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# Axioms, Volume 11, Issue 5 (May 2022) – 60 articles

Cover Story (view full-size image): The affine Hecke algebra of type A has parameters (q,t) and acts on polynomials in N variables. There are two pairwise commuting sets of elements in the algebra: the Cherednik operators and the Jucys–Murphy elements. For certain parameter values, there are polynomials which are simultaneous eigenfunctions with equal corresponding eigenvalues of both sets of operators. These are called singular polynomials. The parameter values are qm =t-n with 2≤n≤N. The singular polynomials span an irreducible module of the Hecke algebra. Previously, it was shown that there are singular polynomials which are specializations of nonsymmetric Macdonald polynomials, and the isotype of the Hecke algebra module is (dn-1,n-1,…,n-1,r) for some d≥1. Here, it is shown that there are no other singular polynomials. View this paper
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Article
Some New Bounds for the Inverse Sum Indeg Energy of Graphs
Axioms 2022, 11(5), 243; https://doi.org/10.3390/axioms11050243 - 23 May 2022
Viewed by 466
Abstract
Let G be a (molecular) graph with n vertices, and ${d}_{i}$ be the degree of its i-th vertex. Then, the inverse sum indeg matrix of G is the $n×n$ matrix $\mathcal{C}\left(G\right)$ with entries [...] Read more.
Let G be a (molecular) graph with n vertices, and ${d}_{i}$ be the degree of its i-th vertex. Then, the inverse sum indeg matrix of G is the $n×n$ matrix $\mathcal{C}\left(G\right)$ with entries ${c}_{ij}=\frac{{d}_{i}{d}_{j}}{{d}_{i}+{d}_{j}}$, if the i-th and the j-th vertices are adjacent and 0 otherwise. Let ${\mu }_{1}\ge {\mu }_{2}\ge \dots \ge {\mu }_{n}$ be the eigenvalues of $\mathcal{C}$ arranged in order. The inverse sum indeg energy of G, ${\epsilon }_{isi}\left(G\right)$ can be represented as ${\sum }_{j=1}^{n}|{\mu }_{i}|$. In this paper, we establish several novel upper and lower sharp bounds on ${\mu }_{1}$ and ${\epsilon }_{isi}\left(G\right)$ via some other graph parameters, and describe the structures of the extremal graphs. Full article
Review
The State of the Art of Data Mining Algorithms for Predicting the COVID-19 Pandemic
Axioms 2022, 11(5), 242; https://doi.org/10.3390/axioms11050242 - 23 May 2022
Viewed by 685
Abstract
Current computer systems are accumulating huge amounts of information in several application domains. The outbreak of COVID-19 has increased rekindled interest in the use of data mining techniques for the analysis of factors that are related to the emergence of an epidemic. Data [...] Read more.
Current computer systems are accumulating huge amounts of information in several application domains. The outbreak of COVID-19 has increased rekindled interest in the use of data mining techniques for the analysis of factors that are related to the emergence of an epidemic. Data mining techniques are being used in the analysis and interpretation of information, which helps in the discovery of patterns, planning of isolation policies, and even predicting the speed of proliferation of contagion in a viral disease such as COVID-19. This research provides a comprehensive study of various data mining algorithms that are used in conjunction with epidemiological prediction models. The document considers that there is an opportunity to improve or develop tools that offer an accurate prognosis in the management of viral diseases through the use of data mining tools, based on a comparative study of 35 research papers. Full article
(This article belongs to the Special Issue Softcomputing: Theories and Applications II)
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Article
Retrial Queuing-Inventory Systems with Delayed Feedback and Instantaneous Damaging of Items
Axioms 2022, 11(5), 241; https://doi.org/10.3390/axioms11050241 - 20 May 2022
Viewed by 512
Abstract
This paper studies a Markov model of a queuing-inventory system with primary, retrial, and feedback customers. Primary customers form a Poisson flow, and if an inventory level is positive upon their arrival, they instantly receive the items. If the inventory level is equal [...] Read more.
This paper studies a Markov model of a queuing-inventory system with primary, retrial, and feedback customers. Primary customers form a Poisson flow, and if an inventory level is positive upon their arrival, they instantly receive the items. If the inventory level is equal to zero upon arrival of a primary customer, then this customer, according to the Bernoulli scheme, either leaves the system or goes into an infinite buffer to repeat their request in the future. The rate of retrial customers is constant, and if the inventory level is zero upon arrival of a retrial customer, then this customer, according to the Bernoulli scheme, either leaves orbit or remains in orbit to repeat its request in the future. According to the Bernoulli scheme, each served primary or retrial customer either leaves the system or feedbacks into orbit to repeat their request. Destructive customers that form a Poisson flow cause damage to items. Unlike primary, retrial, and feedback customers, destructive customers do not require items, since, upon arrival of such customers, the inventory level instantly decreases by one. The system adopted one of two replenishment policies: (s, Q) or (s, S). In both policies, the lead time is a random variable that has an exponential distribution. It is shown that the mathematical model of the system under study was a two-dimensional Markov chain with an infinite state space. Algorithms for calculating the elements of the generating matrices of the constructed chains were developed, and the ergodicity conditions for both policies were found. To calculate the steady-state probabilities, a matrix-geometric method was used. Formulas were found for calculating the main performance measures of the system. The results of the numerical experiments, including the minimization of the total cost, are demonstrated. Full article
(This article belongs to the Special Issue Queueing Theory and Network Applications)
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Article
Fuzzy Caratheodory’s Theorem and Outer ∗-Fuzzy Measure
Axioms 2022, 11(5), 240; https://doi.org/10.3390/axioms11050240 - 20 May 2022
Viewed by 421
Abstract
The goal of this paper is to introduce two new concepts ∗-fuzzy premeasure and outer ∗-fuzzy measure, and to further prove some properties, such as Caratheodory’s Theorem, as well as the unique extension of ∗-fuzzy premeasure. This theorem is remarkable for it allows [...] Read more.
