arXiv subjects
Publications and source records attributed to David L. Farnsworth.
An extensive table of pairs of functions linked by the Legendre transformation is presented. Many special functions and formulas that are used in the sciences are included in the pairs. Formulations are provided for finding the Legendre transformations of analytic functions and of polynomial approximations to other functions.
A differentiable curve y = y(x) is determined by its tangent lines and is said to be the envelope of its tangent lines. The coefficients of the curve's tangent lines form a curve in another space, called the dual space. There is a transformation between the original x,y-space and the dual space, such that points on the original curve and the curve of the coefficients of the tangent lines are transformed into each other.The dual space and the transformation depend upon the form that is used for the tangent lines. One choice is y = mx + b, so that the coordinates in the dual space are m and b, where the curve representing the tangent lines is b = b(m). Each point of the curve b = b(m) in the dual space corresponds to a tangent line to the curve y = y(x) in x,y-space. We present other choices of the form for the tangent lines and explore techniques for finding a curve from the tangent lines, transformations between the original space and a dual space and among dual spaces, the differential equation, whose solutions are exactly the equations of the equations of the tangent lines and the original curve. Geometric constructions and other tools are used to find dual curves.
We analyze extensions of the Poisson process in which any interarrival time that exceeds a fixed value $r$ is counted as an interarrival of duration $r$. In the engineering application that initiated this work, one part is tested at a time, and $N(t)$ is the number of parts that, by time $t$, have either failed, or if they have reached age $r$ while still functioning, have been replaced. We refer to $\{N(t), t \geq 0\}$ as a replace-after-fixed-time process. We extend this idea to the case where the replacement time for the process is itself random, and refer to the resulting doubly stochastic process as a replace-after-random-time process.
Properties and examples of the dual transformation between two planes, which is such that the coordinates of a point in the original plane give the coefficients of a line in the dual plane and the coefficients of a line in the original plane give the coordinates of a point in the dual plane, are examined.