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Canonical Functions: a proof via topological dynamics
[article]

2020
*
arXiv
*
pre-print

In this short note, we present

arXiv:1610.09660v3
fatcat:wsly2lcjcbfkzozkovyvn23vau
*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 ...##
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Canonical functions: a proof via topological dynamics

2020

In this short note, we present

doi:10.11575/cdm.v16i2.71724
fatcat:agugelg5uzemjnpyktcw6r6llu
*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 ...##
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Canonical self-affine tilings by iterated function systems
[article]

2007
*
arXiv
*
pre-print

This paper shows how the action of the

arXiv:math/0606111v2
fatcat:75i6irj45neupmgd7r2bzfaqw4
*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. ...##
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A finiteness theorem for canonical heights attached to rational maps over function fields
[article]

2006
*
arXiv
*
pre-print

Our

arXiv:math/0601046v2
fatcat:lwpi4rpkmnechizp5752mbsluu
*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*. ...##
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Classical Density Functional Theory in the Canonical Ensemble
[article]

2021
*
arXiv
*
pre-print

dimensions and finally

arXiv:2109.05787v3
fatcat:6xn4c7zjtzcmpmkdcj7be75wrq
*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. ...##
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A C*-Dynamical Entropy and Applications to Canonical Endomorphisms

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( } ). ...

##
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Canonical transformations of local functionals and sh-Lie structures

2005
*
Journal of Geometry and Physics
*

If

doi:10.1016/j.geomphys.2004.07.005
fatcat:vhdvtbwg75fw7jyobx33hpyiay
*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. ...##
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Local and global canonical height functions for affine space regular automorphisms
[article]

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

arXiv:0909.3573v1
fatcat:yg26wlujefarnb6m3douax7ud4
*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. ...##
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The resolvent algebra: A new approach to canonical quantum systems

2008
*
Journal of Functional Analysis
*

interesting

doi:10.1016/j.jfa.2008.02.011
fatcat:jkj5jkmkcvb4nhcoolduujbi4u
*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 ). ...##
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Local and global canonical height functions for affine space regular automorphisms

2013
*
Algebra & Number Theory
*

In [Kawaguchi 2006 ], we showed the existence of

doi:10.2140/ant.2013.7.1225
fatcat:4yhskrhkavfddf2jqcb6r2ckpa
*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 . ...##
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Canonical heights for correspondences
[article]

2014
*
arXiv
*
pre-print

The

arXiv:1411.1041v2
fatcat:sqadagzernf2vdlougsmmi642q
*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*. ...##
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Canonical simplicial gravity

2012
*
Classical and quantum gravity
*

The central ingredient is Hamilton's principal

doi:10.1088/0264-9381/29/11/115009
fatcat:k7gkmwd3lzbc5b7upi2gkca7ou
*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*...##
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Canonical Supermartingale Couplings
[article]

2017
*
arXiv
*
pre-print

Two probability distributions μ and ν in second stochastic order can be coupled by

arXiv:1609.02867v2
fatcat:h6yofotpvjfxnce64wptxxp3em
*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*. ...##
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A finiteness theorem for canonical heights attached to rational maps over function fields

2009
*
Journal für die Reine und Angewandte Mathematik
*

Our

doi:10.1515/crelle.2009.008
fatcat:2hoouxftmjcrfhnz4hllzper44
*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*. ...##
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Entropy and the Canonical Height

2001
*
Journal of Number Theory
*

local heights at primes of singular reduction, yielding

doi:10.1006/jnth.2001.2682
fatcat:u67booaiafe2thf36vd6sq6i3y
*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. ...
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