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Shuffles of trees

Eric Hoffbeck, Ieke Moerdijk
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.  ... 
doi:10.1016/j.ejc.2018.02.035 fatcat:flzgkcq3nrehvk4q3iy23xz7bu

Bayesian learning of forest and tree graphical models [article]

Edmund Jones
2021 arXiv   pre-print
I describe how to use 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.  ... 
arXiv:2108.13992v1 fatcat:v34ddpig4jcchoog24viamengy

Polynomial functors and combinatorial Dyson-Schwinger equations [article]

Joachim Kock
2016 arXiv   pre-print
Finally, all constructions can be pushed inside 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  ... 
arXiv:1512.03027v3 fatcat:iwofgmgehzbpbbmnisa3w2qou4

Every planar graph with the Liouville property is amenable

Johannes Carmesin, Agelos Georgakopoulos
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.  ... 
doi:10.1002/rsa.20936 fatcat:jhfo75fzo5gjllur7mtrbgchia

Multicoloured Random Graphs: Constructions and Symmetry [article]

Sam Tarzi
2014 arXiv   pre-print
This is 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.  ... 
arXiv:1406.7870v2 fatcat:othjxv7jzbhq7pupvjfmonunva

Intensive use of computing resources for dominations in grids and other combinatorial problems [article]

Alexandre Talon
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.  ... 
arXiv:2002.11615v1 fatcat:dbgy2taztvbu3ms4qpjxipwnye

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

David A. Cohen, Martin C. Cooper, Peter G. Jeavons, Stanislav Zivny
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  ... 
arXiv:1608.05358v2 fatcat:sgaphwoh6rh4xle6xvvmjcclua

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

Johannes Carmesin, Agelos Georgakopoulos
2020 arXiv   pre-print
We introduce 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.  ... 
arXiv:1502.02542v3 fatcat:4rl26dj55fdolecyxv3i2mcs6y

Locally Checkable Proofs in Distributed Computing

Mika Göös, Jukka Suomela
2016 Theory of Computing  
For example, it is easy to prove that 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.  ... 
doi:10.4086/toc.2016.v012a019 dblp:journals/toc/GoosS16 fatcat:5mjhwq5q65apxg4yxmol33e6gi

A Singular Mathematical Promenade [article]

Etienne Ghys
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]).  ... 
arXiv:1612.06373v4 fatcat:3x2nfhmw3jgo3lfwvyt7njnssq

From dependent type theory to higher algebraic structures [article]

Chaitanya Leena Subramaniam
2021 arXiv   pre-print
We use this to recognise dependently typed algebraic theories for 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.  ... 
arXiv:2110.02804v1 fatcat:bdwbbll3ivct3prpbtworvblo4

A topological characterisation of the Kashiwara-Vergne groups [article]

Zsuzsanna Dancso, Iva Halacheva, Marcy Robertson
2021 arXiv   pre-print
Finally, we provide 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.  ... 
arXiv:2106.02373v2 fatcat:7biub5em4nb2ln6ncqwkfq5k7i

Computing spin networks

Annalisa Marzuoli, Mario Rasetti
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] .  ... 
doi:10.1016/j.aop.2005.01.005 fatcat:4n3izo3ozfdcbozh7bzapvbcya

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

Adrian Celestino, Kurusch Ebrahimi-Fard, Frédéric Patras, Daniel Perales
2021 Foundations of Computational Mathematics  
In this work, we focus on 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).  ... 
doi:10.1007/s10208-021-09512-0 fatcat:5cwndzb2nzd3jgztix4bky7x6a

Lie-Butcher series, Geometry, Algebra and Computation [article]

Hans Z. Munthe-Kaas, Kristoffer K. Føllesdal
2017 arXiv   pre-print
First, we aim at presenting 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.  ... 
arXiv:1701.03654v2 fatcat:eqeitpijofbazf7syxdkjwacci
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