-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 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.

scholarly journals Stability and asymptotic properties of dissipative evolution equations coupled with ordinary differential equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Serge Nicaise
Keyword(s):  
Differential Equations ◽  
Evolution Equations ◽  
Dimensional Space ◽  
Hyperbolic Systems ◽  
Asymptotic Properties ◽  
Strong Stability ◽  
Infinite Dimensional ◽  
Telegraph Equations ◽  
Sufficient And Necessary Conditions ◽  
Stability Results

<p style='text-indent:20px;'>In this paper, we obtain some stability results of systems corresponding to the coupling between a dissipative evolution equation (set in an infinite dimensional space) and an ordinary differential equation. Many problems from physics enter in this framework, let us mention dispersive medium models, generalized telegraph equations, Volterra integro-differential equations, and cascades of ODE-hyperbolic systems. The goal is to find sufficient (and necessary) conditions on the involved operators that garantee stability properties of the system, i.e., strong stability, exponential stability or polynomial one. We also illustrate our abstract statements for different concrete examples, where new results are achieved.</p>

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.

scholarly journals Nonsmooth and multivalued analysis with applications in optimization

1983 ◽  
Author(s):  
Νικόλαος Παπαγεωργίου
Keyword(s):  
Optimization Problems ◽  
Recent Theory ◽  
Subdifferential Calculus ◽  
Infinite Dimensional ◽  
Third Stage ◽  
Existence And Stability ◽  
Nonconvex Analysis ◽  
Vector Valued ◽  
Stability Results ◽  
Convex Operators

The object of this thesis is two - fold . In the first part, we develop analogs of convex and nonconvex analysis for vector - valuedoperators, while in the second part we study the theory of Banach valued multifunctions. In the first part, we start with a study of convex operators. We introduce the notion of algebraic and topological subdifferentials and then derive conditions for those two to be equal. Also we develop a complete subdifferential and e-subdifferential calculus. In the sequence, we deal with the duality theory of convex operators. For that purpose, we introduc a notion of lower semicontinuity of operators and we show that this class is identical with the class of operators that are the upper envelope of continuous affine operators . This allows us to study analogs of the major duality schemes for vector optimization problems. Finally, we conclude our study of convex operators with some probabilistic results on Caratheodory convex integrands. Then we pass to nonconvex operators and introduce the class of locally o-Lipschitz operators. For those operators, we define a generalized subdifferential and develop a corresponding calculus that extends Clarke's theory to a vectorial context. Furthermore, we show that this extension is consistent with the convex theory. Applications to vectorial optimization are given. In the third stage of the process, we consider general operators and using geometric notions we introduce a new subdifferential calculus and provide applications in optimization. We close the first part of the thesis with a detailed study of infinite dimensional Pareto optimization problems and obtain existence and stability results for such problems. In the second part of the thesis, we pass to multivalued analysis. We introduce the vector valued Aumann integral and study its properties.Multifunctions depending on parameters are studied and results are obtained determining which properties of the integrand multifunction are preserved by integration. Then, using a notion of set valued conditional expectation, we introduce set valued martingales and obtain several convergence theorems. Also we study properties of the profile of a multifunction and weakly convergent sequences of multifunctions. Then we consider set valued measures, study their properties and define an integral with respect to a set valued measure and determine its properties. Finally, in the last chapter of our thesis, motivated by Ioffe's recent theory of fans, we introduce the notion of a "normal fan " and develop an integral theory for such fans.

Sign in / Sign up

Export Citation Format

Share Document