Mathematics, Volume 9, Issue 24 (December-2 2021) – 182 articles

Computer-Supported Collaborative Learning tools are exhibiting an increased popularity in education, as they allow multiple participants to easily communicate, share knowledge, solve problems collaboratively, or seek advice. Nevertheless, multi-participant conversation logs are often hard to follow by teachers due to the mixture of multiple and many times concurrent discussion threads, with different interaction patterns between participants. Automated guidance can be provided with the help of Natural Language Processing techniques that target the identification of topic mixtures and of semantic links between utterances in order to adequately observe the debate and continuation of ideas. This paper introduces a method for discovering such semantic links embedded within chat conversations using string kernels, word embeddings, and neural networks. Our approach was validated on two datasets and obtained state-of-the-art results on both. Trained on a relatively small set of conversations, our models relying on string kernels are very effective for detecting such semantic links with a matching accuracy larger than 50% and represent a better alternative to complex deep neural networks, frequently employed in various Natural Language Processing tasks where large datasets are available. Full article

The solution of the so-called mixed-integer optimization problem is an important challenge for modern life sciences. A wide range of methods has been developed for its solution, including metaheuristics approaches. Here, a modification is proposed of the differential evolution entirely parallel (DEEP) method introduced recently that was successfully applied to mixed-integer optimization problems. The triangulation recombination rule was implemented and the recombination coefficients were included in the evolution process in order to increase the robustness of the optimization. The deduplication step included in the procedure ensures the uniqueness of individual integer-valued parameters in the solution vectors. The developed algorithms were implemented in the DEEP software package and applied to three bioinformatic problems. The application of the method to the optimization of predictors set in the genomic selection model in wheat resulted in dimensionality reduction such that the phenotype can be predicted with acceptable accuracy using a selected subset of SNP markers. The method was also successfully used to optimize the training set of samples for such a genomic selection model. According to the obtained results, the developed algorithm was capable of constructing a non-linear phenomenological regression model of gene expression in developing a Drosophila eye with almost the same average accuracy but significantly less standard deviation than the linear models obtained earlier. Full article

The recent exponentiated generalized linear exponential distribution is a generalization of the generalized linear exponential distribution and the exponentiated generalized linear exponential distribution. In this paper, we study some statistical properties of this distribution such as negative moments, moments of order statistics, mean residual lifetime, and their asymptotic distributions for sample extreme order statistics. Different estimation procedures include the maximum likelihood estimation, the corrected maximum likelihood estimation, the modified maximum likelihood estimation, the maximum product of spacing estimation, and the least squares estimation are compared via a Monte Carlo simulation study in terms of their biases, mean squared errors, and their rates of obtaining reliable estimates. Recommendations are made from the simulation results and a numerical example is presented to illustrate its use for modeling a rainfall data from Orlando, Florida. Full article

The main aim of this paper was to examine specific approaches to determining the discount rate for comprehensive computation of investment projects efficiency in the oil and gas industry. The objective of the study was to develop a scientific approach for determining the discount rate for integrated oil and gas projects. The authors analyze dynamic methods for determining the efficiency of investment projects in the oil and gas industry and conclude that they are advisable for oil and gas projects due to the high capital intensity of the projects and their long payback period. Regarding the need to implement dynamic indicators of efficiency, the authors set the task of deter-mining the proper discount rate as a factor having a significant impact on effectiveness evaluation. The discount rate is proposed to be evaluated by solving the equation and finding the break-even point where the NPV (net present value) of the integrated project will be equal to 0 (taking into account the revenue of the subprojects included in the complex). The practical implementation of methodological approaches to assessing the discount rate for integrated projects is relevant due to the execution of large, systemically important and integrated projects. As a result of the study, the authors put forward a methodological algorithm for determining the discount rate of an integrated project which assumes an assessment of cash flows for the subprojects included in the complex; determination of the target rate of return for subprojects; and calculation of prices for products at which a complex project become break-even. The practical implementation of methodological approaches to assessing the discount rate for integrated projects is relevant due to the execution of large systemically important integrated projects. Full article

