On Current Developments in Partial Differential Equations

The first part of this article is an overview on some recent major developments in the field of analysis and partial different equations.It is a brief presentation given by the author at a round table discussion.The second part is a supplement of various details provided by several outstanding researchers on subjects.     

An Ensemble Kalman Filter Approach Based on Level Set Parameterization for Acoustic Source Identification Using Multiple Frequency Information

In this paper, a reconstruction problem of the spatial dependent acoustic source from multiple frequency data is discussed. Suppose that the source function is supported on a bounded domain and the piecewise constant intensities of the source are known on the support. We characterize unknown domain by the level set technique. And the level set function can be modeled by a Hamilton-Jacobi system. We use the ensemble Kalman filter approach to analyze the system state. This method can avoid to deal with the nonlinearity directly and reduce the computation complexity. In addition, the algorithm can achieve the stable state quickly with the Hamilton-Jacobi system. From some numerical examples, we show these advantages and verify the feasibility and effectiveness.     

Engineering nanostructures of CuO-based photocatalysts for water treatment: Current progress and future challenges

Nowadays, increasing extortions regarding environmental problems and energy scarcity have stuck the development and endurance of human society. The issue of inorganic and organic pollutants that exist in water from agricultural, domestic, and industrial activities has directed the development of advanced technologies to address the challenges of water scarcity efficiently. To solve this major issue, various scientists and researchers are looking for novel and effective technologies that can efficiently remove pollutants from wastewater. Nanoscale metal oxide materials have been proposed due to their distinctive size, physical and chemical properties along with promising applications. Cupric Oxide (CuO) is one of the most commonly used benchmark photocatalysts in photodegradation owing to the fact that they are cost-effective, non-toxic, and more efficient in absorption across a significant fraction of solar spectrum. In this review, we have summarized synthetic strategies of CuO fabrication, modification methods with applications for water treatment purposes. Moreover, an elaborative discussion on feasible strategies includes; binary and ternary heterojunction formation, Z-scheme based photocatalytic system, incorporation of rare earth/transition metal ions as dopants, and carbonaceous materials serving as a support system. The mechanistic insight inferring photo-induced charge separation and transfer, the functional reactive radical species involved in a photocatalytic reaction, have been successfully featured and examined. Finally, a conclusive remark regarding current studies and unresolved challenges related to CuO are put forth for future perspectives.     

Increased Stress Resistance and Lifespan in Chaenorhabditis elegans Wildtype and Knockout Mutants—Implications for Depression Treatment by Medicinal Herbs

Depression and anxiety disorders are widespread diseases, and they belong to the leading causes of disability and greatest burdens on healthcare systems worldwide. It is expected that the numbers will dramatically rise during the COVID-19 pandemic. Established medications are not sufficient to adequately treat depression and are not available for everyone. Plants from traditional medicine may be promising alternatives to treat depressive symptoms. The model organism Chaenorhabditis elegans was used to assess the stress reducing effects of methanol/dichlormethane extracts from plants used in traditional medicine. After initial screening for antioxidant activity, nine extracts were selected for in vivo testing in oxidative stress, heat stress, and osmotic stress assays. Additionally, anti-aging properties were evaluated in lifespan assay. The extracts from Acanthopanax senticosus, Campsis grandiflora, Centella asiatica, Corydalis yanhusuo, Dan Zhi, Houttuynia cordata, Psoralea corylifolia, Valeriana officinalis, and Withania somnifera showed antioxidant activity of more than 15 Trolox equivalents per mg extract. The extracts significantly lowered ROS in mutants, increased resistance to heat stress and osmotic stress, and the extended lifespan of the nematodes. The plant extracts tested showed promising results in increasing stress resistance in the nematode model. Further analyses are needed, in order to unravel underlying mechanisms and transfer results to humans.     

Phytochemical Differentiation of Saffron (Crocus sativus L.) by High Resolution Mass Spectrometry Metabolomic Studies

The metabolite profiling of saffron (Crocus sativus L.) from several countries was measured by using ultra-performance liquid chromatography combined with high resolution mass spectrometry (UPLC-HR MS). Multivariate statistical analysis was employed to distinguish among the several samples of C. sativus L. from Greece, Italy, Morocco, Iran, India, Afghanistan and Kashmir. The results of this study showed that the phytochemical content in the samples of C. sativus L. were obviously diverse in the different countries of origin. The metabolomics approach was deemed to be the most suitable in order to evaluate the enormous array of putative metabolites among the saffron samples studied, and was able to provide a comparative phytochemical screening of these samples. Several markers have been identified that aided the differentiation of a group from its counterparts. This can be important for the selection of the appropriate saffron sample, in view of its health-promoting effect which occurs through the modulation of various biological and physiological processes.     

Order-Stability in Complex Biological,Social, and AI-Systems from Quantum Information Theory

This paper is our attempt, on the basis of physical theory, to bring more clarification on the question “What is life?” formulated in the well-known book of Schrödinger in 1944. According to Schrödinger, the main distinguishing feature of a biosystem’s functioning is the ability to preserve its order structure or, in mathematical terms, to prevent increasing of entropy. However, Schrödinger’s analysis shows that the classical theory is not able to adequately describe the order-stability in a biosystem. Schrödinger also appealed to the ambiguous notion of negative entropy. We apply quantum theory. As is well-known, behaviour of the quantum von Neumann entropy crucially differs from behaviour of classical entropy. We consider a complex biosystem S composed of many subsystems, say proteins, cells, or neural networks in the brain, that is, S=(Si). We study the following problem: whether the compound system S can maintain “global order” in the situation of an increase of local disorder and if S can preserve the low entropy while other Si increase their entropies (may be essentially). We show that the entropy of a system as a whole can be constant, while the entropies of its parts rising. For classical systems, this is impossible, because the entropy of S cannot be less than the entropy of its subsystem Si. And if a subsystems’s entropy increases, then a system’s entropy should also increase, by at least the same amount. However, within the quantum information theory, the answer is positive. The significant role is played by the entanglement of a subsystems’ states. In the absence of entanglement, the increasing of local disorder implies an increasing disorder in the compound system S (as in the classical regime). In this note, we proceed within a quantum-like approach to mathematical modeling of information processing by biosystems—respecting the quantum laws need not be based on genuine quantum physical processes in biosystems. Recently, such modeling found numerous applications in molecular biology, genetics, evolution theory, cognition, psychology and decision making. The quantum-like model of order stability can be applied not only in biology, but also in social science and artificial intelligence.