
Daniel LichtblauWolfram Research · Kernel Group
Daniel Lichtblau
Ph.D Mathematics UIUC 1991
About
60
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451
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Introduction
Additional affiliations
August 1998 - May 1999
September 1991 - present
Education
August 1985 - August 1991
University of Illinois
Field of study
- Mathematics
September 1975 - June 1979
Publications
Publications (60)
We are given an equilateral triangle with vertices constrained to lie in each of the three positive octant coordinate planes (colloquially, "a triangle in a corner"). We wish to describe the locus of points covered by the midpoint of the triangle, as the vertices range over configurations allowed by the above constraint. This locus comprises a soli...
It is known that five points in R 3 generically determine a finite number of cylinders containing those points. We discuss ways in which it can be shown that the generic (complex) number of solutions, with multiplicity, is six, of which an even number will be real valued and hence correspond to actual cylinders in R 3 . We partially classify the ca...
We discuss computation of approximate Gröbner bases at finite precision. We show how this can be used to deduce exact results for polynomial greatest common divisors and factorization. In particular we indicate an algorithm for factoring multivariate polynomials over the closure algebraic of the rationals.
Since their invention by Buchberger in 1965, Gröbner bases have become a pervasive tool in computational mathematics. In this paper we show several sorts of Gröbner basis computations in Mathematica. We discuss computations of approximate Gröbner bases, with applications to solving systems and implicitization. We also show how to use the built-in G...
Over the past few decades several variations on a "half GCD" algorithm for obtaining the pair of terms in the middle of a Euclidean sequence have been proposed. In the integer case algorithm design and proof of correctness are complicated by the effect of carries. This paper will demonstrate a variant with a relatively simple proof of correctness....
In this paper, we study the complexity of performing some linear algebra operations such as Gaussian elimination and minimal polynomial computation over an algebraic extension field. For this, we use the theory of Gröbner bases to employ linear algebra methods as well as to work in an algebraic extension. We show that this has good complexity. Fina...
The authorship attribution task assumes the presence of several examples of documents written by various authors and it must be determined who wrote a given anonymous text. For each author, a specific writing style is hypothesized, with characteristics that the authors themselves are not aware of. The writing style acts as a fingerprint, as various...
We consider the entire set of COVID-19 local epidemics in the United States; a broad selection of demographic, population density, and climate factors; and local mobility data, tracking social distancing interventions, to determine the key factors driving the spread and containment of the virus. Assuming first a linear model for the rate of exponen...
We apply a recent alignment-free method of genomic comparison to sequences of SARS-CoV-2 as well as other sequences from the Coronaviridae family. We show that this method, while approximate, can enable fast and accurate classification. We illustrate how it might be applied in the search for the possible intermediary host or hosts. We also use this...
An author unconsciously encodes in the written text a certain style that is often difficult to recognize. Still, there are many computational means developed for this purpose that take into account various features, from lexical and character-based attributes to syntactic or semantic ones. We propose an approach that starts from the character level...
The SARS-CoV-2 pandemic has caused significant mortality and morbidity worldwide, sparing almost no community. As the disease will likely remain a threat for years to come, an understanding of the precise influences of human demographics and settlement, as well as the dynamic factors of climate, susceptible depletion, and intervention, on the sprea...
We consider the problem of analyzing chemical reaction networks that may allow multiple positive steady states. We use tools from “classical” computer algebra (Gröbner bases over a parametrized domain, computation of a discriminant variety, graphical and mathematical analysis of solution sets, cylindrical decomposition) to help determine regions of...
Background:
Alignment-free methods of genomic comparison offer the possibility of scaling to large data sets of nucleotide sequences comprised of several thousand or more base pairs. Such methods can be used for purposes of deducing "nearby" species in a reference data set, or for constructing phylogenetic trees.
Results:
We describe one such me...
One way to compute a GCD of a pair of multivariate polynomials is by finding a certain syzygy. We can weaken this to create an “approximate syzygy”, for the purpose of computing an approximate GCD. The primary tools are Gröbner bases and optimization. Depending on specifics of the formulation, one might use quadratic programming, linear programming...
Recent work by Lemke Oliver and Soundararajan, as well as earlier results by Ko, have brought to light an unexpected asymmetry in
the distribution of last digits of consecutive primes. For example, in the first 10^8 pairs of consecutive primes, around 4.6 million end with
{1,1} respectively, whereas more than 7.4 million end with {1,3}. This dispar...
The current research study is concerned with the automated differentiation between histopathological slides from colon tissues with respect to four classes (healthy tissue and cancerous of grades 1, 2 or 3) through an optimized ensemble of predictors. Six distinct classifiers with prediction accuracies ranging from 87% to 95% are considered for the...
