let $f,g$ be contiunous, bounded on $\Bbb R$. Suppose that  $$f(x+y)=f(x)f(y)-g(x)g(y),\quad g(x+y)=f(x)g(y)+f(y)g(x),\ \forall\ x,y\in\ \Bbb R.$$ $f(0)=1$, $g(0)=0$. Show that for some $a\in\Bbb R$, $f(x)=\cos ax, g(x)=\pm \sin a x$.
This question is a possibly harder version of: .  Question.  Let $f,g :\mathbb R\to\mathbb R$, such that \begin{align} f(x-y)=f(x)\, g(y)-f(y)\, g(x), \tag{1}\\ g(x-y)=g(x)\, g(y)+f(x)\, f(y), \tag{2} \end{align} for all $x,y \in \mathbb{R} $. If $g$ is continuous at $x=0$ and not identically zero, then there exists an $\alpha\in\mathbb R$, such  $$ f(x)=\sin \alpha x\quad\text{and}\quad g(x)=\cos \alpha x. $$ Is there a pair of discontinuous $f$ and $g$ satisfying $(1)$ and $(2)$?  Update. If $\ell :\mathbb R\to\mathbb R$ is a linear functional over $\mathbb Q$ (i.e., $\ell(qx+ry)=q\ell(x)+r\ell(y)$, for all $x,y\in\mathbb R$ and $q,r\in\mathbb Q$), then $$ \sin\big(\ell(x)\big), \quad \cos\big(\ell(x)\big), $$ satisfy $(1)$ and $(2)$. Discontinuous such functionals do exist, and they are obtainable using Zorn's Lemma (equivalently the Axiom of Choice.) This takes care of the second question.
Should I use 'or' or 'nor' after a negative statement? This document does not cover the SDK interfaces nor any other reference material.   I think the above is correct, but my grammatical checker in Microsoft Word underlines nor and suggests or. Why? Tools to get a pictorial function call graph of code Usage and origin of "sister" in expressions like "sister company, sister ship, sister site" etc The term  is often  used figuratively  to refer, for instance,  to a “sister company” for a company  within the same group, or to a “sister  site” for  sites that belong to the same family. This connotation as explained by the   means:         belonging to a pair or group of similar and related things, such as businesses, usually owned or operated by the same person or organization:         our sister company in Australia.          also, from :        You can use sister to describe something that is of the same type or is connected in some way to another thing you have mentioned.    ⇒ ...the International Monetary Fund and its sister organisation, the World Bank.   ⇒ ...Voyager 2 and its sister ship, Voyager 1.       This usage appears to be common with things that are  regarded as feminine and are associated, as if by kinship,  with other similar things that belong to the same group. An early  usage example of this is the "sister" referred to  ships:         the US battleship Missouri and her sister ship, the Wisconsin.      In other instances,  the "feminine" issue is less explicit as in the case of internet sites, so I guess this usage has to do with the fact that English is less gender specific when it comes to things or abstract entities.   Questions:    Does “sister” apply to whatever entity that belongs to the same group irrespective of its real or perceived gender? or could "brother" be used instead?  Where does this usage come from? I just read a book "Linear Algebra and Its Applications 4E" by David C. Lay. The book is (should be) LaTeX typeset. The matrix and equations are so nice, please see the following 3 examples:        It's so beautiful, so I want get the same effect inside my paper. Here is my code:   \usepackage{amsmath} \usepackage{colortbl} \newcommand\y{\cellcolor{clight2}} \definecolor{clight2}{RGB}{212, 237, 244}%  %%% \[A=\left[ \begin{array}{ccccc} \y 0 &amp; -3 &amp; -6 &amp; 4 &amp; 9 \\    -1 &amp; \y -2 &amp; -1 &amp; 3 &amp; 1 \\ -2 &amp; -3 &amp; 0 &amp; \y 3 &amp; -1 \\ 1 &amp; 4 &amp; 5 &amp; -9 &amp; -7      \end{array} \right]\]  \[A=\left[\begin{array}{cccccc} \rowcolor{clight2} 3 &amp; -9 &amp; 12 &amp; -9 &amp; 6 &amp; 15\\ 0 &amp; 2 &amp; -4 &amp; 4 &amp; 2 &amp; -6 \\ 0 &amp; 3 &amp; -6 &amp; 6 &amp; 4 &amp; -5 \end{array} \right]\]   And the result:     But how to add these arrows?? I have a problem when a lot of figures are in question. Some figures tend to "fly around", that is, be a paragraph below, although I placed them before that paragraph. I use code:  \begin{figure}[ht] \begin{center} \advance\leftskip-3cm \advance\rightskip-3cm \includegraphics[keepaspectratio=true,scale=0.6]{slike/visina8} \caption{} \label{visina8} \end{center}\end{figure}   to place my figures. How can I tell latex I REALLY want the figure in that specific place, no matter how much whitespace will be left? I'm using Ubuntu 12.04.3 with NVidia drivers (319) for my Quadro K2100M graphic card, and when I switch my main 3200x1800 laptop screen to mode 1920x1080 (using nvidia-settings), the display goes blurry...  I am using two screens:    My main screen is a LCD with max resolution 1920x1200. My laptop screen (original resolution 3200x1800) is sitting to the right of that.   The main problem is that, on Ubuntu, the font is far too small on my laptop screen. Because of that, I wanted to change the resolution of my laptop to 1920x1080.  Reading , I tried to  and here is the DPI configuration which I should have:   94x94 for my main LCD screen 1920x1200 235x236 for my 3200x1800 laptop screen   Is there a way to have dual DPI resolutions with an extended desktop? Imputation before or after splitting into train and test? Laws and theories What is a , a , and what is the relationship among them?  