An eta-quotient of levelN is a modular form of the shape f(z) = ∏ δ|N η(δz) rδ . We study the problem of determining levels N for which the graded ring of holomorphic modular forms for Γ0(N) is generated by (holomorphic, respectively weakly holomorphic) eta-quotients of level N . In addition, we prove that if f(z) is a holomorphic modular form that is nonvanishing on the upper half plane and ha...

Precipitation estimates derived from the Eta model and from TRMM (Tropical Rainfall Measuring Mission) and CHIRPS (Climate Hazards Group InfraRed Precipitation with Station data) remotely sensed data were compared to the precipitation data of the INMET (National Institute of Meteorology) meteorological stations in the south-southeast region of Minas Gerais state, Brazil, in the period between J...

When we form a finite algebraic extension of Q, we are not guaranteed that the ring of integers, O, in our extension will be a unique factorization domain (UFD). We can obtain a measure of how far O is from being a UFD by computing the class number which is defined as the order of the ideal class group. This paper describes the ideal class group and provides examples of how to compute this grou...

We consider various ways to represent irrational numbers by subrecursive functions: via Cauchy sequences, Dedekind cuts, trace functions, several variants of sum approximations and continued fractions. Let S be a class of subrecursive functions. The set of irrational numbers that can be obtained with functions from S depends on the representation. We compare the sets obtained by the different r...

We use the calculus of adiabatic pseudo-differential operators to study the adiabatic limit behavior of the eta and zeta functions of a differential operator δ constructed from an elliptic family of operators with base S. We show that the regularized values η(δt, 0) and tζ(δt, 0) have smooth limits as t → 0, and we identify the limits with the holonomy of the determinant bundle, respectively wi...

We announce misère-play solutions to several previously-unsolved combinatorial games. The solutions are described in terms of misère quotients—commutative monoids that encode the additive structure of specific misère-play games. We also introduce several advances in the structure theory of misère quotients, including a connection between the combinatorial structure of normal and misère play.