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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]

In recent years, many authors studied the arithmetic, analytic and geometric aspects of roots of characteristic polynomials of some linear recurrence sequences. Many of these polynomials are particular cases of the three-parameter family f k , p , q (x) = x k - p x k - 1 - q x - 1 , for non-negative integers k, p and q, with k ≥ 3 and p ≥ 1 . In this work, we prove some properties related to the roots of the polynomial f k , p , q (x) which, for instance, make their applications possible in a large class of Diophantine problems. [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 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]

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 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, 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 1972, Alniaçik proved that every strong Liouville number is mapped into the set of U m -numbers, for any non-constant rational function with coefficients belonging to an m-degree number field. In this paper, we generalize this result by providing a larger class of Liouville numbers (which, in particular, contains the strong Liouville numbers) with this same property (this set is sharp is a certain sense). [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 aims to generalize results that have appeared in Atanassov (Math Balk New Ser 18(1-2):15-32, 2004). We consider here variants of the Halton sequences in a generalized numeration system, called the Cantor expansion, with respect to arbitrary sequences of permutations of the Cantor base. We first show that they provide a wealth of low-discrepancy sequences by giving an estimate of (star) discrepancy bound of the generalized Halton sequence in bounded Cantor bases. Then we impose certain conditions on the sequences of permutations of the Cantor base which are analogous, but not straightforward, to the modified Halton sequence introduced by E.I. Atanassov. We show that this modified Halton sequence in Cantor bases attains a better estimate of the (star) discrepancy bound than the generalized Halton sequence in Cantor bases. [ABSTRACT FROM AUTHOR]

Let ( F ) be the Fibonacci sequence. For 1 ≤ k ≤ m, the Fibonomial coefficient is defined as . In 2013, Marques, Sellers and Trojovský proved that if p is a prime number such that p ≡ ±1 (mod 5), then p∤ $${\left[ {\begin{array}{*{20}{c}} {{p^{a + 1}}} \\ {{p^a}} \end{array}} \right]_F}$$ for all integers a ≥ 1. In 2010, in particular, Kilic generalized the Fibonomial coefficients for . In this note, we generalize Marques, Sellers and Trojovský result to prove, in particular, that if p ≡ ±1 (mod 5), then $${\left[ {\begin{array}{*{20}{c}} {{p^{a + 1}}} \\ {{p^a}} \end{array}} \right]_{F,m}} \equiv 1$$ (mod p), for all a ≥ 0 and m ≥ 1. [ABSTRACT FROM AUTHOR]

Let ( F ) be the Fibonacci sequence. For 1 ≤ k ≤ m, the Fibonomial coefficient is defined as . In 2013, Marques, Sellers and Trojovský proved that if p is a prime number such that p ≡ ±2 (mod 5), then $$p{\left| {\left[ {\begin{array}{*{20}{c}} {{p^{a + 1}}} \\ {{p^a}} \end{array}} \right]} \right._F}$$ for all integers a ≥ 1. In 2015, Marques and Trojovský worked on the p-adic order of $${\left[ {\begin{array}{*{20}{c}} {{p^{a + 1}}} \\ {{p^a}} \end{array}} \right]_F}$$ for all a ≥ 1 when p ≠ 5. In this paper, we shall provide the exact p-adic order of $${\left[ {\begin{array}{*{20}{c}} {{p^{a + 1}}} \\ {{p^a}} \end{array}} \right]_F}$$ for all integers a, b ≥ 1 and for all prime number p. [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]

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]

Introducing a novel meta-heuristic optimization algorithm, the Flood Algorithm (FLA) draws inspiration from the intricate movement and flow patterns of water masses during flooding events in river basins. FLA mathematically models key phenomena such as the movement of water toward slopes, flow rates over time, soil permeability effects, and periodic increases and decreases in water levels from precipitation and losses. Leveraging these models, the algorithm guides the movement and evolution of a population of potential solutions toward enhanced optimality. The algorithm endeavors to establish an appropriate correlation between the fundamental aspects of natural flood events and the optimization process. Its formulation and working mechanism are described in detail. It operates in two main phases—a regular movement phase, where the population moves naturally toward current best solutions, and a flooding phase, which introduces random disturbances to increase diversity. New solutions are periodically introduced while weaker ones are removed, mirroring the natural cycles of water levels. FLA’s effectiveness is demonstrated through its application on well-known benchmark optimization problems and engineering design problems. Extensive comparisons have been carried out on CEC2005 functions using 16 algorithms in both basic and enhanced modes, as well as on CEC2014 functions with dimensions 30, 50, and 100 using a total of 20 other algorithms. These rigorous studies unequivocally confirm the robustness and strength of the proposed algorithm. Furthermore, the algorithm's performance on 12 constrained engineering problems demonstrates its ability to tackle real-world challenges. The FLA’s source code is publicly available at https://www.optim-app.com/projects/fla. [ABSTRACT FROM AUTHOR]

Optimization techniques, particularly meta-heuristic algorithms, are highly effective in optimizing and enhancing efficiency across diverse models and systems, renowned for their ability to attain optimal or near-optimal solutions within a reasonable timeframe. In this work, the Puma Optimizer (PO) is proposed as a new optimization algorithm inspired from the intelligence and life of Pumas in. In this algorithm, unique and powerful mechanisms have been proposed in each phase of exploration and exploitation, which has increased the algorithm’s performance against all kinds of optimization problems. In addition, a new type of intelligent mechanism, which is a type of hyper-heuristic for phase change, is presented. Using this mechanism, the PO algorithm can perform a phase change operation during the optimization operation and balance both phases. Each phase is automatically adjusted to the nature of the problem. To evaluate the proposed algorithm, 23 standard functions and CEC2019 functions were used and compared with different types of optimization algorithms. Moreover, using the statistical test T-test and the execution time to solve the problem have been discussed. Finally, it has been tested using four machine learning and data mining problems, and the results obtained from all the analysis signifies the excellent performance of this algorithm against all kinds of problems compared to other optimizers. This algorithm has performed better than the compared algorithms in 27 benchmarks out of 33 benchmarks and has obtained better results in solving the clustering problem in 7 data sets out of 10 data sets. Furthermore, the results obtained in the problems of community detection and feature selection and MLP were superior. The source codes of the PO algorithm are publicly available at https://www.mathworks.com/matlabcentral/fileexchange/157231-puma-optimizer-po. [ABSTRACT FROM AUTHOR]