(counting $\sin x$ as a variation of $\cos x$).  They are self-similar in that their derivative is also a function of themselves. Crucially, this sort of feedback means that higher derivatives are also similar, resulting in curves that are smooth in what seems to me to be a unique way. For the examples in the title:  $f'(x) = f(x)$ is true for $f(x)=e^x$  $f'(x) = f(x+\pi/2)$ is true for $f(x)=\cos x$  EDIT: That's for the two functions given in the title. The question is: are there any other functions that are similar to their derivative in some other way? Just one example is enough (the "duplicate" doesn't address this).  BTW It seems to me that these are the only solutions to these differential equations, and e.g. a different offset for the second one just changes the period of the $\cos$ function. But I don't see how to show this - or even how to think about it.  BTW this was inspired by Euler's equation, relating $\cos, \sin,$ and $e$. I think the fundamental connection is that they all are self-similar, and the rest is clever bit-twiddling (like how integers alternate even/odd, and $f(x) = (-1)^x$ alternates positive/negative).
Let $f \colon \mathbb{R} \rightarrow  \mathbb{R}$ be a smooth function such that $$f'(x) = f(1+x)$$ How can we find the general form of $f$? I thought of some differential equations, but not sure how to use them here. Thanks.
What to include in my portfolio? I am a relatively recent graphic design graduate. I have been working at a small printing house for about two years, but will soon be interviewing for a different position in another city.  My question is about what to include in my design portfolio. When I was in school, my professors all insisted that work included in a portfolio should be 100% original: my original photography, my original logos, my original layout, my original written content etc. However, at my current office, we rely heavily on client-supplied logos, type content, and photography—and stock photography (shutterstock, etc).  What do other designers include in their portfolios? Has anyone else had this issue and can I reasonable include these more recent projects in my portfolio if I label them appropriately (something like "layout design")? What is the definition of "creature" and is it used consistently? Tools to get a pictorial function call graph of code What happens if there is a security problem in a package in the universe repository four years after the 12.04 LTS release; will the package be updated from upstream, patched, or left alone?  It's my understanding that the "5 years of support &amp; security updates" applies only to the core of Ubuntu -- anything in Main repository.  Not for things in the Universe repository.    For a more specific example -- if I install Ruby now, and want to use it for the next several years on 12.04 and it has a security vulnerability; while this might be patched in the upstream (so I could always download the latest from their website and compile it myself or use a PPA), will this upstream fix be migrated into the precise package repositories? What about backports? Since a few months, on one of the desktop Ubuntu 14.04 systems I use, CPU has regularly been going up to 80-100% for no apparent reason. htop would usually attribute responsability to file system processes, such as nemo or ntfs. To get CPU back down I would kill these process, which in some cases would lead to loss of access to the file system and a consequent restart.  Days ago I noticed that whenever this situation happened another process was always present also using high CPU: mediascanner-service-2.0. Some research lead me to  that simply advised its removal. So I did.  The end result has been unexpectedly positive: not only are the high CPU load episodes gone, the system is altogether much faster. The speed up of the Lens is dramatic: finding an application like Calc now takes 2 seconds at most when previously it took some 20 seconds. Finding a file takes now some 5 seconds, whereas before it took in the order of 30 seconds. Heavy programmes like Eclipse are starting up faster and workspaces are more fluid. There seems to be also an overall reduction in the frequency of disk access.  What is exactly this mediascanner2.0 package? Is it part of Unity 7 or a dependency that gets installed a posteriori? Is it possible to prevent its installation?  Update: following  it is possible to track this package down to indicator-session:  $ sudo aptitude why mediascanner2.0 i   indicator-session         Recommends indicator-applet (&gt;= 0.2) | indicator-renderer i A unity8                    Provides   indicator-renderer                             i A unity8                    Recommends unity-scope-mediascanner2                      pi  unity-scope-mediascanner2 Depends    mediascanner2.0                                  Apparently, mediascanner2.0 (and the Unity 8 shell) are part of the base 14.04 system. Left to answer is the exact function of this rogue package. Is there a detailed activity log on Mac OS X that lists my recent commands and actions? How can I see what I just did on my computer? For example, an activity log showing that I copied a file from here to there? A set $A \subset l_1$ is compact if and only if closed, bounded, and one other condition A set $A \subset \ell_1$ is compact if and only if $A$ is closed and bounded and given any $\epsilon &gt;0$, there exists $n_0$ such that $\sum_{k=n}^{\infty} |x_k| &lt; \epsilon$ for all $n&gt; n_0$ and $x \in A$.   