigger) drive D: to store all my user data and games, but I have installed the Steam client on drive C:. When I was installing a first game (Warhammer 40000), I've chosen to place my library folder on drive D: and the game went there.  But now as I've got installed the second game (Portal), I can see that it was put on drive C: despite to the choice I've done previously. When I open Steam - Settings - Downloads + Cloud - Steam library folders, I can see both installation places in the list. Can I merge them into one?  I would even agree to use the default drive C: library (as I've replaced it with a symlink already). When I press to remove one library it says it can only remove empty ones. When you tap a row in a UITableView, the row is highlighted and selected. Is it possible to disable this so tapping a row does nothing? I'm using the 1-25-19 version of 2.8 beta. You have to double tap "a" every time you want to deselect something which is very annoying. Is there a way to make "a" work the old way where "a" selects all if nothing is selected and "a" deselects all if something is selected? ie. no double tap and no alt? What are the origins of the word "nice"? $P_n(x):=1+ \sum_{m=1}^n\dfrac{x^m}{m!}$ has no real root for even $n$ and exactly one real root for odd $n$ Is it true that  $$P_n(x):=1+ \sum_{m=1}^n\dfrac{x^m}{m!}$$ has no real root for even $n$ and exactly one real root for odd $n$? I can only prove that the polynomial cannot have any multiple roots. Am stuck , Please Help. Get Absolute Position of element within the window in wpf I would like to get the absolute position of an element in relation to the window/root element when it is double clicked. The element's relative position within it's parent is all I can seem to get to, and what I'm trying to get to is the point relative to the window.  I've seen solutions of how to get a the point of an element on the screen, but not in the window. Imagine that you want to develop a non-trivial end-user desktop (not web) application in Python. What is the best way to structure the project's folder hierarchy?  Desirable features are ease of maintenance, IDE-friendliness, suitability for source control branching/merging, and easy generation of install packages.  In particular:   Where do you put the source? Where do you put application startup scripts? Where do you put the IDE project cruft? Where do you put the unit/acceptance tests? Where do you put non-Python data such as config files? Where do you put non-Python sources such as C++ for pyd/so binary extension modules? Using [Authorize] without SimpleMembershipProvider What's the best way to take a segment out of a text file? Let $\alpha = \frac{1+\sqrt{-19}}{2}$. Let $A = \mathbb Z[\alpha]$. Let's assume that we know that its invertibles are $\{1,-1\}$. During an exercise we proved that:     Lemma: If $(D,g)$ is a Euclidean domain such that its invertibles are $\{1,-1\}$, and $x$ is an element of minimal degree among the elements that are not invertible, then $D/(x)$ is isomorphic to $\mathbb Z/2\mathbb Z$ or $\mathbb Z/3\mathbb Z$.   Now the exercise asks:     Prove that $A$ is not a Euclidean Domain.   Everything hints to an argument by contradiction: let $(A, d)$ be a ED and $x$ an element of minimal degree among the non invertibles we'd like to show that $A/(x)$ is not isomorphic to $\mathbb Z/2\mathbb Z$ or $\mathbb Z/3\mathbb Z$.  How do we do that? My problem is that, since I don't know what this degree function looks like, I don't know how to choose this $x$!  I know that the elements of $A/(x)$ are of the form $a+(x)$, with $a$ of degree less than $x$ or zero. By minimality of $x$ this means that $a\in \{0, 1, -1\}$. Now I'm lost: how do we derive a contradiction from this? Why is the Yang-Mills gauge group assumed compact and semi-simple? What is the motivation for including the compactness and semi-simplicity assumptions on the groups that one gauges to obtain Yang-Mills theories?  I'd think that these hypotheses lead to physically "nice" theories in some way, but I've never, even from a computational perspective. really given these assumptions much thought. Unable to SSH to my virtual machine via Windows Power Shell ISE Naming threads and thread-pools of ExecutorService Access-Control-Allow-Origin Multiple Origin Domains?