Solving the functional equation $f(x + y) + g(x-y) = \lambda g(x) f(y)$
d'Alembert-like functional equation: $f(x+y)+g(x-y)=\lambda f(x)g(y)$
iOS: Custom permission alert view text Improve the editing flow with predefined options for "Edit Summary" Another homework problem from my Group Theory class.  Let $a,b$ be elements of a group, $G$.  Let $\operatorname{ord}(a)=m$ and $\operatorname{ord}(b)=n$. Let $a$ and $b$ commute. Prove: If $m$ and $n$ are relatively prime, then the $\operatorname{ord}(ab)=mn$.  So I am having trouble starting off on this problem.  Here is what I believe I know:  $a^m=e$ and $b^n=e$ [as defined by the order of the elements] which leads $a^m=b^n$  $\operatorname{ord}(ab)=mn=\operatorname{ord}(ba)=nm$ as defined by commuting.  $\gcd(m,n)=1$ as defined by $m,n$ being relatively prime.    I don't know how to start this proof. Do I want to to go the route of $a^mb^n=e$?  Then $(ab)^{mn}=e$?   Any tips to start this would be greatly appreciated!!! How to fix broken taskbar icons in Windows 7 Closed form for $\int_0^\infty\frac{1}{(1+x^2)^s}\,dx$ when $s\in (0.5,\infty)\setminus\mathbb{N}$ Is Double-Annotating HREFLANG always alright? What if I remove one of them? I've been doing some practice questions in the textbook for an upcoming predicate calculus lecture and I think I've managed to get A and B (possibly C) correct but I am clueless on how I can manage D and E.   For A I got: ∃x(R(x)∧¬B)  For B I got: ∀x(¬R(x)∧B)  For C I got: ∃x(R(x)∧(x=y))   - Not sure if this is even remotely correct    R(X) : ”X is rich”  X = Y : ”X equals Y”  B : ”Bill Gates”  a) Someone besides Bill Gates is rich.  b) Bill Gates alone is rich.  c) At least two people are rich.  d) Exactly one person is rich.  e) Exactly two persons are rich.  The universe of discourse is the set of all people. Widget - unable to load a hot question, check your internet connection How can I simulate a slow machine in a VM? Why didn't anybody Summon the Triwizard cup in the maze? Help us test the shiny new "User Activity" page! (Plus a bunch of new features.) "A/An" preceding a parenthetical statement When a/an precedes a parenthetical aside (sometimes seen in informal/conversational writing), should the vowel rule depend on the first word in parentheses, or the next word in the "regular" flow of the sentence?     I need a (memorable) idiom (preceding an m word; use a)   or     I need an (memorable) idiom (preceding an i word; use an) Config files for assemblies in GAC Missing abstract with amsart class I'm not very good at LaTeX but I've been using it to write my dissertation. Have just come to the end and trying to format everything correctly. Whenever I try to put my Abstract paragraph into the form \begin{abstract} (writing here) \end{abstract} and click play the whole abstract section just disappears. What am I doing wrong?  \documentclass[12pt]{amsart}  \begin{document}  \begin{abstract} (my abstract here) \end{abstract} \end{document} sort but keep header line at the top I am getting output from a program that first produces one line that is a bunch of column headers, and then a bunch of lines of data. I want to cut various columns of this output and view it sorted according to various columns. Without the headers, the cutting and sorting is easily accomplished via the -k option to sort along with cut or awk to view a subset of the columns. However, this method of sorting mixes the column headers in with the rest of the lines of output. Is there an easy way to keep the headers at the top? What are the differences between consumer grade SSDs and the much more expensive enterprise SSDs? I am in need of a terabyte or so of storage space for Lucene indices on a testing server.  I'm having a hard time differentiating between disks like the  for about $450 vs  of dollars.  What concerns should I have regarding the 840 evo, and what should I be looking for?  This server will be used strictly for in-house testing purposes by our development team and will have to put up with periodic load testing. I can't crack this one.      Prove: If $\gcd(a,b,c)=1$ then there exists $z$ such that $\gcd(az+b,c) = 1$ (the only constraint is that  $a,b,c,z \in \mathbb{Z}$ and $c\neq 0)$ !! Do not use the future tense after unless   [✓]  I won't go unless you go   [✘] [I won't go] unless you will go. [✘]   Why not? Even if the above is prescriptive, what may be some reasons? In my separate question , I quote: unless = if not. If not can precede future tense; so why can't unless?  Footnote:  motivated this. Remove "Set as desktop background" from Windows context menu SSH failed to start - Missing privilege separation directory: /var/run/sshd