How do we identify if a number can be represented as the sum of squares of $3$ integers?  For eg: $434 = 11^2 + 12^2 + 13^2$ but $432$ is not?
I'm trying to determine if 1317 and 116 can be written as the sum of three squares?  I have the condition that if it is not of the form $4^{\alpha}(8k+7)$ then it can be written as a sum of three squares, but how do I use this condition to work it out?  Thank you
Please help me to answer the following problem:  Let $f(x)=x^3-3x-5\in\mathbb{Q}[x]$  Thanks How to adjust the gap between figure and caption within the same subfloat Renormalized mass I am reading Schwarz QFT and I reached the mass renormalization part. So he introduces, after renormalization, a physical mass, defined as the pole of the renormalized propagator, and a renormalized mass which can be the physical mass, but can also take other values, depending on the subtraction schemes used. Are these masses, other than the physical one, observable in any way experimentally, or are they just ways to do the math easier (using minimal subtraction instead of on-shell scheme, for example)?   Also, in the case of charge renormalization, the explanation was that due to the vacuum polarization, the closer you are to the charge, the more you see of it, so the charge of the particle increases with the momentum you are testing the particle with. However, I am not sure I understand, from a physical point of view, why do you need to renormalize the mass. Is this physical mass (defined as the pole of the renormalized propagator) the same no matter how close you get, or it also changes with energy? And if it changes, what is shielding it at big distances? Why isn't the gamma function defined so that $\Gamma(n) = n! $? As a physics student, I have occasionally run across the gamma function  $$\Gamma(n) \equiv \int_0^{\infty}t^{n-1}e^{-t} \textrm{d}t = (n-1)!$$  when we want to generalize the concept of a factorial.  Why not define the gamma function so that   $$\Gamma(n) = n!$$  instead?  I realize either definition is equally good, but if someone were going to ask me to choose one, I would choose the second option.  Are there some areas of mathematics where the accepted definition looks more natural?  Are there some formulas that work out more cleanly with the accepted definition? Function with varying number of For Loops (python) What happens if a Rainbow Servant takes levels in Mystic Theurge? When a Rainbow Servant hits level 10 they gain access to all nondomain Cleric spells. All spells that do not appear on any other spell list they have are cast as Divine spells. This means that, if they have the skills and have learned 2nd level spells, they qualify for the Mystic Theurge Prestige Class.  But what happens when you actually start to take levels in it? The text says:     When a new mystic theurge level is gained, the character gains new spells per day as if he had also gained a level in any one arcane spellcasting class he belonged to before he added the prestige class and any one divine spellcasting class he belonged to previously.   What it does not say however, here or elsewhere, that these classes have to be different classes. Do you get one level in Wizard/Warmage/whaever for being an Arcane caster, then ANOTHER level for being a Divine caster as well? The text for the Rainbow Servant says the following:     Such spells are cast as divine spells if they don’t appear on the sorcerer/wizard or bard spell lists.   So it comes down to these questions:   Does being able to cast Divine spells qualify you as a Divine Caster, or is there more to it than that? If it is, what happens when you take levels in Mystic Theurge like this? Do you go up one level or two? What happens if this means you go over 20th caster level? Do you keep gaining spell slots like a level 20+ character, or does something else happen? How to connect to Salesforce REST API without OAuth? How does Jitterbit do it? I hear so many terms involving the word "linear". Linear function, linear equation, linear system, linear operator, linear transformation, linear mapping, linear space, linear algebra, linear electrical circuits, linear filters, linear electrical elements, linear approximation, linear optimization.  