arXiv subjects

Asgar Jamneshan

Publications and source records attributed to Asgar Jamneshan.

35 records · Page 2Linked to original sources

arXiv2022

An uncountable Mackey-Zimmer theorem

The Mackey-Zimmer theorem classifies ergodic group extensions $X$ of a measure-preserving system $Y$ by a compact group $K$, by showing that such extensions are isomorphic to a group skew-product $X \equiv Y \rtimes_ρH$ for some closed subgroup $H$ of $K$. An analogous theorem is also available for ergodic homogeneous extensions $X$ of $Y$, namely that they are isomorphic to a homogeneous skew-product $Y \rtimes_ρH/M$. These theorems have many uses in ergodic theory, for instance playing a key role in the Host-Kra structural theory of characteristic factors of measure-preserving systems. The existing proofs of the Mackey-Zimmer theorem require various "countability", "separability", or "metrizability" hypotheses on the group $Γ$ that acts on the system, the base space $Y$, and the group $K$ used to perform the extension. In this paper we generalize the Mackey-Zimmer theorem to "uncountable" settings in which these hypotheses are omitted, at the cost of making the notion of a measure-preserving system and a group extension more abstract. However, this abstraction is partially counteracted by the use of a "canonical model" for abstract measure-preserving systems developed in a companion paper. In subsequent work we will apply this theorem to also obtain uncountable versions of the Host-Kra structural theory.

math.DS↗

arXiv2022

Foundational aspects of uncountable measure theory: Gelfand duality, Riesz representation, canonical models, and canonical disintegration

We collect several foundational results regarding the interaction between locally compact spaces, probability spaces and probability algebras, and commutative $C^*$-algebras and von Neumann algebras equipped with traces, in the "uncountable" setting in which no separability, metrizability, or standard Borel hypotheses are placed on these spaces and algebras. In particular, we review the Gelfand dualities and Riesz representation theorems available in this setting. We also present a canonical model that represents probability algebras as compact Hausdorff probability spaces in a completely functorial fashion, and apply this model to obtain a canonical disintegration theorem and to readily construct various product measures. These tools are useful in applications to "uncountable" ergodic theory (as demonstrated by the authors and others).

math.FA↗

arXiv2022

An uncountable Moore-Schmidt theorem

We prove an extension of the Moore-Schmidt theorem on the triviality of the first cohomology class of cocycles for the action of an arbitrary discrete group on an arbitrary measure space and for cocycles with values in an arbitrary compact Hausdorff abelian group. The proof relies on a "conditional" Pontryagin duality for spaces of abstract measurable maps.

math.DS↗

arXiv2021

Decoupling for fractal subsets of the parabola

We consider decoupling for a fractal subset of the parabola. We reduce studying $l^{2}L^{p}$ decoupling for a fractal subset on the parabola $\{(t, t^2) : 0 \leq t \leq 1\}$ to studying $l^{2}L^{p/3}$ decoupling for the projection of this subset to the interval $[0, 1]$. This generalizes the decoupling theorem of Bourgain-Demeter in the case of the parabola. Due to the sparsity and fractal like structure, this allows us to improve upon Bourgain-Demeter's decoupling theorem for the parabola. In the case when $p/3$ is an even integer we derive theoretical and computational tools to explicitly compute the associated decoupling constant for this projection to $[0, 1]$. Our ideas are inspired by the recent work on ellipsephic sets by Biggs using nested efficient congruencing.

math.CA↗

arXiv2020

Wasserstein Distance to Independence Models

An independence model for discrete random variables is a Segre-Veronese variety in a probability simplex. Any metric on the set of joint states of the random variables induces a Wasserstein metric on the probability simplex. The unit ball of this polyhedral norm is dual to the Lipschitz polytope. Given any data distribution, we seek to minimize its Wasserstein distance to a fixed independence model. The solution to this optimization problem is a piecewise algebraic function of the data. We compute this function explicitly in small instances, we examine its combinatorial structure and algebraic degrees in the general case, and we present some experimental case studies.

math.OC↗

arXiv2020

A Fenchel-Moreau theorem for $\bar L^0$-valued functions

We establish a Fenchel-Moreau type theorem for proper convex functions $f\colon X\to \bar{L}^0$, where $(X, Y, \langle \cdot,\cdot \rangle)$ is a dual pair of Banach spaces and $\bar L^0$ is the space of all extended real-valued functions on a $σ$-finite measure space. We introduce the concept of stable lower semi-continuity which is shown to be equivalent to the existence of a dual representation $$\smash{ f(x)=\sup_{y \in L^0(Y)} \left\{\langle x, y \rangle - f^\ast(y)\right\}, \quad x\in X,} $$ where $L^0(Y)$ is the space of all strongly measurable functions with values in $Y$, and $\langle \cdot,\cdot \rangle$ is understood pointwise almost everywhere. The proof is based on a conditional extension result and conditional functional analysis.

math.FA↗

arXiv2019

Sheaves and conditional sets

The purpose of this note is to record a connection between sheaves on complete Boolean algebras and conditional sets. This connection yields a transfer principle for conditional set theory. On the other hand we use conditional set theory to interpret the internal objects of a respective sheaf topos.

