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A Negative Result for Hearing the Shape of a Triangle: a Computer-Assisted Proof
AN ELECTRONIC JOURNAL OF THE SOCIETAT CATALANA DE MATEMATIQUES` A negative result for hearing the shape of a triangle: a computer-assisted proof ⇤Gerard Orriols Gim´enez Resum (CAT) ETH Z¨urich. En aquest article demostrem que existeixen dos triangles diferents pels quals [email protected] el primer, segon i quart valor propi del Laplaci`a amb condicions de Dirichlet coincideixen. Aix`o resol una conjectura proposada per Antunes i Freitas i Corresponding author ⇤ suggerida per la seva evid`encia num`erica. La prova ´esassistida per ordinador i utilitza una nova t`ecnica per tractar l’espectre d’un l’operador, que consisteix a combinar un M`etode d’Elements Finits per localitzar aproximadament els primers valors propis i controlar la seva posici´oa l’espectre, juntament amb el M`etode de Solucions Particulars per confinar aquests valors propis a un interval molt m´esprec´ıs. Abstract (ENG) We prove that there exist two distinct triangles for which the first, second and fourth eigenvalues of the Laplace operator with zero Dirichlet boundary conditions coincide. This solves a conjecture raised by Antunes and Freitas and suggested by their numerical evidence. We use a novel technique for a computer-assisted proof about the spectrum of an operator, which combines a Finite Element Method, to locate roughly the first eigenvalues keeping track of their position in the spectrum, and the Method of Particular Solutions, to get a much more precise bound of these eigenvalues. Keywords: computer-assisted proof, Laplace eigenvalues, spectral geome- Acknowledgement try, Finite Element Method, Method The author would like to thank Re- of Particular Solutions. -
1 Introduction
数理解析研究所講究録 1169 巻 2000 年 27-56 27 Numerical Verification Methods for Solutions of Ordinary and Partial Differential Equations Mitsuhiro T. Nakao 中尾 充宏 Graduate School of Mathematics, Kyushu University 33 Fukuoka 812-8581, Japan $\mathrm{e}$ -mail: [email protected] Abstract. In this article, we describe on a state of the art of validated numerical computations for solutions of differential equations. A brief overview of the main techniques in self-validating numerics for initial and boundary value problems in ordinary and partial differential equations including eigenvalue problems will be presented. A fairly detailed introductions are given for the author’s own method related to second-order elliptic boundary value problems. Many references which seem to be useful for readers are supplied at the end of the article. Key words. Numerical verification, nonlinear differential equation, computer assisted proof in analysis 1 Introduction If we denote an equation for unknown $u$ by $F(u)=0$ (1.1) then the problem of finding the solution $u$ generally implies an $n$ dimensional simulta- neous system of equations provided $u$ is in some finite dimensional ( $\mathrm{n}$ -dimension) space. Since very large scale, e.g. several thousand, linear system of equations can be easily solved on computers nowadays, it is not too surprising that, even if (1.1) is nonlinear, we can verify the existence and uniqueness of the solution as well as its domain by some numerical computations on the digital computer. However, in the case where (1.1) is a differential equation, it becomes a simultaneous equation which has infinitely many unknowns because the dimension of the potential function space containing $\mathrm{u}$ is infinite. -
Notices of the American Mathematical Society
Notices of the American Mathematical Society April 1981, Issue 209 Volume 28, Number 3, Pages 217- 296 Providence, Rhode Island USA ISSN 0002-9920 CALENDAR OF AMS MEETINGS THIS CALENDAR lists all meetings which have been approved by the Council prior to the date this issue of the Notices was sent to press. The summer and annual meetings are joint meetings of the Mathematical Association of America and the Ameri· 'an Mathematical Society. The meeting dates which fall rather far in the future are subject to change; this is particularly true of meetings to which no numbers have yet been assigned. Programs of the meetings will appear in the issues indicated below. First and second announcements of the meetings will have appeared in earlier