Several midshipmen conducted research projects either as Honors Mathematics Majors or in specially created projects under the guidance of faculty members. Several faculty spent many hours serving as mentors and readers for capstone projects required of all majors.

Professor Peter Andre guided midshipman William Getchius in his Honors Research Project: "Spatially Distributed Prisoner's Dilemma". Professor Carol Crawford continued to lead many midshipmen in their discovery of the applicability of discrete mathematics. Professor W. David Joyner helped midshipmen James McShea, Michael Fourte, and Ann Luers in individualized mathematical investigations as well as Luers' Honors Research Project: "Dodecahedral Faces of M₁₂ and the Pyraminx Project". Professor Charles Mylander directed midshipman Janette Hay's development of a Markov chain model to predict how promotion and retirement policy decisions affect the distribution of naval enlisted personnel. Professor Geoffrey Price supervised Midshipman Glenn Truitt's Honors Research Project: "Symmetries of Nullity Sequences of Toeplitz Matrices". Professor Thomas Sanders worked with Midshipman Mark Tripiano on helping the USMC predict enlisted personnel requirements for HMLA after the UH-1 Huey is replaced by the H-60 Blackhawk. Midshipman Michael Wheeler worked on "Factors Affecting Promotion Rates" for the USMC under Professor John Turner and Major J. Michael Shehan as well as on "Factors Affecting Promotion in the Marine Corps" with the guidance of Professor John Turner. Professor Peter Turner led eight midshipmen in various investigations concerning massively parallel computers applied to linear algebra and to artificial intelligence.

Once again, the USNA Mathematics Department produced a wide range of scholarly work that appeared as technical reports or as publications in refereed journals throughout the world. Nearly thirty articles appeared as applications of mathematics or as pure mathematical research. Topics include: fingerprint identification, mathematical physics and cosmology, parallel computer applications, algorithms for computers, and basic research in areas such as algebra, differential equations, combinatorics, matrices, and number theory.

In addition to over a dozen independent research projects, another two dozen research projects were sponsored by a variety of sources, such as:

Arizona State University

The Johns Hopkins University/Applied Physics Lab

Office of Naval Research

Naval Air Warfare Center

Naval Surface Warfare Center

Naval Academy Research Council

National Science Foundation

National Security Agency

During the past year, members of the USNA Mathematics Department presented the results of their work on over fifty occasions at professional mathematical meetings and colloquia throughout the United States and abroad. This activity, along with publication, enhances the academic stature of the Naval Academy and promotes the professional growth and reputation of those individuals involved. Through research activity, the faculty expand their intellectual horizons and stay vital in their disciplines. They contribute to the discovery of new mathematics. And they develop new materials and ideas that they can share with midshipmen students in their mathematics courses and research projects.

Sponsored Research

When considering sound transmission in a shallow ocean, the acoustic properties of the seabed below must be taken into account. The seabed has been modeled variously as a completely rigid slab, dense fluid, or an elastic slab. A more realistic model needs to allow for the poroelastic nature of the sediment. In the Biot-Stoll sediment model the seabed is taken to consist of a viscoelastic frame with an interstitial pore fluid. For the last four years the researcher has been investigating sound transmission in the ocean over a poroelastic seabed. Recent work has included computing acoustic pressure in the near field over a one layer seabed using a modal solution combined with integrals along certain branch cuts, computing pressure in the far field over a two layer seabed using a model solution, and computing pressure in the far field using a numerical technique called parabolic approximation.

A related problem that is currently being investigated is the inverse problem of determining the nature of the seabed from the measured values of pressure in the far field. The investigator has developed a fairly successful algorithm based on the Nelder-Mead simplex method for determining the five parameters of an elastic seabed. Further investigation will reveal whether a similar approach can successfully find the more numerous (eleven) parameters of a poroelastic seabed.

A two-dimensional connected beam structure is modeled as a network of connected dynamic systems, each characterized by a propagation wavenumber, loss factor, and length. An analytic formal procedure has been developed to calculate the response of the network to an out-of-plane harmonic drive as a function of frequency.

Computer code (using a version of FORTRAN-90 on a Massively Parallel Computer) has been written and tested, implementing the above mentioned formal analytic procedure for large systems. The next step is to generalize the formal analysis and computational programs to include three dimensional systems and other than out of plane harmonic drives.

Also, structures with discrete periodic variations in impedance may exhibit pass and stop bands and the related wave localization and delocalization phenomena in their frequency response. Localization, similar to Anderson localization in atomic systems, occurs in the pass band frequency range when the periodicity is perturbed and waves are thereby inhibited from propagating. Conversely, delocalization occurs in the same systems in the stop band regions where perturbing the strict periodicity allow for relatively more propagation. Localization and delocalization are demonstrated in several systems: specifically, a beaded string, membranes and plates with periodic stiffeners attached, and a "jungle gym," i.e., a connected beam structure. We have demonstrated that these effects depend on the interactions between discontinuities and are studying the implications to passive and active vibration control.

Naval vessels containing ferrous material emit detectable magnetic signals. Naval Surface Warfare Center administers a project that studies and measures this phenomenon. Data is collected from naval vessels under a variety of conditions and from scale models under similar conditions. A goal of the project is to connect these data in a manner that will allow measurements from the scale model to predict measurements on the naval vessel. This analysis is both statistical and mathematical in nature. It includes both the design of the experiments and the analysis of the data.

Combinatorial (or Infinite) Group Theory refers to the theory of group presentations, that is, of groups specified by a set of generators and corresponding defining relations. The theory has its origins in topology and complex analysis and in particular in the theory of the fundamental groups of combinatorial cell complexes. Because of its nature and its origins, combinatorial group theory comes into contact with and uses many different areas of mathematics. Clearly algebra and topology as mentioned above are very significant for the combinatorial group theorist. But also hyperbolic geometry comes into play via the study of Cayley graphs, pure mathematical logic through the study of various decision problems, and last, but not least, computer science through the study of rewriting systems (certain kinds of algorithms). Central to all of these studies is the concept of a free group. This centrality is due to the fact that a free group is the most basic construction of infinite group theory and also that free groups serve as primary motivating examples for both properties and proofs in all the other areas mentioned. The purpose of this project would be to try to bring to bear all these different points of view and focus primarily on the group theoretical properties closely tied to the Tarski conjecture. This conjecture has to do with the relations of combinatorial group theory to pure logic and the logical underpinnings of the whole theory.

The authors are defining and analyzing mixed finite element methods for solving time dependent partial differential equations. Our methods are based on a previous paper "Analysis of Some Moving Space-Time Finite Element Methods," by Todd Dupont and Rafael Santos where one method allows for adding and deleting knots in a continuous fashion and the other allows for discontinuous changes in the mesh. They believe the combination of mixed finite element methods with the moving techniques will bring surprising results for parabolic equations models.

The idea here is to take different meshes and subspaces for different time levels. The author believes a good approximation to the solution using this type of technique would be that the finite element solution at the current time is projected on the next (time) finite element space and then it should adopt the Crank-Nicolson scheme to evaluate the next finite element approximation.

The author believes the treatment of the thermo-elasticity problems with mixed methods could bring pleasant surprises on the rates of convergence compared with traditional numerical methods. She
believes the convergence of the mixed method continuous time scheme for parabolic system is going to be reduced to a question of convergence of the associated elliptic problem.

The researcher is studying mild singularities and Cauchy horizons in spacetime models. Mild singularities include quasiregular and nonscalar curvature singularities. In the former, particle paths may end with no accompanying catastrophes, while in the latter, some particles moving near the singularity will feel infinite tidal forces, but not all do. In most cases the spacetime models examined satisfy Einstein's equations.

