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This study introduces an enhanced self-adaptive wild goose algorithm (SAWGA) for solving economical-environmental-technical optimal power flow (OPF) problems in traditional and modern energy systems. Leveraging adaptive search strategies and robust diversity capabilities, SAWGA distinguishes itself from classical WGA by incorporating four potent optimizers. The algorithm's application to optimize an OPF model on the different IEEE 30-bus and 118-bus electrical networks, featuring conventional thermal power units alongside solar photovoltaic (PV) and wind power (WT) units, addresses the rising uncertainties in operating conditions, particularly with the integration of renewable energy sources (RESs). The inherent complexity of OPF problems in electrical networks, exacerbated by the inclusion of RESs like PV and WT units, poses significant challenges. Traditional optimization algorithms struggle due to the problem's high complexity, susceptibility to local optima, and numerous continuous and discrete decision parameters. The study's simulation results underscore the efficacy of SAWGA in achieving optimal solutions for OPF, notably reducing overall fuel consumption costs in a faster and more efficient convergence. Noteworthy attributes of SAWGA include its remarkable capabilities in optimizing various objective functions, effective management of OPF challenges, and consistent outperformance compared to traditional WGA and other modern algorithms. The method exhibits a robust ability to achieve global or nearly global optimal settings for decision parameters, emphasizing its superiority in total cost reduction and rapid convergence. [ABSTRACT FROM AUTHOR]

In this paper, with motivation from the No Free Lunch theorem, a new human-based metaheuristic algorithm named Preschool Education Optimization Algorithm (PEOA) is introduced for solving optimization problems. Human activities in the preschool education process are the fundamental inspiration in the design of PEOA. Hence, PEOA is mathematically modeled in three phases: (i) the gradual growth of the preschool teacher's educational influence, (ii) individual knowledge development guided by the teacher, and (iii) individual increase of knowledge and self-awareness. The PEOA's performance in optimization is evaluated using fifty-two standard benchmark functions encompassing unimodal, high-dimensional multimodal, and fixed-dimensional multimodal types, as well as the CEC 2017 test suite. The optimization results show that PEOA has a high ability in exploration–exploitation and can balance them during the search process. To provide a comprehensive analysis, the performance of PEOA is compared against ten well-known metaheuristic algorithms. The simulation results show that the proposed PEOA approach performs better than competing algorithms by providing effective solutions for the benchmark functions and overall ranking as the first-best optimizer. Presenting a statistical analysis of the Wilcoxon signed-rank test shows that PEOA has significant statistical superiority in competition with compared algorithms. Furthermore, the implementation of PEOA in solving twenty-two optimization problems from the CEC 2011 test suite and four engineering design problems illustrates its efficacy in real-world optimization applications. [ABSTRACT FROM AUTHOR]

This paper uses enhanced turbulent flow in water-based optimization (TFWO), specifically ETFWO, to achieve optimal power flow (OPF) in electrical networks that use both solar photovoltaic (PV) units and wind turbines (WTs). ETFWO is an enhanced TFWO that alters the TFWO structure through the promotion of communication and collaboration. Individuals in the population now interact with each other more often, which makes it possible to search more accurately in the search area while ignoring local optimal solutions. Probabilistic models and real-time data on wind speed and solar irradiance are used to predict the power output of WT and PV producers. The OPF and solution methods are evaluated using the IEEE 30-bus network. By comparing ETFWO to analogical other optimization techniques applied to the same groups of constraints, control variables, and system data, we can gauge the algorithm's robustness and efficiency in solving OPF. It is shown in this paper that the proposed ETFWO algorithm can provide suitable solutions to OPF problems in electrical networks with integrated PV units and WTs in terms of energy generation costs, improved voltage profiles, emissions, and losses, compared to the traditional TFWO and other proposed algorithms in recent studies. [ABSTRACT FROM AUTHOR]

This paper uses enhanced turbulent flow in water-based optimization (TFWO), specifically ETFWO, to achieve optimal power flow (OPF) in electrical networks that use both solar photovoltaic (PV) units and wind turbines (WTs). ETFWO is an enhanced TFWO that alters the TFWO structure through the promotion of communication and collaboration. Individuals in the population now interact with each other more often, which makes it possible to search more accurately in the search area while ignoring local optimal solutions. Probabilistic models and real-time data on wind speed and solar irradiance are used to predict the power output of WT and PV producers. The OPF and solution methods are evaluated using the IEEE 30-bus network. By comparing ETFWO to analogical other optimization techniques applied to the same groups of constraints, control variables, and system data, we can gauge the algorithm's robustness and efficiency in solving OPF. It is shown in this paper that the proposed ETFWO algorithm can provide suitable solutions to OPF problems in electrical networks with integrated PV units and WTs in terms of energy generation costs, improved voltage profiles, emissions, and losses, compared to the traditional TFWO and other proposed algorithms in recent studies. [ABSTRACT FROM AUTHOR]

