scholarly journals New dimension-theory techniques for constructing infinite-dimensional examples

1979 ◽  
Vol 10 (1) ◽  
pp. 93-102 ◽  
Author(s):  
Leonard R. Rubin ◽  
R.M. Schori ◽  
John J. Walsh
Keyword(s):  
Dimension Theory ◽  
Infinite Dimensional

scholarly journals A Packing Problem for Holomorphic Curves

2009 ◽  
Vol 194 ◽  
pp. 33-68 ◽  
Author(s):  
Masaki Tsukamoto
Keyword(s):  
Packing Density ◽  
Packing Problem ◽  
Characteristic Number ◽  
Holomorphic Curves ◽  
Dimension Theory ◽  
Value Distribution Theory ◽  
New Approach ◽  
Infinite Dimensional ◽  
Dimensional Version ◽  
Mean Dimension

AbstractWe propose a new approach to the value distribution theory of entire holomorphic curves. We define packing density of Brody curves, and show that it has various non-trivial properties. The packing density of Brody curves can be considered as an infinite dimensional version of characteristic number, and it has an application to Gromov’s mean dimension theory.

-modules on rigid analytic spaces III: weak holonomicity and operations

2021 ◽  
Vol 157 (12) ◽  
pp. 2553-2584
Author(s):  
Konstantin Ardakov ◽  
Andreas Bode ◽  
Simon Wadsley
Keyword(s):  
Local Cohomology ◽  
Dimension Theory ◽  
Infinite Length ◽  
Analytic Subset ◽  
Infinite Dimensional ◽  
Algebraic Setting ◽  
Minimal Dimension ◽  
Stability Results ◽  
Open Subspace ◽  
Integrable Connection

Abstract We develop a dimension theory for coadmissible $\widehat {\mathcal {D}}$ -modules on rigid analytic spaces and study those which are of minimal dimension, in analogy to the theory of holonomic $\mathcal {D}$ -modules in the algebraic setting. We discuss a number of pathologies contained in this subcategory (modules of infinite length, infinite-dimensional fibres). We prove stability results for closed immersions and the duality functor, and show that all higher direct images of integrable connections restricted to a Zariski open subspace are coadmissible of minimal dimension. It follows that the local cohomology sheaves $\underline {H}^{i}_Z(\mathcal {M})$ with support in a closed analytic subset $Z$ of $X$ are also coadmissible of minimal dimension for any integrable connection $\mathcal {M}$ on $X$ .

scholarly journals Causal variational principles in the infinite-dimensional setting: Existence of minimizers

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Christoph Langer
Keyword(s):  
Variational Principles ◽  
Polish Space ◽  
Dimension Theory ◽  
Regular Measure ◽  
Lagrange Equations ◽  
Locally Compact ◽  
Locally Finite ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Existence Of Minimizers

Abstract We provide a method for constructing (possibly non-trivial) measures on non-locally compact Polish subspaces of infinite-dimensional separable Banach spaces which, under suitable assumptions, are minimizers of causal variational principles in the non-locally compact setting. Moreover, for non-trivial minimizers the corresponding Euler–Lagrange equations are derived. The method is to exhaust the underlying Banach space by finite-dimensional subspaces and to prove existence of minimizers of the causal variational principle restricted to these finite-dimensional subsets of the Polish space under suitable assumptions on the Lagrangian. This gives rise to a corresponding sequence of minimizers. Restricting the resulting sequence to countably many compact subsets of the Polish space, by considering the resulting diagonal sequence, we are able to construct a regular measure on the Borel algebra over the whole topological space. For continuous Lagrangians of bounded range, it can be shown that, under suitable assumptions, the obtained measure is a (possibly non-trivial) minimizer under variations of compact support. Under additional assumptions, we prove that the constructed measure is a minimizer under variations of finite volume and solves the corresponding Euler–Lagrange equations. Afterwards, we extend our results to continuous Lagrangians vanishing in entropy. Finally, assuming that the obtained measure is locally finite, topological properties of spacetime are worked out and a connection to dimension theory is established.

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