1,479 research outputs found

    Opening the system to the environment: new theories and tools in classical and quantum settings

    Get PDF
    The thesis is organized as follows. Section 2 is a first, unconventional, approach to the topic of EPs. Having grown interest in the topic of combinatorics and graph theory, I wanted to exploit its very abstract and mathematical tools to reinterpret something very physical, that is, the EPs in wave scattering. To do this, I build the interpretation of scattering events from a graph theory perspective and show how EPs can be understood within this interpretation. In Section 3, I move from a completely classical treatment to a purely quantum one. In this section, I consider two quantum resonators coupled to two baths and study their dynamics with local and global master equations. Here, the EPs are the key physical features used as a witness of validity of the master equation. Choosing the wrong master equation in the regime of interest can indeed mask physical and fundamental features of the system. In Section 4, there are no EPs. However I transition towards a classical/quantum framework via the topic of open systems. My main contribution in this work is the classical stochastic treatment and simulation of a spin coupled to a bath. In this work, I show how a natural quantum--to--classical transition occurs at all coupling strengths when certain limits of spin length are taken. As a key result, I also show how the coupling to the environment in this stochastic framework induces a classical counterpart to quantum coherences in equilibrium. After this last topic, in Section 5, I briefly present the key features of the code I built (and later extended) for the latter project. This, in the form of a Julia registry package named SpiDy.jl, has seen further applications in branching projects and allows for further exploration of the theoretical framework. Finally, I conclude with a discussion section (see Sec. 5) where I recap the different conclusions gathered in the previous sections and propose several possible directions.Engineering and Physical Sciences Research Council (EPSRC

    On the use of senders for minimal Ramsey theory

    Full text link
    This thesis investigates problems related to extremal and probabilistic graph theory. Our focus lies on the highly dynamic field of Ramsey theory. The foundational result of this field was proved in 1930 by Franck P. Ramsey. It implies that for every integer t and every sufficiently large complete graph Kn, every colouring of the edges of Kn with colours red and blue contains a red copy or a blue copy of Kt. Let q ā©¾ 2 represent a number of colours, and let H1,..., Hq be graphs. A graph G is said to be q-Ramsey for the tuple (H1,...,Hq) if, for every colouring of the edges of G with colours {1, . . . , q}, there exists a colour i and a monochromatic copy of Hi in colour i. As we often want to understand the structural properties of the collection of graphs that are q-Ramsey for (H1,..., Hq), we restrict our attention to the graphs that are minimal for this property, with respect to subgraph inclusion. Such graphs are said to be q-Ramsey-minimal for (H1,..., Hq). In 1976, Burr, Erdős, and LovĆ”sz determined, for every s, t ā©¾ 3, the smallest minimum degree of a graph G that is 2-Ramsey-minimal for (Ks, Kt). Significant efforts have been dedicated to generalising this result to a higher number of colours, qā©¾3, starting with the ā€˜symmetricā€™ q-tuple (Kt,..., Kt). In this thesis, we improve on the best known bounds for this parameter, providing state-of-the-art bounds in different (q, t)-regimes. These improvements rely on constructions based on finite geometry, which are then used to prove the existence of extremal graphs with certain key properties. Another crucial ingredient in these proofs is the existence of gadget graphs, called signal senders, that were initially developed by Burr, Erdős, and LovĆ”sz in 1976 for pairs of complete graphs. Until now, these senders have been shown to exist only in the two-colour setting, when q = 2, or in the symmetric multicolour setting, when H1,..., Hq are pairwise isomorphic. In this thesis, we then construct similar gadgets for all tuples of complete graphs, providing the first known constructions of these tools in the multicolour asymmetric setting. We use these new senders to prove far-reaching generalisations of several classical results in the area

    Data-driven deep-learning methods for the accelerated simulation of Eulerian fluid dynamics

