Algebraic Computations on Noisy, Measured Data

Size: px
Start display at page:

Download "Algebraic Computations on Noisy, Measured Data"

Transcription

1 Algebraic Computations on Noisy, Measured Data 3/1/06 File Title Copyright: Shell Exploration & Production Ltd. Daniel Heldt, Sebastian Pokutta, Hennie Poulisse Based on Joint On-Going Work: Daniel Heldt,, Martin Kreuzer,, Sebastian Pokutta, Hennie Poulisse

2 Contents Industrial Application of Computer Algebra ApVI Calculated

3 Going out into the Field! The Champions Field, South-Chinese Sea, offshore Brunei

4 transportation tubing well heads headers (test- and bulk) separators

5 Production Relation for an Oil Well WELL MEASUREMENTS Surface and Sub-surface Gas Selected Well on Test THP, DHP, THT, FLP, LGF etc. Test Separator (3 Phase) TEST SEPARATOR MEASUREMENTS imitating production circumstances: deliberate disturbances Oil Water inputs outputs variables of polynomial function output = function of inputs output = polynomial function input physical relation between inputs

6 example of a polynomial well production low degree, large number of indeterminates

7 Motivating Example Gas production well A production well B production well C transportation Line Bulk Separator (3 Phase) total production production well D Oil Water measured constructed polynomials construct polynomial polynomials: interactions between productions

8 construction of well production polynomial indeterminates production polynomial f has to be fitted to a set of points: evaluations of indeterminates at first data point number of points in the order of thousands not uncommon

9 important observation: different polynomials (supports, degrees) evaluations close together

10 relations in the data, among the indeterminates transitivity fails

11 Small Polynomials: Vanishing Ideal

12 example univariate polynomial perturb points polynomial of degree 5 vanishing on points remove the purturbations first before constructing the polynomials construct polynomial allowing it to pass close by prescribed points

13 delta-approximate Vanishing Ideal of course we still have:

14 Best thing one can do: Empirical Ring: Empirical Polynomial:

15 HOW TO CALCULATE THE ApVI

16

17 Classical Version

18 Drawbacks: The found relations are far away from vanishing on the points: numerical error prevent accurate calculation exact solution would result in an O with #O = #P (1000 here!!) The exact relations (for the perturbed data) are usually of very high degree How to overcome these problems: Do not force the relations to pass through every point, but demand passing close by Allow points which lie close together to be melt together Divide the good ones from the bad ones Process blocks rather than single elements to (hopefully) prevent sub-optimal solutions and speed-up computations

19

20

21

22 Approximate Vanishing Ideal with Singular Value Decomposition Truncate below eps = 0.001

23 Approximate Vanishing Ideal with Singular Value Decomposition Truncate below eps = 0.01

24 Approximate Vanishing Ideal with Singular Value Decomposition Truncate below eps = 0.1

25 Approximate Vanishing Ideal with Singular Value Decomposition Truncate below eps = 0.5

26 Another (short) example...

27 Approximate Vanishing Ideal with Singular Value Decomposition Classical Version Truncate below eps = 0.7

28

29 Timings (GB-Version) (2024 points,, 9 indets)

30 Timings (BB-Version) (2024 points,, 9 indets)

31 Application within Shell Calculate relations for the productions... Typical datasize: points in up to 15 indets

32 Application within steel industries Application fields: predict production quality automated quality assurance Typical datasize: points in indets solution time < s #G = 464, #O = 391

33 Acknowledgement: C. Fassino and J. Abbott worked in parallel on an algorithm which also computes almost vanishing ideals, but with a different approach... See: C. Fassino. An Approximation to the Gröbner Basis of Ideals of Perturbed Point. Preprint (2006)

34 Thank you!!

Oil Fields and Hilbert Schemes

Oil Fields and Hilbert Schemes Oil Fields and Hilbert Schemes Lorenzo Robbiano Università di Genova Dipartimento di Matematica Lorenzo Robbiano (Università di Genova) Oil Fields and Hilbert Schemes March, 2008 1 / 35 Facts In the realm

More information

Algebraic Oil. Martin Kreuzer Fakultät für Informatik und Mathematik Universität Passau uni-passau.de. NOCAS Passau December 10, 2008

Algebraic Oil. Martin Kreuzer Fakultät für Informatik und Mathematik Universität Passau uni-passau.de. NOCAS Passau December 10, 2008 Algebraic Oil Martin Kreuzer Fakultät für Informatik und Mathematik Universität Passau martin.kreuzer@ uni-passau.de NOCAS Passau December 10, 2008 1 Contents 1. Some Problems in Oil Production 2. Border

More information

Lecture 5: Ideals of Points

Lecture 5: Ideals of Points Lecture 5: Ideals of Points The Vanishing Lecture Martin Kreuzer Fakultät für Informatik und Mathematik Universität Passau martin.kreuzer@ uni-passau.de Sophus Lie Center Nordfjordeid June 18, 2009 1 Contents