Traditional power generating technologies rely on fossil fuels, which contribute to worldwide environmental issues such as global warming and climate change. As a result, renewable energy sources (RESs) are used for power generation where battery energy storage systems (BESSs) are widely used to store electrical energy for backup, match power consumption and generation during peak hours, and promote energy efficiency in a pollution-free environment. Accurate battery state of health (SOH) prediction is critical because it plays a key role in ensuring battery safety, lowering maintenance costs, and reducing BESS inconsistencies. The precise power consumption forecasting is critical for preventing power shortage and oversupply, and the complicated physicochemical features of batteries dilapidation cannot be directly acquired. Therefore, in this paper, a novel hybrid architecture called ‘CL-Net’ based on convolutional long short-term memory (ConvLSTM) and long short-term memory (LSTM) is proposed for multi-step SOH and power consumption forecasting. First, battery SOH and power consumption-related raw data are collected and passed through a preprocessing step for data cleansing. Second, the processed data are fed into ConvLSTM layers, which extract spatiotemporal features and form their encoded maps. Third, LSTM layers are used to decode the encoded features and pass them to fully connected layers for final multi-step forecasting. Finally, a comprehensive ablation study is conducted on several combinations of sequential learning models using three different time series datasets, i.e., national aeronautics and space administration (NASA) battery, individual household electric power consumption (IHEPC), and domestic energy management system (DEMS). The proposed CL-Net architecture reduces root mean squared error (RMSE) up to 0.13 and 0.0052 on the NASA battery and IHEPC datasets, respectively, compared to the state-of-the-arts. These experimental results show that the proposed architecture can provide robust and accurate SOH and power consumption forecasting compared to the state-of-the-art. Full article

This paper considers an approach towards the building of new classes of symmetric closed curves with two or more focal points, which can be obtained by generalizing classical definitions of the ellipse, Cassini, and Cayley ovals. A universal numerical method for creating such curves in mathematical packages is introduced. Specific aspects of the provided numerical data in computer-aided design systems with B-splines for three-dimensional modeling are considered. The applicability of the method is demonstrated, as well as the possibility to provide high smoothness of the curvature profile at the specified accuracy of modeling. Full article

This study investigated the stability and bifurcation of a nonlinear system model developed by Marhl et al. based on the total Ca²⁺ concentration among three different Ca²⁺ stores. In this study, qualitative theories of center manifold and bifurcation were used to analyze the stability of equilibria. The bifurcation parameter drove the system to undergo two supercritical bifurcations. It was hypothesized that the appearance and disappearance of Ca²⁺ oscillations are driven by them. At the same time, saddle-node bifurcation and torus bifurcation were also found in the process of exploring bifurcation. Finally, numerical simulation was carried out to determine the validity of the proposed approach by drawing bifurcation diagrams, time series, phase portraits, etc. Full article

Hepatitis delta virus (HDV) is an infectious subviral agent that can only propagate in people infected with hepatitis B virus (HBV). HDV/HBV infection is considered to be the most severe form of chronic viral hepatitis. In this contribution, a mathematical model for the interplay between HDV and HBV under anti-HDV treatment is presented. Previous models were not designed to account for the observation that HBV rises when HDV declines with HDV-specific therapy. In the simple model presented here, HDV and HBV kinetics are coupled, giving rise to an improved viral kinetic model that simulates the early interplay of HDV and HBV during anti-HDV therapy. Full article

We propose two Mann-type subgradient-like extra gradient iterations with the line-search procedure for hierarchical variational inequality (HVI) with the common fixed-point problem (CFPP) constraint of finite family of nonexpansive mappings and an asymptotically nonexpansive mapping in a real Hilbert space. Our methods include combinations of the Mann iteration method, subgradient extra gradient method with the line-search process, and viscosity approximation method. Under suitable assumptions, we obtain the strong convergence results of sequence of iterates generated by our methods for a solution to HVI with the CFPP constraint. Full article

This research study investigates the issue of finite-time passivity analysis of neutral-type neural networks with mixed time-varying delays. The time-varying delays are distributed, discrete and neutral in that the upper bounds for the delays are available. We are investigating the creation of sufficient conditions for finite boundness, finite-time stability and finite-time passivity, which has never been performed before. First, we create a new Lyapunov–Krasovskii functional, Peng–Park’s integral inequality, descriptor model transformation and zero equation use, and then we use Wirtinger’s integral inequality technique. New finite-time stability necessary conditions are constructed in terms of linear matrix inequalities in order to guarantee finite-time stability for the system. Finally, numerical examples are presented to demonstrate the result’s effectiveness. Moreover, our proposed criteria are less conservative than prior studies in terms of larger time-delay bounds. Full article