In particular, the researcher is using a conjecture she and T. M. Helliwell first published in 1985 to predict whether various mild singularities and Cauchy horizons are stable. Thus far the conjecture has held true for the quasiregular singularities in Taub-NUT-type cosmologies and in Khan-Penrose spacetime when fields are added. When applied to the quasiregular singularity in Bell-Szekeres spacetime and the nonscalar curvature singularity and Cauchy horizon in a type-V LRS spacetime, a prediction was possible but no exact solutions were available for comparison. A study of the Cauchy horizons in Reissner-Nordstrom spacetime using the conjecture correctly predicted the effects of null dust when compared with exact solutions. The Cauchy horizons in the Kerr spacetime were predicted to be generally unstable to the addition of null dust but no exact solutions were known for comparison.

Last year the Cauchy horizons in anti-deSitter spacetime were shown to be unstable to the addition of null dust. For the first time the conjecture failed -- it predicted correctly the occurrence of a singularity but not the type. Therefore this year we have studied the instability more thoroughly using scalar fields and we have altered the conjecture to account for the discrepance. A paper was recently published on scalar field behavior in anti-deSitter spacetime and the newconjecture. Further studies using the new, improved conjecture are underway.

Work was also done this year to understand the nature and stability of the Geroch and Tod spacetimes which have complete geodesics but an incomplete pathof bounded acceleration. The singularities were found to be quasiregular but stable to the addition of fields. In addition, we studied the stability of Cauchy horizons in single plane wave spacetimes and a paper is being prepared for submission to Physical Review D.

This report extends the ideas behind Bareiss' fraction-free Gauss elimination algorithm in a number of directions. First, in the realm of linear algebra, algorithms are presented for fraction-free LU "factorization" of a matrix and for fraction-free algorithms for both forward and back substitution. These algorithms are valid not just for integer computation but also for any matrix system where the
entries are taken from a unique factorization domain such as a polynomial ring. The second part of the paper applies a fraction-free formulation to resultant algorithms for solving systems of polynomial equations. In particular, the use of fraction-free polynomial arithmetic and triangularization algorithms in computing the Dixon resultant of a polynomial system is discussed in detail.

In this project the researchers developed a series of projects for Calculus I, II and III. Professor Michael wrote the projects for Calculus III while Professor Penn wrote the projects for Calculus I and II. These projects involved applications of Calculus that should be of interest to midshipmen. They used the software,

Maple, which was issued to the midshipmen.The projects are available on the courses web pages maintained by the USNA Mathematics Department.

One of the key problems in the theory of von Neumann algebras is to study and to classify the position of subfactors of a prescribed index in the hyperfinite II₁ factor R. In many ways this problem resembles the analysis of subgroups in group theory: in fact, the group-theoretic notions of index, normality, and conjugacy all have analogues in the theory of subfactors. Over the past few years Price has worked jointly with Professor R. T. Powers to study a family of subfactors in R on which one can define a sort of non-commutative version of the Bernoulli shift of index 2. These shifts are called binary shifts. For each binary shift there is a corresponding bitstream of 0's and 1's which defines the shift. The structure of the shift is reflected in certain properties possessed by the bitstream. For example, the relative commutant of the range ^k(R) of the kth power ^k of the shift is non-trivial for some k, if and only if, the bitstream is eventually periodic. In independent work over the past two years, Price has shown that all binary shifts of commutant index 2 are cocycle conjugate. Subsequently to obtaining this result, Price has successfully addressed the cocycle conjugacy problem for binary shifts of higher relative commutant indices.

This project involved the continued development of a cruise missile and tactical air (TACAIR) effectiveness assessment system that is being done by the Strike and Anti-Surface Warfare Group of the Naval Warfare Analysis Department of the John Hopkins University Applied Physics Laboratory (APL). The purpose of this system is to aid an analyst in scenario development, scenario analysis, survivability analysis, mission planning, and equipment performance prediction. During the summer of 1996, this investigator added options and improved the DTED map program (DTMA). This program was written in C++ and MacApp, and may be used by an analyst to display and manipulate Digital Terrain Elevation Data (DTED) files.

The DTED files are data files generated by the Defense Mapping Agency and are used in aspects of cruise missile mission planning. In particular, they are used by an analyst to assist in scenario analysis to investigate such things as radar site location and masking, and cruise missile flight paths. The DTED map program developed allows for computer generated color displays of the (large) data files quickly, and allows the analyst to use the computer to determine radar site locations and masking, and to plan cruise missile flight paths.

The work on this project this past year has consisted mainly of writing up past results and preparing for a major sea trial in the fall of 1997. The preparations for the coming trial include reviewing the design of experiments and developing analysis methodologies.
Also, software design is required for the onboard, real time, data collection system.

The problem of beamforming is that of tuning an array of antennas so as to maximize the reception in the direction of a desired signal while minimizing the signal strength in the direction of a jammer signal. Over the last several years, this work has focused on the use of the Residue Number System to solve the adaptive beamforming problem using Gaussian elimination. During this last year further progress has been made and the scope of the project broadened.

First, the potential for use of Gauss elimination in an integer computing environment has been significantly enhanced by developing an improved fraction-free integer Gauss elimination algorithm. This algorithm has the further benefit of eliminating all unnecessary common factors in matrix elements without the need for any additional computational effort to find these factors. The modified algorithm is also applicable in the situation where the matrix elements (coefficients) are taken from a more general ring such as a ring of real polynomials.

Within many C³ problems it is necessary (or at least desirable) to be able to identify the rank of a matrix which is highly rank deficient - but is contaminated by noise. This problem is of particular importance in radar processing. Late in the summer of 1995 a new approach was developed for this problem based on least squares approximation of rows of a matrix by other rows of the matrix. This is essentially a modification of the Gramm-Schmidt orthogonalization process performed in an iterative manner. This approach appeared from early testing to have great potential as a much cheaper (and therefore quicker) algorithm than the standard SVD-based approach.

From the previous work for NAWC, the potential for use of Gauss elimination in an integer computing environment is significantly enhanced by an improved fraction-free integer Gauss elimination algorithm. This algorithm has the further benefit of eliminating all unnecessary common factors in matrix elements without the need for any additional computational effort to find these factors. The modified algorithm is also applicable in the situation where the matrix elements (coefficients) are taken from a more general ring such as a ring of real polynomials.

Also within the realm of RNS arithmetic a new algorithm for exact integer division within the RNS system has been developed. This could be combined with the fraction-free algorithm since the divisions required there are known to be exact. This may have the effect of making the RNS-based approach more practical. The complexity analysis of this algorithm has not yet been fully explored.

These two developments combine to make a more practical method for solving integer linear systems with the fast parallel arithmetic of RNS. This has resulted in a paper to be presented to the 13th Symposium on Computer Arithmetic this summer.

The developments in fraction-free algorithms, and their extensions to a fraction-free LU algorithm make it possible to combine these ideas within a computer algebra setting with the earlier work of Nakos on using Dixon resultants for the solution of polynomial systems. These are important within the realms of threat analysis, robot control and object recognition.
This is the subject of continuing research to be sponsored again by NAWC-AD, Patuxent River this year.

This project continues the development of possible schemes for the eventual hardware implementation of SLI arithmetic and the analysis of the algorithms used. We also are gaining more computational experience and evidence of the potential practical value of the system using software implementations of the symmetric level-index, SLI arithmetic system. This was a continuation of previous work on the level-index system.

The principal recent objectives have been to investigate further the implementation and application of LI and SLI arithmetic and the comparison of these with other proposed new computer arithmetics. The primary goal was to begin the implementation of this system on the Math Dept MasPar MP-1 system to investigate the advantages to be derived from a massively parallel implementation. This aspect of the work was enhanced during the summer of 1995 by the visit of Nicolas Schabanel, a graduate student from the Ecole Normale Superieure, Lyon, France who spent his summer internship working here at USNA. His work on implementing SLI arithmetic on the MasPar is summarized in his Technical Report. The methods of investigation included mathematical analysis, the development and use of algorithms for various arithmetic systems and their application to the evaluation of mathematical functions. This included a comparative study of the various schemes. The study also included research into recent hardware design developments and their possible use in eventual implementations of the level-index scheme. The other major areas of activity here have been and are concerned with the use of parallel processors and the implications of the parallelism for the arithmetic system used. The principal output of this research has been in the form of research papers and the development of ideas for further developments and publications.

