Conservation laws for difference equations. Peter Hydon (Surrey) Elizabeth Mansfield (Kent) Olexandr Rasin (Bar-Ilan) Tim Grant (Surrey)
|
|
- James Morgan
- 5 years ago
- Views:
Transcription
1 Conservation laws for difference equations Peter Hydon (Surrey) Elizabeth Mansfield (Kent) Olexandr Rasin (Bar-Ilan) Tim Grant (Surrey)
2 Introduction: What is a conservation law? What is a conservation law?
3 Introduction: What is a conservation law? What is a conservation law? Differential equations: An expression of the form D i (P i (x, [u])) = 0 i that holds on all solutions of the given equation.
4 Introduction: What is a conservation law? What is a conservation law? Differential equations: An expression of the form D i (P i (x, [u])) = 0 i that holds on all solutions of the given equation. Difference equations: Something similar, replacing derivatives by differences? Forwards, backwards, something else?
5 Introduction: What is a conservation law? What is a conservation law? Differential equations: An expression of the form D i (P i (x, [u])) = 0 i that holds on all solutions of the given equation. Difference equations: Something similar, replacing derivatives by differences? Forwards, backwards, something else? Observation: Such expressions can be written in standard form, (S i id) F i (n, [u]) = 0, where S i is the unit forward shift in n i. i
6 Introduction: What is a conservation law? What is a conservation law? Differential equations: An expression of the form D i (P i (x, [u])) = 0 i that holds on all solutions of the given equation. Difference equations: Something similar, replacing derivatives by differences? Forwards, backwards, something else? Observation: Such expressions can be written in standard form, (S i id) F i (n, [u]) = 0, where S i is the unit forward shift in n i. i Question: Why not use equivalent expressions? This one is useful: S i A i (n, [u]) = A i (n, [u]). i i
7 Introduction: What is a conservation law? Answer: The Euler operator E annihilates all divergences; indeed, the variational complex is exact. Λ n 2,0 d H Λ n 1,0 d H Λ n,0 Divergences are the elements of ker(e). E Λ n,1
8 Introduction: What is a conservation law? Answer: The Euler operator E annihilates all divergences; indeed, the variational complex is exact. Λ n 2,0 d H Λ n 1,0 d H Λ n,0 Divergences are the elements of ker(e). E Λ n,1 Difference operators also produce a variational complex; now ker(e) is the set of all standard-form expressions.
9 Quad-graph equations Quad-graph equations u 0,1 α u 1,1 β β u 0,0 α u 1,0 Independent variables: k, l Z; dependent variable u R (or C). Quad-graph equations depend on u i,j = u(k + i, l + j), i, j {0, 1}.
10 Quad-graph equations Quad-graph equations u 0,1 α u 1,1 β β u 0,0 α u 1,0 Independent variables: k, l Z; dependent variable u R (or C). Quad-graph equations depend on u i,j = u(k + i, l + j), i, j {0, 1}. Consistency-on-a-cube implies integrability (ABS classification and others); α = α(k), β = β(l).
11 Quad-graph equations ABS classification Other conditions applied by Adler, Bobenko and Suris (2003): Linearity in each u i,j ;
12 Quad-graph equations ABS classification Other conditions applied by Adler, Bobenko and Suris (2003): Linearity in each u i,j ; D 4 symmetry;
13 Quad-graph equations ABS classification Other conditions applied by Adler, Bobenko and Suris (2003): Linearity in each u i,j ; D 4 symmetry; Tetrahedron property
14 Quad-graph equations ABS classification Other conditions applied by Adler, Bobenko and Suris (2003): Linearity in each u i,j ; D 4 symmetry; Tetrahedron property necessary for 3-leg form variational Toda type system.
15 Quad-graph equations ABS classification Other conditions applied by Adler, Bobenko and Suris (2003): Linearity in each u i,j ; D 4 symmetry; Tetrahedron property necessary for 3-leg form variational Toda type system. ABS classification is up to Möbius transformations of u and point transformations of parameters.
16 Quad-graph equations Q1 : α(u 0,0 u 0,1 )(u 1,0 u 1,1 ) β(u 0,0 u 1,0 )(u 0,1 u 1,1 ) + δ 2 αβ(α β) = 0, Q2 : α(u 0,0 u 0,1 )(u 1,0 u 1,1 ) β(u 0,0 u 1,0 )(u 0,1 u 1,1 ) + αβ(α β)(u 0,0 + u 1,0 + u 0,1 + u 1,1 ) αβ(α β)(α 2 αβ + β 2 ) = 0, Q3 : (β 2 α 2 )(u 0,0 u 1,1 + u 1,0 u 0,1 ) + β(α 2 1)(u 0,0 u 1,0 + u 0,1 u 1,1 ) α(β 2 1)(u 0,0 u 0,1 + u 1,0 u 1,1 ) δ 2 (α 2 β 2 )(α 2 1)(β 2 1)/(4αβ) = 0, Q4 : sn(α)(u 0,0 u 1,0 + u 0,1 u 1,1 ) sn(β)(u 0,0 u 0,1 + u 1,0 u 1,1 ) sn(α β)(u 0,0 u 1,1 + u 1,0 u 0,1 ) +sn(α β)sn(α)sn(β)(1 + K 2 u 0,0 u 1,0 u 0,1 u 1,1 ) = 0, H1 : (u 0,0 u 1,1 )(u 1,0 u 0,1 ) + β α = 0, H2 : (u 0,0 u 1,1 )(u 1,0 u 0,1 ) + (β α)(u 0,0 + u 1,0 + u 0,1 + u 1,1 ) + β 2 α 2 = 0, H3 : α(u 0,0 u 1,0 + u 0,1 u 1,1 ) β(u 0,0 u 0,1 + u 1,0 u 1,1 ) + δ 2 (α 2 β 2 ) = 0, A1 : α(u 0,0 + u 0,1 )(u 1,0 + u 1,1 ) β(u 0,0 + u 1,0 )(u 0,1 + u 1,1 ) δ 2 αβ(α β) = 0, A2 : (β 2 α 2 )(u 0,0 u 1,0 u 0,1 u 1,1 + 1) + β(α 2 1)(u 0,0 u 0,1 + u 1,0 u 1,1 ) α(β 2 1)(u 0,0 u 1,0 + u 0,1 u 1,1 ) = 0.