This article's innovation and novelty are introducing a new metaheuristic method called mother optimization algorithm (MOA) that mimics the human interaction between a mother and her children. The real inspiration of MOA is to simulate the mother's care of children in three phases education, advice, and upbringing. The mathematical model of MOA used in the search process and exploration is presented. The performance of MOA is assessed on a set of 52 benchmark functions, including unimodal and high-dimensional multimodal functions, fixed-dimensional multimodal functions, and the CEC 2017 test suite. The findings of optimizing unimodal functions indicate MOA's high ability in local search and exploitation. The findings of optimization of high-dimensional multimodal functions indicate the high ability of MOA in global search and exploration. The findings of optimization of fixed-dimension multi-model functions and the CEC 2017 test suite show that MOA with a high ability to balance exploration and exploitation effectively supports the search process and can generate appropriate solutions for optimization problems. The outcomes quality obtained from MOA has been compared with the performance of 12 often-used metaheuristic algorithms. Upon analysis and comparison of the simulation results, it was found that the proposed MOA outperforms competing algorithms with superior and significantly more competitive performance. Precisely, the proposed MOA delivers better results in most objective functions. Furthermore, the application of MOA on four engineering design problems demonstrates the efficacy of the proposed approach in solving real-world optimization problems. The findings of the statistical analysis from the Wilcoxon signed-rank test show that MOA has a significant statistical superiority compared to the twelve well-known metaheuristic algorithms in managing the optimization problems studied in this paper. [ABSTRACT FROM AUTHOR]

This paper presents a new evolutionary-based approach called a Subtraction-Average-Based Optimizer (SABO) for solving optimization problems. The fundamental inspiration of the proposed SABO is to use the subtraction average of searcher agents to update the position of population members in the search space. The different steps of the SABO's implementation are described and then mathematically modeled for optimization tasks. The performance of the proposed SABO approach is tested for the optimization of fifty-two standard benchmark functions, consisting of unimodal, high-dimensional multimodal, and fixed-dimensional multimodal types, and the CEC 2017 test suite. The optimization results show that the proposed SABO approach effectively solves the optimization problems by balancing the exploration and exploitation in the search process of the problem-solving space. The results of the SABO are compared with the performance of twelve well-known metaheuristic algorithms. The analysis of the simulation results shows that the proposed SABO approach provides superior results for most of the benchmark functions. Furthermore, it provides a much more competitive and outstanding performance than its competitor algorithms. Additionally, the proposed approach is implemented for four engineering design problems to evaluate the SABO in handling optimization tasks for real-world applications. The optimization results show that the proposed SABO approach can solve for real-world applications and provides more optimal designs than its competitor algorithms. [ABSTRACT FROM AUTHOR]

This paper introduces a new bio-inspired metaheuristic algorithm called Walrus Optimization Algorithm (WaOA), which mimics walrus behaviors in nature. The fundamental inspirations employed in WaOA design are the process of feeding, migrating, escaping, and fighting predators. The WaOA implementation steps are mathematically modeled in three phases exploration, migration, and exploitation. Sixty-eight standard benchmark functions consisting of unimodal, high-dimensional multimodal, fixed-dimensional multimodal, CEC 2015 test suite, and CEC 2017 test suite are employed to evaluate WaOA performance in optimization applications. The optimization results of unimodal functions indicate the exploitation ability of WaOA, the optimization results of multimodal functions indicate the exploration ability of WaOA, and the optimization results of CEC 2015 and CEC 2017 test suites indicate the high ability of WaOA in balancing exploration and exploitation during the search process. The performance of WaOA is compared with the results of ten well-known metaheuristic algorithms. The results of the simulations demonstrate that WaOA, due to its excellent ability to balance exploration and exploitation, and its capacity to deliver superior results for most of the benchmark functions, has exhibited a remarkably competitive and superior performance in contrast to other comparable algorithms. In addition, the use of WaOA in addressing four design engineering issues and twenty-two real-world optimization problems from the CEC 2011 test suite demonstrates the apparent effectiveness of WaOA in real-world applications. The MATLAB codes of WaOA are available in https://uk.mathworks.com/matlabcentral/profile/authors/13903104. [ABSTRACT FROM AUTHOR]

A new metaheuristic algorithm called green anaconda optimization (GAO) which imitates the natural behavior of green anacondas has been designed. The fundamental inspiration for GAO is the mechanism of recognizing the position of the female species by the male species during the mating season and the hunting strategy of green anacondas. GAO's mathematical modeling is presented based on the simulation of these two strategies of green anacondas in two phases of exploration and exploitation. The effectiveness of the proposed GAO approach in solving optimization problems is evaluated on twenty-nine objective functions from the CEC 2017 test suite and the CEC 2019 test suite. The efficiency of GAO in providing solutions for optimization problems is compared with the performance of twelve well-known metaheuristic algorithms. The simulation results show that the proposed GAO approach has a high capability in exploration, exploitation, and creating a balance between them and performs better compared to competitor algorithms. In addition, the implementation of GAO on twenty-one optimization problems from the CEC 2011 test suite indicates the effective capability of the proposed approach in handling real-world applications. [ABSTRACT FROM AUTHOR]

