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This study deals with the one-parameter family {Dq}q∈[0,1]{\left\{{D}_{q}\right\}}_{q\in \left[0,1]} of Bernstein-type operators introduced by Gupta and called the limit qq-Durrmeyer operators. The continuity of this family with respect to the parameter qq is examined in two most important topologies of the operator theory, namely, the strong and uniform operator topologies. It is proved that {Dq}q∈[0,1]{\left\{{D}_{q}\right\}}_{q\in \left[0,1]} is continuous in the strong operator topology for all q∈[0,1]q\in \left[0,1]. When it comes to the uniform operator topology, the continuity is preserved solely at q=0q=0 and fails at all q∈(0,1].q\in \left(0,1]. In addition, a few estimates for the distance between two limit qq-Durrmeyer operators have been derived in the operator norm on C[0,1]C\left[0,1].

In this article, we consider the following Choquard equation with upper critical exponent: −Δu=μf(x)∣u∣p−2u+g(x)(Iα*(g∣u∣2α*))∣u∣2α*−2u,x∈Ω,-\Delta u=\mu f\left(x){| u| }^{p-2}u+g\left(x)({I}_{\alpha }* \left(g{| u| }^{{2}_{\alpha }^{* }})){| u| }^{{2}_{\alpha }^{* }-2}u,\hspace{1.0em}x\in \Omega , where μ>0\mu \gt 0 is a parameter, N>4N\gt 4, 0

Discussions on causal relations in real life often consider variables for which the definition of causality is unclear since the notion of interventions on the respective variables is obscure. Asking “what qualifies an action for being an intervention on the variable XX” raises the question whether the action impacted all other variables only through X or directly, which implicitly refers to a causal model. To avoid this known circularity, we instead suggest a notion of “phenomenological causality” whose basic concept is a set of elementary actions. Then the causal structure is defined such that elementary actions change only the causal mechanism at one node (e.g. one of the causal conditionals in the Markov factorization). This way, the principle of independent mechanisms becomes the defining property of causal structure in domains where causality is a more abstract phenomenon rather than being an objective fact relying on hard-wired causal links between tangible objects. In other words, causal relations between variables get defined by the interface between the system and an external agent (who is able to perform the elementary actions), rather than being an internal property of links between the variables. We describe this phenomenological approach to causality for toy and hypothetical real-world examples and argue that it is consistent with the causal Markov condition when the system under consideration interacts with other variables that control the elementary actions.

Bias in causal comparisons has a correspondence with distributional imbalance of covariates between treatment groups. Weighting strategies such as inverse propensity score weighting attempt to mitigate bias by either modeling the treatment assignment mechanism or balancing specified covariate moments. This article introduces a new weighting method, called energy balancing, which instead aims to balance weighted covariate distributions. By directly targeting distributional imbalance, the proposed weighting strategy can be flexibly utilized in a wide variety of causal analyses without the need for careful model or moment specification. Our energy balancing weights (EBW) approach has several advantages over existing weighting techniques. First, it offers a model-free and robust approach for obtaining covariate balance that does not require tuning parameters, obviating the need for modeling decisions of secondary nature to the scientific question at hand. Second, since this approach is based on a genuine measure of distributional balance, it provides a means for assessing the balance induced by a given set of weights for a given dataset. We demonstrate the effectiveness of this EBW approach in a suite of simulation experiments, and in studies on the safety of right heart catheterization and on three additional studies using electronic health record data.

Conditional effect estimation has great scientific and policy importance because interventions may impact subjects differently depending on their characteristics. Most research has focused on estimating the conditional average treatment effect (CATE). However, identification of the CATE requires that all subjects have a non-zero probability of receiving treatment, or positivity, which may be unrealistic in practice. Instead, we propose conditional effects based on incremental propensity score interventions, which are stochastic interventions where the odds of treatment are multiplied by some factor. These effects do not require positivity for identification and can be better suited for modeling scenarios in which people cannot be forced into treatment. We develop a projection approach and a flexible nonparametric estimator that can each estimate all the conditional effects we propose and derive model-agnostic error guarantees showing that both estimators satisfy a form of double robustness. Further, we propose a summary of treatment effect heterogeneity and a test for any effect heterogeneity based on the variance of a conditional derivative effect and derive a nonparametric estimator that also satisfies a form of double robustness. Finally, we demonstrate our estimators by analyzing the effect of intensive care unit admission on mortality using a dataset from the (SPOT)light study.

