William L. Briggs

University of Colorado | UCD · Department of Mathematical and Statistical Sciences

In the manner of Fechner, this study employed a novel psychophysics methodology and a novel geometric construction task to verify the results of preference judgments for dynamic symmetry (i.e., the golden ratio). The participants were instructed to visually discriminate by forced-choice preference between images with dynamic symmetry and their coun...

Student engagement is considered an important predictor of student achievement, but few researchers have attempted to derive a valid and reliable measure of college student engagement in particular courses. In 2 studies, we developed and explored the validity of a measure of student engagement, the Student Course Engagement Questionnaire (SCEQ). Ex...

Marie and Alex just paid $250,000 for a house. They made a down payment of $50,000 and assumed a 30-year $200,000 mortgage with a fixed annual interest rate of 7.50%. The house will serve as a residence for several years, but Marie and Alex also view it as an investment, as property values in the neighborhood are projected to increase at a rate of...

Preface to the Second Edition Twelve years have passed since the publication of the first edition of A Multigrid Tutorial. During those years, the field of multigrid and multilevel methods has expanded at a tremendous rate, reflecting progress in the development and analysis of algorithms and in the evolution of computing environments. Because of...

Students are aware that germs spread disease. They also know, at least on an intellectual level, that they can avoid catching some diseases by avoiding risky encounters with infected individuals. The definition of risky encounter varies with the illness. For example, such illnesses as the common cold may be spread by an activity as common as shakin...

An accepted practice in screening for prostate cancer involves a needle core biopsy of the prostate gland, which can provide information regarding if, and how much, cancer is present in a gland. This paper documents several investigations into prostate gland biopsy techniques. The first phase of study involves a geometric model of a prostate gland...

Iterative methods are important mechanisms for solving linear systems of equations of the form, Ax = b, when A is a large sparse nonsingular matrix. When an efficient implementation exists, the conjugate gradient (CG) method is a popular technique of this type. This method is implemented via the construction of an orthogonal basis for the underlyin...

While most people rely on numercal methods (most notably the fast Fourier transform) for computing discrete Fourier transforms (DFTs), there it is still an occasional need to have analytical DFTs close at hand. Such a table of analytical DFTs is provided in this paper, along with comments and observations, in the belief that it will serve as a usef...

this paper was done in an exploratory spirit, by investigating the very suggestive similarities between multiresolution analysis and multigrid methods. The results are preliminary and only point to several avenues of future work. Like many mathematical topics that suddenly gain currency, wavelet theory has origins that are not all that recent. Both...

The original Cooley-Tukey FFT was published in 1965 and presented for sequences with length N equal to a power of two. However, in the same paper they noted that their algo- rithm could be generalized to composite N in which the length of the sequence was a pro- duct of small primes. In 1967, Bergland presented an algorithm for composite N and vari...

This paper investigates the implementation of fast direct methods for solving the three-dimensional Poisson equation on loosely coupled hypercube multiprocessors. As a preliminary step, the problem of computing multiple FFTs is considered and two different algorithms are compared. These algorithms are then used to implement two FFT-based fast Poiss...

Apart from the fact that they are both ingenious and remarkably efficient, there would appear to be little kinship between fast Fourier transform (FFT) algorithms and basic multigrid methods. The fast Fourier transform is a powerful direct method for computing the discrete Fourier transform (DFT) of a sequence of complex numbers, whereas multigrid...

A general probabilistic approach is applied to the single-point, single-dose method for estimating individual infusion rates and serum drug concentrations. By using transformations of probability density functions, the effects of variations in the elimination rate constant upon pharmacokinetic variables may be studied and optimal sampling times may...

The methods of cyclic reduction, Fourier analysis and the resulting hybrid algorithm FACR(l) are formulated for a shared memory multiprocessor and applied to the two-dimensional Poisson equation. The complexity of the algorithms is analyzed, performance curves are given and optimal values of the parameter l are determined for various problem sizes...

Given the safety height recommendation of the panel, we have found the maximum height for the salt pile to be approx. 21.7 ft. A model employing a loading strategy was developed to obtain a function of three parameters for maximum height.The function was determined to be stable around the solution. A variation of 10% among the three parameters prod...

It has long been known that an in-place version of the Fast Fourier Transform (FFT) exists for real sequences of data. More recently, in-place FFTs have been devised for real sequences with even, odd, or quarter wave symmetries. All of these symmetric FFTs take the input sequence in scrambled (bit-reversed) order and produce the transform sequence...

Proceeding first by experiment and then by analysis, the problem of implementing FFT algorithms on a shared memory multiprocessor is investigated. Several algorithms for performing a single FFT and multiple FFTs are implemented and compared on the Denelcor HEP computer. These algorithms are then analyzed using performance models which reproduce the...

A mechanism for the destabilization of numerical algorithms for partial differential equations is suggested. The novelty of the work is that it attempts to explain the dynamical process by which noise can localize on a spatial grid and cause finite amplitude instability thresholds to be exceeded at distinct locations.

The second order leapfrog method is used to discretize the linearized KdV equation which is itself a dispersive partial differential equation. The resulting difference equation is solved and analyzed in terms of its dispersion relation and propagation properties. Numerical experiments are included to illustrate some of the results obtained. Finally...

Cooperative growth of off-eutectic composition melts can be achieved if the temperature gradient is sufficiently high and stirring not excessive. The initial transient in this process occurs in two stages. The first, Stage I, raises the concentration of the single-phase solid freezing out to the solid solution concentration at the eutectic isotherm...

This chapter discusses the sparse matrix problems in a finite element open ocean model. The earliest efforts in numerical oceanography were devoted to large-scale models that typically encompassed an entire ocean basin. These models furthered the understanding of the general features of ocean circulation such as the Gulf Stream but left unresolved...

This study considers the numerical results for the airflow pattern to obtain droplet trajectories around and into a leading edge hole of a cloud droplet sampling probe. The probe was initially conceived at MIT and developed into a practical airborne system in the Cloud Physics Program at the National Center for Atmospheric Research. The probe is de...

Mathematics educators have been in the midst of reform for the past twenty years. Driven by a series of influential reports concerning the deteriorating state of mathematics education [including, Everybody counts (National Research Council, 1989) and Moving beyond myths (CMS2000, 1991)], many mathematics teachers have altered the way they teach: th...