issues. ABSTRACTS OF PAPERS presented at a meeting of the Society are published in the journal Abstracts of papers presented to the American Mathematical Society in the issue corresponding to that of the Notices which contains the program of the meet· lng. Abstracts should be submitted on special forms which are available in many departments of mathematics and from the offi'e of the Society in Providence. Abstracts of papers to be presented at the meeting must be received at the headquarters of the Soc:iety in Providence, Rhode Island, on or before the deadline given below for the meeting. Note that the deadline for ab· strKts submitted for consideration for presentation at special sessions is usually three weeks earlier than that specified below. For additional information consult the meeting announcement and the list of organizers of special sessions. -
AMS Short Course on Rigorous Numerics in Dynamics: the Parameterization Method for Stable/Unstable Manifolds of Vector Fields
AMS Short Course on Rigorous Numerics in Dynamics: The Parameterization Method for Stable/Unstable Manifolds of Vector Fields J.D. Mireles James, ∗ February 12, 2017 Abstract This lecture builds on the validated numerical methods for periodic orbits presented in the lecture of J. B. van den Berg. We discuss a functional analytic perspective on validated stability analysis for equilibria and periodic orbits as well as validated com- putation of their local stable/unstable manifolds. Building on this analysis we study heteroclinic and homoclinic connecting orbits between equilibria and periodic orbits of differential equations. We formulate the connecting orbits as solutions of certain projected boundary value problems whcih are amenable to an a posteriori analysis very similar to that already discussed for periodic orbits. The discussion will be driven by several application problems including connecting orbits in the Lorenz system and existence of standing and traveling waves. Contents 1 Introduction 2 2 The Parameterization Method 4 2.1 Formal series solutions for stable/unstable manifolds . .4 2.1.1 Manipulating power series: automatic differentiation and all that . .4 2.1.2 A first example: linearization of an analytic vector field in one complex variable . .9 2.1.3 Stable/unstable manifolds for equilibrium solutions of ordinary differ- ential equations . 15 2.1.4 A more realistic example: stable/unstable manifolds in a restricted four body problem . 18 2.1.5 Stable/unstable manifolds for periodic solutions of ordinary differen- tial equations . 23 2.1.6 Unstable manifolds for equilibrium solutions of partial differential equations . 26 2.1.7 Unstable manifolds for equilibrium and periodic solutions of delay differential equations . -
Book of Open Problems Presented at MTNS 2002
2002 MTNS Problem Book Open Problems on the Mathematical Theory of Systems August 12-16, 2002 Editors Vincent D. Blondel Alexander Megretski Associate Editors Roger Brockett Jean-Michel Coron Miroslav Krstic Anders Rantzer Joachim Rosenthal Eduardo D. Sontag M. Vidyasagar Jan C. Willems Some of the problems appearing in this booklet will appear in a more extensive forthcoming book on open problems in systems theory. For more information about this future book, please consult the website http://www.inma.ucl.ac.be/~blondel/op/ We wish the reader much enjoyment and stimulation in reading the problems in this booklet. The editors ii Program and table of contents Monday August 12. Co-chairs: Vincent Blondel, Roger Brockett. 8:00 PM Linear Systems 75 State and first order representations 1 Jan C. Willems 1 59 The elusive iff test for time-controllability of behaviours 4 Amol J. Sasane 4 58 Schur Extremal Problems 7 Lev Sakhnovich 7 69 Regular feedback implementability of linear differential behaviors 9 H.L. Trentelman 9 36 What is the characteristic polynomial of a signal flow graph? 13 Andrew D. Lewis 13 35 Bases for Lie algebras and a continuous CBH formula 16 Matthias Kawski 16 10 Vector-valued quadratic forms in control theory 20 Francesco Bullo, Jorge Cort´es,Andrew D. Lewis, Sonia Mart´ınez 20 8:48 PM Stochastic Systems 4 On error of estimation and minimum of cost for wide band noise driven systems 24 Agamirza E. Bashirov 24 25 Aspects of Fisher geometry for stochastic linear systems 27 Bernard Hanzon, Ralf Peeters 27 iii 9:00 PM Nonlinear Systems 81 Dynamics of Principal and Minor Component Flows 31 U. -
