This thesis is concerned with a solution theory for quasilinear singular stochastic partial differential equations. We approach the theory of regularity structures, a tool to tackle singular stochastic PDEs, from a new perspective which is well suited for, but not restricted to, quasilinear equations. In the first part of this thesis, we revisit the algebraic aspects of the theory of regularity structures. Although we approach regularity structures from a different perspective than originally done, we show that the same (Hopf-) algebraic structure is underlying. Trees do not play any role in our construction, hence the Hopf algebras underlying rough paths and regularity structures are not at our disposal. Instead, our alternative point of view gives a new (Lie-) geometric interpretation of the structure group, arising from simple actions on the nonlinearity of the equation and a parametrization of the solution manifold. In the second part of this thesis, we revisit the probabilistic aspects of the theory of regularity structures. We construct and stochastically estimate the centered model, which captures the local behaviour of the solution manifold. This is carried out under a spectral gap assumption on the driving noise, and based on a novel application of Malliavin calculus in regularity structures. In deriving the renormalized equation we are guided by symmetries, so that natural invariances of the model are built in. In the third part of this thesis, we make again use of the Malliavin derivative to obtain a robust characterization of the model, which persists for rough noise even as a mollification is removed. This allows for a simple derivation of invariances of the model that are not present at the level of approximations. Furthermore, we give a convergence result of models, which together with the characterization establishes a universality result in the class of noise ensembles satisfying uniformly a spectral gap assumption.

[pt] A estrutura dos mean-filed games foi desenvolvida com o intuito de estudar problemas com um infinito número de jogadores em algum tipo de competição, ao qual pode ser aplicado em diversos problemas. O estudo formalizado desses problemas começou, na comunidade matemática com Lasry and Lions, e mais ou menos na mesma época, porém independentemente, na comunidade de engenharia por P. Caines, Minyi Huang, and Roland Malhamé. Desde então a pesquisa nos mean-field games cresceu exponencialmente, e nesse trabalho apresentaremos regularidade para um caso de mean-field games utilizando tecnicas particulares. Nesse trabalho, estudamos time-dependent mean-field games no caso subquadrático, isto é, mean-field games, o qual é escrito como um sistema de duas equações, uma equação de Hamilton-Jacobi e uma equação do transporte ou uma equação de Fokker-Plank, em que o Hamiltoniano na equação de Hamilton-Jacobi possui um crescimento subquadratico. Começamos em assumir dez suposições, e então sob os mesmos deduzir regularidade Lipschitz para o sistema. / [en] The mean-field games framework was developed to study problems with an infinite number of rational players in competition, which could be applied in many problems. The formalized study of these problems has begun, in the mathematical community by Lasry and Lions, and beside them, but independently close to the same time in the engineering community by P. Caines, Minyi Huang, and Roland Malhamé. Since these seminal contributions, the research in mean-field games has grown exponentially, and in this work we present a regularity to a case of mean-field games using particulars techniques. In this work, we study time-dependent mean-field games in the subquadratic case, that is, mean-field games, which are written as a system of a Hamilton–Jacobi equation and a transport or Fokker–Planck equation, where The Hamiltonian presented on the Hamilton–Jacobi equation has a subquadratic growth. We begin by assuming ten assumptions, and then, under these assumptions derive Lipschitz regularity of the system.

We consider the approach to regularity structures introduced by Otto, Sauer, Smith and Weber to obtain a priori bounds for quasi-linear SPDEs. This approach replaces the index set of trees, used in the original constructions of Hairer et. al., by multi-indices describing products of derivatives of the corresponding nonlinearity. The two tasks of this thesis are: - Construction and estimates of the model. We first provide the construction of a model in the regular, deterministic setting, where negative renormalization can be avoided. We later extend these ideas to the singular case, incorporating BPHZ-renormalization under spectral gap assumptions as a convenient input for an automated proof of the stochastic estimates of the singular model in the full subcritical regime. - Characterization of the algebraic structures generated by the multi-index setting. We consider natural actions on functionals of the nonlinearity and build a (pre-)Lie algebra from them. We use this as the starting point of an algebraic path towards the structure group, which as in the regularity structures literature is based on a Hopf algebra. This approach further allows us to explore the relation between multi-indices and trees, which we express through pre-Lie and Hopf algebra morphisms, in certain semi-linear equations. All the results are based on a series of joint works with Otto, Tempelmayr and Tsatsoulis.

