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Journal articles on the topic 'Integer sequence generation'
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1
Sabri, Ahmad, and Vincent Vajnovszki. "On the exhaustive generation of generalized ballot sequences in lexicographic and Gray code order." Pure Mathematics and Applications 28, no. 1 (2019): 109–19. http://dx.doi.org/10.1515/puma-2015-0035.
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Abstract A generalized (resp. p-ary) ballot sequence is a sequence over the set of non-negative integers (resp. integers less than p) where in any of its prefixes each positive integer i occurs at most as often as any integer less than i. We show that the Reected Gray Code order induces a cyclic 3-adjacent Gray code on both, the set of fixed length generalized ballot sequences and p-ary ballot sequences when p is even, that is, ordered list where consecutive sequences (regarding the list cyclically) differ in at most 3 adjacent positions. Non-trivial efficient generating algorithms for these ballot sequences, in lexicographic order and for the obtained Gray codes, are also presented.
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Massazza, Paolo. "A Dynamical System Approach to Polyominoes Generation*." Fundamenta Informaticae 180, no. 3 (2021): 251–73. http://dx.doi.org/10.3233/fi-2021-2040.
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We describe a method which exploits discrete dynamical systems to generate suitable classes of polyominoes. We apply the method to design an algorithm that uses O(n) space to generate in constant amortized time all polyominoes corresponding to hole-free partially directed animals consisting of n sites on the square grid. By implementing the algorithm in C++ we have obtained a new sequence that does not appear in the On-Line Encyclopedia of Integer Sequences.
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Lee, Suk-Hwan. "Reversible Data Hiding for DNA Sequence Using Multilevel Histogram Shifting." Security and Communication Networks 2018 (2018): 1–13. http://dx.doi.org/10.1155/2018/3530969.
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A large number of studies have examined DNA storage to achieve information hiding in DNA sequences with DNA computing technology. However, most data hiding methods are irreversible in that the original DNA sequence cannot be recovered from the watermarked DNA sequence. This study presents reversible data hiding methods based on multilevel histogram shifting to prevent biological mutations, preserve sequence length, increase watermark capacity, and facilitate blind detection/recovery. The main features of our method are as follows. First, we encode a sequence of nucleotide bases with four-character symbols into integer values using the numeric order. Second, we embed multiple bits in each integer value by multilevel histogram shifting of noncircular type (NHS) and circular type (CHS). Third, we prevent the generation of false start/stop codons by verifying whether a start/stop codon is included in an integer value or between adjacent integer values. The results of our experiments confirmed that the NHS- and CHS-based methods have higher watermark capacities than conventional methods in terms of supplementary data used for decoding. Moreover, unlike conventional methods, our methods do not generate false start/stop codons.
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Liu, Bo, Jiandong Liu, Shuhong Wang, Ming Zhong, Bo Li, and Yujie Liu. "HEVC Video Encryption Algorithm Based on Integer Dynamic Coupling Tent Mapping." Journal of Advanced Computational Intelligence and Intelligent Informatics 24, no. 3 (2020): 335–45. http://dx.doi.org/10.20965/jaciii.2020.p0335.
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A selective encryption algorithm is proposed to improve the efficiency of high efficiency video coding (HEVC) video encryption and ensure the security of HEVC videos. The algorithm adopts the integer dynamic coupling tent mapping optimization model as the pseudo-random sequence generator, and multi-core parallelization is used as the sequence generation mechanism. The binstrings during the process of context adaptive binary arithmetic coding are selected for encryption, which conforms to the features of invariable binstream and compatible format in terms of video encryption. Performance tests for six types of standard videos with different resolutions were performed. The results indicated that the encryption algorithm has a large key space and benefits from a high encryption effect.
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Wang, Qianxue, Simin Yu, Christophe Guyeux, Jacques M. Bahi, and Xiaole Fang. "Theoretical Design and Circuit Implementation of Integer Domain Chaotic Systems." International Journal of Bifurcation and Chaos 24, no. 10 (2014): 1450128. http://dx.doi.org/10.1142/s0218127414501284.
