Robert H Lewis
After my PhD at Cornell, I started out as a pure mathematician doing algebraic topology. Then I got an MS in computer science and became interested in computer algebra. I do symbolic computation of multivariate polynomials and matrices. A few of my papers can be found here.
Short professional biography:
http://home.bway.net/lewis/biog.pdf
I am the author of Fermat, computer algebra system. See
http://home.bway.net/lewis/
Under "talks" here you will find a popular article I wrote about my research recently, called "Polynomials Everywhere." Near the end of that article there is reference to a movie of a new flexible structure. See http://home.bway.net/lewis/flex2.MOV
I have also written an essay on mathematics education. See
http://www.fordham.edu/mathematics/whatmath.html
Phone: 718-817-3226
Address: Department of Mathematics
Fordham University
441 E. Fordham Rd.
Bronx NY 10458 USA
Short professional biography:
http://home.bway.net/lewis/biog.pdf
I am the author of Fermat, computer algebra system. See
http://home.bway.net/lewis/
Under "talks" here you will find a popular article I wrote about my research recently, called "Polynomials Everywhere." Near the end of that article there is reference to a movie of a new flexible structure. See http://home.bway.net/lewis/flex2.MOV
I have also written an essay on mathematics education. See
http://www.fordham.edu/mathematics/whatmath.html
Phone: 718-817-3226
Address: Department of Mathematics
Fordham University
441 E. Fordham Rd.
Bronx NY 10458 USA
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Papers by Robert H Lewis
The great misconception about mathematics -- and it stifles and thwarts more students than any other single thing -- is the
notion that mathematics is about formulas and cranking out computations. It is the unconsciously held delusion that mathematics
is a set of rules and formulas that have been worked out by God knows who for God knows why, and the student's duty is to memorize
all this stuff. Such students seem to feel that sometime in the future their boss will walk into the office and demand "Quick,
what's the quadratic formula?" Or, "Hurry, I need to know the derivative of 3x^2 - 6x + 1." There are no such employers.
The great misconception about mathematics -- and it stifles and thwarts more students than any other single thing -- is the notion that mathematics is about formulas and cranking out computations. It is the unconsciously held delusion that mathematics is a set of rules and formulas that have been worked out by God knows who for God knows why, and the student's duty is to memorize all this stuff. Such students seem to feel that sometime in the future their boss will walk into the office and demand "Quick, what's the quadratic formula?" Or, "Hurry, I need to know the derivative of 3x^2 - 6x + 1." There are no such employers.
The great misconception about mathematics -- and it stifles and thwarts more students than any other single thing -- is the
notion that mathematics is about formulas and cranking out computations. It is the unconsciously held delusion that mathematics
is a set of rules and formulas that have been worked out by God knows who for God knows why, and the student's duty is to memorize
all this stuff. Such students seem to feel that sometime in the future their boss will walk into the office and demand "Quick,
what's the quadratic formula?" Or, "Hurry, I need to know the derivative of 3x^2 - 6x + 1." There are no such employers.
The great misconception about mathematics -- and it stifles and thwarts more students than any other single thing -- is the notion that mathematics is about formulas and cranking out computations. It is the unconsciously held delusion that mathematics is a set of rules and formulas that have been worked out by God knows who for God knows why, and the student's duty is to memorize all this stuff. Such students seem to feel that sometime in the future their boss will walk into the office and demand "Quick, what's the quadratic formula?" Or, "Hurry, I need to know the derivative of 3x^2 - 6x + 1." There are no such employers.
We find new ways that Bricard's quadrilaterals can be flexible. This has ramifications for robotics and computational chemistry.
We will examine the systems of equations that result from two- and three-dimensional configurations of interacting Brusselators. We have up to eight equations in eight variables and up to twelve parameters. We find that all are solvable with Dixon resultant methods. We will describe how Groebner Bases fail on all but the simplest cases. We will show other examples of autocatalytic reactions.
