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Web: http://www.ima.umn.edu | Email: ima-staff@ima.umn.edu | Telephone: (612) 624-6066 | Fax: (612) 626-7370

### March 2007

2006-2007 Program

### Applications of Algebraic Geometry

See http://www.ima.umn.edu/2006-2007 for a full description of the 2006-2007 program on Applications of Algebraic Geometry.

News and Notes

New Directions Research Professors

The IMA is pleased to announce that the New Directions Research Professors for the 2007-2008 thematic program, Mathematics of Molecular and Cellular Biology, will be Tiefeng Jiang, Department of Statistics, University of Minnesota, Debra Knisley, Department of Mathematics, Eastern Tennessee State University, and Zhijun Wu, Department of Mathematics,University of Iowa. New Directions Research Professorships provide an extraordinary opportunity for established mathematicians-typically mid-career faculty at US universities-to spend the academic year at the IMA. The visiting Professors will enjoy an excellent research environment and stimulating scientific program, learning about and contributing to exciting new developments in mathematical biology and a broad range of applications.

New publication on mixed integer programming

A special issue of the journal Discrete Optimization based on the IMA workshop Mixed-Integer Programming, July 25-29, 2005, has just appeared.

Summer programs

This summer the IMA will offer four exciting programs. One is intended for mid-career mathematicians interested in exploring a new research direction, and two are exclusively for graduate students.

The 2007 IMA Summer Program Classical and Quantum Approaches in Molecular Modeling, July 23-August 3, 2007, is devoted to computational molecular modeling both via classical approaches (emphasized in week 1) and quantum approaches (emphasized in week 2). Both weeks will include talks at the introductory level. The lectures during the first week will be given by Giovanni Ciccotti, Ben Leimkuhler, Robert Skeel, and Mark Tuckerman while the lectures during the second week will be given by Eric Cancès, and Youssef Saad.

The New Directions Short Course Compressive Sampling and Frontiers in Signal Processing June 4 - 15, 2007 is taught by Emmanuel Candes, Ron DeVore and Rich Barniuk. This exploding area of research has connections to many areas of mathematics and presents many research opportunities. The IMA is currently accepting applications for this Short Course. No prior background in signal processing is expected. Participants will receive full travel and lodging support during the workshop. Participation is by application only and will be limited to 25 midcareer researchers in the mathematical sciences. The application deadline is April 1.

PI Summer Program for Graduate Students Applicable Algebraic Geometry will be held at Texas A&M University in College Station, Texas, July 23-August 10, 2007. This program is open only to graduate students from IMA Participating Institutions. In order to participate, students need to fill out the application form and need to be nominated by their department chair by April 15, 2007. The course will concentrate on Applicable Algebraic Geometry. If you know of a graduate student at a Participating Institution who would benefit from the program, please encourage them to contact their department chair.

Mathematical Modeling in Industry Mathematical Modeling in Industry XI - A Workshop for Graduate Students, August 8-17, 2007 offers graduate students and qualified advanced undergraduates first hand experience in industrial research. Teams of up to six students will work under the guidance of a mentor from industry, who will guide the students in the modeling and analysis of real-world industrial problems. This year's mentors are Natalia M. Alexandrov (NASA), Radu Balan (Siemens), Gary B. Green (The Aerospace Corporation) John R. Hoffman (Lockheed Martin), Mark A. Stuff (General Dynamics), and Lisa Zhang (Lucent Technologies, Bell Laboratories). The application deadline is April 15.

IMA is seeking a new director: The IMA is looking for a new director to begin in summer 2008.

IMA Events

### Applications in Biology, Dynamics, and Statistics

#### March 5-9, 2007

Organizers: Lior Pachter (University of California), Seth Sullivant (Harvard University)

Schedule

## Thursday, March 1

 9:45a-10:45a Algebraic statistics learning seminar Lind Hall 409 11:15a-12:15p Real algebraic geometry tutorial: Real rings Kenneth R. Driessel (Iowa State University) Lind Hall 409 RAG 3:20p-4:20p From Drosophila and transposable elements to phylogenetic networks and associahedra Lior Pachter (University of California) Vincent Hall 16

## Monday, March 5

 8:15a-9:00a Registration and coffee EE/CS 3-176 W3.5-9.07 9:00a-9:15a Welcome and introduction Douglas N. Arnold (University of Minnesota Twin Cities) EE/CS 3-180 W3.5-9.07 9:15a-10:05a Open problems in algebraic statistics Bernd Sturmfels (University of California) EE/CS 3-180 W3.5-9.07 10:05a-10:40a Coffee EE/CS 3-176 W3.5-9.07 10:40a-11:30a Likelihood ratio tests and singularities Mathias Drton (University of Chicago) EE/CS 3-180 W3.5-9.07 11:30a-1:30p Lunch W3.5-9.07 1:30p-2:20p Stability and instability in polynomial equations arising from complex chemical reaction networks: the big picture Martin Feinberg (Ohio State University) EE/CS 3-180 W3.5-9.07 2:20p-2:50p Coffee EE/CS 3-176 W3.5-9.07 2:50p-3:40p Stability and instability in polynomial equations arising from complex chemical reaction networks: some underlying mathematics Gheorghe Craciun (University of Wisconsin) EE/CS 3-180 W3.5-9.07 3:40p-4:10p Second Chances EE/CS 3-180 W3.5-9.07 4:10p-4:30p Group Photos EE/CS 3-180 W3.5-9.07 4:30p-6:30p IMA Tea and Poster Session Lind Hall 400 W3.5-9.07 A flow that computes the best positive semi-definite approximation of a symmetric matrix Kenneth R. Driessel (Iowa State University) Multiple solutions to the likelihood equations in the Behrens-Fisher problem Mathias Drton (University of Chicago) Metric learning for phylogenetic invariants Nicholas Eriksson (Stanford University) Geometry of rank tests Jason Morton (University of California) Supervised learning artifical neural network algorithms for optimizing mechanical properties of elastin-like polypeptide hydrogels for cartilage repair Sarah Olson (North Carolina State University) Toric ideals of phylogenetic invariants for the general group-based model on claw trees Sonja Petrovic (University of Kentucky) Conditional independence for Gaussian random variables is not finitely axiomatizable Seth Sullivant (Harvard University) Linkage problems and real algebraic geometry Thorsten Theobald (Johann Wolfgang Goethe-Universität Frankfurt) Classifying disease models using regular polyhedral subdivisions Debbie Yuster (Columbia University) Maximum likelihood estimation in latent class Yi Zhou (Carnegie-Mellon University)

## Tuesday, March 6

 8:30a-9:00a Coffee EE/CS 3-176 W3.5-9.07 9:00a-9:50a Phylogenetic models: algebra and evolution Elizabeth S. Allman (University of Alaska) EE/CS 3-180 W3.5-9.07 9:50a-10:30a Coffee EE/CS 3-176 W3.5-9.07 10:30a-11:20a Generalized maximum likelihood estimates for exponential families František Matúš (Institute of Information Theory and Automation) EE/CS 3-180 W3.5-9.07 11:20a-1:30p Lunch W3.5-9.07 1:30p-2:20p Bifurcations in coupled systems Martin Golubitsky (University of Houston) EE/CS 3-180 W3.5-9.07 2:20p-3:00p Coffee EE/CS 3-176 W3.5-9.07 3:00p-3:50p Application of algebraic statistics for statistical disclosure limitation Aleksandra B. Slavković (Pennsylvania State University) EE/CS 3-180 W3.5-9.07 4:00p-4:30p Second Chances EE/CS 3-180 W3.5-9.07

## Wednesday, March 7

 8:30a-9:00a Coffee EE/CS 3-176 W3.5-9.07 9:00a-9:50a Polynomial dynamical systems over finite fields, with applications to modeling and simulation of biological networks Reinhard Laubenbacher (Virginia Polytechnic Institute and State University) EE/CS 3-180 W3.5-9.07 9:50a-10:30a Coffee W3.5-9.07 10:30a-11:20a Gene interactions and the geometry of fitness landscapes Niko Beerenwinkel (Harvard University) EE/CS 3-180 W3.5-9.07 11:30a-12:00p Second Chances EE/CS 3-180 W3.5-9.07 5:00p-6:30p Reception Lind Hall 400 W3.5-9.07 7:00p-8:00p Math matters - IMA public lecture: Patterns patterns everywhere Martin Golubitsky (University of Houston) Willey Hall 125 PUB3.07.07 7:00p-8:00p Math Matters - IMA Public Lecture: Patterns Patterns Everywhere Martin Golubitsky (University of Houston) Willey Hall 125 W3.5-9.07

## Thursday, March 8

 8:30a-9:00a Coffee EE/CS 3-176 W3.5-9.07 9:00a-9:50a Gaussian path diagrams Thomas S. Richardson (University of Washington) EE/CS 3-180 W3.5-9.07 9:50a-10:30a Coffee W3.5-9.07 10:30a-11:20a Using algebraic geometry for phylogenetic reconstruction Marta Casanellas (Polytechnical University of Cataluña (Barcelona)) EE/CS 3-180 W3.5-9.07 11:20a-1:30p Lunch W3.5-9.07 1:30p-2:20p Modelling with mass-action kinetics and beyond Markus Kirkilionis (University of Warwick) EE/CS 3-180 W3.5-9.07 2:20p-3:00p Coffee EE/CS 3-176 W3.5-9.07 3:00p-3:50p Information geometry and algebraic statistics Giovanni Pistone (Politecnico di Torino) EE/CS 3-180 W3.5-9.07 4:00p-4:30p Second Chances W3.5-9.07 6:30p-8:00p Dinner: Caspian Bistro Restaurant 2418 University Ave SE612-623-1133 W3.5-9.07

