I'm searching for applications where the distribution of the eigenvalues of a unitary matrix are important. Example properties of the eigenvalues could be that the eigenvalues are clustered, that they live in some half plane, that, in the case that the matrix has real eigenvalues, that a certain fraction are 1, etc. Are there any well known areas where this is important?
I'm searching for applications where the distribution of the eigenvalues of a unitary matrix are important. Example properties of the eigenvalues could be that the eigenvalues are clustered, that they live in some half plane, that, in the case that the matrix is orthogonal, that a certain fraction are 1, etc.
Is “what’s” a correct short form of “what does”? E.g. “What’s he think?”  Usually ’s is short for “is” so I don’t know. What do I need to consider to choose between dSLR, mirrorless, or a compact as my first "serious" camera? I am thinking of buying a camera to learn photography. Most of the time I'll be shooting portraits and landscapes. But I am not sure where I should start. Most of my friends tell me that I should get an entry level DSLR, but I am not sure if I have the persistence and enthusiasm to carry it all day in a backpack. But the camera on my cellphone doesn't seem to be enough to begin with.  So what I want is some expert advice (preferably with professional experience). Can most compact fixed-lens or mirrorless cameras fit the situations I mentioned? Or do I really need to go to a DSLR to get a tool to learn photography? Which computer science / programming Stack Exchange sites do I post on? List of tables, db schema, dump etc using the Python sqlite3 API example of not surjective isometry on a bounded not closed set. Is there an example of a bounded not closed set $A\subset \mathbb R^n$ for some $n\in \mathbb N$ and an isometry $f:A\rightarrow A$ ($\forall x,y\in A\space;\space d(x,y)=d(f(x),f(y))$ where $d$ is the euclidean metric) such that $f$ isn't surjective? Given $a,b,c,d \in \mathbb{Z}$, there is $x,y$ such that  $$(a^2 + b^2)(c^2 + d^2) = x^2 + y^2$$  One can show this by considering the complex number $a + bi$ and $c+ di$, using complex properties to deduce that $x = ac - bd, y = ad + bc$ is a solution.   However, given that either $a^2, b^2$ are distinct or $c^2,d^2$ are distinct, then how can one can find nonzero solutions $x,y$?  Sorry, I am assuming $a,b,c,d$ are nonzero. Calculate the product of distances for a line perpendicular to hyperbola Let us have a hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ and assume the normal at P meets the transverse (horizontal) axis $AA'$ in $G$ and the conjugate (vertical) axis $BB'$ in $g$. Assume CF be perpendicular to the normal from the center.   Find $\frac{PF \cdot PG}{CB^2}$ Find $PF \cdot Pg$   How can I solve the question in least amount of steps?  My attempt:  I first formed the equation of the normal and then I found out the points where it cuts the conjugate and transverse axis. After finding these points I used distance formula to find the distance required and convert it in the form that the question asked. But its a long method and took lot of steps We're building a feature for a website and trying to think of a hypernym for clients, partners and members so that the feature remains generic enough for re-use later on other websites. Checking out  and  (and some papers) was a bit difficult for understanding Group Action. The closest to understanding was this part:    I understand the definition of homomorphism (a structure-preserving map between two algebraic structures of the same type). And from what I understand the symmetric group is a group of functions (bijective functions) mapping elements of a set $X$ onto itself.  However I get confused at this line:     ...an action of $G$ on $X$ may be formally defined as a group homomorphism $\phi$ from $G$ to the symmetric group of $X$.   That seems to be the definition of group action, but I am having trouble understanding how to apply that.  What that says to me is it is a homomorphism (function) $f \colon X \rightarrow S$ where $S$ is a set of bijective functions on $X$. That itself seems weird, I thought a group action would result in a number or something related to $\{1, 2, 3\}$ (defined next).  Continuing, if we have a set $X = \{1, 2, 3\}$ and a group $(*,\{4, 5, 6\})$, then the symmetric group of $X$ would be something like $S = \{1 \rightarrow 2, 2 \rightarrow 3, \dots \}$, and $\phi$ would be something like $f : \{4, 5, 6\} \rightarrow S$. That doesn't make sense to me and I feel like I'm interpreting something wrong. I don't see how mapping to a set of functions $S$ will result in an action on the set. How to get Ubuntu download for a Mac? I am trying to pass a reference to a two-dimensional array to a function in C++. I know the size of both dimensions at compile time. Here is what I have right now:  const int board_width = 80; const int board_height = 80; void do_something(int[board_width][board_height]&amp; array);  //function prototype   But this doesn't work. I get this error from g++:  error: expected ‘,’ or ‘...’ before ‘*’ token   What does this error mean, and how can I fix it? Terminal : needs full path to execute a command Let's suppose i have an application in /Applications/MacPorts/KDE4/myapp.app/Contents/MacOS/myapp.  When i use cd to reach the directory 'MacOS' and then issue the command myapp it says command not found.  But when i issue the command with the full path like /Applications/MacPorts/KDE4/myapp.app/Contents/MacOS/myapp it works.  I had the same problem with mysql inside xampp.  