17 Three-point conservation laws Three-point conservation laws Definition: A conservation law (CLaw) of a quad-graph equation is an expression (S k I )F + (S l I )G = 0 that holds on all solutions of the equation.
18 Three-point conservation laws Three-point conservation laws Definition: A conservation law (CLaw) of a quad-graph equation is an expression (S k I )F + (S l I )G = 0 that holds on all solutions of the equation. Problem: Find a basis for the vector space of all equivalence classes of CLaws.
19 Three-point conservation laws Three-point conservation laws Definition: A conservation law (CLaw) of a quad-graph equation is an expression (S k I )F + (S l I )G = 0 that holds on all solutions of the equation. Problem: Find a basis for the vector space of all equivalence classes of CLaws. Too hard start small.
20 Three-point conservation laws u 0,1 S l G u 1,1 F S k F u 0,0 G u 1,0 The simplest nontrivial CLaws have F = F (k, l, u 0,0, u 0,1 ), G = G(k, l, u 0,0, u 1,0 ).
21 Three-point conservation laws u 0,1 S l G u 1,1 F S k F u 0,0 G u 1,0 The simplest nontrivial CLaws have F = F (k, l, u 0,0, u 0,1 ), G = G(k, l, u 0,0, u 1,0 ). The determining equation for u 1,1 = ω is F (k + 1, l, u 1,0, ω) F (k, l, u 0,0, u 0,1 ) + G(k, l + 1, u 0,1, ω) G(k, l, u 0,0, u 1,0 ) = 0.
22 Three-point conservation laws Solution Method Apply the commuting operators to get L 1 = u 1,0 ω 1,0 ω 0,0, L 2 = ω 0,1 u 0,0 u 0,1 ω 0,0 u 0,0, L 1 L 2 { F (k, l, u0,0, u 0,1 ) + G(k, l, u 0,0, u 1,0 ) } = 0.
23 Three-point conservation laws Solution Method Apply the commuting operators to get L 1 = u 1,0 ω 1,0 ω 0,0, L 2 = ω 0,1 u 0,0 u 0,1 ω 0,0 u 0,0, L 1 L 2 { F (k, l, u0,0, u 0,1 ) + G(k, l, u 0,0, u 1,0 ) } = 0. This reduces to a PDE of order 5 for F. For ABS equations, D 4 symmetry gives extra information.
24 Three-point conservation laws Solution Method Apply the commuting operators to get L 1 = u 1,0 ω 1,0 ω 0,0, L 2 = ω 0,1 u 0,0 u 0,1 ω 0,0 u 0,0, L 1 L 2 { F (k, l, u0,0, u 0,1 ) + G(k, l, u 0,0, u 1,0 ) } = 0. This reduces to a PDE of order 5 for F. For ABS equations, D 4 symmetry gives extra information. Elimination can be made faster for some quad-graph equations.
25 Three-point conservation laws Example: For the dpkdv equation (H1) u 1,1 = u 0,0 + β α u 1,0 u 0,1, the set of three-point CLaws (up to equivalence) is spanned by F 1 = ( 1) k+l ( 2u 0,0 u 0,1 β ), G 1 = ( 1) k+l ( 2u 0,0 u 1,0 α ), F 2 = ( u 0,0 u 0,1 ) ( u0,0 u 0,1 β ), G 2 = ( u 0,0 u 1,0 ) ( u0,0 u 1,0 α ), F 3 = ( 1) k+l ( u 0,0 + u 0,1 ) ( u0,0 u 0,1 β ), G 3 = ( 1) k+l ( u 0,0 + u 1,0 ) ( u0,0 u 1,0 α ), F 4 = ( 1) k+l ( 2u 0,0 2 u0,1 2 4βu0,0 u 0,1 + β 2), G 4 = ( 1) k+l ( 2u 0,0 2 u1,0 2 4αu0,0 u 1,0 + α 2).
26 Three-point conservation laws Example: For the dpkdv equation (H1) u 1,1 = u 0,0 + β α u 1,0 u 0,1, the set of three-point CLaws (up to equivalence) is spanned by F 1 = ( 1) k+l ( 2u 0,0 u 0,1 β ), G 1 = ( 1) k+l ( 2u 0,0 u 1,0 α ), F 2 = ( u 0,0 u 0,1 ) ( u0,0 u 0,1 β ), G 2 = ( u 0,0 u 1,0 ) ( u0,0 u 1,0 α ), F 3 = ( 1) k+l ( u 0,0 + u 0,1 ) ( u0,0 u 0,1 β ), G 3 = ( 1) k+l ( u 0,0 + u 1,0 ) ( u0,0 u 1,0 α ), F 4 = ( 1) k+l ( 2u 0,0 2 u0,1 2 4βu0,0 u 0,1 + β 2), G 4 = ( 1) k+l ( 2u 0,0 2 u1,0 2 4αu0,0 u 1,0 + α 2). Other ABS equations: 4 CLaws : Q1 δ=0 (cross-ratio), A1 δ=0, H3 δ=0 (dpmkdv); 1 CLaw : Q2, Q3 δ=1, Q4 K 2 1; 2 CLaws : all other equations.
27 Five-point conservation laws The same process can be used to find higher conservation laws for u 1,1 = ω(k, l, u 0,0, u 1,0, u 0,1 ). (1)
28 Five-point conservation laws The same process can be used to find higher conservation laws for u 1,1 = ω(k, l, u 0,0, u 1,0, u 0,1 ). (1) 1 Choose the form of F and G.
29 Five-point conservation laws The same process can be used to find higher conservation laws for u 1,1 = ω(k, l, u 0,0, u 1,0, u 0,1 ). (1) 1 Choose the form of F and G. 2 Use (1) and its shifts to write the conservation law in terms of initial values.
30 Five-point conservation laws The same process can be used to find higher conservation laws for u 1,1 = ω(k, l, u 0,0, u 1,0, u 0,1 ). (1) 1 Choose the form of F and G. 2 Use (1) and its shifts to write the conservation law in terms of initial values. 3 Apply appropriate differential operators to reduce the number of unknown functions.
