How to prove that $ a^2+b^2+c^2=1$ implies $ab+bc+ca \in [-\frac12,1]$?
If $a^2+b^2+c^2=1$ then prove the following.
A 2x2 matrix $M$ exists. Suppose $M^3=0$ show that (I want proof) $M^2=0$ Top of answer covered by sticky top bar after clicking direct link With the introduction of the network-wide sticky top bar , I noticed a bug that's happening on every site (except for Stack Overflow). When you click on an answer link (of the form https://site.stackexchange.com/a/####), the very top of the answer and the upvote button are covered by the top bar:    Stack Overflow shows the correct behavior: What does the explicit keyword mean in C++? If $p,q$ are primes $(pq,a)=1$ and $p-1|q-1$ then prove $a^{q-1}\equiv 1\pmod{pq}$ $p,q$  primes, $(pq,a)=1$ and $p-1|q-1$ then prove $a^{q-1}\equiv 1\pmod{pq}$   Got this in an exam today and had no idea how to do it. In order to show $e^{x+y}=e^{x}e^{y}$ by using Exponential Series, I got the following:   $$e^{x}e^{y}=\Big(\sum_{n=0}^{\infty}{x^n \over n!}\Big)\cdot \Big(\sum_{n=0}^{\infty}{y^n \over n!}\Big)=\sum_{n=0}^{\infty}\sum_{k=0}^n{x^ky^n \over {k!n!}}$$  But, where should I go next to get $e^{x+y}=\sum_{n=0}^{\infty}{(x+y)^n \over n!}$.  Thanks in advance. Chain skip happening seemingly at random with a new chain Printing a .NET DataGridView I am fairly new to .NET and C#, but I have a DataGridView that I would like to print.  What would be the best way to go about doing so? Why bosons have integer spin and fermions have half-integer ones? Due the fact that the fermions are the "block particles" and the bosons are the "carriers" I just came out with the question that, why the "block particle" have half-integer spin and the "carriers" have an integer spin? Can I run CAT5/6 cables parallel to electrical cables? SQL function as default parameter value? I tried changing a default parameter value with this:  ALTER PROCEDURE [dbo].[my_sp] @currentDate datetime = GETDATE()   and all the SQL pre-compiler gave me was this error:     Msg 102, Level 15, State 1, Procedure my_sp, Line 8 Incorrect syntax near '('.   I have already created the procedure. (I'm not sure if that's relevant.) I was using a null default value and checking for that later, but that doesn't seem proper. Can I do this in one line?   Update: I was going off of :      [ = default ] Is a default value for the parameter. If a default value is defined, the function can be executed without specifying a value for that parameter.            Note:     Default parameter values can be specified for CLR functions except for the varchar(max) and varbinary(max) data types.         When a parameter of the function has a default value, the keyword DEFAULT must be specified when the function is called to retrieve the default value. This behavior is different from using parameters with default values in stored procedures in which omitting the parameter also implies the default value.   Am I reading this wrong?  Many thanks. Finding a closed form for a recurrence relation $a_n=3a_{n-1}+4a_{n-2}$ Consider the sequence defined by $$ \begin{cases} a_0=1\\ a_1=2\\ a_n=3a_{n-1}+4a_{n-2} &amp; \text{if }n\ge 2 \end{cases} .$$ Find a closed form for $a_n$.    I tried listing out examples, but I don't see any common pattern between them.  All solutions are greatly appreciated. A real $2 \times 2 $ matrix $M$ such that $M^2 = \tiny \begin{pmatrix} -1&0 \\ 0&-1-\epsilon \\ \end{pmatrix}$ , then : A real $2 \times 2 $ matrix $M$ such that $$M^2 =  \begin{pmatrix} -1&amp;0 \\ 0&amp;-1-\epsilon  \\   \end{pmatrix}$$ (a) exists for all $\epsilon &gt; 0$. (b) does not exist for any $\epsilon &gt; 0$. (c) exists for some $\epsilon &gt; 0$. (d) None of the above is true.  Attempt: I couldn't think of any theory base to prove / disprove the existence of such a matrix. Could someone please give me a hint on how to go about this problem? Thank you very much for your help in this regard. How to force a table into page width? I have the following table:  \begin{table}[htb] \begin{tikzpicture} \node (table) [inner sep=0pt] { \begin{tabular}{ l | l }   {\bf Symptom} &amp; {\bf Metric} \\ \hline Class that has many accessor methods and accesses a lot of external data &amp; ATFD is more than a few\\ Class that is large and complex &amp; WMC is high\\ Class that has a lot of methods that only operate on a proper subset of the instance variable set &amp; TCC is low\\ \end{tabular} }; \draw [rounded corners=.5em] (table.north west) rectangle (table.south east); \end{tikzpicture} \caption{God class symptoms} \label{tbl:god_class} \end{table}   Now I want to force the width of the table to be the same as the \textwidth, either by linewrapping of table text or by scaling. How can I achieve that? Adding parentheses around subfig references Say I have a figure with two subfloats inserted using the subfig package.  When I reference them with \ref{myLabel} I get this     Figure 3.5a and 3.5b.     But what I want is this:     Figure 3.5(a) and 3.5(b).   How can I add the parentheses? What's the nice 'trick' to showing that the following expression is irrational?  $2^{1/3} + 2^{2/3}$ Delete duplicate records in SQL Server? Caption on tabular environment I have got codes like the following (generated by Kile). How can I add a caption. I guess I should have used the table environment. Is there any solution now?  \newcommand{\mc}[3]{\multicolumn{#1}{#2}{#3}} \begin{center} \begin{tabular}{rccll}      &amp;   &amp; \mc{3}{c}{Colin}\\      &amp;   &amp; a &amp; \mc{1}{c}{b} &amp; \mc{1}{c}{c}\\\cline{3-5} Rose &amp; \mc{1}{c|}{A} &amp; \mc{1}{c|}{(1,2)} &amp; \mc{1}{c|}{(2,5)} &amp; \mc{1}{c|}{(4,4)}\\\cline{3-5}      &amp; \mc{1}{c|}{B} &amp; \mc{1}{c|}{(7,4)} &amp; \mc{1}{c|}{(3,5)} &amp; \mc{1}{c|}{(0,6)}\\\cline{3-5} \end{tabular} \end{center} Here is my problem: I've got a dialog with some parameters that the user can change (via a spinner for example). Each time one of these parameters is changed, I launch a thread to update a 3D view according to the new parameter value.  If the user changes another value (or the same value again by clicking many times on the spinner arrow) while the first thread is working, I would like to abort the first thread (and the update of the 3D view) and launch a new one with the latest parameter value.  How can I do something like that?  PS: There is no loop in the run() method of my thread, so checking for a flag is not an option: the thread updating the 3D view basically only calls a single method that is very long to execute. I can't add any flag in this method asking to abort either as I do not have access to its code. Why is it differential equations exist on an interval instead of a domain? I understand a domain is the set of input elements a function is defined for (and can have breaks in it e.g. union of 2 sets) and a interval is a continuous range of real numbers. Why do we speak of the solution to a differential equation over an interval instead of an domain? If the solution to an ODE is just a function, and we normally speak of functions as having a domain, then why wouldn't we use domain? How can I change some pages into document into landscape mode, while leaving the others in portrait mode (or vice versa)?