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In this paper, a new approach for constructing integer domain chaotic systems (IDCS) is proposed, and its chaotic behavior is mathematically proven according to Devaney's definition of chaos. Furthermore, an analog-digital hybrid circuit is also developed for realizing the designed basic IDCS. In the IDCS circuit design, chaos generation strategy is realized through a sample-hold circuit and a decoder circuit so as to convert the uniform noise signal into a random sequence, which plays a key role in circuit implementation. The experimental observations further validate the proposed systematic methodology for the first time.
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Hornfeck, Wolfgang. "Chiral spiral cyclic twins." Acta Crystallographica Section A Foundations and Advances 78, no. 1 (2022): 21–35. http://dx.doi.org/10.1107/s2053273321012237.
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A formula is presented for the generation of chiral m-fold multiply twinned two-dimensional point sets of even twin modulus m > 6 from an integer inclination sequence; in particular, it is discussed for the first three non-degenerate cases m = 8, 10, 12, which share a connection to the aperiodic crystallography of axial quasicrystals exhibiting octagonal, decagonal and dodecagonal long-range orientational order and symmetry.
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Pratihar, Sanjoy, and Partha Bhowmick. "On the Farey sequence and its augmentation for applications to image analysis." International Journal of Applied Mathematics and Computer Science 27, no. 3 (2017): 637–58. http://dx.doi.org/10.1515/amcs-2017-0045.
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AbstractWe introduce a novel concept of theaugmented Farey table(AFT). Its purpose is to store the ranks of fractions of aFarey sequencein an efficient manner so as to return therankof any query fraction in constant time. As a result, computations on the digital plane can be crafted down to simple integer operations; for example, the tasks like determining the extent of collinearity of integer points or of parallelism of straight lines—often required to solve many image-analytic problems—can be made fast and efficient through an appropriate AFT-based tool. We derive certain interesting characterizations of an AFT for its efficient generation. We also show how, for a fraction not present in a Farey sequence, the rank of thenearest fractionin that sequence can efficiently be obtained by theregula falsimethod from the AFT concerned. To assert its merit, we show its use in two applications—one in polygonal approximation of digital curves and the other in skew correction of engineering drawings in document images. Experimental results indicate the potential of the AFT in such image-analytic applications.
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Moftah, Khaled. "Block Format Solves the Collatz Conjecture." Journal of Advances in Mathematics and Computer Science 39, no. 4 (2024): 24–36. http://dx.doi.org/10.9734/jamcs/2024/v39i41880.
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Blocks are unit convergence between two consecutive odd numbers formed according to the three x plus one conjecture rules. The left odd number is the left hook, L, and the right odd number is the right hook, R. They include even numbers between their boundaries. They are divided into families (F1 = 5, 11, 17, … & F2 = 1, 7, 11, … & F3 = 3, 9, 15, …) and groups based on their group length (The number of the middle-even numbers between the two hooks). Blocks are taken individually and placed beside each other, similar to the domino tiles play, which, by their formulation, satisfies the conjecture rules. Formed chains reach number one in the convergence mode or continue generating odd positive numbers infinitely according to the generation mode. The final convergence to number one is reached because these blocks have all the positive integers included as left hooks (L1, L2, L3), and all the F1 and F2 odd positive numbers are included as right hooks (R1 and R2). Block rules mandate that a single left hook produces only one right hook. Accordingly, no looping or entanglement (Joining and consequent splitting) between chain branches would occur. Statistics show that R cannot increase infinitely. Repeated oscillation up and down without reaching number one would also violate the statistics. Statistics reveal that blocks of various lengths have a strict occurrence and repetition sequence along the positive integer series. Block lengths can extend infinitely, and each block length repeats its occurrence infinitely. In the generation mode, blocks are attached in reverse order to the conjecture/convergence rules. According to the rules, all positive integers can be generated starting from number one. Multiple sequences and clusters of specific block lengths occur according to specific rules and cannot continue infinitely.