In this paper we study Hyers-Ulam stability of Euler’s equation in the calculus of variations in two special cases: when $F=F(x,y^{\prime })$ and when $F=F(y,y^{\prime }).$ For the first case we use the direct method and for the second case we use the Laplace transform. In the first Theorem and in the first Example the corresponding estimations for $\left|J\left[y\left(x\right)\right]-J\left[y_0\left(x\right)\right]\right|$ are given. We mention that it is the first time that the problem of Ulam-stability of extremals for functionals represented in integral form is studied. Full article

Machine learning has grown in popularity in recent years as a method for evaluating financial text data, with promising results in stock price projection from financial news. Various research has looked at the relationship between news events and stock prices, but there is little evidence on how different sentiments (negative, neutral, and positive) of such events impact the performance of stocks or indices in comparison to benchmark indices. The goal of this paper is to analyze how a specific banking news event (such as a fraud or a bank merger) and other co-related news events (such as government policies or national elections), as well as the framing of both the news event and news-event sentiment, impair the formation of the respective bank’s stock and the banking index, i.e., Bank Nifty, in Indian stock markets over time. The task is achieved through three phases. In the first phase, we extract the banking and other co-related news events from the pool of financial news. The news events are further categorized into negative, positive, and neutral sentiments in the second phase. This study covers the third phase of our research work, where we analyze the impact of news events concerning sentiments or linguistics in the price movement of the respective bank’s stock, identified or recognized from these news events, against benchmark index Bank Nifty and the banking index against benchmark index Nifty50 for the short to long term. For the short term, we analyzed the movement of banking stock or index to benchmark index in terms of CARs (cumulative abnormal returns) surrounding the publication day (termed as D) of the news event in the event windows of (−1,D), (D,1), (−1,1), (D,5), (−5,−1), and (−5,5). For the long term, we analyzed the movement of banking stock or index to benchmark index in the event windows of (D,30), (−30,−1), (−30,30), (D,60), (−60,−1), and (−60,60). We explore the deep learning model, bidirectional encoder representations from transformers, and statistical method CAPM for this research. Full article

The evaluation of corporation operation efficiency (especially innovation efficiency) has been always a hot topic. The currently popular evaluation methods are data envelopment analysis (DEA) and its improved methods. However, these methods have the following problems: the production process is regarded as a black box, and the actual production relationship between input and output is not analyzed. To solve these problems: (1) the black box theory and production function theory are introduced to uncover the black box of input and output; (2) regression models are used to alleviate the multicollinearity problem of inputs, and the most appropriate model of production relationship is selected; and (3) the results of the production function are compared with the results of the efficiency evaluation from multiple perspectives. Taking rural commercial banks in China as examples to evaluate their innovation efficiency, this article shows the following: (1) with the black box theory and production function theory, the staff, equipment, and intermediate business cost are suitable as innovation input variables, and intermediate business income is suitable as an innovation output variable; (2) the main challenges faced by rural commercial banks are reducing the reliance on human capital investment, strengthening technological innovation, and improving the efficiency of intermediate business cost management, which is hard to reveal with traditional DEA. The method proposed in this article provides an applicable reference for improving DEA method analysis. Full article

This paper is concerned with certain non-linear unperturbed and perturbed systems of integro-delay differential equations (IDDEs). We investigate fundamental properties of solutions such as uniformly stability (US), uniformly asymptotically stability (UAS), integrability and instability of the un-perturbed system of the IDDEs as well as the boundedness of the perturbed system of IDDEs. In this paper, five new and improved fundamental qualitative results, which have less conservative conditions, are obtained on the mentioned fundamental properties of solutions. The technique used in the proofs depends on Lyapunov-Krasovski functionals (LKFs). In particular cases, three examples and their numerical simulations are provided as numerical applications of this paper. This paper provides new, extensive and improved contributions to the theory of IDDEs. Full article

We present some solutions of the three-dimensional Laplace equation in terms of linear combinations of generalized hyperogeometric functions in prolate elliptic geometry, which simulates the current tokamak shapes. Such solutions are valid for particular parameter values. The derived solutions are compared with the solutions obtained in the standard toroidal geometry. Full article