This work is the continuation of earlier work in which the authors considered this question from the viewpoint of arithmetic overflow resulting from addition and subtraction on the basis of the logarithmic distribution of numbers. The assumption of the logarithmic distribution combined with a further assumption that the distribution of numbers should be smooth and independent of the arithmetic base leads to the claim that the exponents of floating-point numbers should be uniformly distributed. It was on this basis that alarming frequencies of overflow and underflow were obtained. The further observation was made there that these results were unrealistically pessimistic for scientific computing.

This departure from realism was explained by stating that the distribution of exponents is not uniform in practical scientific applications because of the choice of units and the scaling of the problem. It is the purpose of this work to examine the distribution of exponents in an attempt to obtain a more realistic model for the occurrence of overflow and underflow failure. Initially, this is applied only to a random process taking no account of the special nature of any particular process.

The basic models used to develop the results are described beginning with a discrete hardware designs for their implementation. The log model which is directly comparable to the floating-point situation. This is followed by a continuous model which can be thought of as modeling the situation which would be encountered in using a logarithmic arithmetic. Such arithmetic systems have been proposed as alternatives to floating-point and extensive work has been carried out in obtaining arithmic number systems are essentially equivalent to level 1 of the level-index and symmetric level-index systems.

It is shown here that the continuous model mirrors very closely the behavior of the discrete model, a fact which makes it suitable for the analysis of the floating-point situation. This analysis shows that, as the number of multiplicative operations increases, the exponent distribution becomes a spline function of increasing degree which mimics more and more closely a normal distribution function. This remains true until exponent spill begins to take over. This is followed up by the presentation of computational evidence on the frequency of exponent spill as a result of an extended sequence of multiplications and divisions. One striking aspect is the marked difference between balanced and unbalanced initial ranges of exponents. In the case of even very slight unbalance - which may simply be the result of good scaling applied within an unbalanced floating-point system - the frequency of exponent spill grows alarmingly. Account is taken here of whether the exponent spill is reported before or after the normalization of the result.

Any polynomial F(X) with coefficients chosen from Z/(m) uniquely determines a polynomial function F:Z/(m) Z/(m) evaluated by substituting elements of Z/(m) into F. The researcher denotes the ring of all polynomial functions over Z/(m) by P (Z/(m)). The purpose of this project is to determine the structure and the cardinality of P (Z/(m)). The researcher has run Maple programs to evaluate the cardinality of P (Z/(m)) for a number of values of m, but has so far not been able to determine a pattern for the cardinality for general m. He is now looking into the generalization to the ring P(R) of polynomial functions over an arbitrary commutative ring R in the hope that the abstract algebraic properties of this ring will give some insight into the original problem.
 
 

This ongoing investigation matches the theories and algorithms of graph theory to the design of neural networks for an important computer vision problem. In particular, this research is applied to the design of fully automated programs for fingerprint identification. Graph representations utilizing a new class of proximity graphs, called "Sphere-of-Influence Graphs," provide a robust representation of fingerprint minutiae maps. These graph representations are then incorporated into the design of graph matching neural networks. This research has received previous funding from The Federal Bureau of Investigation and The Office of Naval Research. Various papers have been published and presentations given, including a paper at Oxford University.

This project is a continuation of research funded by the NARC, in joint work with Prof. P. Milman of the University of Toronto. During the year 16 June 1996 - 15 June 1997, the authors have obtained several additional results which go beyond the original objective of the project and which will be included in the paper in preparation. These results are applied to construct a complete Kahler metric of modified Saper type with a particularly simple local description. However, these results may have many other interesting consequences in the theory of singular spaces.

A generalized Chow's Theorem for ideals was proved using the Direct Image Theorem. The author proved that a tower of blow-ups of a compact complex manifold along smooth centers is equivalent to a single blow-up of the manifold along a product of ideals corresponding to the individual blow-ups and gave a formula for these ideals. In spite of the fact that blow-ups are a fundamental tool in resolution of singularities, this result does not seem to have been known except in the case in which the centers are isolated points.

Explicit local formulas were found for Chern forms of the exceptional line bundles of blow-ups over the inverse image of neighborhoods of points in the base.

The primer will be a set of notes for an introductory course for mathematics majors such as our SM291, "Fundamentals of Mathematics."

The objective of the text is to ease the transition from engineering-oriented mathematics, emphasizing techniques for solving particular problems, to mathematicians' mathematics, emphasizing discovery and proofs of mathematical truths. The text begins with an introduction to basic mathematical concepts--sets, logic, numbers, functions, sequences, and vectors. There follows an extensive discussion of how to understand and construct proofs of simple statements. Finally, the readers are invited to apply these techniques in a deeper discussion of the basic ideas. Additional chapters will consider equivalence relations (applied to modular arithmetic and construction of number systems), vector spaces, sequences and their limits, and an introduction to real analysis.

The Vardiman sedimentation equation for ocean floor sediment is obtained from empirical evidence and data matching. In this research, a derivation from first principles of this equation is obtained by considering a homogeneous mixing problem with reflaxation time incorporated for the period of discontinuous mixing. The differential equation obtained has as unique solution which is the Vardiman sedimentation equation.

The author and his colleague C. Moen have studied the multiple harmonic series

since 1988, primarily in an effort to resolve two conjectures (called the duality and sum conjectures) about them. In recent years these sums have arisen in knot theory in connection with the invariant introduced by Kontsevich. The duality conjecture can be proved easily from this point of view, and recently the sum conjecture has been proven in full generality by Granville and Zagier. The focus of investigation has now shifted to more general questions about these sums. For example, if we define the weight of (i₁,,i_k) to be i₁++i_k, then how many sums of weight n are irreducible(i.e., not expressible in terms of sums of lower weight)? The set of multiple harmonic series also has a natural algebraic structure, and it makes sense to ask what the ideal of relations between such sums looks like. Finally, the class of multiple harmonic series can be generalized to include "Euler sums" of the form

where the _j are roots of unity. Again the set of these sums has an algebraic structure, and the problem of analyzing the ideal of relations looks quite challenging.

The author introduced a "harmonic algebra" to formalize the multiplicative structure on the set of multiple harmonic series. This harmonic algebra turns out to be isomorphic to the algebra of quasi-symmetric functions, which had been studied from other points of view. He was also able to describe some of the known sets of relations in more algebraic terms. Recently he has generalized some of his algebraic results to the class of Euler sums.

Crystal, a Maple package which helps decompose the restriction of an irreducible finite dimensional representation and the tensor product of two irreducible finite dimensional representations of a simple Lie algebra into irreducible constituents using crystal graphs. Basically this amounts to implementing a theorem of Kashiwara: the irreducible constituents of such a tensor product are in a natural 1-1 correspondence with the connected components of the crystal graph of the tensor product.

The crystal package contains programs to : 1) compute the crystal graph of an irreducible; 2) representation, associated to a fundamental weight; 3) compute the crystal graph of the tensor product of Two irreducible representations; 4) display the crystal graph of the tensor product obtained above; 5) display the crystal graph of the restriction of an irreducible component of a tensor product representation to a simple subalgebra; 6) computing all the weights occurring in a given irreducible representation.

Further work on revising a work on comparing the automorphic forms of SL(2) with those of its two-fold metaplectic cover based on discussions with Jeff Adams and Jason Schultz.

Revising work on invariant distributions of p-adic reductive groups.

