New modular equations of signature three in the spirit of Ramanujan

Srinivasa Ramanujan recorded many modular equations in his notebooks, which are useful in the computation of class invariants, continued fractions and the values of theta functions. In this paper, we prove some new modular equations of signature three by using theta function identities of composite degrees.

On some automorphic properties of Galois traces of class invariants from generalized Weber functions of level 5

After the significant work of Zagier on the traces of singular moduli, Jeon, Kang and Kim showed that the Galois traces of real-valued class invariants given in terms of the singular values of the classical Weber functions can be identified with the Fourier coefficients of weakly holomorphic modular forms of weight 3/2 on the congruence subgroups of higher genus by using the Bruinier-Funke modular traces. Extending their work, we construct real-valued class invariants by using the singular values of the generalized Weber functions of level 5 and prove that their Galois traces are Fourier coefficients of a harmonic weak Maass form of weight 3/2 by using Shimura’s reciprocity law.

Preservation of Class Invariants in Refactoring UML Models

In the field of software engineering, the term class invariants is known as a valuable term employed to delineate the semantic of UML class diagram elements (attributes and relationships) and must be held throughout the life-time of instances of the class. Refactoring, the activities of re-distributing classes, attributes and methods across the class hierarchy, is a powerful technique that is used to improve the quality of software systems. Performing refactoring on UML class diagrams obviously requires a special investigation of invariant-preserving on the refactored models. In this paper, we propose an approach to preserve class invariants in refactoring UML models. In order to achieve this aim, we first formalize the class diagram along with class invariants by mathematical notations. We then constitute the rules for five refactoring operations (deal with class hierarchies) in such a way to guarantee class invariants as well as proving correctness of the refactoring rules. Finally, the paper also makes provision of the proposed approach for practical applications in software re-engineering development process.

Construction of ray-class fields by smaller generators and their applications

We generate ray-class fields over imaginary quadratic fields in terms of Siegel–Ramachandra invariants, which are an extension of a result of Schertz. By making use of quotients of Siegel–Ramachandra invariants we also construct ray-class invariants over imaginary quadratic fields whose minimal polynomials have relatively small coefficients, from which we are able to solve certain quadratic Diophantine equations.

Ring class invariants over imaginary quadratic fields

We give some explicit conditions under which the singular values of Δ-quotients generate ring class fields over imaginary quadratic fields.