Abstract. Systems of polynomial/algebraic equations with finitely many
solutions arise in many areas of applied mathematics. I will discuss the
design and implementation of a hybrid symbolic-numeric method based
on the endomorphism matrix approach pioneered by Stetter and others.
It makes use of numeric Gröbner bases and arbitrary-precision eigen-...
The Chaos Game Representation, a method for creating images from nucleotide sequences, is modified to make images from chunks of text documents. Machine learning methods are then applied to train classifiers based on authorship. Experiments are conducted on several benchmark data sets in English, including the widely used Federalist Papers, and one...
The East Coast Computer Algebra Day 2017 (ECCAD 2017) was hosted byWolfram Research on Saturday, April 29, 2017.
The large amount of histopathological images that are produced in hospitals worldwide often request an overwhelming effort from the human pathology experts. In this respect, large efforts are made by scientists in various disciplines and especially in computer science to develop automatic procedures to distinguish between different grades of cancer...
This work is devoted to simplify the inverse-forward kinematics of a parallel manipulator generator of the 3T1R motion. The closure equations of the displacement analysis are formulated based on the coordinates of two points embedded in the moving platform. After, five quadratic equations are solved by means of a novel method based on Gröbner bases...
Talk slides presented September 2016, RISC Linz
Recent work by Lemke Oliver and Soundararajan shows an unexpected asymmetry in distribution of last digits of consecutive primes. For example, in the first 10^8 pairs of consecutive primes, around 4.6 million end with {1,1} respectively, whereas more than 7.4 million end with {1,3}. It is not hard to show that his disparity is not explained by the...
A problem frequently encountered in geometric constraint solving and related settings is to ascertain sensitivity of solutions arising from a well constrained input configuration. This is important for tolerancing and motion planning, for example. An example would be determining lines simultaneously tangent to four given spheres (which originates a...
Buchberger and, independently, Kandri-Rody and Kapur, defined a strong Gröbner basis for a polynomial ideal over a Euclidean domain in a way that gives rise to canonical reductions. This retains what is perhaps the most important property of Gröbner bases over fields. A difficulty is that these can be substantially harder to compute than their fiel...
We discuss computation of Gröbner bases using approximate arithmetic for coefficients. We show how certain considerations of tolerance, corresponding roughly to accuracy and precision from numeric computation, allow us to obtain good approximate solutions to problems that are overdetermined. We provide examples of solving overdetermined systems of...
Strong Grobner bases over Euclidean domains and even more general rings
were first defined in the 1980s. Since that time, efficient ways to compute
them, and a variety of applications, have appeared. In this note we show, via
simple examples, applications to solving equations in quotient rings, Hensel
lifting, Hermite normal form computations, redu...
We address the following question: Given five points in 3 , determine a right circular cylinder containing those points. We obtain algebraic equations for the axial line and radius parameters and show that these give six solutions in the generic case. An even number (0, 2, 4, or 6) will be real valued and hence correspond to actual cylinders in 3 ....
In this supplement we describe various ways to count, compute and cull the data tuples corresponding to Fourier arrangements. Actual computations were performed with Mathematica 8.
We show ways in which differential evolution, a
member of the genetic/evolutionary family of global opti-
mization methods, can be used for the purpose of discrete
optimization. We consider several nontrivial problems aris-
ing from actual practice, using differential evolution as our
primary tool to obtain good results. We indicate why meth-
ods m...
The computation of definite integrals presents one with a variety of choices. There are various methods such as Newton−Leibniz or Slater's convolution method. There are issues such as whether to split or merge sums, how to search for singularities on the path of integration, when to issue conditional results, how to assess (possibly conditional) co...
We will show a number of ways in which Differential Evolution, a member of the genetic/evolutionary family of optimization methods, can be used for the purpose of discrete optimization. Problems under consideration include standard integer linear programming, knapsack problems, set partitioning, set covering, quadratic assignment, mixed (discrete-c...
We introduce some standard types of combinatorial optimization problems, and indicate ways in which one might attack them
using Differential Evolution. Our main focus will be on indexing by relative position (also known as order based representation); we will describe some related approaches as well. The types of problems we will consider, which ar...
We study a question with connections to linear algebra, real algebraic geometry, combinatorics, and complex analysis. Let $p(x,y)$ be a polynomial of degree $d$ with $N$ positive coefficients and no negative coefficients, such that $p=1$ when $x+y=1$. A sharp estimate $d \leq 2N-3$ is known. In this paper we study the $p$ for which equality holds....
Knapsack problems and variants thereof arise in several different fields from operations research to cryptography to really, really serious problems for hard−core puzzle enthusiasts. We discuss some of these and show ways in which one might formulate and solve them using Mathematica.
