667 research outputs found

    Model theory of operator algebras III: Elementary equivalence and II_1 factors

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    We use continuous model theory to obtain several results concerning isomorphisms and embeddings between II_1 factors and their ultrapowers. Among other things, we show that for any II_1 factor M, there are continuum many nonisomorphic separable II_1 factors that have an ultrapower isomorphic to an ultrapower of M. We also give a poor man's resolution of the Connes Embedding Problem: there exists a separable II_1 factor such that all II_1 factors embed into one of its ultrapowers.Comment: 16 page

    A modular description of X0(n)\mathscr{X}_0(n)

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    As we explain, when a positive integer nn is not squarefree, even over C\mathbb{C} the moduli stack that parametrizes generalized elliptic curves equipped with an ample cyclic subgroup of order nn does not agree at the cusps with the Γ0(n)\Gamma_0(n)-level modular stack X0(n)\mathscr{X}_0(n) defined by Deligne and Rapoport via normalization. Following a suggestion of Deligne, we present a refined moduli stack of ample cyclic subgroups of order nn that does recover X0(n)\mathscr{X}_0(n) over Z\mathbb{Z} for all nn. The resulting modular description enables us to extend the regularity theorem of Katz and Mazur: X0(n)\mathscr{X}_0(n) is also regular at the cusps. We also prove such regularity for X1(n)\mathscr{X}_1(n) and several other modular stacks, some of which have been treated by Conrad by a different method. For the proofs we introduce a tower of compactifications Ellm\overline{Ell}_m of the stack EllEll that parametrizes elliptic curves---the ability to vary mm in the tower permits robust reductions of the analysis of Drinfeld level structures on generalized elliptic curves to elliptic curve cases via congruences.Comment: 67 pages; final version, to appear in Algebra and Number Theor

    Galois descent of semi-affinoid spaces

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    We study the Galois descent of semi-affinoid non-archimedean analytic spaces. These are the non-archimedean analytic spaces which admit an affine special formal scheme as model over a complete discrete valuation ring, such as for example open or closed polydiscs or polyannuli. Using Weil restrictions and Galois fixed loci for semi-affinoid spaces and their formal models, we describe a formal model of a KK-analytic space XX, provided that XKLX\otimes_KL is semi-affinoid for some finite tamely ramified extension LL of KK. As an application, we study the forms of analytic annuli that are trivialized by a wide class of Galois extensions that includes totally tamely ramified extensions. In order to do so, we first establish a Weierstrass preparation result for analytic functions on annuli, and use it to linearize finite order automorphisms of annuli. Finally, we explain how from these results one can deduce a non-archimedean analytic proof of the existence of resolutions of singularities of surfaces in characteristic zero.Comment: Exposition improved and minor modifications. 37 pages. To appear in Math.

    A Note on Derived Geometric Interpretation of Classical Field Theories

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    In this note, we would like to provide a conceptional introduction to the interaction between derived geometry and physics based on the formalism that has been heavily studied by Kevin Costello. Main motivations of our current attempt are as follows: (i) to provide a brief introduction to derived algebraic geometry, which can be, roughly speaking, thought of as a higher categorical refinement of an ordinary algebraic geometry, (ii) to understand how certain derived objects naturally appear in a theory describing a particular physical phenomenon and give rise to a formal mathematical treatment, such as redefining a perturbative classical field theory (or its quantum counterpart) by using the language of derived algebraic geometry, and (iii) how the notion of factorization algebra together with certain higher categorical structures come into play to encode the structure of so-called observables in those theories by employing certain cohomological/homotopical methods. Adopting such a heavy and relatively enriched language allows us to formalize the notion of quantization and observables in quantum field theory as well.Comment: 14 pages. This note serves as an introductory survey on certain mathematical structures encoding the essence of Costello's approach to derived-geometric formulation of field theories and the structure of observables in an expository manne

    Development of coarse-grained models for the simulation of soft matter systems

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    This thesis aims to examine the parametrisation of coarse-grained models for the simulation of soft matter systems. The strengths and weaknesses of a range of methods are examined, and suggestions for improvements are made. Initially, two bottom-up methods, iterative Boltzmann inversion (IBI) and hybrid force matching (HFM) are applied to a liquid octane/benzene mixture and compared to a top-down model based on a version of statistical associating fluid theory, the SAFT-γ Mie equation of state. These models are tested for their ability to represent the structure and thermodynamics of the underlying atomistic system, as well as their transferability between temperatures and concentrations. Attempts are then made to address the poor transferability of the bottom-up models using a variant of IBI, multi-state IBI (MS-IBI). MS-IBI allows concentration transferable potentials to be generated but is not successful in improving temperature transferability. The state-point dependence of pair potentials is identified as the cause of poor temperature transferability, and initial attempts to address this are discussed. A range of coarse-grained models of the non-ionic liquid crystal TP6EO2M is examined. HFM is able to give a structurally accurate coarse-grained model; however, the difficulty of sampling all relevant configurations within an atomistic reference system appear to cause problems with calculating accurate association free energies. The new MARTINI 3 top-down force field is shown to improve upon the structural and thermodynamic properties of MARTINI 2, allowing larger system sizes to be studied. The nematic and hexagonal columnar chromonic phases are observed, and the concentration dependence seen in the experimental phase diagram is reproduced. This represents the first simulations of chromonic liquid crystal phases using systematic coarse graining

    Copula-based orderings of multivariate dependence

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    In this paper I investigate the problem of defining a multivariate dependence ordering. First, I provide a characterization of the concordance dependence ordering between multivariate random vectors with fixed margins. Central to the characterization is a multivariate generalization of a well-known bivariate elementary dependence increasing rearrangement. Second, to order multivariate random vectors with non-fixed margins, I impose a scale invariance principle which leads to a copula-based concordance dependence ordering. Finally, a wide family of copula-based measures of dependence is characterized to which Spearman’s rank correlation coefficient belongs.copula, concordance ordering, dependence measures, dependence orderings, multivariate stochastic dominance, supermodular ordering.
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