Extra space in Theorems', Lemmas', etc. title
How to remove white space before reference?
Now, I know that it's (relatively) easy to calculate, say, $r^{a+bi}$ (using the fact that, for  $z_1, z_2\in \mathbb{C}, {z_1}^{z_2}=e^{z_2\ln(z_1)}$ and $\ln(z_1$) can just be found using: $\ln(a+bi)=\ln[\sqrt{a^2+b^2}]+i \cdot \arctan(\frac{b}{a})$  ).  Anyway, how would I go about calculating, say, $i^j, k^i$ etc., or, more generally, $(a_1+b_1i+c_1j+d_1k)^{a_2+b_2i+c_2j+d_2k}$  (I know that exponentiating a complex number (to another non-real complex number) produces a non-unique result, so I assume the same would apply further up the hypercomplex ladder; if that's the case, I'm only concerned with the 'principal' value)?  I obviously don't want a general formula or anything like that; just some intuition and a method by which I could calculate such a thing.  And, finally (because I really like to push my luck), can this method for quaternions be extended to higher number systems (i.e. $\mathbb{O, S},$ etc.) to give an analogous result?   Thanks Simple georeferenced 3D data format for building models What solutions does the equation $4x\equiv2\mod10$ have? (Hint, it will have more that one.) What about solutions to an equation $ax\equiv d\mod n$, where $d=\gcd(a,n)$?  This is the question that appeared in our universities last year paper. But it seems very difficult for me like for first part I am able to solve by simplifications I got $x=5\cdot Z +3$ as solution which makes sense and seems fair but for the second one, I have no idea how can I approach that. Can you please help me out with that one. I tried to simplify that by using Euclidean algorithm but still, I didn't get any idea that what actually the solution will be. My final exam is on 2nd of next month and because of covid I am unable to contact in person for any help. So any help from this site will be appreciated. Thank you for looking at this question! Clean install Windows 8.1 or Windows 8 from OEM key Why is it "description" but "describe" (b and p)? I've just wondered by it is "description", but "to describe". It looks as if "to descripe" would be a more consistant choice. Is there any other explanation besides "this is how it is written for years"?  edit: Or alternatively "to describe" and "describtion". Try to do something vs. Try doing something Is there any differences between      Try to do   And     Try doing   Thank you in advence. It would be very useful to be able to overload the . operator in C++ and return a reference to an object.  You can overload operator-&gt; and operator* but not operator.  Is there a technical reason for this? What do I need to know about temperature and food safety? What general rules do I need to follow to keep my food safe? How do I know what temperature to cook something to, or whether my food is safe at room temperature? Continuity and existence of directional derivatives at all directions of a $f:\mathbb{R}^{2} \to \mathbb{R}$ given by $f(x,y)=\frac{2x^2y}{x^2+y^2}$ Let $f(x,y)=     \begin{cases}       \frac{2x^2y}{x^2+y^2} &amp; if &amp; (x,y) \neq (0,0) \\       0       &amp; if &amp; (x,y)=(0,0) .  \end{cases}$  (i) Prove the directional derivatives of $f$ exist in any direction at the point $(0,0)$.  (ii)¿Is $f$ continuous on (0,0)?  For (i) I took $u=(u_{1},u_{2}) \in \mathbb{R^{2}}$ such $\|u\|=1$ and $0=(0,0)$. So $$\lim_{t \to 0} \frac{f(0+ t u)-f(0)}{t}=\lim_{t \to 0}\frac{2(tu_{1})^2 (tu_{2})}{(tu_{1})^4+(tu_{2})^2}=\lim_{t \to 0}\frac{(t^2 u_{1}^2)(t^2 u_{1}^{2}) tu_{2}}{t^{4}u_{1}^{4}+t^{2} u_{2}^{2}}.$$ But I cannot find the limit when $t$ aproaches to $0$. Which by finding them I would prove the directional derivative exist at every direction right?  For (ii) I got the intuition $f$ is not continuous at $(0,0)$ so I took $\lbrace (\frac{1}{k},\frac{1}{k}) \rbrace_{k \in \mathbb{N}}$ and $\lbrace (\frac{1}{k},0) \rbrace_{k \in \mathbb{N}}$ which are two different sequences in $\mathbb{R}^{2}$ converging to $(0,0)$. However, $\lbrace f(\frac{1}{k},\frac{1}{k}) \rbrace_{k \in \mathbb{N}}=\lbrace \frac{1}{k^{2}} \rbrace_{n \in \mathbb{N}} \to 0$ and $\lbrace f(\frac{1}{k},0) \rbrace_{k \in \mathbb{N}} \to 0$. So maybe my intuition was not right? Can anyone help me end the proof of continuity or not continuity , please? How much sleep do adults actually need? The common consensus is that most adults should get 8 hours of sleep a night.  However, sleep needs are highly variable, with some people needing more to function properly, and some gaining little benefit past just a few hours. I'm trying to learn more about the statistics here.  I'd imagine sleep needs would fit a normal distribution, with a median of around 8 hours and a standard deviation of around an hour. The  would then put 68% of adults in the 7-9 hour range, 95% in the 6-10 hour range, and 99.7% in the 5-11 hour range, but that's just a guess.  Has this distribution been studied, and where? Detect when browser receives file download How can I change the default location for bluetooth file transfers? I have an Android 2.3 device (Samsung Galaxy Y Duos, S6102) and want all files that I receive via bluetooth to save to my SD card.  How can I do that? Strange Recurrence: What is it asymptotic to? So I have the following recurrence relation for the growth rate of an algorithm:  $T(n)$ = time taken by algorithm to solve problem of size n:  $$T(n) = T(n-1) + T(\lceil(n/2)\rceil)$$  Clearly this does not obey any polynomial law but now my question is what does it obey?  I merely want an asymptotic equivalence.  My guess is subexponential since this is faster than polynomial but clearly slower than any exponential form.  What though is the question.... I have created a simple model here   When rendering it the end of it is cut off   How can I get the camera to render a longer distance? Can you get a Windows (AD) username in PHP? I have a PHP web application on an intranet that can extract the IP and host name of the current user on that page, but I was wondering if there is a way to get/extract their Active Directory/Windows username as well. Is this possible? Free UML drawing library for .NET I made some scripts on my Raspberry Pi to automate a daily shot at a specific hour.  My DSLR is plugged in through USB. I thought this would keep its battery charged, but no.  Is there really no way automate a DSLR to shoot every 24 hours without having to pick it up, charge/replace its battery, and put in back on the same spot? Finding files by only knowing extension? I'm somewhat disorganized in my file saving habits, so I wind up using the Search function in Windows Explorer more often than one should. I was wondering, is it possible to have Windows find all files of a certain extension ( like, say for example all .rar files) if it is, how? How to delay command blocks until player respawns? I have a command block line set up so that it detects for a players death, then gives them weakness for 600 seconds. It applies weakness to soon before they get out of the respawn time. If there is not way to delay a command is there a way to detect a player respawn? How to demystify the axioms of propositional logic? How might I go about getting some intuition on the typical axiom schemes given for propositional logic? They seem rather mysterious at first glance.  For example, these are taken from:    $\phi \to \phi$ $\phi \to \left( \psi \to \phi \right)$ $\left ( \phi \to ( \psi \rightarrow \xi \right)) \to \left( \left( \phi \to \psi \right) \to  \left( \phi \to \xi \right) \right)$ $\left ( \lnot \phi \to \lnot \psi \right) \to \left( \psi \to \phi \right)$