A stability result for the cube edge isoperimetric inequality

We prove the following stability version of the edge isoperimetric inequality for the cube: any subset of the cube with average boundary degree within $K$ of the minimum possible is $\varepsilon$-close to a union of $L$ disjoint cubes, where $L \leq L(K,\varepsilon )$ is independent of the dimension. This extends a stability result of Ellis, and can viewed as a dimension-free version of Friedgut's junta theorem.Comment: 12 page

Forbidden vector-valued intersections

We solve a generalised form of a conjecture of Kalai motivated by attempts to improve the bounds for Borsuk's problem. The conjecture can be roughly understood as asking for an analogue of the Frankl-R\"odl forbidden intersection theorem in which set intersections are vector-valued. We discover that the vector world is richer in surprising ways: in particular, Kalai's conjecture is false, but we prove a corrected statement that is essentially best possible, and applies to a considerably more general setting. Our methods include the use of maximum entropy measures, VC-dimension, Dependent Random Choice and a new correlation inequality for product measures.Comment: 40 page

Municipal wastewater treatment with pond technology : historical review and future outlook

Facing an unprecedented population growth, it is difficult to overstress the assets for wastewater treatment of waste stabilization ponds (WSPs), i.e. high removal efficiency, simplicity, and low cost, which have been recognized by numerous scientists and operators. However, stricter discharge standards, changes in wastewater compounds, high emissions of greenhouse gases, and elevated land prices have led to their replacements in many places. This review aims at delivering a comprehensive overview of the historical development and current state of WSPs, and providing further insights to deal with their limitations in the future. The 21st century is witnessing changes in the way of approaching conventional problems in pond technology, in which WSPs should no longer be considered as a low treatment technology. Advanced models and technologies have been integrated for better design, control, and management. The roles of algae, which have been crucial as solar-powered aeration, will continue being a key solution. Yet, the separation of suspended algae to avoid deterioration of the effluent remains a major challenge in WSPs while in the case of high algal rate pond, further research is needed to maximize algal growth yield, select proper strains, and optimize harvesting methods to put algal biomass production in practice. Significant gaps need to be filled in understanding mechanisms of greenhouse gas emission, climate change mitigation, pond ecosystem services, and the fate and toxicity of emerging contaminants. From these insights, adaptation strategies are developed to deal with new opportunities and future challenges

Opportunities and challenges for the sustainability of lakes and reservoirs in relation to the sustainable development goals (SDGs)

Emerging global threats, such as biological invasions, climate change, land use intensification, and water depletion, endanger the sustainable future of lakes and reservoirs. To deal with these threats, a multidimensional view on the protection and exploitation of lakes and reservoirs is needed. The holistic approach needs to contain not just the development of economy and society but also take into account the negative impacts of this growth on the environment, from that, the balance between the three dimensions can be sustained to reach a sustainable future. As such, this paper provides a comprehensive review on future opportunities and challenges for the sustainable development of lakes and reservoirs via a critical analysis on their contribution to individual and subsets of the Sustainable Development Goals (SDGs). Currently, lakes and reservoirs are key freshwater resources. They play crucial roles in human societies for drinking water provision, food production (via fisheries, aquaculture, and the irrigation of agricultural lands), recreation, energy provision (via hydropower dams), wastewater treatment, and flood and drought control. Because of the (mostly) recent intensive exploitations, many lakes and reservoirs are severely deteriorated. In recent years, physical (habitat) degradation has become very important while eutrophication remains the main issue for many lakes and ponds worldwide. Besides constant threats from anthropogenic activities, such as urbanization, industry, aquaculture, and watercourse alterations, climate change and emerging contaminants, such as microplastics and antimicrobial resistance, can generate a global problem for the sustainability of lakes and reservoirs. In relation to the SDGs, the actions for achieving the sustainability of lakes and reservoirs have positive links with the SDGs related to environmental dimensions (Goals 6, 13, 14, and 15) as they are mutually reinforcing each other. On the other hand, these actions have direct potential conflicts with the SDGs related to social and economic dimensions (Goals 1, 2, 3 and 8). From these interlinkages, we propose 22 indicators that can be used by decision makers for monitoring and assessing the sustainable development of lakes and reservoirs