The goal of this paper is to introduce two new concepts ∗-fuzzy premeasure and outer ∗-fuzzy measure, and to further prove some properties, such as Caratheodory’s Theorem, as well as the unique extension of ∗-fuzzy premeasure. This theorem is remarkable for it allows one to construct a ∗-fuzzy measure by first defining it on a small algebra of sets, where its ∗-additivity could be easy to verify, and then this theorem guarantees its extension to a sigma-algebra. Full article
Article
Two Invariants for Geometric Mappings
Axioms 2022, 11(5), 239; https://doi.org/10.3390/axioms11050239 - 20 May 2022
Viewed by 473
Abstract
Two invariants for mappings of affine connection spaces with a special form of deformation tensors are obtained in this paper. We used the methodology of Vesić to obtain the form of these invariants. At the end of this paper, we used these forms [...] Read more.
Two invariants for mappings of affine connection spaces with a special form of deformation tensors are obtained in this paper. We used the methodology of Vesić to obtain the form of these invariants. At the end of this paper, we used these forms to obtain two invariants for third-type almost-geodesic mappings of symmetric affine connection. Full article
(This article belongs to the Special Issue Differential Geometry and Its Application)
Article
An Application of the Super-SBM MAX and LTS(A,A,A) Models to Analyze the Business Performance of Hydropower Suppliers in Vietnam
Axioms 2022, 11(5), 238; https://doi.org/10.3390/axioms11050238 - 20 May 2022
Viewed by 575
Abstract
As Vietnam continues to industrialize and modernize, such economic development and high-tech will require a major electrical energy source to operate the electrical equipment; hence, the hydropower plants are established and growing up to demand. Therefore, the purpose of this study is to [...] Read more.
As Vietnam continues to industrialize and modernize, such economic development and high-tech will require a major electrical energy source to operate the electrical equipment; hence, the hydropower plants are established and growing up to demand. Therefore, the purpose of this study is to evaluate the business performance of Vietnamese hydropower suppliers by integrating the LTS(A,A,A) model of the Additive Holt-winters method in Tableau and a super-slacks-based measure (super-SBM) max model in data envelopment analysis (DEA). The LTS(A,A,A) model is applied to forecast future valuation from 2022 to 2025 based on historical time series from 2012 to 2021. Next, with the actual and predicted data, the researcher uses the super-SBM max model to calculate the business performance of these hydropower suppliers from past to future. The empirical result reveals efficient and inefficient cases to explore which hydropower suppliers can achieve the business performance in their operational process. The position of hydropower suppliers in Vietnam from past to future time is determined particularly based on their scores every year. Further, the empirical result recommends a solution to deal with inefficient cases by deducting the input excesses and raising the output shortages based on the principle of the super-SBM Max model in DEA. The finding results create an overview of the operational process with the continuing variations in each period to equip hydropower suppliers in Vietnam which will determine their future and operational orientation. Full article
(This article belongs to the Special Issue Mathematical Analysis for Financial Modelling)
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Article
Certain Subclasses of Bi-Starlike Function of Complex Order Defined by Erdély–Kober-Type Integral Operator
Axioms 2022, 11(5), 237; https://doi.org/10.3390/axioms11050237 - 20 May 2022
Viewed by 494
Abstract
In the present paper, we introduce new subclasses of bi-starlike and bi-convex functions of complex order associated with Erdély–Kober-type integral operator in the open unit disc and find the estimates of initial coefficients in these classes. Moreover, we obtain Fekete-Szegő inequalities for functions [...] Read more.
In the present paper, we introduce new subclasses of bi-starlike and bi-convex functions of complex order associated with Erdély–Kober-type integral operator in the open unit disc and find the estimates of initial coefficients in these classes. Moreover, we obtain Fekete-Szegő inequalities for functions in these classes. Some of the significances of our results are pointed out as corollaries. Full article
(This article belongs to the Special Issue New Developments in Geometric Function Theory)
Article
Bounds for the Neuman–Sándor Mean in Terms of the Arithmetic and Contra-Harmonic Means
Axioms 2022, 11(5), 236; https://doi.org/10.3390/axioms11050236 - 19 May 2022
Cited by 1 | Viewed by 559
Abstract
In this paper, the authors provide several sharp upper and lower bounds for the Neuman–Sándor mean in terms of the arithmetic and contra-harmonic means, and present some new sharp inequalities involving hyperbolic sine function and hyperbolic cosine function. Full article
Article
Projective Integration for Hyperbolic Shallow Water Moment Equations
Axioms 2022, 11(5), 235; https://doi.org/10.3390/axioms11050235 - 18 May 2022
Viewed by 619
Abstract
In free surface flows, shallow water models simplify the flow conditions by assuming a constant velocity profile over the water depth. Recently developed Shallow Water Moment Equations allow for variations of the velocity profile at the expense of a more complex PDE system. [...] Read more.
