Matérn Gaussian Processes on Graphs.

Gaussian processes are a versatile framework for learning unknown functions in a manner that permits one to utilize prior information about their properties. Although many different Gaussian process models are readily available when the input space is Euclidean, the choice is much more limited for Gaussian processes whose input space is an undirected graph. In this work, we leverage the stochastic partial differential equation characterization of Mat´ern Gaussian processes—a widelyused model class in the Euclidean setting—to study their analog for undirected graphs. We show that the resulting Gaussian processes inherit various attractive properties of their Euclidean and Riemannian analogs and provide techniques that allow them to be trained using standard methods, such as inducing points. This enables graph Mat´ern Gaussian processes to be employed in mini-batch and non-conjugate settings, thereby making them more accessible to practitioners and easier to deploy within larger learning frameworks

Logic and Automata

Mathematical logic and automata theory are two scientific disciplines with a fundamentally close relationship. The authors of Logic and Automata take the occasion of the sixtieth birthday of Wolfgang Thomas to present a tour d'horizon of automata theory and logic. The twenty papers in this volume cover many different facets of logic and automata theory, emphasizing the connections to other disciplines such as games, algorithms, and semigroup theory, as well as discussing current challenges in the field

Field theoretic formulation and empirical tracking of spatial processes

Spatial processes are attacked on two fronts. On the one hand, tools from theoretical and statistical physics can be used to understand behaviour in complex, spatially-extended multi-body systems. On the other hand, computer vision and statistical analysis can be used to study 4D microscopy data to observe and understand real spatial processes in vivo. On the rst of these fronts, analytical models are developed for abstract processes, which can be simulated on graphs and lattices before considering real-world applications in elds such as biology, epidemiology or ecology. In the eld theoretic formulation of spatial processes, techniques originating in quantum eld theory such as canonical quantisation and the renormalization group are applied to reaction-di usion processes by analogy. These techniques are combined in the study of critical phenomena or critical dynamics. At this level, one is often interested in the scaling behaviour; how the correlation functions scale for di erent dimensions in geometric space. This can lead to a better understanding of how macroscopic patterns relate to microscopic interactions. In this vein, the trace of a branching random walk on various graphs is studied. In the thesis, a distinctly abstract approach is emphasised in order to support an algorithmic approach to parts of the formalism. A model of self-organised criticality, the Abelian sandpile model, is also considered. By exploiting a bijection between recurrent con gurations and spanning trees, an e cient Monte Carlo algorithm is developed to simulate sandpile processes on large lattices. On the second front, two case studies are considered; migratory patterns of leukaemia cells and mitotic events in Arabidopsis roots. In the rst case, tools from statistical physics are used to study the spatial dynamics of di erent leukaemia cell lineages before and after a treatment. One key result is that we can discriminate between migratory patterns in response to treatment, classifying cell motility in terms of sup/super/di usive regimes. For the second case study, a novel algorithm is developed to processes a 4D light-sheet microscopy dataset. The combination of transient uorescent markers and a poorly localised specimen in the eld of view leads to a challenging tracking problem. A fuzzy registration-tracking algorithm is developed to track mitotic events so as to understand their spatiotemporal dynamics under normal conditions and after tissue damage.Open Acces

Synchronizing permutation groups and graph endomorphisms

The current thesis is focused on synchronizing permutation groups and on graph endo- morphisms. Applying the implicit classification of rank 3 groups, we provide a bound on synchronizing ranks of rank 3 groups, at first. Then, we determine the singular graph endomorphisms of the Hamming graph and related graphs, count Latin hypercuboids of class r, establish their relation to mixed MDS codes, investigate G-decompositions of (non)-synchronizing semigroups, and analyse the kernel graph construction used in the theorem of Cameron and Kazanidis which identifies non-synchronizing transformations with graph endomorphisms [20]. The contribution lies in the following points: 1. A bound on synchronizing ranks of groups of permutation rank 3 is given, and a complete list of small non-synchronizing groups of permutation rank 3 is provided (see Chapter 3). 2. The singular endomorphisms of the Hamming graph and some related graphs are characterised (see Chapter 5). 3. A theorem on the extension of partial Latin hypercuboids is given, Latin hyper- cuboids for small values are counted, and their correspondence to mixed MDS codes is unveiled (see Chapter 6). 4. The research on normalizing groups from [3] is extended to semigroups of the form , and decomposition properties of non-synchronizing semigroups are described which are then applied to semigroups induced by combinatorial tiling problems (see Chapter 7). 5. At last, it is shown that all rank 3 graphs admitting singular endomorphisms are hulls and it is conjectured that a hull on n vertices has minimal generating set of at most n generators (see Chapter 8)

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum

Syntax-semantics interface: an algebraic model

We extend our formulation of Merge and Minimalism in terms of Hopf algebras to an algebraic model of a syntactic-semantic interface. We show that methods adopted in the formulation of renormalization (extraction of meaningful physical values) in theoretical physics are relevant to describe the extraction of meaning from syntactic expressions. We show how this formulation relates to computational models of semantics and we answer some recent controversies about implications for generative linguistics of the current functioning of large language models.Comment: LaTeX, 75 pages, 19 figure

On Algebraic Singularities, Finite Graphs and D-Brane Gauge Theories: A String Theoretic Perspective

In this writing we shall address certain beautiful inter-relations between the construction of 4-dimensional supersymmetric gauge theories and resolution of algebraic singularities, from the perspective of String Theory. We review in some detail the requisite background in both the mathematics, such as orbifolds, symplectic quotients and quiver representations, as well as the physics, such as gauged linear sigma models, geometrical engineering, Hanany-Witten setups and D-brane probes. We investigate aspects of world-volume gauge dynamics using D-brane resolutions of various Calabi-Yau singularities, notably Gorenstein quotients and toric singularities. Attention will be paid to the general methodology of constructing gauge theories for these singular backgrounds, with and without the presence of the NS-NS B-field, as well as the T-duals to brane setups and branes wrapping cycles in the mirror geometry. Applications of such diverse and elegant mathematics as crepant resolution of algebraic singularities, representation of finite groups and finite graphs, modular invariants of affine Lie algebras, etc. will naturally arise. Various viewpoints and generalisations of McKay's Correspondence will also be considered. The present work is a transcription of excerpts from the first three volumes of the author's PhD thesis which was written under the direction of Prof. A. Hanany - to whom he is much indebted - at the Centre for Theoretical Physics of MIT, and which, at the suggestion of friends, he posts to the ArXiv pro hac vice; it is his sincerest wish that the ensuing pages might be of some small use to the beginning student.Comment: 513 pages, 71 figs, Edited Excerpts from the first 3 volumes of the author's PhD Thesi