31 Five-point conservation laws The same process can be used to find higher conservation laws for u 1,1 = ω(k, l, u 0,0, u 1,0, u 0,1 ). (1) 1 Choose the form of F and G. 2 Use (1) and its shifts to write the conservation law in terms of initial values. 3 Apply appropriate differential operators to reduce the number of unknown functions. 4 Having reached a PDE, back-substitute and solve the resulting linear difference equations.
32 Five-point conservation laws The same process can be used to find higher conservation laws for u 1,1 = ω(k, l, u 0,0, u 1,0, u 0,1 ). (1) 1 Choose the form of F and G. 2 Use (1) and its shifts to write the conservation law in terms of initial values. 3 Apply appropriate differential operators to reduce the number of unknown functions. 4 Having reached a PDE, back-substitute and solve the resulting linear difference equations. 5 Iterate, if necessary.
33 Five-point conservation laws The same process can be used to find higher conservation laws for u 1,1 = ω(k, l, u 0,0, u 1,0, u 0,1 ). (1) 1 Choose the form of F and G. 2 Use (1) and its shifts to write the conservation law in terms of initial values. 3 Apply appropriate differential operators to reduce the number of unknown functions. 4 Having reached a PDE, back-substitute and solve the resulting linear difference equations. 5 Iterate, if necessary. Potential problems: Expression swell,
34 Five-point conservation laws The same process can be used to find higher conservation laws for u 1,1 = ω(k, l, u 0,0, u 1,0, u 0,1 ). (1) 1 Choose the form of F and G. 2 Use (1) and its shifts to write the conservation law in terms of initial values. 3 Apply appropriate differential operators to reduce the number of unknown functions. 4 Having reached a PDE, back-substitute and solve the resulting linear difference equations. 5 Iterate, if necessary. Potential problems: Expression swell, difficulty solving Es.
35 Five-point conservation laws Five-point conservation laws F G u 0,0 F G u 0,0 F F G G For arbitrary u 1,1 = ω, one can use the staircase.
36 Five-point conservation laws Five-point conservation laws F G u 0,0 F G u 0,0 F F G G For arbitrary u 1,1 = ω, one can use the staircase. For ABS, the cross is more efficient.
37 Five-point conservation laws Example: The dpkdv equation has the following five-point CLaws (modulo three-point and trivial CLaws): F 1 = ln (u 0,1 u 1,0 ), G 1 = ln (u 1,0 u 1,0 ), F 2 = ln (u 0,1 u 0, 1 ), G 2 = ln(u 1,0 u 0, 1 ), F 3 = kf 1 + lf 2, G 3 = kg 1 + lg 2.
38 Five-point conservation laws Example: The dpkdv equation has the following five-point CLaws (modulo three-point and trivial CLaws): F 1 = ln (u 0,1 u 1,0 ), G 1 = ln (u 1,0 u 1,0 ), F 2 = ln (u 0,1 u 0, 1 ), G 2 = ln(u 1,0 u 0, 1 ), F 3 = kf 1 + lf 2, G 3 = kg 1 + lg 2. For constant α, β: every ABS equation has something similar (at least).
39 What more can be done? What more can be done? Direct computation of five-point CLaws is the limit (for now).
40 What more can be done? What more can be done? Direct computation of five-point CLaws is the limit (for now). However, the action of symmetries on CLaws may give new ones.
41 What more can be done? What more can be done? Direct computation of five-point CLaws is the limit (for now). However, the action of symmetries on CLaws may give new ones. Example: For dpkdv, apply the mastersymmetry to X 8 = k u 1,0 u 1,0 u0,0 α F 1 = ln (u 0,1 u 1,0 ), G 1 = ln (u 1,0 u 1,0 ), to get (up to equivalence) the new CLaw F = 1 (u 0,0 u 2,0 )(u 0,1 u 1,0 ), G = 1 (u 0,0 u 2,0 )(u 1,0 u 1,0 ).
42 What more can be done? What more can be done? Direct computation of five-point CLaws is the limit (for now). However, the action of symmetries on CLaws may give new ones. Example: For dpkdv, apply the mastersymmetry to X 8 = k u 1,0 u 1,0 u0,0 α F 1 = ln (u 0,1 u 1,0 ), G 1 = ln (u 1,0 u 1,0 ), to get (up to equivalence) the new CLaw F = 1 (u 0,0 u 2,0 )(u 0,1 u 1,0 ), G = 1 (u 0,0 u 2,0 )(u 1,0 u 1,0 ). Warning: apparent novelty must be checked; use characteristics.
43 What more can be done? Another approach An alternative approach to generating possibly infinite hierarchies of CLaws is as follows:
44 What more can be done? Another approach An alternative approach to generating possibly infinite hierarchies of CLaws is as follows: 1 Find symmetries and mastersymmetries for Toda type systems (building on quad-graph results).
45 What more can be done? Another approach An alternative approach to generating possibly infinite hierarchies of CLaws is as follows: 1 Find symmetries and mastersymmetries for Toda type systems (building on quad-graph results). 2 Check which (if any) are variational symmetries.
46 What more can be done? Another approach An alternative approach to generating possibly infinite hierarchies of CLaws is as follows: 1 Find symmetries and mastersymmetries for Toda type systems (building on quad-graph results). 2 Check which (if any) are variational symmetries. 3 Construct CLaws from variational symmetries with the discrete version of Noether s Theorem.
47 What more can be done? Another approach An alternative approach to generating possibly infinite hierarchies of CLaws is as follows: 1 Find symmetries and mastersymmetries for Toda type systems (building on quad-graph results). 2 Check which (if any) are variational symmetries. 3 Construct CLaws from variational symmetries with the discrete version of Noether s Theorem. Drawback: The resulting CLaws found so far are quite messy!
48 Summary and open problems Summary For each of the ABS equations with constant parameters, the direct approach has yielded infinite hierarchies of symmetries;
49 Summary and open problems Summary For each of the ABS equations with constant parameters, the direct approach has yielded infinite hierarchies of symmetries; low-order conservation laws;
50 Summary and open problems Summary For each of the ABS equations with constant parameters, the direct approach has yielded infinite hierarchies of symmetries; low-order conservation laws; the possibility of generating higher-order conservation laws.
