Optimal Quadrature Formula of Markov's and Locher's Type with Weight Function
On the discrepancy of Markov-normal sequences
We construct a Markov normal sequence with a discrepancy of . The estimation of the discrepancy was previously known to be .
On the discrepancy estimate of normal numbers
Circles passing through five or more integer points
We find an improvement to Huxley and Konyagin’s current lower bound for the number of circles passing through five integer points. We conjecture that the improved lower bound is the asymptotic formula for the number of circles passing through five integer points. We generalise the result to circles passing through more than five integer points, giving the main theorem in terms of cyclic polygons with m integer point vertices. Theorem. Let m ≥ 4 be a fixed integer. Let be the number of cyclic polygons...
Monopoles and modifications of bundles over elliptic curves.
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