'%' is a common way of writing 'modulo' (or more strictly 'remainder') in lots of programming languages.  I have a simple formula: $a \% 2=1$  I have to prove that $a$ will always be an odd integer, but not sure how to write it out. I can prove by example, for example, substituting an integer into $a$, but that is not proving it.
I am trying to write the Euclidean algorithm in the following way:  $A = \lfloor A \div B \rfloor \times B + (\text{remainder of}) \: A \div B $   Now is there any symbol I can use  to say "remainder of A $\div$ B"? I know that in the C programming language there is the operator % for modulus; is that a valid symbol in maths? Can I write A % B? Or is there some other way?
Windows 7 task bar at autohide, make it totally disapear? My desktop look is empty so all I have is a nice wallpaper. My Windows taskbar has the "Taskbar Appearance" setting to "Auto-hide the taskbar". Although hidden, it leaves a thin top layer so you can still see it a bit, and where the pinned shortcuts are. Is it possible to totally have it hidden without a trace? This is a serious question. It really bugs m. I have a bathroom door that is quite loud when operated. And when someone goes to the bathroom at night, it innevatibly wakes someone up (and baby room is just across from the bathroom!) there are 3 components of noise, and I was wondering if there is an alternative to fixing them:   door bangs against the jamb (this is fixed with simple foam bumpers) latch mechanism makes a loud CLUNK when door is pushed closed (I was able to dampen it just a bit by putting foam inside strike plate, but it is still quite loud) lock button also makes a CLUNK when the knob is turned and it is released (this can be battled by replacing with a mechanism that needs to be manually opened [twist instead of push] but it is less convenient, and I would prefer something that automatically unlocks)   Any suggestions as to a knob that has a dampening of the latch and lock button? I have only found one youtube video of my ideal handle, but there is no information on how it works, where to buy it, or anything else: How to change inside background color of UISearchBar component on iOS Is the graviton hypothetical? Wikipedia lists the  as a hypothetical particle. I wonder whether graviton is indeed hypothetical or does its existence directly follow from modern physics? Does observation of gravitational waves amount to the discovery of graviton or could there be gravitational waves not composed of gravitons? 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. Theorem style with name as argument Is there some way to get a theorem type which will make a theorem with the name passed as a parameter? In other words, is there a way to define "named" such that   \begin{named}[NAME] text \end{named}   will print   "NAME: text"?  I tried using the definition \newtheorem*{named}{}, but that results in   "(NAME): text", which is quite different from what I am trying to do.  I know that I could also create a new instance of \newtheorem for each named theorem I want to use, but that is clunky and inconvenient. Is there a better way? Why are we in love "with" someone? Is "The empty set is a subset of any set" a convention? Evaluating the integral $\mathop{\lim}\limits_{n \to \infty} \int_{-1}^{1} f(t)\cos^{2}(nt) \ dt $ Given that $f\colon [-1,1] \to \mathbb{R}$ is a continuous function such that $ \int_{-1}^{1} f(t) \ dt =1$, how do I evaluate the limit of this integral: $$\lim_{n \to \infty} \int_{-1}^{1} f(t) \cos^{2}{nt} \,dt$$   What I did was to write $\cos^{2}{nt} = \frac{1+\cos{2nt}}{2}$ and substitute it in the integral so that I can make use of the given hypothesis of $\int_{-1}^{1} f(t) \ dt =1$. So the integral becomes,   \begin{align*} \int_{-1}^{1} f(t)\cos^{2}{nt} \ dt = \int_{-1}^{1} f(t) \biggl[\frac{1+\cos{2nt}}{2}\biggr] \ dt  &amp;  \\  \hspace{3cm} = \frac{1}{2}\int_{-1}^{1}f(t) \ dt + \int_{-1}^{1} \frac{f(t)\cos{2nt}}{2} \ dt \end{align*}  But I don't really know how I can evaluate the second integral and also I can't realize as to why that integral condition on $f$ has been assumed. Moreover without assuming that condition on $f$ is it possible to evaluate this integral? If yes, then what would the answer be? If there are two twins. such that one of them goes on a space voyage maintaining a constant velocity, and that one never returns to earth, and the twins decide to never meet but send information about their age to each other by some light pulse or any other method.  If then one knows that he is younger, won't he know he is the one travelling thus violating the postulate of relativity that all inertial frames are equal w.r.t. all phenomena and that one can never know if he is moving with a constant velocity. Show that the set of all $x$ such that $x \in P$ and $x \notin Q$ exists I have to show that set of all $x$ such that $x \in P$ and $x \notin Q$ exists. So, I think I have to assume that sets $P$ and $Q$ already exist. So, with that, I can use  . This axiom is $$ \forall w_1,\cdots, w_n \, \forall A \,\exists B \,\forall x \left( x \in B \Longleftrightarrow [ x \in A \wedge \varphi(x,w_1, \cdots , w_n, A)] \right) $$ So, I will let $\varphi(x, Q, P) = x \in P \text{ and } x \notin Q $. Also, I will let $A = P$ and  $w_1 = Q $. So using universal instantiation, I get the following $$ \exists B \,\forall x \left( x \in B \Longleftrightarrow [ x \in P \wedge \varphi(x, Q, P)] \right) $$ And using, existential instantiation, there exists a set $B$ such that $$ \forall x \left( x \in B \Longleftrightarrow [ x \in P \wedge \varphi(x, Q, P)] \right) $$ Using the definition of $\varphi(x, Q, P)$, this can be simplified as following $$ \forall x \left( x \in B \Longleftrightarrow [ x \in P \wedge x \in P \wedge x \notin Q] \right) $$ $$ \forall x \left( x \in B \Longleftrightarrow [ x \in P \wedge x \notin Q] \right)$$ $$ \forall x \left( x \in B \Longleftrightarrow x \in \{ x \,| x \in P \text{ and } x \notin Q \} \right)$$ Now, using , it will follow that $$ B = \{ x \,| x \in P \text{ and } x \notin Q \}  $$ And, since $B$ exists, this means that $\{ x \,| x \in P \text{ and } x \notin Q \} $ also exists. Is the proof good ? What does my internet provider know about my online activities? What does my internet service provider know about my online activites in these cases?   