This article introduces a new metaheuristic algorithm called the Serval Optimization Algorithm (SOA), which imitates the natural behavior of serval in nature. The fundamental inspiration of SOA is the serval's hunting strategy, which attacks the selected prey and then hunts the prey in a chasing process. The steps of SOA implementation in two phases of exploration and exploitation are mathematically modeled. The capability of SOA in solving optimization problems is challenged in the optimization of thirty-nine standard benchmark functions from the CEC 2017 test suite and CEC 2019 test suite. The proposed SOA approach is compared with the performance of twelve well-known metaheuristic algorithms to evaluate further. The optimization results show that the proposed SOA approach, due to the appropriate balancing exploration and exploitation, is provided better solutions for most of the mentioned benchmark functions and has superior performance compared to competing algorithms. SOA implementation on the CEC 2011 test suite and four engineering design challenges shows the high efficiency of the proposed approach in handling real-world optimization applications. [ABSTRACT FROM AUTHOR]

Let p, q, k and ℓ be positive integers. The (p , q , k , ℓ) -Fibonacci sequence (F k , ℓ , p , q) n ≥ 0 is the four-parameter sequence defined by the following recurrence F k , ℓ , p , q (n) = k F k , ℓ , p , q (n - p) + ℓ F k , ℓ , p , q (n - q) , with appropriate initial conditions. In this paper, we study the geometric, algebraic, and analytic aspects of the roots of the characteristic polynomial of this sequence, namely, f (x) = x q - k x q - p - ℓ . [ABSTRACT FROM AUTHOR]

In this paper, a new optimization algorithm called hybrid leader-based optimization (HLBO) is introduced that is applicable in optimization challenges. The main idea of HLBO is to guide the algorithm population under the guidance of a hybrid leader. The stages of HLBO are modeled mathematically in two phases of exploration and exploitation. The efficiency of HLBO in optimization is tested by finding solutions to twenty-three standard benchmark functions of different types of unimodal and multimodal. The optimization results of unimodal functions indicate the high exploitation ability of HLBO in local search for better convergence to global optimal, while the optimization results of multimodal functions show the high exploration ability of HLBO in global search to accurately scan different areas of search space. In addition, the performance of HLBO on solving IEEE CEC 2017 benchmark functions including thirty objective functions is evaluated. The optimization results show the efficiency of HLBO in handling complex objective functions. The quality of the results obtained from HLBO is compared with the results of ten well-known algorithms. The simulation results show the superiority of HLBO in convergence to the global solution as well as the passage of optimally localized areas of the search space compared to ten competing algorithms. In addition, the implementation of HLBO on four engineering design issues demonstrates the applicability of HLBO in real-world problem solving. [ABSTRACT FROM AUTHOR]

Let p_n be the nth prime number. In 2000, Papaitopol proved that the inequality p_1\cdots p_n>p_{n+1}^{n-\pi (n)} holds, for all n\geq 2, where \pi (x) is the prime counting function. In 2021, Yang and Liao tried to sharpen this inequality by replacing n-\pi (n) by n-\pi (n)+\pi (n)/\pi (\log n)-2\pi (\pi (n)), however there is a small mistake in their argument. In this paper, we exploit properties of the logarithm error term in inequalities of the type p_1\cdots p_n>p_{n+1}^{k(n,x)}, where k(n,x)=n-\pi (n)+\pi (n)/\pi (\log n)-x\pi (\pi (n)). In particular, we improve Yang and Liao estimate, by showing that the previous inequality at x=1.4 holds for all n\geq 21. [ABSTRACT FROM AUTHOR]

In this paper, a new optimization algorithm called hybrid leader-based optimization (HLBO) is introduced that is applicable in optimization challenges. The main idea of HLBO is to guide the algorithm population under the guidance of a hybrid leader. The stages of HLBO are modeled mathematically in two phases of exploration and exploitation. The efficiency of HLBO in optimization is tested by finding solutions to twenty-three standard benchmark functions of different types of unimodal and multimodal. The optimization results of unimodal functions indicate the high exploitation ability of HLBO in local search for better convergence to global optimal, while the optimization results of multimodal functions show the high exploration ability of HLBO in global search to accurately scan different areas of search space. In addition, the performance of HLBO on solving IEEE CEC 2017 benchmark functions including thirty objective functions is evaluated. The optimization results show the efficiency of HLBO in handling complex objective functions. The quality of the results obtained from HLBO is compared with the results of ten well-known algorithms. The simulation results show the superiority of HLBO in convergence to the global solution as well as the passage of optimally localized areas of the search space compared to ten competing algorithms. In addition, the implementation of HLBO on four engineering design issues demonstrates the applicability of HLBO in real-world problem solving. [ABSTRACT FROM AUTHOR]