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Andreani, Virgile, Eric J. South, and Mary J. Dunlop. "Generating information-dense promoter sequences with optimal string packing." PLOS Computational Biology 20, no. 7 (2024): e1012276. http://dx.doi.org/10.1371/journal.pcbi.1012276.
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Dense arrangements of binding sites within nucleotide sequences can collectively influence downstream transcription rates or initiate biomolecular interactions. For example, natural promoter regions can harbor many overlapping transcription factor binding sites that influence the rate of transcription initiation. Despite the prevalence of overlapping binding sites in nature, rapid design of nucleotide sequences with many overlapping sites remains a challenge. Here, we show that this is an NP-hard problem, coined here as the nucleotide String Packing Problem (SPP). We then introduce a computational technique that efficiently assembles sets of DNA-protein binding sites into dense, contiguous stretches of double-stranded DNA. For the efficient design of nucleotide sequences spanning hundreds of base pairs, we reduce the SPP to an Orienteering Problem with integer distances, and then leverage modern integer linear programming solvers. Our method optimally packs sets of 20–100 binding sites into dense nucleotide arrays of 50–300 base pairs in 0.05–10 seconds. Unlike approximation algorithms or meta-heuristics, our approach finds provably optimal solutions. We demonstrate how our method can generate large sets of diverse sequences suitable for library generation, where the frequency of binding site usage across the returned sequences can be controlled by modulating the objective function. As an example, we then show how adding additional constraints, like the inclusion of sequence elements with fixed positions, allows for the design of bacterial promoters. The nucleotide string packing approach we present can accelerate the design of sequences with complex DNA-protein interactions. When used in combination with synthesis and high-throughput screening, this design strategy could help interrogate how complex binding site arrangements impact either gene expression or biomolecular mechanisms in varied cellular contexts.
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Rawal, Shruti, and Indivar Gupta. "Solution of Large Sparse System of Linear Equations over GF(2) on a Multi Node Multi GPU Platform." Defence Science Journal 72, no. 6 (2022): 836–45. http://dx.doi.org/10.14429/dsj.72.17656.
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We provide an efficient multi-node, multi-GPU implementation of the Block Wiedemann Algorithm (BWA)to find the solution of a large sparse system of linear equations over GF(2). One of the important applications ofsolving such systems arises in most integer factorization algorithms like Number Field Sieve. In this paper, wedescribe how hybrid parallelization can be adapted to speed up the most time-consuming sequence generation stage of BWA. This stage involves generating a sequence of matrix-matrix products and matrix transpose-matrix products where the matrices are very large, highly sparse, and have entries over GF(2). We describe a GPU-accelerated parallel method for the computation of these matrix-matrix products using techniques like row-wise parallel distribution of the first matrix over multi-node multi-GPU platform using MPI and CUDA and word-wise XORing of rows of the second matrix. We also describe the hybrid parallelization of matrix transpose-matrix product computation, where we divide both the matrices row-wise into equal-sized blocks using MPI. Then after a GPU-accelerated matrix transpose-matrix product generation, we combine all those blocks using MPI_BXOR operation in MPI_Reduce to obtain the result. The performance of hybrid parallelization of the sequence generation step on a hybrid cluster using multiple GPUs has been compared with parallelization on only multiple MPI processors. We have used this hybrid parallel sequence generation tool for the benchmarking of an HPC cluster. Detailed timings of the complete solution of number field sieve matrices of RSA-130, RSA-140, and RSA-170 are also compared in this paper using up to 4 NVidia V100 GPUs of a DGX station. We got a speedup of 2.8 after parallelization on 4 V100 GPUs compared to that over 1 GPU.