Validated and Numerically Efficient Chebyshev Spectral Methods For
Validated and numerically efficient Chebyshev spectral methods for linear ordinary differential equations Florent Bréhard, Nicolas Brisebarre, Mioara Joldes To cite this version: Florent Bréhard, Nicolas Brisebarre, Mioara Joldes. Validated and numerically efficient Chebyshev spectral methods for linear ordinary differential equations. ACM Transactions on Mathematical Soft- ware, Association for Computing Machinery, 2018, 44 (4), pp.44:1-44:42. 10.1145/3208103. hal- 01526272v3 HAL Id: hal-01526272 https://hal.archives-ouvertes.fr/hal-01526272v3 Submitted on 5 Jun 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Validated and numerically efficient Chebyshev spectral methods for linear ordinary differential equations Florent Bréhard∗ Nicolas Brisebarrey Mioara Joldeşz Abstract In this work we develop a validated numerics method for the solu- tion of linear ordinary differential equations (LODEs). A wide range of algorithms (i.e., Runge-Kutta, collocation, spectral methods) exist for nu- merically computing approximations of the solutions. Most of these come with proofs of asymptotic convergence, but usually, provided error bounds are non-constructive. However, in some domains like critical systems and computer-aided mathematical proofs, one needs validated effective error bounds. We focus on both the theoretical and practical complexity anal- ysis of a so-called a posteriori quasi-Newton validation method, which mainly relies on a fixed-point argument of a contracting map. -
Focm 2017 Foundations of Computational Mathematics Barcelona, July 10Th-19Th, 2017 Organized in Partnership With
FoCM 2017 Foundations of Computational Mathematics Barcelona, July 10th-19th, 2017 http://www.ub.edu/focm2017 Organized in partnership with Workshops Approximation Theory Computational Algebraic Geometry Computational Dynamics Computational Harmonic Analysis and Compressive Sensing Computational Mathematical Biology with emphasis on the Genome Computational Number Theory Computational Geometry and Topology Continuous Optimization Foundations of Numerical PDEs Geometric Integration and Computational Mechanics Graph Theory and Combinatorics Information-Based Complexity Learning Theory Plenary Speakers Mathematical Foundations of Data Assimilation and Inverse Problems Multiresolution and Adaptivity in Numerical PDEs Numerical Linear Algebra Karim Adiprasito Random Matrices Jean-David Benamou Real-Number Complexity Alexei Borodin Special Functions and Orthogonal Polynomials Mireille Bousquet-Mélou Stochastic Computation Symbolic Analysis Mark Braverman Claudio Canuto Martin Hairer Pierre Lairez Monique Laurent Melvin Leok Lek-Heng Lim Gábor Lugosi Bruno Salvy Sylvia Serfaty Steve Smale Andrew Stuart Joel Tropp Sponsors Shmuel Weinberger 2 FoCM 2017 Foundations of Computational Mathematics Barcelona, July 10th{19th, 2017 Books of abstracts 4 FoCM 2017 Contents Presentation . .7 Governance of FoCM . .9 Local Organizing Committee . .9 Administrative and logistic support . .9 Technical support . 10 Volunteers . 10 Workshops Committee . 10 Plenary Speakers Committee . 10 Smale Prize Committee . 11 Funding Committee . 11 Plenary talks . 13 Workshops . 21 A1 { Approximation Theory Organizers: Albert Cohen { Ron Devore { Peter Binev . 21 A2 { Computational Algebraic Geometry Organizers: Marta Casanellas { Agnes Szanto { Thorsten Theobald . 36 A3 { Computational Number Theory Organizers: Christophe Ritzenhaler { Enric Nart { Tanja Lange . 50 A4 { Computational Geometry and Topology Organizers: Joel Hass { Herbert Edelsbrunner { Gunnar Carlsson . 56 A5 { Geometric Integration and Computational Mechanics Organizers: Fernando Casas { Elena Celledoni { David Martin de Diego . -
Book of Abstracts
