Wikipedia:Reference desk/Archives/Mathematics/2014 November 26

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November 26

Parallelogram adjective

A shape which is a rectangle is rectangular, a shape which is square is square, a shape which is a rhombus is rhombic. What about a shape which is a parallelogram? Is there a word which can be used? MChesterMC (talk) 15:54, 26 November 2014 (UTC) (please ping me when you reply)

See wiktionary:parallelogrammatic. --RDBury (talk) 16:59, 26 November 2014 (UTC)

And in case anyone is interested in the solid figure analog, that's parallelepiped, with adjectival form "parallelepipedal." SemanticMantis (talk) 17:54, 26 November 2014 (UTC)

Linear recurrence raised to a integer power

If $a(n)=\sum _{i=1}^{i=j}x_{i}*a(n-i)$, can I always find an independent linear recurrence $b(n)=\sum _{i=1}^{i=k}y_{i}*b(n-i)=a(n)^{q}$, and what is the relationship of b(n) to a(n)? 70.190.182.236 (talk) 20:18, 26 November 2014 (UTC)

Or to put it a better way: given a(n) and q, how do I calculate b(n)? 70.190.182.236 (talk) 21:52, 26 November 2014 (UTC)

The answer to the second version is b(n) = a(n)^q so you should probably stick with the first version. Also, you probably want to add the assumption q is a non-negative integer as there is no linear recurrence if q is negative or a fraction. Given that though, you can actually state something a bit more general, that if two sequences a(n) and b(n) satisfy a linear recurrence, then the product a(n)b(n) also does. To see this, use the fact that a sequence satisfies a linear recurrence if and only if its generating function is rational. Write the generating function of a(n) as P(t)/Q(t) where the degree of P is less than the degree of Q. Similarly, write the generating function of b(n) as the R(t)/S(t). (There is some hand waving that must be done here to account for case where the degree of P being greater than or equal to the degree of Q.) First consider the case where Q and S have distinct roots. Then partial fractions allows us to write P(t)/Q(t)=Σ_ip_i/(1-q_it) and R(t)/S(t)=Σ_jr_j/(1-s_jt). The nth term in the sequence with generating function p_i/(1-q_it) is p_iq_iⁿ and the nth term in the sequence with generating function r_j/(1-s_jt) is r_js_jⁿ. The product, p_ir_jq_iⁿs_jⁿ has generating function p_ir_j/(1-q_is_jt) and so a(n)b(n) has a generating function which is the sum of terms like this. Its generating function is then rational and therefore has a linear recurrence relation. For the case where Q or S have multiple roots, it's not hard to see that the coefficients of the linear recurrence of a(n)b(n) found above is a continuous function of the coefficients for a(n) and b(n). Since polynomials with distinct roots are dense in the space of all polynomials the general result follows by taking limits (plus a bit more hand waving). Seems like this would be a standard result in enumerative combinatorics but I don't have any books handy to give a reference. I doubt there is a closed form expression for the coefficients in the recurrence relation, but this shows that the relation does exist. --RDBury (talk) 03:12, 27 November 2014 (UTC)

What is the relationship between the orders of a and b (j and k)? 70.190.182.236 (talk) 16:03, 27 November 2014 (UTC)

The order of b isn't determined by the order of a, though it's at most qa (assuming I understand what you mean by order in this context).--RDBury (talk) 05:13, 28 November 2014 (UTC)

distance from a surface

(Maybe this belongs in Computing.)

I've made some models of the Lawson-Klein surface

• $w=\cos(u)\cos(2v)$
• $x=\cos(u)\sin(2v)$
• $y=\sin(u)\cos(v)$
• $z=\sin(u)\sin(v)$

by stereographic projection. To keep the model compact, I'd like to put the projection center as far from the surface as possible. At first I thought, because of symmetries, that the best point would be among the 80 permutations of

• (0,0,0,±1)
• (0,0,±1,±1) /√2
• (0,±1,±1,±1) /√3
• (±1,±1,±1,±1) /2

corresponding to the vertices of a 4-cube, the midpoints of its edges and so on; so I tested each of them against about a million sample points on the surface, to minimize the maximum dot product. I now think there probably are better centers. So: how would you go about searching for the farthest point? —Tamfang (talk) 23:11, 26 November 2014 (UTC)

You could try brute force. First find the distance of an arbitrary point to the surface (a constrained optimization problem). You should be able to get an exact solution with enough work. Then maximize this function on the sphere (numerically). Sławomir Biały (talk) 20:01, 29 November 2014 (UTC)

I'm not as confident as I guess you are that there exists a closed expression for "maximum dot product of L(u,v) with a given vector". —Tamfang (talk) 20:36, 30 December 2014 (UTC)

I'm now thinking about pursuit. Let P be any point in S3 and L(u,v) a point in the surface. At each step adjust (u,v) to increase P·L – the partial derivatives are easy – and simultaneously move P directly away from L. —Tamfang (talk) 04:15, 13 December 2014 (UTC)

That got me nowhere. —Tamfang (talk) 05:43, 27 January 2015 (UTC)

And then I had a simpler idea. The circle of self-intersection is in the wx plane. The orthogonal great circle, in the yz plane, also lies in the surface. In the projection W=w/(1-z), X=x/(1-z), Y=y/(1-z), the surface is asymptotic to X=4WY/(W²+Y²), so the planes X=±2 represent empty spheres tangent to the surface, with centres at (0,±1,0,2)/√5; using that as my projection centre gives a pleasing result, though I can't prove its optimality. —Tamfang (talk) 07:27, 30 December 2014 (UTC)