Oscillations and oscillators appear in various fields and find applications in numerous areas. We present an oscillator with infinite equilibria in this work. The oscillator includes only nonlinear elements (quadratic, absolute, and cubic ones). It is different from common oscillators, in which there are linear elements. Special features of the oscillator are suitable for secure applications. The oscillator’s dynamics have been discovered via simulations and an electronic circuit. Chaotic attractors, bifurcation diagrams, Lyapunov exponents, and the boosting feature are presented while measurements of the implemented oscillator are reported by using an oscilloscope. We introduce a random number generator using such an oscillator, which is applied in biomedical image encryption. Moreover, the security and performance analysis are considered to confirm the correctness of encryption and decryption processes. Full article

Space-filling design selects points uniformly in the experimental space, bringing considerable flexibility to the complex-model-based and model-free data analysis. At present, space-filling designs mostly focus on regular spaces and continuous factors, with a lack of studies into the discrete factors and the constraints among factors. Most of the existing experimental design methods for qualitative factors are not applicable for discrete factors, since they ignore the potential order or spatial distance between discrete factors. This paper proposes a space-filling method, called maximum projection coordinate-exchange (MP-CE), taking into account both the diversity of factor types and the complexity of factor constraints. Specifically, the maximum projection criterion and distance criterion are introduced to capture the “bad” coordinates, and the coordinate-exchange and the optimization of experimental design are realized by solving one-dimensional constrained optimization problem. Meanwhile, by adding iterative perturbations to the traditional coordinate exchange process, the adjacent areas of the local optimal solution are explored and the optimum performances of the current optimal solution are retained, while the shortcomings of random restart are effectively avoided. Experiments in the regular space and constraint space, as well as experimental design for the terminal interception effectiveness of a missile defense system, show that the MP-CE method significantly outperforms existing popular space-filling design methods in terms of space-projection properties, while yielding comparable or superior space-filling properties. Full article

The CQ algorithm is widely used in the scientific field and has a significant impact on phase retrieval, medical image reconstruction, signal processing, etc. Moudafi proposed an alternating CQ algorithm to solve the split equality problem, but he only obtained the result of weak convergence. The work of this paper is to improve his algorithm so that the generated iterative sequence can converge strongly. Full article

Jensen-type inequalities for the recently introduced new class of $(h,g;m)$-convex functions are obtained, and certain special results are indicated. These results generalize and extend corresponding inequalities for the classes of convex functions that already exist in the literature. Schur-type inequalities are given. Full article

To date, it is known that tumor cells respond to attacks of T-cells by producing some PD-1/PD-L1 and other connections. Unfortunately, medical methods for preventing these connections are expensive and sometimes non-effective. In this study, we suggest a new way for reducing these connections by producing some noise in the exchanged information between tumor cells, T-cells, hemoglobin, and controller cells such as those of the heart or brain. In this model, we assume that human cells use spinor waves for exchanging information because the velocity of exchanged information between two spinors, which are located a large distance apart, exceeds the velocity of light. In fact, two spinors could send and receive information from each other instantaneously. In this hypothesis, the DNAs within heart cells, brain cells or any controller are built from some spinors such as electrons, and by their motion, some waves are generated. These spinor waves are received by iron atoms and multi-gonal molecules within hemoglobin and other spinors within the blood vessels. The hemoglobin molecules are located on some blood cells, move along the blood vessels and pass on their information to cells, proteins and RNAs. The spins of the spinors within the hemoglobin and also the spins of the charges and ions within the blood vessels are entangled and could transmit any information between cells. Thus, when a tumor is formed, its spinor waves change, and are transmitted rapidly into the heart cells, brain cells and other controller cells. The heart, brain or other controller cells diagnose these quantum waves, and by using the entanglement between the spinors within the blood vessels and the hemoglobin, send some messages to the T-cells. These messages are received by tumor cells and they become ready to respond to attacks. To prevent the reception of information by tumor cells, we can make use of some extra cells or hemoglobin, which interact with spinors and hemoglobin around tumor cells and produce some noise. Science quantum spinor waves are minute and have minor power and intensity; we cannot detect them by our present electronic devices and for this reason, we suggest using biological cells. This is a hypothesis; however, if experiments show its validity, some types of cancers could be cured or controlled by this method. We formulate the model by considering quantum entanglement between spinors within biological systems. By changing any spin within this system, all spins change and consequently, information is transmitted immediately. Then, we add new spinors to this system mathematically, and show that this causes the correlations between the initial spinors to reduce. Thus, the spinors of the extra hemoglobin or cells could act like noise, and prevent reception of real information by tumor cells. Full article

