Prove that $\sqrt{2}$, $\sqrt{3}$, and $\sqrt{5}$ can not all be terms of a single arithmetic progression
A sequence includes $a_p=\sqrt2$, $a_q=\sqrt3$, $a_r=\sqrt5$ for some $1\leq p<q<r$. Can these be terms of an arithmetic progression? harmonic?
How to tell if a site is using Drupal 7 I know how to check , but is there a way to tell if a site is built with Drupal 7? I know if $\ i\ $  of the following form $\ 3x^2 + (6y-3)x - y\ $ or $\ 3x^2 + (6y-3)x + y - 1, \ \ x,y \in \mathbb{Z}^{+},i \in \mathbb Z_{\ge 0}$,  then $\ 12i + 5\ $ must be composite number, because:  $12(3x^2 + (6y-3)x + y - 1) + 5 = 36x^2 + (72y - 36)x + (12y - 7) = (6x + 12y - 7)(6x  + 1)$  How come if $\ i\ $ not of the following form $\ 3x^2 + (6y-3)x - y\ $ and $\ 3x^2 + (6y-3)x + y - 1\ $,  then $12i + 5$ must be prime? $\ x,y \in \mathbb{Z}^{+},i \in \mathbb Z_{\ge 0}$.  For example: $\ 5 = \ 3*1^2 + (6*1-3)*1 - 1\ $ ,when $\ x = y = 1\ $, as proved before,$\ 5*12+5\ $ must be composite number;  $\ 0, \ 1. \ 2.\ 3.\ 4\ $ can't of the following form $\ 3x^2 + (6y-3)x - y\ $ and $\ 3x^2 + (6y-3)x + y - 1\ $,  $\ 12*0 + 5 = 5\ $, $\ 5 $ is prime,$\ 12*1 + 5 = 17\ $, $\ 17 $ is prime, and so on, Cyclic orbit animation I'm trying to make one object orbit another. But I want it to repeat, rather than stop at 100 frames. It seems there is no option for this. Is there a way to make this happen? Word or phrase that can mean something and its opposite So I was in my New Leaf house and I accidentally put on a pair of pants instead of displaying them. Then I went and clicked on myself and I have no option to take off my pants. Ever. I swapped them with another pair for the display, but I can't get those off either. And the weird bit is when I put on the pants nothing was swapped back to my inventory.  How do I take off my pants? We have a sequence $(x_{n})_{n=1}^\infty$ in a Hilbert space $H$ and we know that:   for every $n \in \mathbb{N}$ holds $\|x_{n}\|\le1$ for every $m\ne n$ holds $\|x_{n}-x_{m}\|\ge r &gt; 0$   How I can show that $r \le \sqrt2$ ? Run command on another(new) terminal window When is a clause not essential? I am having trouble figuring out when to use commas to set off &quot;nonessential&quot; information.  Sometimes it's obvious:  Bob, who is thirty years old, is an alcoholic.  But other times I'm not sure:  The day he quits drinking * he will start a llama farm He has his heart set on owning El Duderino ranch * in New Mexico.  In the first case, my ear says that there should be a comma at *, even though information before it seems essential to me.  In the second case, the stuff after * is not essential, and yet it seems a little much to use a comma. Even that last sentence I wrote confuses me.  &quot;In the second case&quot; seems essential but I used a comma.  Is this correct? Does it depend on personal style and the length of the clause?  Or perhaps I'm misinterpreting the meaning of &quot;essential&quot; in this context? How to construct a relative path in Java from two absolute paths (or URLs)? Given two absolute paths, e.g.  /var/data/stuff/xyz.dat /var/data   How can one create a relative path that uses the second path as its base? In the example above, the result should be: ./stuff/xyz.dat QGIS color polygon according to points I have two layers. Layer 1 contains points (x,y) and layer 2 contains one polygon. All points in layer 1 are positioned within the polygon of layer 2. The points in layer 1 contain an attribute with information on the topography/elevation at the specific point. Can I color the polygon in layer 2 based on attributes from the points in layer 1 in order to create an elevation map? Can we create a subsection of the References Cited section with different reference numbering? I need to add some more publications. I do not want to change the previous reference numbers, so I need to create new citations with difference reference numbering like [p1], [p2], etc. Can LaTeX do that? If rest mass does not change with $v$ then why is infinite energy required to accelerate an object to the speed of light? I know that as the velocity increases, the mass of the object also increases so it becomes tougher and tougher to move the object which ultimately leads to a requirement of infinite energy to accelerate an object to the speed of light. But I have a doubt.  As far as I know only the observable mass (Relativistic mass) increases but not the Proper Mass or Intrinsic Mass, right? The actual mass of the object will remain the same. So if the actual mass ($m_0)$ remains the same and only the observable mass increases, why is more and more energy required? I know that even if the object exceeds the speed of light, we will not be able to say that it is moving faster than the speed of light but is it possible to make it move faster than light?  