Generically a Laplace operator on a compact manifold has only simple eigenvalues. This is also true for elliptic boundary value problems for Laplacians on compact manifolds with boundaries. In research done several years ago, however, the researcher showed that if the manifold has a symmetry of order three or more, then there are infinitely many multiple eigenvalues. Furthermore, if the manifold is in fact homogenous, then all but the first eigenvalue is multiple. During the past year the researcher has been investigating the extent to which some symmetry is necessary for the existence of multiple eigenvalues. Put simply, does the existence of infinitely many multiple eigenvalues imply the existence of a symmetry?

This work dealt with a nonlinear system of singular-hyperbolic partial differential equations involving time and three spatial variables. Analytical methods were used to derive a priori estimates on the derivatives of the solutions and these led to a conservation property (length of an arbitrary sub-curve for one parameter and measure of arbitrary measurable sets for three parameters). A class of traveling solutions was constructed, as well as a class of rotating solutions. Mid-point discretizations of the system were studied and shown to have the same conservation propereties as the continuous problem, as well as the same classes of explicit solutions. Numerical computations were performed on the MasPar computer and dealt with examples of the explicitly known solutions, the behavior of a curve with a knot, the behavior of geodesics on a surface and numerical instabilities when explicit discretizations were used.

The authors have jointly been studying these topics for several years. Their paper entitled, "Sums of Triple Harmonic Series," appeared in the Journal of Number Theory. Another paper on these topics, also written with Michael E. Hoffman, is virtually complete and will soon be submitted.

A major conjecture in graph theory is the tree-packing conjecture. The author is studying various generalizations of the conjecture, both in the context of graph theory and in other situations. He has also been studying various related problems concerning packings of rectangles and squares. He has written one paper on this subject entitled, "Dissecting with near sequares," and is currently writing another.

The author is studying various number-theoretical problems which involve automorphic forms, representation theory and algebraic geometry.

Most of this work was done last year. Rewriting had to be done due to the submission to a new journal (the Mathematica Journal where the manuscript was submitted ceased to exist.)

From last year's introduction: "The Dixon resultant can be used to eliminated a number of unknowns from a system of polynomial equations in one step. To the Mathematica user our code complements and greatly enhances the command Eliminate. Our program also improves the command Resultant, which implements the Sylvester resultant. This is up to constant factor(s) a special case of the Dixon resultant. There are two advantages in using Dixon's resultant of Sylvester's: 1) The end matrix has smaller size; hence, it is often easier to row reduce it or compute its determinant. 2) A whole block of variables can be eliminated in one calculation, instead of the successive eliminations. Finally, as a bi-product - which may also be of general interest - the researchers found a symbolic Gauss elimination that complements Mathematica's RowReduce. This Gauss elimination is done without scaling so that no extra factors are imposed on the pivots."

The authors have written a textbook for SM230. This book will be published by Wiley Custom Publishing for use next year. The innovative ideas for this book include: using the table form of Venn diagrams to eliminate the formulas for conditional probability and Bayes' Theorem; programs for calculators that compute cdf's for 4 basic distributions and eliminate the need for paper tables; presenting the negative binomial as a special version of the binomial; similarly presenting the exponential and Erland distributions as "negative Poisson" distributions; presenting sums of discrete random variables as convolutions, along with a spreadsheet to illustrate the Central Limit Theorem; functions of continuous random variables, with an application to simulation.

This project is a continuation of the researcher's previous work. She is writing Mathematica programs which determine prime numbers and compute the class numbers of certain algebraic number fields. The researcher is also working on Mathematica programs which will compute numerical invariants associated to quadratic mappings.

The study examined the probability of landing in each possible configuration after a long time. The study looked for states and orbits of states that have high probability if the transition probability is allowed to move toward zero.

The study can be extended to a two-dimensional toroidal array of cells with each cell possessing four or eight nearest neighbors.

This projects course was created at student request as a flow up course to the SM342 Discrete Mathematics Course. Students in this Reading/Projects course spent half of the course attending in class lectures by outside speakers as well as those by Professor Crawford. Presentations gave special emphasis to applications of graph theory and discrete mathematics to other areas of mathematics as well as computer vision, pattern recognition and operations analysis. The second half of the semester was devoted to 2-person team projects culminating in oral presentations and 12-20 page papers.

The projects investigated included: Neighborhood Graphs and The Henon Attractor; Applications of Graph Theoretic Tournaments; The Chinese Postman Problem; Classification and Fingerprint Identification; Applications of Sphere-of-Influence Graphs; PERT and CPM Path Algorithms and Applications; Project Planning and Organization Using Graph Theory; Latent Fingerprint Analysis; Graph Representations of Groups; and The Application of Graph Theory to Mass Transit Problems.

We study the group theoretic properties of the collection G of all "words" in the basic moves of the square 1 puzzle which preserve the cube shape. This collection G forms a group which, motivated by a paper of Wilson, we call the homology group of the square 1 puzzle.

We plan to answer (in the negative) a question raised in a paper by C. Bailey and M. Kidwell on whether or not a complete odd king tour exists.

The researchers explored twelve different ways M₁₂ appears in mathematics, hence the pun the "dodecahedral faces" in the title. Specifically, this paper relates M₁₂ to the Mongean Shuffle, the Steiner Hexad, Golay Codes, the Hadamard Matrix of order 12, 5-transitivity, presentations, Crossing the Rubicon, the MOG and the Minimog, the Kitten, Mathematical Blackjack, Sporadic Groups, and the Stabilizer in M₂₄ of a dodecad.

We explored the pyraminx puzzle (a "Rubik tetrahedron") using group theory. We proved an isomorphism between the pyraminx group and a semi-direct product of an alternating group with
a Cartesian product of cyclic groups.

CDR Mara believes the forecasting accuracy of the distribution of Navy enlisted personnel can be improved, especially during periods of restructing. Currently, the Bureau of Naval Personnel is using a statistically based forecasting model. Errors resulting from a purely statistical approach during a period of restructuring have caused significant budgeting problems, both cost over-runs and shortfalls. Midshipman Hay explored the use of a Markov chain model to improve the forecasts. She laid out the difficulties in building a Markov chain model of the personnel structure of the Navy's enlisted force and explored alternatives for dealing with the difficulties.

More work must be done to define the model. Handles must be included to deal with the factors subject to the control of Pentagon decision makers and the model must use values that can only be estimated statistically. A Markov chain model still seems appropriate.

In his studies on shifts on operator algebras, Price has shown that there are useful connections to be made between equivalence classes of certain shifts and Toeplitz matrices which are used to define these shifts. Given a sequence d₁,d₂,... of elements of a finite field, the corresponding Toeplitz matrix T_n is the n x n skew-symmetric matrix of the form T_n =

0 d₁ d ₂ . . . . d _n-1

-d₁ 0 d_{1 . . . .} d_n-2

-d₂ -d₁ 0 _{. . . .} d_n-3

.

.

.

-d_n-1 -d_n-2 -d_n-3 . . . . 0

In work carried out in the summer of 1996 Price noticed that if the sequence is finitely nonzero, and if k is the highest index for which d_k is non-zero, then maximum nullity of T_n is k. He also showed that there are infinitely many n for which the nullity of T_n assumes the value k. With the help of a computer program Price and Midshipman Truitt observed that if null(T_n) = k, then null(T_n-r) = null(T_n+r) for all positive integers r < n. Using some results proved by Price and others about the impulse response sequence corresponding to shift register sequences, Midshipman Truitt obtained a proof of the observation about the nullities of T_n-r and T_n+r. Truitt's proof used linear algebra techniques as well as some ideas from the mathematics of cryptology.

