How to store values in dictionary in decrement order by values in java?
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Basically, here's my MWE, and I need to include "Chapter" before every section title in the ToC (without changing anything else).      \documentclass[12pt]{article} \usepackage[utf8]{inputenc}  \usepackage[english]{babel} \usepackage[titles]{tocloft} \usepackage{titlesec} \usepackage{titletoc}  \addto\captionsenglish{ \renewcommand*\contentsname{\centerline{TABLE OF CONTENTS}}}  %\renewcommand{\thesection}{CHAPTER} %\renewcommand{\cftsecpresnum}{Chapter \thesection} \titleformat{\section}[display]% {\null\vskip1em\sffamily\bfseries\filcenter}{Chapter \thesection}{0em}{}[] \titlespacing*{\section}{0em}{1em}{1em}  \titlecontents{section} [1.5em] % ie, 1.5em (chapter) + 2.3em {} {\contentslabel{1em}} {\hspace*{-1.5em}} {\titlerule*[0.5pc]{.}\contentspage}  \begin{document} \newpage \tableofcontents \newpage  \section{Section One} \section{Section Two} Content \end{document}   The commented out code is what I've already tried. The first one just changes the number to "CHAPTER" and doesn't move the text in the ToC (so it just overlays on top of the name of the section), and if I include "\thesection" after "CHAPTER" it creates an infinite loop definition error. The second one causes no differences, and the code still compiles.  Thanks! If I take the $k \in \mathbb{N}$ power of 10 and mod it by a large prime, I notice that the remainders repeat at some point.  For instance $10^k mod~7$ seems to repeat every $8$th value of $k$.  Given $k$ (the power of ten), and a large prime number $p$, how can I compute the size of the cycle?  I tried writing a python script that computes the first instance of a repeat. However, I don't know if this works since a single instance of a repeat may not necessarily mean a full cycle. How do you square an ideal? For some of you, this question is going to seem extremely basic.  I think I understand what an ideal is. Such as $\langle 2, 1 + \sqrt{-5} \rangle$, it consists of all numbers in this ring of the form $2a + (1 + \sqrt{-5})b$.  But then what is $\langle 2, 1 + \sqrt{-5} \rangle^2$? My first thought was $\langle 4, -4 + 2 \sqrt{-5} \rangle$, but that seems wrong somehow. I also had something in the back of my mind saying $\langle 2 \rangle$, but I'm not sure about that one either.  Then I thought about trying to figure out $\langle 2, 1 + \sqrt{-5} \rangle \langle 2, 1 + \sqrt{-5} \rangle$ when I realized I don't actually understand how to multiply ideals to begin with.  I'm only using $\langle 2, 1 + \sqrt{-5} \rangle$ as an example (though that does draw in one question identified as similar that looks much more relevant than all the questions identified as "Questions that may already have your answer"). In a principal ideal domain, would it be correct to think that $\langle a \rangle \langle b \rangle = \langle ab \rangle$?  Any help would be much appreciated. How can I stream live video on YouTube? Nowadays, I see a lot of users live streaming on YouTube and wanted to try it for myself. However, when I go to my channel I don't see an option for live streaming.  I tried researching how to do this on YouTube and Google, but without luck.  So I was wondering, how can I live stream on YouTube?  I don't have any pro account or any of that, but I've watched livestreams from users who I don't think have pro accounts either.  So livestreaming is open for everyone, right? How to remove the Ubuntu menu of Windows Boot Manager? I have an ASUS K55VM and it came with DOS, and then Windows 7 was installed. I am planning to install Ubuntu on it. My computer had 4 partitions: 240GB, 230GB, 230GB, and 230GB, and I changed this to 5 partitions: 243GB, 230GB, 230GB, 150GB, and 80GB. I installed Ubuntu 12.04 to the 80GB partition through a USB flash drive. It installed fine, however the problem is during the Boot option in Windows Boot Manager. When I choose Ubuntu, this what is displayed:  Windows failed to start. A recent hardware or software change might be the cause. To fix the problem:   1. Insert your Windows installation disc and restart your computer.   