In this article, we first put forward the concept of deferred f-double natural density for double sequences, where f is an unbounded modulus. Then, we combine f-density with deferred statistical convergence for double sequences and investigate deferred f-statistical convergence and strongly deferred Cesàro summability with respect to modulus f. Moreover, we extend these concepts to deferred f-statistical convergence for double sequences of random variables in the Wijsman sense and prove some inclusions. Finally, we consider the concepts of deferred f-statistical convergence of order α\alpha and strongly deferred f-summability of order α\alpha for double sequences and obtain some conclusions.

This research aims to look at the impact and learning outcomes of using LMS and online learning platforms, so it is necessary to analyze learning outcomes and how they impact students, lecturers, and learning organizing institutions. This research shows that the LMS and online learning platforms used can improve student learning outcomes, which cannot be separated from the readiness of lecturers and students to implement it. This research used factor analysis (CFA), with a total of 150 respondents, students, and 15 lecturers. The impact provided by the use of LMS and online platforms in online learning is in the form of GPA and students' views on the learning process they have gone through. From the research results, several findings were obtained, including the ability to use LMS in online learning is not optimal, the GPA of learning outcomes is relatively high, and the readiness of lecturers and students to use LMS in online learning is still low.

Abstract The concept of the Fan product of several M-matrices is introduced. Furthermore, two new lower bounds of the minimum eigenvalue of the Fan product of several M-matrices are proposed. These obtained new lower bounds generalize and improve some earlier findings. One example is presented to illustrate the precision of the given lower bounds.

Abstract This manuscript focuses on a class of stochastic functional differential equations driven by time-changed Brownian motion. By utilizing the Lyapunov method, we capture some sufficient conditions to ensure that the solution for the considered equation is η-stable in the pth moment sense. Subsequently, we present some new criteria of the η-stability in mean square by using time-changed Itô formula and proof by contradiction. Finally, we provide some examples to demonstrate the effectiveness of our main results.

In the era of society 5.0, the reach of student learning resources is increasingly wider, with the internet and free AI-based search engines. The use of music during learning is a way for students to increase learning motivation. This research aims to find out: (1) Whether ChatGPT technology and music used during independent study have an impact on students' Mathematics Self-Regulated Learning (MSRL) and Mathematics Anxiety (MA); (2) Whether MSRL and MA have an association with the study program students choose; and (3) Whether MSRL and MA have an association with students' academic achievement. This research uses a correlational descriptive research method. The data collection technique uses a survey, implementing Google Forms. The respondents of this research were students at several universities in Indonesia. The research results show a significant difference in MSRL between students who use ChatGPT and students who do not use ChatGPT during independent learning. However, there was no significant difference in MSRL between students who listened to music and those who did not listen to music during independent learning. There was no significant difference in MA between students who used ChatGPT and those who did not use ChatGPT during independent study. There was no significant difference in MA between students who listened to music and those who did not listen to music during independent study. There is a significant association between MSRL and the origin of the student's Study Program, but there is no significant association between MA and the origin of the Study Program. There is no significant association between MSRL and Academic Achievement. There is no association between MA and students' Academic Achievement.

Addressing the boundary value problems of fractional-order differential equations hold significant importance due to their applications in various fields. The aim of this paper was to approximate solutions for a class of boundary value problems involving Caputo fractional-order differential equations employing the AA-iterative scheme. Moreover, the stability and data dependence results of the iterative scheme were given for a certain class of mappings. Finally, a numerical experiment was illustrated to support the results presented herein. The results presented in this paper extend and unify some well-known comparable results in the existing literature.

Marburg virus disease poses a significant risk to global health, impacting both humans and non-human primates. This study has yielded an optimal control model for potentially mitigating the transmission of the Marburg infection. The proposed mathematical model includes fractional-order derivatives in the Caputo sense. Initially, we analyzed the model without control measures, examining its key characteristics regarding local and global stabilities. Subsequently, we extended the model by incorporating suitable time-dependent optimal control variables. We have also introduced two time-dependent control measures: $ \Psi_1 $ for the prevention of human-to-human Marburg transmission, and $ \Psi_2 $ to enhance the rate of quarantine of exposed individuals. We performed simulation analysis for both cases i.e., with and without optimal controls using the two-step Newton polynomial approximation method, considering both fractional and classical orders. The numerical findings of the comparative study between classical and fractional cases validate the biological significance of the fractional operator and effectiveness of the proposed optimal control strategies.

In this paper, we proposed some new integral inequalities for subadditive functions and the product of subadditive functions. Additionally, a novel integral identity was established and a number of inequalities of the Hermite-Hadamard type for subadditive functions pertinent to tempered fractional integrals were proved. Finally, to support the major results, we provided several examples of subadditive functions and corresponding graphs for the newly proposed inequalities.

