NIU Department of Mathematical Sciences
NIU Department of Mathematical Sciences Here are brief statements from most members of the graduate faculty about their current research.
(to be added soon...)
My research activities are primarily in numerical linear algebra and scientific computation, with focus on algorithms for problems involving structured matrices that find applications in signal processing and control engineering. The structured problems that I have worked on include:
My research interests are mainly in the numerical analysis of partial differential equations and efficient solutions of the discrete problems.
Selected publications
JOHN BEACHY (Professor) -- Ph.D., Indiana University. Noncommutative Algebra.
My research studies noncommutative rings, from the point of view of their categories of modules. I am primarily interested in extending commutative localization techniques to the noncommutative case. One approach studies injective modules and quotient categories. A second and rather different approach uses the universal localization at a prime ideal. I am also interested in studying noncommutative geometry.
I am interested in the qualitative behavior of solutions to non-linear partial differential equations, more particularly in establishing the existence of singular solutions and providing a precise description of the singularities.
My main interest is in parabolic and elliptic equations with secondary interest in hyperbolic equations. Recently I have been focusing on mathematical problems arising from fluid mechanics.
Selected Publications:
My research deals with noncommutative rings with a particular emphasis on right Noetherian rings and their generalizations. Important examples of these rings include certain rings of differential operators, group rings of polycyclic-by-finite groups, and universal enveloping algebras of finite dimensional Lie algebras. The latter play a major role in recent developments in quantum field theory and atomic particle physics.
The main thrust of my current work centers on the question of whether various Noetherian rings can be embedded into Artinian rings. Although it has been known for some time that not all Noetherian rings are embeddable in Artinian rings, many of the most important classes of Noetherian rings can be embedded in Artinian rings. Rings of fractions and various dimension theories play a central role in this work.
I work in the representation theory of finite groups, a branch of algebra
which involves both group theory and ring theory, and which, due largely to
its connection with symmetry, has applications throughout pure and applied
mathematics. Projects of mine have examined
(1) relations between
representation-theoretic properties of groups and of subgroups,
(2) linear
groups of degree less than the order of a Sylow subgroup, and
(3) the decomposition of products of irreducible characters and of
conjugacy classes.
The
study of (3) has led to my involvement with "table algebras," algebraic systems which have
applications to several branches of combinatorics.
References:
My specialty is computational number theory. Although I have stayed away (somewhat) from the well-worn path of primarily testing and factorization, I have used the computer to investigate several interesting combinatorial and number theoretic problems.
Over the last ten years I have worked with John Brillhart, from the University of Arizona and Irving Gerst, from S.U.N.Y. at Stony Brook (Prof. Gerst passed away in 1996) on infinite product identities, especially those involving the Jacobi triple product functions. These functions have very nice product and series representations. Although no one knows an easy way to tell the parity of the partition function, we have found two infinite classes of restricted partitions whose parity can be immediately determined. The three of us were funded by the N.S.F. for two years to study these functions via computer. Recently we have discovered combinatorial mappings we hope will lead to symbolic computer proofs and ultimately a classification of these interesting identities.
A few years ago Paul Erdos introduced the concept of d-completeness. A sequence of positive integers u1, u2, ... is said to be d-complete if every sufficiently large integer n can be represented as a sum of the ui with the condition that no summand divides another. The sequence {2a3b} is d-complete and it is the only such sequence of that form. Recently J. L. Selfridge and I settled one of Erdos' last conjectures concerning the growth of the smallest term needed in these representations. Selfridge and I were working on a joint project with Erdos involving cluster primes. These are prime numbers p with the property that every even number less than p-3 can be written as a difference of primes no larger than p itself. Computationally, the cluster primes appear to have the same density as the twin primes.
Past efforts were focused on the study of the qualitative behavior of solutions of initial-value and initial-boundary value problems for hyperbolic partial differential equations and systems of equations arising in electromagnetic theory and continuum mechanics.
The problems studied arose in the areas of wave propagation in nonlinear dielectrics, the evolution of current and voltage in a nonlinear transmission line, the deformation of viscoelastic solids with both regular and singular memory kernels and, more recently, in the mathematical modeling of viscous fluid flow with an emphasis on models of incompressible fluids governed by both nonlinear viscosity and higher order velocity gradients.
