Find two pairs where neither is the case ⟨x1,y1⟩⪯⟨x2,y2⟩, ⟨x1,y1⟩⪰⟨x2,y2⟩. can you help me find any two pairs where neither ⟨x1,y1⟩⪯⟨x2,y2⟩,  ⟨x1,y1⟩⪰⟨x2,y2⟩. Neither should be the case.
Is $\le$ defined by $\langle 𝑥_1, 𝑦_1 \rangle \le \langle 𝑥_2, 𝑦_2 \rangle$, if $𝑥_1 \le 𝑥_2 \land𝑦_1 \ge 𝑦_2$ a linear order? I need to tell if the relation $\le$ defined by relationship $\langle 𝑥_1, 𝑦_1 \rangle \le \langle 𝑥_2, 𝑦_2 \rangle$, if $𝑥_1 \le 𝑥_2$ $\land$ $𝑦_1 \ge 𝑦_2$ linear order?   I have already proven that the relation is an order, but I need to decide if it is a linear order and why. Thanks for any help!
Working with large numbers in PHP To use  as you would require when using the  with large numbers (100,000+), it calls for some very large calculations.  When I multiply two large numbers (eg: 62574 and 62574) PHP seems to cast the result to a float. Getting the modulus value of that returns strange values.  $x = 62574 * 62574; var_dump($x);          // float(3915505476) ... correct var_dump($x % 104659); // int(-72945)  ... wtf.   Is there any way to make PHP perform these calculations properly? Alternatively, is there another method for finding modulus values that would work for large numbers? Where in the world is tikzlibraryhobby.code.tex? I was trying to reproduce Claudio Fiandrino's answer to .  However, I do not have the hobby package by Andrew Stacey in my local distribution.  After reading  and , I looked at the latest  (build on December 7, 2012) but could not located the hobby package in it.  I found  but do not know where each file is supposed to go in my distribution.  How do I add the hobby package? In general, what would be the easy way to get any new TikZ package installed in a Mac?  EDIT:  I have used the drag-and-drop (mac) method as described below.  (1) Drag pml3array.sty file to /Users/Username/Library/texmf/tex/latex folder. (2) Drag hobby.code.tex file to /Users/Username/Library/texmf/tex/generic/pgf/math folder. (3) Drag pgflibraryfpu.code.tex to /Users/Username/Library/texmf/tex/generic/pgf/libraries folder. (4) Drag tikzlibraryhobby.code.tex to /Users/Username/Library/texmf/tex/generic/pgf/frontendlayer/tikz/libraries folder.  I no longer get the hobby cannot be found error message.  But, is this correct? How to center window title text? Info Icon vs Question Mark Scientists have found mammoth blood, and are planning to   How does one go from having its blood to a full blown living mammoth? Is it possible?    Why does it matter if the blood is frozen or not? Suppose one found dinosaur blood, could one clone a dinsosaur (as in say 'Jurassic Park')? So when we view reputation, there is graph view. However, it seems to be off by a day. Here's what I mean:   I'm sure 70 + 20 + 15 is not 15. Its 105, which is what we see below:   And 40 + 10 is NOT 105, unless you did  :)     Anyways 40 + 10 is 50:   So yeah. Did you do what that post on codegolf says? Devs better fix that :) Account "last activity time" is not always updated It would appear that the account's "Last activity time" is not updated for every hit.  I've been active on SO all morning, and in the last five minutes have voted on some posts as well as posted an answer.  However,  says:     "Last activity: 2 hours ago from &lt;IP address&gt;"   Is there a certain amount of time that needs to elapse between hits before the activity time is updated again?  This issue could result in Enthusiast badges not being awarded when they should. It looks like we'll be adding  support to Stack Overflow. This is necessary to prevent bots, spammers, and other malicious scripted activity. We only want human beings to post or edit things here!  We'll be using a JavaScript (jQuery) CAPTCHA as a first line of defense:    The advantage of this approach is that, for most people, the CAPTCHA won't ever be visible!  However, for people with JavaScript disabled, we still need a fallback and this is where it gets tricky.  I have written a  which we can re-use.    However, I'd prefer to go with something textual to avoid the overhead of creating all these images on the server with each request.  I've seen things like..   ASCII text captcha: \/\/(_)\/\/ math puzzles: what is 7 minus 3 times 2? trivia questions: what tastes better, a toad or a popsicle?   Maybe I'm just tilting at windmills here, but I'd like to have a less resource intensive, non-image based &lt;noscript&gt; compatible CAPTCHA if possible.  Ideas? Is there a way to load my own custom CSS or Javascript while using a browser?  