2. Choose your language settings, and then click "Next,"   3. Click "Repair your computer." If you don't have this disc, contact your system administrator or computer manufacturer for assistance.    File: \ubuntu\winboot\wubildr.mbr   Status: 0xc000000e   Info: The selected entry could not be loaded because the application is missing or corrupt.   I tried to change the boot option for the OS and made Ubuntu as my first option. It worked, however Windows 7 does not show up in GRUB. In this fear I did the following:     Restarted my PC changed boot option 1 to Windows.   Formatted the drive to which Ubuntu was installed (80GB one) through Windows.   Restarted again.     The problem persists. Windows Boot Manager still shows Ubuntu, and when I choose this it gives two lines one like: grub rescue&gt;, but my Windows 7 is working perfectly fine.  Now I want a solution to remove Ubuntu from the Windows boot option and reinstall completely. trigger AccountReplicate on Account (after delete, after insert, after update) {     if ( trigger.isInsert ) {         for ( Account acc : trigger.new ) {             AccountOne__c ac = new AccountOne__c();             ac.Fax__c = acc.Fax;             ac.Phone__c = acc.Phone;             ac.Name =acc.Name;             insert ac;         }     }      if ( trigger.isUpdate ) {         integer  i = 0;         list&lt;AccountOne__c&gt; a2 = new list&lt;AccountOne__c&gt;([SELECT Id                                                         FROM AccountOne__c                                                         WHERE Name =: trigger.old[i].Name ]);         for ( AccountOne__c c1 : a2 ) {             c1.Name = trigger.new[i].name;             c1.Fax__C = trigger.new[i].Fax;             c1.Phone__C = trigger.new[i].Phone;             i++;         }         update a2;     } }   I tried bulk trigger for the update operation, but it's not working can anyone help me? Transitive subgroup of symmetric group $S_n$ containing an $(n-1)$-cycle and a transposition How to use a variable for a key in a JavaScript object literal? QtCreator and Command Line Arguments In the paper "Beautiful Code Compelling Evidence", by J.R. Heard -- Code is formated in a particular way which includes margin comments. I am looking for a way to do this as I think it is a very clear way to comment comment code in a literate programming style. How can I get the current date and time in UTC or GMT in Java? When I create a new Date object, it is initialized to the current time but in the local timezone. How can I get the current date and time in GMT? Zero integral of measurable $f$ on every interval implies $f=0$? Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a Borel measurable function, such that $\displaystyle \int_{a}^{b}f(x)dx=0$ for every $a&lt;b$ in $\mathbb{R}.$ Is it true that $f(x)=0$ for every $x\in \mathbb{R}$?  Comments: the starting point of this is to prove that the (Borel measurable) density function $f$ (if exists) of a symmetric random variable $X$ (meaning that $X,~-X$ have common cdf's) is even. If my question has a positive answer, it follows easily.   Thanks a lot for any comments! Why is torque not measured in Joules? Recently, I was doing my homework and I found out that Torque can be calculated using $\tau = rF$. This means the units of torque are Newton meters. Work &amp; Energy are also measured in Newton meters which are Joules.  However, torque isn't a measure of energy. I am really confused as why it isn't measured in Joules. Do not hide VLQ flag option from users who already flagged and may wish to retract Blurry Text On Model After Render Preface:  An axis plotted with  is usually using the current math font for its ticklabels. But if they are overwritten by e.g. xticklabels={-8,-6,...,8} then the current text font is used. Also an axis label ylabel={Y} would be in text font unless put explicitly in math mode ylabel={$Y$}. However it would be nice, if  one hadn't to do so.   Question:  I'm aiming to ensure that pgfplots always uses the current math font. But please assume that the classic font commands like \rmfamily are screwed up by the corporate design I'm supposed to use, e.g. assume: \renewcommand{\rmdefault}{\sfdefault}  It works if I explicitly define a font  \newcommand{\pgfmathfont}{\fontfamily{pxfonts}\selectfont}   and pass it to pgfplots together with assume math mode=true:  \begin{axis}[ font = \pgfmathfont,  tick label style={/pgf/number format/assume math mode=true} ]   But it requires that I insert the name of the current math font. How can I do that automatically? How to get the currently used math font to insert it into the macro?    