Can we say that infinity equals to infinity In real analysis, we know that the cardinal number of a certain set can be infinite and these infinite can be compared following continuum hypothesis. But for a function, such as $f(x)= x, g(x)=x^{2}$, when x goes infinity, are these infinity equal to each other?
Which infinity is meant in limits? For example, when we write $\lim_{x\rightarrow \infty} f(x)$ - which infinity is meant and why? Countable? If uncountable - which and why?
Let $f \in C [a,b]$ such that $\forall x\exists y$ such that $|f(y)| \leq \frac{1}{2}|f(x)|$. Prove that $\exists c\in [a,b]$ such that $f(c) = 0$. How to trim whitespace from a Bash variable? How is Dru-Zod/General Zod still alive after this character died? When trying to execute a symbolic link which points to an exe file (on Windows 7), the "Open with" windows appears. When renaming it to a ".exe" file, it says "The specified path does not exist. Check the path, and then try again.".  Symlinks to other destinations seem to work fine.  So is there any way to execute a symlink?  EDIT: In the command prompt, I execute the following and the MyProgram.exe runs both times.  However, if I double-click on the symbolic link in Windows Explorer it fails with the above error message.  C:\Temp&gt;mklink SomeProgram.exe MyProgram.exe symbolic link created for Backup.exe &lt;&lt;===&gt;&gt; Backup-1.0.1.exe  C:\Temp&gt;SomeProgram.exe  C:\Temp&gt;del SomeProgram.exe  C:\Temp&gt;mklink SomeProgram.exe C:\Temp\MyProgram.exe symbolic link created for Backup.exe &lt;&lt;===&gt;&gt; C:\Temp\Backup-1.0.1.exe  C:\Temp&gt;SomeProgram.exe Animations about a young man pulled into a fantasy setting with dragons? I'm looking for an animation (more likely a film not a series) that I watched as a kid, watched early 90's I'm sure it contained a book loving male main character (wears glasses might have even had an amulet) who gets pulled into a fantasy via a magic book  Setting with dragons and an evil wizard, I remember it being on TV as I never had it on VHS.  Same sort of animation technique as visionaries or thundercats.  Thanks. Why do investors pay trillions for minority stakes in companies, when their only potential payback is modest uncertain dividends? I'm struggling to understand the core reason to invest in stocks as a minority shareholder. Majority shareholding makes more sense as you effectively control the company, so you can pay yourself dividends or even sell the entire company at will. You can guarantee your own payback.  However, as a minority shareholder, you don't get to make any of these decisions. You're basically just hoping to eventually, some day get paid, if the majority holders decide so.  Why do so many investors agree to pay now huge sums of money (trillions in total) for something that may eventually pay dividends, and even in the best case, these dividends are only a fraction of the price you paid to acquire these stocks?  The average annual dividend rate for the US stock market is 2%. That means if you bought 100 million worth of shares, you'd have to wait 50 years to recover your investment.  Of course, the most probable way you'd eventually get liquidity for these shares is by selling them to some other starry-eyed investor - which makes the whole thing seem like a giant Ponzi scheme.  Given all that, it's unclear to me why the public is willing to spend so much money to acquire minority shares in companies. At what time intervals does ntpd update the time? I'm running the latest ntpd.   When I start ntpd my system time synchronizes with ntp server's ntp. After synchronizing I changed my system time manually using date command date -s '1997-02-22 12:00:00'   My system time got changed as per the date command.   NTPD is still running, I want to know at what time interval my system time will sync with the internet via ntp. Tried to install both Windows and Ubuntu, now neither will boot Hardware:  I've got an SSD (c),an internal HDD (d), an external HDD (e) and a USB flash drive (f).   How this all went down:   I downloaded Ubuntu and used a program to install it on my (F) drive. I reset my computer, ran the (F) drive for boot, and installed Ubuntu. I reset my computer, and it forcibly automatically booted into Windows 10 instead of Ubuntu. I went into Windows 10, and used a program called Easy BCD, by NeoSmart. I followed their guide, but after step 6, it wasn't the same for me. Their guide said that upon rebooting, I should see a grub screen that lets me choose between Windows/Linux and a few other options, however my computer would only boot into Windows 10.  