When I'm accessing a website or developing a website I want to load my own custom scripts from my computer. I know this can be done by Firebug or Chrome's developer toolbar, but after you refresh your browser your changes will be cleared too. This is exactly what I'm looking for adding CSS and Javascript, but I wish it has an option where you temporarily save your changes locally. Or it has an option to load external CSS or JS files.   Purpose:   When I'm developing the frontend, sometimes it's convenient for me using Chrome's developer toolbar or Firefox's Firebug to change or tweak CSS properties because I can then see the changes in real time without refreshing. One problem I see is that every time I refresh my browser, my changes also get reset. Now I wish there were an option where I could save my changes locally or load my own external .css file or .js file. I want to customize how some websites look when viewed locally, e.g., accessing a blog that has too many ads. I want to apply CSS properties to hide the junk.   Browsers I use often:   Google Chrome (for development: Built-in Dev Toolbar) Mozilla Firefox (for development: Firebug) Conceptual half-life Question So I have this word problem and I’m a bit confused about it. I have the answer and explanation but I still don’t understand: The half-life of carbon-14 is approximately 5730 years, while the half-life of carbon-12 is essentially infinite. If the ratio of carbon-14 to carbon-12 in a certain sample is 25% less than the normal ratio in nature, how old is the sample?     A. Less than 5730 years      B. Approximately 5730 years      C. Significantly greater than 5730 years, but less than 11460 years      D. Approximately 11460 years  Correct Answer: A Explanation: Because the half-life of carbon-12 is essentially infinite, a 25 percent decrease in the ratio of carbon-14 to carbon-12 means the same as a 25 percent decrease in the amount of carbon-14. If less than half of the carbon-14 has deteriorated, then less than one half-life has elapsed. Therefore, the sample is less than 5730 years old. Be careful with the wording here—the question states that the ratio is 25% less than the ratio in nature, not 25% of the ratio in nature, which would correspond to choice (D). How is the ratio of the carbon isotopes relevant to half-lives? What is the purpose of saying the half-life of an isotope is infinite? What is meant by “the normal ratio in nature”? Just based on the answer, it seems like the question said “25% of a carbon sample decayed, how old is this sample?” Obviously it’s younger than 5,730 years (the time for its first half life) because if only 25% decayed that means half a half-life has passed. I don’t see how isotopes and ratios would change this problem at all. Edit: Every answer here was very informative and helpful. It was tough to pick a best answer. I picked Cosma’s because it clicked with me. If you’re reading this and want to understand radiocarbon dating, look at every answer because there is some useful information in every answer Given three matrices $A$, $B$ and $C$ such that $ABA^T C$ is a square matrix, the derivative of the trace with respect to $A$ is:  $$ \nabla_A \operatorname{trace}( ABA^{T}C )  = CAB + C^T AB^T $$  There is a proof . However, I was not able to understand the second line of the proof. it says something like this:  $$ \nabla_A \operatorname{trace}( ABA^T C )= \nabla_\bullet \operatorname{trace}(f(\bullet)A^{T}C) + \nabla_\bullet \operatorname{trace}(f(A) \bullet^T C) $$  Where  $f(A) = AB$  I tried searching the proof or any hint using google, but I couldn't fine anything. Could anyone please help me with this second line. I cannot sleep. I am obsessed with this proof. Is every compact subset of $\Bbb{R}$ the support of some Borel measure? I have tried to prove the exercise 2.12 in Rudin's RCA:     12 Show that every compact subset of $\Bbb{R}$ is the support of a Borel measure.   For perfect (i.e. no isolated point) compact $K$ with non-zero Lebesgue measure, we just take the Lebesgue measure. However I struck on the general case.  My attempt is as follows: For every compact $K$, we can find a decreasing sequence of open set $\langle W_n\rangle_n$ s.t. $\bigcap_n W_n =K$ and $m(W_n-K)&lt;1/n$. Find a continuous function $h_n$ s.t.   $h_n(x) = 1$ for $x\in K$ and $h_n(x) = 0$ outside of $V_n$.   (such function exists by Urysohn's lemma.) For continuous function $f$ with compact support define $$\Lambda f = \lim_{n\to\infty}\frac{\int_\Bbb{R} fh_ndm}{\int_\Bbb{R} h_ndm}.