Ramanujan’s class invariants, Kronecker’s limit formula, and modular equations

Modular Verification of Static Class Invariants

Object invariants describe the consistency of object-oriented data structures and are central to reasoning about the correctness of object-oriented software. But object invariants are not the only consistency conditions on which a program may depend. The data in object-oriented programs consists not just of object fields, but also of static fields, which hold data that is shared among objects. ...

Relationships between Darboux Integrability and Limit Cycles for a Class of Able Equations

We consider the class of polynomial differential equation x&= , 2(,)(,)(,)nnmnmPxyPxyPxy++++2(,)(,)(,)nnmnmyQxyQxyQxy++&=++. For where and are homogeneous polynomials of degree i. Inside this class of polynomial differential equation we consider a subclass of Darboux integrable systems. Moreover, under additional conditions we proved such Darboux integrable systems can have at most 1 limit cycle.

Quadratic Diophantine Equations, the Class Number, and the Mass Formula

with a given q ∈ F×. In particular, in the classical case with F = Q and V = Q, we usually assume that φ is Z-valued on Z and q ∈ Z. The purpose of the present article is to present some new ideas on various arithmetical questions on such an equation. We start with some of our basic symbols and terminology. For a set X we denote by #X or #{X} the number (≤ ∞) of elements of X. For an associativ...

Classical Covariants and Modular Invariants

There is a relationship between the covariants of binary forms, a central topic in classical invariant theory, and the invariants of modular representations of cyclic groups of prime order. This relationship was identified by Gert Almkvist [1] and used implicitly in both [15] and [17]. In this note we investigate the relationship and provide a progress report on an application. Our primary moti...

Modular Forms and Projective Invariants