Homological Processing of Biomedical digital images: automation and certification 1
|
|
|
- Nigel Lyons
- 5 years ago
- Views:
Transcription
1 Homological Processing of Biomedical digital images: automation and certification 1 Jónathan Heras, Gadea Mata, María Poza and Julio Rubio Department of Mathematics and Computer Science University of La Rioja Spain June 27, Partially supported by Ministerio de Educación y Ciencia, project MTM C02-01, and by European Commission FP7, STREP project ForMath, n
2 Table of Contents 1 Motivation 2 3 4
3 Table of Contents 1 Motivation 2 3 4
4 Motivation: Synapses counting Synapses are the points of connection between neurons Relevance: Computational capabilities of the brain The different number of synapses may be an important asset in the treatment of neurological diseases
5 Manual processing to count synapses Apply two different antibody markers, bassoon and synapsin
6 Manual processing to count synapses Process the images in order to count the synapses (ImageJ)
7 Manual processing to count synapses Overlap both images
8 Manual processing to count synapses The synapses are manually counted one by one
9 Goal to count synapses Goal Provide a reliable and automatic method for counting synapses in a neuron
10 Table of Contents Motivation Preprocessing the image Algebraic Topology for digital images analysis General method 1 Motivation 2 3 4
11 Count synapses automatically Preprocessing the image Algebraic Topology for digital images analysis General method New ImageJ plugin called SynapCountJ Improve the interaction with the ImageJ system
12 Count synapses automatically Preprocessing the image Algebraic Topology for digital images analysis General method New ImageJ plugin called SynapCountJ Improve the interaction with the ImageJ system Steps Determine the neuron morphology from one of those pictures (NeuronJ plugin)
13 Motivation Preprocessing the image Algebraic Topology for digital images analysis General method Count synapses automatically New ImageJ plugin called SynapCountJ Improve the interaction with the ImageJ system Steps Determine the neuron morphology from one of those pictures (NeuronJ plugin) Overlap the images with the two markers with the one with the structure (SynapCountJ) J. Heras, G. Mata, M. Poza and J. Rubio Homological Processing of Biomedical digital images
14 Count synapses automatically Preprocessing the image Algebraic Topology for digital images analysis General method New ImageJ plugin called SynapCountJ Improve the interaction with the ImageJ system Steps Determine the neuron morphology from one of those pictures (NeuronJ plugin) Overlap the images with the two markers with the one with the structure (SynapCountJ) Invert the colors to show the synapses as black points
15 The method Motivation Preprocessing the image Algebraic Topology for digital images analysis General method Digital Image algebraic structure interpreting Chain Complex Homology computing
16 The method Motivation Preprocessing the image Algebraic Topology for digital images analysis General method Digital Image algebraic structure interpreting Chain Complex Homology computing
17 The method Motivation Preprocessing the image Algebraic Topology for digital images analysis General method Digital Image algebraic structure interpreting Chain Complex Homology computing
18 The method Motivation Preprocessing the image Algebraic Topology for digital images analysis General method Digital Image algebraic structure interpreting Chain Complex Homology computing
19 Image to Chain Complex Preprocessing the image Algebraic Topology for digital images analysis General method interpretation Digital Image Chain Complex Homology
20 Image to Chain Complex Preprocessing the image Algebraic Topology for digital images analysis General method interpretation Digital Image Chain Complex Homology C 0 = Z [vertices] triangulation d 1 C 1 = Z [edges] d 2 C 2 = Z [triangles]
21 Image to Chain Complex Preprocessing the image Algebraic Topology for digital images analysis General method interpretation Digital Image Chain Complex Homology C 0 = Z [vertices] triangulation d 1 C 1 = Z [edges] d 2 C 2 = Z [triangles]
22 Image to Chain Complex Preprocessing the image Algebraic Topology for digital images analysis General method interpretation Digital Image Chain Complex Homology C 0 = Z [vertices] triangulation d 1 C 1 = Z [edges] d 2 C 2 = Z [triangles]
23 Image to Chain Complex Preprocessing the image Algebraic Topology for digital images analysis General method interpretation Digital Image Chain Complex Homology C 0 = Z [vertices] triangulation d 1 C 1 = Z [edges] d 2 C 2 = Z [triangles] 0 Z 16 d 1 Z 32 d 2 Z 16 0
24 Compute Homology Motivation Preprocessing the image Algebraic Topology for digital images analysis General method interpretation Digital Image Chain Complex Homology Problem of diagonalizing matrices Compute the Smith Normal Form
25 Interpretation from Homology to Image Preprocessing the image Algebraic Topology for digital images analysis General method interpretation Digital Image Chain Complex Homology H 0 measures the number of connected components H 1 measures the number of holes In our case, H 0 counts the number of synapses
26 General method Motivation Preprocessing the image Algebraic Topology for digital images analysis General method properties interpretation Biomedical Image Digital Image Chain Complex Homology
27 General method Motivation Preprocessing the image Algebraic Topology for digital images analysis General method properties interpretation Biomedical Image Digital Image Chain Complex Homology
28 General method Motivation Preprocessing the image Algebraic Topology for digital images analysis General method properties interpretation Biomedical Image Digital Image Chain Complex Homology
29 General method Motivation Preprocessing the image Algebraic Topology for digital images analysis General method properties interpretation Biomedical Image Digital Image Chain Complex Homology