The Marine Corps is currently considering replacing the UH-1 Huey helicopter with a variant of the SH-60 Seahawk. The purpose of this study was to predict the number of enlisted Marines that would be needed for maintenance in the squadron once the change is made. Data was obtained from Naval Aviation Maintenance Office (NAMO), Patuxent River, MD. The data for the corrent configuration of AH-1 Cobras and UH-1 Hueys was from HMLA 367. The data for the SH-60 Seahawk was from Helicopter Anti-submarine Squadron 3. The number of flight hours for the period Jan-Sep 96 was 2096 for the Cobras and 1225 for the Hueys. The unit level maintenance hours for that period were 32,272 hours for the Cobra and 18,580 hours for the Huey. This gives a total of 50,852 maintenance hours for the squadron. The number of flight hours for calendar year 1996 for the Seahawk was 2040, and the unit level maintenance resulted in 31,245 hours. The assumptions were made that the ratio of maintenance hours to flight hours (MH/FH) would, over a long period of time, remain the same for the Seahawk, and that the Seahawks in the HMLA squadron would fly the same number of hours as the Huey. This gives a predicted number of unit level maintenance hours of 45,624 for a 9 month period when the Seahawk replaces the Huey. This is a reduction of about 10% over the observed number of maintenance hours. The prediction is then made that the enlisted maintenance personnel can be reduced by 10% without reducing the readiness of the squadron. For a squadron with 18 Cobras and 9 Hueys, the current Table of Manpower Requirements lists 68 enlisted maintenance billets at the unit level. A reduction of 10% would mean a loss of 7 billets and a total of 61 billets remaining.

A discrete-event simulation using ProModel software was also constructed as a model of the helicopter maintenance. The model exhibited chaotic-like behavior and no reliable estimate was obtained from the model.

The focus of the research was to identify factors affecting promotion from Captain to Major and Major to Lieutenant Colonel in the Marine Corps. Data provided by Headquarters, Marine Corps was analyzed to identify significant factors which increase the probability of promotion to the 0-4 and 0-5 level. Further, a predictive model was developed utilizing classical regression techniques.

This SA412 project examined factors that influence promotion in the Marine Corps. These factors include education, performance in TBS, attendance at various schools, source of commission, sex, population group, etc. Promotion to Major turned out to be much more interesting than promotion to Lieutenant Colonel.Neural Networks

This capstone project investigated neural networks and how they relate to standard statistical methods.

The final stage of this course was devoted to team projects. These were on three different topics in two quite distinct areas.

One was devoted to the issue of finding the most efficient matrix multiplication algorithm for matrices smaller than the processor array. Midn Halman and Dunivan came up with a good idea for spreading the work across the processors by making copies of the matrices onto different parts of the array - with one of the factors permuted differently in each copy. The result is that all component multiplications for the complete product of two 16x16 matrices can be performed simultaneously on a 64x64 processor array.

Two teams (Midn Martin and Lyne, and Midn Mitchell and Hingst) each attacked the problem of solving large systems of linear equations on a 64x64 processor array. (Here large means 128x128 or larger.) Their solutions were essentially similar using a block Gauss elimination algorithm together with Gauss-Jordan inversion of the pivot matrices. This algorithm allows for a natural breakdown of the data and makes good use of the basic building blocks of parallel linear algebra.

The final team, Midn Moore and Kritschgau, decided to attack a problem in artificial intelligence. Specifically they used the MasPar machine to try to teach a neural net to learn to recognize fractals. They worked with many different views of the Mandelbrot set, some of which were artificially scrambled using random transformations of the pixels. They then came up with a quantitative scheme for assessing whether a particular image was a fractal by studying the variability of the pixels on the boundary. The decision process depended on a parameter whose value had to be learned from known cases. By the end of this semester their scheme was reporting around 90% success on the trial set of problems.
 

This paper was first presented at The First International Conference on Mathematics and Applications of Neural Networks, Oxford University, July 1995. The paper presents applications of graph theory and neural networks to automated fingerprint identification. In particular, the paper presents relaxation networks for graph matching and deterministic annealing to design a matching program for matrix representations of fingerprint minutiae maps. Research for this program received funding from The Federal bureau of Investigation and The Office of Naval Research. 

GAGLIONE, Anthony M., Professor, coauthor, "The Commutative Transitive Kernel," Algebra Colloquium, accepted for publication on 27 Nov. 1996.

A group G is commutative transitive provided the relation of commutativity is transitive on the non-identity elements of G. A subgroup T(G) is constructed and the main theorem asserts that (1) T(G) is a characteristic subgroup of G contained in the commutator subgroup of G . (2) G/T(G) is commutative transitive. (3) G is commutative transitive if and only if T(G)=1. More generally, if N is a normal subgroup in G then T(G,N) is constructed as the union of an increasing chain of normal subgroups N=T₀(G,N)\&} T₁(G,N)\&}. T(G) is defined as T(G,1).

GAGLIONE, Anthony M., Professor, coauthor, "Formal Power series Representations of Free Exponential Groups,'" Communications in Algebra, 25(2), (February 1997), 631-648.

A classical result of Magnus asserts that a free group F has a faithful representation in the group of units of a ring of non-commuting formal power series with integral coefficients. We generalize this result to the category of A-groups, where A is an associative ring or an Abelian group. We say that a free A-group F^A has a faithful Magnus representation if there is a ring B containing A as an additive subgroup (or a subring) such that F^A is faithfully represented (exactly as in Magnus' classical result in the case A = Z) in the group of units of the ring of formal power series in non-commuting indeterminates over B. The three principal results are: (I) If A is a torsion free Abelian group and F^A is a free A-group of Lyndon's type, then F^A has a faithful Magnus representation; (II) If A is an -residually Z ring, then F^A has a faithful Magnus representation; (III) for every nontrivial torsion-free Abelian group A, F^A has a faithful Magnus representation in B[[Y]] for a suitable ring B if and only if F^Q has a faithful Magnus representation in Q[[Y]].

HERRMANN, Robert A., Professor, "A Hypercontinuous Hypersmooth Schwarzschild Line Element Transformation," International Journal of Mathematics and Mathematical Science., 20(1) (1997), 201-204.

In this paper, a new derivation for one of the basic black hole line elements is given since the basic derivation is flawed mathematically. This derivation postulates a transformation procedure that utilizes a transformation function that is modeled by an ideal nonstandard physical world transformation process that yields a connection between an exterior Schwarzschild line element and a distinctly different interior line element. The transformation is an ideal transformation in that in the natural world the transformation is conceived of as occurring at an unknown moment in the evolution of a gravitationally collapsing spherical body with radius greater than but near to the Schwarzschild radius. An ideal transformation models this transformation in a manner independent of the object's standard radius. It yields predicted behavior based upon a Newtonian gravitational field prior to the transformation, predicted behavior after the transformation for a field internal to the Schwarzschild surface and predicted behavior with respect to field alteration processes taking place during the transformation period.

HOFFMAN, Michael E., Associate Professor, and Courtney Moen, Associate Professor, "Sums of Triple Harmonic Series," Journal of Number Theory 60 (1996), 329-331.

For positive integers a, b, c, with a 2, let A(a,b,c) denote the triple harmonic series

The authors show that the sum of the A(a,b,c) with a + b + c = n is

A similar identity for double harmonic series goes back to Euler.

JOYNER, W. David, Associate Professor, and Roland Martin, "Decomposing Lie Algebra Representations Using Crystal Graphs," The Symbolic and Algebraic Computation Newsletter, No. 2, June 1997, pg. 2.

The researchers use the theory of crystal graphs to give a simple graph-theoretical algorithm for determining the branching rule for decomposing a representation of a simple Lie Algebra when restricted to a simple subalgebra. They also describe a computer package for determining such decompositions graphically.

KONKOWSKI, Deborah A., Associate Professor, (Co-author), "Improved Cauchy horizon stability conjecture", Phys. Rev. D15,7898-7901 (1996).

