Aspects of Computable Analysis

<p>Computable analysis has been well studied ever since Turing famously formalised the computable reals and computable real-valued function in 1936. However, analysis is a broad subject, and there still exist areas that have yet to be explored. For instance, Sierpinski proved that every real-valued function ƒ : ℝ → ℝ is the limit of a sequence of Darboux functions. This is an intriguing result, and the complexity of these sequences has been largely unstudied. Similarly, the Blaschke Selection Theorem, closely related to the Bolzano-Weierstrass Theorem, has great practical importance, but has not been considered from a computability theoretic perspective. The two main contributions of this thesis are: to provide some new, simple proofs of fundamental classical results (highlighting the role of ∏0/1 classes), and to use tools from effective topology to analyse the Darboux property, particularly a result by Sierpinski, and the Blaschke Selection Theorem. This thesis focuses on classical computable analysis. It does not make use of effective measure theory.</p>

Aspects of Computable Analysis

<p>Computable analysis has been well studied ever since Turing famously formalised the computable reals and computable real-valued function in 1936. However, analysis is a broad subject, and there still exist areas that have yet to be explored. For instance, Sierpinski proved that every real-valued function ƒ : ℝ → ℝ is the limit of a sequence of Darboux functions. This is an intriguing result, and the complexity of these sequences has been largely unstudied. Similarly, the Blaschke Selection Theorem, closely related to the Bolzano-Weierstrass Theorem, has great practical importance, but has not been considered from a computability theoretic perspective. The two main contributions of this thesis are: to provide some new, simple proofs of fundamental classical results (highlighting the role of ∏0/1 classes), and to use tools from effective topology to analyse the Darboux property, particularly a result by Sierpinski, and the Blaschke Selection Theorem. This thesis focuses on classical computable analysis. It does not make use of effective measure theory.</p>

Stability of stochastic differential equations driven by multifractional Brownian motion

Our aim in this paper is to establish some strong stability results for solutions of stochastic differential equations driven by a Riemann–Liouville multifractional Brownian motion. The latter is defined as a Gaussian non-stationary process with a Hurst parameter as a function of time. The results are obtained assuming that the pathwise uniqueness property holds and using Skorokhod’s selection theorem.

A Generalization of the Blackwell–Ryll-Nardzewski Measurable Selection Theorem

One of the main forms of the measurable selection theorem is connected with the existence of the graph of a measurable mapping in a given measurable set 𝑆 in the product of two measurable spaces 𝑋 and 𝑌 . Such a graph enables one to pick a point in the section 𝑆𝑥 for each 𝑥 in 𝑋 such that the obtained mapping will be measurable. The indicated selection is called a measurable selection of the multi-valued mapping associating to the point 𝑥 the section 𝑆𝑥 , which is a set in 𝑌 . The classical theorem of Blackwell and Ryll-Nardzewski states that a Borel set 𝑆 in the product of two complete separable metric spaces contains the graph of a Borel mapping (hence admits a Borel selection) provided that there is a transition probability on this product with positive measures for all sections of 𝑆 . The main result of this paper gives a generalization to the case where only one of the two spaces is complete separable and the other one is a general measurable space whose points parameterize a family of Borel probability measures on the first space such that the sections of the given set 𝑆 in the product have positive measures.

Approximation of solutions of mean-field stochastic differential equations

Our aim in this paper is to establish some strong stability properties of solutions of mean-field stochastic differential equations. These latter are stochastic differential equations where the coefficients depend not only on the state of the unknown process but also on its probability distribution. The results are obtained assuming that the pathwise uniqueness property holds and using Skorokhod’s selection theorem.

On a class of generalized stochastic Browder mixed variational inequalities

In this paper, we introduce a class of stochastic variational inequalities generated from the Browder variational inequalities. First, the existence of solutions for these generalized stochastic Browder mixed variational inequalities (GS-BMVI) are investigated based on FKKM theorem and Aummann’s measurable selection theorem. Then the uniqueness of solution for GS-BMVI is proved based on strengthening conditions of monotonicity and convexity, the compactness and convexity of the solution sets are discussed by Minty’s technique. The results of this paper can provide a foundation for further research of a class of stochastic evolutionary problems driven by GS-BMVI.

On Darboux-Type Differential Inclusions with Uncertainty

In this article we prove the existence results for solutions of the Darboux-type problems in fuzzy partial differential inclusions with local conditions of integral types. We present two problems involving open and closed level sets of a given fuzzy mapping. In the first case fuzzy differential inclusion has been transformed into an equivalent Darboux-type problem for partial differential equations and then using the Tychonoff fixed point theorem we prove the existence result for this crisp case. For the second case we use Nadler’s fixed point theorem and selection theorem of Kuratowski-Ryll-Nardzewski to find the solution of given differential inclusions problem. We furnish an example to validate our results.