30 Table of Contents Motivation Reduce the size: Discrete Morse theory Safety of the results: Certification of the programs 1 Motivation 2 Preprocessing the image Algebraic Topology for digital images analysis General method 3 Reduce the size: Discrete Morse theory Safety of the results: Certification of the programs 4
31 Problems Motivation Reduce the size: Discrete Morse theory Safety of the results: Certification of the programs Size of the images Safety of the results
32 Problems Motivation Reduce the size: Discrete Morse theory Safety of the results: Certification of the programs Size of the images Discrete Morse theory Safety of the results Certification of the programs
33 The method Motivation Reduce the size: Discrete Morse theory Safety of the results: Certification of the programs Digital Image algebraic structure interpreting Chain Complex computing reduction Homology
34 The method Motivation Reduce the size: Discrete Morse theory Safety of the results: Certification of the programs Digital Image algebraic structure interpreting Chain Complex computing reduction Homology
35 The method Motivation Reduce the size: Discrete Morse theory Safety of the results: Certification of the programs Digital Image algebraic structure interpreting Chain Complex computing reduction Homology
36 Reduction of chain complex Reduce the size: Discrete Morse theory Safety of the results: Certification of the programs interpretation Digital Image Chain Complex Homology reduction Reduce information keeping the homological properties Discrete Morse Theory Vector fields are a tool to cancel useless information
37 Reduction of chain complex Reduce the size: Discrete Morse theory Safety of the results: Certification of the programs interpretation Digital Image Chain Complex Homology reduction Reduce information keeping the homological properties Discrete Morse Theory Vector fields are a tool to cancel useless information 0 Z 16 Z 32 Z 16 0
38 Reduction of chain complex Reduce the size: Discrete Morse theory Safety of the results: Certification of the programs interpretation Digital Image Chain Complex Homology reduction Reduce information keeping the homological properties Discrete Morse Theory Vector fields are a tool to cancel useless information 0 Z Z 0 0
39 Discrete Morse Theory Reduce the size: Discrete Morse theory Safety of the results: Certification of the programs Definition Let C = (C p, d p ) p Z be a free chain complex with distinguised Z basis β p C p. A (p 1)-cell σ is a face of a p-cell τ if the coefficient of σ in dτ is non-null. It is a regular face if this coefficient is +1 or 1 Definition A discrete vector field on C is a collection of pairs V = {(σ i, τ i )} i β satisfying the conditions: 1 Every σ i is some element of β p, in which case the other corresponding component τ i β p+1. The degree p depends on i and in general is not constant 2 Every component σ i is a regular face of the corresponding component τ i 3 A generator of C appears at most one time in V
40 Discrete Morse Theory Reduce the size: Discrete Morse theory Safety of the results: Certification of the programs Definition A V -path of degree p is a sequence π = ((σ ik, τ ik )) 0 k<m satisfying: 1 Every pair ((σ ik, τ ik )) is a component of V and the cell τ ik is a p-cell 2 For every 0 < k < m, the component σ ik is a face of τ ik 1, non necessarily regular, but different from σ ik 1 Definition A discrete vector field V is admissible if for every p Z, a function λ p : β p Z is provided satisfying the property: every V -path starting from σ β p has a length bounded by λ p (σ)
41 Discrete Morse Theory Reduce the size: Discrete Morse theory Safety of the results: Certification of the programs Definition A cell χ which does not appear in a discrete vector field V = {(σ i, τ i )} i β is called a critical cell Vector-Field Reduction Theorem Let C = (C p, d p β p ) p be a free chain complex and V = {(σ i, β i )} i β be an admissible discrete vector field on C. Then the vector field V defines a canonical reduction ρ = (f, g, h) : (C p, d p ) = (C c p, d p) where C c p = Z [ β c p] is the free Z module generated by the critical p-cells A. Romero and F. Sergeraert. Discrete Vector Fields and Fundamental Algebraic Topology,
42 Discrete vector field over matrices Reduce the size: Discrete Morse theory Safety of the results: Certification of the programs interpretation Digital Image Chain Complex Homology reduction 2D-images Chain complex associated with an image is finite d 0 C 1 d 0 2 C1 C2 0 Differential maps can be represented by integer matrices Reduction chain complex Reduction matrices
43 Discrete vector field over matrices Reduce the size: Discrete Morse theory Safety of the results: Certification of the programs interpretation Digital Image Chain Complex Homology reduction Definition A vector field V for a matrix M Mat m,n (Z) is a set of integer pairs {(a i, b i )} i satisfying these conditions: 1 1 a i m and 1 b i n 2 The entry M [a i, b i ] is ±1 3 The indices a i (respectively b i ) are pairwise different
44 Reduction of Chain Complex Reduce the size: Discrete Morse theory Safety of the results: Certification of the programs interpretation Digital Image Chain Complex Homology reduction Goal Let M n be a finite matrix which represents the differential map d n of C Compute an admissible discrete vector field V from M n Obtain a new matrix M n from M n and V
45 Reduction of Chain Complex Reduce the size: Discrete Morse theory Safety of the results: Certification of the programs interpretation Digital Image Chain Complex Homology reduction Goal Let M n be a finite matrix which represents the differential map d n of C Compute an admissible discrete vector field V from M n Obtain a new matrix M n from M n and V In our case, we have to reduce two matrices M 1 and M 2. Compute the homology groups of C with M 1 and M 2 can be much faster.
