Reasons fundamentalists assert that normative reality is constituted by exemplifications of the normative reasons relation: an irreducible, sui generis relation that strongly supervenes on non-normative reality. In this dissertation, I argue that reasons fundamentalists cannot explain why exemplifications of the normative reasons relation strongly supervene on non-normative reality. Irreduciblists about normativity can avoid this problem by asserting, contra the reasons fundamentalist, that normative reality is constituted by exemplifications of thick properties, which provide material for a conceptual analysis of normative reasons. The theory that results analyzes normative reasons for action as answers to questions why an action promotes a thick property. Nearly every normative theorist affirms what I call Additive Normative Supervenience (ANS): Normatively discernible worlds must be non-normatively discernible. ANS asserts that, if Edward Snowden is morally good, then Snowden's counterparts in worlds that are indiscernible in all non-normative respects must be good. Reasons fundamentalists struggle to explain why ANS is true. I consider and reject potential explanations of ANS that appeal to conceptual entailment and a posteriori necessity. Rosen has recently offered an argument against ANS. Rejecting ANS, however, problematizes irreduciblist accounts of normative explanation and normative epistemology. Irreduciblists can avoid this dilemma by arguing that ANS is either incoherent or false and adopting an alternative formulation of normative supervenience. Bilgrami's arguments against the intelligibility of normative supervenience doctrines purport to show that ANS is in fact unintelligible, and Merricks' arguments against the supervenience of consciousness on microphysical properties can be extended to show that ANS is false. Neither argument, however, establishes the falsity or unintelligibility of a modified formulation of normative supervenience, Transformative Normative Supervenience (TNS): Normatively discernible worlds must be descriptively discernible, where descriptive discernibility is just discernibility with respect to non-normative properties or thick normative properties. Irreduciblists can explain the truth of TNS by adopting non-centralism about normative reasons--that is to say, by maintaining, contra the reasons fundamentalist, that normative reality is constituted most fundamentally by exemplifications of thick properties. This allows the irreduciblist to provide an account of normative explanation and normative epistemology, analyze normative reasons in terms of thick properties, and preserve buck-passing accounts of thin normative properties. Scanlon has argued that the reasons relation is a four-place relation, relating the facts that are reasons for an agent to perform an action in a given circumstance. I argue that facts are also reasons for an action with respect to a thick property that that action will promote, in contrast to sets of distinct actions that the agent could perform instead. The resulting six-place relation turns out to be an instance of the relation that holds between why-questions and answers. What it is to be a normative reason for an agent to do something is to be a correct answer to a question why that agent's doing that action will promote a thick property. Decades ago, Anscombe had also suggested that reasons were answers to why-questions of a certain kind. The attractiveness of this position has been relatively underappreciated in the philosophy of normative reasons, in part because Anscombe had offered the reasons- as-answers thesis as a thesis about motivating reasons rather than normative reasons. The reasons-as-answers thesis also provides resources for those irreduciblists about reasons who reject my non-centralist conclusions to avoid the wrong kind of reason problem for buck- passing accounts of normativity: they can distinguish between right and wrong kinds of reasons by distinguishing between answers to distinct kinds of why-questions.

Kim criticizes non-reductive physicalism as a suitable metaphysics of mind among things because of its failure on the issue of mental causation. The failure is especially present in the thesis of supervenience physicalism. Kim concludes that the causal powers of mental states can be reduced to the causal powers carried by the physical states realizing them. Such causal reduction might involve identity between mental properties and physical properties. I think this is not a necessary conclusion. I try to clarify some premises behind Kim’s analysis, regarding issues of irreducibility, downward causation and the structure of the physical domain. I think the main reason why Kim doubts the plausibility of non-reductive physicalism is his view that downward causation and non-reductive metaphysics indicate the physical domain being hierarchically divided into levels. It seems like Kim would take the opposite position regarding the structure of the physical: an undivided continuum. Yet, the question is if that position follows from the ontological tenet of physicalism. Finally, I conclude that not necessarily, and I develop some further implications and suggestions.

