Problem regarding Fourier Coefficients.
Convergence of the $\sum_{n=1}^{\infty}n^3b_n^3$ where $b_n=\int_{-\pi}^{\pi}f(t)\sin (nt)\ dt$.
Check if the application is already installed Difference between 'max distance' and 'radius' in Physics.SphereCastAll What is the difference between "getting robbed" and "getting mugged"? I can't seem to find the difference on the internet between "getting robbed" and "getting mugged". I would appreciate it if you could explain it to me. How to control object's path follow speed based on its position The Center of $\operatorname{GL}(n,k)$ The given question:  Let $k$ be a ﬁeld and $n \in \mathbb{N}$. Show that the centre of $\operatorname{GL}(n, k)$ is $\lbrace\lambda I\mid λ ∈ k^∗\rbrace$.  I have spent a while trying to prove this and have succeeded if $ k \subseteq \mathbb{R}$. So I imagine there is a nicer way to go about this. I have seen people saying take matrices where every element except one is zero in proving similar results but such matrices are not in $\operatorname{GL}(n,k)$ as they are not invertible.  What I have done.  I have said let $B \in$ Center then $B$ commutes with everything in $\operatorname{GL}(n,k)$. So take a permutation matrix, $P$ which swaps rows $i,j$ $P \neq I$. From this you can deduce that $B^T = B$ and that $B_{ii} = B_{jj}$. We can use the fact that $B$ is symmetric to find an orthogonal diagonalisation of $B$ by the spectral theorem.  This gives $QBQ^T = D \rightarrow B=D$. And as $B_{ii} = B_{jj}$ we can say $B = k * I$.   But the spectral theorem requires $B$ to be a real matrix.   How do I prove this for a general $k$?. Showing that $\sum_{i=1}^n \frac{1}{i} \geq \log{n}$ Suppose I have a python object x and a string s, how do I set the attribute s on x? So:  &gt;&gt;&gt; x = SomeObject() &gt;&gt;&gt; attr = 'myAttr' &gt;&gt;&gt; # magic goes here &gt;&gt;&gt; x.myAttr 'magic'   What's the magic? The goal of this, incidentally, is to cache calls to x.__getattr__(). What nutrients can humans absorb in the mouth? For instance, I realise being able to absorb simple sugars in the mouth is pivotal in the rapid action of oral glucose gel. Thus I was wondering what nutrients in general can be absorbed directly within the mouth, and at what speed? In Postgres txid_current() should return biginteger but instead an integer is returned In Postgres 11, according to documentation, txid_current() should return biginteger but I'm getting small integer that doesn't look like is extended by epoch counter. My txid looks like this: 1123  "The internal transaction ID type (xid) is 32 bits wide and wraps around every 4 billion transactions. However, these functions  export a 64-bit format that is extended with an “epoch” counter so it will not wrap around during the life of an installation."     Am I interpreting documentation incorrectly? Efficiently selecting a set of random elements from a linked list How to prevent freehub body damage? I feel like this is poor design. I have two wheelsets that are less than a year old which have damaged freehub bodys due to cogs digging in to the splines on the freehub. It's so bad that I can't remove the cassette spacer without filing down the splines. I had to use  pliers and chain whip to remove the cogs that were embedded in the freehub when trying to remove the cassette.  Most of the damage is on the center of the cassette as the largest 3 cogs are held together by carrier. I'm using a shimano 105 cassette and  Novatec disc hub and Velocity ATB convertible disc hubs (which were somewhat damaged in the same fashion, but not as bad as the novatec).  I'm using this bike for cyclocross, offroad exploring, light singletrack use, Rail trails and some road riding.    How can I prevent this damage to the freehub from happening? I know that $re^{i\theta} = x + iy$ for any complex number $x + iy$ by Euler's formula. How do you calculate relative and global phase? How to insert HTML to PHP DOMNode? How to get Url Hash (#) from server side Should I let potential employers know I have a job offer? I've interviewed for a few positions, one of which has made me an offer.  I'm very pleased with the offer, but I prefer some of the other positions.  I have the luxury of being able to take a lot of time to decide whether or not I accept this offer, so I'm certain I will hear back (either positively or negatively) from the other positions before a decision is required for the offer I've received.  I'm trying to decide between two courses of action, and I want to choose the one that will best help my candidacy with the positions I've yet to hear from.  Plan 1 is to do nothing and simply wait to hear back from the other positions for which I've already interviewed.  