Given a formal at meromorphic connection over an excellent scheme field of characteristic zero, in previous paper we established existence good structures and Deligne–Malgrange lattice after suitably blowing up. In this paper, reinterpret re fine these results by introducing some related structures. We consider the turning locus , which is set points one cannot achieve structure without show th...

We recall several results of zero divisor graphs of commutative rings. We then examine the preservation of the diameter of the zero divisor graph of polynomial and power series rings.

Three decades ago Cornalba-Harris proved a fundamental positivity result for divisor classes associated to families of stable curves. In this paper we establish an analogous positivity result for divisor classes associated to families of stable differentials.

We obtain the triple correlations for a truncated divisor sum related to primes. We also obtain the mixed correlations for this divisor sum when it is summed over the primes, and give some applications to primes in short intervals.

In this paper, we verify the diameter of zero divisor graphs with respect to direct product. Keywords—Atomic lattice, complement of graph, diameter, direct product of lattices, 0-distributive lattice, girth, product of graphs, prime ideal, zero divisor graph.

Multiplicative Number Theory 638 27.2 Functions . . . . . . . . . . . . . . . . . 638 27.3 Multiplicative Properties . . . . . . . . . 640 27.4 Euler Products and Dirichlet Series . . . 640 27.5 Inversion Formulas . . . . . . . . . . . . 641 27.6 Divisor Sums . . . . . . . . . . . . . . . 641 27.7 Lambert Series as Generating Functions . 641 27.8 Dirichlet Characters . . . . . . . . . . . . 642...

This paper examines percolation questions in a deterministic setting. In particular, I consider R, the set of elements of Z2 with greatest common divisor equal to 1, where two sites are connected if they are at distance 1. The main result of the paper proves that the infinite component has an asymptotic density. An “almost everywhere” sieve of J. Friedlander is used to obtain the result.