Metal magnetic memory (MMM) is an innovative, nondestructive testing method. It can detect both stress concentrations and macroscopic defects. The three-dimensional force-magnetic coupling model was established by the ANSYS simulation software, the evolution process of different defect depths was studied in detail, and the change of the signal characteristic was analyzed. The results showed that the variation trend and amplitude characteristic of MMM signals resulted in obvious differences among different defect types. Meanwhile, the impacts caused by the defect parameters and the type are complex, which cannot be decoupled or calculated by a certain formula. The accuracy of the simulation data was verified by experiments. To solve the classification prediction problem in MMM detection, the signal peak and valley Hp-v, the signal width W, the gradient Ky, and the peak energy Hy were selected as characteristic parameters to evaluate different defect types according to the change in the signal waveform. Finally, using these vectors as the input variables, the radial basis function neural network (RBFNN) pre-classification test model was established to realize the classification recognition of pit defects, crack defects, and porosity defects. The results show that the accuracy of the training and test sets, and it is feasible to use this model to complete the intelligent classification of defects.

This research investigated the fixed-time (FXT) synchronization of fractional-order fuzzy cellular neural networks (FCNNs) with delays and interactions based on an enhanced FXT stability theorem. By conceiving proper Lyapunov functions and applying inequality techniques, several sufficient conditions were obtained to vouch for the fixed-time synchronization (FXTS) of the discussed systems through two categories of control schemes. Moreover, in terms of another FXT stability theorem, different upper-bounding estimating formulas for settling time (ST) were given, and the distinctions between them were pointed out. Two examples were delivered at length to demonstrate the conclusions.

In this paper, we investigate the numerical solution of the Brusselator system using a meshless method. A numerical scheme is derived employing the formulas of the Generalized Finite Difference Method, and the convergence of the approximate solution to the exact solution is examined. In order to demonstrate the applicability and accuracy of the method, several examples are proposed.

Delayed dynamical system plays a vital role in describing the dynamical phenomenon of neural networks. In this article, we proposed a class of new BAM neural networks involving time delay. The traits of solution and bifurcation behavior of the established BAM neural networks involving time delay were probed into. First, the existence and uniqueness is discussed using a fixed point theorem. Second, the boundedness of solution of the formulated BAM neural networks involving time delay was analyzed by applying an appropriate function and inequality techniques. Third, the stability peculiarity and bifurcation behavior of the addressed delayed BAM neural networks were investigated. Fourth, Hopf bifurcation control theme of the formulated delayed BAM neural networks was explored by virtue of a hybrid controller. By adjusting the parameters of the controller, we could control the stability domain and Hopf bifurcation onset, which was in favor of balancing the states of different neurons in engineering. To verify the correctness of gained major outcomes, computer simulations were performed. The acquired outcomes of this article were new and own enormous theoretical meaning in designing and dominating neural networks.

In this research, we have proposed and investigated a time-delayed free boundary problem concerning tumor growth in the presence of almost periodic nutrient supply with angiogenesis. This study primarily focused on examining the impact of almost periodic nutrient supply, angiogenesis, and time delay on tumor growth dynamics. We analyzed the existence, uniqueness, and exponential stability of almost periodic solutions. Furthermore, we established conditions for the disappearance of almost periodic oscillations in tumors. The existence and uniqueness of almost periodic solutions were proven, while sufficient conditions for the exponential stability of the unique solution were established. Finally, computer simulations were employed to illustrate our results.

This article dealt with further properties of the Renyi entropy and the residual Renyi entropy of $ k $-record values. First, we discussed the Renyi entropy order and its connection with the usual stochastic and dispersive orders. We then addressed the monotonicity properties of the residual Renyi entropy of $ k $-records, focusing on the aging properties of the component lifetimes. We also expressed the residual $ n $th upper $ k $-records in terms of Renyi entropy when the first dataset exceeded a certain threshold, and then studied various properties of the given formula. Finally, we conducted a parametric estimation of the Renyi entropy of the $ n $th upper $ k $-records. The estimation was performed using both real COVID-19 data and simulated data.

Typically, in high dimensional data sets, many covariates are not significantly associated with a response. Moreover, those covariates are highly correlated, leading to a multicollinearity problem. Hence, the model is sparse since the coefficient of most covariates are likely to be zero. The classical frequentist or likelihood-based variable selection via any criterion such as Bayesian Information Criteria (BIC) and Akaike Information Criteria (AIC) or a stepwise subset selection becomes infeasible when the number of variables are large. An alternative solution is a Bayesian variable selection. In this study, we used a variable selection via a Bayesian variable selection and the least absolute shrinkage and selection operator (LASSO) method in the logistic regression model. Moreover, those methods were expanded to be applied to real datasets.