Let (F n) n be the Fibonacci sequence defined by F n + 2 = F n + 1 + F n with F 0 = 0 and F 1 = 1 . In this paper, we prove that for any integer m ≥ 1 there exists a positive constant C m for which lim n → ∞ { (∑ k = n ∞ 1 F m k 2) − 1 − (F m n 2 − F m (n − 1) 2 + (− 1) m n C m) } = 0. Furthermore, we show that C m tends to 2 / 5 as m → ∞ (indeed, we provide quantitative versions of the previous results as well as an explicit form for C m ). This confirms some questions proposed by Lee and Park [J. Inequal. Appl. 2020(1):91 2020]. [ABSTRACT FROM AUTHOR]

In 2021, Mork and Ulness studied the Mandelbrot and Julia sets for a generalization of the well-explored function hl(z) = z2 +l. Their generalization was based on the composition of hl with the Möbius transformation m(z) = 1z at each iteration step. Furthermore, they posed a conjecture providing a relation between the coefficients of (each order) iterated series of m(hl(z)) (at z = 0) and the Catalan numbers. In this paper, in particular, we prove this conjecture in a more precise (quantitative) formulation. [ABSTRACT FROM AUTHOR]

We, the Editor, Institution and Publisher of the journal IETE Journal of Research, have retracted the following article, which is part of the Special Issue titled "Federated Learning for Blockchain Systems and Industrial Internet of Things": Mohammad Dehghani, Pavel Trojovský, Štěpán Hubálovský, Theyab R. Alsenani & Jaswinder Singh (2022) Three-Periods Optimization Algorithm: A New Method for Solving Various Optimization Problems, IETE Journal of Research, DOI: Following publication, concerns were raised about the peer review and decision-making processes for this special issue. After an investigation by the Taylor & Francis Publishing Ethics & Integrity team, in full cooperation with the Editor-in-Chief and the Institution, it was confirmed that the articles included in this special issue were not reviewed appropriately, in line with the Journal's peer review standards and policy. As the stringency of the peer review process is core to the integrity of the publication process, the Editor and Publisher have decided to retract all of the articles within the above-named Special Issue. The journal has not confirmed if the authors were aware of this compromised peer review process. The journal is committed to correcting the scientific record and will fully cooperate with any institutional investigations into this matter. The authors have been informed of this decision. We have been informed in our decision-making by our editorial policies and the COPE guidelines. The retracted articles will remain online to maintain the scholarly record, but they will be digitally watermarked on each page as "Retracted". [ABSTRACT FROM AUTHOR]

Recently, Bednarz introduced a new two-parameter generalization of the Fibonacci sequence, which is called the (k , p) -Fibonacci sequence and denoted by (F k , p (n)) n ≥ 0 . In this paper, we study the geometry of roots of the characteristic polynomial of this sequence. [ABSTRACT FROM AUTHOR]

Let (F n) n ≥ 0 be the Fibonacci sequence. The order of appearance z (n) of a positive integer n is defined as z (n) : = min { k ≥ 1 : n ∣ F k } . In 2013, Marques proved that lim inf n → ∞ z (n) / n = 0 . Let ϵ be a positive real number. In this paper, in particular, we generalized this Marques' result by proving that almost all positive integers satisfy z (n) / n < ϵ . [ABSTRACT FROM AUTHOR]

This paper introduces the Botox Optimization Algorithm (BOA), a novel metaheuristic inspired by the Botox operation mechanism. The algorithm is designed to address optimization problems, utilizing a human-based approach. Taking cues from Botox procedures, where defects are targeted and treated to enhance beauty, the BOA is formulated and mathematically modeled. Evaluation on the CEC 2017 test suite showcases the BOA's ability to balance exploration and exploitation, delivering competitive solutions. Comparative analysis against twelve well-known metaheuristic algorithms demonstrates the BOA's superior performance across various benchmark functions, with statistically significant advantages. Moreover, application to constrained optimization problems from the CEC 2011 test suite highlights the BOA's effectiveness in real-world optimization tasks. [ABSTRACT FROM AUTHOR]

This paper is devoted to a generalization of some previous results, as we completely solve a linear homogeneous difference equation of the second order with an exponential coefficient. [ABSTRACT FROM AUTHOR]

• The Transductive Long Short Term Memory model is proposed. • The proposed result indicates that the T -LSTM is better than previous studies. • The T -LSTM method produces 98.2% accuracy in forecasting the weather. • This proposed hybrid T -LSTM method provides a robust solution. Forecasting climate and the development of the environment have been essential in recent days since there has been a drastic change in nature. Weather forecasting plays a significant role in decision-making in traffic management, tourism planning, crop cultivation in agriculture, and warning the people nearby the seaside about the climate situation. It is used to reduce accidents and congestion, mainly based on climate conditions such as rainfall, air condition, and other environmental factors. Accurate weather prediction models are required by meteorological scientists. The previous studies have shown complexity in terms of model building, and computation, and based on theory-driven and rely on time and space. This drawback can be easily solved using the machine learning technique with the time series data. This paper proposes the state-of-art deep learning model Long Short-Term Memory (LSTM) and the Transductive Long Short-Term Memory (T -LSTM) model. The model is evaluated using the evaluation metrics root mean squared error, loss, and mean absolute error. The experiments are carried out on HHWD and Jena Climate datasets. The dataset comprises 14 weather forecasting features including humidity, temperature, etc. The T -LSTM method performs better than other methodologies, producing 98.2% accuracy in forecasting the weather. This proposed hybrid T -LSTM method provides a robust solution for the hydrological variables. [ABSTRACT FROM AUTHOR]

