Important Mathematical Methods Research Articles

Three-way concept lattice based on Boolean formal context

The theory of formal concept analysis(FCA) is an important mathematical method for knowledge representation and knowledge discovery. The Boolean formal context is proposed by introducing Boolean matrices and logical operations into FCA. Based on the concept lattice of the Boolean formal context and the column vector(row-vector)-oriented concept lattice of the Boolean formal context, this paper proposes the column vector(row vector)-induced three-way concept lattice of the Boolean formal context and the column vector(row vector)-induced three-way column vector(row vector)-oriented concept lattice of the Boolean formal context, and proves their rationality. Then, the isomorphism between the three-way concept lattice of the Boolean formal context and the three-way concept lattice of the general formal context is proved.

Analysis of various methodologies for the TSP problem

Linear algebra is an important mathematical method used to solve real-life problems such as optimization. This paper chose the Travelling Salesman Problem as the topic. TSP Problem had a lot of different methods and algorithms. In this paper, the principle of each method is explained, and the comparison with the test data is shown at the end of the paper. Matlab was used to support the research. The source codes of each method were uploaded to Github. According to the test data, the branch-and-bound method performed the best because it takes the least time to run and can handle the most data. TSP Problem is widely used in real life, such as transportation. The faster the code, the better the performance of the program.

Optimistic multigranulation roughness of fuzzy bipolar soft sets by soft binary relations and its applications

Two important mathematical methods for addressing uncertainty in data processing are multigranulation rough set (MGRS) and fuzzy bipolar soft set (FBSS). This paper describes a certain kind of multigranulation rough set in the context of multiple soft binary relations. We first define the multigranulation roughness of fuzzy bipolar soft sets in the two universes. Moreover, a detailed study of structural properties has been conducted in order to explore this concept. The key characteristics of the traditional MGRS model are completely preserved in this new approach. Following that, we suggest two decision-making algorithms with respect to aftersets and foresets of the soft binary relations over dual universes. This approach appears to be better suited and more adaptable than other available methods, making it a favorable option for addressing decision-making problems. Finally, we provide a practical application of the suggested approach to a real-world problem.

Some novel aggregation operators for cubic picture fuzzy information: application in multi-attribute decision support problem

Cubic picture fuzzy set (CPFS) is the hybrid version of the interval-valued picture fuzzy set (IPVFS) and the picture fuzzy set (PFS) simultaneously. Clearly, the CPFS contains significantly more information than the general IVPFS and PFS in order to address the vulnerability of decision-making problems. Lamentably, the aggregation operators have not been researched so far on the basis of CPFS. In the techniques of group decision-making, aggregation operators are an important mathematical method to resolve ambiguity in decision-making problems. The present paper proposes few novel geometric mean aggregation operators, in particular, cubic picture fuzzy geometric and cubic picture fuzzy geometric star aggregation operators between cubic picture fuzzy numbers to aggregate the different predispositions of the decision maker. Then, based on the proposed operators in the cubic picture fuzzy environment, we developed a decision-making method and illustrated a numerical example. Finally, in order to demonstrate the supremacy of the proposed technique, classify its comparison with the existing approach. The results showed that the established decision-making technique was superior, accurate and effective.

Algorithms for Instance Retrieval and Realization in Fuzzy Ontologies

Fuzzy description logics, the formalism behind fuzzy ontologies, are an important mathematical method with applications in many artificial intelligence scenarios. This paper proposes the first specific algorithms to solve two reasoning tasks with respect to a fuzzy ontology: the instance retrieval and the realization problem. Our algorithms are based on a reduction of the number of optimization problems to solve by merging some of them. Our experimental evaluation shows that the novel algorithm to solve the instance retrieval outperforms the previous algorithm, and that in practice it is common to be able to solve a single optimization problem.

Geometric phase from Aharonov–Bohm to Pancharatnam–Berry and beyond

Whenever a quantum system undergoes a cycle governed by a slow change of parameters, it acquires a phase factor: the geometric phase. Its most common formulations are known as the Aharonov-Bohm, Pancharatnam and Berry phases, but both prior and later manifestations exist. Though traditionally attributed to the foundations of quantum mechanics, the geometric phase has been generalized and became increasingly influential in many areas from condensed-matter physics and optics to high energy and particle physics and from fluid mechanics to gravity and cosmology. Interestingly, the geometric phase also offers unique opportunities for quantum information and computation. In this Review we first introduce the Aharonov-Bohm effect as an important realization of the geometric phase. Then we discuss in detail the broader meaning, consequences and realizations of the geometric phase emphasizing the most important mathematical methods and experimental techniques used in the study of geometric phase, in particular those related to recent works in optics and condensed-matter physics.

Compressibility and fractal dimension analysis in the bituminous coal specimens

Coal is the typical multi-pore media. Numerous experiments have been done to investigate the pore structure of coal. Mercury intrusion porosimetry is still a popular way to analyze the pore characteristics. Compressibility effect on the mercury intrusion porosimetry data cannot be ignored. So N2 adsorption (NA) and mercury intrusion porosimetry (MIP) were applied to analyze the pore structure of coal. In this paper, the mercury intrusion process was divided into three stages including interpore filling, large intrapore filling, and small intrapore filling. The fractal analysis is an important mathematical method, which is widely used to evaluate heterogeneity in the size-dependent distribution of porosity. The fractal characteristics of pore structures in four coal samples with the same rank were also investigated. Three pore fractal dimensions corresponding to the three stages of mercury intrusion, D1C, D2C, and D3C were calculated with the Brooks–Corey capillary pressure model. Df calculated with Farin model is used to estimate an overall fractal dimension. The correlation between the pore fractal dimension and the pore size distribution (PSD) characteristics was further discussed.