## Friday, March 9

 8:30a-9:00a Coffee EE/CS 3-176 W3.5-9.07 9:00a-9:50a On phylogenetic trees – a geometer's view Jaroslaw Wisniewski (University of Warsaw) EE/CS 3-180 W3.5-9.07 9:50a-10:30a Coffee W3.5-9.07 10:30a-11:20a A combinatorial test for significant codivergence between cool-season grasses and their symbiotic fungal endophytes Ruriko Yoshida (University of Kentucky) EE/CS 3-180 W3.5-9.07 11:20a-1:30p Lunch W3.5-9.07 1:30p-2:20p Subspace arrangements in theory and practice Robert M. Fossum (University of Illinois at Urbana-Champaign) EE/CS 3-180 W3.5-9.07 2:20p-3:00p Coffee EE/CS 3-176 W3.5-9.07 3:00p-3:50p Algebraic statistics and the analysis of contingency tables: Old wine in new bottles? Stephen E. Fienberg (Carnegie-Mellon University) EE/CS 3-180 W3.5-9.07 4:00p-4:30p Second Chances and closing remarks. W3.5-9.07

## Monday, March 12

 11:15a-12:15p Learning seminar on algebraic statistics Lind Hall 409 LSAS

## Tuesday, March 13

 11:15a-12:15p Laura Matusevich (Texas A&M) IMA postdoc seminar: Combinatorics of binomial primary decomposition Lind Hall 215 PS

## Wednesday, March 14

 11:15a-12:15p Algebraic geometry and applications seminar: Counting monomials Mordechai Katzman (University of Sheffield) EE/CS 3-180 AGS

## Thursday, March 15

 9:45a-10:45a Learning seminar on algebraic statistics Lind Hall 409 LSAS

## Friday, March 16

 All Day University of Minnesota Floating Holiday. The IMA is closed.

## Monday, March 19

 11:15a-12:15p Learning seminar on algebraic statistics Lind Hall 409 LSAS

## Tuesday, March 20

 11:15a-12:15p IMA postdoc seminar: Janet's algorithm for modules over polynomial rings Daniel Robertz (RWTH Aachen) Lind Hall 215 PS

## Wednesday, March 21

 11:15a-12:15p Algebraic geometry and applications seminar: Symmetries in SDP-based relaxations for constrained polynomial optimization Thorsten Theobald (Johann Wolfgang Goethe-Universität Frankfurt) EE/CS 3-180 AGS

## Thursday, March 22

 9:45a-10:45a Learning seminar on algebraic statistics Lind Hall 409 LSAS 11:15a-12:15p Real algebraic geometry tutorial: Real rings (continued) Kenneth R. Driessel (Iowa State University) Lind Hall 409 RAG

## Friday, March 23

 1:25p-2:25p IMA/MCIM Industrial problems seminar: Optical synthetic aperture imaging Joe Buck (Lockheed Martin Coherent Technologies ) Vincent Hall 1 IPS

## Monday, March 26

 11:15a-12:15p Learning seminar on algebraic statistics Lind Hall 409 LSAS

## Tuesday, March 27

 11:15a-12:15p IMA postdoc seminar: The minimum number of set-theoretic defining equations of algebraic varieties Gennady Lyubeznik (University of Minnesota Twin Cities) Lind Hall 215 PS

## Wednesday, March 28

 11:15a-12:15p Algebraic geometry and applications seminar: Algebraic geometry of Gaussian Bayesian networks Seth Sullivant (Harvard University) EE/CS 3-180 AGS

## Thursday, March 29

 9:45a-10:45a Learning seminar on algebraic statistics Lind Hall 409 LSAS 11:15a-12:15p Real algebraic geometry tutorial: Real rings (continued) Kenneth R. Driessel (Iowa State University) Lind Hall 409 RAG