46 Reduction of Chain Complex Reduce the size: Discrete Morse theory Safety of the results: Certification of the programs interpretation Digital Image Chain Complex Homology reduction Implemented in Haskell Algorithm 1 Input: an integer matrix M n Output: an admissible discrete vector field V Algorithm 2 Input: an integer matrix M n Output: a reduced matrix M n
47 The method Motivation Reduce the size: Discrete Morse theory Safety of the results: Certification of the programs Digital Image algebraic structure Chain Complex reduction Homology computing
48 The method Motivation Reduce the size: Discrete Morse theory Safety of the results: Certification of the programs Digital Image algebraic structure Chain Complex reduction Homology computing
49 Coq/SSReflect Motivation Reduce the size: Discrete Morse theory Safety of the results: Certification of the programs Coq Theorem prover assistant High logic order SSReflect Extension of Coq Introduce new tactics and libraries Used to formalize of the Four Colour Theorem
50 Reduction of chain complex Reduce the size: Discrete Morse theory Safety of the results: Certification of the programs Digital Image Chain Complex Homology reduction Steps 1 Translate our Haskell code into the Coq language 2 Define the test functions to specify the properties which our programs must satisfy 3 State and prove the lemmas which ensure the correctness of our programs
51 Reduction of chain complex Reduce the size: Discrete Morse theory Safety of the results: Certification of the programs Digital Image Chain Complex Homology reduction Steps 1 Translate our Haskell code into the Coq language 2 Define the test functions to specify the properties which our programs must satisfy 3 State and prove the lemmas which ensure the correctness of our programs Example: Let M be an integer matrix, (vectorcvd M) builds an admissible discrete vector field Lemma admissible-vf: forall M, (int-matrix M) -> (admissible (vectorcvd M))
52 Table of Contents 1 Motivation 2 3 4
53 Conclusions Motivation Biomedical image SynapCountJ Processed image fkenzo Chain complex fkenzo Matrix properties interpretation fkenzo Haskell Homology fkenzo Reduced matrix Methodology to study Biomedical images Programs partially verified with Theorem Prover tools Application to count synapses
54 Further work Motivation Verification of our Haskell programs by means of Coq/SSReflect is still an ongoing work Verification of Smith Normal Form of a matrix Find other applications of our homological tools in the Biomedical imaging context
55 Homological Processing of Biomedical digital images: automation and certification 2 Jónathan Heras, Gadea Mata, María Poza and Julio Rubio Department of Mathematics and Computer Science University of La Rioja Spain June 27, Partially supported by Ministerio de Educación y Ciencia, project MTM C02-01, and by European Commission FP7, STREP project ForMath, n
Formalization of Mathematics: why Algebraic Topology?
Formalization of Mathematics: why Algebraic Topology? Julio Rubio Universidad de La Rioja Departamento de Matemáticas y Computación MAP Spring School 2012 Sophia Antipolis, March 12th-16th, 2012 Partially
Certifying homological algorithms to study biomedical images
Certifying homological algorithms to study biomedical images Supervisors: Dr. César Domínguez Pérez Dr. Julio Rubio García Department of Mathematics and Computer Science University of La Rioja Spain June
Neuronal structure detection using Persistent Homology
Neuronal structure detection using Persistent Homology J. Heras, G. Mata and J. Rubio Department of Mathematics and Computer Science, University of La Rioja Seminario de Informática Mirian Andrés March
Constructive Algebra in Type Theory
Constructive Algebra in Type Theory Anders Mörtberg mortberg@chalmers.se September 4, 2012 ( 0 2 1 0 ) T ( x1 x 2 ) = ( 2 0 ) ( 0 2 1 0 ) T ( x1 x 2 ) = ( 2 0 ) ( x1 x 2 ) = ( 0 2 ) Matlab 2009b A=[ 0
Discrete Vector Fields and Fundamental Algebraic Topology.