In free surface flows, shallow water models simplify the flow conditions by assuming a constant velocity profile over the water depth. Recently developed Shallow Water Moment Equations allow for variations of the velocity profile at the expense of a more complex PDE system. The resulting equations can become stiff depending on the friction parameters, which leads to severe time step constraints of standard numerical schemes. In this paper, we apply Projective Integration schemes to stiff Shallow Water Moment Equations to overcome the time step constraints in the stiff regime and accelerate the numerical computations while still achieving high accuracy. In different dam break and smooth wave test cases, we obtain a speedup of up to 55 with respect to standard schemes. Full article
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Article
Investigation into the Explicit Solutions of the Integrable (2+1)—Dimensional Maccari System via the Variational Approach
by
Axioms 2022, 11(5), 234; https://doi.org/10.3390/axioms11050234 - 18 May 2022
Cited by 1 | Viewed by 505
Abstract
In this paper, the integrable (2+1)-dimensional Maccari system (MS), which can model many complex phenomena in hydrodynamics, plasma physics and nonlinear optics, is investigated by the variational approach (VA). This proposed approach that based on the variational theory and Ritz-like method can construct [...] Read more.
In this paper, the integrable (2+1)-dimensional Maccari system (MS), which can model many complex phenomena in hydrodynamics, plasma physics and nonlinear optics, is investigated by the variational approach (VA). This proposed approach that based on the variational theory and Ritz-like method can construct the explicit solutions via the stationary conditions only taking two steps. Finally, the dynamic behaviors of the solutions are exhibited by choosing the appropriate parameters through the 3-D and density plots. It can be seen that the proposed method is concise and straightforward, and can be adopted to study the travelling wave theory in physics. Full article
(This article belongs to the Special Issue Calculus of Variations and Nonlinear Partial Differential Equations)
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Article
Some Upper Bounds for RKHS Approximation by Bessel Functions
Axioms 2022, 11(5), 233; https://doi.org/10.3390/axioms11050233 - 17 May 2022
Viewed by 446
Abstract
A reproducing kernel Hilbert space (RKHS) approximation problem arising from learning theory is investigated. Some K-functionals and moduli of smoothness with respect to RKHSs are defined with Fourier–Bessel series and Fourier–Bessel transforms, respectively. Their equivalent relation is shown, with which the upper [...] Read more.
A reproducing kernel Hilbert space (RKHS) approximation problem arising from learning theory is investigated. Some K-functionals and moduli of smoothness with respect to RKHSs are defined with Fourier–Bessel series and Fourier–Bessel transforms, respectively. Their equivalent relation is shown, with which the upper bound estimate for the best RKHS approximation is provided. The convergence rate is bounded with the defined modulus of smoothness, which shows that the RKHS approximation can attain the same approximation ability as that of the Fourier–Bessel series and Fourier–Bessel transform. In particular, it is shown that for a RKHS produced by the Bessel operator, the convergence rate sums up to the bound of a corresponding convolution operator approximation. The investigations show some new applications of Bessel functions. The results obtained can be used to bound the approximation error in learning theory. Full article
Article
Modeling Preferences through Personality and Satisfaction to Guide the Decision Making of a Virtual Agent
Axioms 2022, 11(5), 232; https://doi.org/10.3390/axioms11050232 - 17 May 2022
Viewed by 547
Abstract
Satisfaction is relevant for decision makers (DM, Decision Makers). Satisfaction is the feeling produced in individuals by executing actions to satisfy their needs, for example, the payment of debts, jobs, or academic achievements, and the acquisition of goods or services. In the satisfaction [...] Read more.
Satisfaction is relevant for decision makers (DM, Decision Makers). Satisfaction is the feeling produced in individuals by executing actions to satisfy their needs, for example, the payment of debts, jobs, or academic achievements, and the acquisition of goods or services. In the satisfaction literature, some theories model the satisfaction of individuals from job and customer approaches. However, considering personality elements to influence satisfaction and define preferences in strategies that optimize decision making provides the unique characteristics of a DM. These characteristics favor the scope of solutions closer to the satisfaction expectation. Satisfaction theories do not include specific elements of personality and preferences, so integrating these elements will offer more efficient decisions in computable models. In this work, a model of satisfaction with personality characteristics that influence the preferences of a DM is proposed. The proposed model is integrated into a preference-based optimizer that improves the decision-making process of a Virtual Decision Maker (VDM) in an optimization context. The optimization context addressed in this work is the product selection process within a food product shopping problem. An experimental design is proposed that compares two configurations that represent the cognitive part of an agent’s decision process to validate the operation of the proposed model in the context of optimization: (1) satisfaction, personality, and preferences, and (2) personality and preferences. The results show that considering satisfaction and personality in combination with preferences provides solutions closer to the interests of an individual, reflecting a more realistic behavior. Furthermore, this work demonstrates that it is possible to create a configurable model that allows adapting to different aptitudes and reflecting them in a computable model. Full article
(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Analysis)
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Article
Taylor’s Formula for Generalized Weighted Fractional Derivatives with Nonsingular Kernels
Axioms 2022, 11(5), 231; https://doi.org/10.3390/axioms11050231 - 15 May 2022
Viewed by 587
Abstract
We prove a new Taylor’s theorem for generalized weighted fractional calculus with nonsingular kernels. The proof is based on the establishment of new relations for nth-weighted generalized fractional integrals and derivatives. As an application, new mean value theorems for generalized weighted fractional operators [...] Read more.
