Progress Report D. Parker, M. Puljiz, J. E. Rowe UoB Maths Misc. - - PowerPoint PPT Presentation

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Progress Report D. Parker, M. Puljiz, J. E. Rowe UoB Maths Misc. - - PowerPoint PPT Presentation

Jan 15, 2024 396 likes •512 views

UoB Maths C. Good, N. Kamaleson, C. Mu, Progress Report D. Parker, M. Puljiz, J. E. Rowe UoB Maths Misc. Problem 1 University of Birmingham Problem 2 Chris Good, Nishanthan Kamaleson, Chunyan Mu, References David Parker, Mate Puljiz,

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SLIDE 1

UoB Maths

  • C. Good,
  • N. Kamaleson,
  • C. Mu,
  • D. Parker,
  • M. Puljiz,
  • J. E. Rowe

Misc. Problem 1 Problem 2 References

Progress Report UoB Maths

University of Birmingham Chris Good, Nishanthan Kamaleson, Chunyan Mu, David Parker, Mate Puljiz, Jonathan E. Rowe Birmingham, 29th April 2015

slide-2
SLIDE 2

UoB Maths

  • C. Good,
  • N. Kamaleson,
  • C. Mu,
  • D. Parker,
  • M. Puljiz,
  • J. E. Rowe

Misc. Problem 1 Problem 2 References

Lattices for Lagrange interpolation

Recall: Finding a valid linear coarse graining = Finding a common invariant subspace of the differential DT|x Jacobi - Evaluate at random points If we know that T is a polynomial then e.g. Ben Yaacov and also Chung, Yao give sets of points that suffices to check

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SLIDE 3

UoB Maths

  • C. Good,
  • N. Kamaleson,
  • C. Mu,
  • D. Parker,
  • M. Puljiz,
  • J. E. Rowe

Misc. Problem 1 Problem 2 References

Frobenius theorem, Integrability, Foliations

Continuous time dynamics on a manifold ˙ x = F(x) regular1 coarse grainings correspond to foliations Theorem (Frobenius’ theorem) A subbundle E is integrable if and only if it arises from a regular foliation of M. (Dimension of leaves + Dimension of E = Dimension of M) Taking F as the subbundle, FT gives a sufficient condition for existence of coarse graining onto a 1 dimensional manifold, projecting perpendicular to F

1level sets are manifolds of constant dimension

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SLIDE 4

UoB Maths

  • C. Good,
  • N. Kamaleson,
  • C. Mu,
  • D. Parker,
  • M. Puljiz,
  • J. E. Rowe

Misc. Problem 1 Problem 2 References

Integrability condition on F is d F ∧ F = 0 E.g. in R3 this amounts to requiring (∇ × F) · F = 0 But what about CG that aren’t perpendicular to F? The extensions seem plausible (and possibly already known).

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SLIDE 5

UoB Maths

  • C. Good,
  • N. Kamaleson,
  • C. Mu,
  • D. Parker,
  • M. Puljiz,
  • J. E. Rowe

Misc. Problem 1 Problem 2 References

Dense periodic points in cellular automata

Alphabet A = {1, 2, . . . , n} The full shift ΣA = AZ The shift map σ : ΣA → ΣA σ((ξi)i) = (ξi+1)i A cellular automaton is a continuous σ-commuting map F : ΣA → ΣA. Equivalently ∃k ∈ N and Φ : A2k+1 → A s.t. (F((ξi)i))j = Φ(ξj−k, . . . ξj+k)

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SLIDE 6

UoB Maths

  • C. Good,
  • N. Kamaleson,
  • C. Mu,
  • D. Parker,
  • M. Puljiz,
  • J. E. Rowe

Misc. Problem 1 Problem 2 References

’It is an open problem whether all surjective CA, or at least all transitive CA, have a dense set of properly periodic points.’ Blanchard Consider the coarse graining π : ΣA → ΣA2k+1 which arranges entries of ξ in groups of 2k + 1 It follows that π conjugates F on ΣA to a cellular automaton G on ΣA2k+1 but G has depth k = 1. F is surjective/has periodic points dense iff G is surjective/has periodic points dense

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SLIDE 7

UoB Maths

  • C. Good,
  • N. Kamaleson,
  • C. Mu,
  • D. Parker,
  • M. Puljiz,
  • J. E. Rowe

Misc. Problem 1 Problem 2 References

Reaction networks

Organisation theory

  • rganisation = closed + self-maintaining

Theorem (Dittrich et al.) Given an ODE ˙ x = Sv(x), the abstraction of the stationary state (which is usually attracting) is an organisation. Can this be extended?

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SLIDE 8

UoB Maths

  • C. Good,
  • N. Kamaleson,
  • C. Mu,
  • D. Parker,
  • M. Puljiz,
  • J. E. Rowe

Misc. Problem 1 Problem 2 References

The replicator equation (see in Fontana, Buss) ˙ xi =

  • j
  • k

ai

j,kxjxk − xi

  • r,s,t

at

r,sxrxs

What about its stationary state? Higher order equations? Relation to the heuristic framework?

slide-9
SLIDE 9

UoB Maths

  • C. Good,
  • N. Kamaleson,
  • C. Mu,
  • D. Parker,
  • M. Puljiz,
  • J. E. Rowe

Misc. Problem 1 Problem 2 References

Connections with other papers

Rabitz et al. [’89-’97] study Kinetic differential equations, their lumpings and stability Equations studied are exactly the replicator equations and higher order analogues

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SLIDE 10

UoB Maths

  • C. Good,
  • N. Kamaleson,
  • C. Mu,
  • D. Parker,
  • M. Puljiz,
  • J. E. Rowe

Misc. Problem 1 Problem 2 References

References I

[1] Walter Fontana and Leo W Buss, The arrival of the fittest: Toward a theory of biological organization, Bulletin of Mathematical Biology 56 (1994), no. 1, 1–64. [2] Peter Dittrich and Pietro Speroni Di Fenizio, Chemical organisation theory, Bulletin of mathematical biology 69 (2007), no. 4, 1199–1231. [3] Alison S Tomlin, Genyuan Li, Herschel Rabitz, and J´ anos T´

  • th, The

effect of lumping and expanding on kinetic differential equations, SIAM Journal on Applied Mathematics 57 (1997), no. 6, 1531–1556. [4] Genyuan Li and Herschel Rabitz, A general analysis of approximate lumping in chemical kinetics, Chemical engineering science 45 (1990),

  • no. 4, 977–1002.
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SLIDE 11

UoB Maths

  • C. Good,
  • N. Kamaleson,
  • C. Mu,
  • D. Parker,
  • M. Puljiz,
  • J. E. Rowe

Misc. Problem 1 Problem 2 References

References II

[5] Genyuan Li, Herschel Rabitz, and J´ anos T´

  • th, A general analysis of

exact nonlinear lumping in chemical kinetics, Chemical Engineering Science 49 (1994), no. 3, 343–361. [6] K. C. Chung and T. H. Yao, On lattices admitting unique Lagrange interpolations, SIAM J. Numer. Anal. 14 (1977), no. 4, 735–743. MR0445158 (56 #3502) [7] Ita¨ ı Ben Yaacov, A multivariate version of the Vandermonde determinant identity, arXiv:1405.0993 (2014). [8] http: //www.encyclopediaofmath.org/index.php/Pfaffian_equation [9] https://www.math.iupui.edu/˜mmisiure/open/FB1.pdf