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AmericaLatina,Caribe: Declaración Especial de la Reunión de Ministros del ALBA-TCP sobre Cambio Climático
Caracas, 16 de septiembre de 2014   1. Las Altas Autoridades de Antigua y Barbuda, Bolivia, Cuba, Dominica, Ecuador, Nicaragua, San Vicente y Las Granadinas y Santa Lucía y Venezuela, miembros de la Alianza Bolivariana para los Pueblos de Nuestra América y el Tratado de Comercio de los Pueblos ALBA–TCP, se reun...
El retorno del movimiento climático de base al escenario internacional
El movimiento ciudadano de acción frente a la crisis climática global ha desarrollado con el tiempo una relación de amor-odio con el trabajo en la campaña internacional en general. Durante años, personas en todo el mundo se han unido sobrepasando las fronteras nacionales, para hacer frente a una crisis que no conoce fronteras.
Transferring Kinetic Energy
Is there a way to transfer kinetic energy between two objects without them touching?
Hawking radiation weirdness
The way I understand Hawking radiation is that black holes decay by sucking in anti particles from the virtual particle pairs that are created right at its event horizon. I also understand that these anti particles reduce the mass of the black hole instantly when crossing the event horizon? And that these things are so in Hawkings theory because it is the consequence of the laws of entropy and the information conservation law. Is this correctly understood?

Suppose we have some rigid support that has a circular disk. In solving the 2-D wave eqn, what are the boundary conditions in this problem?
How to make up for a sub-par undergrad
Posting this here because I know that many of you come from similar backgrounds. Back a few years ago when I was just starting undergrad, I chose my current university because it's near my mom who had been diagnosed with cancer immediately before, and they offered me nearly a full-ride. Needless to say, now I'm regretting it.

Now, don't get me wrong. I love having a small department. I've had research opportunities that I wouldn't have had elsewhere, mainly because there is no graduate program. However, it's frustrating in other ways.
  • I'm the only one going to graduate school. Neither of my advisors even mentioned it as an option. When I asked where I should consider applying and what I should consider specializing in, they told me that they felt going into the industry would get me more money in the long run. I'm pretty sure I want to go to graduate school though.
  • Many core classes aren't offered. Mechanics II is only offered every six semesters - meaning I won't be able to take it before I graduate. I also tried to get an independent study, but the professor (also the department chair) didn't have time.
  • No one even told me about the GRE until a few months ago! But I guess that's my fault for not looking into it.

Now, don't get me wrong. I've tried my best to make up for these sore points on my graduate applications. I've been studying vigorously for the GRE and I have a 4.0, but I know that my application simply won't stack up to the people I'm competing with to get into graduate school. I've had an internship at a national lab (and am continuing to work with them, as long as I can afford to) and hope to do so again next summer. I'm also a Goldwater scholar, but as a female that doesn't mean too much.

Long story short... what can I possibly do to make up for my poor choice in universities? I can be as successful as I want, but I'll never be able to compete with even an average student from an Ivy League. Is there *ANY* chance of being able to apply to a tier 1 school, or am I out of luck?
Close Packing of Spheres in Regular Tetrahedral vs. Square Pyramidal C
The full title of this post is "Close Packing of Spheres in Regular Tetrahedral vs. Square Pyramidal Container"

I'm not sure where this post belongs, but Greg Bernhardt suggested I just post it where I thought best, and he would find a place for it. So here goes:

In 1611, Johannes Kepler proposed that face-centered cubic packing achieves the greatest density. In 1998, Thomas Hales proved he was right.

I took a different approach on this and compared the packing density of equal-sized congruent spheres within a regular tetrahedral container (four similar equilateral triangular faces) vs. a square pyramidal container (four similar equilateral triangular faces with a square base), as shown below:

Attachment 73250 (Figure 1)

Attachment 73251 (Figure 2)

I found that, with a very large number of spheres (>100,000), the "packing efficiency" of spheres (volume of spheres/volume of container) within a regular tetrahedral container approached ≈74.048% ("Hales' density" or ∏/√18 ), whereas the packing efficiency of spheres within a square pyramidal container approached only ≈60.460%.

Oddly enough, the space-efficiency of a tetrahedral ball stack decreases in proportion to the number of balls it contains, whereas the space-efficiency of a pyramidal ball stack increases with the number of balls, as shown below:

Attachment 73252 (Figure 3)

Attachment 73253 (Figure 4)

Here are the formulas I used in my calculations (note that I let the radius of each sphere equal 1).