Discrete Vector Fields and Fundamental Algebraic Topology. Ana Romero, Francis Sergeraert 1 May 9, 2012 [Version 6.2.] 1 Both authors partially supported by Ministerio de Ciencia e Innovación, Spain, Project
Formalizing Elementary Divisor Rings in Coq
Formalizing Elementary Divisor Rings in Coq Cyril Cohen Anders Mörtberg University of Gothenburg May 27, 2014 Introduction Goal: Generalize the theory of linear algebra over fields (vector spaces) to rings
1. Introduction to commutative rings and fields
1. Introduction to commutative rings and fields Very informally speaking, a commutative ring is a set in which we can add, subtract and multiply elements so that the usual laws hold. A field is a commutative
CMPS 6630: Introduction to Computational Biology and Bioinformatics. Structure Comparison
CMPS 6630: Introduction to Computational Biology and Bioinformatics Structure Comparison Protein Structure Comparison Motivation Understand sequence and structure variability Understand Domain architecture
Computing the homology of groups: the geometric way
Computing the homology of groups: the geometric way A. Romero, J. Rubio Departamento de Matemáticas y Computación Universidad de La Rioja Edificio Vives, c/luis de Ulloa s/n 26004 Logroño, Spain. Abstract
Formal verification for numerical methods
Formal verification for numerical methods Ioana Paşca PhD thesis at INRIA Sophia Antipolis, Marelle Team Advisor: Yves Bertot 1 7 1 Robots 2 7 Robots efficiency 2 7 Robots efficiency safety 2 7 Inside
arxiv:math/ v5 [math.ac] 17 Sep 2009
![arxiv:math/ v5 [math.ac] 17 Sep 2009](/thumbs/86/93191167.jpg "arxiv:math/ v5 [math.ac] 17 Sep 2009")
On the elementary symmetric functions of a sum of matrices R. S. Costas-Santos arxiv:math/0612464v5 [math.ac] 17 Sep 2009 September 17, 2009 Abstract Often in mathematics it is useful to summarize a multivariate
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
UNIVERSAL DERIVED EQUIVALENCES OF POSETS
UNIVERSAL DERIVED EQUIVALENCES OF POSETS SEFI LADKANI Abstract. By using only combinatorial data on two posets X and Y, we construct a set of so-called formulas. A formula produces simultaneously, for
Elementary maths for GMT
Elementary maths for GMT Linear Algebra Part 2: Matrices, Elimination and Determinant m n matrices The system of m linear equations in n variables x 1, x 2,, x n a 11 x 1 + a 12 x 2 + + a 1n x n = b 1
1. Introduction to commutative rings and fields
1. Introduction to commutative rings and fields Very informally speaking, a commutative ring is a set in which we can add, subtract and multiply elements so that the usual laws hold. A field is a commutative
A Machine Checked Model of Idempotent MGU Axioms For a List of Equational Constraints
A Machine Checked Model of Idempotent MGU Axioms For a List of Equational Constraints Sunil Kothari, James Caldwell Department of Computer Science, University of Wyoming, USA Machine checked proofs of
Linear Algebra and its Applications
Linear Algebra and its Applications 430 (2009) 579 586 Contents lists available at ScienceDirect Linear Algebra and its Applications journal homepage: www.elsevier.com/locate/laa Low rank perturbation
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
Write your own Theorem Prover
Write your own Theorem Prover Phil Scott 27 October 2016 Phil Scott Write your own Theorem Prover 27 October 2016 1 / 31 Introduction We ll work through a toy LCF style theorem prover for classical propositional
The k-fibonacci matrix and the Pascal matrix
Cent Eur J Math 9(6 0 403-40 DOI: 0478/s533-0-0089-9 Central European Journal of Mathematics The -Fibonacci matrix and the Pascal matrix Research Article Sergio Falcon Department of Mathematics and Institute
Chapter 3: Homology Groups Topics in Computational Topology: An Algorithmic View
Chapter 3: Homology Groups Topics in Computational Topology: An Algorithmic View As discussed in Chapter 2, we have complete topological information about 2-manifolds. How about higher dimensional manifolds?
Least squares problems Linear Algebra with Computer Science Application
Linear Algebra with Computer Science Application April 8, 018 1 Least Squares Problems 11 Least Squares Problems What do you do when Ax = b has no solution? Inconsistent systems arise often in applications
Linearizing Symmetric Matrix Polynomials via Fiedler pencils with Repetition
Linearizing Symmetric Matrix Polynomials via Fiedler pencils with Repetition Kyle Curlett Maribel Bueno Cachadina, Advisor March, 2012 Department of Mathematics Abstract Strong linearizations of a matrix
ADDENDUM TO FROBENIUS AND CARTIER ALGEBRAS OF STANLEY-REISNER RINGS [J. ALGEBRA 358 (2012) ]
![ADDENDUM TO FROBENIUS AND CARTIER ALGEBRAS OF STANLEY-REISNER RINGS [J. ALGEBRA 358 (2012) ]](/thumbs/76/73920978.jpg "ADDENDUM TO FROBENIUS AND CARTIER ALGEBRAS OF STANLEY-REISNER RINGS [J. ALGEBRA 358 (2012) ]")
ADDENDUM TO FROBENIUS AND CARTIER ALGEBRAS OF STANLEY-REISNER RINGS [J. ALGEBRA 358 (2012) 162-177] JOSEP ÀLVAREZ MONTANER AND KOHJI YANAGAWA Abstract. We give a purely combinatorial characterization of
arxiv: v2 [math.ac] 19 Dec 2008
![arxiv: v2 [math.ac] 19 Dec 2008](/thumbs/89/98926841.jpg "arxiv: v2 [math.ac] 19 Dec 2008")
A combinatorial description of monomial algebras arxiv:0810.4836v2 [math.ac] 19 Dec 2008 Ignacio Oeda Universidad de Extremadura Departamento de Matemáticas e-mail: oedamc@unex.es March 22, 2009 Abstract
Remark By definition, an eigenvector must be a nonzero vector, but eigenvalue could be zero.
