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## Filters

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Shuffles of trees

2018
*
European journal of combinatorics (Print)
*

*A*shuffle

*of*S and T is

*a*

*tree*

*A*for which

*the*following

*four*conditions hold: (1)

*The*

*edges*

*of*

*A*are labelled by

*pairs*(s, t) where s and t are

*edges*

*of*S and T , respectively. . . , s m are

*the*incoming ... One

*of*these external

*edges*is selected as

*the*root, and

*the*other external

*edges*are called

*the*leaves

*of*

*the*

*tree*.

*The*chosen root provides an orientation, pictured downwards

*towards*

*the*root. ...

##
###
Bayesian learning of forest and tree graphical models
[article]

2021
*
arXiv
*
pre-print

I describe how to use

arXiv:2108.13992v1
fatcat:v34ddpig4jcchoog24viamengy
*the*Chowx2013Liu algorithm and*the*Matrix*Tree**Theorem*to find*the**MAP*forest and certain quantities in*the*posterior distribution on*trees*. ...*The*main topic*of**the*thesis is Bayesian structure-learning with forests or*trees*. Restricting attention to these graphs can be justified using*theorems*on random graphs. ... I would also like to thank staff and students*of*Bristol University and express my gratitude for*the*funding I have received through*the*Engineering and Physical Sciences Research Council. ...##
###
Polynomial functors and combinatorial Dyson-Schwinger equations
[article]

2016
*
arXiv
*
pre-print

Finally, all constructions can be pushed inside

arXiv:1512.03027v3
fatcat:iwofgmgehzbpbbmnisa3w2qou4
*the*classical Connes-Kreimer Hopf algebra*of**trees*by*the*operation*of*taking core*of*P-*trees*. ...*The*isoclasses*of*P-*trees*generate naturally*a*Connes-Kreimer-like bialgebra, in which*the*abstract Dyson-Schwinger equation can be internalised in terms*of*canonical B_+-operators. ... This paper is an elaboration upon my talk at*the*workshop on Mathematical Aspects*of*Hadron Physics at*the*ECT* in Trento, October 2012, and I thank*the*organisers, Kurusch Ebrahimi-Fard, Frédéric Patras ...##
###
Every planar graph with the Liouville property is amenable

2020
*
Random structures & algorithms (Print)
*

*A*

*planar*

*map*G, also called

*a*plane graph, is

*a*graph endowed with an embedding in

*the*plane. ...

*A*graph G is

*a*

*pair*(V, E) where V is

*the*set

*of*vertices

*of*G, and E is

*a*set

*of*directed

*pairs*

*of*elements

*of*V, called

*the*(directed)

*edges*

*of*G. ...

*a*subdivision T

*of*

*the*infinite

*binary*

*tree*T 2 such that

*edges*at level n are subdivided at most 2 n -times. ...

##
###
Multicoloured Random Graphs: Constructions and Symmetry
[article]

2014
*
arXiv
*
pre-print

This is

arXiv:1406.7870v2
fatcat:othjxv7jzbhq7pupvjfmonunva
*a*research monograph on constructions*of*and group actions on countable homogeneous graphs, concentrating particularly on*the*simple random graph and its*edge*-*coloured*variants. ...*The*large number*of*references should help in making this*a*resource for anyone interested in beginning research in this or allied fields. ... By*Theorem*6.7 almost all*the*lattice vector*pair**colourings*can be put into 1-1 correspondence with R t*edge**colourings*. ...##
###
Intensive use of computing resources for dominations in grids and other combinatorial problems
[article]

2020
*
arXiv
*
pre-print

*The*

*four*-

*colour*

*theorem*states that any

*map*whose countries are connected can be

*coloured*with 4

*colours*such that neighbouring countries have differnt

*colours*. ... It was

*the*first result proved using computers, in 1989. We wished to automatise further its

*proof*. We explain

*the*

*proof*and provide

*a*program which replays

*the*

*proof*. ... We will explain each term in Section 1.1, so for

*the*moment we consider an equivalent version.

*Theorem*1.2. Every

*planar*

*map*can be

*coloured*with

*four*

*colours*. ...

##
###
Binary Constraint Satisfaction Problems Defined by Excluded Topological Minors
[article]

2018
*
arXiv
*
pre-print

*The*

*binary*Constraint Satisfaction Problem (CSP) is to decide whether there exists an assignment to

*a*set

*of*variables which satisfies specified constraints between

*pairs*

*of*variables. ... In this paper we extend

*the*notion

*of*pattern and introduce

*the*notion

*of*

*a*topological minor

*of*

*a*

*binary*CSP instance. ...

*Proof*: One

*of*

*the*well-known results

*of*Robertson and Seymour shows that

*the*class

*of*graphs obtained by excluding G as

*a*minor has bounded treewidth if and only if G is

*planar*[38] (see also [20,

*Theorem*...