## Friday, March 30

 1:25p-2:25p IMA/MCIM Industrial problems seminar: Downhole analysis of hydrocarbons Mariya Ponomarenko (Schlumberger-Doll Research) Vincent Hall 1 IPS
Abstracts
 Laura Matusevich (Texas A&M) IMA postdoc seminar: Combinatorics of binomial primary decomposition Abstract: Using examples, I will illustrate the main elements needed to explicitly describe the primary components of a binomial ideal, emphasizing the connections to combinatorics and (hypergeometric) differential equations. This is joint work with Alicia Dickenstein and Ezra Miller. Elizabeth S. Allman (University of Alaska) Phylogenetic models: algebra and evolution Abstract: Molecular phylogenetics is concerned with inferring evolutionary relationships (phylogenetic trees) from biological sequences (such as aligned DNA sequences for a gene shared by a collection of species). The probabilistic models of sequence evolution that underly statistical approaches in this field exhibit a rich algebraic structure. After an introduction to the inference problem and phylogenetic models, this talk will survey some of the highlights of current algebraic understanding. Results on the important statistical issue of identifiability of phylogenetic models will be emphasized, as the algebraic viewpoint has been crucial to obtaining such results. Niko Beerenwinkel (Harvard University) Gene interactions and the geometry of fitness landscapes Abstract: The relationship between the shape of a fitness landscape and the underlying gene interactions, or epistasis, has been extensively studied in the two-locus case. Epistasis has been linked to biological important properties such as the advantage of sex. Gene interactions among multiple loci are usually reduced to two-way interactions. Here, we present a geometric theory of shapes of fitness landscapes for multiple loci. We investigate the dynamics of evolving populations on fitness landscapes and the predictive power of the geometric shape for the speed of adaptation. Finally, we discuss applications to fitness data from viruses and bacteria. Joe Buck (Lockheed Martin Coherent Technologies ) IMA/MCIM Industrial problems seminar: Optical synthetic aperture imaging Abstract: The spatial resolution of a conventional imaging ladar system is constrained by the diffraction limit of the telescope's aperture. At Lockheed Martin Coherent Technologies (LMCT) we are implementing techniques known as synthetic-aperture imaging laser radar (SAIL), which employs aperture synthesis with coherent laser radar to overcome the diffraction limit and achieve fine-resolution, long-range, two-dimensional imaging with modest aperture diameters. I will discuss the results of my experiments while at The Aerospace Corporation which represent the first optical synthetic aperture images of a fixed, diffusely scattering target with moving aperture, as well as the current research program being developed at LMCT. Marta Casanellas (Polytechnical University of Cataluña (Barcelona)) Using algebraic geometry for phylogenetic reconstruction Abstract: Many statistical models of evolution can be viewed as algebraic varieties. The generators of the ideal associated to a model and a phylogenetic tree are called invariants. The invariants of an statistical model of evolution should allow to determine what is the tree formed by a set of living species. We will present a method of phylogenetic inference based on invariants and we will discuss why algebraic geometry should be considered as a powerful tool for phylogenetic reconstruction. The performance of the method has been studied for quartet trees and the Kimura 3-parameter model and it will be compared to widely known phylogenetic reconstruction methods such as Maximum likelihood estimate and Neighbor-Joining. Gheorghe Craciun (University of Wisconsin) Stability and instability in polynomial equations arising from complex chemical reaction networks: some underlying mathematics Abstract: Chemical reaction network models give rise to polynomial dynamical systems that are usually high dimensional, nonlinear, and have many unknown parameters. Due to the presence of these unknown parameters (such as reaction rate constants) direct numerical simulation of the chemical dynamics is practically impossible. On the other hand, we will show that important properties of these systems are determined only by the network structure, and do not depend on the unknown parameters. Also, we will show how some of these results can be generalized to systems of polynomial equations that are not necessarily derived from chemical kinetics. In particular, we will point out connections with classical problems in algebraic geometry, such as the real Jacobian conjecture. This talk describes joint work with Martin Feinberg, and can be regarded as a continuation of his earlier talk. Kenneth R. Driessel (Iowa State University) Real algebraic geometry tutorial: Real rings (continued) Abstract: Recall that we started to talk about real rings on March 1. The topic was new for us at that time. Also recall that a commutative ring with identity is "real" if the only representation of zero as the sum of squares is the trivial one. Since we did not meet during the last two weeks, I shall include an substantial review as part of our discussion. I shall mainly follow the material in the chapter "Real Rings" in the book "Positive Polynomials" by Prestel and Delzell. Our objective will be a proof of an absrtact version of the real Nullstellensatz. Mathias Drton (University of Chicago) Multiple solutions to the likelihood equations in the Behrens-Fisher problem Abstract: The Behrens-Fisher problem concerns testing the statistical hypothesis of equality of the means of two normal populations with possibly different variances. This problem furnishes one of the simplest statistical models for which the likelihood equations may have more than one real solution. In fact, with probability one, the equations have either one or three real solutions. Using the cubic discriminant, we study the large-sample probability of one versus three solutions. Nicholas Eriksson (Stanford University) Metric learning for phylogenetic invariants Abstract: We introduce new methods for phylogenetic tree construction by using machine learning to optimize the power of phylogenetic invariants. Phylogenetic invariants are polynomials in the joint probabilities which vanish under a model of evolution on a phylogenetic tree. We give algorithms for selecting a good set of invariants and for learning a metric on this set of invariants which optimally distinguishes the different models. Our learning algorithms involve semidefinite programming on data simulated over a wide range of parameters. Simulations on trees with four leaves under the Jukes-Cantor and Kimura 3-parameter models show that our method improves on other uses of invariants and is competitive with neighbor-joining. Our main biological result is that the trained invariants can perform substantially better than neighbor joining on quartet trees with short interior edges. This is joint work with Yuan Yao (Stanford). Martin Feinberg (Ohio State University) Stability and instability in polynomial equations arising from complex chemical reaction networks: the big picture Abstract: In nature there are millions of distinct networks of chemical reactions that might present themselves for study at one time or another. Written at the level of elementary reactions taken with classical mass action kinetics, each new network gives rise to its own (usually large) system of polynomial equations for the species concentrations. In this way, chemistry presents a huge and bewildering array of polynomial systems, each determined in a precise way by the underlying network up to parameter values (e.g., rate constants). Polynomial systems in general, even simple ones, are known to be rich sources of interesting and sometimes wild dynamical behavior. It would appear, then, that chemistry too should be a rich source of dynamical exotica. Yet there is a remarkable amount of stability in chemistry. Indeed, chemists and chemical engineers generally expect homogeneous isothermal reactors, even complex ones, to admit precisely one (globally attractive) equilibrium. Although this tacit doctrine is supported by a long observational record, there are certainly instances of homogeneous isothermal reactors that give rise, for example, to multiple equilibria. The vast landscape of chemical reaction networks, then, appears to have wide regions of intrinsic stability (regardless of parameter values) punctuated by far smaller regions in which instability might be extant (for at least certain parameter values). In this talk, I will present some recent joint work with Gheorghe Craciun that goes a long way toward explaining this landscape — in particular, toward explaining how biological chemistry "escapes" the stability doctrine to (literally) "make life interesting." A subsequent talk by Craciun will emphasize more mathematical detail. Stephen E. Fienberg (Carnegie-Mellon University) Algebraic statistics and the analysis of contingency tables: Old wine in new bottles? Abstract: The past decade has seen considerable interest in the reformulation of statistical models and methods for the analysis of contingency tables using the language and results of algebraic and polyhedral geometry. But as algebraic statistics has developed, new ideas have emerged that have changed how we view a number of statistical problems. This talk reviews some of these recent advances and suggests some challenges for collaborative research, especially those involving large scale databases. Robert M. Fossum (University of Illinois at Urbana-Champaign) Subspace arrangements in theory and practice Abstract: A subspace arrangement is a union of a finite number of subspaces of a vector space. We will discuss the importance of subspace arrangements first as mathematical objects and now as a popular class of models for engineering. We will then introduce some of new theoretical results that were motivated from practice. Using these results we will address the computational issue about how to extract subspace arrangements from noisy or corrupted data. Finally we will turn to the importance of subspace arrangements by briefly discussing the connections to sparse representations, manifold learning, etc... Martin Golubitsky (University of Houston) Math Matters - IMA Public Lecture: Patterns Patterns Everywhere Abstract: Regular patterns appear all around us: from vast geological formations to the ripples in a vibrating coffee cup, from the gaits of trotting horses to tongues of flames, and even in visual hallucinations. The mathematical notion of symmetry is a key to understanding how and why these patterns form. In this lecture Professor Golubitsky will show some of these fascinating patterns and explain how mathematical symmetry enters the picture. Mordechai Katzman (University of Sheffield) Algebraic geometry and applications seminar: Counting monomials Abstract: The contents of this elementary talk grew out of my need to explain to non-mathematicians what I do for a living. I will pose (and solve) two old chessboard enumeration problems and a new problem. We will solve these by counting certain monomials, and this will naturally lead us to the notion of Hilbert functions. With these examples in mind, we will try and understand the simplest of monomial ideals, namely, edge ideals, and discover that these are not simple at all! On the way we will discover a new numerical invariant of forests. Markus Kirkilionis (University of Warwick) Modelling with mass-action kinetics and beyond Abstract: Mass-action kinetics is a powerful tool to describe events created by collission of molecules or individuals in a well-mixed environment giving them locally the same probability to meet each other. Moreover this probability is only dependent on the concentration of the mutual partners. Mass action systems can be found in chemistry, cell biology, but also game theory and economics. Mathematically this gives rise to dynamical systems of a special type, more specific of polynomial type. I will give an overview how this property can be used to determine different types of bifurcations, for example the ocurrence of bistability, or oscillations via a Hopf bifurcation. All tools will be borrowing methods from algebraic geometry. Finally I will give an outlook what usually goes wrong in the modelling part while using mass-action kinetics if biochemical or cellular molecular events are considered. Finally the talk ends with a fresh look on mass-action kinetics applied to a spatial setting. Reinhard Laubenbacher (Virginia Polytechnic Institute and State University) Polynomial dynamical systems over finite fields, with applications to modeling and simulation of biological networks Abstract: Time-discrete dynamical systems with a finite state space have been used as models of biological systems since the use/invention of cellular automata by von Neumann in his attempt to model a self-replicating organism in the 1950s. More recently, they have appeared as models of a variety of biological systems, from gene regulatory networks to large-scale epidemiological networks. This talk will focus on theoretical and computational results about polynomial dynamical systems using tools from computational algebra and algebraic geometry. Gennady Lyubeznik (University of Minnesota Twin Cities) IMA postdoc seminar: The minimum number of set-theoretic defining equations of algebraic varieties Abstract: Given an algebraic variety in affine or projective space, what is the minimum number of equations that define this variety set-theoretically? This is a very difficult problem in general; no algorithm to compute this minimum number is currently known. We will discuss some techniques for upper and lower bounds and consider some interesting specific examples. František Matúš (Institute of Information Theory and Automation) Generalized maximum likelihood estimates for exponential families Abstract: Exponential families underpin numerous models of statistics and information geometry that have significant applications. For a standard full exponential family, or its canonically convex subfamily, if the corresponding likelihood function from a sample has a maximizer t* then, by the maximum likelihood principle, the data are judged to be generated by the probability measure P* from the family that is parameterized by t*. Since the likelihood depends on data only through their mean, in this way the mean is mapped to P*. In a joint work with Imre Csiszar, Budapest, we study an extension of this mapping, the generalized maximum likelihood estimator. It assigns to each point of the space at which the likelihood function is bounded above, a probability measure from the closure of the family in variation distance. A detailed description, complete characterization of domain and range, and additional results will be presented, not imposing any regularity assumptions. Jason Morton (University of California) Geometry of rank tests Abstract: We investigate the polyhedral geometry of conditional probability and undirected graphical models, developing new statistical procedures called convex rank tests. The polytope associated to an undirected graphical conditional independence model is the graph associahedron. The convex rank test defined by the dual semigraphoid to the n-cycle graphical model is applied to microarray data analysis to detect periodic gene expression. Sarah Olson (North Carolina State University) Supervised learning artifical neural network algorithms for optimizing mechanical properties of elastin-like polypeptide hydrogels for cartilage repair Abstract: Joint work with Dana L. Nettles3, Kimberly Trabbic Carlson3, Ashutosh Chilkoti3, Lori A. Setton3,4, Mansoor A. Haider1,2 Elastin-like polypeptide (ELP) hydrogels are a class of biomaterials that have potential utility as a biocompatible scaffold for filling defects due to osteoarthritis and for regenerating cartilage. Because of the facility to genetically engineer elastin sequence, there are almost endless possible configurations of ELPs and conformations of the networks formed after crosslinking. ELP biomaterial function will exhibit a complex dependence on these polymer characteristics that impacts properties expected to affect cartilage regeneration, such as mechanical load support. These complex structure-function relationships for crosslinked ELP hydrogels are not well described. A method for predicting the mechanical properties of ELP hydrogels was developed based on structural properties and Supervised Artificial Neural Network (ANN) modeling. The ANN Model used concentration, molecular weight, crosslink density, and sample number to predict the dynamic shear modulus and loss angle of the hydrogels. The ANN was implemented in a custom compiled code based on the Scaled Conjugate Gradient minimization algorithm and a Monte Carlo Method was used to expand the dataset. The ANN was trained using a varying subsets of the full dataset (22 formulations), with the complementary subset used for validation. Trained networks demonstrated excellent accuracy in prediction of hydrogel dynamic shear modulus at physiological temperature, based on polymer design and predictions were robust with respect to statistical variations. The results are used to show the validity of an intermediate screening process using ANNs to obtain the optimal mechanical properties for the ELP. 1 Biomathematics Graduate Program, North Carolina State University, 2 Department of Mathematics, North Carolina State University, 3 Department of Biomedical Engineering, Duke University, 4 Department of Surgery, Duke Medical Center Lior Pachter (University of California) From Drosophila and transposable elements to phylogenetic networks and associahedra Abstract: We begin with an overview of the Drosophila genome project, whose goal is the sequencing and comparison of 12 fruit fly genomes. In particular, we describe some of the dynamic behavior of transposable elements. These are self-replicating sequences that play a major role in shaping the structure and function of genomes. Our methods for studying transposable elements lead naturally to the analysis of split systems and their associated phylogenetic networks. We explain why the tessellation of \$\overline{M}_0^n\$ by associahedra is a natural candidate for the space of phylogenetic networks, and explain the relevance of this observation to the analysis of the popular neighbor-net algorithm used for studying split systems. We discuss various aspects of the neighbor-net algorithm, including its interpretation as a greedy algorithm for the traveling-salesman problem, how to obtain statistically meaningful parameters, and how to interpret its output. The application of neighbor-net to the split system we obtain from transposable elements in Drosophila reveals interesting insights about a set of species that may have undergone lineage sorting. This is joint work with Anat Caspi and Dan Levy. Sonja Petrovic (University of Kentucky) Toric ideals of phylogenetic invariants for the general group-based model on claw trees Abstract: We address the problem of studying the toric ideals of phylogenetic invariants for a general group-based model on an arbitrary claw tree. We focus on the group 2 and choose a natural recursive approach that extends to other groups. The study of the lattice associated with each phylogenetic ideal produces a list of circuits that generate the corresponding lattice basis ideal. In addition, we describe explicitly a quadratic lexicographic Gröbner basis of the toric ideal of invariants for the claw tree on an arbitrary number of leaves. Combined with a result of Sturmfels and Sullivant, this implies that the phylogenetic ideal of every tree for the group 2 has a quadratic Gröbner basis. Hence, the coordinate ring of the toric variety is a Koszul algebra. This is joint work with Julia Chifman, University of Kentucky. Giovanni Pistone (Politecnico di Torino) Information geometry and algebraic statistics Abstract: Recent presentations of Information Geometry (IG), e.g. Amari and Nagaoka (2000), consider general statistical models and general sample spaces. However, the seminal discussion by Cenkov (transl. 1982) is based on finite sample spaces, as it is in Algebraic Statistics (AS). This talk will first review basic IG from the point of view of AS. In the second part, it discusses the issue of computation IG quantities and presents a few examples. Mariya Ponomarenko (Schlumberger-Doll Research) IMA/MCIM Industrial problems seminar: Downhole analysis of hydrocarbons Abstract: Quick and accurate estimation of the composition of the hydrocarbon fluid in the formation is essential in assessing an oil reservoir value and determining optimal production strategies. This task is complicated by contamination from oil- and synthetic-based drilling mud filtrates. In this talk we will describe the visible - near-infrared spectroscopy technique to estimate the composition of formation fluid and level of contamination from the downhole optical absorption spectroscopy measurements. Thomas S. Richardson (University of Washington) Gaussian path diagrams Abstract: In the 1920's the geneticist Sewall Wright introduced a class of Gaussian statistical models represented by graphs containing directed and bi-directed edges, known as path diagrams. These models have been used extensively in psychometrics and econometrics where they are called structural equation models. I will first describe the subclass of bow-free acyclic path diagrams, which have desirable statistical properties. I will then characterize a subclass of models that are characterized by their Markov properties. Lastly I will outline recent work aimed at characterizing non-Markovian constraints that may arise. (This is joint work with Mathias Drton, Michael Eichler and Masashi Miyamura.) Daniel Robertz (RWTH Aachen) IMA postdoc seminar: Janet's algorithm for modules over polynomial rings Abstract: This talk gives an introduction to Janet bases. Originally developed for the algebraic analysis of systems of partial differential equations in the beginning of the 20th century, the algorithm by Maurice Janet is today an efficient alternative for Buchberger's algorithm to compute Gröbner bases of modules over polynomial rings. In this talk we give a modern description of Janet's algorithm and explain nice combinatorial properties of the resulting Janet bases: separation of the variables into multiplicative and non-multiplicative ones for each Janet basis element allows to read off vector space bases for both the submodule and the residue class module. As a consequence, the Hilbert series and polynomial of a (graded) module as well as a free resolution are easily obtained from the Janet basis. If time admits, some modifications of Janet's algorithm will be addressed which allow to work with polynomial rings over the integers instead of a field resp. generalize the algorithm to certain classes of non-commutative polynomial rings. Aleksandra B. Slavković (Pennsylvania State University) Application of algebraic statistics for statistical disclosure limitation Abstract: Statistical disclosure limitation applies statistical tools to the problems of limiting sensitive information releases about individuals and groups that are part of statistical databases while allowing for proper statistical inference. The limited releases can be in a form of arbitrary collections of marginal and conditional distributions, and odds ratios for contingency tables. Given this information, we discuss how tools from algebraic geometry can be used to give both complete and incomplete characterization of discrete distributions for contingency tables. These problems also lead to linear and non-linear integer optimization formulations. We discuss some practical implication, and challenges, of using algebraic statistics for data privacy and confidentiality problems. Bernd Sturmfels (University of California) Open problems in algebraic statistics Abstract: This talk introduces five or six mathematical problems whose solution would likely be a significant contribution to the emerging interactions between algebraic geometry, statistics, and computational biology. Seth Sullivant (Harvard University) Algebraic geometry and applications seminar: Algebraic geometry of Gaussian Bayesian networks Abstract: Conditional independence models for Gaussian random variables are algebraic varieties in the cone of positive definite matrices. We explore the geometry of these varieties in the case of Bayesian networks, with a view towards generalizing the recursive factorization theorem. When some of the random variables are hidden, non-independence constraints are need to describe the Bayesian networks. These non-independence constraints have potential inferential uses for studying collections of random variables. In the case that the underlying network is a tree, we give a complete description of the defining constraints of the model and show a surprising connection to the Grassmannian. Thorsten Theobald (Johann Wolfgang Goethe-Universität Frankfurt) Algebraic geometry and applications seminar: Symmetries in SDP-based relaxations for constrained polynomial optimization Abstract: We consider the issue of exploiting symmetries in the hierarchy of semidefinite programming relaxations recently introduced in polynomial optimization. After providing the necessary background we focus on problems where either the symmetric or the cyclic group is acting on the variables and extend the representation-theoretical methods of Gatermann and Parrilo to constrained polynomial optimization problems. Moreover, we also propose methods to efficiently compute lower and upper bounds for the subclass of problems where the objective function and the constraints are described in terms of power sums. (Joint work with L. Jansson, J.B. Lasserre and C. Riener) Jaroslaw Wisniewski (University of Warsaw) On phylogenetic trees – a geometer's view Abstract: I will discuss geometric methods of investigating phylogenetic trees. In a joint project with Weronika Buczynska we investigate projective varieties which are binary symmetric models of trivalent phylogenetic trees. They have Gorenstein terminal singularities and are Fano varieties. Moreover any two such varieties which are of the same dimension are deformation equivalent, that is, they are in the same connected component of the Hilbert scheme of the projective space whose coordinates are indexed by subsets of their leaves. Ruriko Yoshida (University of Kentucky) A combinatorial test for significant codivergence between cool-season grasses and their symbiotic fungal endophytes Abstract: Symbioses of grasses and fungal endophytes constitute an interesting model for evolution of mutualism and parasitism. Grasses of all subfamilies can harbor systemic infections by fungi of the family Clavicipitaceae. Subfamily Poöideae is specifically associated with epichloë endophytes (species of Epichloë and their asexual derivatives, the Neotyphodium species) in intimate symbioses often characterized by highly efficient vertical transmission in seeds, and bioprotective benefits conferred by the symbionts to their hosts. These remarkable symbioses have been identified in most grass tribes spanning the taxonomic range of the subfamily. Here we examine the possibility of codivergence in the phylogenetic histories of Poöideae and epichloë. We introduce a method of analysis to detect significant codivergence even in the absence of strict cospeciation, and to address problems in previously developed methods. Relative ages of corresponding cladogenesis events were determined from ultrametric maximum likelihood H (host) and P (parasite = symbiont) trees by an algorithm called MRCALink (most recent common ancestor link), an improvement over previous methods that greatly weight deep over shallow H and P node pairs. We then compared the inferred correspondence of MRCA ages in the H and P trees to the spaces of trees estimated from 10,000 randomly generated H and P tree pairs. Analysis of the complete dataset, which included a broad host-range species and some likely host transfers (jumps), did not indicate significant codivergence. However, when likely host jumps were removed the analysis indicated highly significant codivergence. Interestingly, early cladogenesis events in the Poöideae corresponded to early cladogenesis events in epichloë, suggesting concomitant origins of the Poöideae and this unusual symbiotic system. This is joint work with C. L. Schardl, K. D. Craven, A. Lindstrom, and A. Stromberg. Debbie Yuster (Columbia University) Classifying disease models using regular polyhedral subdivisions Abstract: Genes play a complicated role in how likely one is to get a certain disease. Biologists would like to model how one's genotype affects their likelihood of illness. We propose a new classification of two-locus disease models, where each model corresponds to an induced subdivision of a point configuration (basically a picture of connected dots). Our models reflect epistasis, or gene interaction. This work is joint with Ingileif Hallgrimsdottir. For more information, see our preprint at arXiv:q-bio.QM/0612044. Yi Zhou (Carnegie-Mellon University) Maximum likelihood estimation in latent class Abstract: Latent class models have been used to explain the heterogeneity of the observed relationship among a set of categorical variables and have received more and more attention as a powerful methodology for analyzing discrete data. The central goal of our work is to study the existence and computation of maximum likelihood estimates (MLEs) for these models, which are cardinal for assessment of goodness of fit and model selection. Our study is at the interface between the fields of algebraic statistics and machine learning. Traditionally, the expectation maximization (EM) algorithm has been applied to compute the MLEs of a latent class model. However, the solutions provided by the EM correspond to local maxima only, so, although we are able to compute them effectively, we still lack methods for assessing uniqueness and existence of the MLEs. Another interesting problem in statistics is the identifiability of the model. When a model is unidentifiable, it is necessary to adjust the number of degrees of freedom in order to apply correctly goodness-of-fit tests. In our work, we show that both the existence and identifiability problems are closely related to the geometric properties of the latent class models. Therefore, studying the algebraic varieties and ideals arising from these models is particularly relevant to our problem. We include a number of examples as a way of opening a discussion on a general method for addressing both MLE existence and identifiability in latent class models.
Visitors in Residence


