Pavel Trojovský

University of Hradec Králové

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Štěpán Hubálovský

University of Hradec Králové

Mohammad Dehghani

Shiraz University

Eva Trojovská

University of Hradec Králové

Diego Marques

Universidade de Brasília

K. Venkatachalam

Vellore Institute of Technology University

Mojtaba Ghasemi

Shiraz University of Technology

Nebojša Bačanin

Singidunum University

Mohsen Zare

Jahrom University

Pavel Trojovsk

University of Hradec Králové

R. Venkata Rao

Sardar Vallabhbhai National Institute of Technology Surat

Cooperative Institutions

University of Hradec Králové

Universidade de Brasília

Shiraz University of Technology

Singidunum University

Lebanese American University

Middle East University

University of Pardubice

Islamic Azad University of Urmia

Shiraz University

Islamic Azad University, Marvdasht

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On Some Properties of the Hofstadter–Mertens Function

Complexity (2020)

Pavel Trojovský

Many mathematicians have been interested in the study of recursive sequences. Among them, a class of “chaotic” sequences are named “meta-Fibonacci sequences.” The main example of meta-Fibonacci sequence was introduced by Hofstadter, and it is called the Q -sequence. Recently, Alkan–Fox–Aybar and the author studied the pattern induced by the connection between the Q -sequence and other known sequences. Here, we continue this program by studying a “Mertens’ version” of the Hofstadter sequence , defined (for x>0) by x↦∑n≤xμnQn, where µ ( n ) is the Möbius function. In particular, as we shall see, this function encodes many interesting properties which relate prime numbers to “meta-sequences”.

Sequence (biology)

10.1155/2020/1816756

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Symbolic calculus in calculating characteristic and acyclic polynomials of graphs

Jaroslav Seibert Antonín Slabý Pavel Trojovský

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On a New Human-Based Metaheuristic Optimization Algorithm Based on Imitating Foreign Language Education Process

Research Square (Research Square) (2022)

Pavel Trojovský Mohammad Dehghani Eva Trojovská Eva Milková

Abstract The paper presents a new human-based metaheuristic algorithm called Language Education Optimization (LEO). LEO is inspired by the foreign language education process in which a language teacher trains the students of language schools in the desired language skills and rules. LEO is mathematically modeled in three phases: (i) students selecting their teacher, (ii) students learning from each other, and (iii) individual practice, considering exploration in local search and exploitation in local search. The performance of LEO in optimization tasks has been challenged against twenty-three benchmark functions of a variety of unimodal and multimodal types. The optimization results show that LEO, with its acceptable ability to explore, exploit, and maintain a balance between them, has efficient performance in optimization applications and in solution presentation. LEO efficiency in optimization tasks is compared with ten well-known metaheuristic algorithms. Analyses of the simulation results show that LEO has effective performance in dealing with optimization tasks and is significantly superior and more competitive in combating the compared algorithms.

Benchmark (surveying)

Presentation (obstetrics)

10.21203/rs.3.rs-1781300/v1

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Citations (1)

Peer Review #2 of "A new metaphor-less simple algorithm based on Rao algorithms: a Fully Informed Search Algorithm (FISA) (v0.1)"

Mojtaba Ghasemi Abolfazl Rahimnejad Ebrahim Akbari R. Venkata Rao Pavel Trojovský

Many important engineering optimization problems require a strong and simple optimization algorithm to achieve the best solutions.In 2020, Rao introduced three nonparametric algorithms, known as Rao algorithms, which have garnered significant attention from researchers worldwide due to their simplicity and effectiveness in solving optimization problems.In our simulation studies, we have developed a new version of the Rao algorithm called the Fully Informed Search Algorithm (FISA), which demonstrates acceptable performance in optimizing real-world problems while maintaining the simplicity and non-parametric nature of the original algorithms.We evaluate the effectiveness of the suggested FISA approach by applying it to optimize the shifted benchmark functions, such as those provided in CEC 2005 and CEC 2014, and by using it to design mechanical system components.We compare the results of FISA to those obtained using the original RAO method.The outcomes obtained indicate the efficacy of the proposed new algorithm, FISA, in achieving optimized solutions for the aforementioned problems.

