Twitter research
This is a test whether Twitter can be useful to spot results. As a corollary of a Cauchy-Binet
generalization for pseudo determinants, there is an identity for classical determinants. Its a
special case of a special case but maybe this is known. Can twitter find out?
- Has anybody seen the identity det(1+A^2) = sum_P det^2(A_P) over all minors of a symmetric nxn matrix? (see http://goo.gl/5dQon ).
Twitter proofs
In the summer of 2013, some experiments with
Twitter proofs on
21a twitter.
These are proofs in less than 140 characters (some of which are done using Mathematica).
- The Lagrange identity |v|^2 |w|^2 - (v.w)^2 = |v x w|^2 which the last tweet has proven, also implies Cauchy-Schwarz! Why? You can see it.
- Proof of Lagrange identity (a special case of Cauchy-Binet): v={v1,v2,v3}; w={w1,w2,w3}; u=Cross[v,w]; Simplify[(v.v) (w.w) - (v.w)^2==u.u]
- The isosceles triangles in the 3:4:5 triangle have relative 1,2,3 ratios. Their height/base ratios is 1,2,3 (observation of Elijah Gunther).
- a={0,0};b={4,0};c={0,3}; x={1,0};y={8,9}/5;z={0,1}; {u,v,w}={x+y,y+z,z+x}/2; 2{Norm[a-w]/Norm[x-z],Norm[c-v]/Norm[y-z],Norm[b-u]/Norm[x-y]}
- By defining dot product, proving Cauchy-Schwarz, defining angle, we got Al Kashi and the cos formula. For right angles, we get Pythagoras.
- Can define alpha by cos(alpha) = v.w/(|v| |w|). Then (v-w).(v-w) = |v|^2 + |w|^2 - 2 v.w = |v|^2+|w|^2-2 |v| |w| cos(alpha) (Cos Formula)
- Define a=v.w. Assume |w|=1 without loss of generality. 0 <= (v-a w).(v-a w) = |v|^2-a^2 shows v.w = a <= |v| = |v| |w| (Cauchy-Schwarz)
Twitter in Course websites
It seems that Twitter is not very much used in higher education
yet. I tried it out first in fall 2009 in math21a, then in spring 2010:
In my experience there are advantages and disadvantages:
Advantages:
- The twitter feeds be embedded in a usual website
- One can tweet from a handheld like the ipod and so make announcements to the
class which are visible on the website while being on the road. This has
advantages, because one does not want to have access to the website directly
from a handheld (which one can lose).
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Disadvantages:
- One has to stay on the ball. It is easy to neglect the twitter account.
- Like any blogging or social network service, it is not clear how long the service will be there.
One has also to make backups to make sure that one has a long term backup.
For me, any website is a document which I do not want to lose.
Twitter does well now but it could be that in 5 years it is bought by Google
or Microsoft or Yahoo, then crippled to death since such buyouts are often
hostile with the only intention to kill some competition.
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Links:
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