S. H. Kulkarni

A Very Simple and Elementary Proof of a Theorem of Ingelstam

1. T. M. Apostol, Another elementary proof of Euler’s formula for ζ(2n), this MONTHLY 80 (1973) 425–431. 2. R. Ayoub, Euler and the zeta function, this MONTHLY 81 (1974) 1067–1086. 3. B. C. Berndt, Elementary evaluation of ζ(z), Math. Mag. 48 (1975) 148–154. 4. H. M. Edwards, Riemann’s Zeta Function, Academic Press, New York, 1974. 5. L. Euler, Introduction to Analysis of the Infinite, Book I (trans. J. D. Blanton), Springer-Verlag, New York, 1988. 6. K. Knopp, Theory and Application of Infinite Series, Dover, New York, 1990.

Non invertibility of certain almost mathieu operators

It is shown that the almost Mathieu operators of the type Te n =e n-1 + λsin(2nr)e n + e n+1 where λ is real and r is a rational multiple of π and {e n :n = 1,2,3,....} an orthonormal basis for a Hilbert space, is notinvertible.

Arzela-Ascoli theorem is stable

A quantitative version of the Arzela-Ascoli theorem is proved. This version implies that a closed and bounded subset of C(X) is nearly compact, if and only if, it is nearly equicontinuous.