Crimson Sunbird

I am just about the complete opposite of you – I learn more on my own than by talking to other people. Talking to other people tends to be rather distracting for me whereas my concentration is better when I’m doing and focusing on my own research.

Your solution looks fine to me.

It has to do with the fact that water has a much higher specific heat capacity than land. Water (compared with land) takes a long time to heat up, and once heated up takes a long time to cool down.

Exactly! It’s wrong. Apparently the hind legs of locusts weren’t counted as legs. http://www.tektonics.org/af/buglegs.html Likewise some people think crabs have only eight legs because they don’t count the animals’ claws as legs – but that’s also wrong! Crabs have ten legs, not eight.

Don’t forget that short-term memory and long-term memory are very different types of memory. Information in long-term memory is less easy to forget than in short-term memory.

Maybe the science of taxonomy was not as well developed then as it is today? That might explain why the ancient Hebrews didn’t have different words for avian ouphs and mammalian ouphs. And as for why they thought locusts had four rather than six legs, I suppose it’s the same as why some people think decapod crabs have eight rather than ten legs.

[latex]B[/latex] is a quantity called the bulk modulus of the material through which the sound is travelling; it is a measure of the material’s resistance to uniform compression and is defined by [latex]B=-V\frac{\mathrm dp}{\mathrm dV}[/latex] where [latex]V[/latex] is volume and [latex]p[/latex] is pressure. Think of it as the (infinitesimal) ratio of the increase in the external pressure applied to the relative decrease in volume produced.

I’ve just read about something called ankle jerk. Apparently, if I stretch the Achilles tendon of my foot tight by flexing it at the ankle and then tap my Achilles tendon, I should experience a reflex contraction of my calf muscles. I just tried it on my foot – but didn’t experience any involuntary muscle twitching! Is this abnormal? Or am I doing the ankle-jerk experiment correctly?

What do you mean by [latex]\frac{-2}{0}=-\infty[/latex]?

I think there’s an old discussion of this here: http://www.scienceforums.net/topic/15843-iron-iii-chloride-to-dissolve-copper/

[latex]\begin{pmatrix}1 & 2 & 0 & -2 & 0 \\ 0 & 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & 0 & 1\end{pmatrix}\begin{pmatrix}a \\ b \\ c \\ d \\ e\end{pmatrix}=\begin{pmatrix}a+2b-2d \\ c+2d \\ e\end{pmatrix}=\begin{pmatrix}0 \\ 0 \\ 0\end{pmatrix}[/latex] So, letting [latex]b=t,d=u[/latex], the general solution is [latex]\mathbf{x}\,=\,\begin{pmatrix}-2t+2u \\ t \\ -2u \\ u \\ 0\end{pmatrix}[/latex]

[latex]\sin x+\sin y = 2\sin\frac{x+y}2\cos\frac{x-y}2=\pm2\sin\frac{x+y}2\sqrt{1-\sin^2\frac{x-y}2}[/latex] NB: [latex]\cos\frac{x-y}2=\color{red}\pm\color{black}\sqrt{1-\sin^2\frac{x-y}2}[/latex] as [latex]\cos\frac{x-y}2[/latex] can be negative (e.g. when [latex]x=270^{\circ},y=30^{\circ}[/latex]).

How about … [latex]\pi\,=\,(-\pi)e^{i\pi}[/latex]

The largest currently known prime is [latex]2^{57885161}-1[/latex] – a staggering 17-million-digit Mersenne prime, discovered earlier this month by GIMPS (Great Internet Mersenne Prime Search). http://www.newscientist.com/article/dn23138-new-17milliondigit-monster-is-largest-known-prime.html

Same here. I would recommend the book Quantum: A Guide for the Perplexed by Jim Al-Khalili; I’ve found it to be a most readable introduction to quantum physics.

Sorry. I merely came across the news article and thought it might be of interest for discussion. If the thread is out of place, then please close it. Im still new here so I apologize if I have strayed out of line. PS: Please close this thread. I’ve just noticed that there is already an existing thread on this topic, here. I should have searched before posting. Once again I apologize for the inconvenience.

Why, the atheist church, of course! http://www.bbc.co.uk/news/magazine-21319945

Suppose the car catches up with the bicycle after [latex]t[/latex] seconds. The bicycle will have travelled [latex]12t[/latex] metres from P and the car will have travelled [latex]\frac12(2)t^2[/latex] metres from P. The two distances are the same. Equate them and solve for [latex]t[/latex].