text:
How math is builded up from sets

Tags:
1. Feb 6, 2015

Malna

Hi there!
I would like to understand the basic concepts of how mathematics is builded up from sets of elements to anything else (i.e: an equation)

For example here is a formula: 2-1
So are there different sets of elements like

A=(2) B=(-) C=(1) ?

And maybe there is an operation with sets what put the two, minus and the one together?

Recently I thought the function order two elements together, but I've just realized what function does is transform a set of elements to another one like from 2 it made 2-1
2->2-1 if that's right.

So I don't understand what put elements together

I would be really grateful is anybody could help me with this
Malna

2. Feb 6, 2015

Staff: Mentor

No, I don't think so. Ignoring any set theoretic notions for the time being, you have sets of numbers, and you have four arithmetic operations on them: addition, subtraction, multiplication, and division. This is the basis of arithmetic, which is taught in the early grades of school.

More advanced mathematics, such as abstract algebra, starts with the notion of a group, which is a set of elements of some kind, together with one operation, and a few axioms about the operation.

3. Feb 7, 2015

Fredrik

Staff Emeritus
The most common definition of $\mathbb R$ (the field of real numbers) involves a function from $\mathbb R$ into $\mathbb R$ and a function from $\mathbb R\times\mathbb R$ into $\mathbb R$ that are required to satisfy a bunch of conditions. I will denote the first function by - and the second one by +.

-x is just the conventional notation for -(x), i.e. the output produced by - when the input is x.

x+y is just the conventional notation for +(x,y), i.e. the output produced by + when you give it (x,y) as input.

This definition of $\mathbb R$ doesn't mention subtraction directly, but there's an almost obvious way to define it, using the functions mentioned above. It's the function $S:\mathbb R\times\mathbb R\to\mathbb R$ defined by S(x,y)=x+(-y) for all real numbers x,y. Then we can introduce the notational convention to write x-y instead of S(x,y).

x-y=x+(-y).