51 Summary and open problems Summary For each of the ABS equations with constant parameters, the direct approach has yielded infinite hierarchies of symmetries; low-order conservation laws; the possibility of generating higher-order conservation laws. However, there are no five-point symmetries or CLaws if neither α nor β are constant.
52 Summary and open problems Open Problems Are there any higher symmetries and CLaws when α and β are not constant?
53 Summary and open problems Open Problems Are there any higher symmetries and CLaws when α and β are not constant? Does each of the ABS equations have infinitely many CLaws (up to equivalence)?
54 Summary and open problems Open Problems Are there any higher symmetries and CLaws when α and β are not constant? Does each of the ABS equations have infinitely many CLaws (up to equivalence)? If so, is this a property shared by all integrable quad graphs?
55 Summary and open problems Open Problems Are there any higher symmetries and CLaws when α and β are not constant? Does each of the ABS equations have infinitely many CLaws (up to equivalence)? If so, is this a property shared by all integrable quad graphs? Is there a one-to-one correspondence between variational symmetries and CLaws (up to equivalence)?
56 Summary and open problems The End
Symmetry Reductions of Integrable Lattice Equations
Isaac Newton Institute for Mathematical Sciences Discrete Integrable Systems Symmetry Reductions of Integrable Lattice Equations Pavlos Xenitidis University of Patras Greece March 11, 2009 Pavlos Xenitidis
More informationarxiv: v2 [math-ph] 24 Feb 2016
ON THE CLASSIFICATION OF MULTIDIMENSIONALLY CONSISTENT 3D MAPS MATTEO PETRERA AND YURI B. SURIS Institut für Mathemat MA 7-2 Technische Universität Berlin Str. des 17. Juni 136 10623 Berlin Germany arxiv:1509.03129v2
More informationGEOMETRY OF DISCRETE INTEGRABILITY. THE CONSISTENCY APPROACH
GEOMETRY OF DISCRETE INTEGRABILITY. THE CONSISTENCY APPROACH Alexander I. Bobenko Institut für Mathematik, Fakultät 2, Technische Universität Berlin, Strasse des 17. Juni 136, 10623 Berlin, Germany 1 ORIGIN
More informationSymmetry Methods for Differential and Difference Equations. Peter Hydon
Lecture 2: How to find Lie symmetries Symmetry Methods for Differential and Difference Equations Peter Hydon University of Kent Outline 1 Reduction of order for ODEs and O Es 2 The infinitesimal generator
More informationRelativistic Collisions as Yang Baxter maps
Relativistic Collisions as Yang Baxter maps Theodoros E. Kouloukas arxiv:706.0636v2 [math-ph] 7 Sep 207 School of Mathematics, Statistics & Actuarial Science, University of Kent, UK September 9, 207 Abstract
More informationON THE SYMMETRIES OF INTEGRABLE PARTIAL DIFFERENCE EQUATIONS
Proceedings of the International Conference on Difference Equations, Special Functions and Orthogonal Polynomials, World Scientific (2007 ON THE SYMMETRIES OF INTEGRABLE PARTIAL DIFFERENCE EQUATIONS ANASTASIOS
More informationOn Miura Transformations and Volterra-Type Equations Associated with the Adler Bobenko Suris Equations
Symmetry, Integrability and Geometry: Methods and Applications On Miura Transformations and Volterra-Type Equations Associated with the Adler Bobenko Suris Equations SIGMA 4 (2008), 077, 14 pages Decio
More informationConservation laws and symmetries of difference equations. Submitted for the degree of Doctor of Philosophy at The University of Surrey
Conservation laws and symmetries of difference equations 1 Olexandr G. -Rasin, MSc Hons. Submitted for the degree of Doctor of Philosophy at The University of Surrey Supervisor: Prof. P.E. Hydon Co-supervisor:
More informationFrom discrete differential geometry to the classification of discrete integrable systems
From discrete differential geometry to the classification of discrete integrable systems Vsevolod Adler,, Yuri Suris Technische Universität Berlin Quantum Integrable Discrete Systems, Newton Institute,
More informationMATH SOLUTIONS TO PRACTICE MIDTERM LECTURE 1, SUMMER Given vector spaces V and W, V W is the vector space given by
MATH 110 - SOLUTIONS TO PRACTICE MIDTERM LECTURE 1, SUMMER 2009 GSI: SANTIAGO CAÑEZ 1. Given vector spaces V and W, V W is the vector space given by V W = {(v, w) v V and w W }, with addition and scalar
More informationarxiv:nlin/ v2 [nlin.si] 22 Sep 2006
Classification of integrable Hamiltonian hydrodynamic chains associated with Kupershmidt s brackets E.V. Ferapontov K.R. Khusnutdinova D.G. Marshall and M.V. Pavlov arxiv:nlin/0607003v2 [nlin.si] 22 Sep
More informationA variational perspective on continuum limits of ABS and lattice GD equations
A variational perspective on continuum limits of ABS and lattice GD equations Mats Vermeeren Technische Universität Berlin vermeeren@mathtu-berlinde arxiv:80855v [nlinsi] 5 Nov 08 Abstract A pluri-lagrangian
More informationSymbolic Computation of Lax Pairs of Systems of Partial Difference Equations Using Consistency Around the Cube
Symbolic Computation of Lax Pairs of Systems of Partial Difference Equations Using Consistency Around the Cube Willy Hereman Department of Applied Mathematics and Statistics Colorado School of Mines Golden,
More informationSymbolic Computation of Lax Pairs of Nonlinear Systems of Partial Difference Equations using Multidimensional Consistency
Symbolic Computation of Lax Pairs of Nonlinear Systems of Partial Difference Equations using Multidimensional Consistency Willy Hereman Department of Applied Mathematics and Statistics Colorado School
More informationNOTES ON SPLITTING FIELDS
NOTES ON SPLITTING FIELDS CİHAN BAHRAN I will try to define the notion of a splitting field of an algebra over a field using my words, to understand it better. The sources I use are Peter Webb s and T.Y
More informationClassification of integrable equations on quad-graphs. The consistency approach
arxiv:nlin/0202024v2 [nlin.si] 10 Jul 2002 Classification of integrable equations on quad-graphs. The consistency approach V.E. Adler A.I. Bobenko Yu.B. Suris Abstract A classification of discrete integrable
