Can a value of "length, in meters" be attributed to a pair of ends which are rigid (but not at rest) to each other?
If two ends were a certain "length" apart were they therefore at rest (or at least rigid) to each other?
Any way to have CPU temperature in the menu bar? I tried  and  but it does not update the temperature and I do not wish to change the fan speed. Are the datum and the ellipsoid the same? What is a Datum?     The Earth is shaped like a flattened sphere. This shape is called an   ellipsoid. A datum is a model of the earth that is used in mapping.   The datum consists of a series of numbers that define the shape and   size of the ellipsoid and it's orientation in space. A datum is chosen   to give the best possible fit to the true shape of the Earth.  in this   article.... Possible Duplicate:    It's time once again to support your favorite open source project through our Free Vote-Based Advertising for Open Source Projects. We are clearing the leader boards to start the first half of 2011 anew. Here is your chance to create a . 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You can see all the ads that currently make the threshold and meet our criteria on this page:  You can also view the ads that meet the threshold using 's Explanation on when the possessive should be used instead of an attributive noun How would you explain to a person who is learning English, and whose native language does not have attributive nouns, when the possessive should be used instead of an attributive noun?  In particular, how would you explain it referring to the following list of sentences?   Today's news is bad This week's schedule has been changed Tomorrow's lunch will be at the French restaurant This year's report will be communicated in the reunion room Last summer's vacations were memorable Last year's meetings were a complete disaster Where did Harry's parents get all of their money from? What is behind the definitions of work and energy? Solving multiple congruences by CRT = Chinese Remainder Theorem How do I get the solution given by CRT to match another solution, e.g. the least positive? For example say I have X = 1234. I choose mi as 5, 7, 11, 13. This satisfies the simple requirements of . More precisely given in my example k = n = 4, and the product of any k - 1 is smaller then X how come simply computing the remainder of each won't give equations that solve to X = 1234. In the case of the example, x = 4 mod 5 x = 2 mod 7 x = 2 mod 11 x = 12 mod 13  Which resolves to  (won't CRT produce the smallest?) Any hints?  Below is the solution linked to on &quot;Math Celebrity&quot; (cached to avoid link rot). Sum of nil right ideals Characteristic polynomial of companion matrix The effect of gravitational & Coulomb forces on free electrons in a conductor In a metal, why don’t the free electrons fall to the bottom of the metal due to gravity?  Also, charges in a conductor are supposed to reside on the surface so why don’t the free electrons all go to the surface?   Regarding the first question, my hunch is that it is due to the electric force which is greater than the gravitational force and electric force act along the length of the conductor.  I don't have any intuition for the second. About embryonic and genetic evidence of evolution? My question here is about evidence for evolution from Embryology and from Genetics.  Mammals do have similar Embryos, but is it the case that for each species there is an embryonic development that occurs in earlier species, has such relationship been observed? What I mean for example if we generally conceive of having species$Sp_1,Sp_2,...,Sp_n$ where each $ Sp_{i+1}$ being evolved from $Sp_i,$ then is it the case that the embryonic development stages of $ Sp_{i+1}$ contains some of the embryonic development stages of all of its ancestor species.  Similar question to 1 but regarding Genetics, is it the case that every species has a genetic thumbprint of its ancestors that lie there in the non-functional sections of its DNA, i.e. is it the case that all the genetic history of our evolution is recorded in the non-functional section of our DNA. So do human DNA have some portions of the DNA of some dinosaurs for example? Doubt regarding a limit which is related to MVT Let the function $f(x)$ be differentiable  and $f'(x)$ be continuous in $\left(-\infty,\infty \right)$ with $f'(2)=14$ then evaluate the limit  $$\lim_{x\to 0}\frac{f(2+\sin x)-f(2+x\cos x)}{x-\sin x}$$  My attempt:  $\lim_{x\to 0}\frac{f(2+\sin x)-f(2+x\cos x)}{x-\sin x}$  $\lim_{x\to 0}\left(\frac{f(2+\sin x)-f(2+x\cos x)}{\sin x-x\cos x}\right)$ $\frac{\sin x-x\cos x}{x-\sin x}$$=\left(f'(2)\right)(2)=28$  Is the method used here correct Linearization of PDE: $0$ is an eigenvalue since all translates of travelling waves are also travelling waves Consider the following PDE: $$ u_t=u_{xx}+f(u)-w,~~~~~w_t=\varepsilon (u-\gamma w),~~~~~~~~~(1) $$ where $f(u)=u(u-a)(1-u), 0&lt;a&lt;\frac{1}{2}, \varepsilon,\gamma &gt;0, \varepsilon\ll 1,\gamma\ll 1$.  