A Note on Base Change,Identities Involving τ(n), and a Congruence of Ramanujan

We use the theory of quadratic base change to derive some new identities involving the Ramanujan -function, and show how the Ramanujan congruence (n) ₁₁(n) (mod 691) follows.     

Mach's principle and Newtonian mechanics

The union of Mach's principle and Newtonian mechanics gives rise to Relational Mechanics. We find that the characteristics of the revised mechanics are: (1) freedom from any reference to absolute space; (2) the identity of inertial and gravitational mass; (3) the relative acceleration of a body in a gravitational field dependent on the mass of the body. All these results are valid in the context of a Newtonian mechanics which is being developed in the center-of-mass system of all the particles. The conservation of linear momentum, energy, angular momentum are expressed in relational terms, i.e., no reference is made to absolute space. Relational Mechanics is a classical relativistic theory which can be formulated to satisfy Einsteinian relativistic requirements. The Hamiltonian formalism for Relational Mechanics is discussed. Preliminary report Bull. Am. Phys. Soc.14, 15 (1969)     

The q-Zassenhaus formula

A q-deformed analogue of the Zassenhaus formula, expressing the q-exponential of a sum of two noncommuting operators in terms of an infinite product of q-exponentials, is introduced.     

Monte Carlo simulation of 2D quantum gravity as open dynamically triangulate random surfaces

We describe a Monte Carlo procedure for the simulation of dynamically triangulate random surfaces with a boundary (topology of a disk). The algorithm keeps the total number of triangles fixed, while the length of the boundary is allowed to fluctuate. The algorithm works in the presence of matter fields. We here present results for the pure gravity case. The algorithm reproduces the theoretical expectations.     

A second-order approximation for the variance of a renewal reward process

Let {C(t), t ? 0} be a renewal reward process. We obtain the approximation Var C(t) = ct + d + o(1), and explicitly identify c and d.     

The first example of a recursive function which is not primitive recursive

The first example of a recursive function which is not primitive recursive is usually attributed to W. Ackermann. The authors of the present paper show that such an example can also be found in a paper by G. Sudan, published concomitantly with Ackermann's paper.     

Intensity of radiation of charged particles passing through crystals

We have calculated the intensity of radiation of a relativistic charged particle moving in a crystalline medium, taking into account the interaction of the charge with the crystal as well as with the radiation. Various modifications to the usual Cerenkov radiation are discussed and under certain conditions enhancement occurs.     

THE APPLICATION OF OPTIMAL CONTROL METHODOLOGY TO A WELL-STIRRED BIOREACTOR

We examine a model for a well-stirred bioreactor with dynamics governed by two ordinary differential equations. The equations model bacteria and contaminant growth and decay. We solve two optimal control problems, taking the input nutrient as the control.