Sec 6 Eigenvalues and Eigenvectors Definition An eigenvector of an n n matrix A is a nonzero vector x such that A x λ x for some scalar λ A scalar λ is called an eigenvalue of A if there is a nontrivial
Verified Implementation of Exact Real Arithmetic in Type Theory
Verified Implementation of Exact Real Arithmetic in Type Theory Bas Spitters EU STREP-FET ForMath Dec 9th 2013 Computable Analysis and Rigorous Numerics 1 / 14 Rigorous Numerics through Computable Analysis
NOTES IN COMMUTATIVE ALGEBRA: PART 2
NOTES IN COMMUTATIVE ALGEBRA: PART 2 KELLER VANDEBOGERT 1. Completion of a Ring/Module Here we shall consider two seemingly different constructions for the completion of a module and show that indeed they
Multiplying matrices by diagonal matrices is faster than usual matrix multiplication.
7-6 Multiplying matrices by diagonal matrices is faster than usual matrix multiplication. The following equations generalize to matrices of any size. Multiplying a matrix from the left by a diagonal matrix
Smith theory. Andrew Putman. Abstract
Smith theory Andrew Putman Abstract We discuss theorems of P. Smith and Floyd connecting the cohomology of a simplicial complex equipped with an action of a finite p-group to the cohomology of its fixed
Hierarchical Current Density Verification for Electromigration Analysis in Arbitrarily Shaped Metallization Patterns of Analog Circuits
Hierarchical Current Density Verification for Electromigration Analysis in Arbitrarily Shaped Metallization Patterns of Analog Circuits Göran Jerke Jens Lienig Robert Bosch GmbH Reutlingen, Germany goeran.jerke@ieee.org
HOMOLOGY THEORIES INGRID STARKEY
HOMOLOGY THEORIES INGRID STARKEY Abstract. This paper will introduce the notion of homology for topological spaces and discuss its intuitive meaning. It will also describe a general method that is used
Code Generation for a Simple First-Order Prover
Code Generation for a Simple First-Order Prover Jørgen Villadsen, Anders Schlichtkrull, and Andreas Halkjær From DTU Compute, Technical University of Denmark, 2800 Kongens Lyngby, Denmark Abstract. We
Topological Data Analysis - Spring 2018
Topological Data Analysis - Spring 2018 Simplicial Homology Slightly rearranged, but mostly copy-pasted from Harer s and Edelsbrunner s Computational Topology, Verovsek s class notes. Gunnar Carlsson s
Lecture 3: Tropicalizations of Cluster Algebras Examples David Speyer
Lecture 3: Tropicalizations of Cluster Algebras Examples David Speyer Let A be a cluster algebra with B-matrix B. Let X be Spec A with all of the cluster variables inverted, and embed X into a torus by
series. Utilize the methods of calculus to solve applied problems that require computational or algebraic techniques..
1 Use computational techniques and algebraic skills essential for success in an academic, personal, or workplace setting. (Computational and Algebraic Skills) MAT 203 MAT 204 MAT 205 MAT 206 Calculus I
Quantum Entanglement and the Bell Matrix
Quantum Entanglement and the Bell Matrix Marco Pedicini (Roma Tre University) in collaboration with Anna Chiara Lai and Silvia Rognone (La Sapienza University of Rome) SIMAI2018 - MS27: Discrete Mathematics,
Persistent Homology. 128 VI Persistence
8 VI Persistence VI. Persistent Homology A main purpose of persistent homology is the measurement of the scale or resolution of a topological feature. There are two ingredients, one geometric, assigning
CIS Spring 2018 (instructor Val Tannen)
CIS 160 - Spring 018 (instructor Val Tannen) Lecture 11 Tuesday, February 0 PROOFS: STRONG INDUCTION Example 1.1 Any integer n can be written as the product of one or more (not necessarily distinct) prime
Validating QBF Invalidity in HOL4
Interactive Theorem Proving (ITP) 14 July, 2010 Quantified Boolean Formulae Quantified Boolean Formulae Motivation System Overview Related Work QBF = propositional logic + quantifiers over Boolean variables
implies that if we fix a basis v of V and let M and M be the associated invertible symmetric matrices computing, and, then M = (L L)M and the
Math 395. Geometric approach to signature For the amusement of the reader who knows a tiny bit about groups (enough to know the meaning of a transitive group action on a set), we now provide an alternative
A SURPRISING FACT ABOUT D-MODULES IN CHARACTERISTIC p > Introduction
A SUPISING FACT ABOUT D-MODULES IN CHAACTEISTIC p > 0 JOSEP ÀLVAEZ MONTANE AND GENNADY LYUBEZNIK Abstract. Let = k[x,..., x d ] be the polynomial ring in d independent variables, where k is a field of
can only hit 3 points in the codomain. Hence, f is not surjective. For another example, if n = 4
.. Conditions for Injectivity and Surjectivity In this section, we discuss what we can say about linear maps T : R n R m given only m and n. We motivate this problem by looking at maps f : {,..., n} {,...,
Finding parking when not commuting. Jon McCammond U.C. Santa Barbara
Finding parking when not commuting 7 6 8 1 2 3 5 4 {{1,4,5}, {2,3}, {6,8}, {7}} Jon McCammond U.C. Santa Barbara 1 A common structure The main goal of this talk will be to introduce you to a mathematical
Math 3191 Applied Linear Algebra
Math 9 Applied Linear Algebra Lecture 9: Diagonalization Stephen Billups University of Colorado at Denver Math 9Applied Linear Algebra p./9 Section. Diagonalization The goal here is to develop a useful
Constructive Homological Algebra and Applications
Constructive Homological Algebra and Applications (2006 Genova Summer School) Julio Rubio, Francis Sergeraert Version August 23, 2006 1 Introduction. Standard homological algebra is not constructive, and
NOTES ON LINEAR ODES
NOTES ON LINEAR ODES JONATHAN LUK We can now use all the discussions we had on linear algebra to study linear ODEs Most of this material appears in the textbook in 21, 22, 23, 26 As always, this is a preliminary
12. Perturbed Matrices
MAT334 : Applied Linear Algebra Mike Newman, winter 208 2. Perturbed Matrices motivation We want to solve a system Ax = b in a context where A and b are not known exactly. There might be experimental errors,
Proof Certificates for SMT-based Model Checkers for Infinite State Systems. Alain Mebsout and Cesare Tinelli FMCAD 2016 October 5 th, 2016
Proof Certificates for SMT-based Model Checkers for Infinite State Systems Alain Mebsout and Cesare Tinelli FMCAD 2016 October 5 th, 2016 Motivation Model checkers return error traces but no evidence when
Definition 2.3. We define addition and multiplication of matrices as follows.
