I don't have much idea about inner product space.  So, plz help me to understand this question;     Let $A$ be a $n\times n$ matrix with real entries.   Define $\langle x,y\rangle _A:=\langle Ax,Ay\rangle, \quad x,y\in \mathbb{R}^n$.  Then $\langle x,y\rangle_A$ defines an inner-product if   and only if      (a) Ker A={0}    (b) Rank A=n    (c) All eigenvalues of are positive.    (d) All eigenvalues of are non-negative      More than one option may be true.
Let $A$ be an $n \times n$ matrix with real enteries. Define $\langle x,y\rangle_A = \langle Ax,Ay\rangle, \quad x,y \in \mathbb R^n$ , where $\langle,\rangle$ is a standard inner product on $\mathbb R^n$. Then  $\langle x,y\rangle_A$ is an inner product iff  1)  $\ker A = \{0\}$.  2) $\operatorname{rank} A= n$.  3) all eigenvalues of $A$ are positive.  4) all eigen value of $A$ are non negative.  Clearly if $\ker A \neq \{0\}$, then there exist $x \neq 0$ such that $Ax =0$, then $\langle Ax, Ax\rangle_A = 0$, then $\langle ,\rangle_A$ is not inner product, so $\ker A = \{0\}$, thus rank of $A$ is $n$.  I am unable to check the 3) and 4) option. Please help to check the other option. Thank you
For an isolated system, can the entropy decrease or increase? In any sizable system, the number of equilibrium states are much, much greater then the number of non-equilibrium states. Since each accessible micro state is equally probably, it is overwhelmingly likely that we will find the system in an equilibrium state.   However, for a closed system, one that does not interact with any external system, the number of micro states is fixed. Therefore the entropy is fixed.  At any given time, the system will be in one of its micro states. Even if the system is in a micro state that does not look like equilibrium from a macroscopic point of view, its entropy will remain the same, since entropy is a property of all the micro states, not just a given micro state.   So, are the number of accessible micro states (and hence entropy) of an isolated system constant? I am currently working to design a flash virtual tour, but I know little about how the photographs are taken.  Ideally I'd like to be able to take a set of 6 shots from a single focal point, and be able to align them seamlessly into a cube form. The other solution is to take a special fisheye image, and stretch it dynamically in flash.  I've seen examples of both as virtual tours, and ones formatted as a rotating cube are more responsive and have a higher image quality. The airbus and Cruden Homes are definitely in the cube format, I'm not entirely certain which format the New York VT uses:        I have a set of questions, the first is the most important, the others may become separate questions if I don't get an answer here:   How are those photos taken? Is there a specific mount that can be used? How long does it take to set up? How much post-processing is necessary on the images? I have a DEM layer from ASTER satellite and I want to get the slope information from it.  I use terrain analysis > slope, but the output is so wrong. Why is that?  Here is the DEM:     And here is the "Slope" that is totally wrong. It should stretch the values from 0 - 89.9097 but it does not. Say that I have a vertical thermally insulated cylinder which contains an ideal monoatomic gas under a weightless piston and the system is initially in equilibrium.   Now if a load of weight $W$ is kept on the piston,the piston would fall by some distance and the system would attain a new equilibrium. It is intuitive that the sum of the work done by gravity and the work done by atmospheric pressure would change into internal energy of the gas.   However, my book mentions that only the work done by gravity goes into the increase in internal energy of gas. It does not say anything about where the atmospheric work would go. Why is it so? Let $A$ be an $m \times n$ matrix of rank $r$, let $I$ be a set of row indices such that the corresponding rows of $A$ are independent and let $J$ be a set of $r$ column indices such that the corresponding columns of $A$ are independent.  Let $M$ denote the $r \times r$ submatrix obtained by taking rows from $I$ and columns from $J$.  Then $M$ is invertible.  So far I've got: Let $B$ be the $m \times r$ matrix obtain from taking the rows from $J$.  $B$ is row-reducible to having a pivot in each column or else $Ax = 0$ has more than one solution, a contradiction.  Also, the dimension of the row space of $B$ equals $r$ as well. Securing WMS against unauthorized access? In a project the customer wants the images of the WMS to be secured against unauthorized access.   What are the common solutions to realize that?   Are they compatible with the OGC-standards? How do you set password-hash for OpenLDAP? Hyperref "enquotes" metadata EDIT: As Bill B. and Caramdir point out in their comments after Bill B. produced his results posted below, more information is needed in order to make a better call. To remedy, my output was produced using MiKTeX 2.8 pdflatex with no special command line flags or switches. My pdf viewer is Adobe Acrobat Pro 9.0. The platform is Windows 7. All software including hyperref is fully up to date against the latest rev levels and patches.  As I'm guessing that this is a pretty normal setup, at least in the Windows world, I'd be very interested (and grateful) in learning whether this problem affects other people or whether it's just some kind of rather weird one-off. My gut feel after seeing Bill B's results (and also after having traversed hyperref's code) is that the culprit is Adobe Acrobat or possibly MiKTeX pdflatex.    Can anyone corroborate? I'd be very grateful to know.    Original question:  I have a problem with hyperref; it "enquotes" Author and Keywords (but not Title or Subject) metadata if the data sent to these contain a comma or semi-colon (but not, say, a full stop or a colon).    