We prove a new Taylor’s theorem for generalized weighted fractional calculus with nonsingular kernels. The proof is based on the establishment of new relations for nth-weighted generalized fractional integrals and derivatives. As an application, new mean value theorems for generalized weighted fractional operators are obtained. Direct corollaries allow one to obtain the recent Taylor’s and mean value theorems for Caputo–Fabrizio, Atangana–Baleanu–Caputo (ABC) and weighted ABC derivatives. Full article
(This article belongs to the Special Issue Fractional Calculus and Differential Equations)
Article
Evaluation of Digital Marketing Technologies with Fuzzy Linguistic MCDM Methods
Axioms 2022, 11(5), 230; https://doi.org/10.3390/axioms11050230 - 13 May 2022
Viewed by 584
Abstract
Technology is becoming the tool that changes how people live every day, and the marketing strategies of businesses are also gradually shifting to the industry 4.0 mindset of constant growth and development. Digital marketing has changed human habits of information accessibility, determined their [...] Read more.
Technology is becoming the tool that changes how people live every day, and the marketing strategies of businesses are also gradually shifting to the industry 4.0 mindset of constant growth and development. Digital marketing has changed human habits of information accessibility, determined their interactions, and witnessed the birth of a variety of new marketing technologies. Marketers are creating digital marketing products and services that enhance the experience for consumers, products, and services that are also delivered through high digital marketing networks. As a result, data sources become more abundant and allow consumers to have more choices. All products, services, technologies, and data are increasingly meeting the needs of consumers, thereby confirming the effectiveness of digital marketing in today’s market. However, the evaluation and selection of digital marketing technology is very complex since it has many conflicting criteria and goals. The multi-criteria decision-making model (MCDM) is a powerful technique widely used for solving this type of problem. Thus, the author proposed a fuzzy linguistic MCDM method for evaluation of digital marketing technologies. After determining the evaluation criteria and alternatives, two MCDM methods, including Spherical Fuzzy Analytic Hierarchy Process (SF-AHP) and Technique for Order Preference by Similarity to Ideal Solution (TOPSIS), are used in the evaluation and selection of digital marketing technologies procedure. Finally, an application is present to demonstrate the potential use of the proposed methodology. The main contribution of this study is to propose a Spherical fuzzy MCDM model to support planners and decision makers in the digital marketing technology evaluation and selection processes. A case study is also performed to showcase the feasibility of the proposed approach. Full article
(This article belongs to the Special Issue Communications in Industrial Statistics—Theory and Methods)
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Article
The Possibility of Combining and Implementing Deep Neural Network Compression Methods
Axioms 2022, 11(5), 229; https://doi.org/10.3390/axioms11050229 - 13 May 2022
Viewed by 610
Abstract
In the paper, the possibility of combining deep neural network (DNN) model compression methods to achieve better compression results was considered. To compare the advantages and disadvantages of each method, all methods were applied to the ResNet18 model for pretraining to the NCT-CRC-HE-100K [...] Read more.
In the paper, the possibility of combining deep neural network (DNN) model compression methods to achieve better compression results was considered. To compare the advantages and disadvantages of each method, all methods were applied to the ResNet18 model for pretraining to the NCT-CRC-HE-100K dataset while using CRC-VAL-HE-7K as the validation dataset. In the proposed method, quantization, pruning, weight clustering, QAT (quantization-aware training), preserve cluster QAT (hereinafter PCQAT), and distillation were performed for the compression of ResNet18. The final evaluation of the obtained models was carried out on a Raspberry Pi 4 device using the validation dataset. The greatest model compression result on the disk was achieved by applying the PCQAT method, whose application led to a reduction in size of the initial model by as much as 45 times, whereas the greatest model acceleration result was achieved via distillation on the MobileNetV2 model. All methods led to the compression of the initial size of the model, with a slight loss in the model accuracy or an increase in the model accuracy in the case of QAT and weight clustering. INT8 quantization and knowledge distillation also led to a significant decrease in the model execution time. Full article
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Article
A Two-Stage Multi-Criteria Supplier Selection Model for Sustainable Automotive Supply Chain under Uncertainty
by and
Axioms 2022, 11(5), 228; https://doi.org/10.3390/axioms11050228 - 13 May 2022
Viewed by 719
Abstract
Sustainable supplier selection (SSS) is gaining popularity as a practical method to supply chain sustainability among academics and practitioners. However, in addition to balancing economic, social, and environmental factors, the emergence of the COVID-19 pandemic has affected the selection of long-term suppliers to [...] Read more.