14 Chapter 2 Matrices In this chapter, we review matrix algebra from Linear Algebra I, consider row and column operations on matrices, and define the rank of a matrix. Along the way prove that the row
An Introduction to Proof Assistants
An Introduction to Proof Assistants Patrick Schnider Student Seminar in Combinatorics: Mathematical Software, ETH Zürich 1 Motivation The development of proof assistants was motivated by the use of computers
Interactive Theorem Provers
Interactive Theorem Provers from the perspective of Isabelle/Isar Makarius Wenzel Univ. Paris-Sud, LRI July 2014 = Isabelle λ β Isar α 1 Introduction Notable ITP systems LISP based: ACL2 http://www.cs.utexas.edu/users/moore/acl2
Formalisation and execution of Linear Algebra: theorems and algorithms
Formalisation and execution of Linear Algebra: theorems and algorithms Jose Divasón Mallagaray Dissertation submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy Supervisors:
Quantizations of cluster algebras
Quantizations of cluster algebras Philipp Lampe Bielefeld University International Conference Mathematics Days in Sofia July 10, 2014, Sofia, Bulgaria Ph. Lampe (Bielefeld) Quantum cluster algebras July
Recent progress in Homotopy type theory
July 22nd, 2013 Supported by EU FP7 STREP FET-open ForMATH Most of the presentation is based on the book: CC-BY-SA Topos Homotopy type theory Collaborative effort lead by Awodey, Coquand, Voevodsky at
Study of Topological Surfaces
Study of Topological Surfaces Jamil Mortada Abstract This research project involves studying different topological surfaces, in particular, continuous, invertible functions from a surface to itself. Dealing
Beyond First-Order Logic
Beyond First-Order Logic Software Formal Verification Maria João Frade Departmento de Informática Universidade do Minho 2008/2009 Maria João Frade (DI-UM) Beyond First-Order Logic MFES 2008/09 1 / 37 FOL
Problems in Linear Algebra and Representation Theory
Problems in Linear Algebra and Representation Theory (Most of these were provided by Victor Ginzburg) The problems appearing below have varying level of difficulty. They are not listed in any specific
Again we return to a row of 1 s! Furthermore, all the entries are positive integers.
Lecture Notes Introduction to Cluster Algebra Ivan C.H. Ip Updated: April 4, 207 Examples. Conway-Coxeter frieze pattern Frieze is the wide central section part of an entablature, often seen in Greek temples,
Computer Proof Assistants and Univalent Foundations of Mathematics
Nov. 16, 2014, CMA2014, Kuwait. Computer Proof Assistants and Univalent Foundations of Mathematics by Vladimir Voevodsky from the Institute for Advanced Study in Princeton, NJ. Kepler s Conjecture. In
Proof Certificates for SMT-based Model Checkers. Alain Mebsout and Cesare Tinelli SMT 2016 July 2 nd, 2016
Proof Certificates for SMT-based Model Checkers Alain Mebsout and Cesare Tinelli SMT 2016 July 2 nd, 2016 Motivation Model checkers return error traces but no evidence when they say yes Complex tools 2
Proof assistants as a tool for thought
Proof assistants as a tool for thought Katherine Ye @hypotext Tools for thought workshop, March 16 circa 1700 Disputants unable to agree would not waste much time in futile argument... Leibniz (Harrison)
Deduction and Computation in Algebraic Topology
Deduction and Computation in Algebraic Topology J. Aransay 1, C. Ballarin 2, and J. Rubio 1 1 Departamento de Matemáticas y Computación. Universidad de La Rioja. Edificio Vives. Calle Luis de Ulloa s/n.
Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition
Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2016 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing
UNC Charlotte Super Competition - Comprehensive test March 2, 2015
March 2, 2015 1. triangle is inscribed in a semi-circle of radius r as shown in the figure: θ The area of the triangle is () r 2 sin 2θ () πr 2 sin θ () r sin θ cos θ () πr 2 /4 (E) πr 2 /2 2. triangle
The TLA + proof system
The TLA + proof system Stephan Merz Kaustuv Chaudhuri, Damien Doligez, Leslie Lamport INRIA Nancy & INRIA-MSR Joint Centre, France Amir Pnueli Memorial Symposium New York University, May 8, 2010 Stephan
Introduction. Pedro Cabalar. Department of Computer Science University of Corunna, SPAIN 2013/2014
Introduction Pedro Cabalar Department of Computer Science University of Corunna, SPAIN cabalar@udc.es 2013/2014 P. Cabalar ( Department Introduction of Computer Science University of Corunna, SPAIN2013/2014
Automation of Termination: Abstracting Calling Contexts through Matrix-Weighted Graphs
Automation of Termination: Abstracting Calling Contexts through Matrix-Weighted Graphs Andréia Borges Avelar 1 & Mauricio Ayala-Rincón 1 & César A Muñoz 2 1 Instituto de Ciências Exatas - UnB 2 NASA Langley
Problem 1. Let f n (x, y), n Z, be the sequence of rational functions in two variables x and y given by the initial conditions.
18.217 Problem Set (due Monday, December 03, 2018) Solve as many problems as you want. Turn in your favorite solutions. You can also solve and turn any other claims that were given in class without proofs,
MAT 1302B Mathematical Methods II
MAT 1302B Mathematical Methods II Alistair Savage Mathematics and Statistics University of Ottawa Winter 2015 Lecture 19 Alistair Savage (uottawa) MAT 1302B Mathematical Methods II Winter 2015 Lecture
arxiv: v1 [math.ds] 27 Nov 2012
![arxiv: v1 [math.ds] 27 Nov 2012](/thumbs/90/104440878.jpg "arxiv: v1 [math.ds] 27 Nov 2012")
COMPUTATION OF BALANCED EQUIVALENCE RELATIONS AND THEIR LATTICE FOR A COUPLED CELL NETWORK HIROKO KAMEI AND PETER J A COCK arxiv:26334v [mathds] 27 Nov 202 Abstract A coupled cell network describes interacting
Eigenvalues, Eigenvectors, and Diagonalization
Math 240 TA: Shuyi Weng Winter 207 February 23, 207 Eigenvalues, Eigenvectors, and Diagonalization The concepts of eigenvalues, eigenvectors, and diagonalization are best studied with examples. We will
MTH 215: Introduction to Linear Algebra
MTH 215: Introduction to Linear Algebra Lecture 5 Jonathan A. Chávez Casillas 1 1 University of Rhode Island Department of Mathematics September 20, 2017 1 LU Factorization 2 3 4 Triangular Matrices Definition
Counting Subrings of Z n and Related Questions
Counting Subrings Advisor: Francisco Munoz Yale University 9/4/2015 Introduction Goal of Project Given a ring R which is a free Z-module of finite rank, define f R (k) = #{subrings (with identity) of index
18.312: Algebraic Combinatorics Lionel Levine. Lecture 22. Smith normal form of an integer matrix (linear algebra over Z).
18.312: Algebraic Combinatorics Lionel Levine Lecture date: May 3, 2011 Lecture 22 Notes by: Lou Odette This lecture: Smith normal form of an integer matrix (linear algebra over Z). 1 Review of Abelian
(iv) Whitney s condition B. Suppose S β S α. If two sequences (a k ) S α and (b k ) S β both converge to the same x S β then lim.
0.1. Stratified spaces. References are [7], [6], [3]. Singular spaces are naturally associated to many important mathematical objects (for example in representation theory). We are essentially interested
Math 1553, Introduction to Linear Algebra
Learning goals articulate what students are expected to be able to do in a course that can be measured. This course has course-level learning goals that pertain to the entire course, and section-level
October 7, :8 WSPC/WS-IJWMIP paper. Polynomial functions are renable
International Journal of Wavelets, Multiresolution and Information Processing c World Scientic Publishing Company Polynomial functions are renable Henning Thielemann Institut für Informatik Martin-Luther-Universität
Lecture Notes Introduction to Cluster Algebra
Lecture Notes Introduction to Cluster Algebra Ivan C.H. Ip Updated: April 14, 017 Total Positivity The final and historically the original motivation is from the study of total positive matrices, which
Math 6802 Algebraic Topology II
Math 6802 Algebraic Topology II Nathan Broaddus Ohio State University February 9, 2015 Theorem 1 (The Künneth Formula) If X and Y are spaces then there is a natural exact sequence 0 i H i (X ) H n i (Y
Eigenvalues and Eigenvectors: An Introduction
Eigenvalues and Eigenvectors: An Introduction The eigenvalue problem is a problem of considerable theoretical interest and wide-ranging application. For example, this problem is crucial in solving systems
REGULAR-MT: A Formal Proof of the Computation of Hermite Normal Form In a General Setting
