Projective Limit

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Peter Orbanz - One of the best experts on this subject based on the ideXlab platform.

  • Projective Limit random probabilities on polish spaces
    Electronic Journal of Statistics, 2011
    Co-Authors: Peter Orbanz
    Abstract:

    A pivotal problem in Bayesian nonparametrics is the construction of prior distributions on the space M(V) of probability measures on a given domain V. In principle, such distributions on the infinite-dimensional space M(V) can be constructed from their finite-dimensional marginals—the most prominent example being the construction of the Dirichlet process from finite-dimensional Dirichlet distributions. This approach is both intuitive and applicable to the construction of arbitrary distributions on M(V), but also hamstrung by a number of technical difficulties. We show how these difficulties can be resolved if the domain V is a Polish topological space, and give a representation theorem directly applicable to the construction of any probability distribution on M(V) whose first moment measure is well-defined. The proof draws on a Projective Limit theorem of Bochner, and on properties of set functions on Polish spaces to establish countable additivity of the resulting random probabilities.

Pelletier Fernand - One of the best experts on this subject based on the ideXlab platform.

  • AN INTEGRABILITY CRITERION FOR A Projective Limit OF BANACH DISTRIBUTIONS
    HAL CCSD, 2021
    Co-Authors: Pelletier Fernand
    Abstract:

    We give an integrability criterion for a Projective Limit of Banach distributions on a Fréchet manifold which is a Projective Limit of Banach manifolds. This leads to a result of integrability of Projective Limit of involutive bundles on a Projective sequence of Banach manifolds. This can be seen as a version of Frobenius Theorem in Fréchet setting. As consequence, we obtain a version of the third Lie theorem for a Fréchet-Lie group which is a submersive Projective Limit of Banach Lie groups. We also give an application to a sequence of prolongations of a Banach Lie algebroid

  • Projective Limit of a sequence of compatible weak symplectic forms on a sequence of Banach bundles and Darboux Theorem
    2020
    Co-Authors: Pelletier Fernand
    Abstract:

    Given a Projective sequence of Banach bundles, each one provided with a of weak symplectic form, we look for conditions under which, the corresponding sequence of weak symplectic forms gives rise to weak symplectic form on the Projective Limit bundle. Then we apply this results to the tangent bundle of a Projective Limit of Banach manifolds. This naturally leads to ask about conditions under which the Darboux Theorem is also true on the Projective Limit of Banach manifolds. We will give some necessary and some sufficient conditions so that such a result is true. Then we discuss why, in general, the Moser's method can not work on Projective Limit of Banach weak symplectic Banach manifolds without very strong conditions like Kumar 's results ([17]). In particular we give an example of a Projective sequence of weak symplectic Banach manifolds on which the Darboux Theorem is true on each manifold, but is not true on the Projective Limit of these manifolds

Fernand Pelletier - One of the best experts on this subject based on the ideXlab platform.

  • an integrability criterion for a Projective Limit of banach distributions
    arXiv: Differential Geometry, 2021
    Co-Authors: Fernand Pelletier
    Abstract:

    We give an integrability criterion for a Projective Limit of Banach distributions on a Frechet manifold which is a Projective Limit of Banach manifolds. This leads to a result of integrability of Projective Limit of involutive bundles on a Projective sequence of Banach manifolds. This can be seen as a version of Frobenius Theorem in Frechet setting. As consequence, we obtain a version of the third Lie theorem for a Frechet-Lie group which is a submersive Projective Limit of Banach Lie groups. We also give an application to a sequence of prolongations of a Banach Lie algebroid.

  • Projective Limit of a sequence of compatible weak symplectic forms on a sequence of banach bundles and darboux theorem
    Bulletin Des Sciences Mathematiques, 2021
    Co-Authors: Fernand Pelletier
    Abstract:

    Abstract Given a Projective sequence of Banach bundles, each one provided with of a weak symplectic form, we look for conditions under which, the corresponding sequence of weak symplectic forms gives rise to weak symplectic form on the Projective Limit bundle. Then we apply this results to the tangent bundle of a Projective Limit of Banach manifolds. This naturally leads to ask about conditions under which the Darboux Theorem is also true on the Projective Limit of Banach manifolds. We will give some necessary and some sufficient conditions so that such a result is true. Then we discuss why, in general, the Moser's method can not work on Projective Limit of Banach weak symplectic Banach manifolds without very strong conditions like Kumar's results ( [15] ). In particular we give an analog result of Kumar' s one with weaker assumptions and we give an example for which such weaker conditions are satisfied. More generally, we produce examples of Projective sequence of weak symplectic Banach manifolds on which the Darboux Theorem is true and an example for which the Darboux Theorem is true on each manifold, but is not true on the Projective Limit of these manifolds.