Automatic determination of abnormal animal activities can be helpful for the timely detection of signs of health and welfare problems. Usually, this problem is addressed as a classification problem, which typically requires manual annotation of behaviors. This manual annotation can introduce noise into the data and may not always be possible. This motivated us to address the problem as a time-series forecasting problem in which the activity of an animal can be predicted. In this work, different machine learning techniques were tested to obtain activity patterns for Iberian pigs. In particular, we propose a novel stacking ensemble learning approach that combines base learners with meta-learners to obtain the final predictive model. Results confirm the superior performance of the proposed method relative to the other tested strategies. We also explored the possibility of using predictive models trained on an animal to predict the activity of different animals on the same farm. As expected, the predictive performance degrades in this case, but it remains acceptable. The proposed method could be integrated into a monitoring system that may have the potential to transform the way farm animals are monitored, improving their health and welfare conditions, for example, by allowing the early detection of a possible health problem.

In this study, we present a comprehensive framework for enhancing the temperature control of electric furnaces, integrating three novel components: a proportional-integral-derivative controller with a filter (PID-F), a customized objective function, and a modified electric eel foraging optimization (mEEFO) algorithm. The PID-F controller, introduced for the first time in the literature for temperature control of electric furnaces, leverages a filter coefficient to effectively mitigate the kick effect, improving transient and frequency responses. To further optimize the PID-F controller, we employed the mEEFO, a recently proposed metaheuristic inspired by the social predation behaviors of electric eels, with tailored modifications for electric furnace temperature control. The study also introduces a new objective function, based on the modification of the integral of absolute error (IAE) performance index. The proposed framework was evaluated through extensive comparisons with established metaheuristic algorithms, including statistical analysis, Wilcoxon signed-rank test, and time and frequency domain analyses. Comparative assessments with reported methods, such as genetic algorithms and Ziegler–Nichols-based PID controllers, validated the efficacy of the proposed approach, highlighting its transformative impact on electric furnace temperature regulation. The non-ideal conditions such as measurement noise, external disturbance, and saturation at the output of the controller were also evaluated in order to demonstrate the superior performance of the proposed approach from a wider perspective. Furthermore, the robustness of the proposed approach against variations in system parameters was also demonstrated.

Polycyclic aromatic hydrocarbon (PAH) is a compound composed of carbon and hydrogen atoms. Chemically, large PAHs contain at least two benzene rings and exist in a linear, cluster, or angular arrangement. Hexagonal systems are a typical class of PAHs. The Clar covering polynomial of hexagonal systems contains many important topological properties of condensed aromatic hydrocarbons, such as Kekulé number, Clar number, first Herndon number, which is an important theoretical quantity for predicting the aromatic stability of PAH conjugation systems, and so on. In this paper, we first obtained some recursive formulae for the Clar covering polynomials of double hexagonal chains and proposed a Matlab algorithm to compute the Clar covering polynomial of any double hexagonal chain. Moreover, we presented the characterization of extremal double hexagonal chains with maximum and minimum Clar covering polynomials in all double hexagonal chains with fixed $ s $ naphthalenes.

The iterated function system (IFS) is important in different fields like image compression. An important feature of such systems is that they can be used to generate fractals. Yet, for the obtained fractals, it is difficult to locally control them to generate new ones with desired structures at specific places. In this paper, we gave an attempt to solve this problem based on a nonuniform multiple function system. For this, we first analyzed the multiple function systems needed in the generation of the final desired fractals. Based on such analysis, the final fractals with desired structures at specific places can be generated using the nonuniform multiple function system. Moreover, these two procedures were summarized into two algorithms for convenience. Examples were also given to illustrate the performance of the nonuniform multiple function system and the two algorithms in this paper.

We adopted a non decomposition method to study the existence and stability of Stepanov almost periodic solutions in the distribution sense of stochastic shunting inhibitory cellular neural networks (SICNNs) with mixed time delays. Due to the lack of linear structure in the set composed of Stepanov almost periodic stochastic processes in the distribution sense. Due to the lack of linear structure in the set composed of distributed Stepanov periodic stochastic processes, it poses difficulties for the existence of Stepanov almost periodic solutions in the distribution sense of SICNNs. To overcome this difficulty, we first proved that the network under consideration has a unique solution in a space composed of $ \mathcal{L}^p $ bounded and $ \mathcal{L}^p $ uniformly continuous stochastic processes. Then, using stochastic analysis, inequality techniques, and the definition of Stepanov almost periodic stochastic processes in the distribution sense, we proved that this solution is also a Stepanov almost periodic solution in the distribution sense. Moreover, the result of the global exponential stability of this almost periodic solution is given. It is worth noting that even if the network under consideration degenerated into a real-valued network, our results are novel. Finally, we provided a numerical example to validate our theoretical findings.

