Bernstein Numbers of Embeddings of Isotropic and Dominating Mixed Besov Spaces

The purpose of the present paper is to investigate the decay of Bernstein numbers of the embedding from $B^t_{p_1,q}((0,1)^d)$ into the space $L_{p_2}((0,1)^d)$. The asymptotic behaviour of Bernstein numbers of the identity $id: S_{p_1,p_1}^tB((0,1)^d)\rightarrow L_{p_2}((0,1)^d)$ will be also considered.Comment: 31 pages, 1 figur

Weyl Numbers of Embeddings of Tensor Product Besov Spaces

In this paper we investigate the asymptotic behaviour of Weyl numbers of embeddings of tensor product Besov spaces into Lebesgue spaces. These results will be compared with the known behaviour of entropy numbers.Comment: 54 pages, 2 figure

Social capital, livelihood diversification and households' resilience to floods in the Vietnamese Mekong River Delta

Floods are a familiar and frequent feature of life in the Vietnamese Mekong River Delta (MRD). Although floods bring hardship to people, they also bring environmental benefits forsustaining rural livelihoods. People in the MRD have experienced the impacts of floods for hundreds of years since the sparse population settled in the MRD during the 19th Century. In some years, the flood is 'big' or 'small', but it is 'moderate' in most years. Some people are 'winners' while others are 'losers' due to the impacts of the flood events. The aim of this thesis is to advance understanding about perceptions of the flood events, flood impacts on household livelihoods, and households' capacity to live with floods. The aim of the thesis is characterized into four key objectives: (i) to explore the perception of the flood events in the MRD held by different socio-economic groups, (ii) to examine the impacts of three levels of floods (small, moderate and big) on different households' livelihood activities and assets in the MRD, (iii) to investigate the relationship between livelihood adaptation (diversification or specialization) and households' capacity to live with floods in the MRD and (iv) to examine the relationship between different forms of social capital of households and households' capacity to live with floods in the MRD. The research employs both qualitative and quantitative approaches to address four research questions. The key methods used in the qualitative approach include focus group discussions (FGDs), in-depth interviews with key informants, field observations and document research. The structured household interview is the key method for collecting quantitative data. The stratified sampling approach was used to choose households in three study sites in the MRD. The findings indicate that local people use several different terms to describe the flood events. Government staff and local researchers are more likely to use the term 'flood season' (m{u00F9}a l{u0169}) before1998 whereas local people use the term, 'rising water season' (m{u00F9}a n{u01B0}{u1EDB}c n{u1ED5}i). Similarly, local newspapers mostly use the term 'flood' (l{u0169}) in reporting the events. The findings further confirm that there is a shift from the use of term 'flood' to 'rising water season' by government and local newspapers in recent years. Interestingly, there are significant differences in perceptions of flood depth by gender, socio-economic group and by region. The study identifies that people use different approaches to cope with different levels of flood events and the approaches vary by socio-economic group and region. The findings further confirm that the flood has both benefits and costs. The negative and positive impacts of the flood events are variable by socio-economic group and region. Interestingly, the moderate flood event is perceived as the most 'beautiful one' in terms of its impacts on household livelihoods because it creates fewer negative impacts, but brings more benefits to most socio-economic groups. Three properties of households' resilience to floods are obtained from factor analysis which include (i) households' capacity to secure food, income, health, evacuation, and recovery; (ii) households' capacity to secure their homes in future floods as in the threshold flood in 2000; and (iii) households' capacity in learning to adapt using new flood-based livelihoods. The results further demonstrate that different forms of social capital have different effects on different types of households' resilience to floods. Neighbourhood attachment has a statistically significant effect on a household's ability to secure food, income, health, safe evacuation during the floods, and recovery after floods, and level of interest in learning new flood-based livelihoods, but it does not have a significant effect on the households' capacity to secure their homes in future floods as in the threshold flood in 2000. Similarly, a social supportive network has significant effects on a household's ability to secure homes, but it does not have a significant effect on the households' capacity to secure food, income, health, evacuation, and recovery, and learning to adapt new flood-based livelihoods. Participation in groups and associations does not have a significant effect on improving households' resilience scores in most cases. Socio-economic characteristics of households (household income) are shown to have a significant effect on the three properties of households' resilience. Better-off households are more likely to confident in securing with food, health, income, and housing. However, rich households are less likely to be interested in learning new flood-based livelihoods because they often own large areas of land which discourage them to take up new flood-based livelihood activities instead of growing rice. Older and aged people and women are less likely to learn and adapt new flood-based livelihood activities (raising fish, prawns and growing aquatic vegetables). Housing type also has a significant effect on a household's capacity to secure the home (concrete houses are less vulnerable). Regional flood factors also have a significant effect on the three resilience factors; people in the highest flood-prone region are less likely to be resilient in terms of securing their houses, food and income, but are more likely to learn new ways of living with floods. Surprisingly, the livelihood diversification index has no effect on a household's resilience to floods in most cases. This means that livelihood diversification does not necessarily improve households' resilience to floods. However, in the qualitative data, diversification into off-farm fishing and migration made some people more resilient, but others more vulnerable to floods. Diversification within farming activities such as conducting flood-based farming activities may help some households to improve their income during the flood season. Migration to HCM city or Binh Duong industrial zone may be a useful strategy, but this strategy is problematic due to the high living costs in HCM city. Engaging in new flood-based livelihood activities helps some people to improve their household incomes, but they often face risk from market. Policies for living with floods should try to (i) enhance the use of flood benefits by changing the current flood risk communication by local newspapers, government reports and other media; (ii) make use of the full benefits of the flood season to improve livelihoods for the poor households who lack capacity to seek non-farm jobs; (iii) maintain and develop the social capital of households, especially neighbourhood attachment and social supportive networks that may help rural households to adapt to future floods

Pseudo ss-Numbers of Embeddings of Gaussian Weighted Sobolev Spaces

In this paper, we study the approximation problem for functions in the Gaussian-weighted Sobolev space $W^\alpha_p(\mathbb{R}^d, \gamma)$ of mixed smoothness $\alpha \in \mathbb{N}$ with error measured in the Gaussian-weighted space $L_q(\mathbb{R}^d, \gamma)$. We obtain the exact asymptotic order of pseudo $s$-numbers for the cases $1 \leq q< p < \infty$ and $p=q=2$. Additionally, we also obtain an upper bound and a lower bound for pseudo $s$-numbers of the embedding of $W^\alpha_2(\mathbb{R}^d, \gamma)$ into $L_{\infty}^{\sqrt{g}}(\mathbb{R}^d)$. Our result is an extension of that obtained in Dinh D\~ung and Van Kien Nguyen (IMA Journal of Numerical Analysis, 2023) for approximation and Kolmogorov numbers.Comment: 17 page