##
###
Every planar graph with the Liouville property is amenable
[article]

2020
*
arXiv
*
pre-print

We introduce

arXiv:1502.02542v3
fatcat:4rl26dj55fdolecyxv3i2mcs6y
*a*strengthening*of**the*notion*of*transience for*planar**maps*in order to relax*the*standard condition*of*bounded degree appearing in various results, in particular,*the*existence*of*Dirichlet ... As*a*corollary we obtain that every*planar*non-amenable graph admits Dirichlet harmonic functions. ... Acknowledgement We would like to thank Louigi Addario-Berry for suggesting*the*use*of*what we called*the*plane line graph. ...##
###
Locally Checkable Proofs in Distributed Computing

2016
*
Theory of Computing
*

For example, it is easy to prove that

doi:10.4086/toc.2016.v012a019
dblp:journals/toc/GoosS16
fatcat:5mjhwq5q65apxg4yxmol33e6gi
*a*graph is bipartite:*the*locally checkable*proof*gives*a*2-coloring*of**the*graph, which only takes 1 bit per node. ... We show that*the*local*proof*complexities form*a*natural hierarchy*of*complexity classes: for many classical graph properties,*the*local*proof*complexity is either 0, Θ(1), Θ(log n), or poly(n) bits per ... This work was supported in part by*the*Academy*of*Finland, Grants 132380 and 252018,*the*Finnish Cultural Foundation, and*the*Research Funds*of**the*University*of*Helsinki. ...##
###
A Singular Mathematical Promenade
[article]

2017
*
arXiv
*
pre-print

*The*purpose

*of*this little book is to invite

*the*reader on

*a*mathematical promenade. We pay

*a*visit to Hipparchus, Newton and Gauss, but also to many contemporary mathematicians. ... This is neither an elementary introduction to singularity theory nor

*a*specialized treatise containing many new

*theorems*. ... Check that this gives

*a*bijection between rooted

*planar*

*trees*with n

*edges*and rooted

*planar*

*binary*

*trees*with n + 1 leaves ([M15]). ...

##
###
From dependent type theory to higher algebraic structures
[article]

2021
*
arXiv
*
pre-print

We use this to recognise dependently typed algebraic theories for

arXiv:2110.02804v1
fatcat:bdwbbll3ivct3prpbtworvblo4
*a*number*of*classes*of*algebraic structures, such as small categories, n-categories, strict and weak omega-categories,*planar**coloured*...*The*first part*of*this dissertation defines "dependently typed algebraic theories", which are*a*strict subclass*of**the*generalised algebraic theories (GATs)*of*Cartmell. ... We call*the**colours**of**a**tree*its*edges*and*the*inputs*of**a*node*the*input*edges**of*that node. Let*Tree*(denoted TEmb in [Koc11] ) be*the*full subcategory*of*PolyEnd whose objects are*trees*. ...##
###
A topological characterisation of the Kashiwara-Vergne groups
[article]

2021
*
arXiv
*
pre-print

Finally, we provide

arXiv:2106.02373v2
fatcat:7biub5em4nb2ln6ncqwkfq5k7i
*a*description*of**the*graded Grothendieck-Teichmüller group 𝖦𝖱𝖳_1 as automorphisms*of*arrow diagrams. ... Welded foams are*a*class*of*knotted tubes in ℝ^4, which can be finitely presented algebraically as*a*circuit algebra, or, equivalently,*a*wheeled prop. ...*The*authors would like to thank Dror Bar-Natan, who has been generous with his insights throughout*the*writing*of*this paper, and contributed important ideas to several*proofs*. ...##
###
Computing spin networks

2005
*
Annals of Physics
*

*The*model proposed can be thought

*of*as

*the*non-Boolean generalization

*of*

*the*quantum circuit model, with unitary gates expressed in terms

*of*3nj coefficients connecting inequivalent

*binary*coupling schemes ... We expand

*a*set

*of*notions recently introduced providing

*the*general setting for

*a*universal representation

*of*

*the*quantum structure on which quantum information stands. ... Carfora also for his unexhaustible enthusiasm in helping us with

*the*preparation

*of*all figures, and with V. Aquilanti and C. Coletti for

*the*permission to reproduce Fig. 21 from [27] . ...

##
###
Cumulant–Cumulant Relations in Free Probability Theory from Magnus' Expansion

2021
*
Foundations of Computational Mathematics
*

In this work, we focus on

doi:10.1007/s10208-021-09512-0
fatcat:5cwndzb2nzd3jgztix4bky7x6a
*the*problem*of*expressing with*a*closed formula multivariate monotone cumulants in terms*of*free and Boolean cumulants. ... Relations among cumulants are described in terms*of**the*pre-Lie Magnus expansion combined with results on*the*continuous Baker–Campbell–Hausdorff formula due to*A*. Murua. ...*The*second author is supported by*the*Research Council*of*Norway through Project 302831 "Computational Dynamics and Stochastics on Manifolds" (CODYSMA). ...##
###
Lie-Butcher series, Geometry, Algebra and Computation
[article]

2017
*
arXiv
*
pre-print

First, we aim at presenting

arXiv:1701.03654v2
fatcat:eqeitpijofbazf7syxdkjwacci
*the*algebraic structures underlying LB series in*a**concise*and self contained manner. ... Lie-Butcher (LB) series are formal power series expressed in terms*of**trees*and forests. ... We illustrate*a**tree*with*the*root on*the*bottom and*the*direction*of**the**edges*being down*towards**the*root. Let T C denote*the*set*of*all*coloured*rooted*trees*, e.g. ...
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