I've included some figures and code that show this below.         \documentclass{article}  \newcommand*\metadata{% % uncomment all but one line below %  a  b  c%     &lt;-- no problem    a, b, c%     &lt;-- problem %  a; b; c%     &lt;-- problem %  a: b: c%     &lt;-- no problem %  a* b* c%     &lt;-- no problem }  \usepackage[   pdftitle={\metadata},    % never enquoted   pdfauthor={\metadata},   % enquoted if , or ; is present   pdfsubject={\metadata},  % never enquoted   pdfkeywords={\metadata}, % enquoted if , or ; is present ]{hyperref}  \begin{document}   Press ctl-d (in Acrobat Reader) to see pdf file metadata. \end{document}   Not only is this quite ugly but, more importantly, it interferes with metadata extracting programs used by certain document repositories.   I've worked my way through hyperref's code without much success. Would anyone have a solution for this problem?  Or advice about what I might be doing wrong? I've just installed Ubuntu 14.04 and am trying to install php 5.3. But when I run   apt-get install php5    it's installing php 5.5.9-1ubuntu4. I need to use php 5.3 in the system because the application I'm working on is built with php 5.3.   How can I achieve this? Finding $\int \frac{dx}{a+b \cos x}$ without Weierstrass substitution. I saw somewhere on Math Stack that there was a way of finding integrals in the form $$\int \frac{dx}{a+b \cos x}$$ without using Weierstrass substitution, which is the usual technique.   When $a,b=1$ we can just multiply the numerator and denominator by $1-\cos x$ and that solves the problem nicely.   But I remember that the technique I saw was a nice way of evaluating these even when $a,b\neq 1$. Crossfading Video AND Audio with ffmpeg This Super User answer  has got me so far, but now that I have tried many solutions and seen every SO link out there, it's time to ask you all for help.  I am trying to crossfade 2 videos that are 10 seconds each in ts format with audio and video at the same time.   The steps are:   Add complex filter black screen Fade out video 1 after 9 seconds, with 1 second fade duration Fade in video 2 at 9 seconds, with 1 second fade duration Trim black filter output to 19 seconds   This works fine for just video alone, but when I attempt to add audio, I can't get past this error message:     Output pad "default" with type audio of the filter instance "Parsed_asetpts_4" of asetpts not connected to any destination   It appears I'm not connecting the audio output properly. I think I need to alter the [over] lines to include audio. But how do I do this?  `ffmpeg -i vid1.ts -i vid2.ts -f lavfi -i color=black -filter_complex \ "[0:v]format=pix_fmts=yuva420p,\ fade=t=out:st=9:d=1:alpha=1,setpts=PTS-STARTPTS[v0];\ [0:a]\ afade=t=out:st=9:d=1,asetpts=PTS-STARTPTS[a0];\ [1:v]format=pix_fmts=yuva420p,\ fade=t=in:st=0:d=1:alpha=1,setpts=PTS-STARTPTS+9/TB[v1];\ [1:a]\ afade=t=in:st=0:d=1,asetpts=PTS-STARTPTS+9/TB[a1];\ [a0][a1]amix=inputs=2;\  [2:v]scale=720x406,trim=duration=19[over];\ [over][v0]overlay[over1];\ [over1][v1]overlay=format=yuv420[outv]" -vcodec libx264 -y -map [outv] final.ts` Do electrical devices "take what they need" One concept of electronics I have a hard time understanding is if things like motors, actuators, solenoids etc. use as much power as they need or what you give them.   If a motor needs 12 volts and 500ma and I supply it with 12 volts and 3000ma will it only consume 500ma? Also, if I supply it with 15volts and 500ma what will happen?   It seems logical that an LED and a DC motor are way different when it comes to requiring/using electric where as an led has to be completely regulated and (I assume) a DC motor doesn't.   Is my understanding wrong? SQL - an error occurred during the pre-login handshake Mac mini Mac OS 10.6 server migration to new hardware I have a Mac Mini with Mac OS X 10.6 Server installed. It has two external drives attached to it. One is for Time Machine and other contains data. I have bought the latest Mac Mini with Mac OS X 10.7 and two new hard drives. I want to migrate to the new hardware (the new Mac Mini and the new hard disks). Can Migration Assistant do that for me or do I have to transfer the data in the external drive manually? I just deleted a file on my Nexus S that I really need back. (It was actually part of an app using a sqlite database that did a drop table when upgrading to a new version.)  Presumably I need to root the phone first (which I hear is easy on Nexus phones) but am I in a catch-22 here? Namely, does rooting the phone mean wiping/overwriting the partition where the data lives?  If not, what do you recommend for data recovery tools? Presumably the file was just unlinked and not actively overwritten (per usual when "deleting" a file) so technically the data must be there... If $|f(z)|\le |g(z)|$, can we say $f$ is constant? Let $f$ and $g$ be entire functions and $g(z)\neq 0$ for all $z\in \mathbb{C}$. If $|f(z)|\le |g(z)|$, can we say $f$ is constant?  Liouville theorem says that an entire bounded function is constant, but g is not given bounded here. So I think it should be false. What else can we say about $f$? How should I structure my URLs for both SEO and localization? I'm seeing red dots everywhere On all Stack Exchange sites where I have review privileges, the review queue menu icon has a red dot. No matter what I try, it doesn't go away.     on Meta Stack Overflow might or might not be related. I know why questions, answers and some other aspects of the site are color-coded and what they mean but I can't find why badges are color-coded. I'll include a snapshot to show what I mean. Why have a letter in a word when it’s silent in pronunciation, like the b in debt?  Can anyone please clarify my uncertainty here?