In this approach to real numbers, there are only two binary operations: addition and multiplication. There are also two unary operations (a function $x\mapsto -x$ from $\mathbb R$ into $\mathbb R$ and a function $x\mapsto\frac 1 x$ from $\mathbb R-\{0\}$ to $\mathbb R-\{0\}$). Subtraction is defined as described above. Division is defined similarly.

It's also possible to start with four binary operations (addition, subtraction, multiplication and division), and use them to define the two unary operations.

Last edited: Feb 7, 2015
4. Feb 7, 2015

Malna

5. Feb 13, 2015

onethatyawns

It makes more sense to me that you should start with something that can do everything, and then you find components of it that explain specific things, including sets, lines, circles, algebra, and calculus. Essentially, start with a general case, then make it specific. Unfortunately, schools never function this way. They always teach you special cases, forcing you to memorize numerous concepts that you haven't yet unified, and preventing you from really seeing the big picture.

The attempts to go from set theory to all math have failed. Bertrand Russell tried about 100 years ago, and then about 50 years ago they altered set theory so they could (using the zemelo-fraenkel set theory). A bit more recently, they created category theory, which was kind of a compromise to the previous rigidity. It seems like they said: 'there's no way to do this, so let's just get rid of our rules or make them so flexible such that we can force this to work'. That thought process, in my opinion, is the prevailing reason behind string theory and quantum theory (not discounting its uses, of course), but that's a whole separate topic.

Last edited: Feb 13, 2015
6. Feb 18, 2015

homeomorphic

Disagree. You can't see the point of the general case unless you have lots of examples at your disposal.

"Too many people write papers that are very abstract and at the end they may give some examples. It should be the other way around. You should start with understanding the interesting examples and build up to explain what the general phenomena are. This was you progress from initial understanding to more understanding. This is both for expository purposes and for mathematical purposes. It should always guide you."

--Michael Atiyah

I think my overarching concern here is that presenting things in complete generality is sort of dogmatic. I always think in terms of how I could have come up with the theory myself, in a perfect world. The inspiration usually comes from starting with the special cases. No one would think to do ring theory all of the sudden without anything coming before it. It's only because you have examples, like matrix rings or quaternions that you start caring about rings. Until you have a bunch of examples that you need to unify, so that you don't have to prove things separately for each of them, there's no reason to unify. Doing so would be dogmatic. Besides, proving things separately for different cases is not always bad. Sometimes, it's easier to see how to do it in the special case, so it's more motivated, and it adds extra repetition, if you are aware of it, which can also be a good thing for learning.

In this context, though, it's true that, pedagogically, it doesn't make sense to start with the foundations. You just work a little more informally, rather than trying to build everything from the ground up. But when you're ready, you can learn something about the nature of math by studying the foundations.

It wasn't exactly a failure. It is expressible in terms of Zermelo-Fraenkel axioms, at least in principle. What failed, as per Godel, was Hilbert's program of proving consistency and coming up with an effective procedure to find and prove all theorems.

Actually, category theory is motivated by algebraic topology (fundamental group, homology, etc.) and wasn't originally intended for foundational purposes. I don't really know much about the attempts to use it for that, though.

As to the original question, the way math reduces to set theory is roughly something like this. If you want to build the real numbers, you make them out of rational numbers, but you sort of fill in the holes, like pi or e or the square root of 2 that sort of need to be there (for example, the diagonal of a unit square has length equal to the square root of 2). I don't want to get into the details of that construction (actually, there are two equivalent ways: Dedekind cuts or completion), so let's just take for granted that if you give me the rational numbers, I can build the real numbers out of them.

Rational numbers can be reduced to pairs of integers. How? Just think of 4/5 as the ordered pair (4,5). When you do this, you have to keep in mind that there's some reduncy because 1/2 = 2/4. So the pair (1,2) should be identified with (2,4) and so on. And you have to define addition and all that and make sure that it's well-defined, so there are some more details. So, the upshot is that if you give me the integers, I can build you the rational numbers.

Similarly, if you give me the natural numbers, I can build you the integers. This time, again, you can look at pairs of natural numbers like (9,4) and (12, 18). They represent the difference of the pair of numbers, so (9,4) represents 4-9 = -5, and (12,18) represents 18-12 = 6. Again, you have some redundancy like (10, 5) also representing -5, and the details of defining addition and multiplication. But if you give me the natural numbers, I can build you the integers out of them.

And it's natural numbers that are built directly out of sets. Basically, it's just what you would think it would be. The number n is represented by a set with n elements in it. And what the Zermelo-Fraenkel axioms do is give you enough starting assumptions to be able to carry that out. You have to start with something. So, you take the empty set, which has 0 elements, then you have zero, you take a set with 0 in it for a set with 1 thing in it, then you can get {0,1}, which has 2 things in it, and so on. And the axioms just give you the tools to make this actually work. They may be unsatisfactory in some ways. I'm not a logician, so I don't know. But on some level, it does seem to work.

7. Feb 18, 2015

micromass

Staff Emeritus
No it hasn't. Where do you get this from.

8. Feb 18, 2015

onethatyawns

You have some good responses here. I can't address everything right now, so here's one nugget:

I have a simpler way of looking at addition. Imagine you have a unit circle with radius = 1. Place as many duplicate circles as you want in a row next to this circle, with all of the circles barely touching each other (technically, each circle only intersects at one point with each neighboring circle, and the intersection points if connected will make a straight line). Now, draw a circle around this row of circles, with the only intersection points being at the two most outside circles of the row, at one point on each circle. That radius will be equal to the radius of the sum of every circle's radius within the circle.

This also works if you have circles of various radii inside of the larger circle. They don't have to be duplicates.

Now, how about multiplication? All you do is raise this concept one order. For a x b, you have a=number of connected unit circles of r=b radii. Draw a larger circle around this, and you have a circle with the radius = a x b.

Same goes for exponents. You just raise multiplication as many orders as necessary to satisfy the exponent.

Subtraction, division, and logarithms are merely the inverse functions of the aforementioned concepts. You just need to understand those first three concepts, and then you can predict the inverse. We don't think of subtraction that way because it's so ingrained, but it's logically the same thing as practicing indefinite integrals by imaging the derivative of your solution.

There's only a couple main types of functions left: trig/inverse trig and derivative/antiderivative. I haven't 100% satisfied these two concepts with only circles yet, but I have ideas. I think trig is just a ratio of length (two lengths to be specific) to angle. Therefore, you have circles that can combine at various angles in that model, as apposed to my previous example of arithmetic where they have to be lined up in a row.

With all due respect to the people that don't like new theories on here, I'm not advancing anything new. This is geometry. I'm simply pointing out connections between geometry and other parts of math that people think are separate and less intuitive (but are not necessarily so, according to what I am saying).

I use circles rather than the elementary school version of this concept, which are squares, because circles have many more properties to use. Circles should be able to explain trig and derivatives, but squares fail my "intuition test" for absolute correctness in explaining those functions.

Last edited: Feb 18, 2015
9. Feb 19, 2015

micromass

Staff Emeritus
You are missing the point. Your definitions are not satisfying because it doesn't explain:
1) What a circle is exactly
4) How many circles you have to place for addition or multiplication

Now the ancient greeks would likely have accepted your arguments, since circles were seen as an object that exists in nature. But we have found out that it doesn't. We can never construct a perfect circle and the geometry of the universe is not euclidean. So we cannot use any concepts of euclidean geometry. Euclidean geometry is essentially an invention of mankind, and thus we cannot make sure that Euclidean geometry will give the correct concepts.

The entire crisis in set theory was beginning from recognizing that Euclidean geometry does not exist in nature, and thus we cannot know that it is consistent. This is why mathematicians like Dedekind and Peano based Euclidean geometry and arithmetic on natural numbers. And natural numbers are then based on sets. So we reach the conclusion that if sets are consistent, then so is basic arithmetic and Euclidean geometry. We cannot prove however that sets are consistent. But we did manage to base all (or >99.99% of math) on nothing else but sets. No other concepts are needed (such as geometrical concepts of circles).

10. Feb 19, 2015

homeomorphic

Couple points:

1) The goal of set theory is to formalize things, not make them intuitive. It's sort of a reductionist project in a way. You want to boil everything down to a few simple principles, out of which you can build everything else in a precise, formal way. As such, you are working at cross purposes towards set theory, so the fact that you might have a simpler way of looking at addition and everything else is beside the point.

2) I don't see the point of using circles or squares. If I think geometrically about addition, I think just sticking two line segments together. And multiplication can either be viewed in terms of the area of a rectangle or, for natural numbers, you make one copy of the second number for every unit of length in the first and string it all together. If you read book 5 of Euclid's Elements, he presents a treatment of ancient Greek number theory along these lines (and book 5 also deals with the ancient Greek concept of real numbers, which is very similar to Dedekind cuts--Dedekind credits them for his inspiration). But again, that's just an informal way of looking at it, which doesn't have too much to do with the purposes of set theory and the foundations of mathematics because the goal is totally different.

3) All the geometry you are talking about occurs within the Euclidean plane. Guess what the Euclidean plane is? It's R^2. So, the Euclidean plane can also be built out of real numbers, hence in terms of rationals....all the way down until you get to sets.

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# GMLAS - Geography Markup Language (GML) driven by application schemas¶

New in version 2.2.

Driver short name

GMLAS

This driver can read and write XML files of arbitrary structure, included those containing so called Complex Features, provided that they are accompanied by one or several XML schemas that describe the structure of their content. While this driver is generic to any XML schema, the main target is to be able to read and write documents referencing directly or indirectly to the GML namespace.

The driver requires Xerces-C >= 3.1.

The driver can deal with files of arbitrary size with a very modest RAM usage, due to its working in streaming mode.

## Driver capabilities¶

Supports Georeferencing

This driver supports georeferencing

Supports VirtualIO

This driver supports virtual I/O operations (/vsimem/, etc.)