The second identity should be [latex]\sin(a-b)=-\sin(b-a)[/latex]. Let [latex]t=b-a[/latex]; then the two identities are as abvegto has pointed out: [latex]\cos(-t)=\cos t[/latex] and [latex]\sin(-t)=-\sin t[/latex]. Cosine is an even function and sine is an odd function.

As an illustration, let us consider a particle moving along the x-axis between [latex](-1,0)[/latex] and [latex](1,0)[/latex] in simple harmonic motion, its x-coordinate being given by the equation [latex]x=\sin\frac{\pi t}2[/latex] Its velocity and its acceleration are then given by [latex]v=\frac{\pi}2\cos\frac{\pi t}2[/latex] [latex]a=-\frac{\pi^2}4\sin\frac{\pi t}2[/latex] For this motion, the velocity of the particle is positive when it is moving towards [latex](1,0)[/latex] and negative when it is moving towards [latex](-1,0)[/latex]. Its acceleration is positive when the particle is to the right of the origin ([latex]x>0[/latex]) and negative when it is to the left of the origin ([latex]x<0[/latex]). A summary of the motion is given below. [latex]\begin{array}{c|c|c|c} \text{time} & \text{position} & \text{velocity} & \text{acceleration} \\ \hline 0\leqslant t\leqslant1 & 0\leqslant x\leqslant1 & \mathrm{+ve} & \mathrm{-ve} \\ \hline 1\leqslant t\leqslant2 & 0\leqslant x\leqslant1 & \mathrm{-ve} & \mathrm{-ve} \\ \hline 2\leqslant t\leqslant3 & -1\leqslant x\leqslant0 & \mathrm{-ve} & \mathrm{+ve} \\ \hline 3\leqslant t\leqslant4 & -1\leqslant x\leqslant0 & \mathrm{+ve} & \mathrm{+ve} \\ \end{array}[/latex] Thus you can see that velocity and acceleration are opposite in sign except for [latex]1\leqslant t\leqslant2[/latex] and [latex]3\leqslant t\leqslant4[/latex]. At these time intervals, the particle is increasing in speed whereas it is slowing down in the other two time intervals. The acceleration is negative during the first half of the motion: the particle is experiencing a force directed to the left (whereas force on it is to the right when the acceleration is positive).

Starting at the point [latex](1,0)[/latex] on the circle [latex]x^2+y^2=1[/latex], move the point anticlockwise along the circle through angle [latex]\theta[/latex] to the point P, where [latex]\theta[/latex] is in radians and [latex]0\leqslant\theta<2\pi[/latex]. The x-coordinate and the y-coordinate of the point P are [latex]\cos\theta[/latex] and [latex]\sin\theta[/latex] respectively, and [latex]\tan\theta=\frac{\sin\theta}{\cos\theta}[/latex].

One of the best books I’ve ever read is Fermat’s Last Theorem (1997) by Simon Singh, a thoroughly enjoyable account of Andrew Wiles’s proof of the Taniyama–Shimura conjecture for semistable elliptic curves, which implied the result about the most famous Diophantine equation in the history of mathematics. I recently also enjoyed reading Antimatter (2009) by Frank Close, from which I learned a lot about the topic.

Imagine a planet with a tunnel drilled from a point on the surface through to the centre and all the way to the antipodean point at the other end. Then an object dropped into the tunnel and affected only by the planet’s gravitation will oscillate in the tunnel in simple harmonic motion with a period given by [latex]T=\sqrt{\frac{3\pi}{\rho G}}[/latex] where [latex]G[/latex] is the gravitational constant and [latex]\rho[/latex] the density of the planet. What I find elegant about this formula is that the period of oscillation does not depend on the size or the mass of the planet – only on its density.

A nonabelian simple group is a group that is simple (has no normal subgroups other than the trivial subgroup and itself) and nonabelian (not commutative). The smallest nonabelian finite simple group is [latex]A_5[/latex], the alternating group of degree 5, which has order 60. A finite group that is not a nonabelian simple group is either not simple (has a nontrivial proper normal subgroup) or abelian (commutative) or both. Examples of abelian finite simple groups are cyclic groups of prime orders.