Patrick Popescupampu - One of the best experts on this subject based on the ideXlab platform.

  • the valuative tree is the Projective Limit of eggers wall trees
    Revista De La Real Academia De Ciencias Exactas Fisicas Y Naturales Serie A-matematicas, 2019
    Co-Authors: Evelia Garcia R Barroso, Pedro Daniel Gonzalez Perez, Patrick Popescupampu
    Abstract:

    Consider a germ C of reduced curve on a smooth germ S of complex analytic surface. Assume that C contains a smooth branch L. Using the Newton-Puiseux series of C relative to any coordinate system (x, y) on S such that L is the y-axis, one may define the Eggers-Wall tree$$\Theta _L(C)$$ of C relative to L. Its ends are labeled by the branches of C and it is endowed with three natural functions measuring the characteristic exponents of the previous Newton-Puiseux series, their denominators and contact orders. The main objective of this paper is to embed canonically $$\Theta _L(C)$$ into Favre and Jonsson’s valuative tree $${\mathbb P}(\mathcal {V})$$ of real-valued semivaluations of S up to scalar multiplication, and to show that this embedding identifies the three natural functions on $$\Theta _L(C)$$ as pullbacks of other naturally defined functions on $${\mathbb P}(\mathcal {V})$$. As a consequence, we generalize the well-known inversion theorem for one branch: if $$L'$$ is a second smooth branch of C, then the valuative embeddings of the Eggers-Wall trees $$\Theta _{L'}(C)$$ and $$\Theta _L(C)$$ identify them canonically, their associated triples of functions being easily expressible in terms of each other. We prove also that the space $${\mathbb P}(\mathcal {V})$$ is the Projective Limit of Eggers-Wall trees over all choices of curves C. As a supplementary result, we explain how to pass from $$\Theta _L(C)$$ to an associated splice diagram.

  • the valuative tree is the Projective Limit of eggers wall trees
    arXiv: Algebraic Geometry, 2018
    Co-Authors: Evelia Garcia R Barroso, Pedro Daniel Gonzalez Perez, Patrick Popescupampu
    Abstract:

    Consider a germ $C$ of reduced curve on a smooth germ $S$ of complex analytic surface. Assume that $C$ contains a smooth branch $L$. Using the Newton-Puiseux series of $C$ relative to any coordinate system $(x,y)$ on $S$ such that $L$ is the $y$-axis, one may define the {\em Eggers-Wall tree} $\Theta_L(C)$ of $C$ relative to $L$. Its ends are labeled by the branches of $C$ and it is endowed with three natural functions measuring the characteristic exponents of the previous Newton-Puiseux series, their denominators and contact orders. The main objective of this paper is to embed canonically $\Theta_L(C)$ into Favre and Jonsson's valuative tree $\mathbb{P}(\mathcal{V})$ of real-valued semivaluations of $S$ up to scalar multiplication, and to show that this embedding identifies the three natural functions on $\Theta_L(C)$ as pullbacks of other naturally defined functions on $\mathbb{P}(\mathcal{V})$. As a consequence, we prove an inversion theorem generalizing the well-known Abhyankar-Zariski inversion theorem concerning one branch: if $L'$ is a second smooth branch of $C$, then the valuative embeddings of the Eggers-Wall trees $\Theta_{L'}(C)$ and $\Theta_L(C)$ identify them canonically, their associated triples of functions being easily expressible in terms of each other. We prove also that the space $\mathbb{P}(\mathcal{V})$ is the Projective Limit of Eggers-Wall trees over all choices of curves $C$. As a supplementary result, we explain how to pass from $\Theta_L(C)$ to an associated splice diagram.

Donal Oregan - One of the best experts on this subject based on the ideXlab platform.

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