Could you please help me with the following? I think it is a Lebesgue Dominated Convergence Theorem but I am not sure.  Compute the following: For $1\leq p&lt;\infty$ and $f\in L^p(\mathbb{R})$ $$\lim_{h\rightarrow \infty}\int_{-\infty}^\infty\left|f(x+h)-f(x)\right|^p \, dx$$
Let $f \in L^1(\mathbb{R})$, for $a\in \mathbb{R}$ let $f_a(x)=f(x-a)$, prove that: $$\lim_{a\rightarrow 0}\|f_a -f \|_1=0$$  I know that there exists $g\in C(\mathbb{R})$ s.t $\|f-g\|_1 \leq \epsilon$, this is also true for $f_a$ and $g_a$.  Now I have the next estimation:  \begin{align*} \|f_a-f\|_1 &amp;= \int_{\mathbb{R}} |f(x)-f(x-a)| dx \\ &amp;= \int |f(x)-g(x)+g(x)-g(x-a)+g(x-a)-f(x-a)| \\ &amp;\leq \|f-g\|_1 + \|f_a-g_a\|_1 + \int |g(x)-g(x-a)| \end{align*} my question is: I can argue that $\lim_{a\to0} \int |g(x)-g(x-a)| = \lim_{a\to0} \lim_{T\to\infty} |g(x_0)-g(x_0-a)| 2T$, now I think that I can change the order of the limits here, but I am not sure why?  P.s $x_0$ is some point in $[-T,T]$, and the above is valid from the intermediate integral theorem for continuous functions, right?  Thanks.
White lines between objects in render I am kind of new to the whole animation aspect and in fact still in high school.  I am trying to make a Minecraft character rig and I want to make some 3D hair. I did not know how to quite make it so I kind of when just did something pretty simple. It isn't the greatest in any means, but any how the point is every time I render it white lines show up between objects or joined objects. I don't know how to fix this and would like help (a picture is posted below of the problem). Consider the following snippet:  $$\text{d}_{H}(A,B)  = \max\left\{ \sup_{a\in A} \inf_{b\in B} \text{d}(a,b),\sup_{b\in B} \inf_{a\in A}\text{d}(a,b)\right\}$$     In the output, the text under the inf's is significantly smaller then the one under the sup's. The same happens when I have it in an align environment.  You can see the same problem in the wiki definition of the . How can I have the typeset under the inf's and sup's in the same size? If $m^4+4^n$ is prime, then $m=n=1$ or $m$ is odd and $n$ even I have been stuck on this one for months, really simple to state, really giving me trouble.     Show that if $m^4 + 4^n$ is prime, $m&gt;0$, $n&gt;0$, then $m$ is odd and $n$ is even, except when $m=n=1$.   The case $m=n=1$ gives $5$, so that is why it is excluded. Clearly $m$ is odd, for if it was even, the number would be divisible by at least $4$, but I can't seem to get rid of the case where $n$ must be odd.   The only thing I've managed to try is to write $m$ as an odd number, say $m = 2k+1$, and $n = 2\ell + 1$, then get $$ m^4 + 4^n = (2k+1)^4 + 4^{2\ell+1} = 16k^4 + 32k^3 + 24k^2 + 8k + 1 + 4^{2 \ell + 1} $$ and this is congruent to $0 \mod 5$ unless $k \equiv 2 \mod 5$, and now it gives you an even more ugly polynomial if I write $k = 5j+2$, and it's not working. I've tried other inconclusive approaches, like trying to see if two numbers of some form generate numbers of the same form.. (for instance primes of the form 8k+1 and 8k+7 always generate primes of the same form when multiplied together, stuff like that).  Any ideas? Even just ideas you haven't gave a thought... they're welcome! Can every game of Klondike-Solitaire be solved? I play a lot of Solitaire games on my Android phone and love to keep an eye out for the statistics.  Given that the Solitaire version lets you restart the game endlessly, I usually play until I solve it. But I never managed to solve more than 80% of the games played (1000+).  So now I wonder, is every Solitaire game solvable? Evaluating the integral $\int_0^\infty \frac{\sin x} x \,\mathrm dx = \frac \pi 2$? Which types of particles are affected by the wave-particle duality? Start Mathematica without the menu bar? How to get the environment variables of a subprocess after it finishes running? SBS 2011 - Is Store.exe High Memory Usage Normal? GCD of two numbers divided by their greatest common divisor is 1 Start ubuntu without a desktop environment but start an X application I want to do a small computer just for internet browsing. I dont want to load a desktop environment to keep RAM, CPU and GPU ussage as low as possible. The computer is going to run on a ARM CPU. I dont care with Distro I use, I thought about good old plain Debian. Render not showing UV texture Which spells or abilities are affected by things that care about targeting? Note: this is intended to be a canonical question about whether or not a spell or ability targets so there are multiple sub-questions.   