This paper investigates the complex dynamical behavior of a discrete prey-predator system with a fear factor, a strong Allee effect, and prey refuge. The existence and stability of fixed points in the system are discussed. By applying the central manifold theorem and bifurcation theory, we have established the occurrence of various types of bifurcations, including flip bifurcation and Neimark-Sacker bifurcation. Furthermore, to address the observed chaotic behavior in the system, three controllers were designed by employing state feedback control, OGY feedback control, and hybrid control methods. These controllers serve to control chaos in the proposed system and identify specific conditions under which chaos or bifurcations can be stabilized. Finally, the theoretical analyses have been validated through numerical simulations.

The main objective of our research was to explore and develop a fractional-order derivative within the predator-prey framework. The framework includes prey refuge and selective nonlinear harvesting, where the harvesting progressively approaches a threshold value as the density of the harvested population advances. For memory effect, a non-integer order derivative is better than an integer-order derivative. The solutions to the fractional framework were shown to be existence, uniqueness, non-negativity, and boundedness. Matignon's condition was used for analysing local stability, and a suitable Lyapunov function provided global stability. While discussing the Hopf bifurcation's existence condition, we explored derivative order and refuge as bifurcation parameters. We aimed at redefining the predator-prey framework to incorporate fractional order, refuge, and harvesting. This kind of nonlinear harvesting is more realistic and reasonable than the model with constant yield harvesting and constant effort harvesting. The Adams-Bashforth-Moulton PECE algorithm in MATLAB software was used to simulate the proposed outcomes, investigate the impact on various factors, and analyse harvesting's effect on non-integer order predator-prey interactions.

In this work, we focus our attention on the existence of nontrivial solutions to the following supercritical Schrödinger-Poisson type system with $ (p, q) $-Laplacian: $ \begin{equation*} \begin{cases} -\Delta_{p}u-\Delta_{q}u+\phi|u|^{q-2} u = f\left(x, u\right)+\mu|u|^{s-2} u & \text { in } \Omega, \\ -\Delta \phi = |u|^q & \text { in } \Omega, \\ u = \phi = 0 & \text { on } \partial \Omega, \end{cases} \end{equation*} $ where $ \Omega \subset \mathbb{R}^N $ is a bounded smooth domain, $ \mu > 0, N > 1 $, and $ -\Delta_{{\wp}}\varphi = div(|\nabla\varphi|^{{\wp}-2} \nabla\varphi) $, with $ {\wp}\in \{p, q\} $, is the homogeneous $ {\wp} $-Laplacian. $ 1 < p < q < \frac{q^*}{2} $, $ q^*: = \frac{Nq}{N-q} < s $, and $ q^* $ is the critical exponent to $ q $. The proof is accomplished by the Moser iterative method, the mountain pass theorem, and the truncation technique. Furthermore, the $ (p, q) $-Laplacian and the supercritical term appear simultaneously, which is the main innovation and difficulty of this paper.

The fundamental problem of finding transmission schedules for achieving optimal throughput in time division multiple access (TDMA) communication networks is known to be NP-hard. Let $ \mathcal{N} $ be a scheduled $ k $-time slot TDMA network with $ n $ stations and $ m $ links. We showed that an optimal link schedule for $ \mathcal{N} $ can be computed recursively with a recursion tree of logarithmic depth $ \mathcal{O}(\ln m) $ in expectation. Additionally, we showed that optimal link schedules for those TDMA networks, with recursion trees of depth meeting the expectation, can be found in time $ \mathcal{O}(m^{2+\ln k}) $. Likewise, we discuss analogous results for computing optimal station schedules of TDMA networks.

This paper presents the generalized Pell $ p- $numbers and provides some related results. A new sequence is defined using the characteristic polynomial of the Pell $ p- $numbers and generalized Mersenne numbers. Two algorithms for Diffie-Hellman key exchange are given as an application of these sequences. They are illustrated via numerical examples and shown to be secure against attacks. Thus, these new sequences are practical for encryption and constructing private keys.

Previous research has highlighted the challenges of improving students' mathematical skills, particularly in connecting mathematical concepts to everyday life and developing strong mathematical habits of mind. This research aims to integrate problem-based learning (PBL) into a mathematics module to enhance mathematical connection abilities and mathematical habits of mind. The ADDIE method was employed to achieve this goal, encompassing five stages: Analysis, Design, Development, Implementation, and Evaluation. The findings indicate that media, material, and language experts deemed the module highly suitable. Furthermore, the product was rated as highly practical based on teacher assessments, student responses, and observations during implementation. Effectiveness tests of the product, conducted through MANOVA, t-tests, and N-gain tests, revealed that the module is highly effective in enhancing students' mathematical connection abilities and moderately effective in improving their mathematical habits of mind. These findings underscore the importance of integrating digital learning technologies to increase the module's engagement and accessibility.