More informationOn the Linearization of Second-Order Dif ferential and Dif ference Equations
Symmetry, Integrability and Geometry: Methods and Applications Vol. (006), Paper 065, 15 pages On the Linearization of Second-Order Dif ferential and Dif ference Equations Vladimir DORODNITSYN Keldysh
More informationarxiv: v1 [nlin.si] 17 Oct 2010
INTEGRABILITY TEST FOR DISCRETE EQUATIONS VIA GENERALIZED SYMMETRIES. D. LEVI AND R.I. YAMILOV Abstract. In this article we present some integrability conditions for partial difference equations obtained
More informationDiscrete differential geometry. Integrability as consistency
Discrete differential geometry. Integrability as consistency Alexander I. Bobenko Institut für Mathematik, Fakultät 2, Technische Universität Berlin, Strasse des 17. Juni 136, 10623 Berlin, Germany. bobenko@math.tu-berlin.de
More informationLinearization of Second-Order Ordinary Dif ferential Equations by Generalized Sundman Transformations
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 6 (2010), 051, 11 pages Linearization of Second-Order Ordinary Dif ferential Equations by Generalized Sundman Transformations Warisa
More informationarxiv:nlin/ v1 [nlin.si] 16 Nov 2003
Centre de Recherches Mathématiques CRM Proceedings and Lecture Notes arxiv:nlin/0311029v1 [nlinsi] 16 Nov 2003 Symbolic Computation of Conserved Densities, Generalized Symmetries, and Recursion Operators
More information(3) Let Y be a totally bounded subset of a metric space X. Then the closure Y of Y
() Consider A = { q Q : q 2 2} as a subset of the metric space (Q, d), where d(x, y) = x y. Then A is A) closed but not open in Q B) open but not closed in Q C) neither open nor closed in Q D) both open
More informationCohomology and Vector Bundles
Cohomology and Vector Bundles Corrin Clarkson REU 2008 September 28, 2008 Abstract Vector bundles are a generalization of the cross product of a topological space with a vector space. Characteristic classes
More informationLinear Algebra Review
Chapter 1 Linear Algebra Review It is assumed that you have had a course in linear algebra, and are familiar with matrix multiplication, eigenvectors, etc. I will review some of these terms here, but quite
More informationLinear equations in linear algebra
Linear equations in linear algebra Samy Tindel Purdue University Differential equations and linear algebra - MA 262 Taken from Differential equations and linear algebra Pearson Collections Samy T. Linear
More informationarxiv: v1 [nlin.si] 29 Jun 2009
Lagrangian multiform structure for the lattice KP system arxiv:0906.5282v1 [nlin.si] 29 Jun 2009 1. Introduction S.B. Lobb 1, F.W. Nijhoff 1 and G.R.W. Quispel 2 1 Department of Applied Mathematics, University
More informationHilbert Spaces. Hilbert space is a vector space with some extra structure. We start with formal (axiomatic) definition of a vector space.
Hilbert Spaces Hilbert space is a vector space with some extra structure. We start with formal (axiomatic) definition of a vector space. Vector Space. Vector space, ν, over the field of complex numbers,
More informationChapter 2: Approximating Solutions of Linear Systems
Linear of Chapter 2: Solutions of Linear Peter W. White white@tarleton.edu Department of Mathematics Tarleton State University Summer 2015 / Numerical Analysis Overview Linear of Linear of Linear of Linear
More informationPractice Final Solutions. 1. Consider the following algorithm. Assume that n 1. line code 1 alg(n) { 2 j = 0 3 if (n = 0) { 4 return j
Practice Final Solutions 1. Consider the following algorithm. Assume that n 1. line code 1 alg(n) 2 j = 0 3 if (n = 0) 4 return j } 5 else 6 j = 2n+ alg(n 1) 7 return j } } Set up a recurrence relation
More informationDispersionless integrable systems in 3D and Einstein-Weyl geometry. Eugene Ferapontov
Dispersionless integrable systems in 3D and Einstein-Weyl geometry Eugene Ferapontov Department of Mathematical Sciences, Loughborough University, UK E.V.Ferapontov@lboro.ac.uk Based on joint work with
More informationLinear Equations in Linear Algebra
Linear Equations in Linear Algebra.7 LINEAR INDEPENDENCE LINEAR INDEPENDENCE Definition: An indexed set of vectors {v,, v p } in n is said to be linearly independent if the vector equation x x x 2 2 p
More informationComputational Techniques Prof. Sreenivas Jayanthi. Department of Chemical Engineering Indian institute of Technology, Madras
Computational Techniques Prof. Sreenivas Jayanthi. Department of Chemical Engineering Indian institute of Technology, Madras Module No. # 05 Lecture No. # 24 Gauss-Jordan method L U decomposition method
More informationThe Hamiltonian operator and states
The Hamiltonian operator and states March 30, 06 Calculation of the Hamiltonian operator This is our first typical quantum field theory calculation. They re a bit to keep track of, but not really that
More informationCS481F01 Solutions 8
CS481F01 Solutions 8 A. Demers 7 Dec 2001 1. Prob. 111 from p. 344 of the text. One of the following sets is r.e. and the other is not. Which is which? (a) { i L(M i ) contains at least 481 elements }
More informationSupplement. The Alternating Groups A n are Simple for n 5
A n is Simple for n 5 Supplement 1 Supplement. The Alternating Groups A n are Simple for n 5 Note. Recall that a group is simple if it is nontrivial and has no proper nontrivial normal subgroups. In this
More informationPart II. Geometry and Groups. Year
Part II Year 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2014 Paper 4, Section I 3F 49 Define the limit set Λ(G) of a Kleinian group G. Assuming that G has no finite orbit in H 3 S 2, and that Λ(G),
More informationLECTURE 28: VECTOR BUNDLES AND FIBER BUNDLES
LECTURE 28: VECTOR BUNDLES AND FIBER BUNDLES 1. Vector Bundles In general, smooth manifolds are very non-linear. However, there exist many smooth manifolds which admit very nice partial linear structures.