I know this is a very basic question, so any reference to epistemology will be greatly appreciated :) I just uninstalled TeX Live 2016 and installed TeX Live 2017 and I noticed that the vertical alignment of author and title in any beamer presentation with Warsaw theme has been disrupted. In this example the author is JESTIN and the title is Beamer Intro. How can I vertical center them?  Moreover, I just found the same bug in the beamer guide obtained writing texdoc beamer in the command prompt, in the part of a warsaw presentation.    Here is my code:  \documentclass{beamer}  \usetheme{warsaw}  \title[Beamer Intro]{Beamer - Introduction to \LaTeX{} presentation making} \author[JESTIN]{Jestin Joy}  \begin{document}  \begin{frame}   \titlepage \end{frame}  \end{document} Self-promotion in chat We know from the  that it is accepted only in a limited way in answers and frowned upon if found to be spammy. We know from  that it can be deleted at any time from comments if the comment is not relevant or spammy.  But what about the chat? This leaves chat to be the only legitimate place to promote products or services that the user is affiliated to (albeit in an ad-like way).  What is your opinion? I have already read        and many others   But I still have the following problem:   On one hand I've been told that if pure rotation takes place then the object goes on forever. This must imply no frictional force in any form acting on the body. On the second hand I've been told that during pure rotation(no external forces like wind) on a plane surface friction acts opposite to velocity of center of mass (accompanied by change of normal) and slows it down.     Both seem logical enough: The point of contact doesn't slip therefore no friction is required; So no retarding force and it must go on. Things don't go on forever; Friction must be present to slow it down.    So, which hand is correct? Or is there a third hand altogether? Is logic subjective? If logic is constructed from axioms, and axioms are depended on observation which in term could be subjective, does this means that logic could be limited to our observation, and not really absolute and fundamental? Linethrough/strikethrough a whole HTML table row When does a .com domain become available after it expires? (With status clientTransferProhibited) How to reorder /etc/resolv.conf at load time I have a CentOS system that retrieves its upstream DNS servers via DHCP.  I want to run DNSMasq on this box and use it as a server to resolve some hostnames for development.  The problem is when my system starts, the upstream DNS servers are loaded into /etc/resolv.conf and THEN the DNS1 entry from my ifcfg-enp0s3 setup gets loaded.   That's a problem because when I query for the internal dev names, it tries to go out to the upstream DNS server instead of checking DNSMasq first.  I need the DNSMasq server to be at the TOP of the /etc/resolv.conf and the dhcp loaded ones at the bottom of the resolv.conf so that DNSMasq will work properly.  Any simple way to do that? How do I iterate over a range of numbers defined by variables in Bash? Light "diode" and 2nd law of thermodynamics If I had a light "diode" - an object that only allowed light (at least for a range of frequencies) to travel through it in one direction, would this necessarily allow violations of the 2nd Law of Thermodynamics?  Or would it be possible to restrict other properties of the device such that it could be okay.  Why I think it should violate 2nd law:   I can place such devices at temperature T in a bath of thermal radiation at temperature T, such that it forms a box which only lets the light in or out.  This would appear to always allow one to create a thermal gradient.  Once a thermal gradient is formed, useful energy can be extracted.  This seems to allow useful energy to always be extracted from a thermal bath and thus give perpetual motion. Release one such device at temperature T in a bath of thermal radiation at temperature T.  Since the photons are absorbed or reflected in one direction, and undisturbed in the other, the device will start moving.  It can use a thermal bath to directly generate useful motion.   Why I think I'm missing something and such a device could potentially co-exist with the 2nd Law:   The above arguments seem valid if the diode was only 'partially' rectifying as well.  However there exist sunglasses which largely reflect on one side, but not from the other side. Researchers have succeeded in using meta-materials to create a "blackhole" for microwaves.   Can anyone help sort this out? Number of finite simple groups of given order is at most $2$ - is a classification-free proof possible? I'm using soul to letterspace headings in a Hungarian document and compiling it with lualatex.  For some reason soul drops some of the accented letters at the end of a word, but not all. In the example below, only ő and ű are affected:  \documentclass{minimal} \usepackage{fontspec} \setmainfont[Ligatures = TeX]{TeX Gyre Pagella} \usepackage{soul} \begin{document} \noindent \so{áll llá ÁLL LLÁ}\\ \so{íll llí ÍLL LLÍ}\\ \so{űll llű ŰLL LLŰ}\\ \so{őll llő ŐLL LLŐ}\\ \so{üll llü ÜLL LLÜ}\\ \so{öll llö ÖLL LLÖ}\\ \so{úll llú ÚLL LLÚ}\\ \so{óll lló ÓLL LLÓ}\\ \so{éll llé ÉLL LLÉ} \end{document} How to transfer minecraft pocket edition world between iPads? My daughter has built a minecraft world on our iPad. As of Christmas 2013 she has an iPad mini of her own. Is there any way to transfer her world from the old machine?   Even better...is there any way to synchronise the 2 machines?