More information

On The Belonging Of A Perturbed Vector To A Subspace From A Numerical View Point

On The Belonging Of A Perturbed Vector To A Subspace From A Numerical View Point Applied Mathematics E-Notes, 7(007), 65-70 c ISSN 1607-510 Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/ On The Belonging Of A Perturbed Vector To A Subspace From A Numerical View

More information

Stable Border Bases for Ideals of Points arxiv: v2 [math.ac] 16 Oct 2007

Stable Border Bases for Ideals of Points arxiv: v2 [math.ac] 16 Oct 2007 Stable Border Bases for Ideals of Points arxiv:0706.2316v2 [math.ac] 16 Oct 2007 John Abbott, Claudia Fassino, Maria-Laura Torrente Abstract Let X be a set of points whose coordinates are known with limited

More information

Overview of Computer Algebra

Overview of Computer Algebra Overview of Computer Algebra http://cocoa.dima.unige.it/ J. Abbott Universität Kassel J. Abbott Computer Algebra Basics Manchester, July 2018 1 / 12 Intro Characteristics of Computer Algebra or Symbolic

More information

From Oil Fields to Hilbert Schemes

From Oil Fields to Hilbert Schemes Chapter 1 From Oil Fields to Hilbert Schemes Martin Kreuzer, Hennie Poulisse, and Lorenzo Robbiano Abstract New techniques for dealing with problems of numerical stability in computations involving multivariate

More information

WORKING WITH MULTIVARIATE POLYNOMIALS IN MAPLE

WORKING WITH MULTIVARIATE POLYNOMIALS IN MAPLE WORKING WITH MULTIVARIATE POLYNOMIALS IN MAPLE JEFFREY B. FARR AND ROMAN PEARCE Abstract. We comment on the implementation of various algorithms in multivariate polynomial theory. Specifically, we describe

More information

Cylindrical Algebraic Decomposition in Coq

Cylindrical Algebraic Decomposition in Coq Cylindrical Algebraic Decomposition in Coq MAP 2010 - Logroño 13-16 November 2010 Assia Mahboubi INRIA Microsoft Research Joint Centre (France) INRIA Saclay Île-de-France École Polytechnique, Palaiseau

More information

Simple Approximate Varieties for Sets of Empirical Data

Simple Approximate Varieties for Sets of Empirical Data Simple Approximate Varieties for Sets of Empirical Data Maria-Laura Torrente (joint work with Claudia Fassino) Dipartimento di Matematica Università di Genova SAGA Fall School Shapes, Geometry, and Algebra

More information

nonlinear simultaneous equations of type (1)

nonlinear simultaneous equations of type (1) Module 5 : Solving Nonlinear Algebraic Equations Section 1 : Introduction 1 Introduction Consider set of nonlinear simultaneous equations of type -------(1) -------(2) where and represents a function vector.

More information

Introduction to D-module Theory. Algorithms for Computing Bernstein-Sato Polynomials. Jorge Martín-Morales

Introduction to D-module Theory. Algorithms for Computing Bernstein-Sato Polynomials. Jorge Martín-Morales Introduction to D-module Theory. Algorithms for Computing Bernstein-Sato Polynomials Jorge Martín-Morales Centro Universitario de la Defensa de Zaragoza Academia General Militar Differential Algebra and

More information

From Gauss. to Gröbner Bases. John Perry. The University of Southern Mississippi. From Gauss to Gröbner Bases p.

From Gauss. to Gröbner Bases. John Perry. The University of Southern Mississippi. From Gauss to Gröbner Bases p. From Gauss to Gröbner Bases p. From Gauss to Gröbner Bases John Perry The University of Southern Mississippi From Gauss to Gröbner Bases p. Overview Questions: Common zeroes? Tool: Gaussian elimination

More information

Commuting birth-and-death processes

Commuting birth-and-death processes Commuting birth-and-death processes Caroline Uhler Department of Statistics UC Berkeley (joint work with Steven N. Evans and Bernd Sturmfels) MSRI Workshop on Algebraic Statistics December 18, 2008 Birth-and-death

More information

Computing the Bernstein-Sato polynomial

Computing the Bernstein-Sato polynomial Computing the Bernstein-Sato polynomial Daniel Andres Kaiserslautern 14.10.2009 Daniel Andres (RWTH Aachen) Computing the Bernstein-Sato polynomial Kaiserslautern 2009 1 / 21 Overview 1 Introduction 2

More information

Ensemble Consistency Testing for CESM: A new form of Quality Assurance

Ensemble Consistency Testing for CESM: A new form of Quality Assurance Ensemble Consistency Testing for CESM: A new form of Quality Assurance Dorit Hammerling Institute for Mathematics Applied to Geosciences National Center for Atmospheric Research (NCAR) Joint work with

More information

Robustness of Principal Components

Robustness of Principal Components PCA for Clustering An objective of principal components analysis is to identify linear combinations of the original variables that are useful in accounting for the variation in those original variables.

