Following is mwe of what I'm doing. I want to get a couple of things.    Instead of bold 1, 2, I need it to be replaced by Chapter 1 and Chapter 2 respectively, followed by the name. I want the References appeared in the table of contents, separately not under a Chapter, with the heading References.   Thank you for any suggestions.   \documentclass[11 pt]{article}  \usepackage{amsmath, mathrsfs} \usepackage{amssymb}  \usepackage[utf8]{inputenc} \usepackage[english]{babel} \usepackage{blindtext}    \linespread{1.4} \pagestyle{myheadings}      \title{This is the title} \author{Author A and Author B} \date{} \begin{document} \let\conjugatet\overline \maketitle  \tableofcontents   \section{This is section 1} \subsection{This is a subsection}  \section{This is section 2} \subsection{This is a subsection}   \bibliographystyle{abbrv} \bibliography{References}{}  \end{document}
I'm attempting to modify the Table of Contents in my document. Currently, chapter one in my document is listed as   1   Introduction   What I would like to see is  Chapter 1   Introduction   Seems like an easy enough thing to ask for, but can't seem to find a good way to do this. Any feedback would be most appreciative.
$g^\frac{p-1}{2} \equiv -1 \ (mod \ p)$ Let $p$ be an odd prime, and let $g$ be a primitive root modulo $p$. Prove that $g^\frac{p-1}{2} \equiv -1 \ (mod \ p)$.  I've seen this solution on the math stack exchange but most of the solutions use quadratic reciprocity. Unfortunately, I skipped that section because it seemed very hard, but the proof that  $g^\frac{p-1}{2} \equiv -1 \ (mod \ p)$ seems easy with it.  Is there another way to solve this This question is proposed in the chapter of Finite Abelian Groups.  DEF:In elementary number theory, an integer $g$ is called a primitive root for the modulus $n$ if $\mathbb{Z}_n^\times$ is a cyclic group and $[g]_n$, is a generator for $\mathbb{Z}_n^\times$. In order to animate a character so that it grabs an object an throws it, I added a Child-Of constrained and keyframed its influence. In the F-curve Editor I can't set the curve which controls the influence to interpolation-mode Constant without changing the other curves (Which should be linear or bezier).   How can the interpolation mode of a curve changed independently? My question relates to Chapter 3, Exercise 8 in &quot;Baby Rudin&quot;. It states: If $\sum_n a_n$ converges, and if $\{b_n\}$ is monotonic and bounded, prove that $\sum_n a_n b_n$ converges. My attempt would have been: Since $\{b_n\}$ is monotonic and bounded, $\{b_n\}$ converges and it exists $\inf \{b_n\}$ as well as $\sup \{b_n\}$. But then we have $$ \left| \sum_n a_n \inf \{b_n\} \right| \leq \left| \sum_n a_n b_n \right| \leq  \left| \sum_n a_n \sup \{b_n\} \right| \leq \max \left( \left| \sup \{b_n\} \right|, \left| \inf \{b_n \} \right| \right)\varepsilon \leq \tilde{\varepsilon} $$ since $\sum_n a_n$ converges and $\max \left( \left| \sup \{b_n\} \right|, \left| \inf \{b_n \} \right| \right)$ is a finite number. This would imply, by the comparison test, that $\sum_n a_n b_n$ converges as well. $\quad \Box$ But as there was a way longer, more rigorous proof chosen in this , I'm a bit suspicious that my proof is not complete. Am I missing something?  EDIT: Thanks everyone! @hermes: The last part of your proof gave me the following idea. As $\lim_{n \to \infty} b_n = c$ and $\{b_n\}$ is monotonic, couldn't we just set $b_n = c - c_n$ with a monotonically decreasing sequence $\{c_n\}$ which has $\lim_{n \to \infty} c_n = 0$. Then we have $$\sum_n a_n b_n = \underbrace{c \sum_n a_n}_{\text{converges by assumption}} - \underbrace{\sum_n a_n c_n}_{\text{converges by Theorem 3.42}} \leq \varepsilon_1 - \varepsilon_2 = \varepsilon $$ since  Theorem 3.42 Suppose  the partial sums of $\sum_n a_n$ form a bounded space $\quad \checkmark$ $c_0 \geq c_1 \geq c_2 \geq \dots \quad \checkmark$ $\lim_{n \to \infty} c_n = 0 \quad \checkmark$  Then $\sum_n a_n c_n$ converges.  Thus $\sum_n a_n b_n$ converges as well. $\quad \Box$ Now that should hold, I think. So I wouldn't need to go through all the estimates. If photon energies are continuous and atomic energy levels are discrete, how can atoms absorb photons? If photon energies are continuous and atomic energy levels are discrete, how can atoms absorb photons? The probability of a photon having just the right amount of energy for an atomic transition is $0$. Though tangentially related to another post on MathOverflow (), the questions below are mainly out of curiosity. They may be very-well known ones with very well-known answers, but...    Suppose $\Sigma$ is a sigma-algebra over a set, $X$. For any given topology, $\tau$, on $X$ denote by $\mathfrak{B}_X(\tau)$ the Borel algebra over $X$ generated by $\tau$.     Question 1. Does there exist a topology, $\tau$, on $X$ such that $\Sigma = \mathfrak{B}_X(\tau)$?   