How can I make the spacing of the two symbols, \otimes and \tildeotimes, identical?  \documentclass{article} \usepackage{amsmath} \DeclareMathOperator{\tildeotimes}{\tilde{\otimes}} \begin{document}     $a \otimes b$      $a \tildeotimes b$ \end{document}
I'm trying to define \conjunction by means of \textbullet instead of \bullet, since the latter looks too big in logical formulas.  How can I make the spacing around \conjunction to behave like the spacing around \bullet when there is an opening parenthesis just after it?  Compare (2) and (2'):     \documentclass[12pt]{article} \usepackage[paperwidth=10cm,paperheight=7cm,margin=1em]{geometry} \usepackage{amsmath} \usepackage{amssymb} \usepackage{ebgaramond-maths}  \DeclareMathOperator\conjunction{\raisebox{-0.25ex}{\textbullet}}  \begin{document}\begin{align} \intertext {Good spacing but bad size:} (\exists x(\forall y(\phi(y)\equiv y\!=\!x)&amp;\bullet Gx)\supset\psi)\\ \exists  x(\forall y(\phi(y)\equiv y\!=\!x)&amp;\bullet (Gx\supset\psi))\\ \intertext {Good size and spacing:} (\exists x(\forall y(\phi(y)\equiv y\!=\!x)&amp;\conjunction Gx)\supset\psi)\\ \intertext {Good size but bad spacing:} \exists  x(\forall y(\phi(y)\equiv y\!=\!x)&amp;\conjunction (Gx\supset\psi)) \end{align}\end{document}   This won't do:  \DeclareMathOperator\conjunction{\raisebox{-0.25ex}{\textbullet\ }}   because it adds an unwanted extra space to (1').
Compute the period of a decimal number a priori I noticed that , given an operation like $\frac{n}{m},\;n,m \in N$ that result in a periodic decimal number, computes really fast the length of the period.  E.g. $\frac{3923}{6173}$ has a period of 3086: .  I was wondering how this computation is done: is there some method to do this (except the trivial one of executing the division and looking for a sequence repetition) ? Finding the limit of $\left(\frac{n}{n+1}\right)^n$ Solve for $A,B$: $\mathrm{LCM}(A,B)=168$, $\mathrm{HCF(A,B)}=12$ How can I copy the path to a folder or a file? The Info Dialog doesn't allow me to copy the information that stands there ...    Is there an easy way to do this? I agree that every page in a site should have a unique TITLE, and that every page should have one and only one H1 element, but should the text of each H1 be completely unique too?  Instinctively one would think so, but there are potentially many cases where (IMHO) mind this doesn't make sense and attempting to do so will impair the user experience &amp; presentation of the page - for example, where pages are part of a closely related cluster, e.g. a multi-step process/tool/wizard.  Surely the process name should go in the H1 (e.g. "Pregnancy Planner"), and the current step you're on should be marked up as a H2 (e.g. "Step 1 - Trying for a baby"), rather than shoehorning both into a single H1 together (e.g. "Pregnancy Planner - Step 1 - Trying for a baby")?  Not only does this make the H1 rather long and potentially impact the presentation of the page, I would argue that in combining both into one you also lose semantic goodness, as the page is surely semantically richer by keeping parent (process) and child (step) as separate, nested headings?  Furthermore, the current proposal I have read is that on the first page the H1 should be the name of the process and the first step an H2, but on subsequent pages the same text should not be a H1 at all (just text styled exactly the same as an H1) and the current step should become the H1 instead. For me this is introducing unnecessary complexity to the page authoring/production process, and inconsistency in the semantic structure of the pages within the stepped process.  So it all comes back to the basic assertion that each H1 must be completely unique, which I question. Can you convince me that this should be so and that the cost/complexity of implementing this (especially in situations such as that above) is worth it?  Your thoughts please... If $a \cdot a = 1$ then $a = 1$ or $a = -1$ Let $F$ be a field.   And let $a \in F$ such that $a \cdot a = 1$.  Claim: $a = 1$ or $ a = -1$  Before, I begin with the proof I cannot understand what $ a = -1$ represents...  