I decided to follow the rest of the guide regardless, however by the end it would let me boot 'windows' or 'linux', but the linux option would always break. It went to a screen with lots of commands, that I thought -may- be a barebones Linux or something, but it definitely wasn't Ubuntu.  I decided to reinstall Ubuntu using the USB flash drive. Upon resetting, no matter what hard drive or format I select for booting (besides the F drive), it says An operating system wasn't found. Try disconnecting any drives that don't contain an operating system. I can boot the F drive and run Ubuntu, but it's a one time version that doesn't save any of my data. When I went into the installation, it still says that I already have Ubuntu installed on the 4TB HDD and that I have Windows 10 installed on the SSD, however I cannot seem to boot into either of them.   TLDR: Windows 10/Ubuntu are both installed, Windows on the internal SSD and Ubuntu on the external HDD. Ubuntu's installation program on the (F) drive recognizes them, however, when I try to boot anything it says An operating system wasn't found.   When I press F11 and force it to boot from the SSD it says:  error: no such device: 83b6c64d-9a2c0467b-a02c-6374ee6ac3dc.  error: unknown filesystem. Entering rescue mode. . .  grub rescue&gt; _   and it will allow me to type some commands. "At" vs. "in" before verb In a document I found the following sentence:     listeners are more accurate at understanding speech spoken in their own accent...   Would it be an error to use "in" instead of "at"?  Actually in this case "at" sounds better than "in", but in general before verb is "at" always used? Nth derivative of $x^x$ A few days ago, I was wondering about a series expansion for $$f(x) = \  x^x$$ so I tried taking a couple derivatives. From these I was able to guess at some formulas for the general coefficients of the first few terms of the nth derivative as well as the last term (where I put the terms of the derivative in the order  $$f^{(n)}(x) = c_1x^x + c_2x^{x-1} + c_3x^{x-2}\ + \ ... \ +\ c_nx^{x-(n-1)}$$  Please let me know if there is a general formula for the coefficients as a function of n. In my own fooling around, I found that $$\ln(x)+1$$ appears quite a bit, so I called this $a$ in my work. Since I would like to center my expansion at $1$, and this expression evaluated to $1$, at $x=0$, I was interested in writing my formulas as functions of $n$ and $a$. I found that you get something of the form (this could very well be completely incorrect):  $$f^{(n)}(x) = ax^x + a^{n-2}{\binom n2}x^{x-1} + a^{n-4}(3{\binom n4}-a{\binom n3})x^{x-2}\\ + a^{n-6}(15{\binom n6}-10{\binom n5}a+2{\binom n4}a^2)x^{x-3}\ + \ ... \ +\ (-1)^n(n-2)! \ x^{x-(n-1)}$$  An interesting property that seemed to recur for the various coefficients was  that once I found a recursion formula for the coefficient in question and plugged in my initial value, it always seemed to be the same as taking the integral of the previous coefficient with respect to $a$. That is to say, if I plugged in the nth coefficient (a polynomial of $a$) into my recursion formula for the $(n+1)$th coefficient, this gave me the same result as integrating. In general, as reflected by the few coefficient formulas above,  the formula for the coefficient of $$x^{x-c}$$ where c is an integer such that $$0\le c&lt;n$$ "Stabilized" to a polynomial of a with c terms after the first few applications of the recursion formula. This stabilization occurred when $$n=2c$$ Also, as a direct result of the tentative and incomplete formula I have written above, the first nonzero-value for the coefficient of $$x^{x-c}$$ is $$(-1)^{c+1}(c-1)!$$ and appears in $$f^{(c+1)}(x)$$ If these various properties are indeed true, can anyone explain why this would be the case? I am aware that this function can be expanded using the expansion for $$e^x$$ with $$x\ln(x)$$ in the exponent. This does not strike me as particularly enlightening (let me know if I'm missing something). I am most interested in the coefficients on the binomial coefficients, since these are the only things that stand between me and a general formula for the coefficients (since the form is quite standard). These are: $$1$$ then $$3,-1$$ then $$15, -10,2$$The sum of these coefficients should also give the nth derivative evaluated at zero since $$f(1) = 1$$ and $$a(1) = 1$$  Anyway, please let me know what you think! I am currently stuck due to an inability to brute force my way any further and a complete lack of cleverness. My knowledge of mathematics is quite elementary, so please explain any higher level concepts if they are necessary to explain this problem. A feature of C++ is the ability to create unnamed (anonymous) namespaces, like so:  namespace {     int cannotAccessOutsideThisFile() { ... } } // namespace   You would think that such a feature would be useless -- since you can't specify the name of the namespace, it's impossible to access anything within it from outside. But these unnamed namespaces are accessible within the file they're created in, as if you had an implicit using-clause to them.  