MWE  \documentclass{beamer}  \usetheme{Warsaw} \usefonttheme{professionalfonts} \usepackage{pxfonts} \usepackage{pgfplots}  % screwed up by Corporate Design \renewcommand{\rmdefault}{\sfdefault}  % I'd like this line to be generic! \newcommand{\pgfmathfont}{\fontfamily{pxfonts}\selectfont} %           put current math font here   X  \begin{document}  \begin{frame}  This is a basic equation:    \[ x + y = 42\]  \begin{tikzpicture}     \begin{axis}[ %   font = \rmfamily, % makes no sense here     font = \pgfmathfont,      xlabel={$x$-label: I want to be in math font!},     ylabel={Y},     xtick={-4,-3,...,4},     xticklabels={-8,-6,...,8}, %   xticklabels={$-8$,$-6$,...,$8$}, % does not work     tick label style={/pgf/number format/assume math mode=true}     ]      \addplot {x^2};      \end{axis} \end{tikzpicture}  \end{frame}  \end{document}     Everything related to  is no option at all!  Thank you! The boundary of union is the union of boundaries when the sets have disjoint closures An integration question to be solved without using differentiation under the integral sign. What is the relationship between symmetry and degeneracy in quantum mechanics? Let me remind you about the following classical  examples in quantum mechanics.     Example 1. Bound states in 1-dim potential V(x).   Let $V(x)$ be a symmetric potential i.e.   $$V(x) = V(-x)$$   Let us introduce the parity operator $\hat\Pi$ in the following way:   $$\hat\Pi f(x) = f(-x).$$   It is obvious that    $$[\hat H,\hat\Pi] = 0.$$   Therefore, for any eigenfunction of $\hat H$ we have:   $$\hat H|\psi_E(x)\rangle = E|\psi_E(x)\rangle = E\hat\Pi|\psi_E(x)\rangle,$$   i.e. state $\hat\Pi|\psi_E(x)\rangle$ is eigenfunction with the same eigenvalue. Is $E$ a degenerate level? No, because of linear dependence of $|\psi\rangle$ and $\hat\Pi|\psi\rangle.$   Consider the second example.     Example 2. Bound states in 3-dim a potential $V(r)$. Where $V(r)$ possesses central symmetry, i.e. depends only on distance to center.   In that potential we can choose eigenfunction of     angular momentum $\hat L^2$ for basis    $$|l,m\rangle,$$   where $l$ is total angular momentum and $m$ - its projection on chosen axis (usually $z$). Because of isotropy eigenfunction with different $m$ but the same $l$ correspond to one energy level and linearly independent. Therefore, $E_l$ is a degenerate level.      My question is if there is some connection between symmetries and degeneracy of energy levels. Two cases are possible at the first sight:   Existence of symmetry $\Rightarrow$ Existence of degeneracy Existence of degeneracy $\Rightarrow$ Existence of symmetry     It seems like the first case is not always fulfilled as shown in the first example. I think case 1 may be fulfilled if there is continuous symmetry. I think the second case is always true. I am trying to understand capitalization rules with Mom and Dad. I believe I have it correct below, but please let me know if I do not.     The one thing I learned from my dad was that it was good to earn the trust of one's children. Dad has shown me how good that a dad can be. One day when I went to his house, and Dad wasn't there, I ran outside....   The above passage is just made-up just so that you would see what I mean about capitalization. Let's put numbers by each one, like so:     The one thing I learned from my dad(1) was that it was good to earn the trust of one's children. Dad(2) has shown me how good that a dad(3) can be. One day when I went to his house, and Dad(4) wasn't there, I ran outside....    This is one where I'm not certain. I believe in this sense I'm showing possession here with "my", and it just didn't seem right saying "my Dad". I don't know what the grammar rule is here, or if I even have it right. This one is obvious—it starts the sentence. In this case, I'm using Dad like a name instead of saying "Larry". Therefore, it should be capitalized. Again, just like 3.