Why is that? How to install TP-LINK T2UH Wireless adapter Driver (Ralink mt7610u) Automorphism group of the quaternion group Let $Q_8$ be the quaternion group. How do we determine the automorphism group $Aut(Q_8)$ of $Q_8$ algebraically? I searched for this problem on internet. I found some geometric proofs that $Aut(Q_8)$ is isomorphic to the rotation group of a cube, hence it is isomorphic to the symmetric group $S_4$. I would like to know an algebraic proof that $Aut(Q_8)$ is isomorphic to $S_4$. How do I change the computer name? How to diagnose slow booting or logon in Windows? When I start my PC, it boots into Windows (Windows 7 32-bit) quite happily, I login and the main Windows screen appears. At that point, however, the system is extremely unresponsive for up to 5 minutes - if I click on Firefox to start it, nothing appears to happen for a long time, etc, etc. The mouse and Windows UI is responsive, but programs take forever to start, webpages seem to hang (or at least load extremely slowly) and so on.  I imagine that the issue is with one or more programs that run on startup, but I don't know quite how to identify which one it might be. The event log shows nothing of any obvious interest. I can't see any high CPU activity in task manager or Process Explorer, and I can't hear any high disk activity.  I'd rather understand what's happening and then work out how to address it, than have to go for a blind "switch things off and see what helps" approach, but maybe that's being optimistic :-)  I suspect disk bottlenecks or network, but I don't know how I could set something up to diagnose this (and in particular, given that I can't get programs to start in a timely manner, I'd prefer something that automatically starts and logs data for later analysis than something like Process Explorer, which always leaves me worrying that by the time it started the problem had gone away :-)  I guess my ideal would be 2 things:   Something that gave me an overall "what are the current key bottlenecks in this system" summary (ideally, which could be set to run in the background and collect data for later analysis). Some means of starting a program (say, Firefox) and have it record what was going on in terms of waits and delays in the system.   Unfortunately, I've never really found anything like either of these.  I'm not a professional systems admin - my background is database admin - so if there are "obvious" tools to look at or try, then I'd be grateful to know of them (particularly if the information they give is easy for an amateur to analyze) :-) I am trying to balance underbraces in the equation below, how can I ensure that both underbraces at the same depth?  \begin{equation} y = \underbrace{ \left [ x^2 + r^2 (a^2+\frac{a^{0.3}}{b}) \right]}_\text{growth rate} \times  \underbrace{ C_0}_\text{initial} \end{equation} Let us say we have the following Makefile  include config.mk  # Generate summary table. results.txt : $(ZIPF_SRC) isles.dat abyss.dat last.dat     $(ZIPF_EXE) *.dat &gt; $@  # Count words. .PHONY : dats dats : isles.dat abyss.dat last.dat  %.dat : books/%.txt $(COUNT_SRC)     $(COUNT_EXE) $&lt; $*.dat  .PHONY : clean clean :     rm -f *.dat     rm -f results.txt   where all the variables are defined in config.mk  LANGUAGE=python COUNT_SRC=countwords.py COUNT_EXE=$(LANGUAGE) $(COUNT_SRC) ZIPF_SRC=testzipf.py ZIPF_EXE=$(LANGUAGE) $(ZIPF_SRC)   Is there any tool available that can expand all the include lines within the makefile so as to produce a bigMakefile? If such a tool does not currently exist, can a python/perl/awk/bash script be cooked up to achieve this goal? This is from  .      Let $\mathcal{A}$ be a locally small category, and let $A,A' \in \mathcal{A}$ with $\operatorname{Hom}(-,A) \cong \operatorname{Hom}(-,A')$. Prove directly that $A \cong A'$.     My thoughts:      Let $\eta$ denote the natural isomoprhism between the two functors.    We have    $$H(A,A) \xrightarrow{ \eta_A} H(A,A')$$   So $\eta_A(id_A): A \rightarrow A'$ is a candidate. Now I want to construct inverse.    $$ H(A',A') \xrightarrow{\eta_{A'}} H(A',A)$$   Then the map $\eta^{-1}_{A'} (id_{A'}):A' \rightarrow A$.    So I'd like to show that the composition is identity. By the naturality condition of $\eta$, from $H_A(A) \rightarrow H_{A'}(A')$ (and similarly on other direction)    $$ \eta_A(id_A) \circ \eta^{-1}_{A'}(id_{A'}) = \eta_{A'}(\eta_{A'}^{-1}(id_{A'}))= id_{A'} $$   We deduce these two maps are inverses. Hence $A' \cong A$.    Is this correct? Or is there a neat way to see this? When I was a kid we had a VHS tape of some Disney live action (not animated) sci-fi movie. It may have been a mini-series or a made-for-tv movie, but I'm not sure. I'm also not sure about the year but it would probably have been late 80s or early 90s based on when I would have seen it. I know it was Disney because they had some live action Goofy and Mickey running around the movie's set as an introduction discussing the plot with a host before the movie.  The plot is basically as follows:   A starship is loaded with mostly children and teenagers as a cross-galaxy human settling effort. Accidents (which are not really accidents and later revealed) cause adult crew members to be killed.  They travel to another planet and face all sorts of troubles. It arrives at the new planet after overcoming obstacles.   Other memorable scenes that may help:   Navigating out of earth's asteroid belt made out of space junk. One of the adults was killed via an airlock malfunction. The ship was supposed to be peaceful with no weapons. At one point they bring on board some crazy guy who puts together a rail gun. There's a memorable line like "Do you want no weapons at all or a well aimed rock?"  There's a planet or space station (I can't remember which) that has a fight scene with the evil bad guy. This scene takes place on what looks to be a carnival or fairground. The bad guys are trying to steal the ship the kids are on because it has some kind of really good futuristic space drive.   Hopefully you remember it better than I do! I've tried looking online for it before with no luck.