Parametric Estimation

The exponential distribution is often used in reliability work to describe the distribution of time to “chance” failure and is characterized by a constant failure rate. In this paper the small sample powers are compared for four test statistics for the hypothesis of constant failure rate vs. the hypothesis of non-constant failure rate. The tests are compared for samples of size n = 10(5)50 using the Weibull distribution for the alternative distribution. The shape parameter of the Weibull is varied from 0.5 to 2.5. For the two test statistics which involve arbitrary grouping of the data the effect of group size and number was also examined.     

Geometric invariance

 We discuss changes of variables deriving from changes in the placement of reference entities in the modeling of rigid bodies, beams and shells. We show that standard numerical procedures in general fail to be invariant with respect to these particular changes of variables, and we characterize the nature of the error. Finally, we show how the fixed pole form of the equations of equilibrium automatically guarantees invariant numerical processes, being based on global kinematic and co-kinematic quantities. Received 10 March 2001 / Accepted 30 April 2002     

Roton Parametric Resonance

Parametric excitation of rotons by oscillating electric field exhibits a narrow resonance at the roton minimum frequency. The resonance region width is in good agreement with experimental results on the microwave absorption in superfluid helium.     

The Geometric Phase

Abstract

The change in the phase of a beam of light produced by a cyclic change in its state of polarization is an example of the geometric phase that can be verified by interferometric measurements. This paper presents a quantum-field theoretical analysis of the geometric phase interferometer in the limit of a small photon number, as well as some experimental results that confirm that the optical effects due to the geometric phase persist down to the single-photon level.     

Parametric light generation

Since its invention more than 40 years ago, the laser has become an indispensable optical tool, capable of transforming light from its naturally incoherent state to a highly coherent state in space and time. Yet, due to fundamental limitations, operation of the laser remains confined to restricted spectral and temporal regions. Nonlinear optics can overcome this limitation by allowing access to new spectral and temporal regimes through the exploitation of suitable dielectric materials in combination with the laser. In particular, optical parametric oscillators are versatile coherent light sources with unique flexibility that can provide optical radiation across an entire spectral range from the ultraviolet to the far-infrared and over all temporal scales from continuous wave to the ultrafast femtosecond domain.     

Parametric time warping

A parametric model is proposed for the warping function when aligning chromatograms. A very fast and stable algorithm results that consumes little memory and avoids the artifacts of dynamic time warping. The parameters of the warping function are useful for quality control. They also are easily interpolated, allowing alignment of batches of chromatograms based on warping functions for a limited number of calibration samples.     

Applicability of Optical Geometric Differentiation for Time-average Geometric Moiré

Abstract:  Optical experimental method for analysis of derivatives of dynamic displacements from patterns of time-average moiré fringes is proposed in this paper. Derived analytical relationships for one-dimensional optical model are validated for two-dimensional elastic structures performing in-plane vibrations. The fact that digital image filtering techniques are normally required for the identification of time-averaged super-fringes does not lessen the practical value of the proposed method.     

Geometric quantum computation

Abstract

We describe in detail a general strategy for implementing a conditional geometric phase between two spins. Combined with single-spin operations, this simple operation is a universal gate for quantum computation, in that any unitary transformation can be implemented with arbitrary precision using only single-spin operations and conditional phase shifts. Thus quantum geometrical phases can form the basis of any quantum computation. Moreover, as the induced conditional phase depends only on the geometry of the paths executed by the spins it is resilient to certain types of errors and offers the potential of a naturally fault-tolerant way of performing quantum computation.     

Geometric phase lens

The design of a lens that modulates the geometric phase of an optical beam by manipulating its polarization is presented. To produce such a geometric phase element with a spatially varying phase function, one needs a wave plate with varying orientation. One can use subwavelength grooves to produce form birefringence, but the variation in orientation generally leads to branch points in the groove pattern. These branch points do not affect the phase of the traversing beam directly because the grooves are subwavelength. However, they do produce errors in the groove orientation, which indirectly leads to errors in the geometric phase function that is implemented. A design procedure is provided to compute the groove pattern for such a rotationally symmetric geometric phase element; and, with the aid of a numerical simulation, the effect of the branch points in the groove pattern on its performance is investigated.     

Parametric identification problems

An approach to the generation of stopping rules in parametric identification problems is proposed on the basis of the computation of a statistic of the difference between two successive estimates. This statistic is also used for fault detection in the Kalman filter. Translated from Izmeritel'naya Tekhnika, No. 3, pp. 20–22, March, 1994.     

Remarks on the Computation of Minimal Finite Free Resolutions

Several improvements to the computation of the minimal free resolution of finite modules have been made also recently ([5], [15]) and some of them concerning ideals of polynomials are Hilbert driven or depend on the knowledge of the regularity of the ideal. Here I show that some of the calculations usually made to determine the minimal free resolution of a homogeneous polynomial ideal I are still redundant, provided that we know a set of minimal homogeneous generators, the regularity, and the Hilbert function of I. More precisely, I show that the construction of some syzygies can be avoided. As result, the known methods for the computation of a minimal free resolution are improved. The underlying idea of this work can be also used to predict if certain points in generic position on general rational curves satisfy the minimal resolution conjecture. Received: May 30, 2000; revised version: March 8, 2001