Wikipedia:Reference desk/Archives/Mathematics/2015 January 25

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January 25

Factorization of $\sum _{n=0}^{M}~(-1)^{n}~n^{k}$

For k ≥ 2 and even values of M = 2N we have

$\sum _{n=0}^{2N}~(-1)^{n}~n^{k}~=~N^{(k{\bmod {2}})+1}~(2N+1)^{(k+1){\bmod {2}}}~P_{_{k}}(N)$

For k ≥ 2 and odd values of M = 2N + 1 we have

$\sum _{n=0}^{2N+1}~(-1)^{n}~n^{k}~=~(N+1)^{(k{\bmod {2}})+1}~(2N+1)^{(k+1){\bmod {2}}}~Q_{_{k}}(N)$

where P_k and Q_k are polynomials of degree k - 2 in N. My conjecture, based on computer aided verification for all values of $k\leq 10^{3},$ is that P_k and Q_k are irreducible over the rationals. What are your opinions on the subject, and how might one prove (or disprove) such a conjecture ? Is there any literature or research on this particular topic ? Thank you. — 79.113.210.27 (talk) 11:20, 25 January 2015 (UTC)

I don't know about irreducibility, but the polynomials seem to be closely related to the Euler polynomials; see Faulhaber's formula for a possible explanation. --RDBury (talk) 13:03, 25 January 2015 (UTC)

One can express them easily as the difference of two of those Faulhaber's formulae, for instance leaving out the N=0 case we get $F_{2N}(k)-2^{1+k}F_{N}(k)$ for the first one where $F_{p}(k)=\sum _{i=1}^{k}i^{p}$ so that gives an expression in terms of either Bernoulli polynomials or Bernoulli numbers as desired. Or one can get a generating function but I don't know if any of that advances towards the target. It might be interesting to see how the polynomials behave in modular arithmetic. Dmcq (talk) 18:50, 25 January 2015 (UTC)

Have you had a look at s few small cases and at Factorization of polynomials to see how the polynomials fail to be factorizable? Dmcq (talk) 11:52, 27 January 2015 (UTC)

Here are the explicit formulas for the first ten P's and Q's. (As can be seen, even their lengths have a fixed form). — 79.113.193.206 (talk) 02:18, 28 January 2015 (UTC)

Simplex coordinates starting with (0,0,0,...),(1,0,0,...)

I'm looking for a reasonably easy way to calculate the coordinates for an n dimensional simplex where the first two coordinates are at the origin and (1,0,0...) and all-coordinates are non-negative (essentially each time the next dimension is added on, it gets added on in the positive value of the next dimension). So after origin and (1,0,0,0...) the next is (1/2,sqrt(3)/2,....). and the next is (1/2,sqrt(3)/6,?,0,0,0,0), etc. I'd like a formula where I can put in d for dimension of the simplex and n for which coordinate in the the simplex. So for d=3, n=2, I'm getting the sqrt(3)/6.Naraht (talk) 16:07, 25 January 2015 (UTC)

Did you look over here? YohanN7 (talk) 16:15, 25 January 2015 (UTC)

Yes, I already did. Most of the coordinate systems are definitely not what I want. The only one that I'm not sure of is "Increasing Coordinates" section which I simply don't understand.Naraht (talk) 15:02, 26 January 2015 (UTC)

Try this: Let

$a_{i}={\sqrt {\frac {i+1}{2i}}},b_{i}={\sqrt {\frac {1}{2i(i+1)}}}.$

then the kth vector is

$(b_{1},b_{2},\dots ,b_{k-1},a_{k},0,\dots )$

starting with the the 0th vector as

$(0,0,0,\dots ).$

The requirements are that each vector (after the 0th) has length 1 and the dot product of any two is 1/2 = cos π/3. --RDBury (talk) 22:41, 26 January 2015 (UTC)

Question in applied linear algebra

At the Science Desk, I asked a linear algebra question regarding how to best remove instabilities from a linear system. I thought the Science Desk was a slightly better fit because the instabilities are a consequence of measurement uncertainties; however, if anyone here has any insights, they would also be welcome. Please reply at the Science Desk. Dragons flight (talk) 19:50, 25 January 2015 (UTC)

A general method for such problems is the singular value decomposition. The directions corresponding to the small singular values of A are the bad directions. So one way to force stability is to project away those directions. In the case of interest, $Ax=b$ with A a matrix with more rows than columns (an overdetermined system). Multiply by $A^{T}$, giving $A^{T}Ax=b$. Now, $A^{T}A$ is a symmetric matrix. Orthogonally diagonalize that. Say $A^{T}A=UDU^{T}$ where the columns of D are orthonormal, and D is a real diagonal matrix. The subspace on which this system is unstable is spanned by the columns of the matrix U corresponding to the small eigenvalues of $A^{T}A$ (fix some cutoff ε, and this gives a subspace for the singular values of A less than ε). Then P be the projection onto the orthogonal complement of this space. The problem then becomes to solve $PA^{T}Ax=PA^{T}b$. This system is now under-determined: the instability of the original system has been absorbed into the kernel of the system. Sławomir Biały (talk) 11:51, 26 January 2015 (UTC)