More information

A novel method for estimating the distribution of convective heat flux in ducts: Gaussian Filtered Singular Value Decomposition

A novel method for estimating the distribution of convective heat flux in ducts: Gaussian Filtered Singular Value Decomposition A novel method for estimating the distribution of convective heat flux in ducts: Gaussian Filtered Singular Value Decomposition F Bozzoli 1,2, L Cattani 1,2, A Mocerino 1, S Rainieri 1,2, F S V Bazán 3

More information

Ranks of Real Symmetric Tensors

Ranks of Real Symmetric Tensors Ranks of Real Symmetric Tensors Greg Blekherman SIAM AG 2013 Algebraic Geometry of Tensor Decompositions Real Symmetric Tensor Decompositions Let f be a form of degree d in R[x 1,..., x n ]. We would like

More information

Rational Univariate Representation

Rational Univariate Representation Rational Univariate Representation 1 Stickelberger s Theorem a rational univariate representation (RUR) 2 The Elbow Manipulator a spatial robot arm with three links 3 Application of the Newton Identities

More information

ON THE CONNECTION OF THE SHERALI-ADAMS CLOSURE AND BORDER BASES

ON THE CONNECTION OF THE SHERALI-ADAMS CLOSURE AND BORDER BASES ON THE CONNECTION OF THE SHERALI-ADAMS CLOSURE AND BORDER BASES SEBASTIAN POKUTTA AND ANDREAS S. SCHULZ ABSTRACT. The Sherali-Adams lift-and-project hierarchy is a fundamental construct in integer programming,

More information

Fast Straightening Algorithm for Bracket Polynomials Based on Tableau Manipulations

Fast Straightening Algorithm for Bracket Polynomials Based on Tableau Manipulations Fast Straightening Algorithm for Bracket Polynomials Based on Tableau Manipulations Changpeng Shao, Hongbo Li KLMM, Chinese Academy of Sciences July 18, 2018 1 / 30 Outline Background: Bracket Polynomials

More information

Groebner bases under composition and multivariate matrix factorization

Groebner bases under composition and multivariate matrix factorization Groebner bases under composition and multivariate matrix factorization Mingsheng Wang Institute of Software, Chinese Academy of Sciences Contents of the talk In this talk, I will introduce some of my work

More information

Project One: C Bump functions

Project One: C Bump functions Project One: C Bump functions James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University November 2, 2018 Outline 1 2 The Project Let s recall what the

More information

Algorithms for Tensor Decomposition via Numerical Homotopy

Algorithms for Tensor Decomposition via Numerical Homotopy Algorithms for Tensor Decomposition via Numerical Homotopy Session on Algebraic Geometry of Tensor Decompositions August 3, 2013 This talk involves joint work and discussions with: Hirotachi Abo Dan Bates

More information

COMPARISON OF TWO METHODS TO SOLVE PRESSURES IN SMALL VOLUMES IN REAL-TIME SIMULATION OF A MOBILE DIRECTIONAL CONTROL VALVE

COMPARISON OF TWO METHODS TO SOLVE PRESSURES IN SMALL VOLUMES IN REAL-TIME SIMULATION OF A MOBILE DIRECTIONAL CONTROL VALVE COMPARISON OF TWO METHODS TO SOLVE PRESSURES IN SMALL VOLUMES IN REAL-TIME SIMULATION OF A MOBILE DIRECTIONAL CONTROL VALVE Rafael ÅMAN*, Heikki HANDROOS*, Pasi KORKEALAAKSO** and Asko ROUVINEN** * Laboratory

More information

While using the input and output data fu(t)g and fy(t)g, by the methods in system identification, we can get a black-box model like (In the case where

While using the input and output data fu(t)g and fy(t)g, by the methods in system identification, we can get a black-box model like (In the case where ESTIMATE PHYSICAL PARAMETERS BY BLACK-BOX MODELING Liang-Liang Xie Λ;1 and Lennart Ljung ΛΛ Λ Institute of Systems Science, Chinese Academy of Sciences, 100080, Beijing, China ΛΛ Department of Electrical

More information

MCS 563 Spring 2014 Analytic Symbolic Computation Friday 31 January. Quotient Rings

MCS 563 Spring 2014 Analytic Symbolic Computation Friday 31 January. Quotient Rings Quotient Rings In this note we consider again ideals, but here we do not start from polynomials, but from a finite set of points. The application in statistics and the pseudo code of the Buchberger-Möller

More information

Technical Report

Technical Report The Exact Rational Univariate Representation and its Application Koji Ouchi, John Keyser J. Maurice Rojas kouchi@cs.tamu.edu, keyser@cs.tamu.edu rojas@math.tamu.edu Department of Computer Science, Department

More information

Rational Univariate Reduction via Toric Resultants

Rational Univariate Reduction via Toric Resultants Rational Univariate Reduction via Toric Resultants Koji Ouchi 1,2 John Keyser 1 Department of Computer Science, 3112 Texas A&M University, College Station, TX 77843-3112, USA Abstract We describe algorithms

More information

Finer rook equivalence: Classifying Ding s Schubert varieties

Finer rook equivalence: Classifying Ding s Schubert varieties Finer rook equivalence: Classifying Ding s Schubert varieties Mike Develin Jeremy Martin Victor Reiner (AIM) (University of Minnesota) (University of Minnesota Preprint: arxiv:math.ag/4353 math.umn.edu/