This study proposes the One-to-One-Based Optimizer (OOBO), a new optimization technique for solving optimization problems in various scientific areas. The key idea in designing the suggested OOBO is to effectively use the knowledge of all members in the process of updating the algorithm population while preventing the algorithm from relying on specific members of the population. We use a one-to-one correspondence between the two sets of population members and the members selected as guides to increase the involvement of all population members in the update process. Each population member is chosen just once as a guide and is only utilized to update another member of the population in this one-to-one interaction. The proposed OOBO's performance in optimization is evaluated with fifty-two objective functions, encompassing unimodal, high-dimensional multimodal, and fixed-dimensional multimodal types, and the CEC 2017 test suite. The optimization results highlight the remarkable capacity of OOBO to strike a balance between exploration and exploitation within the problem-solving space during the search process. The quality of the optimization results achieved using the proposed OOBO is evaluated by comparing them to eight well-known algorithms. The simulation findings show that OOBO outperforms the other algorithms in addressing optimization problems and can give more acceptable quasi-optimal solutions. Also, the implementation of OOBO in six engineering problems shows the effectiveness of the proposed approach in solving real-world optimization applications. [ABSTRACT FROM AUTHOR]

For k ≥ 2, the k-generalized Fibonacci sequence ( F ) is defined by the initial values 0 ,0 ,..., 0 , 1 ( k terms) so that each term afterward is the sum of the k preceding terms. In this paper, we prove that the only solution of the Diophantine equation F = k = k with t > 1 and m > k+ 1 ≥ 4 is F = 3. [ABSTRACT FROM AUTHOR]

With the advancement of science and technology, new complex optimization problems have emerged, and the achievement of optimal solutions has become increasingly important. Many of these problems have features and difficulties such as non-convex, nonlinear, discrete search space, and a non-differentiable objective function. Achieving the optimal solution to such problems has become a major challenge. To address this challenge and provide a solution to deal with the complexities and difficulties of optimization applications, a new stochastic-based optimization algorithm is proposed in this study. Optimization algorithms are a type of stochastic approach for addressing optimization issues that use random scanning of the search space to produce quasi-optimal answers. The Selecting Some Variables to Update-Based Algorithm (SSVUBA) is a new optimization algorithm developed in this study to handle optimization issues in various fields. The suggested algorithm's key principles are to make better use of the information provided by different members of the population and to adjust the number of variables used to update the algorithm population during the iterations of the algorithm. The theory of the proposed SSVUBA is described, and then its mathematical model is offered for use in solving optimization issues. Fifty-three objective functions, including unimodal, multimodal, and CEC 2017 test functions, are utilized to assess the ability and usefulness of the proposed SSVUBA in addressing optimization issues. SSVUBA's performance in optimizing real-world applications is evaluated on four engineering design issues. Furthermore, the performance of SSVUBA in optimization was compared to the performance of eight well-known algorithms to further evaluate its quality. The simulation results reveal that the proposed SSVUBA has a significant ability to handle various optimization issues and that it outperforms other competitor algorithms by giving appropriate quasi-optimal solutions that are closer to the global optima. [ABSTRACT FROM AUTHOR]

Optimization is an important and fundamental challenge to solve optimization problems in different scientific disciplines. In this paper, a new stochastic nature-inspired optimization algorithm called Pelican Optimization Algorithm (POA) is introduced. The main idea in designing the proposed POA is simulation of the natural behavior of pelicans during hunting. In POA, search agents are pelicans that search for food sources. The mathematical model of the POA is presented for use in solving optimization issues. The performance of POA is evaluated on twenty-three objective functions of different unimodal and multimodal types. The optimization results of unimodal functions show the high exploitation ability of POA to approach the optimal solution while the optimization results of multimodal functions indicate the high ability of POA exploration to find the main optimal area of the search space. Moreover, four engineering design issues are employed for estimating the efficacy of the POA in optimizing real-world applications. The findings of POA are compared with eight well-known metaheuristic algorithms to assess its competence in optimization. The simulation results and their analysis show that POA has a better and more competitive performance via striking a proportional balance between exploration and exploitation compared to eight competitor algorithms in providing optimal solutions for optimization problems. [ABSTRACT FROM AUTHOR]