More informationGroup Actions and Cohomology in the Calculus of Variations
Group Actions and Cohomology in the Calculus of Variations JUHA POHJANPELTO Oregon State and Aalto Universities Focused Research Workshop on Exterior Differential Systems and Lie Theory Fields Institute,
More informationOn universality of critical behaviour in Hamiltonian PDEs
Riemann - Hilbert Problems, Integrability and Asymptotics Trieste, September 23, 2005 On universality of critical behaviour in Hamiltonian PDEs Boris DUBROVIN SISSA (Trieste) 1 Main subject: Hamiltonian
More informationWeek 5-6: Lectures The Charged Scalar Field
Notes for Phys. 610, 2011. These summaries are meant to be informal, and are subject to revision, elaboration and correction. They will be based on material covered in class, but may differ from it by
More informationFrom Yang-Baxter maps to integrable quad maps and recurrences
From Yang-Baxter maps to integrable quad maps and recurrences arxiv:1206.1217v1 [nlin.si] 6 Jun 2012 B. Grammaticos IMNC, Universités Paris VII & XI, CNRS UMR 8165 Bât. 440, 91406 Orsay, France A. Ramani
More informationMA106 Linear Algebra lecture notes
MA106 Linear Algebra lecture notes Lecturers: Diane Maclagan and Damiano Testa 2017-18 Term 2 Contents 1 Introduction 3 2 Matrix review 3 3 Gaussian Elimination 5 3.1 Linear equations and matrices.......................
More informationChapter 7 Network Flow Problems, I
Chapter 7 Network Flow Problems, I Network flow problems are the most frequently solved linear programming problems. They include as special cases, the assignment, transportation, maximum flow, and shortest
More informationSymbolic Computation of Lax Pairs of Integrable Nonlinear Partial Difference Equations on Quad-Graphs
Symbolic Computation of Lax Pairs of Integrable Nonlinear Partial Difference Equations on Quad-Graphs Willy Hereman & Terry Bridgman Department of Mathematical and Computer Sciences Colorado School of
More information2 of PU_2015_ of PU_2015_ of PU_2015_118
PU Ph D Mathematics Error! Not a valid embedded object.error! Not a valid embedded object.error! Not a valid embedded object.error! Not a valid embedded object.error! Not a valid embedded object.error!
More informationUNIT 3 REASONING WITH EQUATIONS Lesson 2: Solving Systems of Equations Instruction
Prerequisite Skills This lesson requires the use of the following skills: graphing equations of lines using properties of equality to solve equations Introduction Two equations that are solved together
More informationAutomata on linear orderings
Automata on linear orderings Véronique Bruyère Institut d Informatique Université de Mons-Hainaut Olivier Carton LIAFA Université Paris 7 September 25, 2006 Abstract We consider words indexed by linear
More informationDesign and Analysis of Algorithms
CSE 101, Winter 2018 Design and Analysis of Algorithms Lecture 5: Divide and Conquer (Part 2) Class URL: http://vlsicad.ucsd.edu/courses/cse101-w18/ A Lower Bound on Convex Hull Lecture 4 Task: sort the
More informationSuperintegrability? Hidden linearity? Classical quantization? Symmetries and more symmetries!
Superintegrability? Hidden linearity? Classical quantization? Symmetries and more symmetries! Maria Clara Nucci University of Perugia & INFN-Perugia, Italy Conference on Nonlinear Mathematical Physics:
More informationSection Gaussian Elimination
Section. - Gaussian Elimination A matrix is said to be in row echelon form (REF) if it has the following properties:. The first nonzero entry in any row is a. We call this a leading one or pivot one..
More informationSIGNATURES OVER FINITE FIELDS OF GROWTH PROPERTIES FOR LATTICE EQUATIONS
SIGNATURES OVER FINITE FIELDS OF GROWTH PROPERTIES FOR LATTICE EQUATIONS JOHN A. G. ROBERTS 1, DINH T. TRAN 1,2 Abstract. We study integrable lattice equations and their perturbations over finite fields.
More informationLinear Equations in Linear Algebra
1 Linear Equations in Linear Algebra 1.7 LINEAR INDEPENDENCE LINEAR INDEPENDENCE Definition: An indexed set of vectors {v 1,, v p } in n is said to be linearly independent if the vector equation x x x
More informationA Brief Introduction to Numerical Methods for Differential Equations
A Brief Introduction to Numerical Methods for Differential Equations January 10, 2011 This tutorial introduces some basic numerical computation techniques that are useful for the simulation and analysis
More informationSymmetries and Polynomials
Symmetries and Polynomials Aaron Landesman and Apurva Nakade June 30, 2018 Introduction In this class we ll learn how to solve a cubic. We ll also sketch how to solve a quartic. We ll explore the connections
More informationTo hand in: (a) Prove that a group G is abelian (= commutative) if and only if (xy) 2 = x 2 y 2 for all x, y G.
Homework #6. Due Thursday, October 14th Reading: For this homework assignment: Sections 3.3 and 3.4 (up to page 167) Before the class next Thursday: Sections 3.5 and 3.4 (pp. 168-171). Also review the
More informationSymmetries and Group Invariant Reductions of Integrable Partial Difference Equations
Proceedings of 0th International Conference in MOdern GRoup ANalysis 2005, 222 230 Symmetries and Group Invariant Reductions of Integrable Partial Difference Equations A. TONGAS, D. TSOUBELIS and V. PAPAGEORGIOU
More informationME Computational Fluid Mechanics Lecture 5
ME - 733 Computational Fluid Mechanics Lecture 5 Dr./ Ahmed Nagib Elmekawy Dec. 20, 2018 Elliptic PDEs: Finite Difference Formulation Using central difference formulation, the so called five-point formula
More informationThe Gauss-Jordan Elimination Algorithm
The Gauss-Jordan Elimination Algorithm Solving Systems of Real Linear Equations A. Havens Department of Mathematics University of Massachusetts, Amherst January 24, 2018 Outline 1 Definitions Echelon Forms
More informationPaths, cycles, trees and sub(di)graphs in directed graphs
Paths, cycles, trees and sub(di)graphs in directed graphs Jørgen Bang-Jensen University of Southern Denmark Odense, Denmark Paths, cycles, trees and sub(di)graphs in directed graphs p. 1/53 Longest paths
More informationOutline Inverse of a Relation Properties of Relations. Relations. Alice E. Fischer. April, 2018
Relations Alice E. Fischer April, 2018 1 Inverse of a Relation 2 Properties of Relations The Inverse of a Relation Let R be a relation from A to B. Define the inverse relation, R 1 from B to A as follows.