More information

Lecture 1: Gröbner Bases and Border Bases

Lecture 1: Gröbner Bases and Border Bases Lecture 1: Gröbner Bases and Border Bases The Schizophrenic Lecture Martin Kreuzer Fakultät für Informatik und Mathematik Universität Passau martin.kreuzer@ uni-passau.de Sophus Lie Center Nordfjordeid

More information

Linear Least-Squares Data Fitting

Linear Least-Squares Data Fitting CHAPTER 6 Linear Least-Squares Data Fitting 61 Introduction Recall that in chapter 3 we were discussing linear systems of equations, written in shorthand in the form Ax = b In chapter 3, we just considered

More information

Almost polynomial Complexity for Zero-dimensional Gröbner Bases p.1/17

Almost polynomial Complexity for Zero-dimensional Gröbner Bases p.1/17 Almost polynomial Complexity for Zero-dimensional Gröbner Bases Amir Hashemi co-author: Daniel Lazard INRIA SALSA-project/ LIP6 CALFOR-team http://www-calfor.lip6.fr/ hashemi/ Special semester on Gröbner

More information

MHV Diagrams and Tree Amplitudes of Gluons

MHV Diagrams and Tree Amplitudes of Gluons Institute for Advanced Study, Princeton, NJ 08540 U.S.A. E-mail: cachazo@ias.edu Institute for Advanced Study, Princeton, NJ 08540 U.S.A. E-mail: cachazo@ias.edu Recently, a new perturbation expansion

More information

Bertini: A new software package for computations in numerical algebraic geometry

Bertini: A new software package for computations in numerical algebraic geometry Bertini: A new software package for computations in numerical algebraic geometry Dan Bates University of Notre Dame Andrew Sommese University of Notre Dame Charles Wampler GM Research & Development Workshop

More information

Degeneration of Orlik-Solomon algebras and Milnor bers of complex line arrangements

Degeneration of Orlik-Solomon algebras and Milnor bers of complex line arrangements Degeneration of Orlik-Solomon algebras and Milnor bers of complex line arrangements Pauline Bailet Hokkaido University Computational Geometric Topology in Arrangement Theory July 6-10, 2015 1 / 15 Reference

More information

A Study of Numerical Elimination for the Solution of Multivariate Polynomial Systems

A Study of Numerical Elimination for the Solution of Multivariate Polynomial Systems A Study of Numerical Elimination for the Solution of Multivariate Polynomial Systems W Auzinger and H J Stetter Abstract In an earlier paper we had motivated and described am algorithm for the computation

More information

Greatest Common Divisor

Greatest Common Divisor Greatest Common Divisor Graeme Taylor February 8, 2005 In a computer algebra setting, the greatest common divisor is necessary to make sense of fractions, whether to work with rational numbers or ratios

More information

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 41

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 41 FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 41 RAVI VAKIL CONTENTS 1. Normalization 1 2. Extending maps to projective schemes over smooth codimension one points: the clear denominators theorem 5 Welcome back!

More information

Introduction to Data Mining

Introduction to Data Mining Introduction to Data Mining Lecture #21: Dimensionality Reduction Seoul National University 1 In This Lecture Understand the motivation and applications of dimensionality reduction Learn the definition

More information

Wooldridge, Introductory Econometrics, 4th ed. Chapter 2: The simple regression model

Wooldridge, Introductory Econometrics, 4th ed. Chapter 2: The simple regression model Wooldridge, Introductory Econometrics, 4th ed. Chapter 2: The simple regression model Most of this course will be concerned with use of a regression model: a structure in which one or more explanatory

More information

Math 1314 Lesson 4 Limits

Math 1314 Lesson 4 Limits Math 1314 Lesson 4 Limits Finding a it amounts to answering the following question: What is happening to the y-value of a function as the x-value approaches a specific target number? If the y-value is

More information

Localization. Introduction. Markus Lange-Hegermann

Localization. Introduction. Markus Lange-Hegermann Localization Markus Lange-Hegermann Introduction This talk deals with localisation of holonomic Weyl algebra modules and their localisation. Consider left modules an d left ideals for this talk. Instead

More information

Introduction to Gröbner Bases for Geometric Modeling. Geometric & Solid Modeling 1989 Christoph M. Hoffmann

Introduction to Gröbner Bases for Geometric Modeling. Geometric & Solid Modeling 1989 Christoph M. Hoffmann Introduction to Gröbner Bases for Geometric Modeling Geometric & Solid Modeling 1989 Christoph M. Hoffmann Algebraic Geometry Branch of mathematics. Express geometric facts in algebraic terms in order

More information

Exploring the Exotic Setting for Algebraic Geometry

Exploring the Exotic Setting for Algebraic Geometry Exploring the Exotic Setting for Algebraic Geometry Victor I. Piercey University of Arizona Integration Workshop Project August 6-10, 2010 1 Introduction In this project, we will describe the basic topology