This article deals with issues related to the use of mathematical methods of cash deficit probability predictions. A number of objective and subjective factors are described that prevent the wide integration of mathematical methods in the practical activities of economists. It is justified that, due to the large number of external and internal factors affecting the economic system state, the values of indicators of an economic system state are often random. The possibility of using probability theory methods to predict the occurrence of cash deficits is proved. Using empirical data including the results of thousands of observations, the possibility of using the normal distribution density function for the purpose of predicting insufficient funds for payment is illustrated. The essence of the proposed model is that it contains a prediction of a macrotrend—i.e., the risk of a cash gap—based on high-frequency microlevel data. At the same time, a prediction of the probability of a cash deficit, and not its estimation for a specific date, was made. This is the main difference between the described model and common scoring estimates. This article proposes an approach to estimate the probability of a cash deficit based on data from a specific business entity, rather than aggregated data from other organizations. Full article

By the iteration of the theta-formula and treating the neighbors explicitly such as the unconditionally positive finite difference (UPFD) methods, we construct a new 2-stage explicit algorithm to solve partial differential equations containing a diffusion term and two reaction terms. One of the reaction terms is linear, which may describe heat convection, the other one is proportional to the fourth power of the variable, which can represent radiation. We analytically prove, for the linear case, that the order of accuracy of the method is two, and that it is unconditionally stable. We verify the method by reproducing an analytical solution with high accuracy. Then large systems with random parameters and discontinuous initial conditions are used to demonstrate that the new method is competitive against several other solvers, even if the nonlinear term is extremely large. Finally, we show that the new method can be adapted to the advection–diffusion-reaction term as well. Full article

Linking pensions to longevity developments at retirement age has been one of the most common policy responses of pension schemes to aging populations. The introduction of automatic stabilizers is primarily motivated by cost containment objectives, but there are other dimensions of welfare restructuring in the politics of pension reforms, including recalibration, rationalization, and blame avoidance for unpopular policies that involve retrenchments. This paper examines the policy designs and implications of linking entry pensions to life expectancy developments through sustainability factors or life expectancy coefficients in Finland, Portugal, and Spain. To address conceptual and specification uncertainty in policymaking, we propose and apply a Bayesian model averaging approach to stochastic mortality modeling and life expectancy computation. The results show that: (i) sustainability factors will generate substantial pension entitlement reductions in the three countries analyzed; (ii) the magnitude of the pension losses depends on the factor design; (iii) to offset pension cuts and safeguard pension adequacy, individuals will have to prolong their working lives significantly; (iv) factor designs considering cohort longevity markers would have generated higher pension cuts in countries with increasing life expectancy gap. Full article

In this paper, Clifford-valued fuzzy neural networks with proportional delays, whose leakage term coefficients are also Clifford numbers, are considered. Based on the Banach fixed point theorem and differential inequality technique, we use a direct method to obtain the existence, uniqueness, and global attractivity of pseudo almost periodic solutions for the considered networks. Finally, we provide a numerical example to illustrate the feasibility of our results. Our results are new. Full article

This paper proposes a class of time-delay fractional complex $L\ddot{u}$ system and utilizes the adomian decomposition algorithm to study the dynamics of the system. Firstly, the time chaotic attractor, coexistence attractor and parameter space are studied. The bifurcation diagram and complexity are used to analyze the dynamic characteristics of the system. Secondly, the definition of modified fractional projective difference function synchronization (MFPDFS) is introduced. The corresponding synchronous controller is designed to realize the MFPDFS of the time-delay fractional complex $L\ddot{u}$ system. Thirdly, based on the background of wireless speech communication system (WSCs), the MFPDFS controller is used to realize the secure speech transmission. Finally, the effectiveness of the controller is verified by numerical simulation. The signal-noise ratio (SNR) analysis of speech transmission is given. The performance of secure communication is verified by numerical simulation. Full article