This thesis is a contribution to determine the absolute irreducibility of polynomials via their Newton polytopes. For any field F / a polynomial f in F[x1, x2,..., xk] can be associated with a polytope, called its Newton polytope. If the polynomial f has integrally indecomposable Newton polytope, in the sense of Minkowski sum, then it is absolutely irreducible over F / i.e. irreducible over every algebraic extension of F. We present some new results giving integrally indecomposable classes of polytopes. Consequently, we have some new criteria giving infinitely many types of absolutely irreducible polynomials over arbitrary fields.

This thesis is an examination of the structure of irreducible continua, with a particular emphasis on local connectedness and monotone maps. A continuum is irreducible if there exist a pair of points such that no proper subcontinuum contains both, with the arc being the most basic example. Being irreducible has a number of interesting implications for a continuum, both locally and globally, and it is these consequences we shall focus on. As mentioned above, the arc is the most straightforward example of an irreducible continuum. Indeed, an intuitive understanding of an irreducible continuum would be that it is structured like an arc, with the points of irreducibility at either end joined by a subspace with no loops or offshoots. In Chapter 2 we will see that for a certain class of continua this intuition is well founded by constructing a monotone map from an irreducible continuum onto an arc. This monotone map will preserve much of the structure of our continuum and as such will provide an insight into that structure. We will next examine a generalisation of irreducibility which considers finite sets of points rather than just pairs. A number of classical results will be re-examined in this light in Chapter 3. While the majority of these theorems will be shown to have close parallels in higher finite and infinite irreducibility there will be several which do not hold without further conditions on the continuum. Such anomalies will be particularly prevalent in continua which have indecomposable subcontinua dominating their structure. In Chapter 4 monotone maps will be constructed for finitely irreducible continua similar to the map to an arc mentioned previously. Chapters 7 and 8 will generalise irreducibility further to the infinite case and we will again construct monotone maps preserving the structure of our continuum. Along with the arc, another highly significant irreducible continuum is the sin 1 x continuum. Chapter 5 will focus on this continuum, which will be the basis for a nested sequence of continua. A number of results concerning continuous images of these continua will be presented before using the sequence of continua to define an indecomposable continuum. This continuum will be investigated, and it will be shown that the union of our nested continua form a composant of the indecomposable continuum. In Chapter 6 we will turn to the question of compactifications. If a space X is connected then any metric compactification of X will be a continuum. This chapter will answer the question of when a compactification is an irreducible continuum, with the remainder of the compactification consisting of all of the irreducible points. A list of properties will given such that a continuum has such a compactification if and only if it has each property on the list. It will also be demonstrated that each of these properties is independent of the others. Finally, in Chapter 9 we will revisit the idea of structure-preserving monotone maps, but this time in continua which are not irreducible. Motivated by the fibres of the maps in previous chapters, we will introduce two categories of subcontinua of a continuum X. The first will be nowhere dense subcontinua which are maximal with this property and the second will be subcontinua about which X is locally connected and which are minimal with this property. Continua in which every point lies in a maximal nowhere dense subcontinuum will be examined, as well as spaces in which every point lies in a unique minimal subcontinuum about which X is locally connected. We will also look at the properties of monotone maps arising from partitions of X into such subcontinua, and will prove that if every point of X lies in a maximal nowhere dense subcontinuum then the resulting quotient space will be one dimensional.

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The concept of emergence stands in need of an update, and I propose that ontologically emergent phenomena are characterized by four necessary features: relationality, novelty, irreducibility and broken symmetry. ‘Emergence’ is a useful term to denote the varied qualitative changes that spontaneously arise as the scale and complexity of related phenomena increases. Moreover, emergent phenomena share a unique relationship with the phenomena from which they emerge, namely the emergent relation. This relation is distinct from other types of relations (i.e., identity, composition, supervenience, etc.) and moreover is not beset by the problems of causal exclusion or downward causation. Lastly, I advance this account of emergence partly as an empirical hypothesis. The epistemic resources in dynamical systems theory are uniquely suited to describe the evolution of systems that manifest emergent phenomena. This is primarily because features like novelty and broken symmetry can be given mathematically precise descriptions in dynamical systems terms. The advantage of this updated concept of emergence is its compatibility with ideas of explanation, prediction and reduction. / Thesis / Master of Philosophy (MA) / The concept of emergence is a useful one to succinctly describe the relatedness of a variety of complex phenomena in our universe. The concept of emergence however stands in need of an update. Emergent phenomena, as some would argue, are not unexplainable brute facts nor are they wholly unpredictable. I propose that ontologically emergent phenomena are characterized by four necessary features: relationality, novelty, irreducibility, and broken symmetry.