Early detection of rail defects is necessary for preventing derailments and costly damage to the train and railway infrastructure. A rail surface flaw can quickly propagate from a small fracture to a broken rail after only a few train cars have passed over it. Rail defect detection is typically performed by using an instrumented car or a separate railway monitoring vehicle. Rail surface irregularities can be measured using accelerometers mounted to the bogie side frames or wheel axles. Typical signal processing algorithms for detecting defects within a vertical acceleration signal use a simple thresholding routine that considers only the amplitude of the signal. As a result, rail surface defects that produce low amplitude acceleration signatures may not be detected, and special track components that produce high amplitude acceleration signatures may be flagged as defects. The focus of this research is to develop an intelligent signal processing algorithm capable of detecting and classifying various rail surface irregularities, including defects and special track components. Three algorithms are proposed and validated using data collected from an instrumented freight car. For the first two algorithms, one uses a windowed Fourier Transform while the other uses the Wavelet Transform for feature extraction. Both of these algorithms use an artificial neural network for feature classification. The third algorithm uses the Wavelet Transform to perform a regularity analysis on the signal. The algorithms are validated with the collected data and shown to out-perform the threshold-based algorithm for the same data set. Proper training of the defect detection algorithm requires a large data set consisting of operating conditions and physical parameters. To generate this training data, a dynamic wheel-rail interaction model was developed that relates defect geometry to the side frame vertical acceleration signature. The model was generated by using combined systems dynamic modeling, and the system was solved with a developed combined lumped and distributed parameter system numerical approximation. The broken rail model was validated with real data collected from an instrumented freight car. The model was then used to train and validate the defect detection methodologies for various train and rail physical parameters and operating conditions. / Ph. D.

Introduction: Adverse Childhood Experiences (ACEs) are known to increase risk of mental health challenges throughout development, and sleep is known to decrease risk of mental health challenges. These have not been studied in tandem in younger cohorts. We investigated whether interactions between sleep duration and sleep regularity would moderate the impact of ACE exposure on risk for the development of mental health disorders. Methods: We conducted secondary cross-sectional analyses on the 2020-2021 waves of the National Survey of Children's Health (NSCH) (n = 92,669). We used logistic and ordinal regression to replicate known main effects of ACEs (total, household, community, and single) and sleep (duration and irregularity) on mental health diagnostic status and symptom severity, and we examined the interaction of ACEs and sleep on mental health diagnostic status. To correct for multiple comparisons, all original models were performed with one half of the dataset and then replicated in the second half. Follow-up analyses by age cohort (0-5, 6-11, 12-17 years) further examined interaction effects across development. Poverty level, parental education status, child age, child sex, neighborhood safety, neighborhood support, and race/ethnicity were included as covariates, as indicated in a priori acyclic graph (DAG) modeling. Results: Known main effects for ACE and sleep on mental health diagnoses were replicated across all models. Interactions between ACE exposure and adequate sleep duration or increased sleep irregularity were not clinically significant, although some were statistically significant due to large sample size, such that adequate sleep duration was associated with marginally increased risk of mental health diagnosis (Omnibus B = 0.048, p < 0.0001) and greater bedtime irregularity was associated with marginally decreased risk of mental health diagnosis (Omnibus B = -0.030, p < 0.001). Discussion: Main effects in this analysis are consistent with previous literature on ACEs, sleep, and mental health. However, interaction effects were largely small and clinically insignificant. Dichotomous and categorical parent-report items assessing sleep health may not be sensitive to interaction effects, compared with continuous data or physiological measurements. Further, examining mental health symptoms (rather than diagnosis status) may also allow for more nuanced understanding of potential interaction effects.

