arXiv subjects

R. Padmanabhan

Publications and source records attributed to R. Padmanabhan.

4 recordsLinked to original sources

arXiv2021

When do we have 1 + 1 = 11 and 2 + 2 =5?

The phrase "$ 2+2=5 $" is a cliché or a slogan used in political speeches, propaganda, or literature, most notably in the novel "$ 1984 $" by George Orwell. More recently, we came across a You-Tube short film comedy, Alternative Math, produced by IdeaMan Studios (see \cite{Danny}). It is a hilarious exaggeration of a teacher who is dragged through the mud for teaching that $ 2+2=4 $ and not $ 22 $, as Danny, a young student, kept on insisting. In the movie, Danny and the whole community sincerely believe $ 1+1=11 $ and $ 2+2=22 $. Jokes aside, we ask the question whether a polynomially defined group law "$\oplus$" defined over the field of rationals such that $ 1\oplus1=u $ and $ 2 \oplus2=v $ can simultaneously be satisfied for arbitrary integers $u$ and $ v $. Answer to this question takes us through a fascinating journey from Brahmagupta all the way to the modern works of Louis Joel Mordell and Ramanujan!

math.HO↗

arXiv2020

Orchards in elliptic curves over finite fields

Consider a set of $ n $ points on a plane. A line containing exactly $ 3 $ out of the $ n $ points is called a $ 3 $-rich line. The classical orchard problem asks for a configuration of the $ n $ points on the plane that maximizes the number of $ 3 $-rich lines. In this note, using the group law in elliptic curves over finite fields, we exhibit several (infinitely many) group models for orchards wherein the number of $ 3 $-rich lines agrees with the expected number given by Green-Tao (or, Burr, Grünbaum and Sloane) formula for the maximum number of lines. We also show, using elliptic curves over finite fields, that there exist infinitely many point-line configurations with the number of $ 3 $-rich lines exceeding the expected number given by Green-Tao formula by two, and this is the only other optimal possibility besides the case when the number of $ 3 $-rich lines agrees with the Green-Tao formula.

math.NT↗

arXiv2019

Means Compatible with Semigroup Laws

A binary mean operation m(x,y) is said to be compatible with a semigroup law *, if * satisfies the Gauss' functional equation m(x,y) * m(x,y) = x * y for all x, y. Thus the arithmetic mean is compatible with the group addition in the set of real numbers, while the geometric mean is compatible with the group multiplication in the set of all positive real numbers. Using one of Jacobi's theta functions, Tanimoto has constructed a novel binary operation * corresponding to the arithmetic-geometric mean agm(x,y) of Gauss. Tanimoto shows that it is only a loop operation, but not associative. A natural question is to ask if there exist a group law * compatible with arithmetic-geometric mean. In this paper we prove that there is no semigroup law compatible with agm and hence, in particular, no group law either. Among other things, this explains why Tanimoto's novel operation * using theta functions must be non-associative.

math.GR↗

arXiv2012

A 2-base for inverse semigroups

An open problem in the theory of inverse semigroups was whether the variety of such semigroups, when viewed as algebras with a binary operation and a unary operation, is 2-based, that is, has a base for its identities consisting of 2 independent axioms. In this note, we announce the affirmative solution to this problem: the identities \[ \quad x(x'x) = x \qquad \quad x (x' (y (y' ((z u)' w')'))) = y (y' (x (x' ((w z) u)))) \] form a base for inverse semigroups where ${}'$ turns out to be the natural inverse operation. We recount here the history of the problem including our previous efforts to find a 2-base using automated deduction and the method that finally worked. We describe our efforts to simplify the proof using \textsc{Prover9}, present the simplified proof itself and conclude with some open problems.

math.GR↗