The Cauchy-Mirimanoff Polynomials are a class of polynomials that naturally arise in various classical studies of Fermat's Last Theorem. Originally conjectured to be irreducible over 100 years ago, the irreducibility of the Cauchy-Mirimanoff polynomials is still an open conjecture. This dissertation takes a new approach to the study of the Cauchy-Mirimanoff Polynomials. The reciprocal transform of a self-reciprocal polynomial is defined, and the reciprocal transforms of the Cauchy-Mirimanoff Polynomials are found and studied. Particular attention is given to the Cauchy-Mirimanoff Polynomials with index three times a power of a prime, and it is shown that the Cauchy-Mirimanoff Polynomials of index three times a prime are irreducible.

<p dir="ltr">This thesis consists of four parts that study different topics in commutative algebra. The main results of the first part of the dissertation are in Chapter 3, which is based on the author’s paper [1]. Let R be a commutative Noetherian ring graded by a torsionfree abelian group G. We introduce the notion of G-graded irreducibility and prove that G-graded irreducibility is equivalent to irreducibility in the usual sense. This is a generalization of a result by Chen and Kim in the Z-graded case. We also discuss the concept of the index of reducibility and give an inequality for the indices of reducibility between any radical non-graded ideal and its largest graded subideal. The second topic is developed in Chapter 4 which is based on the author’s paper [2]. In this chapter, we prove that if P is a prime ideal of inside a polynomial ring S with dim S/P = r, and adjoining s general linear forms to the prime ideal changes the (r − s)-th Hilbert coefficient of the quotient ring by 1 and doesn’t change the 0th to (r − s − 1)-th Hilbert coefficients where s ≤ r, then the depth of S/P is n − s − 1. This criterion also tells us about possible restrictions on the generic initial ideal of a prime ideal inside a polynomial ring. The third part of the thesis is Chapter 5 which is based on the author’s paper [3]. Let R be a polynomial ring over a field. We introduce the concept of sequentially almost Cohen-Macaulay modules, describe the extremal rays of the cone of local cohomology tables of finitely generated graded R-modules which are sequentially almost Cohen-Macaulay, and also describe some cases when the local cohomology table of a module of dimension 3 has a nontrivial decomposition. The last part is Chapter 6 which is based on the author’s paper [4]. We introduce the notion of strongly Lech-independent ideals as a generalization of Lech-independent ideals defined by Lech and Hanes, and use this notion to derive inequalities on multiplicities of ideals. In particular, we prove a new case of Lech’s conjecture, namely, if (R, m) → (S, n) is a flat local extension of local rings with dim R = dim S, the completion of S is the completion of a standard graded ring over a field k with respect to the homogeneous maximal ideal, and the completion of mS is the completion of a homogeneous ideal, then e(R) ≤ e(S).</p>

Dans la thèse présente, nous nous intéressons à l'étude de la relative hyperbolicité des produits semi-direct des produits libres, ainsi que le problème de conjugaison pour certains automorphismes de ces produits libres.Plus précisement, pour un produit libre $$G=G_1astdotsast G_past F_k$$ un automorphisme $phi$ est intitulé atoroidal s'il ne fixe pas (ni aucune de ses puissances) la classe de conjugaison d'un élément hyperbolique de $G$. Cet automorphisme est appelé completement irréductible si le système de facteurs libres est le plus grand qui est fixé par toutes les puissances de cet automorphisme. Il est appelé toral si pour tous les $i$, il existe $g_iin G$ tel que ${rm ad}_{g_i}circ phi|_{G_i}$ est identité sur le facteur libre $G_i$. Nous disons qu'il a la condition centrale si pour chaque $i$, il existe $g_iin G$ conjugue $phi(G_i)$ à $G_i$, et s'il existe un élément non trivial de $G_irtimes_{{rm ad}_{g_i} circ phi|_{G_i}} mathbb{Z}$ qui est central dans $G_irtimes_{{rm ad}_{g_i} circ phi|_{G_i}} mathbb{Z}$.Nous prouvons, dans le Théorème 4.28, que si $phi$ est atoroidal et completement irréductible, et si le produit libre est non-elementaire ($kgeq 