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Nadhir, Ibrahim Abdulkhaleq, Abdul Jalil Salih Nahla, Salih Mohammed Hasan Rajaa, and Jabbar Hasan Ihsan. "A Simulink model for modified fountain codes." TELKOMNIKA (Telecommunication, Computing, Electronics and Control) 21, no. 1 (2023): 18–25. https://doi.org/10.12928/telkomnika.v21i1.23274.
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This paper introduces a Simulink model design for a modified fountain code. The code is a new version of the traditional Luby transform (LT) codes. The design constructs the blocks required for generation of the generator matrix of a limited-degree-hopping-segment Luby transform (LDHS-LT) codes. This code is especially designed for short length data files which have assigned a great interest for wireless sensor networks. It generates the degrees in a predetermined sequence but random generation and partitioned the data file in segments. The data packets selection has been made serialy according to the integer generated from both degree and segment generators. The code is tested using Monte Carlo simulation approach with the conventional code generation using robust soliton distribution (RSD) for degree generation, and the simulation results approve better performance with all testing parameter.
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Boguslavsky, I. A. "A Bayes Estimator of Parameters of Nonlinear Dynamic Systems." Mathematical Problems in Engineering 2009 (2009): 1–21. http://dx.doi.org/10.1155/2009/801475.
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A new multipolynomial approximations algorithm (the MPA algorithm) is proposed for estimating the state vectorθof virtually any dynamical (evolutionary) system. The input of the algorithm consists of discrete-time observationsY. An adjustment of the algorithm is required to the generation of arrays of random sequences of state vectors and observations scalars corresponding to a given sequence of time instants. The distributions of the random factors (vectors of the initial states and random perturbations of the system, scalars of random observational errors) can be arbitrary but have to be prescribed beforehand. The output of the algorithm is a vector polynomial series with respect to products of nonnegative integer powers of the results of real observations or some functions of these results. The sum of the powers does not exceed some given integerd. The series is a vector polynomial approximation of the vectorE(θ∣Y), which is the conditional expectation of the vector under evaluation (or given functions of the components of that vector). The vector coefficients of the polynomial series are constructed in such a way that the approximation errors uniformly tend to zero as the integerdincreases. These coefficients are found by the Monte-Carlo method and a process of recurrent calculations that do not require matrix inversion.
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Zhang, Dandan, and Canrong Zhang. "Study on column generation for the lot-sizing and scheduling problem with sequencedependent setup time." MATEC Web of Conferences 189 (2018): 06002. http://dx.doi.org/10.1051/matecconf/201818906002.
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The capacitated lot-sizing and scheduling problem with sequence-dependent setup time and carryover setup state is a challenge problem in the semiconductor assembly and test manufacturing. For the problem, a new mixed integer programming model is proposed, followed by exploring its relative efficiency in obtaining optimal solutions and linearly relaxed optimal solutions. On account of the sequence-dependent setup time and the carryover of setup states, a per-machine Danzig Wolfe decomposition is proposed. We then build a statistical estimation model to describe correlation between the optimal solutions and two lower bounds including the linear relaxation solutions, and the pricing sub-problem solutions of Danzig Wolfe decomposition, which gives insight on the optimal values about information regarding whether or not the setup variables in the optimal solution take the value of 1, and the information is further used in the branch and select procedure. Numerical experiments are conducted to test the performance of the algorithm.
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Desaulniers, Guy, Timo Gschwind, and Stefan Irnich. "Variable Fixing for Two-Arc Sequences in Branch-Price-and-Cut Algorithms on Path-Based Models." Transportation Science 54, no. 5 (2020): 1170–88. http://dx.doi.org/10.1287/trsc.2020.0988.
