Elementary Proofs of Algebraic Relationships for the Exponential and Logarithm Functions

This paper uses elementary algebraic methods to obtain new proofs for theorems on algebraic relationships between the logarithmic and exponential functions. The main result is multivariate version of a special case of the Structure Theorem due to Risch that gives in a very explicit fashion the possible algebraic relationships between the exponential and logarithm functions. In addition there are some more results that give new information about the forms of elementary integrals of elementary functions as well as a new treatment of some algebraic dependence theorems previously discussed by Ostrowski, Kolchin and Ax

Government Debt, Reputation and Creditors’ Protections: The Tale of San Giorgio

San Giorgio (1407–1805) was a formal association aimed at protecting creditors’ rights and reducing the risk of debt repudiation by the Republic of Genoa. The behavior of this institution is broadly consistent with debt models that predict lending if lenders can impose big penalties on debtors, and models in which lenders can differentiate between excusable and inexcusable defaults. San Giorgio shareholders enjoyed low credit risk but also lower returns on capital than those prevailing on comparable foreign assets for which creditors’ protection mechanisms were lacking. The Republic’s quid pro quo was a low cost of financing. Differences in credit risk were an important explanation of differences in long-term interest rates across countries in 16th and 17th century Europe, a point not sufficiently emphasized by the literature. Copyright Oxford University Press Science+Business Media, LLC 2006