A travelling wave for (1) is a solution that is a function of the single variable $\xi=x-ct$, i.e., $(u(\xi),w(\xi))$ satisfies $$ -cu'=u''+f(u)-w,~~~~~~-cw'=\varepsilon (u-\gamma w)~~~~~~~('=d/d\xi).~~~~~~(2) $$  If (1) is recast in a moving coordinate frame, i.e., in terms of variables $\xi=x-ct$ and $t$, it becomes $$ u_t=u_{\xi\xi}+cu_{\xi}+f(u)-w,~~~~~~w_t=cw_{\xi}+\varepsilon (u-\gamma w).~~~~~~~~(3) $$  One can see by (2), that a travelling wave is an equilibrium solution of (3). So one nearby question is to ask whether a travelling wave is stable.    For a fixed $\varepsilon$, let $U_{\varepsilon}(\xi)=(u_{\varepsilon}(\xi),w_{\varepsilon}(\xi))$ be a travelling wave for (1), i.e., an equilibrium solution for (3).          A standard technique for determining stability of $U_{\varepsilon}(\xi)$ is to linearize: If the right-hand side of (3) is linearized along $U_{\varepsilon}(\xi)$, the resulting operator is     $$ L\begin{pmatrix}p\\r\end{pmatrix}=\begin{pmatrix}p_{\xi\xi}+cp_{\xi}+f'(u_{\varepsilon})p-r\\cr_{\xi}+\varepsilon (p-\gamma r)\end{pmatrix}, $$     where $\begin{pmatrix}p\\r\end{pmatrix}(\xi)\in\text{BC}(\mathbb{R},\mathbb{R}^2):=\left\{u\colon\mathbb{R}\to\mathbb{R}^2 | u\text{ is bounded and uniformly continuous}\right\}$.      (The linearized criterion for stability of the travelling wave is that the spectrum of $L$ (except for $0$) lies in a left half-plane and $0$ is a simple eigenvalue.)  Thats the background.  Now, there is a statement that I could not understand yet.          Note, that $0$ must be in the spectrum because the translate of a travelling wave is another travelling wave.      I know that a translate of a travelling wave is again a travelling wave. But why does this imply that $0$ is in the spectrum of $L$?  Edit  After some looks at definitions, I have an idea:  Consider a solution $U(\xi,t)$ that is a translate of the travelling wave $U_{\varepsilon}(\xi)=(u_{\varepsilon}(\xi),w_{\varepsilon}(\xi))$, i.e.  $$ U(\xi,t)=U_{\varepsilon}(\xi-k)=(u_{\varepsilon}(\xi-k),w_{\varepsilon}(\xi -k)) $$ for some $k$. Then $U$ is an equilibrium solution of (3), too.  If I am not totally wrong, then we have  $$ p=u_{\varepsilon}(\xi-k)-u_{\varepsilon}(\xi),~~~~r=w_{\varepsilon}(\xi-k)-w_{\varepsilon}(\xi) $$ and  $$ p_t=0,~~~w_t=0, $$ because travelling waves are equilibria for (3) and a translate of a travelling wave is a travelling wave.  Hence $$ L\begin{pmatrix}p\\r\end{pmatrix}=\begin{pmatrix}0\\0\end{pmatrix}, $$ meaning that $$ \text{ker}(L-0\text{ id })\neq\left\{\begin{pmatrix}0\\0\end{pmatrix}\right\}, $$ because for each translate of the travelling wave, $p,r\neq 0$ and  $\begin{pmatrix}p\\r\end{pmatrix}\in\text{ker}(L-0\text{ id})$.  Hence $0$ is in the spectrum (point spectrum) of $L$.  Is this what was meant? Is there a way to convert an image to grayscale? I do have a longer paper. In one chapter I got several multiline equations, which I wrote using the align environment from amsmath-package. Actually those are quite long computations which should break anywhere for avoiding to much empty space on a page.  I want to use something like \allowdisplaybreaks, which does exactly what I want. The problem is, that I do not want to allow displaybreaks anywhere else in the document, where I have no control about it. I tried to use \allowdisplaybreaks directly before \begin{align}, hoping it would only apply on the next following align-enviroment. But it applies to every align-environment later in the text.  Is there any way to "stop" the command again?  And yes, I could use \displaybreak for single breaks, but I had to use it in many equations and I needed to put it in every single line, since the pagebreak should move, if I add a few more lines anywhere in the chapter.   Thanks for your help, Greyfox Inexact measurement and wavefunction collapse As is usually said, measurement of an observable $q$ leads to collapse of wavefunction to an eigenstate of the corresponding operator $\hat q$. That is, now the wavefunction in $q$ representation is $\psi(q)=\delta(q-q_0)$ where $q_0$ is result of measurement.  Now, in reality measurements are never exact, so the wavefunction should not have that sharp peak. Instead it should be broadened depending on precision of measurement. In this case, can we still introduce an operator, an eigenstate of which would the new wavefunction appear? Is it useful in any way? Or does the new wavefunction depend too much on the way it was measured so that each instrument would have its own operator? How would such opeartor look e.g. for a single-slit experiment? Radiation– white vs black house, hot or cool? How to boot from non-Ubuntu live ISO images like Fedora or CentOS? Android reject incoming call I have the following problem:  Let $a, b \in\mathbb{Z}$. Show that $\,\{ ax + by\ :\ x, y \in \mathbb{Z}\} = \{ n \gcd(a,b)\ :\ n\in \mathbb{Z} \}$   I understand that the Bezout's lemma says that $gcd(a,b) = ax +by$, so Im not really how you would go about proving the above, it doesn't really make sense to me. Any help is appreciated1