I thought about Stirling equation for the solution but I couldn't get the answer.  How i need to approach in order to solve that?
I have been trying to show these two inequalities hold for all positive integers n, but I don't know how to proceed at all... I have tried playing around with them but I didn't find anything helpful. I have also tried induction but I couldn't make it work.
CUDA Driver API vs. CUDA runtime When writing CUDA applications, you can either work at the driver level or at the runtime level as illustrated on this image (The libraries are CUFFT and CUBLAS for advanced math):   (source: )    I assume the tradeoff between the two are increased performance for the low-evel API but at the cost of increased complexity of code. What are the concrete differences and are there any significant things which you cannot do with the high-level API?  I am using CUDA.net for interop with C# and it is built as a copy of the driver API. This encourages writing a lot of rather complex code in C# while the C++ equivalent would be more simple using the runtime API. Is there anything to win by doing it this way? The one benefit I can see is that it is easier to integrate intelligent error handling with the rest of the C# code. How can I encrypt and decrypt a string in C#? How to solve the equation $\phi(n) = k$? The stress tensor can be written as: $$\sigma_{ij}=-p \delta_{ij}+\sigma'_{ij}\label{1}\tag{1}$$ where $\sigma_{ij}'$ is called the extra stress tensor.   From what I understand pressure is pressure is force per unit area, so that the force acting on the surface $d\vec A$ is given by:  $$\vec F=-p d\vec A\label{2}\tag{2}$$ (This has the effect that the force on an infinitesimally thin surface due to a continuous pressure is zero). However, $\sigma_{ij}$ represent the force in the $i$ direction on a surface with normal in the $\hat e_j$ direction. So the force acting on a surface $d \vec A$ must be: $$F_i=\sigma_{ij}dA_j\label{3}\tag{3}$$ Equations (2) and (3) clearly do not agree (even if we look soley at the force in the $d\vec A$ direction they don't agree). So my question is what is the exact meaning of pressure in expression (1) and why do we not have: $$\sigma_{ii}=-p$$ I'm stuck with the proof of the following:  Suppose that $(s_n)$ converges to $s$, $(t_n)$ converges to $t$, and $s_n \leq t_n \: \forall \: n$. Prove that $s \leq t$.   I've tried starting with $s_n \leq t_n \: \forall : n$ and the definitions of each limit (i.e. $|s_n - s| \leq \epsilon \: \forall \: n &gt; N_1$), but I'm not really getting very far. Any help is appreciated! How to merge two arrays in JavaScript and de-duplicate items This is Ubuntu server 10.04 64 and samba 3.4.7.  I have a shared directory /home/mit/share and another one /home/temp that I link into the shared one:  ln -s /home/temp /home/mit/share/temp   But on windows, after using internet, I cannot open S:/temp, but on Linux it is possible to access /home/mit/share/temp like expected.   This works if I link directories inside /home/mit/share/temp, so I guess samba is restricting to jump with a link outside/above the shared directory.   EDIT:  See also this question titled .  It seems best to put unix extensions = no into the global section and follow symlinks = yes and wide links = yes only into the shares section, where you really need it.  The unix extension flag has to live in the global section and not in the individual shares sections. But for security reasons it is better to use the other options only where you need it, and not globally. How can I merge properties of two JavaScript objects dynamically? Is it OK to borrow a ground wire from a different circuit? I've got a circuit in the basement carrying a handful of outlets.  This circuit has no ground wire.  I want to add a couple of outlets and want them to be grounded.  I may even want gfci outlets on them in the future.  I have no reasonable way of getting a ground wire to the panel but I do have easy access to another circuit that does have a ground wire.  Is it acceptable to borrow or share a ground wire between circuits?  If not, why not?  Both circuits are 20A with 12 AWG wire. Database does not update automatically with MySQL and Python I'm having some trouble updating a row in a MySQL database. Here is the code I'm trying to run:  import MySQLdb  conn=MySQLdb.connect(host="localhost", user="root", passwd="pass", db="dbname") cursor=conn.cursor()  cursor.execute("UPDATE compinfo SET Co_num=4 WHERE ID=100") cursor.execute("SELECT Co_num FROM compinfo WHERE ID=100") results = cursor.fetchall()  for row in results:     print row[0]  print "Number of rows updated: %d" % cursor.rowcount  cursor.close() conn.close()   The output I get when I run this program is:     4Number of rows updated: 1   It seems like it's working but if I query the database from the MySQL command line interface (CLI) I find that it was not updated at all. However, if from the CLI I enter UPDATE compinfo SET Co_num=4 WHERE ID=100; the database is updated as expected.  What is my problem? I'm running Python 2.5.2 with MySQL 5.1.30 on a Windows box. Replacing non-English characters in attribute tables using ArcPy and Python? I have a few shapefiles where some of the attributes contain the non-English characters ÅÄÖ. Since some queries doesn't work with these characters (specifically ), I tried to change them in advance with a simple script and add the new strings to another field.  However, change in characters works fine but not updating the field with arcpy.UpdateCursor.   What is an appropriate way of solving this?  I have also tried to do this via the Field Calculator while posting "code" to the codeblock, with the same error.  