GAMM International Association of Applied Mathematics and Mechanics IMACS International Association for Mathematics and Computers in Simulation 13th International Symposium on Scienti¯c Computing Computer Arithmetic and Veri¯ed Numerical Computations SCAN'2008 El Paso, Texas September 29 { October 3, 2008 ABSTRACTS Published in El Paso, Texas July 2008 1 2 Contents Preface....................................................................................... 8 Mohammed A. Abutheraa and David Lester, \Machine-E±cient Chebyshev Approximation for Standard Statistical Distribution Functions" . 9 GÄotzAlefeld and Guenter Mayer, \On the interval Cholesky method" . 11 GÄotzAlefeld and Zhengyu Wang, \Bounding the Error for Approximate Solutions of Almost Linear Complementarity Problems Using Feasible Vectors" . 12 Rene Alt, Jean-Luc Lamotte, and Svetoslav Markov, "On the accuracy of the CELL processor" . 13 Roberto Araiza, \Using Intervals for the Concise Representation of Inverted Repeats in RNA" .................................................................................... 15 Ekaterina Auer and Wolfram Luther, \Veri¯cation and Validation of Two Biomechanical Problems" . 16 Ali Baharev and Endre R¶ev,\Comparing inclusion techniques on chemical engineering problems" . 17 Neelabh Baijal and Luc Longpr¶e,\Preserving privacy in statistical databases by using intervals instead of cell suppression" . 19 Bal¶azsB¶anhelyiand L¶aszl¶oHatvani, \A computer-assisted proof for stable/unstable behaviour of periodic solutions for the forced damped pendulum" . 20 Moa Belaid, \A Uni¯ed Interval Constraint Framework for Crisp, Soft, and Parametric Constraints" . 21 Moa Belaid, \Box Satisfaction and Unsatisfaction for Solving Interval Constraint Systems" . 22 Frithjof Blomquist, \Staggered Correction Computations with Enhanced Accuracy and Extremely Wide Exponent Range" . 24 Gerd Bohlender and Ulrich W. Kulisch, \Complete Interval Arithmetic and its Implementation on the Computer" . 26 Chin-Yun Chen, \Bivariate Product Cubature Using Peano Kernels For Local Error Estimates" . -
Final Catalogue
Price Sub Price Qty. Seq. ISBN13 Title Author Category Year After Category USD Needed 70% 1 9780387257099 Accounting and Financial System Robert W. McGee, Barry University, Miami Shores, FL, USA; Galina Accounting Auditing 2006 125 38 Reform in Eastern Europe and Asia G. Preobragenskaya, Omsk State University, Omsk, Russia 2 9783540308010 Regulatory Risk and the Cost of Capital Burkhard Pedell, University of Stuttgart, Germany Accounting Auditing 2006 115 35 3 9780387265971 Economics of Accounting Peter Ove Christensen, University of Aarhus, Aarhus C, Denmark; Accounting Auditing 2005 149 45 Gerald Feltham, The University of British Columbia, Faculty of Commerce & Business Administration 4 9780387238470 Accounting and Financial System Robert W. McGee, Barry University, Miami Shores, FL, USA; Galina Accounting Auditing 2005 139 42 Reform in a Transition Economy: A G. Preobragenskaya, Omsk State University, Omsk, Russia Case Study of Russia 5 9783540408208 Business Intelligence Techniques Murugan Anandarajan, Drexel University, Philadelphia, PA, USA; Accounting Auditing 2004 145 44 Asokan Anandarajan, New Jersey Institute of Technology, Newark, NJ, USA; Cadambi A. Srinivasan, Drexel University, Philadelphia, PA, USA (Eds.) 6 9780387239323 Economics of Accounting Peter O. Christensen, University of Southern Denmark-Odense, Accounting Special 2003 85 25 Denmark; G.A. Feltham, University of British Columbia, Vancouver, Topics BC, Canada 7 9781402072291 Economics of Accounting Peter O Christensen, University of Southern Denmark-Odense, Accounting Auditing 2002 185 56 Denmark; G.A. Feltham, Faculty of Commerce and Business Administration, University of British Columbia, Vancouver, Canada 8 9780792376491 The Audit Committee: Performing Laura F. Spira Accounting Special 2002 175 53 Corporate Governance Topics 9 9781846280207 Adobe® Acrobat® and PDF for Tom Carson, New Economy Institute, Chattanooga, TN 37403, USA.; Architecture Design, 2006 85 25 Architecture, Engineering, and Donna L. -
Deterministic Global Optimization for Dynamic Systems Using Interval Analysis
