Properties of solutions of stochastic differential equations driven by the G-Brownian motion

In this paper, we study the differentiability of solutions of stochastic differential equations driven by the $G$-Brownian motion with respect to the initial data and the parameter. In addition, the stability of solutions of stochastic differential equations driven by the $G$-Brownian motion is obtained

Semantics-based selection of everyday concepts in visual lifelogging

Concept-based indexing, based on identifying various semantic concepts appearing in multimedia, is an attractive option for multimedia retrieval and much research tries to bridge the semantic gap between the media’s low-level features and high-level semantics. Research into concept-based multimedia retrieval has generally focused on detecting concepts from high quality media such as broadcast TV or movies, but it is not well addressed in other domains like lifelogging where the original data is captured with poorer quality. We argue that in noisy domains such as lifelogging, the management of data needs to include semantic reasoning in order to deduce a set of concepts to represent lifelog content for applications like searching, browsing or summarisation. Using semantic concepts to manage lifelog data relies on the fusion of automatically-detected concepts to provide a better understanding of the lifelog data. In this paper, we investigate the selection of semantic concepts for lifelogging which includes reasoning on semantic networks using a density-based approach. In a series of experiments we compare different semantic reasoning approaches and the experimental evaluations we report on lifelog data show the efficacy of our approach

Harmonic maps of finite uniton type into non-compact inner symmetric spaces

Due to the efforts of many mathematicians, there has been a classification of harmonic two-spheres into compact (semi-simple) Lie groups as well as compact inner symmetric spaces. Such harmonic maps have been shown by Uhlenbeck, Burstall-Guest, Segal to have a finite uniton number. Moreover, the monodromy representation was shown to be trivial and to be polynomial in the loop parameter. We will introduce a general definition according to which such maps are called to be of finite uniton type. This paper aims to generalize results of [2] to harmonic maps of finite uniton type into a non-compact inner symmetric space. For this purpose, we first recall some basic results about harmonic maps of finite uniton type. Then we interpret the work of Burstall and Guest on harmonic maps of finite uniton type into compact (semi-simple) Lie groups in terms of the language of the DPW method. Moreover, to make the work of Burstall and Guest applicable to our setting we show that a harmonic map into a non-compact inner symmetric space $G/K$ shares the normalized potential as well as the meromorphic extended framing with a harmonic map into $U/(U\cap K^{\mathbb{C}})$, the compact dual of $G/K$. Thus we reduce the description of harmonic maps of finite uniton type into a non-compact inner symmetric space to the description of harmonic maps of finite uniton type into a compact inner symmetric space. Our main goal for the study of such harmonic maps is to provide a classification of Willmore two-spheres (whose conformal Gauss maps take value in the non-compact symmetric space $SO^+(1,n+3)/SO(1,3)\times SO(n)$). We will finish this paper by presenting the coarse classification of Willmore two-spheres in terms of their conformal Gauss maps [28] as well as examples of Willmore surfaces constructed by using [13] and the results of this paper.Comment: 33 pages. We divide the original paper into several papers due to the length and the confusions of topics. The present paper is to characterize all harmonic maps of finite uniton type into non-compact inner symmetric spaces by their normalized potential