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Canonical Functions: a proof via topological dynamics [article]

Manuel Bodirsky, Michael Pinsker
2020 arXiv   pre-print
In this short note, we present a proof of the existence of canonical functions in certain sets using topological dynamics, providing a shorter alternative to the original combinatorial argument.  ...  Canonical functions are a powerful concept with numerous applications in the study of groups, monoids, and clones on countable structures with Ramsey-type properties.  ...  It is therefore natural to ask for a perhaps more elegant proof of the existence of canonical functions via topological dynamics, reminiscent of the numerous proofs of combinatorial statements obtained  ... 
arXiv:1610.09660v3 fatcat:wsly2lcjcbfkzozkovyvn23vau

Canonical functions: a proof via topological dynamics

Michael Pinsker, Manuel Bodirsky
2020
In this short note, we present a proof of the existence of canonical functions in certain sets using topological dynamics, providing a shorter alternative to the original combinatorial argument.  ...  Canonical functions are a powerful concept with numerous applications in the study of groups, monoids, and clones on countable structures with Ramsey-type properties.  ...  It is therefore natural to ask for a perhaps more elegant proof of the existence of canonical functions via topological dynamics, reminiscent of the numerous proofs of combinatorial statements obtained  ... 
doi:10.11575/cdm.v16i2.71724 fatcat:agugelg5uzemjnpyktcw6r6llu

Canonical self-affine tilings by iterated function systems [article]

Erin P. J. Pearse
2007 arXiv   pre-print
This paper shows how the action of the function system naturally produces a tiling T of the convex hull of the attractor.  ...  An iterated function system Φ consisting of contractive similarity mappings has a unique attractor F ⊆R^d which is invariant under the action of the system, as was shown by Hutchinson [Hut].  ...  I would also like to thank Bob Strichartz and Steffen Winter for comments on a preliminary version of the paper, and several excellent references.  ... 
arXiv:math/0606111v2 fatcat:75i6irj45neupmgd7r2bzfaqw4

A finiteness theorem for canonical heights attached to rational maps over function fields [article]

Matthew Baker
2006 arXiv   pre-print
Our proof is essentially analytic, making use of potential theory on Berkovich spaces to prove some new results about the dynamical Green's functions g_v(x,y) attached to f at each place v of K.  ...  Let K be a function field, let f be a rational function of degree d at least 2 defined over K, and suppose that f is not isotrivial.  ...  The following result gives a useful functional equation for the dynamical Green's function.  ... 
arXiv:math/0601046v2 fatcat:lwpi4rpkmnechizp5752mbsluu

Classical Density Functional Theory in the Canonical Ensemble [article]

James F. Lutsko
2021 arXiv   pre-print
dimensions and finally a system of two hard-spheres in one dimension (hard rods) in a small cavity.  ...  Although the tools of standard, grand-canonical density functional theory are often used in an ad hoc manner to study closed systems, their formulation directly in the canonical ensemble has so far not  ...  A corollary of this proof is the existence of a functional of the one-body density which is minimized by the equilibrium density distribution.  ... 
arXiv:2109.05787v3 fatcat:6xn4c7zjtzcmpmkdcj7be75wrq

A C*-Dynamical Entropy and Applications to Canonical Endomorphisms

Marie Choda
2000 Journal of Functional Analysis  
Proof. It is clear by Corollary 5.2.7 and Lemma 7.2.1. K 7.3. Von Neumann Dynamical Entropy for #. In this section, we compute the Conne Narnhofer Thirring entropy of Longo's canonical #.  ...  Here, as a slight modification of Voiculescu's topological entropy ht( } ) a C*-dynamical entropy ht , ( } ) is given. We show relations among the three entropies h , ( } ), ht , ( } ), and ht( } ).  ... 
doi:10.1006/jfan.2000.3564 fatcat:ccd45liayrhi7n7dvglqqunx5i

Canonical transformations of local functionals and sh-Lie structures

Samer Al-Ashhab, Ron Fulp
2005 Journal of Geometry and Physics  
If a Lie group acts on the bundle via canonical automorphisms, there are induced actions on the space of local functionals and consequently on the corresponding sh-Lie algebra.  ...  In many Lagrangian field theories, there is a Poisson bracket on the space of local functionals. One may identify the fields of such theories as sections of a vector bundle.  ...  Acknowledgements The authors would like to thank Jim Stasheff for his useful remarks on a draft of this paper.  ... 
doi:10.1016/j.geomphys.2004.07.005 fatcat:vhdvtbwg75fw7jyobx33hpyiay

Local and global canonical height functions for affine space regular automorphisms [article]

Shu Kawaguchi
2009 arXiv   pre-print
Next we introduce for f the notion of good reduction at v, and using this notion, we show that the sum of v-adic Green functions over all v gives rise to a canonical height function for f that satisfies  ...  For each place v of K, we construct the v-adic Green functions G_f,v and G_f^-1,v (i.e., the v-adic canonical height functions) for f and f^-1.  ...  In [7] , we showed the existence of canonical height functions for affine plane polynomial automorphisms of dynamical degree ≥ 2.  ... 
arXiv:0909.3573v1 fatcat:yg26wlujefarnb6m3douax7ud4