More informationNumerical algorithms for one and two target optimal controls
Numerical algorithms for one and two target optimal controls Sung-Sik Kwon Department of Mathematics and Computer Science, North Carolina Central University 80 Fayetteville St. Durham, NC 27707 Email:
More informationVector Spaces. Vector space, ν, over the field of complex numbers, C, is a set of elements a, b,..., satisfying the following axioms.
Vector Spaces Vector space, ν, over the field of complex numbers, C, is a set of elements a, b,..., satisfying the following axioms. For each two vectors a, b ν there exists a summation procedure: a +
More informationLecture 1: Systems of linear equations and their solutions
Lecture 1: Systems of linear equations and their solutions Course overview Topics to be covered this semester: Systems of linear equations and Gaussian elimination: Solving linear equations and applications
More informationLinear Algebra Review
Chapter 1 Linear Algebra Review It is assumed that you have had a beginning course in linear algebra, and are familiar with matrix multiplication, eigenvectors, etc I will review some of these terms here,
More informationarxiv:math/ v1 [math.qa] 8 May 2006
Yang-Baxter maps and symmetries of integrable equations on quad-graphs Vassilios G. Papageorgiou 1,2, Anastasios G. Tongas 3 and Alexander P. Veselov 4,5 arxiv:math/0605206v1 [math.qa] 8 May 2006 1 Department
More informationFractals and Fractal Dimensions
Fractals and Fractal Dimensions John A. Rock July 13th, 2009 begin with [0,1] remove 1 of length 1/3 remove 2 of length 1/9 remove 4 of length 1/27 remove 8 of length 1/81...................................
More information118 PU Ph D Mathematics
118 PU Ph D Mathematics 1 of 100 146 PU_2016_118_E The function fz = z is:- not differentiable anywhere differentiable on real axis differentiable only at the origin differentiable everywhere 2 of 100
More informationLecture III: Tensor calculus and electrodynamics in flat spacetime
Lecture III: Tensor calculus and electrodynamics in flat spacetime Christopher M. Hirata Caltech M/C 350-17, Pasadena CA 91125, USA (Dated: October 5, 201 I. OVERVIEW In this lecture we will continue to
More informationContinuous and Discrete Homotopy Operators with Applications in Integrability Testing. Willy Hereman
Continuous and Discrete Homotopy Operators with Applications in Integrability Testing Willy Hereman Department of Mathematical and Computer Sciences Colorado School of Mines Golden, Colorado, USA http://www.mines.edu/fs
More informationDiffeomorphism Invariant Gauge Theories
Diffeomorphism Invariant Gauge Theories Kirill Krasnov (University of Nottingham) Oxford Geometry and Analysis Seminar 25 Nov 2013 Main message: There exists a large new class of gauge theories in 4 dimensions
More information(x 1 +x 2 )(x 1 x 2 )+(x 2 +x 3 )(x 2 x 3 )+(x 3 +x 1 )(x 3 x 1 ).
CMPSCI611: Verifying Polynomial Identities Lecture 13 Here is a problem that has a polynomial-time randomized solution, but so far no poly-time deterministic solution. Let F be any field and let Q(x 1,...,
More informationModern Algebra Prof. Manindra Agrawal Department of Computer Science and Engineering Indian Institute of Technology, Kanpur
Modern Algebra Prof. Manindra Agrawal Department of Computer Science and Engineering Indian Institute of Technology, Kanpur Lecture - 05 Groups: Structure Theorem So, today we continue our discussion forward.
More informationIf every Bounded Input produces Bounded Output, system is externally stable equivalently, system is BIBO stable
1. External (BIBO) Stability of LTI Systems If every Bounded Input produces Bounded Output, system is externally stable equivalently, system is BIBO stable g(n) < BIBO Stability Don t care about what unbounded
More informationConstraints on Fluid Dynamics From Equilibrium Partition Func
Constraints on Fluid Dynamics From Equilibrium Partition Function Nabamita Banerjee Nikhef, Amsterdam LPTENS, Paris 1203.3544, 1206.6499 J. Bhattacharya, S. Jain, S. Bhattacharyya, S. Minwalla, T. Sharma,
More informationarxiv:nlin/ v1 [nlin.si] 17 Sep 2006
Seed and soliton solutions for Adler s lattice equation arxiv:nlin/0609044v1 [nlin.si] 17 Sep 2006 1. Introduction James Atkinson 1, Jarmo Hietarinta 2 and Frank Nijhoff 1 1 Department of Applied Mathematics,
More informationChapter 1. Vectors, Matrices, and Linear Spaces
1.6 Homogeneous Systems, Subspaces and Bases 1 Chapter 1. Vectors, Matrices, and Linear Spaces 1.6. Homogeneous Systems, Subspaces and Bases Note. In this section we explore the structure of the solution
More informationSoliton solutions to the ABS list
to the ABS list Department of Physics, University of Turku, FIN-20014 Turku, Finland in collaboration with James Atkinson, Frank Nijhoff and Da-jun Zhang DIS-INI, February 2009 The setting CAC The setting
More informationChapter 2 Notes, Linear Algebra 5e Lay
Contents.1 Operations with Matrices..................................1.1 Addition and Subtraction.............................1. Multiplication by a scalar............................ 3.1.3 Multiplication
More informationMATH45061: SOLUTION SHEET 1 V
1 MATH4561: SOLUTION SHEET 1 V 1.) a.) The faces of the cube remain aligned with the same coordinate planes. We assign Cartesian coordinates aligned with the original cube (x, y, z), where x, y, z 1. The
More informationNotes on graduate algebra. Robert Harron
Notes on graduate algebra Robert Harron Department of Mathematics, Keller Hall, University of Hawai i at Mānoa, Honolulu, HI 96822, USA E-mail address: rharron@math.hawaii.edu Abstract. Graduate algebra