An improved stability conjecture for Cauchy horizons is presented. The conjecture predicts the stability of Cauchy horizons based upon the behavior of test fields, and in the case of instability it also predicts the nature of the singularity produced. The results for Cauchy horizons in Reissner-Nordstrom, Kerr, Reissner-Nordstrom-de Sitter, Kerr-de Sitter, anti-De Sitter, and a type V LRS spacetime are reviewed. The improved conjecture agrees with the stability and singularity types in all cases for which an exact back reaction solution has been found.

MARUSZEWSKI, Richard F., Associate Professor, "Scoring via Galois Equations," DOD Technical Report, October 1996.

MARUSZEWSKI, Richard F., Associate Professor, "Spreadsheets in a Differential Equations Course," AMATC Review, Volume 18, #1, pp 40-44, Fall 1996.

MARUSZEWSKI, Richard F., Associate Professor, "A Note on Benford's Law," with J. Huddle, MACE Journal, Volume 31, #1, pp 66-69, Winter 1997.

MCCOY, Peter A., Professor, "A Classical Theorem on the Singularities of Legendre Series in C³ and Associated System of Hyperbolic Partial Differential Equations," SIAM J. Math Analysis, vol. 28, issue 3, May 1997.

A classical theorem of Z. Nehari relates the singularities of Legendre series expansions with those of Taylor's series. The generalization of Nehari's theorem is known for Legendre series in two complex variables. In this paper, function theoretic methods develop the analogous relationships between the singularities of series expanded as triple products of Legendre series and those of the associated Taylor's series. The singularities of these generalized Legendre series are determined by certain elliptic curves. Moreover, these series satisfy a system of hyperbolic partial differential equations in three complex variables that are connected to S. Bochner's study of Poisson processes in two real variables.

MEYERSON, Mark D., "The x^x Spindle," Mathematics Magazine, June 1996, pp. 198-206.

MEYERSON, Mark D., Professor, One problem solution published by American Mathematical Monthly (10328), November 1996.

PRICE, Geoffrey L., Professor, co-author, "On the Ranks of Skew-Centrosymmetric Matrices over Finite Fields," Linear Algebra and Application, 248 (1996) 317-325.

This work is an extension of an honors project by (then) Midshipman Kristen W. Culler under the supervision of Professor Price. If F is a finite field, and if d₁,d₂,... is a sequence of elements of F, then for each positive integer n one can form the n by n skew-symmetric matrix T_n of the form

0 d ₁ d ₂ . . . . d _n-1

-d₁ 0 d₁ . . . . d_n-2

-d₂ -d₁ 0 . . . . d_n-3

.

.

-d _n-1 -d _n-2 -d _n-3 . . . . 0

Let _n be the nullity of this matrix. Then the authors showed that the sequence {_n} of nullities is the concatenation of infinitely many strings of the form 1,2,...,m-1,m,m-1,...,1,0. Using this result the authors were able to compute the number of n by n matrices of skew centrosymmetric form of a prescribed rank.

PRICE, Geoffrey L., Professor, co-author, "Endomorphisms of B(H)," Proceedings of Symposia in Pure Mathematics, 59 (1996) 93-138.

In this paper the authors study unital shifts on the algebra of bounded operators on an infinite-dimensional Hilbert space H. The authors show that each unital shift on B(H), with Jones subfactor index [B(H): (B(H))] = n² , n = ,2,3,... is implemented by a representation of the Cuntz algebra O_n. Using this result the corresponding conjugacy classes of shifts are identified. The authors also use a construction due to von Neumann to show that there exist shifts on B(H) which have no invariant normal states.

SHEHAN, J. Michael, Major, USMC, "Benchmarking for Better Mathematics at a Service Academy," Mathematica Militaris, Vol. 6, issue 3, Fall 1997.

Benchmarking is a process that involves seeing which procedures enhanced the success rate at other institutions and then applying those procedures at one's own organization. Responses to a questionnaire given to Math Department faculty members at the Naval Academy regarding benchmarking are explored. The author argues that while this approach to organizational improvement has merit, service academies must exercise considerable care when selecting for implementation things which have worked for civilian colleges and universities.

TURNER, Peter R., Professor and Alan Feldstein,(ASU), "Overflow and Underflow in Multiplication and Division," Applied Numerical Mathematics, 21 (1996) 221-239.

The logarithmic distribution combined with a further assumption that the distribution of numbers should be smooth and independent of the arithmetic base leads to alarming frequencies of overflow and underflow. It is the purpose of this work to examine the distribution of exponents in an attempt to obtain a more realistic model for the occurrence of overflow and underflow failure. The basic models used to develop the results are described beginning with a discrete model which is directly comparable to the floating-point situation. This is followed by a continuous model which can be thought of as modeling the situation which would be encountered in using a logarithmic arithmetic. It is shown here that the continuous model mirrors very closely the behavior of the discrete model. This analysis shows that, as the number of multiplicative operations increases, the exponent distribution becomes a spline function of increasing degree which mimics more and more closely a normal distribution function. This remains true until exponent spill begins to take over.

TURNER, Peter R., Professor, Daniel W. Lozier, (NIST) and Michael A. Anuta (Cray Research), "The MasPar MP-1 as a Computer Arithmetic Laboratory," NIST J Research 101 (1996) 165-174.

This paper describes the use of a massively parallel SIMD computer architecture for the simulation of various forms of computer arithmetic. The particular system used is a DEC/MasPar MP-1 with 4096 processors in a square array. This architecture has many advantages for such simulations due largely to the simplicity of the individual processors. Arithmetic operations can be spread across the processor array to simulate hardware. Alternatively they may be performed on individual processors to allow simulation of a massively parallel implementation of the arithmetic. Compromises between these extremes permits speed-area trade-offs to be examined. The paper includes a description of the architecture and its feature. It then summarizes some of the arithmetic systems which have been, or are to be, implemented. The implementation of the level-index and symmetric level-index, SLI, systems is described in some detail. An extensive bibliography is included.

TURNER, Peter R., "Fraction-Free RNS Algorithms for solving Linear Systems ARITH13," 13th IEEE

Symposium on Computer Arithmetic, IEEE Computer Society, Washington DC, (1997) 218-224.

This paper is concerned with overcoming the arithmetic problems which arise in the solution of linear systems with integer coefficients. Specifically, solving such systems using (integer) Gauss elimination or its variants usually results in severe growth in the dynamic range of the integers that must be represented. To alleviate this problem, a Residue Number System (RNS) can be utilized so that large integers can be represented by a vector of residues which require only short wordlengths. RNS arithmetic, however, cannot easily handle any divisions that are needed in the solution process. This paper presents fraction-

free algorithms for the solution of integer systems. This does involve divisions --- but only divisions where the result is known to be an exact integer. The other principal contribution of this paper is the presentation of an RNS division algorithm for exact integer division which does not require any conversion to standard binary form. It uses entirely modular arithmetic, perhaps including a step equivalent to RNS base extension.

TURNER, Peter R., Professor and LOZIER, Daniel W (NIST), "Error-bounding in Level-Index Computer Arithmetic," Numerical Methods and Error Bounds (G. Alefeld & J. Herzberger, eds) Akademie Verlag, Berlin, 1996, pp 138-145.

The main purpose of this paper is to compare the new SLI arithmetic with the old (floating-point) with particular reference to interval arithmetic and other error-bounding techniques. This yields advantages such as immunity to considerations of underflow and overflow, a unified error analysis and a natural means for increasing precision within any algorithm. The secondary purpose is to describe the Computer Arithmetic Laboratory which is being developed on a MasPar MP-1 system. The MasPar built-in arithmetic is built up from 4-bit operations and the new arithmetic systems will be similarly constructed. The MasPar setting is ideal for demonstrating and comparing different computer arithmetics.

TURNER, Peter R., and D. W. Lozier, "Parallel Algorithms for Basic Linear Algebra in SLI Arithmetic," Proceedings of Computational and Applied Mathematics, Leuven, Belgium, July (1996) ID2.