The order of appearance (in the Fibonacci sequence) function z : Z ≥ 1 → Z ≥ 1 is an arithmetic function defined for a positive integer n as z (n) = min { k ≥ 1 : F k ≡ 0 (mod n) } . A topic of great interest is to study the Diophantine properties of this function. In 1992, Sun and Sun showed that Fermat's Last Theorem is related to the solubility of the functional equation z (n) = z (n 2) , where n is a prime number. In addition, in 2014, Luca and Pomerance proved that z (n) = z (n + 1) has infinitely many solutions. In this paper, we provide some results related to these facts. In particular, we prove that lim sup n → ∞ (z (n + 1) − z (n)) / (log n) 2 − ϵ = ∞ , for all ϵ ∈ (0 , 2) . [ABSTRACT FROM AUTHOR]

The current population worldwide extensively uses social media to share thoughts, societal issues, and personal concerns. Social media can be viewed as an intelligent platform that can be augmented with a capability to analyze and predict various issues such as business needs, environmental needs, election trends (polls), governmental needs, etc. This has motivated us to initiate a comprehensive search of the COVID-19 pandemic-related views and opinions amongst the population on Twitter. The basic training data have been collected from Twitter posts. On this basis, we have developed research involving ensemble deep learning techniques to reach a better prediction of the future evolutions of views in Twitter when compared to previous works that do the same. First, feature extraction is performed through an N-gram stacked autoencoder supervised learning algorithm. The extracted features are then involved in a classification and prediction involving an ensemble fusion scheme of selected machine learning techniques such as decision tree (DT), support vector machine (SVM), random forest (RF), and K-nearest neighbour (KNN). all individual results are combined/fused for a better prediction by using both mean and mode techniques. Our proposed scheme of an N-gram stacked encoder integrated in an ensemble machine learning scheme outperforms all the other existing competing techniques such unigram autoencoder, bigram autoencoder, etc. Our experimental results have been obtained from a comprehensive evaluation involving a dataset extracted from open-source data available from Twitter that were filtered by using the keywords "covid", "covid19", "coronavirus", "covid-19", "sarscov2", and "covid_19". [ABSTRACT FROM AUTHOR]

In this paper, a new stochastic optimization algorithm is introduced, called Driving Training-Based Optimization (DTBO), which mimics the human activity of driving training. The fundamental inspiration behind the DTBO design is the learning process to drive in the driving school and the training of the driving instructor. DTBO is mathematically modeled in three phases: (1) training by the driving instructor, (2) patterning of students from instructor skills, and (3) practice. The performance of DTBO in optimization is evaluated on a set of 53 standard objective functions of unimodal, high-dimensional multimodal, fixed-dimensional multimodal, and IEEE CEC2017 test functions types. The optimization results show that DTBO has been able to provide appropriate solutions to optimization problems by maintaining a proper balance between exploration and exploitation. The performance quality of DTBO is compared with the results of 11 well-known algorithms. The simulation results show that DTBO performs better compared to 11 competitor algorithms and is more efficient in optimization applications. [ABSTRACT FROM AUTHOR]

For r ≥ 2 and a ≥ 1 integers, let (t n (r , a)) n ≥ 1 be the sequence of the (r , a) -generalized Fibonacci numbers which is defined by the recurrence t n (r , a) = t n − 1 (r , a) + ⋯ + t n − r (r , a) for n > r , with initial values t i (r , a) = 1 , for all i ∈ [ 1 , r − 1 ] and t r (r , a) = a . In this paper, we shall prove (in particular) that, for any given r ≥ 2 , there exists a positive proportion of positive integers which can not be written as t n (r , a) for any (n , a) ∈ Z ≥ r + 2 × Z ≥ 1 . [ABSTRACT FROM AUTHOR]

Let (F n) n ≥ 0 be the Fibonacci sequence. The order of appearance function (in the Fibonacci sequence) z : Z ≥ 1 → Z ≥ 1 is defined as z (n) : = min { k ≥ 1 : F k ≡ 0 (mod n) } . In this paper, among other things, we prove that z (n) is an even number for almost all positive integers n (i.e., the set of such n has natural density equal to 1). [ABSTRACT FROM AUTHOR]

Abstract: Some new identities for the Fibonomial coefficients are derived. These identities are related to the generating function of the kth powers of the Fibonacci numbers. Proofs are based on manipulation with the generating function of the sequence of “signed Fibonomial triangle”. [Copyright &y& Elsevier]

Population-based optimization algorithms are one of the most widely used and popular methods in solving optimization problems. In this paper, a new population-based optimization algorithm called the Teamwork Optimization Algorithm (TOA) is presented to solve various optimization problems. The main idea in designing the TOA is to simulate the teamwork behaviors of the members of a team in order to achieve their desired goal. The TOA is mathematically modeled for usability in solving optimization problems. The capability of the TOA in solving optimization problems is evaluated on a set of twenty-three standard objective functions. Additionally, the performance of the proposed TOA is compared with eight well-known optimization algorithms in providing a suitable quasi-optimal solution. The results of optimization of objective functions indicate the ability of the TOA to solve various optimization problems. Analysis and comparison of the simulation results of the optimization algorithms show that the proposed TOA is superior and far more competitive than the eight compared algorithms. [ABSTRACT FROM AUTHOR]

