David Shelupsky

Limitless Integrals and a New Definition of the Logarithm

We show by a clever choice of the evaluation points in the partition intervals used to define the Riemann sums for these integrals that we can arrange for these sums to agree exactly with the value of the integral. Thus the integrals can be evaluated without having to take limits. We also show how the inequalities used here lead to a new definition of the natural logarithm that does not involve limits, series or integrals, or even the exponential function. A Riemann sum for the integral in (1) is

The principle of equivalence and theories of gravity

We construct classical theories of gravity on the basis of special relativity and the Einstein-Infeld accelerating-elevator thought experiment. The resulting theories share most of the main features of general relativity, namely the nonlinear character of the theory, the metrical significance of the gravitational potentials and the geodesic equation of particle motion. They differ from general relativity in at most nonlinear terms in the gravitational constant G in their equations of particle motion and field equations.