Adding Arabic text to English document
Adding Arabic text to an English document
What makes frictional force the centripetal force of a car turning along a curve? As friction is the opposing force and acts anti-parallel so there is no component of frictional force towards the center,right? Then how can frictional force be centripetal force? Does borrowing from my 401(k) make sense in my specific circumstance? Replace Builtin commands with Custom Commands for CMD.exe If a ring element is right-invertible, but not left-invertible, then it has infinitely many right-inverses. How can I fix apt error "W: Target Packages ... is configured multiple times"? Is full wave rectifier better than half wave one? Image of the tensor product of strict maps of Banach spaces On finding polynomials that approximate a function and its derivative (extensions of Stone-Weierstrass?) The Stone-Weierstrass Theorem tells us that we can approximate any continuous $f:\mathbb{R}^n\to\mathbb{R}$ arbitrary well on a compact subset of $\mathbb{R}^n$ by some polynomial. Suppose that $f$ is continuously differentiable, under what conditions can we guarantee we can find a polynomial $p$ such that $p$ approximates arbitrarily well $f$ and the partial derivatives approximate of $p$ arbitrarily well those of $f$ (on a compact subset of $\mathbb{R}^n$)? References are welcome. See below for my (probably naive) approach to the question.    It is straightforward to find conditions if $n=1$: Let $K:=[a,b]\subseteq\mathbb{R}$ and let $K_+$ denote some open cover of $K$. Suppose that $f:K_+\to\mathbb{R}$ is continuously differentiable and let $f'$ denote its derivative. Then, by Weierstrass's approximation Theorem we have that there exists a polynomial $q$ such that   $$||f'(x)-q(x)||\leq \varepsilon$$  for all $x\in K$. Choose some $y\in K$. For any $x\in K$ define  $$p(x):=f(y)+\int_y^xq(t)dt.\quad\quad(*)$$  By the Fundamental Theorem of calculus we have that $p'=q$. So for any $x\in K$  $$||f(x)-p(x)||=\left|\left|\int_y^x (f'(t)-q(t))dt\right|\right|\leq \int_y^x ||f'(t)-q(t)||dt\leq \varepsilon||x-y||\leq \varepsilon (a-b).$$  However, the above argument doesn't work in higher dimensions because the analogue of $(*)$ does not imply that $\nabla p = q$ (since not every vector of polynomials spans a gradient field). Can anyone suggest a way around this? That is, a way to to show that for continuously differentiable $f:\mathbb{R}^n\to\mathbb{R}$ and for any $\varepsilon&gt;0$, we can always find a polynomial $p$ such that   $$||f(x)-p(x)||+||\nabla f(x)-\nabla p(x)||\leq \varepsilon$$  for all $x$ in some compact set $K$? Feel free to add extra conditions on $K$, if it makes life easier. A friend has been kind enough to lend me a DSLR for a few months.  It's my first foray into DSLR photography, or indeed any kind of photography beyond simply aiming my Canon IXUS at something and clicking the button.    Over the last few weeks, I've been busy studying, practising and trying to learn as much as possible ahead of a planned trip to Norway where I hope to see, and get some photos of, the Aurora Borealis.  I currently have:   Canon Rebel Xsi (450D) Canon 50mm f/1.4 L  Canon 18-55m kit lens Aluminium full height tripod with ball head Cable release Spare battery &amp; memory cards   I'm prepared to spend the night wrapped up outdoors, in the hope that they will make an appearance. I'd like some advice on how to get the most out of this kit for the trip:   How should I go about setting up to get pics of the Aurora?  How might I achieve optimum focus? From what I've read, I'll need to use the tripod and cable release, with IS turned off, on continuous shooting mode with long exposure times. But I'd like some more specific tips in terms of which aperture and ISO settings might work best? Should I use one of the lenses above, or would it be worthwhile renting a better, wide angle lens; if so, which would you recommend? Finding roots/zeros for collision detection? For the most simple of 2D games, I have implemented a posteriori collision detection (overlapping rectangles) on the x/y Cartesian plane, but am now interested in understanding the basics of a priori collision detection...  In the link , I noticed the reference to Newton’s Method.  So, I want to better understand the link b/w finding zeros of an equation and detecting a posteriori collision.    Let's start with the most obvious case of finding roots of an equation:  If you have a projectile's trajectory modeled as a quadratic function, you can find the roots to predict at what time the height will equal zero (or any other constant height, actually, by just solving f(x)=C to be f(x)-C=0 and then finding roots of that expression).    So, as I understand, finding the root can tell me the time at which the object will be at any chosen height.  More broadly, we basically know the (x,y) location of the object for any given time.  But how does root finding, in particular, translate to detecting collisions with another object?  (The other object may be stationary, or be moving.  If the latter, do you also determine roots/location of this moving object as well? Calculate the (x,y) trajectory of both objects and determine where they will intersect?)  Again, the basic question is where does root finding come into play?  Any links to a basic primer on this topic, with a simple example would also be greatly appreciated. My Ubuntu version 18.04 brings me a GNU GRUB version 2.02 and I cannot access my entry password panel? Friendly_Id and Reserved Words -- How can i replace the reserved word? For some time, Microsoft has supported an option to remove the limit of 260 symbols for NTFS file name length.  