Sustainable supplier selection (SSS) is gaining popularity as a practical method to supply chain sustainability among academics and practitioners. However, in addition to balancing economic, social, and environmental factors, the emergence of the COVID-19 pandemic has affected the selection of long-term suppliers to ensure sustainable supply chains, recover better from the pandemic and effectively respond to any future unprecedented crises. The purpose of this study is to assess and choose a possible supplier based on their capability to adapt to the COVID-19 epidemic in a sustainable manner. For this assessment, a framework based on multi-criteria decision making (MCDM) is provided that integrates spherical fuzzy Analytical Hierarchical Process (SF-AHP) and grey Complex Proportional Assessment (G-COPRAS), in which spherical fuzzy sets and grey numbers are used to express the ambiguous linguistic evaluation statements of experts. In the first stage, the evaluation criteria system is identified through a literature review and experts’ opinions. The SF-AHP is then used to determine the criteria weights. Finally, the G-COPRAS method is utilized to select sustainable suppliers. A case study in the automotive industry in Vietnam is presented to demonstrate the proposed approach’s effectiveness. From the SF-AHP findings, “quality”, “use of personal protective equipment”, “cost/price”, “safety and health practices and wellbeing of suppliers”, and “economic recovery programs” have been ranked as the five most important criteria. From G-COPRAS analysis, THACO Parts (Supplier 02) is the best supplier. A sensitivity study was also conducted to verify the robustness of the proposed model, in which the priority rankings of the best suppliers are very similar. For long-term development and increased competitiveness, industrial businesses must stress the integration of response mechanisms during SSS implementation in the COVID-19 epidemic, according to the findings. This will result in significant cost and resource savings, as well as reduced environmental consequences and a long-term supply chain, independent of the crisis. Full article
(This article belongs to the Special Issue Soft Computing with Applications to Decision Making and Data Mining)
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Article
Some Generalizations of the Jensen-Type Inequalities with Applications
Axioms 2022, 11(5), 227; https://doi.org/10.3390/axioms11050227 - 13 May 2022
Cited by 1 | Viewed by 527
Abstract
Motivated by some results about reverses of the Jensen inequality for positive measure, in this paper we give generalizations of those results for real Stieltjes measure $d\lambda$ which is not necessarily positive using several Green functions. Utilizing these results we define some [...] Read more.
Motivated by some results about reverses of the Jensen inequality for positive measure, in this paper we give generalizations of those results for real Stieltjes measure $d\lambda$ which is not necessarily positive using several Green functions. Utilizing these results we define some new mean value theorems of Lagrange and Cauchy types, and derive some new Cauchy-type means. Full article
(This article belongs to the Special Issue Current Research on Mathematical Inequalities)
Article
A Note on the Reverse Order Law for g-Inverse of Operator Product
Axioms 2022, 11(5), 226; https://doi.org/10.3390/axioms11050226 - 12 May 2022
Viewed by 487
Abstract
The generalized inverse has many important applications in aspects of the theoretical research of matrices and statistics. One of the core problems of the generalized inverse is finding the necessary and sufficient conditions of the reverse order laws for the generalized inverse of [...] Read more.
The generalized inverse has many important applications in aspects of the theoretical research of matrices and statistics. One of the core problems of the generalized inverse is finding the necessary and sufficient conditions of the reverse order laws for the generalized inverse of the operator product. In this paper, we study the reverse order law for the g-inverse of an operator product ${T}_{1}{T}_{2}{T}_{3}$ using the technique of matrix form of bounded linear operators. In particular, some necessary and sufficient conditions for the inclusion ${T}_{3}\left\{1\right\}{T}_{2}\left\{1\right\}{T}_{1}\left\{1\right\}\subseteq \left({T}_{1}{T}_{2}{T}_{3}\right)\left\{1\right\}$ is presented. Moreover, some finite dimensional results are extended to infinite dimensional settings. Full article
Article
Tensor Product of Operators Satisfying Zariouh’s Property (gaz), and Stability under Perturbations
Axioms 2022, 11(5), 225; https://doi.org/10.3390/axioms11050225 - 12 May 2022
Viewed by 1993
Abstract
For bounded linear operators defined on complex infinite-dimensional Banach space, H. Zariouh, in an article [1] introduced and studied the property $\left(gaz\right)$. In this study, through techniques using the local spectral theory of operators, we discover the sufficient [...] Read more.