$$  I can not certain that the $\Lambda$ is well-defined. But if it is well-defined, then it gives an regular Borel measure (by Riesz representation theorem) $\mu$ s.t. $\mu(K)=1$. I guess that the $\mu$ has a support $K$ but I don't know how to get it. Thanks for any help! Is it ok to leave a slow cooker unattended? Ubuntu not detecting second monitor I'm running Ubuntu 12.04 on a Lenovo x61s Thinkpad. As the screen's rather small and I want to do some video editing, I thought I'd plug in a monitor and use that. The monitor is Relisys JM777 (quite old).  When I plug it into my other computer, which is running Windows 7, it immediately mirrors the display; but when plugged into the Lenovo the monitor screen remains blank. The graphics card on the Lenovo is a "VGA compatible controller" according to SysInfo.   Anybody got any suggestions for getting this monitor to work? I'm quite new to Linux. Taking screenshot of a specific area from the command line For a graph $G$, let $W$ be the (random) vertex occupied at the ﬁrst time the random walk has visited every vertex. That is, $W$ is the last new vertex to be visited by the random walk. Prove the following remarkable fact:      For the   random walk on an $n$-cycle, $W$ is uniformly distributed over all vertices different   from the starting vertex. Are large data sets inappropriate for hypothesis testing? Make rigid body collide with hair particles Why do clocks measure arc-length? Apologies in advance for the long question.  My understanding is that in GR, massive observers move along timelike curves $x^\mu(\lambda)$, and if an observer moves from point $x^\mu(\lambda_a)$ to $x^\mu(\lambda_b)$, then his clock will measure that an amount of time $t_{ba}$ given by the curve's arc length; $$   t_{ba} = \int_{\lambda_a}^{\lambda_b}d\lambda \sqrt{-g_{\mu\nu}(x(\lambda))\dot x^\mu(\lambda)\dot x^\nu(\lambda)} $$ will have elapsed where $g_{\mu\nu}$ is a metric on spacetime with signature $(-,+,+,+)$.    Why is this so?     Here is how I would attempt to justify this fact in special relativity with $g_{\mu\nu} = \eta_{\mu\nu}$.  Consider an inertial observer $O$ in $\mathbb R^{3,1}$, and suppose that this observer sees a clock, which I'll call observer $O'$ moving around on a curve $x^\mu(\lambda)$.  If $O'$ were also an inertial observer, then given any event with coordinates $x^\mu$ as measured by $O$, observer $O'$ would measure the coordinates of the event to be $x'^\mu = \Lambda^\mu_{\phantom\mu\nu} x^\nu + x_0^\mu$ for some Lorentz transformation $\Lambda$.  If $O'$ is not inertial, then this is no longer true, and there is some more complicated family of transformations, say $T_\lambda$ between events as seen by both observers.  I would argue, however, that if we were to partition the interval $[\lambda_a, \lambda_b]$ into a large number $N$ of intervals $I_1=[\lambda_a, \lambda_i], \dots, I_N=[\lambda_{N-1}, \lambda_b]$ with $\lambda_n = \lambda_a+n\epsilon_N$ and $\epsilon_N=(\lambda_b-\lambda_a)/N$, then on each interval $I_n$, $O'$ is approximately an inertial observer in the sense that $$   T_{\lambda_n} = P_n + \mathcal O(\epsilon_N), \qquad (\star) $$ for some Poincare transformation $P_n$.  Then we would note that since $O'$ is stationary in his own reference frame, he measures his worldline to have the property $\dot x'^\mu(\lambda) = (\dot t(\lambda), \mathbf 0)$ so that $$   I_{ba}=\int_{\lambda_a}^{\lambda_b}d\lambda \,\sqrt{-\eta_{\mu\nu}\dot x'^\mu\dot x'^\mu} = \int_{\lambda_a}^{\lambda_b} d\lambda \, \sqrt{\dot t^2} = t(\lambda_b) - t(\lambda_a) = t_{ba} $$ On the other hand the integral on the left can be written as a Riemann sum using the partition above, and we can invoke ($\star$) above to get \begin{align}   I_{ba}  &amp;= \lim_{N\to\infty}\left[\sum_{n=1}^N \epsilon_N\sqrt{-\eta_{\mu\nu}\dot x'^\mu(\lambda_n)\dot x'^\nu(\lambda_n)}\right] \notag\\ &amp;= \lim_{N\to\infty}\left[\sum_{n=1}^N \epsilon_N\sqrt{-\eta_{\mu\nu}\dot x^\mu(\lambda_n)\dot x^\nu(\lambda_n)} + \mathcal{O}(\epsilon_N^2)\right] \notag\\ &amp;= \int_{\lambda_a}^{\lambda_b}d\lambda \,\sqrt{-\eta_{\mu\nu}\dot x^\mu\dot x^\mu} \end{align} Combining these two computations gives the desired result.  How do others feel about this argument?  I'm not completely comfortable with it because of the assumption $(\star)$ I made on $T_\lambda$.  I imagine that in GR a similar argument could be made by invoking local flatness of the metric. How to sync bookmarks across Google Chrome and Mozilla Firefox bookmarks?