Let (t n (r)) n ≥ 0 be the sequence of the generalized Fibonacci number of order r, which is defined by the recurrence t n (r) = t n − 1 (r) + ⋯ + t n − r (r) for n ≥ r , with initial values t 0 (r) = 0 and t i (r) = 1 , for all 1 ≤ i ≤ r . In 2002, Grossman and Luca searched for terms of the sequence (t n (2)) n , which are expressible as a sum of factorials. In this paper, we continue this program by proving that, for any ℓ ≥ 1 , there exists an effectively computable constant C = C (ℓ) > 0 (only depending on ℓ), such that, if (m , n , r) is a solution of t m (r) = n ! + (n + 1) ! + ⋯ + (n + ℓ) ! , with r even, then max { m , n , r } < C . As an application, we solve the previous equation for all 1 ≤ ℓ ≤ 5 . [ABSTRACT FROM AUTHOR]

This study presents a powerful and new modeling variant of bio-inspired algorithms, namely, the Ivy algorithm (IVYA), drawn from the growth patterns of Ivy plants. The algorithm simulates the coordinated and ordered population growth and the spreading and evolution of Ivy plants. The growth rate of Ivy plants is modeled using a differential equation and a data-intensive experimental process. The algorithm utilizes the knowledge of nearby Ivy plants to determine the direction of growth. Additionally, the algorithm mimics the behavior of Ivy plants in nature by choosing the closest and most vital neighbor for self-improvement. The IVYA's unique characteristics of preserving population diversity and its simplicity and flexibility allow for easy modification and extension, thus enabling researchers and practitioners to explore various modifications and techniques to enhance its performance and capabilities. These are the basic needs in optimizing engineering problems. The IVYA is compared with ten other algorithms on 26 classical test functions, demonstrating superior performance. Furthermore, the effectiveness of IVYA is shown by solving 12 engineering optimization problems and comparing the results with various optimization algorithms. The experimental results highlight the efficacy and competitiveness of the IVYA in solving optimization problems. Consider that the source code of the IVYA is publicly available at https://www.optim-app.com/projects/ivya. [ABSTRACT FROM AUTHOR]

We give three consequences of Schanuel's Conjecture. The first is that P (e) Q (e) and P (π) Q (π) are transcendental, for any non-constant polynomials P (x) , Q (x) ∈ Q ¯ [ x ] . The second is that π ≠ α β , for any algebraic numbers α and β. The third is the case of the Gelfond's conjecture (about the transcendence of a finite algebraic power tower) in which all elements are equal. [ABSTRACT FROM AUTHOR]

For any integer k ≥ 2 , the sequence of the k-generalized Fibonacci numbers (or k-bonacci numbers) is defined by the k initial values F − (k − 2) (k) = ⋯ = F 0 (k) = 0 and F 1 (k) = 1 and such that each term afterwards is the sum of the k preceding ones. In this paper, we search for repdigits (i.e., a number whose decimal expansion is of the form a a ... a , with a ∈ [ 1 , 9 ] ) in the sequence (F n (k) F n (k + m)) n , for m ∈ [ 1 , 9 ] . This result generalizes a recent work of Bednařík and Trojovská (the case in which (k , m) = (2 , 1) ). Our main tools are the transcendental method (for Diophantine equations) together with the theory of continued fractions (reduction method). [ABSTRACT FROM AUTHOR]

In this paper, a new stochastic optimization algorithm is introduced, called Driving Training-Based Optimization (DTBO), which mimics the human activity of driving training. The fundamental inspiration behind the DTBO design is the learning process to drive in the driving school and the training of the driving instructor. DTBO is mathematically modeled in three phases: (1) training by the driving instructor, (2) patterning of students from instructor skills, and (3) practice. The performance of DTBO in optimization is evaluated on a set of 53 standard objective functions of unimodal, high-dimensional multimodal, fixed-dimensional multimodal, and IEEE CEC2017 test functions types. The optimization results show that DTBO has been able to provide appropriate solutions to optimization problems by maintaining a proper balance between exploration and exploitation. The performance quality of DTBO is compared with the results of 11 well-known algorithms. The simulation results show that DTBO performs better compared to 11 competitor algorithms and is more efficient in optimization applications. [ABSTRACT FROM AUTHOR]

In this paper, we use Baker's theory for nonzero linear forms in logarithms of algebraic numbers and a Baker-Davenport reduction procedure to find all repdigits (i.e., numbers with only one distinct digit in its decimal expansion, thus they can be seen as the easiest case of palindromic numbers, which are a "symmetrical" type of numbers) that can be written in the form F n + T n , for some n ≥ 1 , where (F n) n ≥ 0 and (T n) n ≥ 0 are the sequences of Fibonacci and Tribonacci numbers, respectively. [ABSTRACT FROM AUTHOR]