More informationLinear Algebra Lecture Notes
Linear Algebra Lecture Notes Lecturers: Inna Capdeboscq and Damiano Testa Warwick, January 2017 Contents 1 Number Systems and Fields 3 1.1 Axioms for number systems............................ 3 2 Vector
More informationLP relaxation and Tree Packing for Minimum k-cuts
LP relaxation and Tree Packing for Minimum k-cuts Chandra Chekuri 1 Kent Quanrud 1 Chao Xu 1,2 1 UIUC 2 Yahoo! Research, NYC k-cut Graph G = (V, E). c : E R + a capacity function. A k-cut is the set of
More informationThe classification of root systems
The classification of root systems Maris Ozols University of Waterloo Department of C&O November 28, 2007 Definition of the root system Definition Let E = R n be a real vector space. A finite subset R
More informationConvolutional networks. Sebastian Seung
Convolutional networks Sebastian Seung Convolutional network Neural network with spatial organization every neuron has a location usually on a grid Translation invariance synaptic strength depends on locations
More informationApplication of Euler and Homotopy Operators to Integrability Testing
Application of Euler and Homotopy Operators to Integrability Testing Willy Hereman Department of Applied Mathematics and Statistics Colorado School of Mines Golden, Colorado, U.S.A. http://www.mines.edu/fs
More informationPart III Symmetries, Fields and Particles
Part III Symmetries, Fields and Particles Theorems Based on lectures by N. Dorey Notes taken by Dexter Chua Michaelmas 2016 These notes are not endorsed by the lecturers, and I have modified them (often
More informationb 0 + b 1 z b d z d
I. Introduction Definition 1. For z C, a rational function of degree d is any with a d, b d not both equal to 0. R(z) = P (z) Q(z) = a 0 + a 1 z +... + a d z d b 0 + b 1 z +... + b d z d It is exactly
More informationFinal Exam Practice Problems Answers Math 24 Winter 2012
Final Exam Practice Problems Answers Math 4 Winter 0 () The Jordan product of two n n matrices is defined as A B = (AB + BA), where the products inside the parentheses are standard matrix product. Is the
More informationDiscrete Differential Geometry: Consistency as Integrability
Discrete Differential Geometry: Consistency as Integrability Yuri SURIS (TU München) Oberwolfach, March 6, 2006 Based on the ongoing textbook with A. Bobenko Discrete Differential Geometry Differential
More informationLinear Algebra (Part II) Vector Spaces, Independence, Span and Bases
Linear Algebra (Part II) Vector Spaces, Independence, Span and Bases A vector space, or sometimes called a linear space, is an abstract system composed of a set of objects called vectors, an associated
More informationSYMBOLIC COMPUTATION OF LAX PAIRS OF NONLINEAR PARTIAL DIFFERENCE EQUATIONS
SYMBOLIC COMPUTATION OF LAX PAIRS OF NONLINEAR PARTIAL DIFFERENCE EQUATIONS by Terry J. Bridgman c Copyright by Terry J. Bridgman, 2018 All Rights Reserved A thesis submitted to the Faculty and the Board
More informationAutomorphism groups of wreath product digraphs
Automorphism groups of wreath product digraphs Edward Dobson Department of Mathematics and Statistics Mississippi State University PO Drawer MA Mississippi State, MS 39762 USA dobson@math.msstate.edu Joy
More informationComputability Theoretic Properties of Injection Structures
Computability Theoretic Properties of Injection Structures Douglas Cenzer 1, Valentina Harizanov 2 and Jeffrey B. Remmel 3 Abstract We study computability theoretic properties of computable injection structures
More informationAlexei F. Cheviakov. University of Saskatchewan, Saskatoon, Canada. INPL seminar June 09, 2011
Direct Method of Construction of Conservation Laws for Nonlinear Differential Equations, its Relation with Noether s Theorem, Applications, and Symbolic Software Alexei F. Cheviakov University of Saskatchewan,
More informationGraph Transformations T1 and T2
Graph Transformations T1 and T2 We now introduce two graph transformations T1 and T2. Reducibility by successive application of these two transformations is equivalent to reducibility by intervals. The
More informationOrder 3 elements in G 2 and idempotents in symmetric composition algebras. Alberto Elduque
Order 3 elements in G 2 and idempotents in symmetric composition algebras Alberto Elduque 1 Symmetric composition algebras 2 Classification 3 Idempotents and order 3 automorphisms, char F 3 4 Idempotents
More informationPETER A. CHOLAK, PETER GERDES, AND KAREN LANGE
D-MAXIMAL SETS PETER A. CHOLAK, PETER GERDES, AND KAREN LANGE Abstract. Soare [23] proved that the maximal sets form an orbit in E. We consider here D-maximal sets, generalizations of maximal sets introduced
More informationLinear Independence. Consider a plane P that includes the origin in R 3 {u, v, w} in P. and non-zero vectors
Linear Independence Consider a plane P that includes the origin in R 3 {u, v, w} in P. and non-zero vectors If no two of u, v and w are parallel, then P =span{u, v, w}. But any two vectors determines a
More informationMIT Algebraic techniques and semidefinite optimization May 9, Lecture 21. Lecturer: Pablo A. Parrilo Scribe:???
MIT 6.972 Algebraic techniques and semidefinite optimization May 9, 2006 Lecture 2 Lecturer: Pablo A. Parrilo Scribe:??? In this lecture we study techniques to exploit the symmetry that can be present
More informationPhysics 342 Lecture 2. Linear Algebra I. Lecture 2. Physics 342 Quantum Mechanics I
Physics 342 Lecture 2 Linear Algebra I Lecture 2 Physics 342 Quantum Mechanics I Wednesday, January 27th, 21 From separation of variables, we move to linear algebra Roughly speaking, this is the study
More information