We focus on the following fractional (p, q)-Choquard problem: (−Δ)psu+(−Δ)qsu+V(εx)(|u|p−2u+|u|q−2u)=1|x|μ*F(u)f(u) in RN,u∈Ws,p(RN)∩Ws,q(RN),u>0 in RN, $\begin{cases}{\left(-{\Delta}\right)}_{p}^{s}u+{\left(-{\Delta}\right)}_{q}^{s}u+V\left(\varepsilon x\right)\left(\vert u{\vert }^{p-2}u+\vert u{\vert }^{q-2}u\right)=\left(\frac{1}{\vert x{\vert }^{\mu }}{\ast}F\left(u\right)\right)f\left(u\right) \,\text{in}\,{\mathbb{R}}^{N},\quad \hfill \\ u\in {W}^{s,p}\left({\mathbb{R}}^{N}\right)\cap {W}^{s,q}\left({\mathbb{R}}^{N}\right), u{ >}0\,\text{in}\,{\mathbb{R}}^{N},\quad \hfill \end{cases}$ where ɛ > 0 is a small parameter, 0 < s < 1, 1

This work investigates the asymptotic behavior of energy solutions to the focusing nonlinear Schrödinger equation of Choquard type i∂tu+Δu+∣u∣p−2(Iα*∣u∣p)u=0,p=1+2+αN−2,N≥3.i{\partial }_{t}u+\Delta u+{| u| }^{p-2}\left({I}_{\alpha }* {| u| }^{p})u=0,\hspace{1.0em}p=1+\frac{2+\alpha }{N-2},\hspace{1.0em}N\ge 3. Indeed, in the energy-critical spherically symmetric regime, one proves a global existence and scattering versus finite time blow-up dichotomy. Precisely, if the data have an energy less than the ground state one, two cases are possible. If the kinetic energy of the radial data is less than the ground state one, then the solution is global and scatters. Otherwise, if the data have a finite variance or is spherically symmetric and have a finite mass, then the solution is nonglobal. The main difficulty is to deal with the nonlocal source term. The argument is the concentration-compactness-rigidity method introduced by Kenig and Merle (Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math. 166 (2006), no. 3, 645–675). This note naturally complements the work by Saanouni (Scattering theory for a class of defocusing energy-critical Choquard equations, J. Evol. Equ. 21 (2021), 1551–1571), where the scattering of the defocusing energy-critical generalized Hartree equation was obtained.

Over any GCD (greatest common divisor) commutative domain we show that the nontrivial 2×22\times 2 idempotent matrices are products of two nilpotent matrices. In order to find explicitly such decompositions, two procedures are described. Assisted by computer, we were able to find an example of idempotent 2×22\times 2 matrix over Z[−5]{\mathbb{Z}}\left[\sqrt{-5}], which shows that the GCD condition is (sufficient but) not necessary. Finally, a generalization is discussed and some open questions stated.

Abstract In this study, we introduce a method for finding common fixed points of a finite family of ( η i , k i ) $(\eta _{i}, k_{i})$ -enriched strictly pseudocontractive (ESPC) maps and ( η i , β i ) $(\eta _{i}, \beta _{i})$ -enriched strictly pseudononspreading (ESPN) maps in the setting of real Hilbert spaces. Further, we prove the strong convergence theorem of the proposed method under mild conditions on the control parameters. Our main results are also applied in proving strong convergence theorems for η i $\eta _{i}$ -enriched nonexpansive, strongly inverse monotone, and strictly pseudononspreading maps. Some nontrivial examples are given, and the results obtained extend, improve, and generalize several well-known results in the current literature.

Abstract The objective of this research is to present new fixed point theorems for two separate families of fuzzy-dominated mappings. These mappings must satisfy a unique locally contraction in a complete b-multiplicative metric space. Also, we have obtained novel results for families of fuzzy-dominated mappings on a closed ball that meet the requirements of a generalized locally contraction. This research introduces new and challenging fixed-point problems for families of ordered fuzzy-dominated mappings in ordered complete b-multiplicative metric spaces. Moreover, we demonstrate a new concept for families of fuzzy graph-dominated mappings on a closed ball in these spaces. Additionally, we present novel findings for graphic contraction endowed with graphic structure. These findings are groundbreaking and provide a strong foundation for future research in this field. To demonstrate the uniqueness of our novel findings, we provide evidence of their applicability in obtaining the common solution of integral and fractional differential equations. Our findings have resulted in modifications to several contemporary and classical results in the research literature. This provides further evidence of the originality and impact of our work.