For bounded linear operators defined on complex infinite-dimensional Banach space, H. Zariouh, in an article [1] introduced and studied the property $\left(gaz\right)$. In this study, through techniques using the local spectral theory of operators, we discover the sufficient conditions that allow the transfer of the property $\left(gaz\right)$ from two tensor factors $\mathcal{T}$ and $\mathcal{S}$ to their tensor product $\mathcal{T}\otimes \mathcal{S}$. The stability of the property $\left(gaz\right)$ in the tensor product under perturbations is also investigated. The theory is exemplified by considering suitable classes of operators such as shift operators, convolution operators, and m-invertible contractions. Full article
Article
Krein’s Theorem in the Context of Topological Abelian Groups
Axioms 2022, 11(5), 224; https://doi.org/10.3390/axioms11050224 - 12 May 2022
Viewed by 741
Abstract
A topological abelian group G is said to have the quasi-convex compactness property (briefly, qcp) if the quasi-convex hull of every compact subset of G is again compact. In this paper we prove that there exist locally quasi-convex metrizable complete groups G which [...] Read more.
A topological abelian group G is said to have the quasi-convex compactness property (briefly, qcp) if the quasi-convex hull of every compact subset of G is again compact. In this paper we prove that there exist locally quasi-convex metrizable complete groups G which endowed with the weak topology associated to their character groups ${G}^{\wedge }$, do not have the qcp. Thus, Krein’s Theorem, a well known result in the framework of locally convex spaces, cannot be fully extended to locally quasi-convex groups. Some features of the qcp are also studied. Full article
Article
Co-Training Method Based on Semi-Decoupling Features for MOOC Learner Behavior Prediction
Axioms 2022, 11(5), 223; https://doi.org/10.3390/axioms11050223 - 11 May 2022
Viewed by 524
Abstract
Facing the problem of massive unlabeled data and limited labeled samples, semi-supervised learning is favored, especially co-training. Standard co-training requires sufficiently redundant and conditionally independent dual views; however, in fact, few dual views exist that satisfy this condition. To solve this problem, we [...] Read more.
Facing the problem of massive unlabeled data and limited labeled samples, semi-supervised learning is favored, especially co-training. Standard co-training requires sufficiently redundant and conditionally independent dual views; however, in fact, few dual views exist that satisfy this condition. To solve this problem, we propose a co-training method based on semi-decoupling features, that is, semi-decoupling features based on a known single view and then constructing independent and redundant dual views: (1) take a small number of important features as shared features of the dual views according to the importance of the features; (2) separate the remaining features one by one or in small batches according to the correlation between the features to make “divergent” features of the dual views; (3) combine the shared features and the “divergent” features to construct dual views. In this paper, the experimental dataset was from the edX dataset jointly released by Harvard University and MIT; the evaluation metrics adopted F1, Precision, and Recall. The analysis methods included three experiments: multiple models, iterations, and hyperparameters. The experimental results show that the effect of this model on MOOC learner behavior prediction was better than the other models, and the best prediction result was obtained in iteration 2. These all verify the effectiveness and superiority of this algorithm and provide a scientific and feasible reference for the development of the future education industry. Full article
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Article
Exact Criteria for the Existence of a Moving Singular Point in a Complex Domain for a Nonlinear Differential Third-Degree Equation with a Polynomial Seventh-Degree Right-Hand Side
Axioms 2022, 11(5), 222; https://doi.org/10.3390/axioms11050222 - 10 May 2022
Viewed by 509
Abstract
Earlier, the authors formulated and proved interval and point criteria for the existence of moving singular points of a third-degree nonlinear differential equation with a polynomial seventh-degree right-hand side for a real domain. For the complex domain, these criteria are associated with specificity [...] Read more.
Earlier, the authors formulated and proved interval and point criteria for the existence of moving singular points of a third-degree nonlinear differential equation with a polynomial seventh-degree right-hand side for a real domain. For the complex domain, these criteria are associated with specificity of transition to phase spaces. Necessary as well as necessary and sufficient conditions for the existence of moving singular points are obtained. The results of these studies are the basis of the algorithm for obtaining moving singular points of nonlinear differential equations in the complex domain. The results of this paper allow us to expand the classes of the studied nonlinear differential equations with moving peculiarities. Full article
Article
Aspects of Differential Calculus Related to Infinite-Dimensional Vector Bundles and Poisson Vector Spaces
Axioms 2022, 11(5), 221; https://doi.org/10.3390/axioms11050221 - 09 May 2022
Viewed by 529
Abstract
We prove various results in infinite-dimensional differential calculus that relate the differentiability properties of functions and associated operator-valued functions (e.g., differentials). The results are applied in two areas: (1) in the theory of infinite-dimensional vector bundles, to construct new bundles from given ones, [...] Read more.
We prove various results in infinite-dimensional differential calculus that relate the differentiability properties of functions and associated operator-valued functions (e.g., differentials). The results are applied in two areas: (1) in the theory of infinite-dimensional vector bundles, to construct new bundles from given ones, such as dual bundles, topological tensor products, infinite direct sums, and completions (under suitable hypotheses); (2) in the theory of locally convex Poisson vector spaces, to prove continuity of the Poisson bracket and continuity of passage from a function to the associated Hamiltonian vector field. Topological properties of topological vector spaces are essential for the studies, which allow the hypocontinuity of bilinear mappings to be exploited. Notably, we encounter ${k}_{\mathbb{R}}$-spaces and locally convex spaces E such that $E×E$ is a ${k}_{\mathbb{R}}$-space. Full article
Article
New Fuzzy Extensions on Binomial Distribution
Axioms 2022, 11(5), 220; https://doi.org/10.3390/axioms11050220 - 09 May 2022
Viewed by 544
Abstract
The use of discrete probabilistic distributions is relevant to many practical tasks, especially in present-day situations where the data on distribution are insufficient and expert knowledge and evaluations are the only instruments for the restoration of probability distributions. However, in such cases, uncertainty [...] Read more.
