Dynamic programming in cricket: optimizing batting order for a sticky wicket

In cricket, a rain-affected pitch can make batting more difficult than normal. Several other conditions such as poor light or an initially lively pitch, may also result in difficulties for the batsmen. In this note, we refer to all of them as ‘sticky wickets’. On sticky wickets, lower order batsmen are often sent into ‘hold the fort’ until conditions improve. In this paper, a stochastic dynamic programming model is used to examine the appropriateness of this policy. The model suggests that the tactic is often optimal when the sticky wicket persists until the end of the day's play, but not often when the sticky wicket is transitory. In some circumstances, it is worthwhile, on a normal wicket near the end of the day, to send in a lower order batsman to hold the fort (a night watchman): when the wicket is sticky, this tactic is even more worthwhile.     

Characteristics of transmission of electrons in a quantum wire tangentially attached to a superconductor ring threaded by magnetic flux

We present numerical investigations of the transmission properties of electrons in a normal quantum wire tangentially attached to a superconductor ring threaded by magnetic flux. A point scatterer with a δ -function potential is placed at node to model scattering effect. We find that the transmission characteristics of electrons in this structure strongly depend on the normal or superconducting state of the ring. The transmission probability as a function of the energy of incident electrons, in the case of a superconductor ring threaded by one quantum magnetic flux, emerges one deep dip, imposed upon the first broad bump in spectrum. This intrinsic conductance dip originates from the superconductor state of the ring. When increasing the magnetic flux from one quantum magnetic flux to two, the spectrum shifts toward higher energy region in the whole. This conductance dip accordingly shifts and appears in the second bump. In the presence of a point-scatterer at the node, the spectrum is substantially modified. Based on the condition of the formation of the standing wave functions in the ring and the broken of the time-reserve symmetry of Schr?dinger equation after switching magnetic flux, the characteristics of transmission of electrons in this structure can be well understood. Received 6 November 2001     

Phase Conjugation in Saturable Absorbing Dye Films by Degenerate Four-Wave Mixing and Holographic Processes Using Nanosecond Pulse and CW Lasers

We report on the simultaneous generation of phase-conjugate signals by degenerate four-wave mixing (DFWM) and holographic processes using a nanosecond pulse and a CW lasers in polyvinyl alcohol (PVA) films doped with four kinds of saturable absorbing dyes. For the pulse laser, of the four kinds of dye-doped PVA films, the erythrosine B-doped PVA and uranine doped-PVA films generate PC signals only by the DFWM process, while the other dye-doped PVA films generate PC signals simultaneously by not only the DFWM process but also the holographic process. Especially, the safranin T-doped PVA film generates strong PC signals by the holographic process. In contrast, all of the dye-doped PVA films generate the two types of PC signals for the CW laser. The fading of dye molecules is found to result in the generation of the holographic component of PC signals which governs the temporal behavior of the total PC signals.     

Controlling global stochasticity of Hamiltonian systems by nonlinear perturbation

A method of controlling global stochasticity in Hamiltonian systems by applying nonlinear perturbation is proposed. With the well-known standard map we demonstrate that this control method can convert global stochasticity into regular motion in a wide chaotic region for arbitrary initial condition, in which the control signal remains very weak after a few kicks. The system in which chaos has been controlled approximates to the original Hamiltonian system, and this approach appears robust against small external noise. The mechanism underlying this high control efficiency is intuitively explained. Received 15 January 2002 Published online 6 June 2002     

Improving the Fundamental Theorem of Algebra

Theorems 2 and 3 establish the minimum algebraic conditions necessary for a field to be algebraically closed, and they can therefore be said to “optimize” the Fundamental Theorem of Algebra. But each specific“degree implication” is a first-order consequence of the axioms for fields, and could have been discovered two centuries ago; the existence of these finitary relationships appears to have been unsuspected by practically everyone, with one important exception.     

Ultrashort-pulse dual-wavelength source for digital holographic two-wavelength contouring

Digital holographic shape measurements using femtosecond laser pulses are reported. For contouring of very fast moving objects, the simultaneous generation of at least two spectrally separated ultrashort pulses is required. To deliver this particular spectral signature at high pulse energies, a chirped-pulse Ti:sapphire laser amplifier was modified to emit two spectrally separated pulses with energies above 1 μJ each. The wavelength separation of these pulses was adjustable within the 50 nm gain bandwidth, cutting out two distinct wavelength peaks by a variable double-slit assembly in a prism sequence. A Michelson-type interferometer was employed to perform the two-wavelength contouring. The phases of the holograms and the phase differences are calculated numerically, which allow us to deduce the contour lines of the topology of the object. The suitability of the light source for digital holography is demonstrated with contouring of stationary objects and the potential for high-speed applications is indicated. PACS 42.40.-i; 42.60.By     

A TVD Type Wavelet-Galerkin Method for Hamilton-Jacobi Equations

In this paper,we use Daubechies scaling functions as test functions for the Galerkin method,and discuss Wavelet-Galerkin solutions for the Hamilton-Jacobi equations.It can be proved that the schemesare TVD schemes.Numerical tests indicate that the schemes are suitable for the Hamilton-Jacobi equations.Furthermore,they have high-order accuracy in smooth regions and good resolution of singularities.     

Network externalities in a dynamic monopoly

We reconsider the role of network externalities in a dynamic spatial monopoly where the firm must invest in order to accumulate capacity, while consumers may have either linear or quadratic preferences. We (1) characterise saddle point equilibria, (2) prove that the extent of market coverage is increasing in the network effect and (3) unlike the existing static literature on the same problem, the monopolist may not make introductory price offers. Then, we briefly deal with the socially optimal solution, showing that, in general, a planner would serve more consumers than a profit-seeking monopolist.