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Professor Lee Rubel’s fundamental theorem of analogue computation (under construction)

Rubel, L.A. "A Universal Differential Equation." Bull. Amer. Math. Soc.  4 , 345-349, 1981.

Rubel, L.A. "Some Research Problems About Algebraic Differential Equations." Trans. Amer. Math. Soc.  280 , 43-52, 1983.

Rubel, L.A. "Some Research Problems About Algebraic Differential Equations II." Illinois J. Math.  36 , 659-680, 1992.

Rubel, L.A. "Uniform Approximation by Rational Functions All of Which Satisfy the Same Algebraic Differential Equation." J. Approx. Th.  84 , 123-128, 1996.

THEOREM .  There exists a non trivial fourth-order algebraic differential equation

 P(y',y",y'",y"") = 0, *

where P is a polynomial in four variables, with integer coefficients, such that for any continuous function f on R and for any positive continuous function e(t) on R ,  there exists a C-infinity solution y of  *  such that on R for all t

 |y(t) - f(t)| < e(t) 

One such specific equation (homogeneous of degree seven, with seven terms

of weight  14)  is

 3 y’^4 y’’ y'"’^2 - 4 y’^4y’’’ y’’’’ + 6 y’3 y’’^2 y’’’ y’’’’

 + 24 y’^2 y"^4y’’’’ - 12 y’^3 y" y’’’^3 - 29 y’^2 y"^3y’’’^2 + 12 y’’^7 = 0

Every continuous function is a weak solution to this equation.

Professor Rubel’s letter: Proposal for a New Mathematical Society, published in Notices AMS  42 , 220 (1995) 

The following comments apply to any field.

Proposal for a New Mathematical Society

 If I had the time (which I don’t), I would found a new organization with a title like “ The Research Mathematical Society ” to counter the lamentable state of the AMS, which has become like a professional society of grocers or insurance salesmen, profaning the sacred vocation of mathematics (witness “Mathfest”). 

 The purpose of the new organization would be to further research in mathematics as single-mindedly as humanly possible, leaving to other organizations (or individuals) peripheral things such as politics, funding, prizes, education, job placement, and so on. These may be worthy areas influencing research mathematics, but paying attention to them dilutes the pure pursuit of mathematics, whose aim is to discover, prove, and understand theorems. I long ago stopped attending AMS meetings (especially national ones) when it became obvious that it was no longer possible to discuss real mathematics at them. In the real old days, people would get together, at meetings over coffee or beer and exchange theorems and ideas endlessly. No more!

 First, a partial list of what the proposed society would avoid.

1. Tie clips with integral signs

2. Barbecues

3. Employment registers

4. Prizes (which corrupt and distort the subject)

5. Questions of education, particularly calculus reform

6. Riverboat rides

7. Politics

8. Funding questions (especially involving the NSF)

9. Musical performances at meetings

10. Lobbying

11. Publishers’ receptions

 Lest the tenor of this letter be too negative, let me stress the main positive functions of the proposed organization.

A.  To hold meetings where mathematical lectures are given and mathematical discussions of all kinds are encouraged.

B.  To publish a Notices where meetings are announced and described. Possibly to publish some books and research journals. Finally, as though this letter were not inflammatory enough,

C.  To set up some minimum standards for membership―too many people join the AMS solely because it looks good on their credentials. Such a standard might be having published at least one mathematical paper every two years for the last ten years, with some sort of exceptions for young mathematicians and Godel. 

 Since we can’t drive the money changers from the temple, let’s build a new temple!

 Lee A. Rubel University of Illinois at Urbana–Champaign