Current work centers on problems of pulse combustion and jet impingement as well as on bubble growth in non-Newtonian fluids.
My research centers around using combinatorics and analysis for attacking problems in special functions and, in particular, number theory.
The topics which interest me most are continued fractions, multiple polylogarithms (and their special values), q-series, and orthogonal polynomials. I am also interested in some aspects of diophantine approximation.
My research is interdisciplinary in nature, blending theoretical and applied and computational linear algebra with application areas such as control and systems theory.
The control theory is a major source of beautiful linear algebra problems. The design and analysis of linear control systems give rise to well-known linear algebra problems such as Eigenvalue and Eigen-Structure Assignment Problems, Frequency Response Problems, Controllability and Observability Problems, Matrix Equations Problems, Stability and Inertia Problems, etc. The development of numerically effective algorithms for these problems, especially algorithms for Large Problems and those suitable for implementation on existing vector and parallel machines are of utmost importance. Numerical algorithms for control problems are still in their infancy. The control theory is lagging behind in this respect compared to other areas of applied sciences and engineering. Yet, there are control problems which are so large that they can be termed as ``Super Computer Problems''. An outstanding example is that of Large Space Structures (LSS).
My current research centers around understanding and analysis of the existing algorithms and developing new numerically viable algorithms, both sequential and parallel, for linear algebra problems in control. A particular attention is being given to the development of algorithms for large-scale solutions of problems arising especially from second order differential equations associated with large space structure problems. In the design of parallel algorithms for control problems, we make use of the existing sophisticated parallel algorithms for matrix computations and the associated software libraries presently being built both for distributed and shared-memory computers such as CRAY XMP and Hypercubes. My research addresses the urgent need clearly pointed out in the recent NSF panel report on ``FUTURE DIRECTIONS IN CONTROL THEORY.''
My research interests are in optimization along with related areas of nonsmooth analysis. Currently I am working on
My main area of research is in Reliability and Life Testing. My recent interest is to model the lifetime of a system through several stochastic processes and use the model to estimate its reliability.
I work in Abstract Harmonic Analysis. This involved the analysis of functions, measures, and related structures in the setting of locally compact groups. My work often involves substantial amounts of functional analysis, as well as topology.
My specific interests concern generalizing certain relatively well-known results from classical Fourier analysis to the setting of locally compact abelian groups.
My research mainly concerns incomplete block designs, block designs with nested rows and columns and row-column designs. My work has been on designs for factorial experiments, biological experiments, and experiments for test versus control comparisons. I am also interested in the design of mixture experiments. Currently I am also working on statistical design of medical experiments.
Ordinary differential equations and asymptotic analysis. In particular linear equations and questions relating to the spectral theory. I also work on questions involving the qualitative theory of ordinary differential equations.
My personal interest in attaining a deeper understanding of how students learn mathematics has been a driving factor in my research interests. I am interested in the processes used by students at a variety of levels to understand the symbols and procedures of mathematics and in how classroom teachers might become aware of and build instruction on the learning processes shown by their students. I am particularly interested in the processes used by middle school and high school students to develop meaning for symbolic variables so as to support their understanding of algebra in increasingly symbolic learning situations.
Professor Holland's research interests are in model theory, a branch of mathematical logic. She is particularly interested in the study of strongly minimal theories through their associated combinatorial geometries. Her recent work has centered on fusion, a means of constructing new strongly minimal sets from known ones.
My research interest is in complex matrices. Currently, I am working on the properties of nonsingular matrices under $\Phi$-maps, i.e., equivalences, canonical form, invariances, spectral structures, generalized singular or polar decomposition under a $\Phi$-map, and so on, Rank revealing QR factorization, and other problems in (numerical) matrix theory.
Recent Publications:
The statistical methodology is being developed to assess the non-inferiority testing of ECG parameters in both cross-over trials and parallel studies with baseline covariates. This methodology is applied to study the safety of every new chemical entity (new drug) that is being developed by pharmaceutical industry.
1. A Simulation Study of Power in 'Thorough' QT/QTc Studies and a Normal Approximation for Planning Purposes (with Charles Locke), Drug Information Journal. 2005.