2$ ou $ p+k geq 3$), le groupe $Grtimes_phi mathbb{Z}$ est relativement hyperbolique (relativement a des suspensions de chaque $G_i$). Après, dans le Théorème 6.10, nous prouvons le même résultat si $phi$ est atoroidal avec la condition centrale. Nous prouvons aussi dans le Théorème 7.21 que si tous les $G_i$ sont abelien, le problème de conjugaison est solvable pour les automorphismes atoroidaux, toraux. Ces sont des analogues du résultat de Brinkmann [7] (celui qui a donné le résultat d'hyperbolicité pour les groupes libres), et du résultat de Dahmani [12] (celui qui a résolu le problème de conjugaison des automorphismes hyperboliques). / In this thesis, we are interested in the study of the relative hyperbolicity of the suspensions of free products, as well as the conjugacy problem of certain automorphisms of free products.To be more precise, given a free product $$G=G_1astdotsast G_past F_k$$ an automorphism $phi$ is said atoroidal if no power fixes the conjugacy class of an hyperbolic element. It is called fully irreducible if the given free factor system $[G_1],dots,[G_p]$ is the largest one that is fixed by every power of the automorphism. It is said toral if for all $i$, there exists $g_iin G$ such that ${rm ad}_{g_i}circ phi|_{G_i}$ is the identity on the free factor $G_i$. It is said to have central condition if for each $i$, there exists $g_iin G$ conjugating $phi(G_i)$ to $G_i$, and if there exists a non-trivial element of $G_irtimes_{{rm ad}_{g_i} circ phi|_{G_i}} mathbb{Z}$ that is central in $G_irtimes_{{rm ad}_{g_i} circ phi|_{G_i}} mathbb{Z}$.We prove, in Theorem 4.28, that if $phi$ is atoroidal and fully irreducible, and if the free product is non-elementary ($kgeq 2$ or $ p+k geq 3$), the group $Grtimes_phi mathbb{Z}$ is relatively hyperbolic (relative to the mapping torus of each $G_i$). Then in Theorem 6.10 we prove the same result holds if $phi$ is atoroidal with central condition. We also prove in Theorem 7.21 that if all $G_i$ are abelian, the conjugacy problem is solvable for toral atoroidal automorphisms. These are analogue of the result of Brinkmann [7] (which gave the hyperbolicity result for free groups) and the result of Dahmani [12] (which solved the conjugacy problem of hyperbolic automorphisms).

Cette thèse contient des preuves de convergence ou de divergence d'algorithmes d'optimisation appelés stratégies d'évolution (ESs), ainsi que le développement d'outils mathématiques permettant ces preuves.Les ESs sont des algorithmes d'optimisation stochastiques dits ``boîte noire'', i.e. où les informations sur la fonction optimisée se réduisent aux valeurs qu'elle associe à des points. En particulier, le gradient de la fonction est inconnu. Des preuves de convergence ou de divergence de ces algorithmes peuvent être obtenues via l'analyse de chaînes de Markov sous-jacentes à ces algorithmes. Les preuves de convergence et de divergence obtenues dans cette thèse permettent d'établir le comportement asymptotique des ESs dans le cadre de l'optimisation d'une fonction linéaire avec ou sans contrainte, qui est un cas clé pour des preuves de convergence d'ESs sur de larges classes de fonctions.Cette thèse présente tout d'abord une introduction aux chaînes de Markov puis un état de l'art sur les ESs et leur contexte parmi les algorithmes d'optimisation continue boîte noire, ainsi que les liens établis entre ESs et chaînes de Markov. Les contributions de cette thèse sont ensuite présentées:o Premièrement des outils mathématiques généraux applicables dans d'autres problèmes sont développés. L'utilisation de ces outils permet d'établir aisément certaines propriétés (à savoir l'irreducibilité, l'apériodicité et le fait que les compacts sont des small sets pour la chaîne de Markov) sur les chaînes de Markov étudiées. Sans ces outils, établir ces propriétés était un processus ad hoc et technique, pouvant se montrer très difficile.o Ensuite différents ESs sont analysés dans différents problèmes. Un (1,\lambda)-ES utilisant cumulative step-size adaptation est étudié dans le cadre de l'optimisation d'une fonction linéaire. Il est démontré que pour \lambda > 2 l'algorithme diverge log-linéairement, optimisant la fonction avec succès. La vitesse de divergence de l'algorithme est donnée