Selling Passes to Strategic Customers

Many service providers offer a prepaid package of credits that can be redeemed for future use, often called passes, in conjunction with regular individual sales. In dynamic pricing situations, customers can strategize on the purchase, redemption, and renewal of the pass by optimizing the timing and choices between passes and individual items based on future prices and their own changing needs. In “Selling Passes to Strategic Customers,” Wang, Levin, and Nediak integrate dynamic choice modeling and optimal control theory to study how to jointly price the passes and individual items in a dynamic setting. They endogenize an individual customer’s purchase/redemption decisions in their model and find that the seemingly complex problem has a simple (approximate) solution. The optimal prices remain nearly constant most of the time, except near the beginning and end of the sales horizon, exhibiting so-called turnpike properties. The pass, as a form of advance purchase, allows the seller to capitalize on the customer’s forward-looking behavior by exploiting the uncertainty of customer valuations.

DOUBLE WELL POTENTIAL: PERTURBATION THEORY, TUNNELING, WKB (BEYOND INSTANTONS)

A simple approximate solution for the quantum-mechanical quartic oscillator V = m2x2 + gx4 in the double-well regime m2 < 0 at arbitrary g ≥ 0 is presented. It is based on a combining of perturbation theory near true minima of the potential, semi-classical approximation at large distances and a description of tunneling under the barrier. It provides 9-10 significant digits in energies and gives for wavefunctions the relative deviation in real x-space less than ≲ 10-3.

The Top With a Blunted Vertex Slipping With Light Friction at a High Rate of Spin

The problem of determining the motion of a top is a classic example of a complex analysis in analytical dynamics. Adding a blunt tip to the top and setting it spinning on a surface with sliding friction might be thought to render it intractable for simple analysis. However, if it is set in motion with a high rate of spin it is possible to find a simple approximate solution for the case of approximate steady precession. For this pseudo-steady motion it will be noted that the rate of diminution of the nutation will also be almost constant. Further, the ratio of these rates (latter over former) will be equal to the negative of the coefficient of friction for the top slipping on the surface. As a consequence the mass center of the top will tend to proceed around a steady circle above the plane. These results will first be observed by writing the full Lagrange’s equations for the problem and reducing them prior to integration by observing appropriate approximations by deleting relatively smaller terms. The above results will then follow directly. Further, the full Lagrange’s equations will be numerically integrated to show that the analytically developed approximate results are appropriate. Once these results are known, it is observed that a subsequent intuitive analysis based on time rate of change of angular momentum leads to the same results, if only the angular momentum about the spin axis is considered with other relevant assumptions. [S0021-8936(00)00203-8]

Approximate Biaxial Stress Solution for Orthotropic Open-Hole Composite Laminates

A simple approximate solution has been derived for the stress distribution near a circular hole applicable to any orthotropic composite laminate subjected to biaxial loading. The degree of accuracy of this solution was found to be overall acceptable, but strongly dependent upon the laminate lay-up and biaxiality ratio.