2. Using lower trophic level factors to predict outcomes in classical biological control of insect pests (with Gross, et. al), Basic and Applied Ecology. 2005.
3. Test for misspecification of link in dependent binary regression using generalized estimating equations (with Calachan-Molefe), Journal of Statistical Computation and Simulation. 2005.
I am interested in developing efficient algorithms for solving theoretical and applied problems in optimization. I am also interested in error bounds and perturbation analysis in optimization.
Examples of my research:
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The focus of my mathematics education research has been on investigating students' mathematical learning with the intention of attempting to improve mathematical instruction. In specific, my research emphasis has been on understanding students' intellectual and constructive mathematical development. I have worked with students, in groups and individually, and I have observed and analyzed the reasoning strategies students apply in specific mathematical problem-solving situations, as well as the connections they form with their mathematical and othe domains of knowledge.
In my research, I have investigated the mathematical reasoning strategies which students use as they solve problems related to any of the following multiplicative-structured concepts: place-value, rational numbers and proportions, equation-solving, functions, probability, correlations, and conditional logic. I have worked with students of various ages: elementary school students, secondary school students, and college level students including preservice school teachers. In my research, I have applied both quantitative and qualitative research methodologies depending upon the questions that were investigated. Currently, along with the identification and analyses of students' developmental reasoning and problem-solving strategies, I am also investigating students' structures of conceptual units and their flexibility of reasoning during problem solving.
My research interests cover a wide range that includes spectral theory of linear operators and linear relations as well as abstract and applied harmonic analysis. I am investigating problems in frame theory, wavelet and time-frequency analysis, sampling theory, causal pseudo-differential operators, etc. A usually up-to-date list of my publications and preprints is available on my home page.
My research interests are mainly in the qualitative analysis of differential equations and its applications which include:
My field of research is in Non-linear Partial Differential Equations. It can be mainly divided into 2 parts:
My main area of research is Bayesian Statistical Inference from Stochastic Processes. I also have research interest in Statistical Computing and Actuarial Modeling. Currently, Bayesian estimation, hypotheses testing and model selection issues relating to random processes are being pursued.
References:
My current research interests are related to problems regarding periodic extremals. Many lasting non-chaotic physical phenomena can be viewed as extremals of this kind. By the use of essentially infinite dimensional methods it is possible to represent all periodic functions as a nonflat subset of the space of all functions. The extremals appear at points where the projected gradient vector field vanishes. In order to find these elusive extremals one attempts to follow the trajectories of the gradient vector field.
My methods have proven to be very useful when applied to so-called nonlinear splines in approximation theory. The techniques used involve differential geometry, global analysis, calculus of variations and optimal control. I also use Sobolev spaces, convexity, tensor analysis, numerical analysis, Mathematica, the C-language, computer graphics (real time and animated) and occasionally theories of physics. In the future some of this work will lead to computer implementations of new algorithms.
A good reference is my paper ``Curve straightening'' which appeared in ``Proceedings of Symposia in Pure Mathematics'' by the American Mathematical Society. This volume covers the AMS summer research institute in differential geometry at UCLA and it gives the state of the art as of 1990.
My research interests concern the algebraic structure of semigroups, in particular regular and inverse semigroups. Most of my work has been based on the program of describing a given semigroup as a ``nice'' homomorphic image of a much simpler semigroup for which a tractable structural model exists. That is, it is based on the idea of finding nice covers for semigroups. Investigations of this type lead naturally to questions involving partially ordered sets, automata, formal languages, and effective computability.
For the last few years I have been investigating several methods of performing adaptive tests of significance and adaptive confidence intervals. The advantage of adaptive tests is that, if at least 20 observations are used in the analysis, they are usually more powerful than the traditional tests with non-normal error distributions. In 2002 I developed and published a method that can be used to perform adaptive tests for a subset of coefficients in a linear model. Consequently, an adaptive two-sample test, an adaptive test for a one-way layout, and an adaptive test for any subset of coefficients in a regression model can easily be performed. It is also possible to construct adaptive confidence intervals that are based on the corresponding test. By carefully constructing adaptive confidence intervals we find that they are often narrower than the traditional confidence intervals, while they maintain their nominal coverage probabilities.