My question is, why or when would this be preferable to using static functions? Or are they essentially two ways of doing the exact same thing? I have a hazy notion of some stuff in differential geometry and a better, but still not quite rigorous understanding of basics of differential topology.  I have decided to fix this lacuna once for all. Unfortunately I cannot attend a course right now. I must teach myself all the stuff by reading books.  Towards this purpose I want to know what are the most important basic theorems in differential geometry and differential topology. For a start, for differential topology, I think I must read Stokes' theorem and de Rham theorem with complete proofs.   Differential geometry is a bit more difficult. What is a connection? Which notion should I use? I want to know about parallel transport and holonomy. What are the most important and basic theorems here? Are there concise books which can teach me the stuff faster than the voluminous Spivak books?  Also finally I want to read into some algebraic geometry and Hodge/Kähler stuff.  Suggestions about important theorems and concepts to learn, and book references, will be most helpful. I would like to make a sum of all dropdown values into a label text box. I am using following formula, but I do not understand where am I going wrong. Can anyone assist? I have also added a screenshot if that helps anyhow. Note: scores is the table name the dropdown values feed from. Sum(scores, Dropdown3.Selected.Value + Dropdown3_1.Selected.Value + Dropdown3_2.Selected.Value) I have the following recurrence that I've been pounding on:  $$ a(0)=1\\ a(1)=1\\ a(2)=2\\ a(2n)=a(n)+a(n+1)+n\ \ \ (\forall n&gt;1)\\ a(2n+1)=a(n)+a(n-1)+1\ \ \ (\forall n\ge 1) $$  I don't have much background in solving these things, so I've been googling around looking for different techniques. Unfortunately, I haven't been able to correctly apply them yet. So far I've tried:   Characteristic equations Generating functions Plugging it into Wolfram Alpha Telescoping Observation   I'm sure that one or more of these is the way to go, but my biggest problem right now is figuring out how to deal with the idea that there are different equations for odd and even values of $n$. I'm not necessarily looking for a solution (though it would be gratefully accepted), but any nudges in the right direction would be much appreciated.    To follow up, it turned out that the speed problem described in my comment, below was related to the Decimal implementation in Python 2.7. Switching to Python 3.3 (and Java) made everything many orders of magnitude better. Right-hand sidebar covers the footer on the "ask question" page and markdown help page What's the difference between "I love singing" and "I love to sing"? I would like to know what the difference is between "I love doing something" and "I love to do something".  For example:        “I love singing.”      and,        “I love to sing.”      When I was young, I was told that "love doing something" is right, but one day, I found Jessie J wrote "I love to sing" on her Instagram. Does anybody know how to move the keyboard caret in a textbox to a particular position?  For example, if a text-box (e.g. input element, not text-area) has 50 characters in it and I want to position the caret before character 20, how would I go about it?  This is in differentiation from this question:  , which requires jQuery. Need rtl8814au driver for kernel 5.3 on Ubuntu 19.10 Change all file type associations from one program to another Is it possible to easily change all file type associations that are currently associated with one program to another program in one batch?  In my case, I am trying to change all files types currently associated with Notepad to instead use Notepad++ in Windows 7.  It's a pain to go through Default Programs &gt; Set Associations because after each type the entire list re-populates, taking several seconds.  Is there a way I can do a search and replace in the registry, or something even easier? Why can't I destroy or place blocks?