More information

computing an irreducible decomposition of A B looking for solutions where the Jacobian drops rank

computing an irreducible decomposition of A B looking for solutions where the Jacobian drops rank Diagonal Homotopies 1 Intersecting Solution Sets computing an irreducible decomposition of A B 2 The Singular Locus looking for solutions where the Jacobian drops rank 3 Solving Systems restricted to an

More information

A Bayesian Model for Root Computation

A Bayesian Model for Root Computation A Bayesian Model for Root Computation Josef Schicho, RICAM joint work with Hanna K. Pikkarainen, RICAM Approximate Algebraic Computation Observation: choose any computational problem in algebra that involves

More information

Severe slugging: modeling, simulation and stability criteria

Severe slugging: modeling, simulation and stability criteria Severe slugging: modeling, simulation and stability criteria Jorge Luis Baliño jlbalino@usp.br Departamento de Engenharia Mecânica Escola Politécnica Outline Introduction Air-water model Simulation results

More information

Zero Decomposition Algorithms for System of Polynomial Equations 1)

Zero Decomposition Algorithms for System of Polynomial Equations 1) MM Research Preprints, 200 207 No. 18, Dec. 1999. Beijing Zero Decomposition Algorithms for System of Polynomial Equations 1) D.K. Wang 2) Abstract. This paper will give the zero decomposition algorithms

More information

Gröbner Complexes and Tropical Bases

Gröbner Complexes and Tropical Bases Gröbner Complexes and Tropical Bases Jan Verschelde University of Illinois at Chicago Department of Mathematics, Statistics, and Computer Science http://www.math.uic.edu/ jan jan@math.uic.edu Graduate

More information

On the BMS Algorithm

On the BMS Algorithm On the BMS Algorithm Shojiro Sakata The University of Electro-Communications Department of Information and Communication Engineering Chofu-shi, Tokyo 182-8585, JAPAN Abstract I will present a sketch of

More information

On Conversions from CNF to ANF

On Conversions from CNF to ANF On Conversions from CNF to ANF Jan Horáček Martin Kreuzer Faculty of Informatics and Mathematics University of Passau, Germany Jan.Horacek@uni-passau.de Martin.Kreuzer@uni-passau.de Background ANF is XOR

More information

19. TAYLOR SERIES AND TECHNIQUES

19. TAYLOR SERIES AND TECHNIQUES 19. TAYLOR SERIES AND TECHNIQUES Taylor polynomials can be generated for a given function through a certain linear combination of its derivatives. The idea is that we can approximate a function by a polynomial,

More information

MATH 115, SUMMER 2012 LECTURE 12

MATH 115, SUMMER 2012 LECTURE 12 MATH 115, SUMMER 2012 LECTURE 12 JAMES MCIVOR - last time - we used hensel s lemma to go from roots of polynomial equations mod p to roots mod p 2, mod p 3, etc. - from there we can use CRT to construct

More information

Exact Arithmetic on a Computer

Exact Arithmetic on a Computer Exact Arithmetic on a Computer Symbolic Computation and Computer Algebra William J. Turner Department of Mathematics & Computer Science Wabash College Crawfordsville, IN 47933 Tuesday 21 September 2010

More information

Chapter 8. Rigid transformations

Chapter 8. Rigid transformations Chapter 8. Rigid transformations We are about to start drawing figures in 3D. There are no built-in routines for this purpose in PostScript, and we shall have to start more or less from scratch in extending

More information

Lifting to non-integral idempotents

Lifting to non-integral idempotents Journal of Pure and Applied Algebra 162 (2001) 359 366 www.elsevier.com/locate/jpaa Lifting to non-integral idempotents Georey R. Robinson School of Mathematics and Statistics, University of Birmingham,

More information

On the Computation of the Adjoint Ideal of Curves with Ordinary Singularities

On the Computation of the Adjoint Ideal of Curves with Ordinary Singularities Applied Mathematical Sciences Vol. 8, 2014, no. 136, 6805-6812 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.49697 On the Computation of the Adjoint Ideal of Curves with Ordinary Singularities

More information

(Reprint of pp in Proc. 2nd Int. Workshop on Algebraic and Combinatorial coding Theory, Leningrad, Sept , 1990)

(Reprint of pp in Proc. 2nd Int. Workshop on Algebraic and Combinatorial coding Theory, Leningrad, Sept , 1990) (Reprint of pp. 154-159 in Proc. 2nd Int. Workshop on Algebraic and Combinatorial coding Theory, Leningrad, Sept. 16-22, 1990) SYSTEMATICITY AND ROTATIONAL INVARIANCE OF CONVOLUTIONAL CODES OVER RINGS

More information

TABLE OF CONTENTS INTRODUCTION, APPROXIMATION & ERRORS 1. Chapter Introduction to numerical methods 1 Multiple-choice test 7 Problem set 9