The web is full of articles, describing what should be done.   Hit the Windows key, type gpedit.msc and press Enter. Navigate to Local Computer Policy > Computer Configuration > Administrative Templates > System > Filesystem > NTFS. Double click the Enable NTFS long paths option and enable it.   However, I don't have such option. Please see the screenshot below:    Am I missing some Windows update, or is it something else? 1) "The book 'The Three Musketeers' is a wonderful example of..." Here we have a proper noun, a title that happens to end in a plural, and I have no sense that the verb should be plural. "Musketeers"/is seems proper/natural/correct, and 'are' not at all.   However, such certainty begins to wane in other examples: 2) "'The Heidi Chronicles' is a wonderful example of..." doesn't completely shake my confidence, but it moves the needle, at least a little bit. But, practically speaking, I wouldn't have a second thought about correcting a student who used 'are' instead of 'is'.  Here's the one that really bothers me: 3a) " 'The Basement Tapes' are a wonderful example of..." I don't think this sounds any less proper/natural/correct than 3b) " 'The Basement Tapes' is..." In fact, 3a seems less awkward than 3b.  Why? A friend asked me last week how to enumerate or list all variables within a program/function/etc. for the purposes of debugging (essentially getting a snapshot of everything so you can see what variables are set to, or if they are set at all). I looked around a bit and found a relatively good way for Python:   #!/usr/bin/python                                                                                                                                                                                                                            foo1 = "Hello world" foo2 = "bar" foo3 = {"1":"a",         "2":"b"} foo4 = "1+1"  for name in dir():     myvalue = eval(name)     print name, "is", type(name), "and is equal to ", myvalue   which will output something like:   __builtins__ is &lt;type 'str'&gt; and is equal to  &lt;module '__builtin__' (built-in)&gt; __doc__ is &lt;type 'str'&gt; and is equal to  None __file__ is &lt;type 'str'&gt; and is equal to  ./foo.py __name__ is &lt;type 'str'&gt; and is equal to  __main__ foo1 is &lt;type 'str'&gt; and is equal to  Hello world foo2 is &lt;type 'str'&gt; and is equal to  bar foo3 is &lt;type 'str'&gt; and is equal to  {'1': 'a', '2': 'b'} foo4 is &lt;type 'str'&gt; and is equal to  1+1   I have so far found a partial way in PHP (courtesy of ) but it only lists all variables and their types, not the contents:   &lt;?php // create a few variables $bar = 'foo'; $foo ='bar'; // create a new array object $arrayObj = new ArrayObject(get_defined_vars()); // loop over the array object and echo variables and values for($iterator = $arrayObj-&gt;getIterator(); $iterator-&gt;valid(); $iterator-&gt;next())         {         echo $iterator-&gt;key() . ' =&gt; ' . $iterator-&gt;current() . '&lt;br /&gt;';         } ?&gt;   So I put it to you: how do you list all variables and their contents in your favorite language?    Edit by : I propose this question follows the spirit of a little "". If you do not agree, just edit and remove the tag and the link. When I reinstall Windows XP by mistake I deleted all five partitions. I did not format them yet though.  How can I recover the partitions? When I attach the hard disk to another PC it just says that the partition is unformated and asks "Would you like to format it?". How to include graphics with spaces in their path? I have a COM dll written in vb6. When I try to create a new object of a class module from this dll I get a Run timer error 430: Class does not support automation or does not support expected interface. The interesting thing is that this happens only from outside the IDE, when I am debugging from within the IDE there is no error thrown and the new object of the class is created successfully. What can be the cause?  In general I occasionally get these kind of errors in COM dlls. What is the best way to debug COM issues? How can I know the path of the dll that is being used when a program is running? Writing to NTFS drive in Linux I've got an external USB drive that is formatted with NTFS mounted on ~/tmp.  I can read it just fine, but cannot write to it, even as root (temp is a folder on the USB drive):  [root@Plugbox temp]# touch test touch: cannot touch `test': Permission denied [root@Plugbox temp]#   However, if I do a cat /proc/mounts, it looks like I do have NTFS read/write support:  [root@Plugbox temp]# grep /dev/sdb1 /proc/mounts /dev/sdb1 /root/tmp ntfs rw,relatime,uid=0,gid=100,umask=02,nls=utf8,errors=continue,mft_zone_multiplier=1 0 0 [root@Plugbox temp]#   Finally, if I go to the folder above temp, it looks like I should have plenty of permissions to write to the folder:  [root@Plugbox tmp]# ls -al total 61500764 ... drwxrwxr-x 1 root users           0 Apr 19  2010 temp ... [root@Plugbox tmp]#   What am I missing? How to change desktop background from command line in Unity? I have a script that updates my desktop background every day with the NASA APOD. Since upgrading to 11.10, that script stopped working, presumably because I'm using gconftool-2 to actually update the desktop background. Is there a new way to do this under Unity?