I'm getting crazy trying to understand the application of the "linear concept" to all this aspects (function, equation, mapping, system, operator, transformation, algebra, etc.) and I wish to know the one essence that is to be linear. What something has to be to be linear? If I say something is linear, what do I know for sure about that something (no matter what the something is)?       I heard a definition of linearity is by homogeinity (scaling the input results in a scaled output) and addition (summing the inputs results in summing the outputs). Can I apply this simple definition to all the branches (operator, mapping, system, transformation, algebra, ...) I mentioned ? Do they all behave like a line ?  y = ax + b, for example, is a line but doesn't behave like a line because y is not linear. I have just updated (through Software Update) to Ubuntu 16.04 LTS from 15.10. There are random instances of screen flickering throughout usage. After noticing this issue, I installed Compiz Config Settings Manager, selecting Force fulls screen redraws (buffer swap) on repaint, which did nothing.   Is there any way this could be related to compiz settings or is this another issue?    EDIT: not fixed after clean install, upgrade to kernel 4.5.2, however screen flickering is reduced, it is every now and then (10 minutes apart). I will see if upgrading to kernel 4.6.0-994-generic will fix this.     EDIT: fixed as of kernel 4.6.2 generic,  currently running 4.7.2 without any issues. translation invariant of integral on $\mathbb{R}$ Problem 1: Let $f\in L^1(\mathbb{R})$, show $\int_\mathbb{R}f(t)dt=\int_{\mathbb{R}}f(x+t)dt,\forall x\in (-\infty, \infty)$.  Problem 2: Let $f\in L^1(\mathbb{R})$,show $\displaystyle \lim_{t\rightarrow 0}\int_{\mathbb{R}}|f(x+t)-f(x)|dx=0$.  My try for Problem 1:  Consider $\displaystyle \int_{\mathbb{R}}f(t)dt=\lim_{n\rightarrow \infty}\int_{-n}^nf(t)dt=\lim_{n\rightarrow \infty}\int_{\mathbb{R}}f_n(t)dt$, where $f_n(t)=f(t)\chi_{[-n,n]}$.  And since $dt=d(x+t)$, we have $\displaystyle \int_{-n}^nf(x+t)dt=\int_{-n+x}^{n+x}f(t)dt$.  So for each $x\in (-\infty,\infty)$,  $$ \begin{split} \int_{\mathbb{R}}f(t)-f(t+x)dt &amp;=\lim_{n\rightarrow \infty}\int_{\mathbb{R}}f(t)(\chi_{[-n,n]}-\chi_{[-n+x,n+x]})dt\\ &amp;=\lim_{n\rightarrow \infty}\int_{\mathbb{R}}f(t)(\chi_{[-n,-n+x]}-\chi_{[n,n+x]})dt\\ &amp;=\lim_{n\rightarrow \infty}\lim_{k\rightarrow \infty}\int_\mathbb{R}\varphi_k(t)(\chi_{[-n,-n+x]}-\chi_{[n,n+x]})dt\\ &amp;=\lim_{k\rightarrow \infty}\lim_{n\rightarrow \infty}\int_\mathbb{R}\varphi_k(t)(\chi_{[-n,-n+x]}-\chi_{[n,n+x]})dt \end{split} $$ where $\varphi_k(t)$ is simple function pointwise converges to $f(t)$, simple function vanish out of set of finite measure, so $\varphi_k(t)(\chi_{[-n,-n+x]}-\chi_{[n,n+x]})$ vanishes when $n$ big enough(is this right?)    Another try:  Since $f\in L^1$, $f$ is bounder a.e. so can be approximated by simple functions, and $\chi_{A+t}(x)=\chi_A(x-t)$,thus $$\int f(t)=\lim_n\int\varphi_n=\lim_n \sum_{i=1}^nc_i\mu(E_i)=\lim_n \sum_{i=1}^nc_i\mu(E_i-x)=\int f(t+x)$$  For problem 2, I guess they share the same method.   Any help or correction about these two problems? Thanks. Why is glass transparent? Why can't a Federation Starship's transporter history be used to bring someone back to life? How can we unlock an iPhone backup file if we don't know the password? I've recently noticed an issue with Ubuntu Software Center which I have tried to resolve following the answers provided in  without result and  which partially worked but didn't persist. The Issue After installing 20.04 I opened Ubuntu Software Center and installed a number of apps without any problem - one of which was Stacer which I will use as an example.  In Show Applications the app is clearly present and right clicking on the icon selecting Show Details provides the image above and it works perfectly well. However, if I open Ubuntu Software Center and type in the search bar stacer it reports that no application is found:  Nor is it listed when selecting the Installed tab. This is the setup for for Software and Updates.  