It's easy to prove the part assuming that $A \subset l_1$ is compact, but I am finding difficulty to prove the other part. I need to view the saved WiFi passwords on my non-rooted Android phone I have an LG Optimus 7 (not-rooted), and all my WiFi passwords are there. I want to view the saved WiFi passwords to transfer thm to my new Samsung Galaxy S3, but I don't know how.  I tried to search on the Internet and Google play for an application that would allow me to view this information, but I can't find one. Is there any way for me to accomplish this? How are unbalanced forces even possible, given Newton's 3rd law? I want to install the latest Firefox on my Ubuntu (64bit) box.  What is the best way to do this? Is there a specific update site or should I download the binaries manually? If doing manually, in which directory should I put the files? In my home directory, or is there a better place? Suppose there are two face down cards each with a positive real number and with one twice the other. Each card has value equal to its number. You are given one of the cards (with value $x$) and after you have seen it, the dealer offers you an opportunity to swap without anyone having looked at the other card.  If you choose to swap, your expected value should be the same, as you still have a $50\%$ chance of getting the higher card and $50\%$ of getting the lower card.  However, the other card has a $50\%$ chance of being $0.5x$ and a $50\%$ chance of being $2x$. If we keep the card, our expected value is $x$, while if we swap it, then our expected value is: $$0.5(0.5x)+0.5(2x)=1.25x$$ so it seems like it is better to swap. Can anyone explain this apparent contradiction? From the Immediate Window in Visual Studio:   &gt; Path.Combine(@"C:\x", "y") "C:\\x\\y" &gt; Path.Combine(@"C:\x", @"\y") "\\y"   It seems that they should both be the same.    The old FileSystemObject.BuildPath() didn't work this way... How do I set exposure with a manual flash? I recently got an old Canon AE-1 film camera back in working order. I'd like to use my modern 430EXII Speedlight flash with it, however since there is no TTL metering, I have no idea how much the flash will affect the exposure of my shot (note that I have confirmed that the AE-1 will trigger the flash). I'd rather not discover that my shots were either blown out or way to dark after going through several rolls of film.  Are there any basic rules-of-thumb or basic calculations for anticipating how much a burst from the 430EXII will affect the exposure of a subject at various distances? Let $f$ be an analytic function such that $\operatorname{Im}(f(z))=\operatorname{Re}(f(z))^2$. Prove that $f$ is constant. How to solve this question? Also it would be great if someone could tell me from which book this question is from? How to find Mac admin password not reset Mac admin password The Drupal 8 REST UI configuration page asks me to select Authentication providers for GET method; otherwise, it doesn't allow me to export the REST request.  How can I provide anonymous user to access my REST GET API without providing any auth information? How to enable assembly bind failure logging (Fusion) in .NET How do I enable assembly bind failure logging (Fusion) in .NET? How can I render an edge only mesh as solid lines? I have a schematic of a raspberry pi that is completely flat, no solid faces, just connected vertices.  Is it possible to render these lines somehow, so that it looks like a 2D drawing?  Maybe with freestyle? How can I change pdf page size? I would like to split pages in pdf so that each page would be divided into 2 pages ( e.g. form 100 pages I would go into 200 but smaller - exactly half size).  I need this to read pdfs on my ebook reader.  Anybody knows how can this be done ? Recall the definition of the Weyl Chambers: A Weyl Chamber is a region of $V \setminus \bigcup_{\alpha \in \Phi} H_{\alpha}$, where $V$ is underlying Euclidean space, and $H_\alpha$ the hyperplane that is perpendicular to $\alpha$ the elements of  $\Phi$ in $V$.   I want to show that:   The Weyl chambers are open, convex and connected.   My attempts:   A single hyperplane $H_\alpha$ separates the space $V$ into two half-spaces: positive half-space $H_\alpha^+=\{x \in V \mid (\alpha,x)&gt;0\}$ and negative half-space $H_\alpha^-=\{x \in V \mid (\alpha,x)&lt;0\}$. It is clear that $V \setminus H_\alpha = H_\alpha^+\cup H^-_\alpha.$ As the result, the Weyl chambers can be regarded as components of  $$V \setminus \bigcup_{\alpha \in \Phi} H_{\alpha}= \bigcap_{\alpha \in \Phi} \left( V \setminus H_{\alpha} \right)= \bigcap_{\alpha \in \Phi} \left( H_\alpha^+\cup H_\alpha^- \right).$$ By the discussion above, a Weyl chamber is simply a $\textbf{intersection}$ of finite half-spaces. It is the case since one may write $$\left( H_\alpha^+\cup H_\alpha^- \right) \cap \left( H_\beta^+\cup H_\beta^- \right) =\left( H_\alpha^+\cap H_\beta^+ \right)  \cup\left( H_\alpha^+\cap H_\beta^- \right)  \cup \left( H_\alpha^-\cap H_\beta^+ \right)  \cup\left( H_\alpha^-\cap H_\beta^- \right).$$ Therefore openness and convexity of the Weyl chambers are straightforward since half-spaces are open and convex, so are their intersections. Moreover, we would replace $V$ by $\Bbb R^{\dim V}$ since they are isomorphic. Now we can prove that the Weyl chamber is connected by showing that every convex set in $\Bbb R^{\dim V}$ is connected. And I'm sure that proof of this statement existed. So am I done?