Suppose $f: [0,1]\to \mathbb R$ is continuous and $$\int_{0}^{1} f(x)e^{nx} \mathsf dx=0$$ for every $n$. Prove that $f(x)=0$ for all $x \in[0,1]$.   Since $f$ is continuous on $[0,1]$, by Stone-Weierstrass theorem, we can say that there exists a sequence of polynomials $P_n$ that converges uniformly to $f$ on $[0,1]$. What the next step?
I'm trying to show that if $f$ is a continuous function on $[0,1]$ and $\int_0^{1} f(x)e^{nx}\,{\rm d}x = 0$ for all $n = 0, 1, 2, \dots$, then $f(x) = 0$.  I'd like to use Weierstrass approximation theorem to find a sequence of polynomials $p_m$ that converge uniformly to $f(x)$. Then we could say $\lim\limits_{m\to \infty} \int p_m(x)e^{nx}\,{\rm d}x = \int f(x)e^{nx}\,{\rm d}x = 0$, but I'm struggling to deduce that then all the $p_m$ are zero which would give the result.
How can I use the subdivision surface modifier to create more high poly meshes without ruining sharp edges? For example this mesh  The sphere should be subdivided but the rim of the cylinder should stay the same because it is a sharp edge. This is the result of the subdevision  How can I get sharp edges to not be affected by the sub surf mod? How to define orbital angular momentum in other than three dimensions? In classical mechanics with 3 space dimensions the orbital angular momentum is defined as  $$\mathbf{L} = \mathbf{r} \times \mathbf{p}.$$  In relativistic mechanics we have the 4-vectors $x^{\mu}$ and $p^{\mu}$, but the cross product in only defined for 3 dimensions. So how to define orbital angular momentum e.g. in special relativity in terms of 4-vectors? Or more generally in $d$ dimensions? What is this element on the far right of schematic described as "270 OHM"? Is this some kind of inductor? What is its purpose in this device (this is a reference design of dc-dc power supply of sim900 taken from its datasheet) and what would be an exemplary part number? Showing that $\displaystyle\lim_{s \to{1+}}{(s-1)\zeta(s)}=1$ I need prove the following: ($\zeta(s)$ is the Riemann zeta function)  $\displaystyle\lim_{s \to{1+}}{(s-1)\zeta(s)}=1$  I really don't know, I have tried hard, but have obtained nothing for now. Movie Identification - 1990s VHS Sci-Fi/Horror I feel like someone made a comment as an answer on a question with a bounty just so they could reap the "auto bounty" since they are the only answer on it.   I've flagged the bad/useless answer (as a non-answer) for moderator review but the bounty will be assigned automatically in less than 24hrs and I feel like this guy/gal is just trying to cheat the system.   All I'm trying to do is ensure the bounty simply expires and the guy/gal that was trying to cheat the system by putting down a useless guess doesn't get it.   I've answered my own question (properly) and marked it as the accepted answer, not sure if that makes a difference though in this person still cheating the system since I'm the OP and created the accepted answer.   Here is the post: How to increase JAVA Heap Size? Define additional math operators to be typeset in roman Mathematical operators, such as function names, should be set in roman type, not italics. Latex already has commands for some operators, including \max, \min, and \log.  How can I define additional such commands? I just enabled the Blenderkit addon but cannot seem to figure out how to edit the objects. I looked in this reddit thread and amazingly none of the solutions worked for me.  Making it a single user object, data, and materials did not work. I do not see any option to append when adding a model form the Blenderkit asset bar (I even tried saving it and appending the object in another instance of Blender and it still would not let me edit it). And when I right click I do not see an option to make local on outliner collection. I am using Blender 2.83 if that makes a difference but can use 2.9 if need be. Any other solutions? Is the empty set a subset of itself? Clipping LAS data using shapefile polygons and open source software? I am trying to clip or filter lidar files (.las, .xyz) using a shapefile that contains multiple polygons.  Is there open source software that can do this spatial operation?  My .las files are enormous, and I should say that I am very new to DEM data and analysis so any help would be very welcome. Alignment numbered apalike bibliography style I prefer the apalike bibliography style with the number in brackets in front (see also  and ). However what I am curious about is how to align the first and subsequent text lines neatly next to the number in brackets.    Does anyone know how to do this? No more updates for Google Chrome 32-bit Cardinal number subtraction We all know that $|\mathbb{N}| = \aleph_0$. Since $|\{-1\} \cup \mathbb{N}| = \aleph_0$ as well, I guess you could say that $\aleph_0 + 1 = \aleph_0$.  You can go on to derive that $\aleph_0 + \aleph_0 = \aleph_0$, and by induction $n \times \aleph_0 = \aleph_0$ (assuming that $n$ is finite, anyway).  Now, here's a thing: What is $\aleph_0 - \aleph_0$?  Well, $|\mathbb{P}| = \aleph_0$, and $|\mathbb{P} \backslash \mathbb{N}| = \aleph_0$, so perhaps $\aleph_0 - \aleph_0 = \aleph_0$?  No, wait. Consider the set $S = \{n \in \mathbb{N}: n &gt; 5\}$. Now we have $|S| = \aleph_0$, and yet $|\mathbb{N} \backslash S| = 5$. So maybe $\aleph_0 - \aleph_0 = 5$?  But that is absurd, since we can redefine $S$ to make the result any finite number we wish. Or we can define $S$ such that the result is countably infinite.  Does this mean that the notion of $\aleph_0 - \aleph_0$ simply has no definite answer? Or am I just being too simplistic here? Making separate Python installation that can call ArcPy? I see a lot of Q&amp;As asking how you can create a separate Python installation (e.g. the Anaconda or Python XY distribution) that can use arcpy, while not 'breaking' the original ArcGIS python installation.  I had this same problem, and I think I have found a 'universal' solution. So far we have done this on about 8 computers at work and it all works fine.   How to install a separate Python distribution that can use ArcGIS? Can prematurely powering down a computer cause bad sectors in a hard drive? What is the reason behind "non-static method cannot be referenced from a static context"? The cube of any number not a multiple of $7$, will equal one more or one less than a multiple of $7$ Yeah so I'm kind of stuck on this problem, and I have two questions.   Is there a way to define a number mathematically so that it cannot be a multiple of $7$? I know $7k+1,\ 7k+2,\ 7k+3,\ \cdots$, but that will take ages to prove for each case.  Is this a proof? $$(7k + a)^3 \equiv (0\cdot k + a)^3 \equiv a^3 \bmod 7$$    Thank you. Why doesn't bash expand this variable when I prefix a command with a "one time variable assignment" If I run this bash command and prefix the statement, so that the variable fruit should exist, but only for the duration of this command:  $ fruit=apple echo $fruit  $   The result is an empty line. why?  To quote a comment from wildcard on :     parameter expansion is done by the shell, and the "fruit" variable is   not a shell variable; it's only an environment variable within the   environment of the "echo" command   An environment variable is still a variable, so surely this should still be available to the echo command? What is the best way to restore the state of ViewController