Plan 2 is to be in touch with the other positions to let them know I have an offer, and ask where I stand with them.  Would Plan 2 make me a more desirable candidate with the other positions, as they might see my offer as further validation of my credentials?  Or, might initiating with the interviewers in this way be detrimental to my candidacy, perhaps making me seem pushy or self-centered?  If so, I would rather just wait for them to run their course of action (Plan 1).  As I've said, I have plenty of time, so I don't want impatience to hurt me.  If there is a good chance that my meddling will be detrimental, then I'd rather be hands-off here.  However, if it's more likely that informing the other interviewers of my offer will give me a boost, then I'd want to do so.  Thanks for any clarity you can provide!  Edit: For a bit more information and context, the position I've been offered is a non-academic research position that my advisor has described as "prestigious."  The other positions for which I've interviewed (and which I prefer) are mostly academic teaching-centered positions (visiting assistant professors, tenure-track at smaller four year institutions). InDesign: How do I automatically align the bottoms of different columns of text boxes to the same height as the one with the most text? In InDesign, I have some filled text boxes, sized by columns, with the one on the left being three columns wide.   The wide text box is most likely to contain the most text (it's an invoice template).  I will use the 'Fit frame to content' button on the Selection Tool toolbar for automatically vertically resize it to fit the containing text.  How do I make the other three smaller text frames automatically adjust to the same height as the big (and longer) text box? How will old M42 lenses work on a Nikon camera? Memory allocation for member functions in C++ Showing OSM source acknowledgment on the map in QGIS? I am using OpenStreetMap as a base layer via XYZ tiles in QGIS 3.01.  I can't seem to find a way of having source acknowledgement appear on the lower right corner of the map.  Can somebody point me in the right direction? In thinking how to ask this question (somewhat) succinctly, I keep coming back to a Microwave Oven.  A Microwave Oven has a grid of holes over the window specifically designed to be smaller in diameter than the wavelength of the microwaves it produces, yet larger than the wavelengths of the visible light spectrum - this is so you can watch your food being heated without getting an eye full of microwaves.  The "realness" of electromagnetic waves seems indisputable - both from the microwave example above, and also because if I want to broadcast a radio wave with a certain wavelength, then I need to make sure I have an antenna of corresponding length to produce the wave I'm looking for.  Furthermore, we discuss and treat these waves as real, measurable "objects" that exist and can be manipulated.     Now, if I want to describe the behavior of my Microwave Oven in the framework of QM (let's pretend my oven is going to only produce 1 photon of energy corresponding to a wavelength of the microwave spectrum for simplicity) then I'll describe the behavior of that photon as a wavefunction that evolves over time and gives a probability distribution within my microwave that similarly does not allow the photon to pass through the safety grid and exit the oven cavity giving me a retinal burn.  The difference is, the wavefunction is never treated as something "real" in this description.  When the safety grid is described as working to protect you because it has holes smaller than the wavelength of the classical waves it's blocking, this is a useful description that seems to describe "real" objects/be-ables.  While it is possible to describe why an individual photon has low probability of passing through the same grid but extended physical properties such as wavelength (in space) are treated as non-real because we're dealing with a point particle and with behavior described by something we also treat as non-real (the wavefunction); it seems unclear to me why we insist this wave function which predicts behavior of physical measurements so well is somehow "non-real."    Put another way, if we have no problem treating EM waves as "Real," then why do we insist on treating the wavefunction that describes the same behavior as "unreal?"  I understand there is recent research (Eric Cavalcanti and his group for one) trying to argue this point, but as every respectable physics professor, I've ever encountered, has treated the wavefunction as an indisputably non-real mathematical tool, I needed to ask this community for an answer.