I have large image with emoji and use it like this:  Css:  .emoticon {   width: 26px;   height: 26px;   display: inline-block;   vertical-align: top;   background-image: url(path/to/image.png);   background-repeat: no-repeat;   background-size: auto;  }  emoticon.emoticon_1 { background-position: 0 -869px }   Html:   &lt;div class='emoticon emoticon_1'&gt;&lt;/div&gt;   Problem is that original size of emoticons is 26x26 px, it is too lage. Are there some ways to transform it in 16x16 px image via css?
Is there a way to get a background in CSS to stretch or scale to fill its container?
Solving the recurrence $T(n) = \sqrt n T(\sqrt{n}) + \sqrt{n}$ A former student of mine was TA-ing an algorithms class last quarter and asked students to solve this famous recurrence relation:  $$T(n) = \sqrt n T(\sqrt{n}) + n$$  There are  and this question has already been asked here.  I was talking to my student about this problem and mentioned that it's quite hard to solve and is pretty dependent on the particular choices being made. As an example, I suggested this variant of a recurrence relation that was less simple to solve:  $$T(n) = \sqrt n T(\sqrt{n}) + \sqrt{n}$$  The TA and I tried solving this recurrence for about an hour and a half without making any progress. Here are a few things we tried:   My approach to solving the initial recurrence relation was to draw out a recursion tree and notice that each level of the tree contributes $n$ to the total and that there are $O(\log \log n)$ layers, so the recurrence solves to $O(n \log \log n)$. When I tried doing this here, I noticed that the work per layer was no longer constant; instead, the top layer sums to $\sqrt{n}$, the second layer to $n^{3/4}$, the third to $n^{7/8}$, etc. We got stuck working with the summation $n^{1/2} + n^{3/4} + n^{7/8} + ...$. We tried using the iteration method to unroll the recurrence. With the original recurrence relation, this works out nicely; here, we got stuck at the same summation given above.   I'm completely stuck trying to figure out how to solve this recurrence relation. There's nothing riding on it per se - I don't need to solve it for any particular reason - but the fact that we arrived at it by a straightforward modification of a common algorithms problem set question makes it all the more enticing.  Any idea how to solve this recurrence?  Thanks! I have installed Ubuntu on, but now I am trying to install NVIDIA CUDA Toolkit for the GPU. When I install it, it says that there is no driver found, so I have tried to install the driver, but I don't know why after installation, the system not coming up. Maybe it is because of the wrong version of the driver.   When I tried to detect the driver version using sudo lshw -C display , it showed this information:    But here it doesn't say that what is the product name of the driver, and also when I go Software &amp; Updates -> Additional Drivers, then it says NVIDIA Corporation: Unknown.    What should I do? $$\lim_{n\to\infty} (1+\frac{2}{n})^{n^2} e^{-2n} = ? $$         $e$   $e^2$   $e^{-1}$   $e^{-2}$      My answer is 1. Since $$\lim_{n \to \infty} \left(1+\frac{2}{n}\right)^{n} =e^{+2}.$$  Therefore $$\lim_{n\to\infty} \left(1+\frac{2}{n}\right)^{n^2} =e^{2n}.$$  Where am I wrong? What are the salary and perks of Captain America? Brooklyn is Captain America's home, but times have changed since he was kid 8-9 decades ago. He can't afford a house there, now. This thing really troubled me when he said it in Avengers: Age of Ultron. I mean, I really became sympathetic, despite it being intended for fun.  What's the salary Captain America collects from (fallen) S.H.I.E.L.D. and/or Tony's Avengers organization? Can't they gift him a house in Brooklyn? I mean, he is one of Earth's mightiest heroes who help saved the world lots of times. Where is attribute table in ArcGIS Desktop when not visible? I've just updated to ArcGIS 10.1 for Desktop and when I go to open an attribute table, I can not see it anywhere.   I'm wondering if it is off my screen somewhere or if its just super tiny that I can't see it.   I'm using dual monitors and it's not popping up on either.  Is there a way I can find my attribute table? Formula for prime counting function I saw this formula on  page 2  $$\pi (n)=\sum_{j=2}^{n}\frac{\sin^{2}\left(\pi \frac{(j-1)!^{2}}{j}\right)}{\sin^{2}(\frac{\pi }{j})}$$  Where $\pi(n)$ is the prime counting function. Is this true? How to prove it? I just read a description of CVE-2020-1971, which indicates that a patch was released today for openSSL, the patched version is 1.1.1i -- my version of Ubuntu 20.04.1 LTS is running openSSL version 1.1.1f.  From last March. Software Updater says that my system is up to date.  