TABLE OF CONTENTS INTRODUCTION, APPROXIMATION & ERRORS 1. Chapter Introduction to numerical methods 1 Multiple-choice test 7 Problem set 9 TABLE OF CONTENTS INTRODUCTION, APPROXIMATION & ERRORS 1 Chapter 01.01 Introduction to numerical methods 1 Multiple-choice test 7 Problem set 9 Chapter 01.02 Measuring errors 11 True error 11 Relative

More information

Visual Tracking via Geometric Particle Filtering on the Affine Group with Optimal Importance Functions

Visual Tracking via Geometric Particle Filtering on the Affine Group with Optimal Importance Functions Monday, June 22 Visual Tracking via Geometric Particle Filtering on the Affine Group with Optimal Importance Functions Junghyun Kwon 1, Kyoung Mu Lee 1, and Frank C. Park 2 1 Department of EECS, 2 School

More information

EVALUATION OF (UNSTABLE) NON-CAUSAL SYSTEMS APPLIED TO ITERATIVE LEARNING CONTROL

EVALUATION OF (UNSTABLE) NON-CAUSAL SYSTEMS APPLIED TO ITERATIVE LEARNING CONTROL EVALUATION OF (UNSTABLE) NON-CAUSAL SYSTEMS APPLIED TO ITERATIVE LEARNING CONTROL M.G.E. Schneiders, M.J.G. van de Molengraft and M. Steinbuch Eindhoven University of Technology, Control Systems Technology,

More information

Practical Linear Algebra: A Geometry Toolbox

Practical Linear Algebra: A Geometry Toolbox Practical Linear Algebra: A Geometry Toolbox Third edition Chapter 12: Gauss for Linear Systems Gerald Farin & Dianne Hansford CRC Press, Taylor & Francis Group, An A K Peters Book www.farinhansford.com/books/pla

More information

Latent Semantic Indexing (LSI) CE-324: Modern Information Retrieval Sharif University of Technology

Latent Semantic Indexing (LSI) CE-324: Modern Information Retrieval Sharif University of Technology Latent Semantic Indexing (LSI) CE-324: Modern Information Retrieval Sharif University of Technology M. Soleymani Fall 2016 Most slides have been adapted from: Profs. Manning, Nayak & Raghavan (CS-276,

More information

An Efficient Algorithm for Computing Parametric Multivariate Polynomial GCD

An Efficient Algorithm for Computing Parametric Multivariate Polynomial GCD An Efficient Algorithm for Computing Parametric Multivariate Polynomial GCD Dong Lu Key Laboratory of Mathematics Mechanization Academy of Mathematics and Systems Science, CAS Joint work with Deepak Kapur,

More information

DIVISIBILITY OF BINOMIAL COEFFICIENTS AT p = 2

DIVISIBILITY OF BINOMIAL COEFFICIENTS AT p = 2 DIVISIBILITY OF BINOMIAL COEFFICIENTS AT p = 2 BAILEY SWINFORD Abstract. This paper will look at the binomial coefficients divisible by the prime number 2. The paper will seek to understand and explain

More information

On a matrix product question in cryptography

On a matrix product question in cryptography On a matrix product question in cryptography Mei-Chu Chang Department of Mathematics University of California, Riverside mcc@math.ucr.edu Abstract Let A, B be invertible n n matrices with irreducible characteristic

More information

Local Parametrization and Puiseux Expansion

Local Parametrization and Puiseux Expansion Chapter 9 Local Parametrization and Puiseux Expansion Let us first give an example of what we want to do in this section. Example 9.0.1. Consider the plane algebraic curve C A (C) defined by the equation

More information

1 xa 2. 2 xan n. + c 2 x α 2

1 xa 2. 2 xan n. + c 2 x α 2 Operations Research Seminar: Gröbner Bases and Integer Programming Speaker: Adam Van Tuyl Introduction In this talk I will discuss how to use some of the tools of commutative algebra and algebraic geometry

More information

A Hierarchy of Proof Rules for Checking Differential Invariance of Algebraic Sets

A Hierarchy of Proof Rules for Checking Differential Invariance of Algebraic Sets A Hierarchy of Proof Rules for Checking Differential Invariance of Algebraic Sets Khalil Ghorbal 1 Andrew Sogokon 2 André Platzer 1 1. Carnegie Mellon University 2. University of Edinburgh VMCAI, Mumbai,

More information

Modular Algorithms for Computing Minimal Associated Primes and Radicals of Polynomial Ideals. Masayuki Noro. Toru Aoyama

Modular Algorithms for Computing Minimal Associated Primes and Radicals of Polynomial Ideals. Masayuki Noro. Toru Aoyama Modular Algorithms for Computing Minimal Associated Primes and Radicals of Polynomial Ideals Toru Aoyama Kobe University Department of Mathematics Graduate school of Science Rikkyo University Department

More information

The Calculation of Radical Ideals in Positive Characteristic

The Calculation of Radical Ideals in Positive Characteristic J. Symbolic Computation (2002) 34, 229 238 doi:10.1006/jsco.2002.0560 Available online at http://www.idealibrary.com on The Calculation of Radical Ideals in Positive Characteristic GREGOR KEMPER IWR, Universität

