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May 31

Linear independence of partial derivatives

Consider the general solution y(x,c₁,c₂) of a 2nd-order linear homogeneous ODE, where c₁ and c₂ are two parameters, and y is assumed smooth in x, c₁, and c₂. Consider the partial derivatives

{\frac {\partial y}{\partial c_{1}}}(x,c_{1},c_{2}),\quad {\frac {\partial y}{\partial c_{2}}}(x,c_{1},c_{2})

Now fix some (c₁,c₂) = (k₁,k₂). Is is true that the functions of x defined by

u_{1}(x)={\frac {\partial y}{\partial c_{1}}}(x,k_{1},k_{2}),\quad u_{2}(x)={\frac {\partial y}{\partial c_{2}}}(x,k_{1},k_{2})

are linearly independent? If not, what extra assumptions are necessary? Does it depend on the choice of (k₁,k₂), and if so, in what way? Thanks for any help. —Anonymous Dissident^Talk 00:40, 31 May 2013 (UTC)[reply]

A condition is clearly needed to ensure that c1 and c2 are independendent parameters (they could otherwise be the same parameter, for instance). Then the answer would presumably be yes, just by definition of independence of the parameters. Sławomir Biały (talk) 13:22, 31 May 2013 (UTC)[reply]

Two questions related to sphere packing...

These are both questions just to start getting my head around sphere packing.

Let A and B both be n dimensional balls. Color any point on the surface of A if it is between the center of A and any point of B. For dimension n, What fraction of the surface of A is colored. (The answer for 2 dimensions is exactly 1/6.
Let A be an n dimensional ball and let B₁, B₂, ...B_n additional n dimensional balls with the centers making an n-dimensional simplex. Mark the n points (p_n on the surface of A where the lines from the center of A to B_i touch the surface of A. Now connect all p_n the "short way" round to get a n-1 dimensional simplex on the surface of A. Color the surface inside A if it is between the center of A and any point in any of the B_i. For each dimension n, what fraction of the area of the triangle is *uncolored*.

For the three dimensional version of this, the first question is what percentage of the surface area of the sphere is "covered" by a touching sphere. The second question is if you have three spheres all touching each other and the main sphere, how much of a "hole" is there between them.Naraht (talk) 00:41, 31 May 2013 (UTC)[reply]

Borsuk's question

In recent edits two anonymous users added an information about new bound for the Borsuk's problem [1]. However it is not clear from the linked abstract, that the claimed bound is true: is says 'We found a two-distance set consisting of 416 points on the unit sphere in the dimension 65 which cannot be partitioned into 83 parts of smaller diameter. This also reduces the smallest dimension in which Borsuk's conjecture is known to be false.', however the Borsuk's question concerns dividing sets in d–space into $(d+1)$ parts. Bondarenko says his set can not be partitioned into 83 subsets, but that does not obviously imply it can't be partitioned into 66 subsets of required size.

Additionally he says 'This reduces the smallest dimension...', but it's not clear in what manner it reduces that dimension (possibly the two dimensions mentioned are related somehow, but not necessarily equal). Anybody has access to the full text, and can verify the 65 is an actual new limit for Borsuk's question, please? --CiaPan (talk) 06:24, 31 May 2013 (UTC)[reply]

Is it correct to equate the 2nd equation?

Consider the following two equations-
A) 2^x * 2^y = 2² * 2⁴ From this equation, we can simply write x = 2 and y = 4.
B) 2^x + 2^y = 2² + 2⁴ From this equation, can I write x = 2 and y = 4?
I know, I am right for my 1st equation, here, I just equated LHS to RHS. In the second case also I did the same thing, but I am not sure whether I am right or wrong. So, correct me if I am wrong. Scientist456 (talk) 15:57, 31 May 2013 (UTC)[reply]

x=2 and y=4 are *one* solution to the second equation, but they are by no means the only ones. An additional answer to both equations is x=4 and y=2. For the first you can also have x=3 and y=3 (and x=1, y=5, etc.). The second is harder to come up with alternative integer solutions to, but there are plenty of non-integer solutions, e.g. x=3 and ~~y=ln(2.5)/ln(2)~~ y=ln(12)/ln(2) (correct thinko). To be equal, the two sides have to be the same, either before or after algebraic manipulation. If you substitute x=2 and y=4 in both equations, you'll end up with a true statement (both sides are the same), but that can also happen with a number of other values for x and y. -- 71.35.97.37 (talk) 16:13, 31 May 2013 (UTC)[reply]

Actually, I was confused in this problem- 3^x + 7^y = 3² + 7⁴
I had to find the value of x and y. Here, can I write x = 2 and y = 4 by equating LHS to RHS. Scientist456 (talk) 17:04, 31 May 2013 (UTC)[reply]

Did you make up this problem or does it come from some source? If it comes from some source, there is a high probability that you have left out important information. Looie496 (talk) 17:14, 31 May 2013 (UTC)[reply]

There are an infinite number of possible solutions to both equations, unless perhaps you are restricted to integers only, but even so, there's likely no single solution. For example, if x=0 in the second equation, then y=log₇2409.--Jasper Deng (talk) 17:18, 31 May 2013 (UTC)[reply]

Choose x yourself and then compute y to satisfy your equation.
A) 2^x * 2^y = 2² * 2⁴ From this equation write y = log((2² * 2⁴) / 2^x)/log(2)
B) 2^x + 2^y = 2² + 2⁴ From this equation write y = log((2² + 2⁴) - 2^x)/log(2)
Bo Jacoby (talk) 13:35, 3 June 2013 (UTC).[reply]