## Opening syntax¶

The connection string is GMLAS:/path/to/the.gml. Note the GMLAS: prefix. If this prefix it is omitted, then the GML driver is likely to be used.

It is also possible to only used “GMLAS:” as the connection string, but in that case the schemas must be explicitly provided with the XSD open option.

## Mapping of XML structure to OGR layers and fields¶

The driver scans the XML schemas referenced by the XML/GML to build the OGR layers and fields. It is strictly required that the schemas, directly or indirectly used, are fully valid. The content of the XML/GML file itself is marginally used, mostly to determine the SRS of geometry columns.

XML elements declared at the top level of a schema will generally be exposed as OGR layers. Their attributes and sub-elements of simple XML types (string, integer, real, …) will be exposed as OGR fields. For sub-elements of complex type, different cases can happen. If the cardinality of the sub-element is at most one and it is not referenced by other elements, then it is “flattened” into its enclosing element. Otherwise it will be exposed as a OGR layer, with either a link to its “parent” layer if the sub-element is specific to its parent element, or through a junction table if the sub-element is shared by several parents.

By default the driver is robust to documents non strictly conforming to the schemas. Unexpected content in the document will be silently ignored, as well as content required by the schema and absent from the document.

Consult the GMLAS mapping examples page for more details.

By default in the configuration, swe:DataRecord and swe:DataArray elements from the Sensor Web Enablement (SWE) Common Data Model namespace will receive a special processing, so they are mapped more naturally to OGR concepts. The swe:field elements will be mapped as OGR fields, and the swe:values element of a swe:DataArray will be parsed into OGR features in a dedicated layer for each swe:DataArray. Note that those conveniency exposure is for read-only purpose. When using the write side of the driver, only the content of the general mapping mechanisms will be used.

Three special layers “_ogr_fields_metadata”, “_ogr_layers_metadata”, “_ogr_layer_relationships” and “_ogr_other_metadata” add extra information to the basic ones you can get from the OGR data model on OGR layers and fields.

Those layers are exposed if the EXPOSE_METADATA_LAYERS open option is set to YES (or if enabled in the configuration). They can also be individually retrieved by specifying their name in calls to GetLayerByName(), or on as layer names with the ogrinfo and ogr2ogr utility.

Consult the GMLAS metadata layers page for more details.

## Configuration file¶

A default configuration file gmlasconf.xml file is provided in the data directory of the GDAL installation. Its structure and content is documented in gmlasconf.xsd schema.

This configuration file enables the user to modify the following settings:

• whether the local cache of schemas is enabled. Enabled by default.

• the path of the local cache. By default, \$HOME/.gdal/gmlas_xsd_cache

• whether validation of the document against the schemas should be enabled. Disabled by default.

• whether validation error should cause dataset opening to fail. Disabled by default.

• whether the metadata layers should be exposed by default. Disabled by default.

• whether a ‘ogr_pkid’ field should always be generated. Disabled by default. Turning that on can be useful on layers that have a ID attribute whose uniqueness is not guaranteed among various documents. Which could cause issues when appending several documents into a target database table.

• whether layers and fields that are not used in the XML document should be removed. Disable by default.

• whether OGR array data types can be used. Enabled by default.

• whether the XML definition of the GML geometry should be reported as a OGR string field. Disabled by default.

• whether only XML elements that derive from gml:_Feature or gml:AbstractFeature should be considered in the initial pass of the schema building, when at least one element in the schemas derive from them. Enabled by default.

• several rules to configure if and how xlink:href should be resolved.

• a definition of XPaths of elements and attributes that must be ignored, so as to lighten the number of OGR layers and fields.

This file can be adapted and modified versions can be provided to the driver with the CONFIG_FILE open option. None of the elements of the configuration file are required. When they are absent, the default value indicated in the schema documentation is used.

Configuration can also be provided through other open options. Note that some open options have identical names to settings present in the configuration file. When such open option is provided, then its value will override the one of the configuration file (either the default one, or the one provided through the CONFIG_FILE open option).

## Geometry support¶

XML schemas only indicate the geometry type but do not constraint the spatial reference systems (SRS), so it is theoretically possible to have object instances of the same class having different SRS for the same geometry field. This is not practical to deal with, so when geometry fields are detected, an initial scan of the document is done to find the first geometry of each geometry field that has an explicit srsName set. This one will be used for the whole geometry field. In case other geometries of the same field would have different SRS, they will be reprojected.

By default, only the OGR geometry built from the GML geometry is exposed in the OGR feature. It is possible to change the IncludeGeometryXML setting of the configuration file to true so as to expose a OGR string field with the XML definition of the GML geometry.

## Performance issues with large multi-layer GML files.¶

Traditionnaly to read a OGR datasource, one iterate over layers with GDALDataset::GetLayer(), and for each layer one iterate over features with OGRLayer::GetNextFeature(). While this approach still works for the GMLAS driver, it may result in very poor performance on big documents or documents using complex schemas that are translated in many OGR layers.

It is thus recommended to use GDALDataset::GetNextFeature() to iterate over features as soon as they appear in the .gml/.xml file. This may return features from non-sequential layers, when the features include nested elements.

## Open options¶

• XSD=filename(s): to specify an explicit XSD application schema to use (or a list of filenames, provided they are comma separated). “http://” or “https://” URLs can be used. This option is not required when the XML/GML document has a schemaLocation attribute with valid links in its root element.

• CONFIG_FILE=filename or inline XML definition: filename of a XML configuration file conforming to the gmlasconf.xsd schema. It is also possible to provide the XML content directly inlined provided that the very first characters are <Configuration.

• VALIDATE=YES/NO: whether the document should be validated against the schemas. Validation is done at dataset opening. Default is NO.

• FAIL_IF_VALIDATION_ERROR=YES/NO: Whether a validation error should cause dataset opening to fail. (only used if VALIDATE=YES) Default is NO.

• REFRESH_CACHE=YES/NO: Whether remote schemas and documents pointed by xlink:href links should be downloaded from the server even if already present in the local cache. If the cache is enabled, it will be refreshed with the newly downloaded resources. Default is NO.

• SWAP_COORDINATES=AUTO/YES/NO: Whether the order of the x/y or long/lat coordinates should be swapped. In AUTO mode, the driver will determine if swapping must be done from the srsName. If the srsName is urn:ogc:def:crs:EPSG::XXXX and that the order of coordinates in the EPSG database for this SRS is lat,long or northing,easting, then the driver will swap them to the GIS friendly order (long,lat or easting,northing). For other forms of SRS (such as EPSG:XXXX), GIS friendly order is assumed and thus no swapping is done. When SWAP_COORDINATES is set to YES, coordinates will be always swapped regarding the order they appear in the GML, and when it set to NO, they will be kept in the same order. The default is AUTO.

• REMOVE_UNUSED_LAYERS=YES/NO: Whether unused layers should be removed from the reported layers. Defaults to NO

• REMOVE_UNUSED_FIELDS=YES/NO: Whether unused fields should be removed from the reported layers. Defaults to NO

• HANDLE_MULTIPLE_IMPORTS=YES/NO: Whether multiple imports with the same namespace but different schema are allowed. Defaults to NO

• SCHEMA_FULL_CHECKING=YES/NO: Whether to be pedantic with XSD checking or to be forgiving e.g. if the invalid part of the schema is not referenced in the main document. Defaults to NO

## Creation support¶

The GMLAS driver can write XML documents in a schema-driven way by converting a source dataset (contrary to most other drivers that have read support that implement the CreateLayer() and CreateFeature() interfaces). The typical workflow is to use the read side of the GMLAS driver to produce a SQLite/Spatialite/ PostGIS database, potentially modify the features imported and re-export this database as a new XML document.

The driver will identify in the source dataset “top-level” layers, and in those layers will find which features are not referenced by other top-level layers. As the creation of the output XML is schema-driver, the schemas need to be available. There are two possible ways:

• either the result of the processing of the schemas was stored as the 4 _ogr_* metadata tables in the source dataset by using the EXPOSE_METADATA_LAYERS=YES open option when converting the source .xml),

• or the schemas can be specified at creation time with the INPUT_XSD creation option.

By default, the driver will “wrap” the features inside a WFS 2.0 wfs:FeatureCollection / wfs:member element. It is also possible to ask the driver to create instead a custom wrapping .xsd file that declares the ogr_gmlas:FeatureCollection / ogr_gmlas:featureMember XML elements.

Note that while the file resulting from the export should be XML valid, there is no strong guarantee that it will validate against the additional constraints expressed in XML schema(s). This will depend on the content of the features (for example if converting from a GML file that is not conformant to the schemas, the output of the driver will generally be not validating)

If the input layers have geometries stored as GML content in a _xml suffixed field, then the driver will compare the OGR geometry built from that XML content with the OGR geometry stored in the dedicated geometry field of the feature. If both match, then the GML content stored in the _xml suffixed field will be used, such as to preserve particularities of the initial GML content. Otherwise GML will be exported from the OGR geometry.

To increase export performance on very large databases, creating attribute indexes on the fields pointed by the ‘layer_pkid_name’ attribute in ‘_ogr_layers_metadata’ might help.

### ogr2ogr behaviour¶

When using ogr2ogr / GDALVectorTranslate() to convert to XML/GML from a source database, there are restrictions to the options that can be used. Only the following options of ogr2ogr are supported:

• dataset creation options (see below)

• layer names

• spatial filter through -spat option.

• attribute filter through -where option

The effect of spatial and attribute filtering will only apply on top-levels layers. Sub-features selected through joins will not be affected by those filters.

### Dataset creation options¶

The supported dataset creation options are:

• INPUT_XSD=filename(s): to specify an explicit XSD application schema to use (or a list of filenames, provided they are comma separated). “http://” or “https://” URLs can be used. This option is not required when the source dataset has a _ogr_other_metadata with schemas and locations filled.

• CONFIG_FILE=filename or inline XML definition: filename of a XML configuration file conforming to the gmlasconf.xsd schema. It is also possible to provide the XML content directly inlined provided that the very first characters are <Configuration>.