For a regular tetrahedral container:

Volume of each sphere = 4/3∏r³ (r = 1) = 4.1888
Number of spheres in a regular tetrahedron Tt = n(n+1)(n+2)/6 where n = number of spheres along base (n=5 in Figure 1), so the number of spheres in a tetrahedron with 5 spheres along the base would equal 35.
Volume of spheres in a regular tetrahedron = Vst = n(n+1)(n+2)/6*4/3*∏r^3 ≈ 146.608 (for n=5)
Length of side of regular tetrahedral container = b = ((2n)-2)+2*3^0.5 = 11.464 (for n=5)
Volume of regular tetrahedral container = Vt = b^3/6*2^0.5 = (((2n)-2)+2*√3)^3/(6*(2^0.5)) = 177.564 (for n=5)
"Packing efficiency" = Vst/Vt ≈ 82.566% (for n=5)

The packing efficiency of a regular tetrahedral container decreases with the number of spheres until it reaches a limit of ≈74.048% (Figure 3). Note that the packing efficiency actually increases between 1 sphere to 4 spheres. This is unexpected, and could be an error in my calculations, or it could be accurate. I've checked it numerous times. Feel free to check it for yourself.

For a square pyramidal container:

Volume of each sphere = 4/3∏r³ (r = 1) ≈ 4.189
Number of spheres in a square pyramid = Tp = n(n+1)(2n+1)/6 where n = number of spheres along base (n=4 in Figure 2), so the number of spheres in a square pyramid with 4 spheres along the base would equal 30. With 5 spheres along the base, the total number of spheres would equal 55. (I couldn't find an illustration that showed 5 spheres along the base, so I had to use the one with 4 spheres along the base).
Volume of spheres in a square pyramid = Vsp = n(n+1)(2n+1)/6*4/3*∏r^3 ≈ 230.383 (for n=5)
Length of side of square pyramidal container = b = ((2n)-2)+2*3^0.5 ≈ 11.464 (for n=5)
Height of square pyramidal container = h = (((n*2-2)+(2*(3^0.5)))*3^0.5/2) ≈ 9.928 (for n=5)
Volume of square pyramidal container = Vp = b^2*h/3 = ((((2*n)-2)+2*3^0.5)^2*(2n)-2) + 2*(3^0.5)*2/3^0.5)/3 ≈ 434.94 (for n=5)
"Packing efficiency" = Vsp/Vp ≈ 52.969% (for n=5)

The packing efficiency of a square pyramidal container increases with the number of spheres until it reaches a limit of ≈60.460% (Figure 4).

There may be practical applications for this (if my calculations are right) for the packing of spherical objects (oranges, tennis balls, baseballs, cannonballs, etc.). If so, it would mean that the more space efficient container for packing equal-sized spheres would be a regular tetradhedral container rather than a square pyramidal container.

Your comments are welcome!


Calculating steradians (solid angle)
1. The problem statement, all variables and given/known data
For a sphere of radius r, find the solid angle Ω in steradians defined by spherical angles
of: a.) 0°≤θ≤ 20°, 0°≤ø≤360°;

2. Relevant equations
dA = r2 sin dθ dø (m2)
dΩ = dA / r2 = sin dθ dø (sr)

3. The attempt at a solution
I think I understand what a steradian (sr) is, on a sphere, and I need to find what ratio of a 4∏ is formed by the above limits, but I can't make the connection on how to form that ratio. ?
help with beam deflection problem
I need help with this beam deflection problem for 1 the book my professor gave me almost everything is hand written including the beam stuff... SO I cant read it at all and its really frustrating trynna do this without being able to read the formulas or anything please helpp!! I attached the problem please link whatever formula you used ty!!

Attached Images
File Type: png beam.png (30.6 KB)
Contact Vector Fields. "Flow Preserves Contact Structure?
Hi All, I am going over a definition of a Contact Vector Field defined on a 3-manifold: this is defined as " a vector field v whose flow preserves the contact structure " .
1) Background (sorry if this is too simple) A contact structure ## xi ##( let's stick to 3-manifolds for now ) is a nowhere-integrable plane bundle on a 3-manifold M^3, i.e., we have a 2-plane distribution so that there are no submanifolds N < M^3 (i.e., surfaces here) so that TN = ## xi ## , i.e., there are no submanifolds N of M^3 whose tangent bundle coincides with the contact distribution (this is related to one of Frobenius' theorems and involutivity).

Now ,does the statement " the flow of the vector field v preserves the contact structure" mean that the tangent space T_C(t) along any flow curve C(t) (local or global) coincides with the contact plane at C(t) ?

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