math.CT↗

arXiv2019

A transfer principle for second-order arithmetic, and applications

In the theory of conditional sets, many classical theorems from areas such as functional analysis, probability theory or measure theory are lifted to a conditional framework, often to be applied in areas such as mathematical economics or optimization. The frequent experience that such theorems can be proved by `conditionalizations' of the classical proofs suggests that a general transfer principle is in the background, and that formulating and proving such a transfer principle would yield a wealth of useful further conditional versions of classical results, in addition to providing a uniform approach to the results already known. In this paper, we formulate and prove such a transfer principle based on second-order arithmetic, which, by the results of reverse mathematics, suffices for the bulk of classical mathematics, including real analysis, measure theory and countable algebra, and excluding only more remote realms like category theory, set-theoretical topology or uncountable set theory, see e.g. the introduction of \cite{simpson2009subsystems}.This transfer principle is then employed to give short and easy proofs of conditional versions of central results in various areas of mathematics, including theorems which have not been proven by hand previously such as Peano existence theorem, Urysohn's lemma and the Markov-Kakutani fixed point theorem. Moreover, we compare the interpretation of certain structures in a conditional model with their meaning in a standard model.

math.LO↗

arXiv2018

Parameter-dependent Stochastic Optimal Control in Finite Discrete Time

We prove a general existence result in stochastic optimal control in discrete time where controls take values in conditional metric spaces, and depend on the current state and the information of past decisions through the evolution of a recursively defined forward process. The generality of the problem lies beyond the scope of standard techniques in stochastic control theory such as random sets, normal integrands and measurable selection theory. The main novelty is a formalization in conditional metric space and the use of techniques in conditional analysis. We illustrate the existence result by several examples including wealth-dependent utility maximization under risk constraints with bounded and unbounded wealth-dependent control sets, utility maximization with a measurable dimension, and dynamic risk sharing. Finally, we discuss how conditional analysis relates to random set theory.

math.OC↗

arXiv2018

A perturbational duality approach in vector optimization

A perturbational vector duality approach for objective functions $f\colon X\to \bar{L}^0$ is developed, where $X$ is a Banach space and $\bar{L}^0$ is the space of extended real valued functions on a measure space, which extends the perturbational approach from the scalar case. The corresponding strong duality statement is proved under a closedness type regularity condition. Optimality conditions and a Moreau-Rockafellar type formula are provided. The results are specialized for constrained and unconstrained problems. Examples of integral operators and risk measures are discussed.

math.OC↗

arXiv2018

Measures and integrals in conditional set theory

The aim of this article is to establish basic results in a conditional measure theory. The results are applied to prove that arbitrary kernels and conditional distributions are represented by measures in a conditional set theory. In particular, this extends the usual representation results for separable spaces.

math.PR↗

arXiv2017

On compactness in $L^0$-modules

Several results in functional analysis are extended to the setting of $L^0$-modules, where $L^0$ denotes the ring of all measurable functions $x\colon Ω\to \mathbb{R}$. The focus is on results involving compactness. To this end, a notion of stable compactness is introduced, and it is argued that the conventional notion of compactness does not allow to establish a functional analytic discourse in $L^0$-modules. Several characterizations of stable compactness are discussed, and its importance in applications is highlighted.

math.FA↗

arXiv2017

Vector duality via conditional extension of dual pairs

A Fenchel-Moreau type duality for proper convex and lower semi-continuous functions $f\colon X\to \overline{L^0}$ is established where $(X,Y,\langle \cdot,\cdot \rangle)$ is a dual pair of Banach spaces and $\overline{L^0}$ is the set of all extended real-valued measurable functions. We provide a concept of lower semi-continuity which is shown to be equivalent to the existence of a dual representation in terms of elements in the Bochner space $L^0(Y)$. To derive the duality result, several conditional completions and extensions are constructed. This is an earlier version of arXiv e-print 1708.03127, where the main results were formulated in an abstract setting of conditional completions, conditional extensions and conditional real numbers.

math.FA↗

arXiv2017

Multidimensional quadratic BSDEs with separated generators

We consider multidimensional quadratic BSDEs with bounded and unbounded terminal conditions. We provide sufficient conditions which guarantee existence and uniqueness of solutions. In particular, these conditions are satisfied if the terminal condition or the dependence in the system are small enough.

math.PR↗

arXiv2017

Solvability of multidimensional quadratic BSDEs

math.PR↗

arXiv2016

The algebra of conditional sets and the concepts of conditional topology and compactness

The concepts of a conditional set, a conditional inclusion relation and a conditional Cartesian product are introduced. The resulting conditional set theory is sufficiently rich in order to construct a conditional topology, a conditional real and functional analysis indicating the possibility of a mathematical discourse based on conditional sets. It is proved that the conditional power set is a complete Boolean algebra, and a conditional version of the axiom of choice, the ultrafilter lemma, Tychonoff's theorem, the Borel-Lebesgue theorem, the Hahn-Banach theorem, the Banach-Alaoglu theorem and the Krein-Šmulian theorem are shown.

math.LO↗

arXiv2016

Conditional Preference Orders and their Numerical Representations

We provide an axiomatic system modeling conditional preference orders which is based on conditional set theory. Conditional numerical representations are introduced, and a conditional version of the theorems of Debreu on the existence of numerical representations is proved. The conditionally continuous representations follow from a conditional version of Debreu's Gap Lemma the proof of which relies on a conditional version of the axiom of choice, free of any measurable selection argument. We give a conditional version of the von Neumann and Morgenstern representation as well as automatic conditional continuity results, and illustrate them by examples.

econ.GN↗