Dempster-Shafer Evidence Theory and Study of Some Key Problems

As one of the most important mathematical methods, the Dempster-Shafer (D-S) evidence theory has been widely used in date fusion, risk assessment, target identification, knowledge reasoning, and other fields. This paper summarized the development and recent studies of the explanations of D-S model, evidence combination algorithms, and the improvement of the conflict during evidence combination, and also compared all explanation models, algorithms, improvements, and their applicable conditions. We are trying to provide a reference for future research and applications through this summarization.

Multiscale modeling of brain dynamics: from single neurons and networks to mathematical tools.

The extreme complexity of the brain naturally requires mathematical modeling approaches on a large variety of scales; the spectrum ranges from single neuron dynamics over the behavior of groups of neurons to neuronal network activity. Thus, the connection between the microscopic scale (single neuron activity) to macroscopic behavior (emergent behavior of the collective dynamics) and vice versa is a key to understand the brain in its complexity. In this work, we attempt a review of a wide range of approaches, ranging from the modeling of single neuron dynamics to machine learning. The models include biophysical as well as data-driven phenomenological models. The discussed models include Hodgkin-Huxley, FitzHugh-Nagumo, coupled oscillators (Kuramoto oscillators, Rössler oscillators, and the Hindmarsh-Rose neuron), Integrate and Fire, networks of neurons, and neural field equations. In addition to the mathematical models, important mathematical methods in multiscale modeling and reconstruction of the causal connectivity are sketched. The methods include linear and nonlinear tools from statistics, data analysis, and time series analysis up to differential equations, dynamical systems, and bifurcation theory, including Granger causal connectivity analysis, phase synchronization connectivity analysis, principal component analysis (PCA), independent component analysis (ICA), and manifold learning algorithms such as ISOMAP, and diffusion maps and equation-free techniques. WIREs Syst Biol Med 2016, 8:438-458. doi: 10.1002/wsbm.1348 For further resources related to this article, please visit the WIREs website.

Predicting biofilm thickness and biofilm viability based on the concentration of carbon-nitrogen-phosphorus by support vector regression.

Current tools to predict biofilm thickness and viability in spatial distribution are poor, especially those based on chemical oxygen demand (COD), total nitrogen (TN), and total phosphate (TP) due to their limited data and complex calculations. Here, support vector regression (SVR) was used to predict biofilm thickness and viability in a reactor filled with carriers of crushed stone globular aggregates. Analyses combined confocal laser scanning microscopy and flow cytometry with Kriging interpolation revealed that biofilm thickness varied from 22 to 31 μm, and biofilm viability decreased from 80 to 30% in the flow direction of the reactor. The biofilm thickness at the bottom was thicker than that in the upper layer, but biofilm viability contrasted with biofilm thickness in the vertical distribution. The values of biofilm thickness and viability were predicted at a layer 35 cm from the bottom of the reactor with mean squared error values of 0.014 and 0.011, respectively. Correlation coefficients were 0.996 and 0.997 between carbon-nitrogen-phosphorus (C-N-P) removal with biofilm thickness and viability in spatial distribution, respectively. This study provided an important mathematical method to predict biofilm thickness and viability in spatial distribution based on the concentration of C-N-P.

The elusive concept of brain network: Comment on “Understanding brain networks and brain organization” by Luiz Pessoa

As the poet John Donne said of man -“No man is an island entire of itself; every man is a piece of the continent, a part of the main.” - so the neuroscience research community now says of brain areas. This is the topic that Luiz Pessoa expands upon in his thorough review of the paradigm shift that has occurred in much of brain research, especially in cognitive neuroscience [1]. His key point is made explicitly in the Abstract: “I argue that a network perspective should supplement the common strategy of understanding the brain in terms of individual regions.” In his review, Pessoa covers a large range of topics, including how the network perspective changes the way in which one views the structure-function relationship between brain and behavior, the importance of context is ascertaining how a brain region functions, and the notion of emergent properties as a network feature. Also discussed is graph theory, one of the important mathematical methods used to analyze and describe network structure and function.

Fitting physiological models to data.

Methods of fitting models to experimental data obtained from biological systems are reviewed. The Michaelis-Menten model is used as the working example of a familiar and well-studied biological model. Criteria for selecting models include goodness of fit, freedom from systematic errors, and simplicity. A given model is usually fitted and the optimal parameters determined by minimization of an objective function, usually the sum of the squared errors. Freedom from systematic errors is best judged graphically. The important mathematical methods of fitting models are derived from the calculus and include linear and quadratic programming. The latter leads to minimization of the sum of squares of errors. Optimal search procedures, which also perform this minimization, are surveyed. Other properties of models, such as the proper number of parameters and whether linearization is appropriate, are discussed. The specialized problems of biological models and data are considered.