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Variable fixing by reduced costs is a popular technique for accelerating the solution process of mixed-integer linear programs. For vehicle-routing problems solved by branch-price-and-cut algorithms, it is possible to fix to zero the variables associated with all routes containing at least one arc from a subset of arcs determined according to the dual solution of a linear relaxation. This is equivalent to removing these arcs from the network used to generate the routes. In this paper, we extend this technique to routes containing sequences of two arcs. Such sequences or their arcs cannot be removed directly from the network because routes traversing only one arc of a sequence might still be allowed. For some of the most common vehicle-routing problems, we show how this issue can be overcome by modifying the route-generation labeling algorithm in order to remove indirectly these sequences from the network. The proposed acceleration strategy is tested on benchmark instances of the vehicle-routing problem with time windows (VRPTW) and four variants of the electric VRPTW. The computational results show that it yields a significant speedup, especially for the most difficult instances.
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Shakhmuratov, Rustem. "Generation of pulses and multiplying their repetition rate using the temporal fractional Talbot effect." Laser Physics 34, no. 10 (2024): 105103. http://dx.doi.org/10.1088/1555-6611/ad6d4f.
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Abstract The generation of pulses from a periodic phase-modulated continuous wave (CW) laser field, which is transmitted through a group-delay-dispersion (GDD) circuit, is considered. A time lens (TL), consisting of a quadratic phase modulator and a GDD circuit is proposed in combination with temporal array illuminators (TAI) using another GDD circuit. The time lens producing field compression into pulses is realized for a particular value of the normalized fractional Talbot length (NFTL) L / L T = P 1 / Q 1 , where L is the physical length of the GDD circuit, L T is the Talbot length, P 1 = 1 , and Q 1 is an integer. The length of the GDD circuit is selected to convert a given parabolic phase-modulated CW laser field into short pulses repeated with a phase modulation period T in accordance with the chirp radar concept. If NFTL is increased by 1 / Q 2 , where Q 2 = 4 , 6, or 8, the pulse sequence period is shortened as T / 2 , T / 3 , and T / 4 , respectively. This is because the additional GDD circuit with NFTL 1 / Q 2 , performs repetition rate multiplication of the initially prepared pulse sequence as TAI does. The maximum multiplication number considered in this paper is 12, which makes it possible to reduce the time interval between pulses by a factor of 12 and obtain a repetition rate 120 GHZ of picosecond pulses generated by phase modulation with frequency f = 1 / T = 10 GHz. The proposed method of pulse sequence generation with a discretely tunable period provides a new tool for optical signal processing in optical communication.
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Jallouli Khlif, Rim, Ahmed Abid, Pierre Melchior, and Nabil Derbel. "UM Shaper Command Inputs for CRONE Control: Application on a DC Motor Bench." Mathematical Problems in Engineering 2021 (May 29, 2021): 1–11. http://dx.doi.org/10.1155/2021/9935875.
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This study proposes an approach to synthesize a three-impulse sequence input shaper with a negative impulse, known as Unity Magnitude (UM) shaper. The corresponding analytic model has been already achieved for undamped and low-damped systems. In this paper, the analytic design of UM shaper is demonstrated for the generalized case of damped systems for both types: integer and fractional orders. Hence, the UM shaper model has been designed for second-order systems with damped dynamics, associating a graphical fitting and an analytical procedure; then, it has been extended to explicit fractional derivative systems. Moreover, the feasibility and the effectiveness of the proposed on-off profile prefilter applied on a second-generation controller have been substantiated by experimental results on an instrumented DC motor bench.
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Benda, Frank, Roland Braune, Karl F. Doerner, and Richard F. Hartl. "A machine learning approach for flow shop scheduling problems with alternative resources, sequence-dependent setup times, and blocking." OR Spectrum 41, no. 4 (2019): 871–93. http://dx.doi.org/10.1007/s00291-019-00567-8.