This is an extended version of a paper presented at the Internat...
The Gröbner walk is a useful method for conversion from a "simple" Gröbner basis to a different one in a desired term order.Various issues along the way include coefficient swell (similar to that seen in the classical Buchberger algorithm), polynomials with many initial elements in the "end-game" phase, and the like.We take as benchmark the implici...
In this talk I will consider the following problem. We are given a "units monomial", that is, a product of (possibly negative) integer powers of physical units, e.g. meters 2 volts farads seconds 2 . We might try to make sense of this by finding all equivalent monomials subject to a minimality condition. Good candidates for such a condition involve...
The Frobenius number g(A) of a set A = (a1, a2,..., an) of positive integers is the largest integer not representable as a nonnegative linear combination of the ai. We interpret the Frobenius number in terms of a discrete tiling of the integer lattice of dimension n−1 and obtain a fast algorithm for computing it. The algorithm appears to run in ave...
In this talk I will present several problems that have caught my attention over the past few years. We will go over Mathematica formulations and solutions. Along the way we will meet with a branch- and- bound loop in its natural habitat, some rampaging Gröbner bases, a couple of tamed logic puzzles, and at least a dozen wild beasts (well...would yo...
We will discuss knapsack problems that arise in certain computational number theory settings. A common theme is that the search
space for the standard real relaxation is large; in a sense this translates to a poor choice of variables. Lattice reduction
methods have been developed in the past few years to improve handling of such problems. We show...
Over the past few decades several variations on a "half GCD" algorithm for obtaining the pair of terms in the middle of a Euclidean sequence have been proposed. In the integer case algorithm design and proof of correctness are complicated by the effect of carries. This paper will demonstrate a variant with a relatively simple proof of correctness....
A few years ago S-H Kim investigated some problems at the boundary of number theory, optimization, and geometry. One question
regarded an optimal packing of certain “triangular oval” planar curves and another looked at some related transformations
of ℝ2 to ℝ2. These were investigated primarily using tools from calculus but it turns out that computa...
Buchberger and Kandri−Rody and Kapur defined a strong Gröbner basis for a polynomial ideal over a Euclidean domain in a way that gives rise to canonical reductions. This retains what is perhaps the most important property of Gröbner bases over fields. A difficulty is that these can be substantially harder to compute than their field counterparts. W...
Many important classes of optimization problems are discrete in nature. Examples are the standard problems of integer pro-gramming (including the ever important "knapsack problems"), permutation assignment problems (e.g., the notorious "traveling salesman problem"), set coverings, set partitioning, and so on. Problems of practical interest are freq...
Systems of algebraic equations with finitely many solutions arise in many areas of applied mathematics. Among these are motion planning, robotics, computer−aided design, and graphics. We will discuss the design and implementation of a hybrid symbolic− numeric method, and a Mathematica implementation thereof, that finds all solutions to an algebraic...
We begin with a function expressed as a certain infinite product. It is a twice-mutated variation of another product that has its origins in counting irreducible factors of univariate polynomials over Galois fields. Knopfmacher's limit is taken as we approach 1 from below in this product. We derive and execute an algorithm that finds a good approxi...
We will discuss several problems that appear from time to time as questions to tech-support or the Mathematica users' news group comp.soft-sys.math.mathematica. While not all are frequently asked, those that are not fall under the heading of "general interest." For example, we have some that involve essential computer-science ideas such as bit vect...
In: The Mathematica Journal 6(4): 81−88. 1996. Gröbner bases are used heavily throughout computational mathematics. This article demonstrates new functionality of Mathemati− ca's GroebnerBasis command and describes two new related functions. As examples, we show how GroebnerBasis can be used to rationalize denominators, and how it can be extended f...
We determine precisely for which spherical space forms there are nontrivial smooth CR mappings to spheres. Equivalently we
determine for which fixed point free finite unitary groups ⌈ there exists a ⌈-invariant proper holomorphic rational map between
balls. The answer is that the group must be cyclic and essentially only two classes of representati...
In this paper we consider proper holomorphic maps between balls that are invariant under the action of finite fixed point free unitary groups. Forstnerič showed that given any such group (representation) there exists such a map. He showed that if we also require the map to be smooth to the boundary, then many available groups are ruled out. In this...
We consider the following two conjectures concerning intersecting families of a finite
set X. Conjecture 1. Let S be a collection of subsets X, (|X|=n) such that for all
A, B. in S, 1<=|A intersect b|<=k (k fixed). Then |S|<=t[n,k] where
t[n,k] = Sum{binomial n-1,i}, {i,1,k}.
Conjecture 2. Let T = {l-1,...,l_k} be a collection of k positive integer...