Error message: Runtime error  Traceback (most recent call last):   File "", line 1, in    File "c:/gis/python/teststring.py", line 28, in      val = code(str(prow.Typkod)) UnicodeEncodeError: 'ascii' codec can't encode character u'\xc4' in position 3: ordinal not in range(128)  Code:  # -*- coding: cp1252 -*- def code(infield):     data = ''     for i in infield: ##        print i         if i == 'Ä':             data = data + 'AE'         elif i == 'ä':             data = data + 'ae'         elif i == 'Å':             data = data + 'AA'         elif i == 'å':             data = data + 'aa'         elif i == 'Ö':             data = data + 'OE'         elif i == 'ö':             data = data + 'oe'         else:             data = data + i     return data   shp = r'O:\XXX\250000\DB\ArcView\shape.shp'  prows = arcpy.UpdateCursor(shp)  for prow in prows:     val = code(unicode(str(prow.Typkod), "utf-8"))     prow.Typkod_U = val     print val     prows.updateRow(prow)   The values of Typkod are of the type: [D, D, S, DDRÄ, TRÄ] etc.  I use ArcMap Basic (10.1) on Windows 7.    New Error message:  Runtime error  Traceback (most recent call last):   File "", line 1, in    File "c:/gis/python/teststring.py", line 29, in      val = code(unicode(str(row.Typkod), "utf-8")) UnicodeEncodeError: 'ascii' codec can't encode character u'\xc4' in position 3: ordinal not in range(128)  &gt;&gt;&gt; val 'DDRÄ' &gt;&gt;&gt; type(val) type 'str'      It appears as if the output from the function is wrong somehow. When there's ÅÄÖ involved it returns data = u'DDR\xc4' and not (as was my intention) data = 'DDRAE'. Any suggestions on what might cause this? Is there a grammatical term for moving a word to the front or back of a sentence? There is a function in Arabic grammar where you may bring forward in sentence order a word - as well as deferring it.  For example: if the sentence order is Subject - Verb - Object, you can bring the object forward to be before the subject,  yielding Object - Verb - Subject or Object - Subject - Verb.  Similarly, if you have Noun Phrase - Verb Phrase, you can move the verb phrase forward so you get Verb Phrase - Noun Phrase.  This is used to add emphasis to the item brought forward. The same applies to moving the item to the end of the sentence.  Is there a grammatical term in English that represents this function or functions? Can Minecraft Pocket Edition play with a desktop Minecraft? My kids are playing Minecraft on 2 PCs, creative mode.  They are connected by the way of one of them starts a single player game on the other one than joins that game over the LAN.  Can I, using Minecraft PE on my iPad, join their game as well? Removed Python 3 and now Ubuntu Software Center, terminal and other applications don't work I am a new Ubuntu user running Ubuntu 13.04.  I've been trying to install and use a DeDRM tool, but I've been having problems doing so. I was following some directions that said I needed a version of Python 2.7, but not anything Python 3.0 or later (because 3.0 versions are missing some necessary libraries?). I tried to install Python 2.7.5.6, but it didn't work. I thought it was probably because I had the later version of Python, so I went through terminal and removed Python 3.3, so I could install the earlier version of Python.  Now that I've uninstalled Python 3.3, a lot of applications no longer work, including the terminal and the Ubuntu Software Center. I have no idea how to fix this problem now. I have an element that already has a class: &lt;div class=&quot;someclass&quot;&gt;     &lt;img ... id=&quot;image1&quot; name=&quot;image1&quot; /&gt; &lt;/div&gt;  Now, I want to create a JavaScript function that will add a class to the div (not replace, but add). How can I do that? SCP through multiple hosts describes how to SSH into a machine that can only be accessed through one or more hosts (that can also be SSH'd into).  Is there an equivalent way of using scp? That is, copy a file to my local machine from the far machine (assuming I have password or key access to all machines along the way)? I deleted the "/etc/passwd" file and cannot log in I was trying to change my username and also my home directory (/home/username) and my system started to crash. I deleted the passwd file but I had a backup named passwd_bkp. I tried to rename this passwd_bkp as passwd and it did not work. No commands were being executed... I was in a terminal window.   I restarted my system and now I cannot log in. GRUB gives two options: Linux and recovery mode.   I tried to open a sessions as root but it says the filesystem is corrupted. I cannot access my files.  Did I lose all my files? I'm writing a web app using Twitter Bootstrap with PHP on server side. My app makes regular ajax requests via jQuery and i want to disable certain parts of the UI until the server completes processing and hands back the control to Client-Side.  I wonder what would be best practices for the same? I want to determine $$\lim\limits_{(x,y) \to (0,0)} \frac{x^2·y^3}{x^4+y^6}$$  I'm sure the limit exists, it's zero because I tried to find other different limits in line and parabola points ($(x,mx)$ and $(x,mx^2)$) but every time I got zero. So, I have to use the definition or sequences to confirm. What is the command to add newline linebreak in \sum \prod? I am sure this question or similar has been asked but I cannot find it by searching so adding it to make it more accessible: what is the command/environment to add newline linebreak in \sum \prod so the expression is on two rows instead of one line hard to read?    MWE  \documentclass{article} \usepackage{amsmath,amsfonts}  \begin{document}  $$\sum^{N_{cut}}_{i=1,\newline e_1\in C_i,e_2\not\in C_i} \prod_{j\in C_i} p_j-\sum^{N_{cut}}_{i=1,e_2\in C_i,e_1\not\in C_i} \prod_{j\in C_i} p_j\geq 0$$  \end{document}