Computer algebra and operators

The symbolic computation of operator expansions is discussed. Some of the capabilities that prove useful when performing computer algebra computations involving operators are considered. These capabilities may be broadly divided into three areas: the algebraic manipulation of expressions from the algebra generated by operators; the algebraic manipulation of the actions of the operators upon other mathematical objects; and the development of appropriate normal forms and simplification algorithms for operators and their actions. Brief descriptions are given of the computer algebra computations that arise when working with various operators and their actions

Contingent Capital and Bank Risk-Taking among British Banks before World War I

The recent financial turmoil highlights the incentive of highly leveraged financial institutions to take excessive risk, given the protection of limited liability. During the nineteenth and early twentieth century, many banks operated under liability rules which obligated shareholders to bear larger costs of bank insolvency in the form of contingent, or even unlimited liability. This paper examines the empirical relationship between the size of banks’ contingent liability and their risk-taking behavior using data on British banks from 1878-1912. We find that banks with more contingent liability appear to have taken less risk. We also find evidence that the risk-reducing effects of contingent liability were larger for banks with higher leverage, suggesting that contingent capital mitigated moral hazard problem at banks.Contingent Capital, Bank Risk-Taking, British Banks

The realization of input-output maps using bialgebras

The theory of bialgebras is used to prove a state space realization theorem for input/output maps of dynamical systems. This approach allows for the consideration of the classical results of Fliess and more recent results on realizations involving families of trees. Two examples of applications of the theorum are given

Trees, bialgebras and intrinsic numerical algorithms

Preliminary work about intrinsic numerical integrators evolving on groups is described. Fix a finite dimensional Lie group G; let g denote its Lie algebra, and let Y(sub 1),...,Y(sub N) denote a basis of g. A class of numerical algorithms is presented that approximate solutions to differential equations evolving on G of the form: dot-x(t) = F(x(t)), x(0) = p is an element of G. The algorithms depend upon constants c(sub i) and c(sub ij), for i = 1,...,k and j is less than i. The algorithms have the property that if the algorithm starts on the group, then it remains on the group. In addition, they also have the property that if G is the abelian group R(N), then the algorithm becomes the classical Runge-Kutta algorithm. The Cayley algebra generated by labeled, ordered trees is used to generate the equations that the coefficients c(sub i) and c(sub ij) must satisfy in order for the algorithm to yield an rth order numerical integrator and to analyze the resulting algorithms

International Aspects of the Great Depression and the Crisis of 2007: Similarities, Differences, and Lessons

We focus on two international aspects of the Great Depression--financial crises and international trade—and try to discern lessons for the current economic crisis. Both downturns featured global banking crises which were generated by boom-slump macroeconomic cycles. During both crises, world trade collapsed faster than world incomes and the trade decline was highly synchronized across countries. In the Depression, income losses and rises in trade barriers explain trade’s collapse. Due to vertical specialization and more intense trade in durables today’s trade collapse is due to uncertainty and small shocks to trade costs hitting international supply chains. So far, the global economy has avoided the global trade wars and banking collapses of the Depression perhaps due to improved policy. Even so, the global economy remains susceptible to large shocks due to financial innovation and technological change as recent events illustrate.Great Depression, Crisis of 2007