My research interests are in the general area of nonparametric Statistics. Specifically, my research focuses on the following areas:
My general area of research is in probability and statistics. More specifically, I am interested in prediction theory of stochastic processes, and theoretical aspects of time series analysis and signal processing. In these areas it is of interest to study the evolution of a system (such as weather, radio signal, path of an airplane) over time by a stochastic process so that its future can be predicted and possibly controlled.
By using probability theory such problems can be formulated properly, and viewed as certain approximation or optimization problems in Hilbert or Banach spaces. Solution of such problems requires techniques from functional and harmonic analysis, linear algebra, etc. In recent years I have been interested in theoretical, computational and statistical aspects of such problems.
I am a finite group theorist by training but considered a topologist on the basis of my work; lately I've felt more like a computer programmer. My research interests lie at the overlap of these areas, especially in the cohomology of groups. Group cohomology is a process of calculating rings and other algebraic invariants associated to groups, and using this information to describe the group, its subgroups, and related information. But the first definitions and intuitive descriptions of these invariants are topological by and large. (Given a small group we create a large topological space and calculate homological data from it.)
My recent work in this area has been explicitly computational. I exploit the fact that these easily-described invariants are a real bear to compute in any but trivial examples to gain my fame and fortune. My paper ``The Cohomology of the Groups of Order 32'' is about where the state of the art is. (Math. of Computation Vol. 53 (1989) pp. 359-385.)
I am a problem solver, sometimes by computation. I prefer exact solutions, rather than estimates. In 1953 I found the first factors of Fermat numbers 10 and 16. The larger was conjectured prime in 1828. "The product of consecutive integers is never a power" IL J. Math 1975 (with Paul Erdos). Papers on aliquot sequences with Richard Guy, 1970's. "Factors of bn ± 1" (with Brillhart, Lehmer, Wagstaff) third ed. 1998. Current problem: The 392 conjecture (with Aaron Meyerowitz).
The focus of my mathematics education research has been on studying teachers' pedagogical decisions and classroom assessment practices and investigating the impact of Standards-based mathematics curricula. I am co-editor of a forthcoming book on the results of NSF-funded research projects on the impact of Mathematics in Context on student achievement. In these studies, I identified and scaled important dimensions of classroom instruction, classroom assessment, and students' opportunity to learn mathematics with understanding. My publications have focused on classroom assessment and studying the impact of reform mathematics curricula.
I am interested in the following areas of Functional analysis and Operator theory.
* Geometry of Banach spaces (Hereditarily Indecomposable Banach spaces; Operators with and without invariant subspaces; Daugavet equation);
* Positive operators in Banach lattices (Invariant subspace problem for positive operators; Compact-friendly operators).
Professor Sons' research interests are in classical complex analysis. More specifically they revolve about value distribution studies for meromorphic functions in the complex plane or in the unit disk. She has done studies involving normal families, normal functions, automorphic functions, power series, polynomial growth along arcs, and differential equations. She has been interested in values assumed by functions or combinations of functions and their derivatives for functions of unbounded, but ``slow'' growth in the unit disk. She has considered questions about characteristics of classes of frictions and the growth of solutions of ordinary differential equations defined in the unit disk whose coefficients are functions of low growth in the unit disk. Finally, her interests also include the connections between Nevanlinna theory and number theory.
My research interests include investigating how children learn mathematics, how they communicate about mathematics, and how teachers use this information in their teaching. In research publications I have described how teachers may use Vygotsky's idea of the "zone of proximal development" to help children construct knowledge and understanding of mathematics. I have found that when teachers enter into a child's ZPD, they can help the child think in ways that are needed for understanding in the discipline of mathematics. Recently, the focus of my research has been on investigating students' algebraic thinking. I have worked with middle school students, in groups and individually, in a teaching experiment in which I analyzed the reasoning strategies students apply in algebraic problem situations, specifically geometric growth problems. I have explored the qualities and types of problem-solving schemas students developed as they generalize algebraic problems. Findings from this research indicate that there is a link between the types of generalizations students construct and the schemas they are forming. For this research focus, I am now analyzing the representations students develop as they solve algebraic problems. I have applied both quantitative and qualitative research methodologies, depending upon the questions that I investigated. Currently, I am using both of these research methodologies to investigate a multi-faceted women's calculus course designed around the ways women learn. I am analyzing in what ways the participation in this course affected their mathematics learning or willingness to take additional mathematics courses.