    Get PDF
    Deep-learning (DL) methods for the fast inference of the temporal evolution of ļ¬‚uid-dynamics systems, based on the previous recognition of features underlying large sets of ļ¬‚uid-dynamics data, have been studied. Speciļ¬cally, models based on convolution neural networks (CNNs) and graph neural networks (GNNs) were proposed and discussed. A U-Net, a popular fully-convolutional architecture, was trained to infer wave dynamics on liquid surfaces surrounded by walls, given as input the system state at previous time-points. A term for penalising the error of the spatial derivatives was added to the loss function, which resulted in a suppression of spurious oscillations and a more accurate location and length of the predicted wavefronts. This model proved to accurately generalise to complex wall geometries not seen during training. As opposed to the image data-structures processed by CNNs, graphs oļ¬€er higher freedom on how data is organised and processed. This motivated the use of graphs to represent the state of ļ¬‚uid-dynamic systems discretised by unstructured sets of nodes, and GNNs to process such graphs. Graphs have enabled more accurate representations of curvilinear geometries and higher resolution placement exclusively in areas where physics is more challenging to resolve. Two novel GNN architectures were designed for ļ¬‚uid-dynamics inference: the MuS-GNN, a multi-scale GNN, and the REMuS-GNN, a rotation-equivariant multi-scale GNN. Both architectures work by repeatedly passing messages from each node to its nearest nodes in the graph. Additionally, lower-resolutions graphs, with a reduced number of nodes, are deļ¬ned from the original graph, and messages are also passed from ļ¬ner to coarser graphs and vice-versa. The low-resolution graphs allowed for eļ¬ƒciently capturing physics encompassing a range of lengthscales. Advection and ļ¬‚uid ļ¬‚ow, modelled by the incompressible Navier-Stokes equations, were the two types of problems used to assess the proposed GNNs. Whereas a single-scale GNN was suļ¬ƒcient to achieve high generalisation accuracy in advection simulations, ļ¬‚ow simulation highly beneļ¬ted from an increasing number of low-resolution graphs. The generalisation and long-term accuracy of these simulations were further improved by the REMuS-GNN architecture, which processes the system state independently of the orientation of the coordinate system thanks to a rotation-invariant representation and carefully designed components. To the best of the authorā€™s knowledge, the REMuS-GNN architecture was the ļ¬rst rotation-equivariant and multi-scale GNN. The simulations were accelerated between one (in a CPU) and three (in a GPU) orders of magnitude with respect to a CPU-based numerical solver. Additionally, the parallelisation of multi-scale GNNs resulted in a close-to-linear speedup with the number of CPU cores or GPUs.Open Acces

    A machine learning approach to constructing Ramsey graphs leads to the Trahtenbrot-Zykov problem.

    Get PDF
    Attempts at approaching the well-known and difficult problem of constructing Ramsey graphs via machine learning lead to another difficult problem posed by Zykov in 1963 (now commonly referred to as the Trahtenbrot-Zykov problem): For which graphs F does there exist some graph G such that the neighborhood of every vertex in G induces a subgraph isomorphic to F? Chapter 1 provides a brief introduction to graph theory. Chapter 2 introduces Ramsey theory for graphs. Chapter 3 details a reinforcement learning implementation for Ramsey graph construction. The implementation is based on board game software, specifically the AlphaZero program and its success learning to play games from scratch. The chapter ends with a description of how computing challenges naturally shifted the project towards the Trahtenbrot-Zykov problem. Chapter 3 also includes recommendations for continuing the project and attempting to overcome these challenges. Chapter 4 defines the Trahtenbrot-Zykov problem and outlines its history, including proofs of results omitted from their original papers. This chapter also contains a program for constructing graphs with all neighborhood-induced subgraphs isomorphic to a given graph F. The end of Chapter 4 presents constructions from the program when F is a Ramsey graph. Constructing such graphs is a non-trivial task, as Bulitko proved in 1973 that the Trahtenbrot-Zykov problem is undecidable. Chapter 5 is a translation from Russian to English of this famous result, a proof not previously available in English. Chapter 6 introduces Cayley graphs and their relationship to the Trahtenbrot-Zykov problem. The chapter ends with constructions of Cayley graphs Ī“ in which the neighborhood of every vertex of Ī“ induces a subgraph isomorphic to a given Ramsey graph, which leads to a conjecture regarding the unique extremal Ramsey(4, 4) graph

    A triangle process on graphs with given degree sequence

    Full text link
    The triangle switch Markov chain is designed to generate random graphs with given degree sequence, but having more triangles than would appear under the uniform distribution. Transition probabilities of the chain depends on a parameter, called the activity, which is used to assign higher stationary probability to graphs with more triangles. In previous work we proved ergodicity of the triangle switch chain for regular graphs. Here we prove ergodicity for all sequences with minimum degree at least 3, and show rapid mixing of the chain when the activity and the maximum degree are not too large. As far as we are aware, this is the first rigorous analysis of a Markov chain algorithm for generating graphs from a a known non-uniform distribution.Comment: 35 page