explicitement, ce qui peut être utilisé pour calculer une valeur optimale pour \lambda dans le cadre de la fonction linéaire. De plus, la variance du step-size de l'algorithme est calculée, ce qui permet de déduire une condition sur l'adaptation du paramètre de cumulation avec la dimension du problème afin d'obtenir une stabilité de l'algorithme. Ensuite, un (1,\lambda)-ES avec un step-size constant et un (1,\lambda)-ES avec cumulative step-size adaptation sont étudiés dans le cadre de l'optimisation d'une fonction linéaire avec une contrainte linéaire. Avec un step-size constant, l'algorithme résout le problème en divergeant lentement. Sous quelques conditions simples, ce résultat tient aussi lorsque l'algorithme utilise des distributions non Gaussiennes pour générer de nouvelles solutions. En adaptant le step-size avec cumulative step-size adaptation, le succès de l'algorithme dépend de l'angle entre les gradients de la contrainte et de la fonction optimisée. Si celui ci est trop faible, l'algorithme convergence prématurément. Autrement, celui ci diverge log-linéairement.Enfin, les résultats sont résumés, discutés, et des perspectives sur des travaux futurs sont présentées. / In this dissertation an analysis of Evolution Strategies (ESs) using the theory of Markov chains is conducted. Proofs of divergence or convergence of these algorithms are obtained, and tools to achieve such proofs are developed.ESs are so called "black-box" stochastic optimization algorithms, i.e. information on the function to be optimized are limited to the values it associates to points. In particular, gradients are unavailable. Proofs of convergence or divergence of these algorithms can be obtained through the analysis of Markov chains underlying these algorithms. The proofs of log-linear convergence and of divergence obtained in this thesis in the context of a linear function with or without constraint are essential components for the proofs of convergence of ESs on wide classes of functions.This dissertation first gives an introduction to Markov chain theory, then a state of the art on ESs and on black-box continuous optimization, and present already established links between ESs and Markov chains.The contributions of this thesis are then presented:o General mathematical tools that can be applied to a wider range of problems are developed. These tools allow to easily prove specific Markov chain properties (irreducibility, aperiodicity and the fact that compact sets are small sets for the Markov chain) on the Markov chains studied. Obtaining these properties without these tools is a ad hoc, tedious and technical process, that can be of very high difficulty.o Then different ESs are analyzed on different problems. We study a (1,\lambda)-ES using cumulative step-size adaptation on a linear function and prove the log-linear divergence of the step-size; we also study the variation of the logarithm of the step-size, from which we establish a necessary condition for the stability of the algorithm with respect to the dimension of the search space. Then we study an ES with constant step-size and with cumulative step-size adaptation on a linear function with a linear constraint, using resampling to handle unfeasible solutions. We prove that with constant step-size the algorithm diverges, while with cumulative step-size adaptation, depending on parameters of the problem and of the ES, the algorithm converges or diverges log-linearly. We then investigate the dependence of the convergence or divergence rate of the algorithm with parameters of the problem and of the ES. Finally we study an ES with a sampling distribution that can be non-Gaussian and with constant step-size on a linear function with a linear constraint. We give sufficient conditions on the sampling distribution for the algorithm to diverge. We also show that different covariance matrices for the sampling distribution correspond to a change of norm of the search space, and that this implies that adapting the covariance matrix of the sampling distribution may allow an ES with cumulative step-size adaptation to successfully diverge on a linear function with any linear constraint.Finally, these results are summed-up, discussed, and perspectives for future work are explored.