Let A be an Euclidean open Lipschitz set. This dissertation aims to discuss some results concerning the boundary regularity for almost-minimizers of the relative perimeter in A. An almost-minimizer of the relative perimeter in A is a measurable set E that minimizes the perimeter functional P(E;A), roughly speaking the (n-1)-area of the boundary of E in A, among local competitors for E. Important examples of almost-minimizers in A are given, for instance, by the solutions to relative isoperimetric problems like min { P(E;A) : E is contained in A and |E| = m}. While when A is smooth the theory of the boundary regularity for almost-minimizers is well-established, little is known when the boundary of A contains singular points such as edges, vertices, cusps, etc. In particular, we prove a boundary Monotonicity Formula, holding under a so-called visibility condition on A at a point x on the boundary of A, and a Vertex-skipping Theorem, valid when n = 3 and A is convex. This latter result establishes that the closure of the boundary of an almost-minimizer of the relative perimeter in a 3-dimensional open, convex set A cannot contain vertex-type singularities of the boundary of A. The optimality of the dimensional restriction n = 3 is also examined in the thesis.

We examine different methods of generalize Van der Waerden's Theorem, the Multidimensional Van der Waerden Theorem, the Canonical Van der Waerden Theorem, and other Variants. / Master of Science / Ramsey Theory is a subfield of mathematics in which randomness is studied from the perspective of partition regularity. We say that a structure <i>A</i> is partition regular within some space, if for any partition of the space into to some finite number of pieces, one of the pieces contains a copy of <i>A</i>. The simplest example of this, is letting <i>A</i> be the collection of 2 points sets, then no matter how you partition the integers into a finite number of pieces, at least one of the pieces must contain some 2 point set. If we replace 2 in the previous example with some fixed number <i>n</i>, then we obtain what is commonly referred to as the pigeon hole principle, which is one of the earliest results of combinatorics. To be more precise, the pigeon hole principle tells us that given any number <i>n</i>, and any finite partition of the positive integers, at least one of the pieces contains some <i>n</i> point set. However, the pigeon hole principle does not tell us anything about the <i>n</i> point set other than its size. Ramsey Theory seeks to generalize the pigeon hole principle by imposing further restrictions, by asking questions such as if we can always find an <i>n</i> point set consisting of consecutive integers, even integers, perfect squares, and so on. One of the resulting generalizations is known as Van der Waerden’s Theorem, which deals with structures known as arithmetic progressions. An arithmetic progression is a set of integers in which the difference between consecutive elements is the same, such as {3, 7, 11, 15, 19, 23, 27, 31}, or {a + jd}_j=0^k is its most general form. Van der Waerden’s Theorem states that we can generalize the pigeon hole principle by assuming that the <i>n</i> point sets we are finding are also arithmetic progressions. To be more precise, Van der Waerden’s Theorem states that for any partition of the positive integers into a finite number of pieces, and any positive integer <i>n</i>, at least one of the pieces of the partition contains an arithmetic progression of <i>n</i> numbers. In this thesis, we will be examining how to further refine Van der Waerden’s Theorem and its generalizations.

We examine the Rayleigh–Bénard convection as modelled by the Boussinesq equation. Our aim is at deriving bounds for the heat enhancement factor in the vertical direction, the Nusselt number, which reproduce physical scalings. In the first part of the dissertation, we examine the the simpler model when the acceleration of the fluid is neglected (Pr=∞) and prove the non-optimality of the temperature background field method by showing a lower bound for the Nusselt number associated to it. In the second part we consider the full model (Pr<∞) and we prove a new upper bound which improve the existing ones (for large Pr numbers) and catches a transition at Pr~Ra^(1/3).