Present activity:
My research interests are in Diophantine equations, Diophantine approximation, arithmetic geometry, and the geometry of numbers. The study of Diophantine equations deals with finding integral or rational solutions to polynomial equations. This is closely related to Diophantine approximation, where one studies approximations to real numbers by rational numbers. Arithmetic geometry, broadly speaking, deals with arithmetic properties (i.e., properties concerning the integers) of geometric objects, usually affine or projective varieties. This is an area where number theory and algebraic geometry come together. The geometry of numbers deals with points with integer coordinates in regions of real n-space (under what condition will a region have such a point? how many? etc...), sphere packing, and related subjects.
Möbius transformations of Euclidean n- space (n>2) arise as the most general conformal (angle preserving) transformations. The unit ball may be given a non-Euclidean metric for which those Möbius transformations preserving the ball comprise the full group of isometries. Studying these groups involves the interplay between complex analysis, infinite group theory and geometry.
Reference:
``Purely elliptic Möbius groups.'' In Holomorphic Functions and Moduli II ed. D. Drasin. Springer-Verlag. N.Y., 1988.
(to be added soon...)
My research uses ideas drawn from functional analysis, measure theory, and topology. It is focused on an interesting new area of topological measure theory, the study of quasi-linear functionals and quasi-measures, initiated recently by the Norwegian functional analyst Johan Aarnes.
If X is a completely regular Hausdorff space, and C(X) is the vector space of bounded continuous real-valued functions on X , then a quasi-linear functional p on C(X) is a map p from C(X) to R such that (i) p(f) is non-negative whenever f is a non-negative function; and (ii) the restriction of p to any supremum norm closed, singly generated subalgebra of C(X) is linear. Such functionals need not be linear on the entire space C(X) , for spaces X of dimension at least two.
Aarnes (and John Boardman, in his NIU dissertation) have established a correspondence between quasi-linear functionals on C(X) and certain set functions on X called quasi-measures, via a process of (non-linear) integration. A quasi-measure has virtually every property of an ordinary finitely additive measure, except that it need not be finitely sub-additive, for spaces X with dimension at least two.
The study of quasi-linear functionals and quasi-measures is quite interesting for its own sake, but has also been shown to have potentially significant connections to algebraic topology, point-set topology, probability theory, and fractals. It is likely to be an active area of research for many years to come.
References:
I work mainly in algebraic number theory at a relatively elementary level, dealing with problems of classnumbers and units in algebraic number fields. I also make an occasional sideline into diphantine equations, that is, equations in which integer solutions are required. Some recent references are the following:
My recent research is in four areas in mathematics, with a common theme---the interplay of geometric ideas and analytic methods.
We apply refined techniques from the soliton theory to obtain representations of such geometric objects using holomorphic or meromorphic data. These representations are then used in constructions and classifications of the geometric objects. Computer visualizations of the objects are also among our interests.
We introduce geometric structures onto spaces of spectral problems, especially onto spaces of boundary conditions, and study the dependence of the spectrum on the problem, especially the dependence on the boundary condition. Also, based on these theoretic results, we develop computer codes for numerical computations of spectra.
We classify such transformations and use them to generate new solutions. Also, we develop computer codes for automated searches of such transformations.
We develop computer codes for drawing the regions in the space of constant terms in which the number of solutions is a given number. Theoretic results are then formulated and proven. These two aspects stimulate each other.
Motivated by P. Vojta's dictionary of number-theory-Nevanlinna-theory and related S. Lang's questions, my current research interests are finding analogies of Diophantine approximation in Nevanlinna theory of meromorphic mappings between two complex manifolds.
I am interested in virtually any problem involving linear ordinary differential equations.
Dr. Zollman's scholarship interests focus on research-based, classroom-tested curriculum innovations in mathematics education. His current investigations are in the areas of performance-based assessments and student-work mathematics portfolios at the primary and middle school levels, including their implementation implications for teachers and students. Previous work has included publications on teachers' beliefs concerning mathematics education reform.