Abstract In this paper, we introduce new types of contraction mappings named S-Pata-type contraction mapping and Generalized S-Pata-type contraction mapping in the framework of S-metric space. Then, we prove some new fixed-point results for S-Pata-type contraction mappings and Generalized S-Pata-type contraction mappings. To support our results, we provide examples to illustrate our findings and also apply these results to the ordinary differential equation to strengthen our conclusions.

The process of identifying the optimal unknown variables for the creation of a precision fuel-cell performance forecasting model using optimization techniques is known as parameter identification of the proton exchange membrane fuel cell (PEMFC). Recognizing these factors is crucial for accurately forecasting and assessing the fuel cell's performance, as they may not always be included in the manufacturer's datasheet. Six optimization algorithms—the Walrus Optimizer (WO), the Tunicate Swarm Algorithm (TSA), the Harris Hawks Optimizer (HHO), the Heap Based Optimizer (HBO), the Chimp Optimization Algorithm (ChOA), and the Osprey Optimization Algorithm (OOA) were used to compute six unknown variables of a PEMFC. Also, the proposed WO method was compared with other published works' methods such as the Equilibrium Optimizer (EO), Manta Rays Foraging Optimizer (MRFO), Neural Network Algorithm (NNA), Artificial Ecosystem Optimizer (AEO), Slap Swarm Optimizer (SSO), and Vortex Search Approach with Differential Evolution (VSDE). Minimizing the sum squares error (SSE) between the estimated and measured cell voltages requires treating these six parameters as choice variables during optimization. The WO algorithm yielded an SSE of 1.945415603, followed by HBO, HHO, TSA, ChOA, and OOA. Given that WO accurately forecasted the fuel cell's performance, it is appropriate for the development of digital twins for fuel cell applications and control systems for the automobile industry. Furthermore, it was shown that the WO convergence speed was faster than the other approaches studied.

Finite-time synchronization is a critical problem in the study of neural networks. The primary objective of this study was to construct feedback controllers for various models based on fuzzy shunting inhibitory cellular neural networks (FSICNNs) and find out the sufficient conditions for the solutions of those systems to reach synchronization in finite time. In particular, by imposing global assumptions of Lipschitz continuous and bounded activation functions, we prove the existence of finite-time synchronization for three basic FSICNN models that have not been studied before. Moreover, we suggest both controllers and Lyapunov functions that would yield a feasible convergence time between solutions that takes into account the chosen initial conditions. In general, we consecutively explore models of regular delayed FSICNNs and then consider them in the presence of either inertial or diffusion terms. Using criteria derived by means of the maximum-value approach in its different forms, we give an upper bound of the time up to which synchronization is guaranteed to occur in all three FSICNN models. These results are supported by 2D and 3D computer simulations and two respective numerical examples for $ 2\times 2 $ and $ 2\times 3 $ cases, which show the behavior of the solutions and errors under different initial conditions of FSICNNs in the presence and absence of designed controllers.

Joule dissipation has an important role in the conversion of mechanical energy to heat within a fluid due to the internal friction and viscosity. Moreover, Darcy friction is a measure of the resistance to flow in a porous medium. In response to the efficient heat transfer performance, a robust statistical approach was established to optimize the heat transfer rate in a two-dimensional flow of a nanofluid over a permeable surface embedded with a porous matrix. The electrically conducive fluid affected the flow phenomena to include a carbon nanotube nanoparticle in the conventional liquid water for the enhanced heat transfer properties; additionally, the power-law heat flux model was considered. Appropriate transformation rules were adopted to obtain a non-dimensional system that brought a developed model equipped with several factors. The traditional numerical technique (i.e., shooting based Runge-Kutta) was proposed to handle the coupled nonlinear system. Furthermore, the statistical response surface methodology (RSM) was adopted to obtain an efficient optimized model for the heat transportation rate of the considered factors. An analysis of variance (ANOVA) was utilized to validate the result of the regression analysis. However, it was evident that the nanoparticle concentrations were useful to augment the fluid velocity and the temperature distributions; the statistical approach adopted for the heat transfer rate displayed an optimized effect as compared to a conventional effect.

In this article, the Montgomery identity and Ostrowski inequality are established for univariate first-order diamond-alpha differentiable functions. We also investigate the generalization of Ostrowski-type inequalities for bivariate functions with bounded second-order diamond-alpha derivatives by applying integration by parts for $ \diamondsuit_\alpha $-integrals. Moreover, some extensions of dynamic trapezoid- and Grüss-type inequalities are also obtained by using the Montgomery identity.