Deterministic Global Optimization for Dynamic Systems Using Interval Analysis Youdong Lin and Mark A. Stadtherr Department of Chemical and Biomolecular Engineering University of Notre Dame, Notre Dame, IN, USA 12th GAMM–IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics Duisburg, Germany, 26–29 September 2006 Outline Background • Tools • – Interval Analysis – Taylor Models – Validated Solutions of Parametric ODE Systems Algorithm Summary • Computational Studies • Concluding Remarks • 2 Background Many practically important physical systems are modeled by ODE systems. • Optimization problems involving these models may be stated as • min φ [xµ(θ); θ; µ = 0; 1; : : : ; r] θ;xµ s:t: x_ = f(x; θ) x0 = x0(θ) xµ(θ) = x(tµ; θ) t [t ; tr] 2 0 θ Θ 2 Sequential approach: Eliminate xµ using parametric ODE solver, obtaining • an unconstrained problem in θ May be multiple local solutions – need for global optimization • 3 Deterministic Global Optimization of Dynamic Systems Much recent interest, mostly combining branch-and-bound and relaxation techniques, e.g., Esposito and Floudas (2000): α-BB approach • – Rigorous values of α not used: no theoretical guarantees Chachuat and Latifi (2003): Theoretical guarantee of -global optimality • Papamichail and Adjiman (2002, 2004): Theoretical guarantee of -global • optimality Singer and Barton (2006): Theoretical guarantee of -global optimality • – Use convex underestimators and concave overestimators to construct two bounding IVPs, which are then solved to obtain -
Validated Numerics for Pedestrians
4ECM Stockholm 2004 c 2005 European Mathematical Society Validated Numerics for Pedestrians Warwick Tucker Abstract. The aim of this paper is to give a very brief introduction to the emerg- ing area of validated numerics. This is a rapidly growing field of research faced with the challenge of interfacing computer science and pure mathematics. Most validated numerics is based on interval analysis, which allows its users to account for both rounding and discretization errors in computer-aided proofs. We will il- lustrate the strengths of these techniques by converting the well-known bisection method into a efficient, validated root finder. 1. Introduction Since the creation of the digital computer, numerical computations have played an increasingly fundamental role in modeling physical phenomena for science and engineering. With regards to computing speed and memory capacity, the early computers seem almost amusingly crude compared to their modern coun- terparts. Nevertheless, real-world problems were solved, and the speed-up due to the use of machines pushed the frontier of feasible computing tasks forward. Through a myriad of small developmental increments, we are now on the verge of producing Peta-flop/Peta-byte computers – an incredible feat which must have seemed completely unimaginable fifty years ago. Due to the inherent limitations of any finite-state machine, numerical computations are almost never carried out in a mathematically precise manner. As a consequence, they do not produce exact results, but rather approximate values that usually, but far from always, are near the true ones. In addition to this, external influences, such as an over-simplified mathematical model or a discrete approximation of the same, introduce additional inaccuracies into the calculations. -
Mafelap 2016
MAFELAP 2016 Conference on the Mathematics of Finite Elements and Applications 14–17 June 2016 Abstracts MAFELAP 2016 The organisers of MAFELAP 2016 are pleased to ac- knowledge the financial support given to the confer- ence by the Institute of Mathematics and its Appli- cations (IMA) in the form of IMA Studentships, and the financial support from Brunel University London for the Tuesday evening showcase event. Contents of the MAFELAP 2016 Abstracts Alphabetical order by the speaker Finite element approximations for a fractional Laplace equation Gabriel Acosta and Juan Pablo Borthagaray Mini-Symposium: Elliptic problems with singularities . ....1 A Mixed-Method B-Field Finite-Element Formulation for Incompressible, Resistive Magnetohydrodynamics James H. Adler, Thomas Benson and Scott P. MacLachlan Mini-Symposium: Advances in Finite Element Methods for Nonlinear Mate- rials ................................................ ...............................2 Fitted ALE scheme for Two-Phase Navier–Stokes Flow Marco Agnese and Robert N¨urnberg Parallel session....................................... ....................3 An isogeometric approach to symmetric Galerkin boundary element method Alessandra Aimi, Mauro Diligenti, Maria Lucia Sampoli, and Alessandra Sestini Mini-Symposium: Recent developments in isogeometric analysis . .....3 High order finite elements: mathematician’s playground or practical engineering tool? Mark Ainsworth INVITED LECTURE..................................... ...............4 High-Order Discontinuous Galerkin methods