In 2006, A. Stakhov introduced a new coding/decoding process based on generating matrices of the Fibonacci p-numbers, which he called the Fibonacci coding/decoding method. Stakhov's papers have motivated many other scientists to seek certain generalizations by introducing new additional coefficients into recurrence of Fibonacci p-numbers. In 2013, I. Włoch et al. studied (2 , q) -distance Fibonacci numbers F 2 (q , n) and found some of their combinatorial properties. In this paper, we state a new coding theory based on the sequence (T q (n)) n = − ∞ ∞ , which is an extension of Włoch's sequence (F 2 (q , n)) n = 0 ∞ . [ABSTRACT FROM AUTHOR]

In 2008, I. Włoch introduced a new generalization of Pell numbers. She used special initial conditions so that this sequence describes the total number of special families of subsets of the set of n integers. In this paper, we prove some results about the roots of the characteristic polynomial of this sequence, but we will consider general initial conditions. Since there are currently several types of generalizations of the Pell sequence, it is very difficult for anyone to realize what type of sequence an author really means. Thus, we will call this sequence the generalized k-distance Tribonacci sequence (T n (k)) n ≥ 0 . [ABSTRACT FROM AUTHOR]

Let k ≥ 1 be an integer and denote (F k , n) n as the k-Fibonacci sequence whose terms satisfy the recurrence relation F k , n = k F k , n − 1 + F k , n − 2 , with initial conditions F k , 0 = 0 and F k , 1 = 1 . In the same way, the k-Lucas sequence (L k , n) n is defined by satisfying the same recursive relation with initial values L k , 0 = 2 and L k , 1 = k . The sequences (F k , n) n ≥ 0 and (L k , n) n ≥ 0 were introduced by Falcon and Plaza, who derived many of their properties. In particular, they proved that F k , n 2 + F k , n + 1 2 = F k , 2 n + 1 and F k , n + 1 2 − F k , n − 1 2 = k F k , 2 n , for all k ≥ 1 and n ≥ 0 . In this paper, we shall prove that if k > 1 and F k , n s + F k , n + 1 s ∈ (F k , m) m ≥ 1 for infinitely many positive integers n, then s = 2 . Similarly, that if F k , n + 1 s − F k , n − 1 s ∈ (k F k , m) m ≥ 1 holds for infinitely many positive integers n, then s = 1 or s = 2 . This generalizes a Marques and Togbé result related to the case k = 1 . Furthermore, we shall solve the Diophantine equations F k , n = L k , m , F k , n = F n , k and L k , n = L n , k . [ABSTRACT FROM AUTHOR]

The sequence of the k-generalized Fibonacci numbers (F n (k)) n is defined by the recurrence F n (k) = ∑ j = 1 k F n − j (k) beginning with the k terms 0 , ... , 0 , 1 . In this paper, we shall solve the Diophantine equation F n (k) = (F m (l)) 2 + 1 , in positive integers m , n , k and l. [ABSTRACT FROM AUTHOR]

The order of appearance z : Z > 0 → Z > 0 is an arithmetic function related to the Fibonacci sequence (F n) n . This function is defined as the smallest positive integer solution of the congruence F k ≡ 0 (mod n) . In this paper, we shall provide lower and upper bounds for the functions ∑ n ≤ x z (n) / n , ∑ p ≤ x z (p) and ∑ p r ≤ x z (p r) . [ABSTRACT FROM AUTHOR]

Symmetry and elementary symmetric functions are main components of the proof of the celebrated Hermite–Lindemann theorem (about the transcendence of e α , for algebraic values of α) which settled the ancient Greek problem of squaring the circle. In this paper, we are interested in similar results, but for powers such as e γ log   n . This kind of problem can be posed in the context of arithmetic functions. More precisely, we study the arithmetic nature of the so-called γ-th arithmetic zeta function ζ γ (n) : = n γ ( = e γ log   n ), for a positive integer n and a complex number γ. Moreover, we raise a conjecture about the exceptional set of ζ γ , in the case in which γ is transcendental, and we connect it to the famous Schanuel's conjecture. [ABSTRACT FROM AUTHOR]

In this paper, we prove that F 22 = 17711 is the largest Fibonacci number whose decimal expansion is of the form a b ... b c ... c . The proof uses lower bounds for linear forms in three logarithms of algebraic numbers and some tools from Diophantine approximation. [ABSTRACT FROM AUTHOR]

Let F n be the nth Fibonacci number. Order of appearance z (n) of a natural number n is defined as smallest natural number k, such that n divides F k . In 1930, Lehmer proved that all solutions of equation z (n) = n ± 1 are prime numbers. In this paper, we solve equation z (n) = n + ℓ for | ℓ | ∈ { 1 , ... , 9 } . Our method is based on the p-adic valuation of Fibonacci numbers. [ABSTRACT FROM AUTHOR]

In this paper, we study a higher order generalization of the Jacobsthal sequence, namely, the (k , c) -Jacobsthal sequence (J n (k , c)) for any integers n, k ≥ 2 and a real number c > 0 . In particular, we find information about roots of its characteristic polynomial. For that purpose, we combine some powerful tools such as Marden's method, the Perron–Frobenius theorem, and the Eneström–Kakeya theorem. [ABSTRACT FROM AUTHOR]