There are other applications I have previously installed which are also now reported as No Application found such as gLabels, gscan2pdf and others. Has something changed since I initially installed 20.04.1 and more importantly, how can I restore good order to the Ubuntu Software Center? Stopping a Thread in Java? I'm in the process of writing a piece of code that connects to a server spawns a bunch of threads using that connection and does a bunch of "stuff".  There are certain instances where the connection fails and I need to stop everything and start from scratch with a new object.  I wanted to clean up after the object but calling thread.stop on the threads, but this method is seemingly deprecated.  What is the recommended alternative to doing this? Should I write my own cleanup and exit method for each of the threads? Set the thread to null? or something else? Cannot invoke method or retrieve property from null object. Object returned by the following call stack is null. \"File\r\n\" Time Dilation inside a hollow shell Assuming I have a hollow shell with total mass $M$ and radius $r$. On the surface, the gravitational time dilation would be  $$\tau=t \cdot \sqrt{1-\frac{v_{esc}^2}{c^2}}$$  where  $$v_{esc} = \sqrt{\frac{2 \cdot G \cdot M}{r}}$$  but inside the shell there would be no gravitational field (Newton's shell theorem and Birkhoff's theorem).  But still, the escape velocity required to escape to infinity would be the same as on the surface, since inside the shell you could move without any accelerating or decelerating forces acting on you until you reach the surface, from where you would get pulled backwards.  So is the time dilation inside the hollow shell relative to a field free observer at infinity    zero (I assume it's not) or  the same as on the surface (my best guess), or something completely different?   I found some not really duplicate but related threads on the interior of black holes which did not really focus on the math, but I am more behind the calculations in terms of $M$ and $r$. I have been given the task to rewrite a function as a sum of an even and an odd function. But when I went to analyze the parts of the original function I noticed that one part was neither even nor odd.   The original function was: $g(x) = (x + 1) / (x^2 - 3x + 4$)  After analyzing I realized that when using f(-x) to determine symmetry it produced a 'neither' result. How would I go about writing a function that is neither even nor odd as an even function. (It would have to be an even function as the (x+1) part of the original function is odd and since I need to write it as a sum of even and odd, the denominator would need to be even. What am I doing when I separate the variables of a differential equation? I see an equation like this:  $$y\frac{\textrm{d}y}{\textrm{d}x} = e^x$$  and solve it by "separating variables" like this:  $$y\textrm{d}y = e^x\textrm{d}x$$ $$\int y\textrm{d}y = \int e^x\textrm{d}x$$ $$y^2/2  = e^x + c$$  What am I doing when I solve an equation this way?  Because $\textrm{d}y/\textrm{d}x$ actually means  $$\lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}$$  they are not really separate entities I can multiply around algebraically.    I can check the solution when I'm done this procedure, and I've never run into problems with it.  Nonetheless, what is the justification behind it?  What I thought of to do in this particular case is write  $$\int y \frac{\textrm{d}y}{\textrm{d}x}\textrm{d}x = \int e^x\textrm{d}x$$ $$\int \frac{\textrm{d}}{\textrm{d}x}(y^2/2)\textrm{d}x = e^x + c$$  then by the fundamental theorem of calculus  $$y^2/2 = e^x + c$$  Is this correct?  Will such a procedure work every time I can find a way to separate variables? Is there a way to force Google Photos to recognize new faces? I have "Face grouping" enabled in my Google account and I can see around 10 faces successfully tagged when I go to my Albums. However, there's a lot of other people who are not tagged, as well as some unrecognized photos of people who are already tagged.   Is there a way to force Google Photos to run the facial recognition algorithm on a given photo, so that it adds new faces or maps a new photo to an existing face?