The resolvent algebra: A new approach to canonical quantum systems

Detlev Buchholz, Hendrik Grundling
2008 Journal of Functional Analysis  
interesting dynamical laws nor does it incorporate pertinent physical observables such as (bounded functions of) the Hamiltonian.  ...  All proofs for our results are collected in Section 10.  ...  Let V ∈ C 0 (R) be any real function. Then the corresponding selfadjoint Hamil- tonian H = P 2 + V (Q) induces a dynamics on R(X, σ ).  ... 
doi:10.1016/j.jfa.2008.02.011 fatcat:jkj5jkmkcvb4nhcoolduujbi4u

Local and global canonical height functions for affine space regular automorphisms

Shu Kawaguchi
2013 Algebra & Number Theory  
In [Kawaguchi 2006 ], we showed the existence of canonical height functions for affine plane polynomial automorphisms of dynamical degree at least 2.  ...  Néron's construction is via a local method and gives deeper properties of the canonical height functions. Both constructions are useful in studying arithmetic properties of abelian varieties.  ...  Proof. For each v ∈ M K , we have estimates of Green functions for f at v as in Lemmas 1.3 and 2.8. We use the suffix v when we work over the absolute value v ∈ M K .  ... 
doi:10.2140/ant.2013.7.1225 fatcat:4yhskrhkavfddf2jqcb6r2ckpa

Canonical heights for correspondences [article]

Patrick Ingram
2014 arXiv   pre-print
The canonical height associated to a polarized endomporhism of a projective variety, constructed by Call and Silverman and generalizing the N\'eron-Tate height on a polarized Abelian variety, plays an  ...  important role in the arithmetic theory of dynamical systems.  ...  The functionĥ X,C,L : P(K) → R is continuous with respect to the tree topology, and each fibre P a (K) = π −1 (a) ⊆ P(K) is compact. Proof.  ... 
arXiv:1411.1041v2 fatcat:sqadagzernf2vdlougsmmi642q

Canonical simplicial gravity

Bianca Dittrich, Philipp A Höhn
2012 Classical and quantum gravity  
The central ingredient is Hamilton's principal function which generates canonical time evolution and ensures that the canonical formalism reproduces the dynamics of the covariant formulation following  ...  The end result is a general and fully consistent formulation of canonical Regge calculus, thereby removing a longstanding obstacle in connecting covariant simplicial gravity models to canonical frameworks  ...  Canonical discrete dynamics The central idea [10, 16, 22, [24] [25] [26] for generating a canonical discrete dynamics is to employ Hamilton's principal functionS as a generating function for a canonical  ... 
doi:10.1088/0264-9381/29/11/115009 fatcat:k7gkmwd3lzbc5b7upi2gkca7ou

Canonical Supermartingale Couplings [article]

Marcel Nutz, Florian Stebegg
2017 arXiv   pre-print
Two probability distributions μ and ν in second stochastic order can be coupled by a supermartingale, and in fact by many. Is there a canonical choice?  ...  transport and its symmetric counterpart, the antitone coupling, these can be characterized by order-theoretic minimality properties, as simultaneous optimal transports for certain classes of reward (or cost) functions  ...  There is a Polish topology τ on R such that f is τ ⊗ τ -continuous. Moreover, τ refines the Euclidean topology and induces the same Borel sets. Proof.  ... 
arXiv:1609.02867v2 fatcat:h6yofotpvjfxnce64wptxxp3em

A finiteness theorem for canonical heights attached to rational maps over function fields

Matthew Baker
2009 Journal für die Reine und Angewandte Mathematik  
Our proof is essentially analytic, making use of potential theory on Berkovich spaces to study the dynamical Green's functions g ϕ,v (x, y) attached to ϕ at each place v of K.  ...  In an appendix, we use a similar method to give a new proof of the Mordell-Weil theorem for elliptic curves over K.  ...  The following result gives a useful functional equation for the dynamical Green's function.  ... 
doi:10.1515/crelle.2009.008 fatcat:2hoouxftmjcrfhnz4hllzper44

Entropy and the Canonical Height

M Einsiedler, G Everest, T Ward
2001 Journal of Number Theory  
local heights at primes of singular reduction, yielding a dynamical interpretation of singular reduction.  ...  The proofs use transcendence theory, a strong form of Siegel's theorem, and an elliptic analogue of Jensen's formula.  ...  In [5] , [11] and [12] , attempts have been made to define dynamical systems whose topological entropy is given by ĥ(Q), the global canonical height of Q.  ... 
doi:10.1006/jnth.2001.2682 fatcat:u67booaiafe2thf36vd6sq6i3y
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