If rest mass does not change with $v$ then why is infinite energy required to accelerate an object to the speed of light?  I quote Igor Ivanov (a stackexchange user from this question ) to give an insight into my question.     The mass (the true mass which physicists actually deal with when they calculate something concerning relativistic particles) does not change with velocity. The mass (the true mass!) is an intrinsic property of a body, and it does not depends on the observer's frame of reference. I strongly suggest to  popular article by Lev Okun, where he calls the concept of relativistic mass a "pedagogical virus".   EDIT  After reading the answers I have a doubt.  SO can I say that only the overall energy of the system increases while the mass remains constant? But then if mass remains constant, then why is more and more energy gradually required? I mean there should be a reason for that requirement of infinite energy. In the proof of the lemma     For every limit ordinal $\alpha$, $\textrm{cf } \alpha$ is a regular cardinal    Jech says that     It is easy to see that if $\alpha$ is not a cardinal, then using a mapping of $|\alpha|$ onto $\alpha$ one can construct a cofinal sequence in $\alpha$ of length $\leq |\alpha|$, and therefore $\textrm{cf } \alpha &lt; \alpha$.    I've spent a while trying to construct this cofinal sequence and was wondering if I was on the right path:   From a bijection $f$ between $\kappa = |\alpha|$ and $\alpha$ take $S = \{\beta \in \kappa : \ f(\gamma) &lt; f(\beta)$ for all $\gamma &lt; \beta\}$ then if $\xi$ is the order type of $S$ and $g$ is the isomorphism from $\xi$ to $S$ the function $f\circ g$ is a cofinal sequence in $\alpha$ and $\xi &lt; \alpha$.   Is this on the right track, or is there an easier way of doing it? How to upload an mp3 to Facebook? I know it's possible to upload videos to Facebook. Is it possible to upload an mp3 to Facebook using their user interface? Can we express a trigonometric function for the product of two angles as a function of trigonometric functions of its factors?  For example: Is there a formula for $\sin(xy)$ as a function of $\sin x$ and $\sin y$ or other trigonometric functions of $x$ and $y$. I'm studying abstract algebra for the first time.  let $F$ be a field. I learned that polynomials over field $F$ means polynomials with coefficients in $F$ (denoted by $F[x]$). But when I keep studying, I get to confused the word, 'polynomial functions from $F$ to $F$'.  Firstly, I consider this as induced functions from $F[x]$, so I guess all coefficients of these functions are in $F$. But let $F_{4}$ be a finite field consisting of four elements $\{0, 1, a, b\}$. and let $f: \{0,1\} \rightarrow \{0,1\}$ be a function defined by $$ f(x)=ax^2-ax$$ Since $\{0,1\}$ is a subfield of $F_{4}$ and $f(0)=0, f(1)=0$, I think $f$ is a example polynomial functions from $F$ to $F$ of which coefficients are not in $F$. Am I wrong? if so, please help me.  I encounter this problem while solving 'the set of all polynomial functions from $\Bbb{Z_{3}}$ to $\Bbb{Z_{3}}$ cannot possibly be isomorphic to $\Bbb{Z_{3}}[x]$'. What is the difference between $* and $@? I need to calculate the Cramer Rao lower bound of variance for the parameter $\theta$ of the distribution $$f(x)=\frac{1}{\pi(1+(x-\theta)^2)}$$   How do I proceed I have calculated $$4 E\frac{(X-\theta)^2}{1+X^2+\theta^2-2X\theta}$$  Can somebody help Anyone know of a custom GPS ROM for Samsung Galaxy 5? Possible Duplicate:          I recently bought a Samsung Galaxy 5 (model number GT-I5500) and I'm new to the android world.   I'm not satisfied with the performance of the GPS on my cell phone so I thought it would be interesting to use a custom ROM to see if it improves a bit.   Anyone knows of a ROM that was tested in this model? Why can't we use entanglement to defy Heisenberg's Uncertainty Principle? In principle, it is possible to entangle any property of two particles, including speed and momentum. Surely then, this could be used to defy the Uncertainty Principle, which states that the momentum and position of a particle cannot be together measured with accuracy, but the more accurate one gets, the less accurate the other becomes:  $$\Delta x\Delta p=\dfrac{\hbar}{2}.$$  If one was to set up an experiment where two particles are entangled, one could then measure the position of one particle with great accurary at the same time as measuring the momentum of the other particle with great accuracy, which would mean that, as the two particles are entangled, one would then know the momentum of the particle they measured the position of.  Therefore, the Uncertainty Principle would have been defied, as both momentum and position have been measured with a high accuracy.  However, this can't be true, as someone would have thought of this before.  So why wouldn't this work?