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Bibliography: p. 335-337

The research under this project was aimed at developing numerical methods for fitting stochastic partial differential equations to irregularly spaced spatial data. This is related to two dimensional smoothing spline fitting where the partial differential equation is the Laplacian driven by white noise. A class of continuous two dimensional spatial autoregressive, moving average (ARMA) models were investigated and numerical methods developed to implement fitting these models to spatial data. The spatial ARMA models provide a complete class of covariance structures rather than a very limited set of covariance functions that are typically used in Kriging. Since maximum likelihood methods are used to fit the models, methods such as likelihood ratio tests and Akaike's Information Criterion (AIC) can be used for model selection. Prediction maps can then be calculated at a grid of points, and contour maps drawn. Also maps can be drawn of the standard deviation of the predicted fields giving indications of the variability of the predictions. Applications include aquifer heights, coal field depth and thickness and snowfall amounts. Results have been presented in a number of presentations and publications.

The research under this project was aimed at developing numerical methods for fitting stochastic partial differential equations to irregularly spaced spatial data. This is related to two dimensional smoothing spline fitting where the partial differential equation is the Laplacian driven by white noise. A class of continuous two dimensional spatial autoregressive, moving average (ARMA) models were investigated and numerical methods developed to implement fitting these models to spatial data. The spatial ARMA models provide a complete class of covariance structures rather than a very limited set of covariance functions that are typically used in Kriging. Since maximum likelihood methods are used to fit the models, methods such as likelihood ratio tests and Akaike's Information Criterion (AIC) can be used for model selection. Prediction maps can then be calculated at a grid of points, and contour maps drawn. Also maps can be drawn of the standard deviation of the predicted fields giving indications of the variability of the predictions. Applications include aquifer heights, coal field depth and thickness and snowfall amounts. Results have been presented in a number of presentations and publications.

Thesis advisor(s): Lucas, Thomas W

This thesis extends previous research on validating Lanchester's equations with real data. The quality of the available, historical data for validation of attrition models is poor. Most accessible battle data contain only starting sizes and casualties, sometimes only for one side. A detailed database of the Battle of Kursk of World War II, the largest tank battle in history, has recently been developed. The data, were collected from military archives in Germany and Russia by the Dupuy Institute (TDI) and were reformatted into a computerized data base, designated as the Kursk Data Base (KDB), and recently made available and documented in the KOSAVE (Kursk Operation Simulation and Validation Exercise of the U.S. Army) study. The data are two-sided, time phased (daily), and highly detailed. They cover 15 days of the campaign. This thesis examines how the various derivatives of Lanchester's equations fit the newly compiled database on the Battle of Kursk. In addition, other functional forms are fit. These results are contrasted with earlier studies on the Ardennes campaign. It turns out that a wide variety of models fit the data about as well. Unfortunately, none of the basic Lanchester models fit the data, bringing into question their use in combat modeling.

This paper presents and documents a FORTRAN program FINLIE for fitting a system of ordinary differential equations (or a system of algebraic or transcendental equations) to observed data. FINLIE determines those values of the possibly nonlinear system parameters and initial conditions that yield a best fit - in the least squares sense - of the solution curves to measurements of one or more of the dependent variables. The basic fitting technique is Chapman-Kirk, with the Marquardt algorithm aiding convergence. The data from more than one experiment can be handled simultaneously to obtain one common set of parameters and set of initial conditions for each experiment. For each computer run, the value of any parameter or initial condition can be held fixed or adjusted by FINLIE.

Much of mathematical analysis is devoted to the problem of predicting the future behavior of a system, given a descriptive equation and the current state. This is surprising since a basic scientific problem in such fields as physics, engineering, biology and economics is that of determining the structure of a system, given various observations over time. Many types of functional equations may be converted into systems of ordinary differential equations. This means that wide classes of direct problems can be solved as initial-value problems. This also means that a great many inverse problems may be computationally resolved. Let the equations which describe a particular process be a system of differential equations. The unknown structure of the process is reflected in unknown system parameters which appear in the differential equations or in the initial conditions. These parameters are to be estimated on the basis of observations of the process. The identification problem takes the form of a nonlinear boundary-value problem. This can be solved by a variety of methods. One that has been shown to be quite effective is described in this report.

This article is from BMC Systems Biology , volume 8 . Abstract Background: The miRNAs are small non-coding RNAs of roughly 22 nucleotides in length, which can bind with and inhibit protein coding mRNAs through complementary base pairing. By degrading mRNAs and repressing proteins, miRNAs regulate the cell signaling and cell functions. This paper focuses on innovative mathematical techniques to model gene interactions by algorithmic analysis of microarray data. Our goal was to elucidate which mRNAs were actually degraded or had their translation inhibited by miRNAs belonging to a very large pool of potential miRNAs. Results: We proposed two chemical kinetics equations (CKEs) to model the interactions between miRNAs, mRNAs and the associated proteins. In order to reduce computational cost, we used a non linear profile clustering method named minimal net clustering and efficiently condensed the large set of expression profiles observed in our microarray data sets. We determined unknown parameters of the CKE models by minimizing the discrepancy between model prediction and data, using our own fast non linear optimization algorithm. We then retained only the CKE models for which the optimized fit to microarray data is of high quality and validated multiple miRNA-mRNA pairs. Conclusion: The implementation of CKE modeling and minimal net clustering reduces drastically the potential set of miRNA-mRNA pairs, with a high gain for further experimental validations. The minimal net clustering also provides good miRNA candidates that have similar regulatory roles.

Delia (1592) is a cycle of Petrarchan love sonnets written by Renaissance poet Samuel Daniel (1562-1619). He was also a noted playwright and historian, and a close contemporary of Ben Jonson and William Shakespeare. Delia may have influenced Shakespeare’s sonnets. This project contains the first 30 sonnets from the collection "Delia". (Summary by Dr Alan Weber)