Let $n=2^k$ and $T(2)=1$. Then, if we define $T(n)=\sqrt{n}(T(\sqrt{n}))+n$ for $n&gt;2$, then what would be the exact solution of the recurrence?  The master theorem might not work here, and guessing a solution seems hard. In addition, I suspect $n=2^{2^k}$, for otherwise, we get irrational values. I think $T(n)=O(n\log n)$. But, is a better bound possible? Thanks beforehand.
$$T(n) = \sqrt{n} T \left(\sqrt n \right) + n$$  Master method does not apply here. Recursion tree goes a long way. Iteration method would be preferable.  The answer is $Θ (n \log \log n)$.  Can anyone arrive at the solution.
What are the Alternatives For RemasterSys? Recently my wife and I traveled to Italy and before we left we got about 85% of our money in travelers checks and the other 15% we carried on in cash. When we arrived there, no one would take our travelers checks and many places did not accept credit cards. Eventually, we found some currency exchange places that would take the travelers checks and give us Euro's, but at a 7-15% fee, which was not really a very fun surprise.   While I was under the impression that it is safer to use travelers checks, it seems that they were not really accepted (the usual explanation being fraud). So the question is, is it best to just exchange your currency at home and carry all of it with you when you travel or are there other alternatives to carrying cash that are safer and cost effective? As per title, I wish to know if I can, after a successful 32bit setup and consequent software installations, update to a 64bit version.  I know how to partition (actually one of the solutions is to set /, /etc, /home, /var/www, and /opt as separate partitions) and I know that a clean install is way better than a dirty one, yet I would like to know if/how it's possible to do that. All of my Firefox extensions were spontaneously and simultaneously disabled (converted to Legacy status) a few moments ago.  How can I re-enable them? Firefox 66.0.3 (64-bit) Change Folder Permissions Why can't I update my Nexus 4 to Lollipop? As I understand it, Android Lollipop (5.0) is now generally available over-the-air. My Nexus 4 briefly offered for an hour or two to install it. I said yes, and watched it start to download. When I next went to unlock my phone, it obviously wasn't installed. Now, when I go into the Settings app, and manually check for updates, it says my phone is up-to-date. It's still on Android 4.4.4.  I don't particularly want to install the developer image, as that'll wipe all the data from my phone. Is there any way I can encourage it to update?  I'm in the UK, if it's relevant. How do we escape the cycle of bad college eating? Mouseover text: And yet I never stop thinking, 'sure, these ingredients cost more than a restaurant meal, but think how many meals I'll get out of them! Especially since each one will have leftovers!'    Randall Munroe really hit the nail on the head with this comic and especially the mouseover text. There's this sort of cycle that I seem to follow when I try cooking, and I need to get my way out of it. It goes like this:   Decide I should cook something. Thumb through my recipe books, find one or two recipes I like. Write down the ingredients for the recipes in a list. Go to the grocery store, stumble around for a while and eventually find exactly those ingredients. Go back home; at some point in time soon after, cook each recipe. Whew, that was a lot of work! Spend the next week or two eating junk again, until I decide I should cook something. (repeat)   Very similar to the comic; sometimes I too let the leftovers go bad (usually not). And every time I rack up a huge bill at the supermarket and think to myself it's justified because of the leftovers or the amount of food. Not to mention, that's a lot of work for just a couple of meals, and a long list of ingredients.  I love cooking fresh meals but that's just so much overhead and so time consuming! So I always resort back to Easy Mac, Ramen, frozen dinners and frozen pizza. Yuck.  How can I escape this cycle and cut down on the overhead? How can I focus on more than 2 recipes at a time without buying giant lists of ingredients and spending tons of money? Is it really worth the trouble or should I just chock it up to being a college student and wonder about the years ahead of me where I'll have much more time to spend cooking one meal at a time? Let $R$ be a commutative ring with identity, and $S$ a multiplicative subset of $R$ not containing $0$. Which step in my reasoning is wrong?   