In order to approximate the common solution of quasi-nonexpansive fixed point and pseudo-monotone variational inequality problems in real Hilbert spaces, this paper presented three new modified sub-gradient extragradient-type methods. Our algorithms incorporated viscosity terms and double inertial extrapolations to ensure strong convergence and to speed up convergence. No line search methods of the Armijo type were required by our algorithms. Instead, they employed a novel self-adaptive step size technique that produced a non-monotonic sequence of step sizes while also correctly incorporating a number of well-known step sizes. The step size was designed to lessen the algorithms' reliance on the initial step size. Numerical tests were performed, and the results showed that our step size is more effective and that it guarantees that our methods require less execution time. We stated and proved the strong convergence of our algorithms under mild conditions imposed on the control parameters. To show the computational advantage of the suggested methods over some well-known methods in the literature, several numerical experiments were provided. To test the applicability and efficiencies of our methods in solving real-world problems, we utilized the proposed methods to solve optimal control and image restoration problems.

We have studied initial value problems for Caputo fractional differential equations with singular nonlinearities involving the p-Laplacian operator. We have given a precise mathematical analysis of the equivalence of the fractional differential equations and Volterra integral equations studied in this paper. A theorem for the global existence of the solution was proven. In addition, an example was given at the end of the article as an application of the results found in this paper.

We considered a swelling porous-elastic system characterized by two nonlinear variable exponent damping and logarithmic source terms. Employing the Faedo-Galerkin method, we established the local existence of weak solutions under suitable assumptions on the variable exponents functions. Furthermore, we proved the global existence utilizing the well-depth method. Finally, we established several decay results by employing the multiplier method and the Logarithmic Sobolev inequality. To the best of our knowledge, this represents the first study addressing swelling systems with logarithmic source terms.

In an active magnetic bearings (AMBs) model, every pair of opposite poles is aligned at an angle with the horizontal axis. In some configurations, there is a pair of poles which is in line with the horizontal axis. In other configurations, the same pair of poles might make a nonzero angle with the horizontal axis. This paper focused on the effect of changing such a configuration angle on the control process of an oscillatory rotor in an 8-pole active magnetic bearings model. Adopting the proportional-derivative (PD) control algorithm, the radial or Cartesian control techniques were applied. It was found that the rotor's oscillation amplitudes were not affected by the change in the configuration angle, even if its rotation speed and eccentricity were varied in the radial control scheme. However, the amplitudes were severely affected by the change in the configuration angle except at a specific angle in the Cartesian control scheme. The approximate modulating amplitudes and phases of the rotor's oscillations were extracted by the method of multiple-scales and a stability condition was tested based on the eigenvalues of the corresponding Jacobian matrix.

A new class of nonlocal boundary value problems consisting of multi-term delay fractional differential equations and multipoint-integral boundary conditions is studied in this paper. We derive a more general form of the solution for the given problem by applying a fractional integral operator of an arbitrary order $ \beta_{\xi} $ instead of $ \beta_{1} $; for details, see Lemma 2. The given problem is converted into an equivalent fixed-point problem to apply the tools of fixed-point theory. The existence of solutions for the given problem is established through the use of a nonlinear alternative of the Leray-Schauder theorem, while the uniqueness of its solutions is shown with the aid of Banach's fixed-point theorem. We also discuss the stability criteria, icluding Ulam-Hyers, generalized Ulam-Hyers, Ulam-Hyers-Rassias, and generalized Ulam-Hyers-Rassias stability, for solutions of the problem at hand. For illustration of the abstract results, we present examples. Our results are new and useful for the discipline of multi-term fractional differential equations related to hydrodynamics. The paper concludes with some interesting observations.

Our main purpose of this article was using the analytic methods and the properties of Dirichlet $ L $-functions to study the properties of Dedekind sums and give a new reciprocity formula for it. As its applications, some exact calculating formula for one kind mean square value of Dirichlet $ L $-fuctions with the weight of the character sums were obtained.

In this paper, we propose a way to partition any constacyclic code over a finite field in its equivalence classes according to the algebraic structure of the code. Such a method gives the generalization of cyclically permutable codes (CPCs), which are called constacyclically permutable codes (CCPCs), and it is useful to derive a CCPC from a given constacyclic code. Moreover, we present an enumerative formula for the code size of such a CCPC, with all of the terms being positive integers, and we provide an algebraic method to produce such a CCPC.

In this paper, we studied the existence of solutions for a coupled system of nonlinear sequential proportional $ \psi $-Hilfer fractional differential equations with multi-point boundary conditions. By using a Burton's version of the Krasnosel'ski$\breve{{\rm{i}}}$'s fixed-point theorem we established sufficient conditions for the existence result. An example illustrating our main result was also provided.