I am hoping that this openSSL update is released by Canonical by tomorrow... can anyone confirm they are working on this? Order of an element in a finite cyclic group Whenever I access Stack Exchange the tab appears busy Javascript get date in format Difference between standard beta and unstandard beta distributions? What is the difference between standard beta and unstandard beta distributions? How to understand in an article if it is not described if it is standard or not? At work we are being asked to create XML files to pass data to another offline application that will then create a second XML file to pass back in order to update some of our data. During the process we have been discussing with the team of the other application about the structure of the XML file.   The sample I came up with is essentially something like:  &lt;INVENTORY&gt;    &lt;ITEM serialNumber="something" location="something" barcode="something"&gt;       &lt;TYPE modelNumber="something" vendor="something"/&gt;     &lt;/ITEM&gt; &lt;/INVENTORY&gt;   The other team said that this was not industry standard and that attributes should only be used for meta data. They suggested:  &lt;INVENTORY&gt;    &lt;ITEM&gt;       &lt;SERIALNUMBER&gt;something&lt;/SERIALNUMBER&gt;       &lt;LOCATION&gt;something&lt;/LOCATION&gt;       &lt;BARCODE&gt;something&lt;/BARCODE&gt;       &lt;TYPE&gt;          &lt;MODELNUMBER&gt;something&lt;/MODELNUMBER&gt;          &lt;VENDOR&gt;something&lt;/VENDOR&gt;       &lt;/TYPE&gt;    &lt;/ITEM&gt; &lt;/INVENTORY&gt;   The reason I suggested the first is that the size of the file created is much smaller. There will be roughly 80000 items that will be in the file during transfer. Their suggestion in reality turns out to be three times larger than the one I suggested. I searched for the mysterious "Industry Standard" that was mentioned, but the closest I could find was that XML attributes should only be used for meta data, but said the debate was about what was actually meta data.  After the long winded explanation (sorry) how do you determine what is meta data, and when designing the structure of an XML document how should you decide when to use an attribute or an element? Incompatibility between TikZ and babel $\mathbb{F}_p[X]/(X^2+X+1)$ is a field iff $p \equiv 2 \bmod 3$ Let $p$ be a prime. Prove that $\mathbb{F}_p[X]/(X^2+X+1)$ is a field iff $p \equiv 2\bmod3$.    So:   If $p \equiv 2 $ mod $ 3$, I have to show that every element of $\mathbb{F}_p[X]/(X^2+X+1)$ has an inverse.  If $\mathbb{F}_p[X]/(X^2+X+1)$ is a field I have to show $p \equiv 2 $ mod $ 3$. I was having a discussion with a friend of mine about some normal group properties and then came up with the question "if G is not commutative, then is there always a subgroup that is not a normal subgroup?" It's probably more easy to solve this in the following form:$$\forall H \leq G : H  \lhd G \Rightarrow \forall a,b \in G : ab=ba$$ My question is, can anybody give a proof, or a counter-example (because I don't think it holds) of this theorem? Thanks! What is the best way to return an error from a function when I'm already returning a value? Make an object invisible to a specific camera? (Cycles) In an hydrogen-like atoms the orbitals are solutions to the Schrodinger equation suitable for the problem.  They describe the regions where an electron can be found.  So, why don't they have spherical symmetric?  As in this picture:    Why, in a certain distance from the nucleus, are there points where an electron can be found, and points where it can't?  EDIT: Is there any intuation about the symmetry breaking? Should you declare methods using overloads or optional parameters in C# 4.0? I was watching , and it got me thinking about when optional parameters are available in C# what is going to be the recommended way to declare methods that do not need all parameters specified?  For example something like the FileStream class has about fifteen different constructors which can be divided into logical 'families' e.g. the ones below from a string, the ones from an IntPtr and the ones from a SafeFileHandle.  FileStream(string,FileMode); FileStream(string,FileMode,FileAccess); FileStream(string,FileMode,FileAccess,FileShare); FileStream(string,FileMode,FileAccess,FileShare,int); FileStream(string,FileMode,FileAccess,FileShare,int,bool);   It seems to me that this type of pattern could be simplified by having three constructors instead, and using optional parameters for the ones that can be defaulted, which would make the different families of constructors more distinct [note: I know this change will not be made in the BCL, I'm talking hypothetically for this type of situation].  What do you think? From C# 4.0 will it make more sense to make closely related groups of constructors and methods a single method with optional parameters, or is there a good reason to stick with the traditional many-overload mechanism? how to replace multiple strings together in Oracle