Global optimization is an imperative development of local optimization because there are many problems in artificial intelligence and machine learning requires highly acute solutions over entire domain. There are many methods to resolve the global optimization, which can be classified into three groups such as analytic methods (purely mathematical methods), probabilistic methods, and heuristic methods. Especially, heuristic methods like particle swarm optimization and ant bee colony attract researchers because their effective and practical techniques which are easy to be implemented by computer programming languages. However, these heuristic methods are lacking in theoretical mathematical fundamental. Fortunately, minima distribution establishes a strict mathematical relationship between optimized target function and its global minima. In this research, I try to study minima distribution and apply it into explaining convergence and convergence speed of optimization algorithms. Especially, weak conditions of convergence and monotonicity within minima distribution are drawn so as to be appropriate to practical optimization methods.

Excitotoxicity related to the dysfunction of the N-methyl-d-aspartate receptor (NMDAR) has been indicated to play an integral role in the pathophysiology of multiple disease states, including neurodegenerative disorders such as Parkinson’s disease. There is a notable gap in the market for novel NMDAR antagonists, however current methods to analyze potential antagonists rely on indirect measurements of calcium flux and hazardous radioligand binding assays. Recently, a fluorescent NMDAR ligand, N-adamantan-1-yl-dimethylamino-1-naphthalenesulfonic acid, known as AM-DAN was developed by our group. Additional studies on this ligand is necessary to evaluate its potential as a biological tool in NMDAR research. Therefore, this study was aimed at conducting structural analyses, fluorescence experiments, high-accuracy NMDAR molecular modelling and NMDAR phencyclidine (PCP) site competition binding studies using AM-DAN. Results revealed that AM-DAN has appropriate structural properties, significant fluorescent ability in various solvents and is able to bind selectively and compete for the PCP-binding site of the NMDAR. Therefore, AM-DAN holds promise as a novel fluorescent ligand to measure the affinity of prospective drugs binding at the NMDAR PCP-site and may circumvent the use of radioligands.

Invasive macrophytes are considered problematic in natural environments and hydroelectric reservoirs. Climate changes, the occurrence of watercourses, and biotic interactions influence the biological invasion of macrophytes. Native species can correlate with invasives positively or negatively. Urochloa subquadripara is an invasive in natural or disturbed habitats co-occurring with the natives Eichhornia crassipes and Salvinia minima. Aquatic plant communities can be altered by climate change, so species distribution models (SDMs) are important tools for predicting in-vaded areas. This work aimed to make an SDM for U. subquadripara correlating with the poten-tial distribution of native species E. crassipes and S. minima. Occurrence data for U. subquadripara, E. crassipes, and S. minima were collected from databases and in consultation with the published literature. Parameters encompassing biological information of the species were entered into the CLIMEX software and used to generate the Ecoclimatic Index (EI). The species co-occurrence was performed based on multicriteria decision-making (MCDM), and weights were assigned using the analytical hierarchy process (AHP). It was observed that U. subquadripara, E. crassipes, and S. minima have a higher occurrence in tropical and subtropical regions; however, it is predicted that, from climatic changes, these species may move to high latitudes. According to climate change, the risk of invasion by U. subquadripara in the northern hemisphere is mainly in lakes, while in the southern hemisphere, the areas conducive to invasions are rivers and reservoirs. In general, emerging and floating macrophyte species such as U. subquadripara, E. crassipes, and S. minima will be favored, causing suppression of submerged species. Therefore, identifying the potential distribution of these species allows the creation of pre-invasion intervention strategies.

This paper investigates the randomly stopped sums, minima and maxima of heavy- and light-tailed random variables. The conditions on the primary random variables, which are independent but generally not identically distributed, and counting random variable are given in order that the randomly stopped sum, random minimum and maximum is heavy/light tailed. The results generalize some existing ones in the literature. The examples illustrating the results are provided.

This study scrutinizes the complexities of designing and exploring the potential energy surface (PES) of systems containing more than twenty atoms with planar tetracoordinate carbons (ptCs). To tackle this issue, we utilized an established design rule to design a Naphtho [1,2-b:4,3-b′:5,6-b′′:8,7-b′′′]tetrathiophene derivative computationally. This process began with substituting S atoms with CH– units, then replacing three sequential protons with two Si2+ units in the resultant polycyclic aromatic hydrocarbon polyanion. Despite not representing the global minimum, the newly designed Si8C22 system with four ptCs provided valuable insights into strategic design and PES exploration. Our results underscore the importance of employing adequate methodologies to confirm the stability of newly designed molecular structures containing planar hypercoordinate carbons.

We have defined M cycles as modified form of Gleissberg cycles to study the quasi-periodic secular changes in sunspot activity. Using direct and proxy observations for more than 1200 years we have provided evidence for the influence of the above M cyles on the monsoon rainfall variability and occurrences of major droughts in India. The solar cycle averages of All Indian summer monsoon rainfall (AISMR) and probability of observation of below or above normal AISMR is found to show correlated variations with changes in amplitude of sunspot cycles during the years 1901-2018 AD. Major droughts in India show a preference to occur during minima and declining phases of the M cycles during the years 650-2018 AD . We could generally find low probability of occurrence of droughts in India during the medieval solar maximum epoch ( 1100-1250 AD) and during most of the prolonged sunspot minima periods including the Maunder minima. The evolution of mean Indian summer monsoon rainfall during 650-1900 AD is studied using proxy data from multiple sources with maxima during the 12^th century and minima during 14^th and 19^th centuries. The association of Indian droughts with El-Nino and the possibilities of long term prediction of Indian summer monsoon rainfall variability will be also discussed

The Kwee van Woerden (KvW) method for the determination of eclipse minimum times has been a staple in eclipsing binary research for decades, due its simplicity and independence of external input parameters. However, its estimates of the timing error have been known to be of low reliability. During the analysis of very precise photometry of CM Draconis eclipses from TESS space mission data, KvW’s original equation for the timing error estimate produced numerical errors, which evidenced a fundamental problem in this equation. This contribution introduces an improved way to calculate the timing error with the KvW method. A code that implements this improved method, together with several further updates over the original method is presented as well. An example application on the CM Draconis light curves from TESS is given, where we show that its timing error estimates of about 1 second are in excellent agreement with error estimates obtained by other means.

The notion of Critical Point Theory has extensive application in the filed of Partial Differential Equations, one of which is studying the existence and commenting on the uniqueness and multiplicity (in case when the solution is not unique) of weak solutions to Elliptic PDEs under certain boundary conditions.\par This article provides a brief survey about specific concepts like : differentiation on Banach Spaces, maxima and minima and its applications to PDEs. Furthermore, we've discussed the notion of defining a weak topology on Banach Spaces, before introducing the Variational Principle and its applications.\par The epilogue of the article primarily deals with results which studies the existence of weak solutions to Dirichlet Boundary Value Problems under specific conditions. The most notable aspect of this article is the proof of \textit{Rabinowitz's Saddle Point Theorem} using the method of \textit{Brouwer Degree}. Readers who are highly motivated in pursuing research in any of the topics relevent to the contents of this paper will surely find the \texttt{References} section to be extremely resourceful.