This paper reports on a continuing project to develop, implement and apply parallel algorithms for the SLI arithmetic system. In this paper the arithmetic algorithms for SLI are reexamined with a view to SIMD and possible eventual hardware implementation. The algorithms are extended to parallel SLI BLAs such as scalar products and saxpys. The parallel computational complexity of these algorithms is seen to be similar to a single scalar SLI arithmetic operation.

TURNER, Peter R., Professor and M. A. Anuta, D. W. Lozier, N. Schabanel, "Basic Linear Algebra in SLI Arithmetic," EuroPar96 Parallel Processing, LNCS 1124, Springer, 1996, pp 193-202.

Symmetric level-index arithmetic was introduced to overcome recognized limitations of the floating-point system. The original recursive algorithms for SLI operations could be parallelized to some extent and a SIMD implementation of these algorithms is described. The main purpose of the paper is to present parallel SLI algorithms for arithmetic and basic linear algebra operations.

TURNER, Peter R., Professor and D. W. Lozier, "Parallel BLAs in SLI Arithmetic," (Extended Abstract) Proceedings of 33rd Annual Meeting of Soc of Engineering Science, Tempe, AZ, October 1996 ID2.

This paper reports on a continuing project to develop, implement and apply parallel algorithms for the SLI arithmetic system. Algorithms for SLI are developed for a SIMD parallel computer. Vector code is also being developed using MATLAB. The standard arithmetic algorithms are extended to parallel SLI BLAs such as scalar products and saxpys. The application of these to solving linear systems using Gauss elimination and Gauss-Jordan reduction is discussed.

TURNER, Peter R., Professor, "Gauss elimination: Workhorse of Linear Algebra," Tech Rep NAWCADPAX-96-194-TR, 1996.

This report brings together many different aspects of Gauss elimination. The basic GE algorithm is a fundamental tool of linear algebra for solving systems, computing determinants and determining rank. All of these are discussed in various contexts. These include different arithmetic or algebraic settings such as integer arithmetic or polynomial rings as well as conventional real (floating-point) arithmetic. Both accuracy and complexity are considered. The impact of parallelism is also discussed. Finally, GE is considered in the context of noisy matrices.

TURNER, Peter R., Professor, "A simplified Fraction-Free Integer Gauss Elimination Algorithm," Tech Rep NAWCADPAX-96-196-TR, 1996

This report presents a new version of Gauss elimination for integer arithmetic. This new algorithm allows fraction-free integer computation without requiring any calls to a greatest common divisor routine. It does however keep the growth in the integer dynamic range to a minimum. The algorithm is based on a careful comparison of the divisionless integer GE and the normal algorithm using divisions in a real arithmetic setting. From this analysis, we identify common factors which are necessarily present throughout the active part of the matrix. These can then be removed by exact integer division. A further consequence of this analysis is that the diagonal entries of the final triangular factor are precisely the determinants of the principal minors of the original matrix. In a parallel processing environment, the additional cost of these integer divisions is minimal since, at each stage, the whole active array is being divided by the same integer.

TURNER, Peter R., Professor, "Low Rank Determination using Least Squares," Tech Rep NAWCADPAX-96-195-TR, 1996.

This report discusses a technique for determining the rank of a matrix of special type. The matrix is assumed to be composed of a matrix of very low rank relative to its size, a "noise" component. The objective is to determine the rank of the underlying matrix. The idea behind this algorithm is to approximate each row of the matrix (in a least squares sense) by linear combinations of "significant rows" in an iterative manner until the effective rank is revealed.

TURNER, Peter R., Professor, with George Nakos and Robert M. Williams (NAWCADPAX), "Fraction-Free Algorithms for Linear and Polynomial Equations," Tech Rep NAWCADPAX, 1997.

This report extends the ideas behind Bareiss' fraction-free Gauss elimination algorithm in a number of directions. First, in the realm of linear algebra, algorithms are presented for fraction-free LU "factorization" of a matrix and for fraction-free forward and backward substitution. These algorithms are valid not just for integer computation, but also for any matrix system where the entries are taken from a unique factorization domain such as a polynomial ring. The second part of the paper applies a fraction-free formulation to resultant algorithms for solving polynomial systems. In particular, the use of fraction-free polynomial arithmetic and triangularization algorithms in computing the Dixon resultant of a polynomial system is discussed in detail.

WARDLAW, William P., Professor (with G. M. Benkart and M. D. Meyerson), Problem 600, The College Mathematics Journal, vol. 28 no.2 (March 1997) 146.

Let R be a commutative ring with unit element 1. Prove or disprove: If a, b R are multiples of one another, then they are unit multiples of one another; that is, there is an invertible element u R such that a = ub.

WARDLAW, William P., Professor, Problem 601, The College Mathematics Journal, vol. 28 no. 3 (May 1997) 231.

Prove that if the sum of finitely many consecutive terms from the harmonic sequence is written as a fraction in lowest terms, then the numerator is odd.

WITHERS, W. Douglas, Associate Professor, "A Rapid Entropy-Coding Algorithm," Dr. Dobb's Journal, 264 (April 1997), 38-43.

The researcher describes a new entropy-coding algorithm which he calls the ELS-coder. The ELS-coder is extremely simple to implement, and yet combines rapid execution with near-optimum compression performance. It is competitive in both speed and compression with the Q-coder and QM-coder

BUCHANAN, James L., Professor, R. P. GILBERT, and Z. LIN, "Determination of Seabed Parameters from Far Field Acoustic Data," World Congress for Nonlinear Analysis, Athens, Greece, 11 July 1996.

BUCHANAN, James L., Professor, R. P. GILBERT, and Z. LIN, "Determination of Seabed Parameters from Far Field Acoustic Data," AMS Sectional Meeting, Chattanooga, TN, 11 October 1996.

BUCHANAN, James L., Professor, "Direct and Inverse Problems for a Poroelastic Seabed," Conference on Generalized Analytic Functions, Graz, Austria, 8 January 1997.

BUCHANAN, James L., Professor, and R. P. GILBERT, "Transmission Loss over a Two-layer Seabed," ISAAC Conference, Newark, DE, 5 June 1997.

CRAWFORD, Carol G., Professor, Panel Member, "National Science Foundation Grants Review for Undergraduate Course and Curriculum Development Program", Washington, DC, 22-25 July, 1996.

CRAWFORD, Carol G., Professor, Team Member, "Middle States Evaluation of East Stroudsburg State University", Report of Team Evaluation to East Stroudsburg, Pennsylvania, 6-10 April 1997.

CRAWFORD, Carol G., Professor, Conference Chair, "Maryland-DC-Virginia Regional Meeting Introductory Remarks", Hood College, Frederick, Maryland, 1-2 November 1996.

CRAWFORD, Carol G., Professor, Conference Chair, "Maryland-DC-Virginia Regional Meeting Introductory Remarks", The College of William and Mary, Williamsburg, VA, 18-19 April 1997.

GAGLIONE, Anthony M., Professor, "Logic and the first-order theory of the non-Abelian free groups," AMS meeting Special Session, Ryder College, Lawrenceville, NJ, 4 October 1996.

GAGLIONE, Anthony M., Professor, "Surface Groups and Logic," McGill Univ., Montreal, Canada, Conference on Geometric Group Theory, 19 October 1996.

GAGLIONE, Anthony M. Professor, "Residual Properties of Groups and Rings," Zassenhaus Group Theory Conference, Univ. of S. FL, Sarasota, FL, 10 January 1997.

GAGLIONE, Anthony M. Professor, "Surface Groups and Logic," NY Group Theory Seminar, 14 February 1997.

GAGLIONE, Anthony M. Professor, "Solving Equations in Groups," Fairfield Mathematics Seminar, Fairfield Univ., Fairfield, CT, 11 April 1997

GRANT, Caroline G., Associate Professor, "Kahler Metrics for Singular Algebraic Varieties," The Fields Institute for Research in Mathematical Sciences, Toronto, Canada, 26 February 1997.

