By Doron Zeilberger
Posted: Sept. 22, 2010.
This lecture, delivered on Sept. 17, 2010 in the Rutgers University Experimental Mathematics Seminar is a mathematical eulogy to my great mentor and influencer Leon Ehrenpreis (24 Iyyar, 5690- 6 Elul, 5770) who was not only one of the greatest mathematicians of our time, and a great rabbi (see, e.g., his insightful commentary on Genesis) (and a very accomplished piano player, and a Marathon runner) but, most important, a great mensch.
In order to find the truncated
generating Laurent formal power series for the symmetrized FUNDAMENTAL solution of the 2D Discete Laplacian
truncated to order 20
the input
gives the output.
The table of values of the above is:
the input
gives the output.
In order to find
the generating-function for the symmetrized FUNDAMENTAL solution of the 2D
Duffin-Discete Laplacian,
the input
gives the output.
In order to find
the table of values for the symmetrized FUNDAMENTAL solution of the 2D
Duffin-Discete Laplacian,
the input
gives the output.
In order to find
the generating-function for the symmetrized FUNDAMENTAL solution of the 3D Discete Laplacian,
truncated to order 5 is:
the input
gives the output.
In order to see
the first 10 terms of the sequence of dimensions of
the space of polynomials of degree ≤ d, annihilated by the differential operators
D12+D22+D3 and D13+D23+D33 ,
where D1, D2, D3 are differentiations w.r.t. to x,y,z resp.
the input
gives the output.
In order to see
the first 10 terms of the sequence of dimensions of
the space of homog. polynomials of degree d, annihilated by the differential operators
D12+D22+D32 and D13+D23+D33 ,
where D1, D2, D3 are differentiations w.r.t. to x,y,z resp.
the input
gives the output.
In order to find
the multipliticity variety for the two operators, D12-D2,D22 ,
the input
gives the output.
In order to
find a basis for the polynomials annihilated by the operators
D12+D22,D1+D2, of degree ≤ 2,
the input
gives the output.
In order to find
a basis for the homogeneous polynomials annihilated by the operators
D14+D24+D34,D13+D23+D33 of degree ≤ 4
the input
gives the output.
In order to find
a basis for the polynomials in x[1], ..., x[5]
of degree 7, that are totally harmonic,
the input
gives the output.
In order to see
a basis for the discrete analytic polynomials of degree ≤ 10,
the input
gives the output.
In order to see a basis for the discrete harmonic polynomials, in the plane, with
variables m and n,
the input
gives the output.
In order to see a basis for the discrete harmonic polynomials in three dimensions, with
variables m1,m2,m3
the input
gives the output.
In order to find the first ten terms of an Ehrenpreis-inspired basis for discrete analytic
polynomials
the input
gives the output.
In order to find
a basis for the linear vector space consisting of polynomials in m,n, of degree ≤ 5
viewed as functions on the discrete lattice with the curious property that for every 1 by 4 discrete
rectangle in the plane, the value of the polynomial equals the average of its value at the other 9 locations,
the input
gives the output.