10.7287/peerj-cs.1431v0.1/reviews/2

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On factorization of the Fibonacci and Lucas numbers using tridiagonal determinants

Mathematica Slovaca (2012)

Jaroslav Seibert Pavel Trojovský

Abstract The aim of this paper is to give new results about factorizations of the Fibonacci numbers F n and the Lucas numbers L n. These numbers are defined by the second order recurrence relation a n+2 = a n+1+a n with the initial terms F 0 = 0, F 1 = 1 and L 0 = 2, L 1 = 1, respectively. Proofs of theorems are done with the help of connections between determinants of tridiagonal matrices and the Fibonacci and the Lucas numbers using the Chebyshev polynomials. This method extends the approach used in [CAHILL, N. D.—D’ERRICO, J. R.—SPENCE, J. P.: Complex factorizations of the Fibonacci and Lucas numbers, Fibonacci Quart. 41 (2003), 13–19], and CAHILL, N. D.—NARAYAN, D. A.: Fibonacci and Lucas numbers as tridiagonal matrix determinants, Fibonacci Quart. 42 (2004), 216–221].

Lucas number

Lucas sequence

Recurrence relation

Pisano period

10.2478/s12175-012-0020-2

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Botox Optimization Algorithm: A New Human-Based Metaheuristic Algorithm for Solving Optimization Problems

Biomimetics (2024)

Marie Hubálovská Štěpán Hubálovský Pavel Trojovský

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.

Benchmark (surveying)

Test suite

Optimization algorithm

10.3390/biomimetics9030137

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Citations (13)

Some Diophantine Problems Related to k-Fibonacci Numbers

Mathematics (2020)

Pavel Trojovský Štěpán Hubálovský

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 .

Lucas number

Sequence (biology)

10.3390/math8071047

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Citations (1)

An Efficient Gabor Walsh-Hadamard Transform Based Approach for Retrieving Brain Tumor Images From MRI

IEEE Access (2021)

K. Venkatachalam Siuly Siuly Nebojša Bačanin Štěpán Hubálovský Pavel Trojovský

Brain tumors are a serious and death-defying disease for human life. Discovering an appropriate brain tumor image from a magnetic resonance imaging (MRI) archive is a challenging job for the radiologist. Most search engines retrieve images on the basis of traditional text-based approaches. The main challenge in the MRI image analysis is that low-level visual information captured by the MRI machine and the high-level information identified by the assessor. This semantic gap is addressed in this study by designing a new feature extraction technique. In this paper, we introduce Content-Based Medical Image retrieval (CBMIR) system for retrieval of brain tumor images from the large data. Firstly, we remove noise from MRI images employing several filtering techniques. Afterward, we design a feature extraction scheme combining Gabor filtering technique (which is mainly focused on specific frequency content at the image region) and Walsh-Hadamard transform (WHT) (conquer technique for easy configuration of image) for discovering representative features from MRI images. After that, for retrieving the accurate and reliable image, we employ Fuzzy C-Means clustering Minkowski distance metric that can evaluate the similarity between the query image and database images. The proposed methodology design was tested on a publicly available brain tumor MRI image database. The experimental results demonstrate that our proposed approach outperforms most of the existing techniques like Gabor, wavelet, and Hough transform in detecting brain tumors and also take less time. The proposed approach will be beneficial for radiologists and also for technologists to build an automatic decision support system that will produce reproducible and objective results with high accuracy.

Feature (linguistics)

Gabor wavelet

Content-Based Image Retrieval

10.1109/access.2021.3107371

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Citations (23)

Subtraction-Average-Based Optimizer: A New Swarm-Inspired Metaheuristic Algorithm for Solving Optimization Problems

Biomimetics (2023)

Pavel Trojovský Mohammad Dehghani

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.

Benchmark (surveying)

10.3390/biomimetics8020149

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Citations (113)

The p-adic Order of Some Fibonomial Coefficients.

J. Integer Seq. (2015)

Diego Marques Pavel Trojovský

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Citations (6)