More information

ECON2228 Notes 2. Christopher F Baum. Boston College Economics. cfb (BC Econ) ECON2228 Notes / 47

ECON2228 Notes 2. Christopher F Baum. Boston College Economics. cfb (BC Econ) ECON2228 Notes / 47 ECON2228 Notes 2 Christopher F Baum Boston College Economics 2014 2015 cfb (BC Econ) ECON2228 Notes 2 2014 2015 1 / 47 Chapter 2: The simple regression model Most of this course will be concerned with

More information

Integration of Rational Functions by Partial Fractions

Integration of Rational Functions by Partial Fractions Integration of Rational Functions by Partial Fractions Part 2: Integrating Rational Functions Rational Functions Recall that a rational function is the quotient of two polynomials. x + 3 x + 2 x + 2 x

More information

Analysis of Ordinary Differential Equations

Analysis of Ordinary Differential Equations Analysis of Ordinary Differential Equations J. M. Cushing Department of Mathematics Interdisciplinary Program in Applied Mathematics University of Arizona, Tucson, AZ Version 5 August 208 Copyright 208,

More information

Intro to Rings, Fields, Polynomials: Hardware Modeling by Modulo Arithmetic

Intro to Rings, Fields, Polynomials: Hardware Modeling by Modulo Arithmetic Intro to Rings, Fields, Polynomials: Hardware Modeling by Modulo Arithmetic Priyank Kalla Associate Professor Electrical and Computer Engineering, University of Utah kalla@ece.utah.edu http://www.ece.utah.edu/~kalla

More information

Time Series Models and Inference. James L. Powell Department of Economics University of California, Berkeley

Time Series Models and Inference. James L. Powell Department of Economics University of California, Berkeley Time Series Models and Inference James L. Powell Department of Economics University of California, Berkeley Overview In contrast to the classical linear regression model, in which the components of the

More information

Global optimization, polynomial optimization, polynomial system solving, real

Global optimization, polynomial optimization, polynomial system solving, real PROBABILISTIC ALGORITHM FOR POLYNOMIAL OPTIMIZATION OVER A REAL ALGEBRAIC SET AURÉLIEN GREUET AND MOHAB SAFEY EL DIN Abstract. Let f, f 1,..., f s be n-variate polynomials with rational coefficients of

More information

Intrinsic geometry and the invariant trace field of hyperbolic 3-manifolds

Intrinsic geometry and the invariant trace field of hyperbolic 3-manifolds Intrinsic geometry and the invariant trace field of hyperbolic 3-manifolds Anastasiia Tsvietkova University of California, Davis Joint with Walter Neumann, based on earlier joint work with Morwen Thistlethwaite

More information

Math 1314 Lesson 4 Limits

Math 1314 Lesson 4 Limits Math 1314 Lesson 4 Limits What is calculus? Calculus is the study of change, particularly, how things change over time. It gives us a framework for measuring change using some fairly simple models. In

More information

Solving Parametric Polynomial Systems by RealComprehensiveTriangularize

Solving Parametric Polynomial Systems by RealComprehensiveTriangularize Solving Parametric Polynomial Systems by RealComprehensiveTriangularize Changbo Chen 1 and Marc Moreno Maza 2 1 Chongqing Key Laboratory of Automated Reasoning and Cognition, Chongqing Institute of Green

More information

1.2 Finite Precision Arithmetic

1.2 Finite Precision Arithmetic MACM Assignment Solutions.2 Finite Precision Arithmetic.2:e Rounding Arithmetic Use four-digit rounding arithmetic to perform the following calculation. Compute the absolute error and relative error with

More information

Latent Semantic Indexing (LSI) CE-324: Modern Information Retrieval Sharif University of Technology

Latent Semantic Indexing (LSI) CE-324: Modern Information Retrieval Sharif University of Technology Latent Semantic Indexing (LSI) CE-324: Modern Information Retrieval Sharif University of Technology M. Soleymani Fall 2014 Most slides have been adapted from: Profs. Manning, Nayak & Raghavan (CS-276,

More information

arxiv: v2 [math.ag] 30 Apr 2014

arxiv: v2 [math.ag] 30 Apr 2014 HODGE DIAMONDS OF ADJOINT ORBITS BRIAN CALLANDER AND ELIZABETH GASPARIM arxiv:3.265v2 [math.ag] 30 Apr 204 ABSTRACT. We present a Macaulay2 package that produces compactifications of adjoint orbits and

More information

ALGEBRAIC CHARACTERIZATION OF UNIQUELY VERTEX COLORABLE GRAPHS

ALGEBRAIC CHARACTERIZATION OF UNIQUELY VERTEX COLORABLE GRAPHS ALGEBRAIC CHARACTERIZATION OF NIQELY VERTEX COLORABLE GRAPHS CHRISTOPHER J. HILLAR AND TROELS WINDFELDT Abstract. The study of graph vertex colorability from an algebraic perspective has introduced novel

More information

Numerical Integration (Quadrature) Another application for our interpolation tools!