Die Regularitätsmethode für Graphen wurde vor über 30 Jahren von Szemerédi, für den Beweis seines Dichteresultates über Teilmengen der natürlichen Zahlen, welche keine arithmetischen Progressionen enthalten, entwickelt. Grob gesprochen besagt das Regularitätslemma, dass die Knotenmenge eines beliebigen Graphen in konstant viele Klassen so zerlegt werden kann, dass fast alle induzierten bipartiten Graphen quasi-zufällig sind, d.h. sie verhalten sich wie zufällige bipartite Graphen mit derselben Dichte. Das Regularitätslemma hatte viele weitere Anwendungen, vor allem in der extremalen Graphentheorie, aber auch in der theoretischen Informatik und der kombinatorischen Zahlentheorie, und gilt mittlerweile als eines der zentralen Hilfsmittel in der modernen Graphentheorie. Vor wenigen Jahren wurden Regularitätslemmata für andere diskrete Strukturen entwickelt. Insbesondere wurde die Regularitätsmethode für uniforme Hypergraphen und dünne Graphen verallgemeinert. Ziel der vorliegenden Arbeit ist die Weiterentwicklung der Regularitätsmethode und deren Anwendung auf Probleme der theoretischen Informatik. Im Besonderen wird gezeigt, dass vererbbare (entscheidbare) Hypergrapheneigenschaften, das sind Familien von Hypergraphen, welche unter Isomorphie und induzierten Untergraphen abgeschlossen sind, testbar sind. D.h. es existiert ein randomisierter Algorithmus, der in konstanter Laufzeit mit hoher Wahrscheinlichkeit zwischen Hypergraphen, welche solche Eigenschaften haben und solchen die „weit“ davon entfernt sind, unterscheidet. / About 30 years ago Szemerédi developed the regularity method for graphs, which was a key ingredient in the proof of his famous density result concerning the upper density of subsets of the integers which contain no arithmetic progression of fixed length. Roughly speaking, the regularity lemma asserts, that the vertex set of every graph can be partitioned into a constant number of classes such that almost all of the induced bipartite graphs are quasi-random, i.e., they mimic the behavior of random bipartite graphs of the same density. The regularity lemma had have many applications mainly in extremal graph theory, but also in theoretical computer science and additive number theory, and it is considered one of the central tools in modern graph theory. A few years ago the regularity method was extended to other discrete structures. In particular extensions for uniform hypergraphs and sparse graphs were obtained. The main goal of this thesis is the further development of the regularity method and its application to problems in theoretical computer science. In particular, we will show that hereditary, decidable properties of hypergraphs, that are properties closed under isomorphism and vertex removal, are testable. I.e., there exists a randomised algorithm with constant running time, which distinguishes between Hypergraphs displaying the property and those which are “far” from it.

Na presente dissertação faremos um estudo dos conjuntos algébricos, semialgébricos, analíticos, semianalíticos e subanalíticos, real e complexo, através das condições de regularidade da estratificação destes conjuntos. A idéia básica em estratificação é decompor um espaço singular em variedades regulares; e as condições de regularidade, são um controle de como essas variedades se reencontram. Faremos uma abordagem geral das principais condições de regularidade. As condições (a) e (b) de H. Whitney, a (c)-regularidade de K. Bekka, a condição Whitney fraca, definida por D. Trotman e K. Bekka, o teste da razão de Kuo e a (w)-regularidade de Verdier, apresentando suas principais propriedades, teoremas e condições de existência / In the present dissertation we do a study of algebraic, semialgebraic, analytic, semianalytic and subanalytic sets, real and complex, through the regularity conditions of the stratification of these sets. The basic idea in stratification is to decompose a singular space into manifolds; and the regularity conditions, is a control of how these manifolds fit together. We do a general approach of the main regularity conditions. The conditions (a) and (b) of H. Whitney, the (c)-regularity of K. Bekka, the condition weakly Whitney, defined for D. Trotman and K. Bekka, the Kuo ratio test and the (w)-regularity of Verdier, presenting their main properties, theorems and conditions of existence