• LAYERS=layers. Comma separated list of layers to export as top-level features. The special value “{SPATIAL_LAYERS}” can also be used to specify all layers that have geometries. When LAYERS is not specified, the driver will identify in the source dataset “top-level” layers, and in those layers will find which features are not referenced by other top-level layers.

• SRSNAME_FORMAT=SHORT/OGC_URN/OGC_URL (Only valid for GML 3 output) Defaults to OGC_URL. If SHORT, then srsName will be in the form AUTHORITY_NAME:AUTHORITY_CODE If OGC_URN, then srsName will be in the form urn:ogc:def:crs:AUTHORITY_NAME::AUTHORITY_CODE If OGC_URL, then srsName will be in the form http://www.opengis.net/def/crs/AUTHORITY_NAME/0/AUTHORITY_CODE For OGC_URN and OGC_URL, in the case the SRS is a SRS without explicit AXIS order, but that the same SRS authority code imported with ImportFromEPSGA() should be treated as lat/long or northing/easting, then the function will take care of coordinate order swapping.

• INDENT_SIZE=[0-8]. Number of spaces for each indentation level. Default is 2.

• COMMENT=string. Comment to add at top of generated XML file as a XML comment.

• LINEFORMAT=CRLF/LF. End-of-line sequence to use. Defaults to CRLF on Windows and LF on other platforms.

• WRAPPING=WFS2_FEATURECOLLECTION/GMLAS_FEATURECOLLECTION. Whether to wrap features in a wfs:FeatureCollection or in a ogr_gmlas:FeatureCollection. Defaults to WFS2_FEATURECOLLECTION.

• TIMESTAMP=XML date time. User-specified XML dateTime value for timestamp to use in wfs:FeatureCollection attribute. If not specified, current date time is used. Only valid for WRAPPING=WFS2_FEATURECOLLECTION.

• WFS20_SCHEMALOCATION=Path or URL to wfs.xsd. Only valid for WRAPPING=WFS2_FEATURECOLLECTION. Default is “http://schemas.opengis.net/wfs/2.0/wfs.xsd

• GENERATE_XSD=YES/NO. Whether to generate a .xsd file that has the structure of the wrapping ogr_gmlas:FeatureCollection / ogr_gmlas:featureMember elements. Only valid for WRAPPING=GMLAS_FEATURECOLLECTION. Default to YES.

• OUTPUT_XSD_FILENAME=string. Wrapping .xsd filename. If not specified, same basename as output file with .xsd extension. Note that it is possible to use this option even if GENERATE_XSD=NO, so that the wrapping .xsd appear in the schemaLocation attribute of the .xml file. Only valid for WRAPPING=GMLAS_FEATURECOLLECTION

## Examples¶

Listing content of a data file:

ogrinfo -ro GMLAS:my.gml

Converting to PostGIS:

ogr2ogr -f PostgreSQL PG:'host=myserver dbname=warmerda' GMLAS:my.gml -nlt CONVERT_TO_LINEAR

Converting to Spatialite and back to GML

ogr2ogr -f SQLite tmp.sqlite GMLAS:in.gml -dsco SPATILIATE=YES -nlt CONVERT_TO_LINEAR -oo EXPOSE_METADATA_LAYERS=YES
ogr2ogr -f GMLAS out.gml tmp.sqlite


text:
Converted document
$\newcommand{\lyxlock}{}$
Subsubsection 3.11.2.6: Rigid Lock Joint Up Subsection 3.11.2: Rigid Connectors Subsubsection 3.11.2.8: Rigid Damper

3.11.2.7 Rigid Spring

The rigid spring applies a linear spring that connects two rigid bodies and at arbitrary points (not necessarily nodes).
In addition to the shared parameters above, it defines the following parameters.
 parameter Description k spring constant (in units of force per length) insertion_a Insertion point of the spring on body a insertion_b Insection point of the spring on body b free_length The length of the unloaded spring (1)
Comments:
1. If the free_length parameter is zero, then the initial length of the spring, as defined by the two insertion points, is taken as the free length.
Example:
<constraint type="rigid spring">
<body_a>1</body_a>
<body_b>2</body_b>
<insertion_a>0,0,1</insertion_a>
<insertion_b>0,0,3</insertion_b>
<k>1</k>
</constraint>

Subsubsection 3.11.2.6: Rigid Lock Joint Up Subsection 3.11.2: Rigid Connectors Subsubsection 3.11.2.8: Rigid Damper


text:
Results (1-50 of at least 1000)