If the answer to the previous question is affirmative, it makes sense to ask for the following too:     Question 2. Denote by ${\frak{T}}_X(\Sigma)$ the family of all topologies $\tau$ on $X$ such that $\Sigma = \mathfrak{B}_X(\tau)$ and let $\tau_X(\Sigma) := \bigcap_{\tau \in {\frak{T}}_X(\Sigma)} \tau$. Is $\Sigma = \mathfrak{B}_X({\frak{T}}_X(\Sigma))$?   Updates. Q2 was answered in the negative by Mike (). I have attempted a proof of this, but upon looking at my notes, I feel it might be incorrect: it is noticeably simpler than the one in my notes.  Proposition: any permutation in $S_n$ can be written as a product of disjoint cycles.  Proof (attempt): $s\in S_n$. Let $\langle s\rangle$ act on $X=\{1,\cdots,n\}$. Define $x\sim x'\iff x'\in\text{Orb}(x)$ for $x,x'\in X:$ this is clearly an equivalent relation, hence the orbits partition $X$. Each orbit corresponds a cycle in $s$, so we are done.  Does this hold, or have I missed something vital? This is an example from a question sheet (non-assessed) of a university class. If M is a non-orientable, closed, connected 3 manifold, prove $H_1(M;\mathbb{Z})$ is an infinite group. I know that since $M$ is non-orientable it follows that $H_3(M;\mathbb{Z}) = 0$ and since $M$ is connected $H_0(M;\mathbb{Z}) = \mathbb{Z}$. I want to apply a combination of Poincare duality and Universal coefficient theorem to try and find a contradiction if $H_1(M;\mathbb{Z})$ has no free part. However poincare duality doesn't apply here so I'm stuck.  I would appreciate any hints as to how to proceed. I am reading Boyd &amp; Vandenberghe's Convex Optimization. The authors use curved greater than or equal to (\succcurlyeq)  $$f(x^*) \succcurlyeq \alpha$$  and curved less than or equal to (\preccurlyeq)  $$f(x^*) \preccurlyeq \alpha$$  Can someone explain what they mean? Unable to send mail to hotmail from rackspace cloud Concatenating two std::vectors Incorrect notation in math? Does math have an incorrect notation / syntax? I don't mean writing misaligned notation (google), but when you take something like a number to powers to powers to powers, $${{2^2}^2}^3$$ (I was told this is incorrect notation by a teacher). Is it really incorrect, or does it just need to be simplified with parentheses? Do people write maths like this?  a radical expression with the root being a radical expression? $$\sqrt[\sqrt{2^3}]{2}$$ Word for feeling conflicting emotions simultaneously Let $S$ be an arbitrary amount and define the relation $R \subseteq \mathcal{P}(S) \times \mathcal{P}(S)$ so that $(A,B) \in R$ if and only if $A \supseteq B$. Here $\mathcal{P}(S)$ is a spelling for the power set to $S$. Show that $(\mathcal{P}(S),R)$ is a partially ordered quantity. How do I clone a generic List in Java? Is “all women who are mothers of daughters” right? Is the expression      All women who are mothers of daughters are here.   correct? I mean every woman who has at least one daughter is here. How do you show that $e^{-1/x^2}$ is differentiable at $0$? Define the real valued function $f(x)$ by $f(x) = e^{-1/x^2}$ if $x \ne 0$ and $0$ if $x = 0$.    How do you show that $e^{-1/x^2}$ is differentiable at $0$? How do you show that this function is infinitely differentiable? How can I use Photoshop to imitate a photo taken with a low-quality camera? How can I create an image in Photoshop that imitates a photo taken with a low-quality phone camera? Examples of qualities I want to simulate:   blurring in some areas caused by a dirty lens low quality exposure handling in the darkroom noise   Some examples: Find and replace text within multiple files In this sentence:  "We would also accept an international transfer, but in that case we would ask you to pay your bank's charges at the time of the instruction to ensure that we received the full amount"  is it correct to use "received" or should it be in present tense? Journals selection - alternative criteria to impact factor While trying to select the journals I can submit my present work and future work, I realize the time between the date the authors send the article to the journal and when the revised article is submitted. Usually this is something between 6 months and more than 12 months.  Althought I am focusing on the Impact Factor (IF) to select journals, I think that 12 months to have a feedback that might include "add more variables analysis","this article will be more valuable if you test also model x,y and z", can have a huge impact on you work. It will make you stop what you are currently doing and possible spend up to 2 months making the changes. This might make a big mess on your work if you have deadlines to respect, however this happens to everyone in academia.  My question is, besides the IF journal based selection, how can we select a journal with enought quality with lower IF? What alternative criteria should one use to select the journal? Should we choose newly created Journals?