Protein flexibility is a major research topic in computational chemistry. In general, a polypeptide backbone can be modeled as a polygonal line whose edges and angles are fixed while some of the dihedral angles can vary freely. It is well known that a segment of backbone with fixed ends will be (generically) flexible if it includes more than six free torsions. Resultant methods have been applied successfuly to this problem. In this work we focus on non-generically flexible structures (like a geodesic dome) that are rigid but become continuously movable under certain relations. The subject has a long history: Cauchy (1812), Bricard (1896), Connelly (1978).
In a previous work, we began a new approach to understanding flexibility, using not numeric but symbolic computation. We describe the geometry of the object with a set of multivariate polynomial equations, which we solve with resultants. Resultants were pioneered by Bezout, Sylvester, Dixon, and others. The resultant appears as a factor of the determinant of a matrix containing multivariate polynomials. We describe a method to find these factors "early". Given the resultant, we described an algorithm, Solve, that examines it and determines relations for the structure to be flexible. We discovered in this way the conditions of flexibility for an arrangement of quadrilaterals in Bricard, which models molecules. Here we significantly extend the algorithm and the molecular structures. We consider the cylo-octane molecule.
Secondly, we use Dixon-EDF to solve several sets of equations that arise from the study of Nash equilibria. This is an important topic in economic game theory. We examine the cases of three or four players with two pure strategies each. The latter produces a set of 8 equations with 8 variables and 32 parameters. Then we look at a classic problem due to Nash, simplified three-man poker (with 4 equations, 4 variables, 44 parameters), and lastly at a "cube game" (8, 8, 4). These are found in the book by Sturmfels and the papers by Datta. Apparently we are the first to provide fully symbolic solutions to these games.
All of these problems are solvable with Dixon-EDF. Apparently all are intractable with other methods. We report on failed attempts to solve these with Maple12, using both its builtin Groebner bases command and its implementation of Faugere's fgb algorithm.
There is another common thread in these two apparently disparate subjects: all the equations are of degree one in each variable. That is, in every equation no variable is squared. In only one of the equations is any parameter squared. Indeed, often we find that every equation is of total degree two in the variables. These are in some sense the simplest non-linear equations; we call them "almost linear." Yet Groebner bases methods fail repeatedly as Dixon-EDF succeeds.
I think the general public does not have a very good idea of what mathematicians do, so let me start with some generalities. Before roughly 1985, almost all mathematicians in academic positions in the United States and Europe were pure mathematicians. ...
Exact symbolic computation with polynomials and matrices over polynomial rings has wide applicability to many fields. By "exact symbolic" we mean computation with polynomials whose coefficients are integers (of any size), rational numbers, or finite fields, as opposed to coefficients that are "floats" of a certain precision. Such computation is part of most computer algebra systems ("CA systems"). Over the last dozen years several large CA systems have become widely available, such as Axiom, Derive, Macsyma, Magma, Maple, Mathematica, and Reduce. They tend to have great breadth, be produced by profit-making companies, and be relatively expensive. However, most if not all of these systems have difficulty computing with the polynomials and matrices that arise in actual research. Real problems tend to produce large polynomials and large matrices that the general CA systems cannot handle.
In the last few years several smaller CA systems focused on polynomials have been produced at universities by individual researchers or small teams. They run on Macs, PCs, and workstations. They are freeware or shareware. Several claim to be much more efficient than the large systems at exact polynomial computations. The list of these systems includes CoCoA, Fermat, MuPAD, Pari-GP, and Singular.
One important topic addressed here with a broad range of applications is the solution of multivariate polynomial systems by means of resultants and Groebner bases. But that’s barely the beginning, as the authors proceed to discuss genetic algorithms, integer programming, symbolic regression, parallel computing, and many other topics.
The book is strictly goal-oriented, focusing on the solution of fundamental problems in the geosciences, such as positioning and point cloud problems. As such, at no point does it discuss purely theoretical mathematics.