The use of discrete probabilistic distributions is relevant to many practical tasks, especially in present-day situations where the data on distribution are insufficient and expert knowledge and evaluations are the only instruments for the restoration of probability distributions. However, in such cases, uncertainty arises, and it becomes necessary to build suitable approaches to overcome it. In this direction, this paper discusses a new approach of fuzzy binomial distributions (BDs) and their extensions. Four cases are considered: (1) When the elementary events are fuzzy. Based on this information, the probabilistic distribution of the corresponding fuzzy-random binomial variable is calculated. The conditions of restrictions on this distribution are obtained, and it is shown that these conditions depend on the ratio of success and failure of membership levels. The formulas for the generating function (GF) of the constructed distribution and the first and second order moments are also obtained. The Poisson distribution is calculated as the limit case of a fuzzy-random binomial experiment. (2) When the number of successes is of a fuzzy nature and is represented as a fuzzy subset of the set of possible success numbers. The formula for calculating the probability of convolution of binomial dependent fuzzy events is obtained, and the corresponding GF is built. As a result, the scheme for calculating the mathematical expectation of the number of fuzzy successes is defined. (3) When the spectrum of the extended distribution is fuzzy. The discussion is based on the concepts of a fuzzy-random event and its probability, as well as the notion of fuzzy random events independence. The fuzzy binomial upper distribution is specifically considered. In this case the fuzziness is represented by the membership levels of the binomial and non-binomial events of the complete failure complex. The GF of the constructed distribution and the first-order moment of the distribution are also calculated. Sufficient conditions for the existence of a limit distribution and a Poisson distribution are also obtained. (4) As is known, based on the analysis of lexical material, the linguistic spectrum of the statistical process of word-formation becomes two-component when switching to vocabulary. For this, two variants of the hybrid fuzzy-probabilistic process are constructed, which can be used in the analysis of the linguistic spectrum of the statistical process of word-formation. A fuzzy extension of standard Fuchs distribution is also presented, where the fuzziness is reflected in the growing numbers of failures. For better representation of the results, the examples of fuzzy BD are illustrated in each section. Full article
(This article belongs to the Special Issue 10th Anniversary of Axioms: Logic)
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Article
Characterizing Forbidden Pairs for the Edge-Connectivity of a Connected Graph to Be Its Minimum Degree
Axioms 2022, 11(5), 219; https://doi.org/10.3390/axioms11050219 - 09 May 2022
Cited by 1 | Viewed by 599
Abstract
Let $\mathcal{H}$ be a class of given graphs. A graph G is said to be $\mathcal{H}$-free if G contains no induced copies of H for any $H\in \mathcal{H}$. In this article, we characterize all connected subgraph pairs [...] Read more.
Let $\mathcal{H}$ be a class of given graphs. A graph G is said to be $\mathcal{H}$-free if G contains no induced copies of H for any $H\in \mathcal{H}$. In this article, we characterize all connected subgraph pairs $\left\{R,S\right\}$ guranteeing the edge-connectivity of a connected $\left\{R,S\right\}$-free graph to have the same minimum degree. Our result is a supplement of Wang et al. Furthermore, we obtain a relationship of forbidden sets when those general parameters have the recurrence relation. Full article
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Article
On the Group of Absolutely Summable Sequences
Axioms 2022, 11(5), 218; https://doi.org/10.3390/axioms11050218 - 07 May 2022
Viewed by 499
Abstract
For an abelian topological group G, the sequence group ${\ell }^{1}\left(G\right)$ of all absolutely summable sequences in G is studied. It is shown that ${\ell }^{1}\left(G\right)$ is a Pontryagin reflexive group in case G is a [...] Read more.
For an abelian topological group G, the sequence group ${\ell }^{1}\left(G\right)$ of all absolutely summable sequences in G is studied. It is shown that ${\ell }^{1}\left(G\right)$ is a Pontryagin reflexive group in case G is a reflexive metrizable group or an LCA group. Further, ${\ell }^{1}\left(G\right)$ has the Schur property if and only if G has it and ${\ell }^{1}\left(G\right)$ is a Schwartz group if and only if G is linearly topologized. Full article
Article
A Symplectic Algorithm for Constrained Hamiltonian Systems
Axioms 2022, 11(5), 217; https://doi.org/10.3390/axioms11050217 - 07 May 2022
Viewed by 644
Abstract
In this paper, a symplectic algorithm is utilized to investigate constrained Hamiltonian systems. However, the symplectic method cannot be applied directly to the constrained Hamiltonian equations due to the non-canonicity. We firstly discuss the canonicalization method of the constrained Hamiltonian systems. The symplectic [...] Read more.