This study relies on a theoretical perspective to provide a model that can be used to observe the educational practice of mathematics teachers. To this end, various existing observation instruments are studied, which, in the case of mathematics, aim to observe the teaching practice employed in the classroom, without considering the before and after of the implementation, which is what characterizes the professional task of teaching. Indicators that emerge from the teaching practice are proposed, together with the teacher’s knowledge and reflection constructs, based on three phases: for, in, and on the educational practice. As a result, understanding the educational practice of mathematics teachers would allow various educational stakeholders (teachers, administrators, instructors of teachers, and others) to focus their attention on what elements develop so as to improve how students are taught, and consequently learn, mathematics. Full article

The problems of synthesis and analysis of multidimensional controlled models of population dynamics are of both theoretical and applied interest. The need to solve numerical optimization problems for such a class of models is associated with the expansion of ecosystem control requirements. The need to solve the problem of stochastization is associated with the emergence of new problems in the study of ecological systems properties under the influence of random factors. The aim of the work is to develop a new approach to studying the properties of population dynamics systems using methods of numerical optimization, stochastization and machine learning. The synthesis problems of nonlinear three-dimensional models of interconnected species number dynamics, taking into account trophic chains and competition in prey populations, are studied. Theorems on the asymptotic stability of equilibrium states are proved. A qualitative and numerical study of the models is carried out. Using computational experiments, the results of an analytical stability and permanent coexistence study are verified. The search for equilibrium states belonging to the stability and permanent coexistence region is made using the developed intelligent algorithm and evolutionary calculations. The transition is made from the model specified by the vector ordinary differential equation to the corresponding stochastic model. A comparative analysis of deterministic and stochastic models with competition and trophic chains is carried out. New effects are revealed that are characteristic of three-dimensional models, taking into account the competition in populations of prey. The formulation of the optimal control problem for a model with competition and trophic chains is proposed. To find optimal trajectories, new generalized algorithms for numerical optimization are developed. A methods for the synthesis of controllers based on the use of artificial neural networks and machine learning are developed. The results on the search for optimal trajectories and generation of control functions are presented.The obtained results can be used in modeling problems of ecological, demographic, socio-economic and chemical kinetics systems. Full article

Recently, a quasi-fractional order gradient descent (QFGD) algorithm was proposed and successfully applied to solve system identification problem. The QFGD suffers from the overparameterization problem and results in estimating the redundant parameters instead of identifying only the actual parameters of the system. This study develops a novel hierarchical QFDS (HQFGD) algorithm by introducing the concepts of hierarchical identification principle and key term separation idea. The proposed HQFGD is effectively applied to solve the parameter estimation problem of input nonlinear autoregressive with exogeneous noise (INARX) system. A detailed investigation about the performance of HQFGD is conducted under different disturbance conditions considering different fractional orders and learning rate variations. The simulation results validate the better performance of the HQFGD over the standard counterpart in terms of estimation accuracy, convergence speed and robustness. Full article

This review paper combs through reports that have enhanced antenna gain for ultra-wideband (UWB) frequencies using frequency-selective surface (FSS) techniques. The FSS techniques found across the research landscape were mapped onto a taxonomy in order to determine the most effective method for improving antenna gain. Additionally, this study looked into the motivation behind using FSS as a reflector in UWB frequencies to obtain directional radiation. The FSS suits multiple applications due to its exceptional ability to minimize power loss in undesired transmission areas in the antenna, as well as to hinder the interference that may occur from undesirable and wasted radiation. An efficient way to obtain constant gain over a wide range of frequencies is also elaborated in this paper. Essentially, this paper offers viable prescription to enhance antenna gain for UWB applications. Methods: A comprehensive study was performed using several imminent keywords, such as “high gain using FSS”, “gain enhancement using FSS”, “high gain UWB antennas”, and “gain enhancement of UWB antennas”, in different modifications to retrieve all related articles from three primary engines: Web of Science (WoS), IEEE Xplore, and Science Direct. Results: The 41 papers identified after a comprehensive literature review were classified into two categories. The FSS single- and multi-layer reflectors were reported in 25 and 16 papers, respectively. New direction: An effective method is proposed for FSS miniaturization and for obtaining constant gain over UWB frequencies while maintaining the return loss at −10 dB. Conclusion: The use of FSS is indeed effective and viable for gain enhancement in UWB antennas. This systematic review unravels a vast range of opportunities for researchers to bridge the identified gaps. Full article