REGULAR-MT: A Formal Proof of the Computation of Hermite Normal Form In a General Setting Jose Divasón and Jesús Aransay Universidad de La Rioja, Logroño, La Rioja, Spain {jose.divason,jesus-maria.aransay}@unirioja.es
Performance Analysis of ARQ Protocols using a Theorem Prover
Performance Analysis of ARQ Protocols using a Theorem Prover Osman Hasan Sofiene Tahar Hardware Verification Group Concordia University Montreal, Canada ISPASS 2008 Objectives n Probabilistic Theorem Proving
On the variety of two dimensional real associative algebras
Int. J. Contemp. Math. Sciences, Vol. 2, 2007, no. 26, 1293-1305 On the variety of two dimensional real associative algebras José María Ancochea Bermúdez 1 Dpto. Geometría y Topología. Facultad de Matemáticas
Neural Codes and Neural Rings: Topology and Algebraic Geometry
Neural Codes and Neural Rings: Topology and Algebraic Geometry Ma191b Winter 2017 Geometry of Neuroscience References for this lecture: Curto, Carina; Itskov, Vladimir; Veliz-Cuba, Alan; Youngs, Nora,
Wigner s semicircle law
CHAPTER 2 Wigner s semicircle law 1. Wigner matrices Definition 12. A Wigner matrix is a random matrix X =(X i, j ) i, j n where (1) X i, j, i < j are i.i.d (real or complex valued). (2) X i,i, i n are
A Randomized Algorithm for the Approximation of Matrices
A Randomized Algorithm for the Approximation of Matrices Per-Gunnar Martinsson, Vladimir Rokhlin, and Mark Tygert Technical Report YALEU/DCS/TR-36 June 29, 2006 Abstract Given an m n matrix A and a positive
Gramians based model reduction for hybrid switched systems
Gramians based model reduction for hybrid switched systems Y. Chahlaoui Younes.Chahlaoui@manchester.ac.uk Centre for Interdisciplinary Computational and Dynamical Analysis (CICADA) School of Mathematics
Dimension. Eigenvalue and eigenvector
Dimension. Eigenvalue and eigenvector Math 112, week 9 Goals: Bases, dimension, rank-nullity theorem. Eigenvalue and eigenvector. Suggested Textbook Readings: Sections 4.5, 4.6, 5.1, 5.2 Week 9: Dimension,
Automated Generation of Formal and Readable Proofs of Mathematical Theorems ongoing work
Automated Generation of Formal and Readable Proofs of Mathematical Theorems ongoing work Sana Stojanović Predrag Janičić Faculty of Mathematics University of Belgrade SVARM 2013 Rome, Italy, January 20-21,
Math 315: Linear Algebra Solutions to Assignment 7
Math 5: Linear Algebra s to Assignment 7 # Find the eigenvalues of the following matrices. (a.) 4 0 0 0 (b.) 0 0 9 5 4. (a.) The characteristic polynomial det(λi A) = (λ )(λ )(λ ), so the eigenvalues are
Towards Lightweight Integration of SMT Solvers
Towards Lightweight Integration of SMT Solvers Andrei Lapets Boston University Boston, USA lapets@bu.edu Saber Mirzaei Boston University Boston, USA smirzaei@bu.edu 1 Introduction A large variety of SMT
Topological Simplification in 3-Dimensions
Topological Simplification in 3-Dimensions David Letscher Saint Louis University Computational & Algorithmic Topology Sydney Letscher (SLU) Topological Simplification CATS 2017 1 / 33 Motivation Original
A GLIMPSE OF ALGEBRAIC K-THEORY: Eric M. Friedlander
A GLIMPSE OF ALGEBRAIC K-THEORY: Eric M. Friedlander During the first three days of September, 1997, I had the privilege of giving a series of five lectures at the beginning of the School on Algebraic
Monomial equivariant embeddings of quasitoric manifolds and the problem of existence of invariant almost complex structures.
Monomial equivariant embeddings of quasitoric manifolds and the problem of existence of invariant almost complex structures. Andrey Kustarev joint work with V. M. Buchstaber, Steklov Mathematical Institute
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
The P versus NP Problem. Ker-I Ko. Stony Brook, New York
The P versus NP Problem Ker-I Ko Stony Brook, New York ? P = NP One of the seven Millenium Problems The youngest one A folklore question? Has hundreds of equivalent forms Informal Definitions P : Computational
Advanced Engineering Mathematics Prof. Pratima Panigrahi Department of Mathematics Indian Institute of Technology, Kharagpur
Advanced Engineering Mathematics Prof. Pratima Panigrahi Department of Mathematics Indian Institute of Technology, Kharagpur Lecture No. #07 Jordan Canonical Form Cayley Hamilton Theorem (Refer Slide Time:
CHEVALLEY-EILENBERG HOMOLOGY OF CROSSED MODULES OF LIE ALGEBRAS IN LOWER DIMENSIONS
Homology, Homotopy and Applications, vol 152, 2013, pp185 194 CHEVALLEY-EILENBERG HOMOLOGY OF CROSSED MODULES OF LIE ALGEBRAS IN LOWER DIMENSIONS GURAM DONADZE and MANUEL LADRA communicated by Graham Ellis
Ripser. Efficient Computation of Vietoris Rips Persistence Barcodes. Ulrich Bauer TUM
Ripser Efficient Computation of Vietoris Rips Persistence Barcodes Ulrich Bauer TUM March 23, 2017 Computational and Statistical Aspects of Topological Data Analysis Alan Turing Institute http://ulrich-bauer.org/ripser-talk.pdf