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In this article we use techniques from algebraic geometry and homological algebra, together with ideas from string theory to construct a class of 3-dimensional Calabi-Yau algebras. The Calabi-Yau property appears throughout geometry and string theory and is increasingly being studied in algebra. We further show that the algebras constructed are examples of non-commutative crepant resolutions (NCCRs), in the sense of Van den Bergh, of Gorenstein affine toric threefolds. Dimer models, first studied in theoretical physics, give a way of writing down a class of non-commutative algebras, as the path algebra of a quiver with relations obtained from a ‘superpotential’. Some examples are Calabi-Yau and some are not. We consider two types of ‘consistency’ conditions on dimer models, and show that a ‘geometrically consistent’ dimer model is ‘algebraically consistent’. We prove that the algebras obtained from algebraically consistent dimer models are 3-dimensional Calabi-Yau algebras. This is the key step which allows us to prove that these algebras are NCCRs of the Gorenstein affine toric threefolds associated to the dimer models.

This book studies generalized Donaldson-Thomas invariants $\\bar{DT}{}^\\alpha(\\tau)$. They are rational numbers which `count' both $\\tau$-stable and $\\tau$-semistable coherent sheaves with Chern character $\\alpha$ on $X$; strictly $\\tau$-semistable sheaves must be counted with complicated rational weights. The $\\bar{DT}{}^\\alpha(\\tau)$ are defined for all classes $\\alpha$, and are equal to $DT^\\alpha(\\tau)$ when it is defined. They are unchanged under deformations of $X$, and transform by a wall-crossing formula under change of stability condition $\\tau$. To prove all this, the authors study the local structure of the moduli stack $\\mathfrak M$ of coherent sheaves on $X$. They show that an atlas for $\\mathfrak M$ may be written locally as $\\mathrm{Crit}(f)$ for $f:U\\to{\\mathbb C}$ holomorphic and $U$ smooth, and use this to deduce identities on the Behrend function $\\nu_\\mathfrak M$. They compute the invariants $\\bar{DT}{}^\\alpha(\\tau)$ in examples, and make a conjecture about their integrality properties. They also extend the theory to abelian categories $\\mathrm{mod}$-$\\mathbb{C}Q\\backslash I$ of representations of a quiver $Q$ with relations $I$ coming from a superpotential $W$ on $Q$.

For $M$ a closed manifold or the Euclidean space $\\mathbb{R}^n$, the authors present a detailed proof of regularity properties of the composition of $H^s$-regular diffeomorphisms of $M$ for $s >\\frac{1}{2}\\dim M 1$.

Nous étudions la fonction zêta des hauteurs anticanonique d’une variété torique (non nécessairement déployée) définie sur un corps global de caractéristique positive. Nous nous inspirons pour cela de la méthode utilisée par Batyrev et Tschinkel pour traiter la situation analogue en caractéristique zéro, méthode que nous rappelons d’ailleurs en détail. We investigate the anticanonical height zeta function of a (not necessarily split) toric variety defined over a global field of positive characteristic, drawing our inspiration from the method used by Batyrev and Tschinkel to deal with the analogous problem over a number field. By the way, we give a detailed account of their method.

We study the Lyapunov exponents and their associated invariant subspaces for infinite dimensional random dynamical systems in a Banach space, which are generated by, for example, stochastic or random partial differential equations. We prove a multiplicative ergodic theorem. Then, we use this theorem to establish the stable and unstable manifold theorem for nonuniformly hyperbolic random invariant sets.

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We develop a theory of Nöbeling manifolds similar to the theory of Hilbert space manifolds. We show that it reflects the theory of Menger manifolds developed by M. Bestvina [Mladen Bestvina, We define the The following theorem was proved by D. W. Henderson and R. Schori [David W. Henderson and R. Schori, We also establish the open embedding theorem, the

In this paper the author classifies symplectic actions of $2$-tori on compact connected symplectic $4$-manifolds, up to equivariant symplectomorphisms. This extends results of Atiyah, Guillemin-Sternberg, Delzant and Benoist. The classification is in terms of a collection of invariants of the topology of the manifold, of the torus action and of the symplectic form. The author constructs explicit models of such symplectic manifolds with torus actions, defined in terms of these invariants.

The study of relativizers in the Spanish language has not been quite explored from a sociolinguistic point of view. A few research papers have analyzed varieties found in Santiago de Chile, Mexico City, Sevilla, and Madrid. Nevertheless, none of them has addressed a Colombian variety in depth. This variationist study aims to fill the gap by reporting the results of a correlational study focused on the use and variation of relativizers in the preseea-Medellín corpus. An anova test, a Games-Howell test, multiple Tukey’s tests, and pairwise comparisons t-tests with Bonferroni correction were run to identify the variables with significant effects (e.g., geographical position, gender, level of education, and social class). Results suggest that, while there is significant diatopic variation in the selection of relativizers, diastratic variables have a minor role in their frequency of use. It was also noted that while the use of the pronoun que tends to spread in all varieties of Spanish, the relative adjectives cuyo and cuanto continue their trend towards disuse. Further studies are necessary to determine if this tendency to simplification is related to psycholinguistic constraints (i.e. mental load) or functional aspects of Spanish. Finally, this study opens the doors to the analysis of relativizers in other varieties of Colombian Spanish from a sociolinguistic perspective.

Although the field of studies in Industrial Relations (IR) in Brazil was born along with the Brazilian Academy of Management itself, contributions regarding the models used to compare Industrial Relations Systems (IRSs) from different countries in Brazilian literature are still very rare. On the other hand, models to compare IRSs from different countries are quite present and can be considered traditional in the international literature produced in Europe, the USA, Australia and more recently in Asia. This work presents the trajectory of IRSs theories and the debate surrounding them, from the founders of this field in the USA and England to the VoC model – Varieties of Capitalism, a model that has been widely used internationally. Criticism of this VoC model, initially composed of two types, liberal and regulated, for being excessively focused on the most industrialized countries, generated other types within VoC, such as the Hierarchical Model of Capitalism (HME). This HME type is understood here as a possible starting model for the analysis of Latin American SRTs, including the Brazilian one, although this variant has also suffered criticism, which, in the understanding of this article, does not preclude its use.