Is $a = -1$ representing the inverse of $a$?   I am capable of getting to $a = 1$ with no trouble, but how do you get the $-$ sign to show up in your proof?   Proof: This is my attempt at getting to $a=-1$.  If $a = 0$ then we have $0 \cdot 0 = 0 \neq 1$. Then this implies that $a \neq 0$ which implies that $ \exists b \in F,$ s.t. $a \cdot b = 1$, i.e. $b$ is the multiplicative inverse of $a$.  Knowing this fact prove as follows...:  $a \cdot a = 1 \implies$ $ (a \cdot a) \cdot b = 1 \cdot b \implies$ $a \cdot (a \cdot b) = 1 \cdot b \implies$ $a \cdot 1 = 1 \cdot b \implies$ $ 1 \cdot a =1 \cdot b \implies$ $a = b$.   But since $b$ is the multiplicative inverse of $a$ is this enough to say that $a = -1$? Java is one of my programming languages of choice.  I always run into the problem though of distributing my application to end-users.  Giving a user a JAR is not always as user friendly as I would like and using Java WebStart requires that I maintain a web server.  What's the best way to distribute a Java application?  What if the Java application needs to install artifacts to the user's computer?  Are there any good Java installation/packaging systems out there? Why can't I log into minecraft with a mojang account? How do you disable autocomplete in the major browsers for a specific input (or form field)? The pronunciation of words which begins 'con' and 'com' I understand that the Istari - the only other Maiar besides the Balrog to come into close proximity to the Ring - are constrained within a mortal body and subject to its mortal weaknesses, which renders them vulnerable to the Ring's domination and corruption. However, what is to stop Gandalf from returning to his true form, picking up the Ring, and just walking all the way to Mount Doom? Sauron is too weak at the time; he would have been able to do nothing but watch helplessly and pray his mortal armies can defeat a Ring-bearing Gandalf.  Is Gandalf's decision to abide by the Valar's will to the end the only thing holding him back from this particular option? Are there enchantments in place to disable his ability to revert to Maia form, such that the option was never there to begin with? Or would he, even in Maia form and free of mortal weaknesses, still be vulnerable to domination and corruption by the Ring?  The One Ring is forged with the essence of Sauron to dominate and rule the minds of lesser beings. However, Sauron is only a Maia - albeit the highest-ranking in Morgoth's following. Fellow Maiar would be his equal (at least in rank), while the Valar are certainly superior in power to him. If an Ainu becomes a Ringbearer, will he/she eventually fall to Sauron's will too, or at least succumb to the corruption of the Ring's dark power? Sum of reciprocals of numbers with certain terms omitted How to remove the left part of a string? In Xubuntu 16.04.3 (and several previous versions aswell) it is not possible to have no power icon in the notification area.  A power icon is displayed even though 'Show system tray icon' is unchecked in Xfce Power Manager.  When 'Show system tray icon' is checked, there are two power icons displayed.  (See screenshots)  I've searched and I can't find any information related to this in Xubuntu/Xfce.  Is there a way to fix or work around this bug? Any idea what kind of tool should I but to fix the loose left crank on my bike? I found this  but not sure if it's gonna suit. Or is it enough to buy only nut driver? IP tables - changes not persisting Short story about a group of sci-fi writers sitting around discussing their profession I am a newbie in linux and I don't know how to open the Ubuntu Software Center via Terminal!! Please help I have the following code in my theme's archive.php:  &lt;?php the_archive_title( '&lt;h1 class="page-title"&gt;', '&lt;/h1&gt;' ); ?&gt;   This gives me titles like "Category: Russia", "Tag: America", "Author: John".  I would like to remove the "Category:", "Tag:" and "Author:" part and just display the category, tag and author names.  Does anyone know how to accomplish this?  Thank you. How do I deal with "unauthenticated sources" errors in the Software Center?