In this paper, a symplectic algorithm is utilized to investigate constrained Hamiltonian systems. However, the symplectic method cannot be applied directly to the constrained Hamiltonian equations due to the non-canonicity. We firstly discuss the canonicalization method of the constrained Hamiltonian systems. The symplectic method is used to constrain Hamiltonian systems on the basis of the canonicalization, and then the numerical simulation of the system is carried out. An example is presented to illustrate the application of the results. By using the symplectic method of constrained Hamiltonian systems, one can solve the singular dynamic problems of nonconservative constrained mechanical systems, nonholonomic constrained mechanical systems as well as physical problems in quantum dynamics, and also available in many electromechanical coupled systems. Full article
(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Physics)
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Article
Interval-Valued Pseudo Overlap Functions and Application
Axioms 2022, 11(5), 216; https://doi.org/10.3390/axioms11050216 - 06 May 2022
Cited by 1 | Viewed by 572
Abstract
A class of interval-valued OWA operators can be constructed from interval-valued overlap functions with interval-valued weights, which plays an important role in solving multi-attribute decision making (MADM) problems considering interval numbers as attribute values. Among them, when the importance of multiple attributes is [...] Read more.
A class of interval-valued OWA operators can be constructed from interval-valued overlap functions with interval-valued weights, which plays an important role in solving multi-attribute decision making (MADM) problems considering interval numbers as attribute values. Among them, when the importance of multiple attributes is different, it can only be calculated by changing the interval-valued weights. In fact, we can directly abandon the commutativity and extend the interval-valued overlap functions (IO) to interval-valued pseudo overlap functions (IPO) so that function itself implies the weights of the attributes, thus there is no need to calculate the OWA operator, which is more flexible in applications. In addition, the similar generalization on interval-valued pseudo t-norms obtained from interval-valued t-norms further enhances the feasibility of our study. In this paper, we mainly present the notion of interval-valued pseudo overlap functions and a few their qualities, including migrativity and homogeneity, and give some construction theorems and specific examples. Then, we propose the definitions of residuated implications induced by interval-valued pseudo overlap functions, give their equivalent forms, and prove some properties satisfied by them. Finally, two application examples about IPO to interval-valued multi-attribute decision making (I-MADM) are described. The results show that interval-valued pseudo overlap functions can not only be used to obtain the same rankings, but also be more flexible, simple and widely used. Full article
Article
Countermeasures of Double Carbon Targets in Beijing–Tianjin–Hebei Region by Using Grey Model
Axioms 2022, 11(5), 215; https://doi.org/10.3390/axioms11050215 - 05 May 2022
Viewed by 579
Abstract
In this paper, by combining the development characteristics of the Beijing–Tianjin–Hebei region, the fractional accumulation GM (1,1) model was used to predict the peak time of the Beijing–Tianjin–Hebei region, and the carbon peak year was predicted to be 2044. Then, according to the [...] Read more.
In this paper, by combining the development characteristics of the Beijing–Tianjin–Hebei region, the fractional accumulation GM (1,1) model was used to predict the peak time of the Beijing–Tianjin–Hebei region, and the carbon peak year was predicted to be 2044. Then, according to the urbanization level and the proportion of the added value of the secondary industry in different regions in 2018, regions were divided into four categories: the first to reach the peak, the peak on schedule (easy), the peak on schedule (general), and the peak may be delayed. The Beijing–Tianjin–Hebei region plans to achieve a carbon peak by 2044 and proposes specific suggestions to achieve carbon neutrality by 2060 to achieve coordinated development of Beijing–Tianjin–Hebei and high-quality development. Full article
(This article belongs to the Special Issue Grey System Theory and Applications in Mathematics)
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Article
Cheap Control in a Non-Scalarizable Linear-Quadratic Pursuit-Evasion Game: Asymptotic Analysis
Axioms 2022, 11(5), 214; https://doi.org/10.3390/axioms11050214 - 05 May 2022
Viewed by 601
Abstract
In this work, a finite-horizon zero-sum linear-quadratic differential game, modeling a pursuit-evasion problem, was considered. In the game’s cost function, the cost of the control of the minimizing player (the minimizer/the pursuer) was much smaller than the cost of the control of the [...] Read more.
In this work, a finite-horizon zero-sum linear-quadratic differential game, modeling a pursuit-evasion problem, was considered. In the game’s cost function, the cost of the control of the minimizing player (the minimizer/the pursuer) was much smaller than the cost of the control of the maximizing player (the maximizer/the evader) and the cost of the state variable. This smallness was expressed by a positive small multiplier (a small parameter) of the square of the ${L}_{2}$-norm of the minimizer’s control in the cost function. Parameter-free sufficient conditions for the existence of the game’s solution (the players’ optimal state-feedback controls and the game value), valid for all sufficiently small values of the parameter, were presented. The boundedness (with respect to the small parameter) of the time realizations of the optimal state-feedback controls along the corresponding game’s trajectory was established. The best achievable game value from the minimizer’s viewpoint was derived. A relation between solutions of the original cheap control game and the game that was obtained from the original one by replacing the small minimizer’s control cost with zero, was established. An illustrative real-life example is presented. Full article
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