Numerical Integration (Quadrature) Another application for our interpolation tools! Numerical Integration (Quadrature) Another application for our interpolation tools! Integration: Area under a curve Curve = data or function Integrating data Finite number of data points spacing specified

More information

Notes for Lecture 15

Notes for Lecture 15 COS 533: Advanced Cryptography Lecture 15 (November 8, 2017) Lecturer: Mark Zhandry Princeton University Scribe: Kevin Liu Notes for Lecture 15 1 Lattices A lattice looks something like the following.

More information

Journal of Symbolic Computation. The Gröbner basis of the ideal of vanishing polynomials

Journal of Symbolic Computation. The Gröbner basis of the ideal of vanishing polynomials Journal of Symbolic Computation 46 (2011) 561 570 Contents lists available at ScienceDirect Journal of Symbolic Computation journal homepage: www.elsevier.com/locate/jsc The Gröbner basis of the ideal

More information

UNCERTAINTY MODELING VIA FREQUENCY DOMAIN MODEL VALIDATION

UNCERTAINTY MODELING VIA FREQUENCY DOMAIN MODEL VALIDATION AIAA 99-3959 UNCERTAINTY MODELING VIA FREQUENCY DOMAIN MODEL VALIDATION Martin R. Waszak, * NASA Langley Research Center, Hampton, Virginia Dominick Andrisani II, Purdue University, West Lafayette, Indiana

More information

Reverse engineering using computational algebra

Reverse engineering using computational algebra Reverse engineering using computational algebra Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4500, Fall 2016 M. Macauley (Clemson)

More information

Principal Component Analysis

Principal Component Analysis Machine Learning Michaelmas 2017 James Worrell Principal Component Analysis 1 Introduction 1.1 Goals of PCA Principal components analysis (PCA) is a dimensionality reduction technique that can be used

More information

Complex Valued Nonlinear Adaptive Filters

Complex Valued Nonlinear Adaptive Filters Complex Valued Nonlinear Adaptive Filters Noncircularity, Widely Linear and Neural Models Danilo P. Mandic Imperial College London, UK Vanessa Su Lee Goh Shell EP, Europe WILEY A John Wiley and Sons, Ltd,

More information

When does T equal sat(t)?

When does T equal sat(t)? When does T equal sat(t)? Wei Pan joint work with François Lemaire, Marc Moreno Maza and Yuzhen Xie MOCAA M 3 workshop UWO May 7, 2008 1 Introduction Given a regular chain T, the saturated ideal sat(t)

More information

4 Hilbert s Basis Theorem and Gröbner basis

4 Hilbert s Basis Theorem and Gröbner basis 4 Hilbert s Basis Theorem and Gröbner basis We define Gröbner bases of ideals in multivariate polynomial rings and see how they work in tandem with the division algorithm. We look again at the standard

More information

AUTOMATIC CONTROL. Andrea M. Zanchettin, PhD Spring Semester, Introduction to Automatic Control & Linear systems (time domain)

AUTOMATIC CONTROL. Andrea M. Zanchettin, PhD Spring Semester, Introduction to Automatic Control & Linear systems (time domain) 1 AUTOMATIC CONTROL Andrea M. Zanchettin, PhD Spring Semester, 2018 Introduction to Automatic Control & Linear systems (time domain) 2 What is automatic control? From Wikipedia Control theory is an interdisciplinary

More information

Primary Decomposition of Ideals Arising from Hankel Matrices

Primary Decomposition of Ideals Arising from Hankel Matrices Primary Decomposition of Ideals Arising from Hankel Matrices Paul Brodhead University of Wisconsin-Madison Malarie Cummings Hampton University Cora Seidler University of Texas-El Paso August 10 2000 Abstract

More information

The Kakeya Problem Connections with Harmonic Analysis Kakeya sets over Finite Fields. Kakeya Sets. Jonathan Hickman. The University of Edinburgh

The Kakeya Problem Connections with Harmonic Analysis Kakeya sets over Finite Fields. Kakeya Sets. Jonathan Hickman. The University of Edinburgh The University of Edinburgh The Kakeya Problem Definition A Kakeya set K R n is a compact subset which contains a unit line segment in every direction. The Kakeya Problem Examples of Kakeya subsets of

More information

In a series of papers that N. Mohan Kumar and M.P. Murthy ([MK2], [Mu1], [MKM]) wrote, the final theorem was the following.

In a series of papers that N. Mohan Kumar and M.P. Murthy ([MK2], [Mu1], [MKM]) wrote, the final theorem was the following. EULER CYCLES SATYA MANDAL I will talk on the following three papers: (1) A Riemann-Roch Theorem ([DM1]), (2) Euler Class Construction ([DM2]) (3) Torsion Euler Cycle ([BDM]) In a series of papers that

More information

arxiv: v1 [math.ac] 10 Feb 2011

arxiv: v1 [math.ac] 10 Feb 2011 STABLE COMPLETE INTERSECTIONS LORENZO ROBBIANO AND MARIA LAURA TORRENTE arxiv:1102.2158v1 [math.ac] 10 Feb 2011 Abstract. A complete intersection of n polynomials in n indeterminates has only a finite

More information