Next
Label $\alpha$ $A$ $d$ $N$ $\chi$ $\mu$ $\nu$ $w$ prim arith $\mathbb{Q}$ self-dual $\operatorname{Arg}(\epsilon)$ $r$ First zero Origin
2-536-536.171-c0-0-0 $0.517$ $0.267$ $2$ $2^{3} \cdot 67$ 536.171 $$0.0 0 -0.0997 0 0.638299 Artin representation 2.536.6t5.b.b Modular form 536.1.o.a.171.1 2-536-536.155-c0-0-0 0.517 0.267 2 2^{3} \cdot 67 536.155$$ $0.0$ $0$ $-0.174$ $0$ $1.53947$ Modular form 536.1.ba.a.155.1
2-536-536.483-c0-0-0 $0.517$ $0.267$ $2$ $2^{3} \cdot 67$ 536.483 $$0.0 0 -0.281 0 0.775036 Modular form 536.1.t.a.483.1 2-536-536.475-c0-0-0 0.517 0.267 2 2^{3} \cdot 67 536.475$$ $0.0$ $0$ $-0.159$ $0$ $1.44849$ Modular form 536.1.ba.a.475.1
2-536-536.19-c0-0-0 $0.517$ $0.267$ $2$ $2^{3} \cdot 67$ 536.19 $$0.0 0 -0.270 0 0.471914 Modular form 536.1.ba.a.19.1 2-536-536.427-c0-0-0 0.517 0.267 2 2^{3} \cdot 67 536.427$$ $0.0$ $0$ $0.0264$ $0$ $1.15307$ Modular form 536.1.t.a.427.1
2-536-536.133-c0-0-4 $0.517$ $0.267$ $2$ $2^{3} \cdot 67$ 536.133 $$0.0 0 0 0 2.24411 Artin representation 2.536.7t2.a.b Modular form 536.1.h.b.133.2 2-536-536.339-c0-0-0 0.517 0.267 2 2^{3} \cdot 67 536.339$$ $0.0$ $0$ $0.0299$ $0$ $1.67418$ Modular form 536.1.ba.a.339.1
2-536-536.83-c0-0-0 $0.517$ $0.267$ $2$ $2^{3} \cdot 67$ 536.83 $$0.0 0 0.174 0 2.47084 Modular form 536.1.ba.a.83.1 2-536-536.315-c0-0-0 0.517 0.267 2 2^{3} \cdot 67 536.315$$ $0.0$ $0$ $0.263$ $0$ $2.65059$ Modular form 536.1.ba.a.315.1
2-536-536.435-c0-0-0 $0.517$ $0.267$ $2$ $2^{3} \cdot 67$ 536.435 $$0.0 0 -0.129 0 2.10281 Modular form 536.1.ba.a.435.1 2-536-536.35-c0-0-0 0.517 0.267 2 2^{3} \cdot 67 536.35$$ $0.0$ $0$ $-0.461$ $0$ $3.47338$ Modular form 536.1.ba.a.35.1
2-536-536.491-c0-0-0 $0.517$ $0.267$ $2$ $2^{3} \cdot 67$ 536.491 $$0.0 0 -0.307 0 1.06229 Modular form 536.1.t.a.491.1 2-536-536.531-c0-0-0 0.517 0.267 2 2^{3} \cdot 67 536.531$$ $0.0$ $0$ $-0.397$ $0$ $1.11566$ Modular form 536.1.t.a.531.1
2-536-536.411-c0-0-0 $0.517$ $0.267$ $2$ $2^{3} \cdot 67$ 536.411 $$0.0 0 -0.153 0 1.87483 Modular form 536.1.t.a.411.1 2-536-536.123-c0-0-0 0.517 0.267 2 2^{3} \cdot 67 536.123$$ $0.0$ $0$ $0.159$ $0$ $1.58151$ Modular form 536.1.ba.a.123.1
2-536-536.91-c0-0-0 $0.517$ $0.267$ $2$ $2^{3} \cdot 67$ 536.91 $$0.0 0 0.281 0 1.80226 Modular form 536.1.t.a.91.1 2-536-536.227-c0-0-0 0.517 0.267 2 2^{3} \cdot 67 536.227$$ $0.0$ $0$ $-0.104$ $0$ $2.15532$ Modular form 536.1.ba.a.227.1
2-536-536.323-c0-0-0 $0.517$ $0.267$ $2$ $2^{3} \cdot 67$ 536.323 $$0.0 0 -0.255 0 1.41551 Modular form 536.1.ba.a.323.1 2-536-536.131-c0-0-0 0.517 0.267 2 2^{3} \cdot 67 536.131$$ $0.0$ $0$ $0.307$ $0$ $2.10136$ Modular form 536.1.t.a.131.1
2-536-536.133-c0-0-5 $0.517$ $0.267$ $2$ $2^{3} \cdot 67$ 536.133 $$0.0 0 0 0 2.42865 Artin representation 2.536.7t2.a.c Modular form 536.1.h.b.133.3 2-536-536.371-c0-0-0 0.517 0.267 2 2^{3} \cdot 67 536.371$$ $0.0$ $0$ $-0.129$ $0$ $1.54038$ Modular form 536.1.ba.a.371.1
2-536-536.211-c0-0-0 $0.517$ $0.267$ $2$ $2^{3} \cdot 67$ 536.211 $$0.0 0 -0.263 0 1.64022 Modular form 536.1.ba.a.211.1 2-536-536.133-c0-0-2 0.517 0.267 2 2^{3} \cdot 67 536.133$$ $0.0$ $0$ $0$ $0$ $1.54283$ Artin representation 2.536.7t2.a.a Modular form 536.1.h.b.133.1
2-536-536.133-c0-0-0 $0.517$ $0.267$ $2$ $2^{3} \cdot 67$ 536.133 $$0.0 0 0 0 1.36077 Modular form 536.1.h.a.133.2 2-536-536.467-c0-0-0 0.517 0.267 2 2^{3} \cdot 67 536.467$$ $0.0$ $0$ $0.129$ $0$ $2.06538$ Modular form 536.1.ba.a.467.1
2-536-536.419-c0-0-0 $0.517$ $0.267$ $2$ $2^{3} \cdot 67$ 536.419 $$0.0 0 -0.0299 0 1.40486 Modular form 536.1.ba.a.419.1 2-536-536.107-c0-0-0 0.517 0.267 2 2^{3} \cdot 67 536.107$$ $0.0$ $0$ $0.397$ $0$ $2.47055$ Modular form 536.1.t.a.107.1
2-536-536.307-c0-0-0 $0.517$ $0.267$ $2$ $2^{3} \cdot 67$ 536.307 $$0.0 0 0.255 0 2.59319 Modular form 536.1.ba.a.307.1 2-536-536.133-c0-0-3 0.517 0.267 2 2^{3} \cdot 67 536.133$$ $0.0$ $0$ $0$ $0$ $2.18569$ Modular form 536.1.h.a.133.3
2-536-536.133-c0-0-1 $0.517$ $0.267$ $2$ $2^{3} \cdot 67$ 536.133 $$0.0 0 0 0 1.37045 Modular form 536.1.h.a.133.1 2-536-536.451-c0-0-0 0.517 0.267 2 2^{3} \cdot 67 536.451$$ $0.0$ $0$ $0.104$ $0$ $2.44789$ Modular form 536.1.ba.a.451.1
2-536-536.395-c0-0-0 $0.517$ $0.267$ $2$ $2^{3} \cdot 67$ 536.395 $$0.0 0 0.270 0 1.60152 Modular form 536.1.ba.a.395.1 2-536-536.163-c0-0-0 0.517 0.267 2 2^{3} \cdot 67 536.163$$ $0.0$ $0$ $0.0997$ $0$ $1.68280$ Artin representation 2.536.6t5.b.a Modular form 536.1.o.a.163.1
2-536-536.283-c0-0-0 $0.517$ $0.267$ $2$ $2^{3} \cdot 67$ 536.283 $$0.0 0 0.153 0 2.46488 Modular form 536.1.t.a.283.1 2-536-536.523-c0-0-0 0.517 0.267 2 2^{3} \cdot 67 536.523$$ $0.0$ $0$ $0.129$ $0$ $2.35190$ Modular form 536.1.ba.a.523.1
2-536-536.59-c0-0-0 $0.517$ $0.267$ $2$ $2^{3} \cdot 67$ 536.59 $$0.0 0 -0.0264 0 1.58534 Modular form 536.1.t.a.59.1 2-536-536.291-c0-0-0 0.517 0.267 2 2^{3} \cdot 67 536.291$$ $0.0$ $0$ $0.461$ $0$ $1.05623$ Modular form 536.1.ba.a.291.1
2-536-536.21-c1-0-49 $2.06$ $4.27$ $2$ $2^{3} \cdot 67$ 536.21 $$1.0 1 0.0511 0 1.99931 Modular form 536.2.bf.a.21.65 2-536-536.21-c1-0-46 2.06 4.27 2 2^{3} \cdot 67 536.21$$ $1.0$ $1$ $0.385$ $0$ $1.91527$ Modular form 536.2.bf.a.21.34
2-536-536.21-c1-0-5 $2.06$ $4.27$ $2$ $2^{3} \cdot 67$ 536.21 $$1.0 1 0.398 0 0.297515 Modular form 536.2.bf.a.21.19 2-536-536.21-c1-0-43 2.06 4.27 2 2^{3} \cdot 67 536.21$$ $1.0$ $1$ $0.273$ $0$ $1.82527$ Modular form 536.2.bf.a.21.7
2-536-536.21-c1-0-41 $2.06$ $4.27$ $2$ $2^{3} \cdot 67$ 536.21 $$1.0 1 0.125 0 1.81768 Modular form 536.2.bf.a.21.52 2-536-536.21-c1-0-44 2.06 4.27 2 2^{3} \cdot 67 536.21$$ $1.0$ $1$ $0.400$ $0$ $1.83474$ Modular form 536.2.bf.a.21.30
2-536-536.21-c1-0-50 $2.06$ $4.27$ $2$ $2^{3} \cdot 67$ 536.21 $$1.0 1 0.279 0 2.00080 Modular form 536.2.bf.a.21.42 2-536-536.21-c1-0-38 2.06 4.27 2 2^{3} \cdot 67 536.21$$ $1.0$ $1$ $0.135$ $0$ $1.62283$ Modular form 536.2.bf.a.21.29
2-536-536.21-c1-0-35 $2.06$ $4.27$ $2$ $2^{3} \cdot 67$ 536.21 $$1.0 1 -0.241 0 1.56918 Modular form 536.2.bf.a.21.57 2-536-536.21-c1-0-39 2.06 4.27 2 2^{3} \cdot 67 536.21$$ $1.0$ $1$ $-0.0497$ $0$ $1.76158$ Modular form 536.2.bf.a.21.54
2-536-536.21-c1-0-31 $2.06$ $4.27$ $2$ $2^{3} \cdot 67$ 536.21 $$1.0 1 -0.0730 0 1.22577 Modular form 536.2.bf.a.21.28 2-536-536.205-c1-0-42 2.06 4.27 2 2^{3} \cdot 67 536.205$$ $1.0$ $1$ $-0.162$ $0$ $1.53309$ Modular form 536.2.bf.a.205.61


text:
## linear distribution isotherm

in chromatography
https://doi.org/10.1351/goldbook.LT06935
A @DT06932-1@ which can be approximated as $$c_{\text{s}} = K_{\text{C}}.c_{\text{M}}$$ where $$K_{\text{C}}$$ is a constant.
Note:
At low concentrations, all @DT06932-2@ tend towards being linear. $$K_{\text{C}}$$ is the @D01813@.
Source:
PAC, 1996, 68, 1591. (Nomenclature for non-linear chromatography (IUPAC Recommendations 1996)) on page 1593 [Terms] [Paper]


text:
# Enemy movement in straight line to player in chase game

I'm creating a game where the enemies spawn randomly on a map then move towards the player every frame at a random speed. The map has no obstacles so the enemies should always move in a straight line. I wrote the movement function a few times but no matter what the enemies always hit 0, 45, 90, 135, 180, 225, 270, 315 angles but never a straight line. Here is an example of the code:

    base_speed = random();

diff_x = abs(enemy_y_pos - player_x_pos);
diff_y = abs(enemy_x_pos - player_y_pos);
if (diff_x > diff_y) {
y_speed = base_speed;
} else if (diff_y > diff_x) {
x_speed = base_speed;
}

if (enemy_x_pos < player_x_pos) {
velocity.x = x_speed;
} else if (enemy_x_pos > player_x_pos) {
velocity.x = -x_speed;
} else {
velocity.x = 0;
}

if (enemy_y_pos < player_y_pos) {
velocity.y = y_speed;
} else if (enemy_y_pos > player_y_pos) {
velocity.y = -y_speed;
} else {
velocity.y = 0;
}

enemy_x_pos = enemy_x_pos + velocity.x;
enemy_y_pos = enemy_y_pos + velocity.y;

This is my first attempt at game programming. I guessing it's should use algorithm like Bresenham’s Line( http://en.wikipedia.org/wiki/Bresenham%27s_line_algorithm) but my attempts to implement have the same issue. How do I make enemies move in a straight line?

Bresenham's line algorithm was developed for fast drawing lines without using floating point operations (integer operations are faster). It is very interesting historicaly but not very good solution for a game (especialy because you cannot specify speed).

When doing a 2D movement use always vectors. This will do all the stuff for you. Sometimes is algebra just wonderful.

Vec2d playerPos;
Vec2d direction; // always normalized
float velocity;

update()
{
direction = normalize(playerPos - enemyPos);
playerPos = playerPos + direction * velocity;
}

There will be some vector2D class for your target language for sure.

• That worked perfectly. Normalize was the function I needed. Thanks. Jan 21, 2011 at 0:24

Are you calculating every frame? Your code sets either x_speed or y_speed to a random number and the other to zero (or the default). Let's say the enemy is at 0, 0 and player is at 20, 10. This looks like it will set x_speed to some random number each frame and move the enemy to the right until he is at 10, 0. Then the enemy will move either x or y each frame so it will seem he moves in a 45 degree angle up to 10, 10.

Notabene's vector explanation is right but here's how it works. The Vec2d will have an x and y component. In the example above, playerPos - enemyPos will give you a vector of (20, 10). normalize() changes the vector so that the length is 1. Currently the length is sqrt(20 * 20 + 10 * 10). That is about 22.36068. normalize divides both X and Y by this number, giving you about (0.89443, 0.44721) for the direction, which is a vector pointing from the enemy to the player with a length of 1. Multiplying this by your random speed does what you want.

However I would not calculate a new random speed each frame. You probably just want to set the random speed once, or have an initial speed and a target speed that you get closer to each frame. It doesn't make sense for your enemy to go 100 mph initially, then 50 mph for one frame and 200 mph for the next.

float targetSpeed = random(); // do this only rarely, or each frame if you don't mind slowing down and speeding up constantly

float acceleration = max(abs(targetSpeed - speed), 0.5) * sgn(targetSpeed - speed); // cap accel/decel at 0.5 per frame
speed = speed + acceleration;

• Nice explanations and additions Jason, thanks. I like yours acceleration function Jan 20, 2011 at 15:25
• random() in that scope was a typo. Your explanation and code with what notabene wrote help me get the game working. Thanks. Jan 21, 2011 at 0:27