Can Hexproof or Shroud save my creature from ? What about a  that was cast for it's Overload cost? Does shroud prevent me from using the Monstrosity ability of a creature, or from giving it more counters with Proliferate? Can I cast an Aura Enchantment on one of my creatures with Shroud, or an opponent's creature with Hexproof? Or can I use the Equip ability to attach an equipment to one of my creatures with Shroud? Will 's ability trigger when my opponent plays ? And will I have to sacrifice my  if my opponent casts ? Will Shroud or Hexproof prevent an opponent from choosing me with 's ability? Will Hexproof keep one of my creatures from being affected by ? If an Aura or Equipment is attached to a creature with Hexproof, does the Hexproof protect the attached permanent? broadcast receiver for missed call in android Do we have negative prime numbers?   $..., -7, -5, -3, -2, ...$ 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. meaning of so much as In the following sentence, I want to know if I've understand the correct meaning or not:     This version holds that the wave function does not so much describe the state of the physical system as the state of our knowledge of the system.    Can we say that it means the wave function describes the state of our knowledge  of the system more than it describes the state of the physical system  ?  It is worth noting that in the Cambridge Dictionary, we read:  not so much sth as sth ​ If you say that something is not so much one thing as something else, you mean it is more the second thing: They're not so much lovers as friends. I don't feel angry so much as sad. Close and flag links do nothing on mobile site This morning (it's about six AM here) I find that I can no longer submit close votes; the link just jumps to the beginning of the page.  I'm on iOS, using a third-party browser called Grazing, but a brief check with Mobile Safari shows the same problem there.   Here on Meta, where my score is much lower, I only get a "delete" link on my own question, and that works fine. (Oops :-))   Update: As per comments, this affects the "flag" link, too, which doesn't even scroll to the top of the page. Update 2: Latest comment suggests this affects Android, too. Update 3: This also affects desktop computers when viewing mobile (when you click "mobile" in the footer) Update 4:  mentions Windows 8, too; simply deleting platform-specific tags Using right-hand Riemann sum to evaluate the limit of $ \frac{n}{n^2+1}+ \cdots+\frac{n}{n^2+n^2}$ I'm asked to prove that $$\lim_{n \to \infty}\left(\frac{n}{n^2+1}+\frac{n}{n^2+4}+\frac{n}{n^2+9}+\cdots+\frac{n}{n^2+n^2}\right)=\frac{\pi}{4}$$ This looks like it can be solved with Riemann sums, so I proceed:  \begin{align*} \lim_{n \to \infty}\left(\frac{n}{n^2+1}+\frac{n}{n^2+4}+\frac{n}{n^2+9}+\cdots+\frac{n}{n^2+n^2}\right)&amp;=\lim_{n \to \infty} \sum_{k=1}^{n}\frac{n}{n^2+k^2}\\ &amp;=\lim_{n \to \infty} \sum_{k=1}^{n}(\frac{1}{n})(\frac{n^2}{n^2+k^2})\\ &amp;=\lim_{n \to \infty} \sum_{k=1}^{n}(\frac{1}{n})(\frac{1}{1+(k/n)^2})\\ &amp;=\lim_{n \to \infty} \sum_{k=1}^{n}f(\frac{k}{n})(\frac{k-(k-1)}{n})\\ &amp;=\int_{0}^{1}\frac{1}{1+x^2}dx=\frac{\pi}{4} \end{align*}  where $f(x)=\frac{1}{1+x^2}$. Is this correct, are there any steps where I am not clear? Effect Of Frequency on Photocurrent The situation is this.. I am performing a photoelectric effect experiment - just the usual - evacuated glass tube containing two metal electrodes connected to an external source of DC voltage (say an alkaline cell) and ammeter in series. I shine a light on one of the electrodes - the frequency is more than enough to cause eject photoelectrons, and the voltage of the cell opposes the electrons which are trying to get to the other electrode. The question is this - what happens to the current on increasing the intensity of light while keeping no. of photons falling on the electrode constant (in other words increasing the frequency)?   Does the current remain the same because the no. of photons falling on the electrode is constant and so no. of electrons being ejected is constant? Or does it increase because the electrons have more kinetic energy, ie, they are moving faster towards the other electrode so that the $t$ (time) term in $I$ = $q$/$t$ is less ($q$ remains constant because no. of electrons being ejected is still the same)?   This question appeared in one of my tests and the answer key says that the current remains constant but think that case 2 is correct and current would increase, which is why I asked this question.  Thanks.