Every element of $S$ is a unit in $S^{-1}R$. Every element of $S$ that's a zero divisor in $R$ is a zero divisor in $S^{-1}R$. But zero divisors can't be units. Therefore it's not possible to localize $R$ at $S$. I have three partitions on my drive, arranged something like the following:   ___________ | 1 |  2 | 3| -------------   I wish to merge partitions 1 and 3.  Is there a way to do this, or do I have to manually "slide" partition 2 over to partition 3, and then merge the newly open space where partition 2 used to be?  Note: I do not have the space to copy partition 2 to partition 3 (2 is 200GB and 3 is 65GB). ... or should I offer them a separate formular/window where the user can fill the data to the related person?  Its about entering marks/grades of pupils for a certain subject. So sliding to the wrong line could be bad or good for the one or other pupil...  A separate Window could help, but the productivity is smaller.  What do you think? Is there an equivalent of computation of physical processes in nature? I was watching a waterfall in the Austrian Alps. There were thousands of water droplets falling down, splattering on the stones below. I thought - how does nature find out so quickly where each droplet of water should go? To find out what happens to a falling droplet of water, one can use the laws of motion and calculate the trajectory. To calculate, one needs some amount of time, some machine (the brain, computer) and some energy to feed the machine. Does it make sense to ask what is the equivalent of this computation in nature? How does nature find out so quickly how things should move? More generally, to find out how anything should happen? Where is the &quot;calculation&quot; in nature? There's no room or energy for a machine in the particles that make up things. Why is the mean function in Gaussian Process uninteresting? I uninstalled and re-installed steam. I didn't uninstall my games, and they still exist on my hard drive (I checked).  I have an SSD and a HDD, and to save space I install programs (including steam) to C:\Program Files (x86)\Steam drive (SSD) and games to D:\Games\SteamApps (HDD). How to I let steam know the programs are installed there? I need to find the intersection point of 2 line segments (lines are finite, i.e., they have end points).   e.g. segment 1 from $(x_1, y_1)$ to $(x_2, y_2)$ -- segment 2 from $(x_3, y_3)$ to $(x_4, y_4)$  you can assume $m_1$ and $m_2$ are the gradients of segment 1 and segment 2 respectively  similarly, $c_1$ and $c_2$ being the y-intercepts of segment 1 and segment 2 respectively  Using $y=mx+c$ I can easily find the equations of both lines and then derive the intersection point from those equations and check if that point actually lies on both the line segments (intersection point may not lie between the end points).  My problem is when I calculate the gradients $m_1$ and $m_2$ I do $\frac{y_2-y_1}{x_2-x_1}$ there for if one of the lines is vertical then I am going to have problems. How can I deal with this?  Another problem is that if the two segments intersect at a point which is also the same as one of the end points then I want to assume that they are not intersecting.  e.g. if segment 1 is from $(2, 3)$ to $(3, 7)$ and segment 2 is from $(3, 7)$ to $(7, 3)$ then I want to assume they don't intersect. Proving that the given set has at most $2^{n-1}$ elements Since there are  complex numbers (2 dimensions) and quaternions (4 dimensions), it follows intuitively that there ought to be something in between for 3 dimensions ("triernions").   Yet no one uses these. Why is this? I am using PHP and JavaScript. My JavaScript code contains a function, get_data():  function get_Data(){     var name;     var job;     .....      return buffer; }   Now I have PHP code with the following.  &lt;?php     $i=0;     $buffer_data;      /* Here I need to get the value from JavaScript get_data() of buffer;        and assign to variable $buffer_data. */ ?&gt;   How do I assign the JavaScript function data into the PHP variable? How do I extract all the external links of a web page and save them to a file? It seems that the question has eventually become a series of questions....    An example goes as:     The 1st and (the) 2nd paragraphs of the article are extremely long.   Another example:     What are the situation, (the) task and (the) result of your story?   Is it necessary to use the in ()? How to work with "FIFO" in C# .NET?