GRANT, Caroline G., Associate Professor, "Generating Functions for Kahler Metrics on Singular Algebraic Varieties," Algebraic Geometry Seminar, University of Toronto, Toronto, Canada, 19 March 1997.

GRANT, Caroline G., Associate Professor,"Chow's Theorem in Algebraic Geometry," U. S. Naval Academy, Annapolis, Maryland, 9 April 1997.

HOFFMAN, Michael E., Associate Professor, "Recent Progress on Multiple Harmonic Series," Mathematics Department Colloquium, U.S. Naval Academy, Annapolis, MD, 2 October 1996.

KAPLAN, Harold M., Professor, "A neglected virtue of the Cox-Stuart tests for trend," Mid-Atlantic Probability and Statistics Day, Department of Mathematics and Statistics, University of Maryland Baltimore County, 20 October 1990.

KONKOWSKI, Deborah A., Associate Professor, "Stability Tests for Mild Singularities and Cauchy Horizons", Texas Symposium on Relativistic Astrophysics, Chicago, Illinois, December 1990.

KONKOWSKI, Deborah A., Associate Professor, "The Stability of Plane-Wave Cauchy Horizons," London Relativity Seminar, Queen Mary and Westfield College, University of London, London, England, March 1990.

KONKOWSKI, Deborah A., Associate Professor, "The Stability of Plane-Wave Cauchy Horizons," Relativity Seminar, University of Southampton, Southampton, England, March 1990.

KONKOWSKI, Deborah A., Associate Professor, "The Stability of Cauchy Horizons in Plane Wave Spacetimes," Spring APS Meeting, Washington, D.C., April 1990.

LOCKHART, Robert B., Associate Professor, "Symmetry and Multiple Eigenvalues for Laplacians," Special Session on Partial Differential Equations at the Eastern Regional Meeting of the AMS, College Park, MD, 12 April 1990.

MARUSZEWSKI, Richard F., Associate Professor, "Introduction to the Mathematics Department Computer Systems," Mathematics Department Computer Seminar, U.S. Naval Academy, Annapolis, MD, Fall 1990.

MARUSZEWSKI, Richard F., Associate Professor, "Uploading/Downloading," Mathematics Department Computer Seminar, U.S. Naval Academy, Annapolis, MD, Fall 1990.

MARUSZEWSKI, Richard F., Associate Professor, "Using Spreadsheets In and Out of Class," Mathematics Department Computer Seminar, U.S. Naval Academy, Annapolis, MD, Fall 1990.

MCCOY, Peter A., Professor, "Sampling Theorems for Initial-boundary Value Problems in Several Variables," SIAM Annual Meeting, Kansas City, MO, 24 July 1990.

MCCOY, Peter A., Professor, "Wavelet Sampling Associated with Fractional Partial Differential Operators, Solution of Boundary Value Problems via Sampling Theory," American Mathematical Society Annual Meeting #103, San Diego, CA, 8 January 1990.

MCCOY, Peter A., Professor, "Solution of Boundary Value Problems via Sampling Theory," Special Session on Harmonic Analysis and Applications, American Mathematical Society Meeting #920, College Park, MD, 12 April 1990.

MCCOY, Peter A., Professor, "United States Naval Academy: History and Function," Introduction to the Conference on "High Performance Micro Wave Technology," NATO Conference held at the U.S. Naval Academy, 20-29 May 1990. NRL Condensed Matter Division sponsor.

MICHAEL, T.S., Associate Professor, "Rigidity and Rank: Report from Annapolis," Wisconsin Centennial Conference, May 1997.

NAKOS, George, Professor, Presented a Poster titled, "Fraction-Free Algorithms for Linear and Polynomial Equations," ECCAD 98, Boston, MA.

PENN, Howard L., Professor, "A Comparison of Grades in Follow on Courses for Students who took Reformed Calculus vs Students who Took Traditional Calculus," Arizona State Conference on Calculus Reform, invited talk, Scottsdale, AZ, June 1996.

PENN, Howard L., Professor, "A Comparison of Grades in Follow on Courses for Students who took Reformed Calculus vs Students who took Traditional Calculus," 5th Annual Conference on the Teaching of Mathematics, invited talk, Baltimore, MD, June 1996.

PENN, Howard L., Professor, "Military Application Projects for Calculus," Mathematical Association of America Summer National Meeting, Seattle, WA, August 1996.

PENN, Howard L., Professor, "A Comparison of Grades in Follow on Courses for Students who took Reformed Calculus vs Students who took Traditional Calculus," MAA Sectional Meeting, Fredericksburg, MD, November 1996.

PRICE, Geoffrey L., Professor, "Shifts on the Hyperfinite II₁ Factor," University of Heidelberg, Heidelberg, Germany, 23 January 1997.

PRICE, Geoffrey L., Professor, "Toeplitz Matrices over Finite Fields," MAA Regional Meeting, College of William and Mary, Williamsburg, VA, 19 April 1997.

PRICE, Geoffrey L., Professor, "Toeplitz Matrices and Shift Equivalence," National Security Agency, Fort George G. Meade, MD, 14 May 1997.

TURNER, John C., Professor, "Electromagnetic Signature Reduction," Mathematics Majors open house, U.S. Naval Academy, Annapolis, MD, March 1997.

TURNER, John C., Professor, "Using Calculators in SM230," Service Academies Student Mathematics Conference, U.S. Air Force Academy, Colorado Springs, CO, 18 April 1997.

TURNER, Peter R., Professor and D. W. Lozier, "Parallel Algorithms for Basic Linear Algebra in SLI Arithmetic," Computational and Applied Mathematics, Leuven, Belgium, July 1996.

TURNER, Peter R., Professor and M. A. Anuta, D. W. Lozier, N. Schabanel, "Basic Linear Algebra in SLI Arithmetic," EuroPar 96, Lyon, France, August 1996.

TURNER, Peter R., Professor and M. A. Anuta, D. W. Lozier, "Parallel Solution of Linear Systems using SLI Arithmetic," SIAM National Meeting, Kansas City, MO, July 1996.

TURNER, Peter R., Professor, "A Simplified Fraction-Free Integer Gauss Elimination Algorithm," SIAM National Meeting, Kansas City, MO, July 1996.

TURNER, Peter R., Professor, and M. A. Anuta, D. W. Lozier, "Parallel BLAs in SLI Arithmetic," Society of Engineering Science National Meeting, Tempe, AZ, October 1996.

TURNER, Peter R., Professor and G. Nakos, R. M. Williams, "Fraction-Free Algorithms for Linear and Polynomial Equations," ECCAD, East Coast Computer Algebra Days, Boston, MA, May 1997.

TURNER, Peter R., Professor, "Fraction-Free Algorithms for Solving Linear Systems," ARITH13, Asilomar, CA, July 1997.

TURNER, Peter R., Professor, "Fraction-Free Algorithms for Solving Linear and Polynomial Systems," SIAM National Meeting, Stanford, CA, July 1997.

WARDLAW, William P., Professor, "Principal Ideals and Associates in Commutative Rings," fall meeting of the Maryland-District of Columbia-Virginia Section of the Mathematical Association of America at Hood College in Frederick, MD, 2 November 1996.

WARDLAW, William P., Professor, "Linear Independence in Sets and Sequences," spring meeting of the Maryland-District of Columbia-Virginia Section of the Mathematical Association of America at the College of William and Mary, Williamsburg, VA, 19 April 1997.

WITHERS, W. Douglas, Associate Professor, "The ELS-coder and Augmented ELS-coder," AT&T Research Labs, Holmdel, NJ, 5 March 1997.

WITHERS, W. Douglas, Associate Professor, "The ELS-coder: a Rapid Entropy-Coder with Applications to Image Compression," ImageTech '97, Atlanta, GA, 12 April 1997.