plain server (www, ftp, whatever) connection and provider's DNS secured (e.g. TLS) server connection and provider's DNS plain server (www, ftp, whatever) connection and an non-provider non-logging DNS secured (e.g. TLS) server (www, ftp, whatever) connection and non-provider non-logging DNS   In case 4) does my provider know anything more about my online behaviour than "IP w.x.y.z was assigned to Foo Bar between connect-time and disconnect-time" (i.e. only the duration my router was connected)?  What traffic does need to go through the provider's infrastructure to make my connection work at all? Limit point versus limit and implications for convergence I understand that similar questions to this have been asked  and  but I believe my question is different because with regards to (1) I am not asking about the set $X$ that contains the points of my sequence and with regard to (2) because I have the distinct definitions of limit and limit point in mind. That said:  My Question: My textbook, Fundamental Ideas of Analysis by Michael Reed, defines a limit point as:     Definition: Let $(a_n)$ be a sequence of real numbers. A number $d$ is called a limit point if, given $\epsilon &gt; 0$ and an integer $N$, there exists $\color{red}{\mathrm{ an }} \ n \ge N$ so that $|a_n - d| \le \epsilon$.    The definition of a limit is as follows:     Definition: We say that a sequence $(a_n)$ converges to a limit $a$ if, for every $\epsilon \ge 0$, there is an integer $N(\epsilon)$ so that $|a_n - a| \le \epsilon \color{red}{\text{  for all  }} n \ge N$.    From these definitions, and given this question:     If a sequence converges it has exactly one limit point. Is the converse true? If not, provide a counterexample   Can I conclude that the converse is indeed true simply by appealing to the definitions, or do I need a tighter proof. And if I do, I'm a little stuck on how to refine this proof. My company has purchased a product that renders an ASP.NET control on the page. This control uses jQuery 1.2.3 and adds a script tag to the page to reference it. The developers of the control will not support use of the control if it modified in any way (including modification to reference a different version of jQuery).  I'm about to start development of my own control and would like to use the features and speed improvements of jQuery 1.3. Both of these controls will need to exist on the same page.  How can I allow the purchased control to use jQuery 1.2.3 and new custom development to use jQuery 1.3? Also out of curiosity, what if we were to use an additional control that needed to reference yet another version of jQuery? We need a hard rate limit on suggested edits, to limit the damage someone can do with blasting the review queue with minor edit suggestions.  Case in point:  peppered the site with a whopping 133 tag-only edit suggestions in 2 hours and 15 minutes. That's one edit every minute on average, but towards the end he was entering a new suggestion every 20 seconds. He only stopped because a  drew enough rejections to have him banned from entering more.  If there was a limit to the number of suggested edits you can enter in a time period we could have stopped this guy earlier. No &lt;2k user needs to edit hundreds of tags across that many questions in 2 hours time.  Considerations   There are cases where a retag drive or tag burninate justifies editing a larger number of posts, but I'd be fine if such tasks were limited to people with 2k+ reputation. A retag or burninate job requires a minimal engagement with the community to know when they are justified anyway. The suggested edit review process, by itself, was clearly not enough to stop this barrage. It cannot be seen as a rate limiting measure, at least not as a sufficiently efficient one.  The vast majority of the edits made in the above example were too minor, leaving spelling, grammar and formatting mistakes in the post untouched. The user was going too fast to pay attention to such details anyway. Yet this user has 98 approved edit suggestions, but all but 6 or 7 I'd have rejected as too minor, so the review process itself is broken.   What should be limited   The number of review votes reviewers have is already limited; I burned through 20 reviews in short order just reviewing the suggestions from just this one user. Certainly the number of unreviewed suggestions should be limited to let reviewers catch up with a user like this. I'd say having 20 outstanding suggested edits pending review is plenty. Suggested edits should be made with some consideration; we don't want minor edits, so if you are suggesting more than 20 in an hour (that's 1 every 3 minutes) you are not improving posts anymore, you are gaming the system to gain reputation. How long does the Truffle take to move in after building the mushroom biome? I currently find myself in a pickle because even after I've built the mushroom biome and met all other requirements, the Truffle NPC won't move in.   I've defeated not only the Wall of Flesh but Ocram already. Created a surface mushroom biome. Built a house made out of mushrooms with mushroom furniture.   I have no idea how long it takes for the NPC to move in or if I'm wrong in any of the steps. SCJP6 regex issue What does the word treats mean in this context? I don't think it means entertainment, food, drink etc.. Maybe it is an old usage of this word? Does it mean influencing factors or something like that?  This is the first part of the title of the first chapter of the novel "Oliver Twist" by Charles Dickens. The whole title reads:     Treats of the Place Where Oliver Twist was Born and of the Circumstances Attending His Birth Chapter on the right side - How to turn it off? I have a large file and I would like print from each sequential 50 lines , the 15th and 25th lines.  sed -n '15,25p' inputfile   How to modify this command to print only lines 15 and 25 and to loop over each 50 lines in the file.