    Planar hypohamiltonian oriented graphs

    Get PDF
    In 1978 Thomassen asked whether planar hypohamiltonian oriented graphs exist. Infinite families of such graphs have since been described but for infinitely many it remained an open question whether planar hypohamiltonian oriented graphs of order exist. In this paper we develop new methods for constructing hypohamiltonian digraphs, which, combined with efficient graph generation algorithms, enable us to fully characterise the orders for which planar hypohamiltonian oriented graphs exist. Our novel methods also led us to discover the planar hypohamiltonian oriented graph of smallest order and size, as well as infinitely many hypohamiltonian orientations of maximal planar graphs. Furthermore, we answer a question related to a problem of Schiermeyer on vertex degrees in hypohamiltonian oriented graphs, and characterise all the orders for which planar hypotraceable oriented graphs exist.Research Foundation Flanders; VSC(Flemish Supercomputer Center);DSTā€NRF Centre of Excellence in Mathematical and Statistical Sciences.http://wileyonlinelibrary.com/journal/jgthj2023Mathematics and Applied Mathematic

    Oriented Spanners

    Get PDF
    Given a point set P in the Euclidean plane and a parameter t, we define an oriented t-spanner as an oriented subgraph of the complete bi-directed graph such that for every pair of points, the shortest cycle in G through those points is at most a factor t longer than the shortest oriented cycle in the complete bi-directed graph. We investigate the problem of computing sparse graphs with small oriented dilation. As we can show that minimising oriented dilation for a given number of edges is NP-hard in the plane, we first consider one-dimensional point sets. While obtaining a 1-spanner in this setting is straightforward, already for five points such a spanner has no plane embedding with the leftmost and rightmost point on the outer face. This leads to restricting to oriented graphs with a one-page book embedding on the one-dimensional point set. For this case we present a dynamic program to compute the graph of minimum oriented dilation that runs in ?(n?) time for n points, and a greedy algorithm that computes a 5-spanner in ?(nlog n) time. Expanding these results finally gives us a result for two-dimensional point sets: we prove that for convex point sets the greedy triangulation results in an oriented ?(1)-spanner

    Coloring Tournaments with Few Colors: Algorithms and Complexity

    Get PDF
    A k-coloring of a tournament is a partition of its vertices into k acyclic sets. Deciding if a tournament is 2-colorable is NP-hard. A natural problem, akin to that of coloring a 3-colorable graph with few colors, is to color a 2-colorable tournament with few colors. This problem does not seem to have been addressed before, although it is a special case of coloring a 2-colorable 3-uniform hypergraph with few colors, which is a well-studied problem with super-constant lower bounds. We present an efficient decomposition lemma for tournaments and show that it can be used to design polynomial-time algorithms to color various classes of tournaments with few colors, including an algorithm to color a 2-colorable tournament with ten colors. For the classes of tournaments considered, we complement our upper bounds with strengthened lower bounds, painting a comprehensive picture of the algorithmic and complexity aspects of coloring tournaments

    Prime Graph over Cartesian Product over Rings and Its Complement

    Get PDF
    Graph theory is a branch of algebra that is growing rapidly both in concept and application studies. This graph application can be used in chemistry, transportation, cryptographic problems, coding theory, design communication network, etc. There is currently a bridge between graphs and algebra, especially an algebraic structures, namely theory of graph algebra. One of researchs on graph algebra is a graph that formed by prime ring elements or called prime graph over ring R. The prime graph over commutative ring R (PG(R))) is a graph construction with set of vertices V(PG(R))=R and two vertices x and y are adjacent if satisfy xRy={0}, for xā‰ y. Girth is the shortest cycle length contains in PG(R) or can be written gr(PG(R)). Order in PG(R) denoted by |V(PG(R))| and size in PG(R) denoted by |E(PG(R))|. In this paper, we discussed prime graph over cartesian product over rings Z_mƗZ_n and its complement. We focused only for m=p_1, n=p_2 and m=p_1, n=怖p_2怗^2, where p_1 and p_2 are prime numbers. Then, we obtained some properties related to order and size, degree, and girth. We also observe some examples. Moreover, we found that a correction in the statement of (Pawar & Joshi, 2019) about the complement graph over prime graph over a ring and gave a counter example for that.
    • ā€¦
    corecore