Axioms

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

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

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

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

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

For bounded linear operators defined on complex infinite-dimensional Banach space, H. Zariouh, in an article (Property $\left(gz\right)$ for bounded linear operators, (2013)) 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

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

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

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

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 $E\times E$ is a $k_{\mathbb{R}}$-space. Full article

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

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 $\{R,S\}$ guranteeing the edge-connectivity of a connected $\{R,S\}$-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

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

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

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

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

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

Modeling the number of individuals in different states is a principal tool in the event of an epidemic. The natural transition of individuals between possible states often includes deliberate interference such as isolation or vaccination. Thus, the mathematical model may need to be re-calibrated due to various factors. The model considered in this paper is the SIRD epidemic model. An additional parameter is the moment of changing the description of the phenomenon when the parameters of the model change and the change is not pre-specified. Detecting and estimating the moment of change in real time is the subject of statistical research. A sequential (online) approach was applied using the Bayesian shift point detection algorithm and trimmed exact linear time. We show how methods of analysis behave in different instances. These methods are verified on simulated data and applied to pandemic data of a selected European country. The simulation is performed with a social network graph to obtain a practical representation ability. The epidemiological data used come from the territory of Poland and concern the COVID-19 epidemic in Poland. The results show satisfactory detection of the moments where the applied model needs to be verified and re-calibrated. These show the effectiveness of the proposed combination of methods. Full article

In this work, we consider a class of initial boundary value problems for fourth-order dispersive wave equations with superlinear damping and non-local source terms as well as time-dependent coefficients in $\Omega \times (t>0)$, where $\Omega$ is a bounded domain in ${\mathbb{R}}^N$ and $N\ge 2$. We prove that there exists a safe time interval of existence in the solution $[0,T]$, with T being a lower bound of the blowup time $t^{*}$. Moreover, we find an explicit lower bound of $t^{*}$, assuming the coefficients are positive constants. Full article