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AbstractIn proposing a machine learning approach for a flow shop scheduling problem with alternative resources, sequence-dependent setup times, and blocking, this paper seeks to generate a tree-based priority rule in terms of a well-performing decision tree (DT) for dispatching jobs. Furthermore, generating a generic DT and RF that yields competitive results for instance scenarios that structurally differ from the training instances was another goal of our research. The proposed DT relies on high quality solutions, obtained using a constraint programming (CP) formulation. Novel aspects include a unified representation of job sequencing and machine assignment decisions, as well as the generation of random forests (RF) to counteract overfitting behaviour. To show the performance of the proposed approaches, different instance scenarios for two objectives (makespan and total tardiness minimisation) were implemented, based on randomised problem data. The background of this approach is a real-world physical system of an industrial partner that represents a typical shop floor for many production processes, such as furniture and window construction. The results of a comparison of the DT and RF approach with two priority dispatching rules, the original CP solutions and tight lower bounds retrieved from a strengthened mixed-integer programming (MIP) formulation show that the proposed machine learning approach performs well in most instance sets for the makespan objective and in all sets for the total tardiness objective.
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Pardo-Martínez, Ricardo Andrés, Jesús M. López-Lezama, and Nicolás Muñoz-Galeano. "Optimal Generation Start-Up Methodology for Power System Restoration Considering Conventional and Non-Conventional Renewable Energy Sources." Applied Sciences 11, no. 17 (2021): 8246. http://dx.doi.org/10.3390/app11178246.
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Power system restoration must be accomplished as soon as possible after a blackout. In this process, available black-start (BS) units are used to provide cranking power to non-black-start (NBS) units so as to maximize the overall power system generation capacity. This procedure is known as the generation start-up problem, which is intrinsically combinatorial with complex non-linear constraints. This paper presents a new mixed integer linear programming (MILP) formulation for the generation start-up problem that integrates non-conventional renewable energy sources (NCRES) and battery energy storage systems (BESS). The main objective consists of determining an initial starting sequence for both BS and NBS units that would maximize the generation capacity of the system while meeting the non-served demand of the network. The nature of the proposed model leads to global optimal solutions, clearly outperforming heuristic and enumerative approaches, since the latter may take higher computational time while the former do not guarantee global optimal solutions. Several tests were carried out on the IEEE 39-bus test system considering BESS as well as wind and solar generation. The results showed the positive impact of NCRES in the restoration processes and evidenced the effectiveness and applicability of the proposed approach. It was found that including NCRES and BESS in the restoration process allows a reduction of 24.4% of the objective function compared to the classical restoration without these technologies.
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Zhu, Hongwei, Zhiqiang Lu, Chenyao Lu, and Yifei Ren. "Modeling and algorithm for resource-constrained multi-project scheduling problem based on detection and rework." Assembly Automation 41, no. 2 (2021): 174–86. http://dx.doi.org/10.1108/aa-09-2020-0132.
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Purpose To meet the requirement of establishing an effective schedule for the assembly process with overall detection and rework, this paper aims to address a new problem named resource-constrained multi-project scheduling problem based on detection and rework (RCMPSP-DR). Design/methodology/approach First, to satisfy both online and offline scheduling, a mixed integer programming model is established with a weighted bi-objective minimizing the expected makespan and the solution robustness. Second, an algorithm that combines a tabu search framework with a critical chain-based baseline generation scheme is designed. The tabu search framework focuses on searching for a reasonable resource flow representing the execution sequence of activities, while the critical chain-based baseline generation scheme establishes a buffered baseline schedule by estimating the tradeoff between two aspects of bi-objective. Findings The proposed algorithm can get solutions with gaps from −4.45% to 2.33% when compared with those obtained by the commercial MIP solver CPLEX. Moreover, the algorithm outperforms four other algorithms in terms of both objective performance and stability over instances with different weighting parameters, which reveals its effectiveness. Originality/value The represented RCMPSP-DR considering the overall detection and rework is an extension of the scheduling problem for large-scale equipment. An effective algorithm is proposed to establish the baseline schedule and determine the execution sequence of activities for the assembly process, which is significant for practical engineering applications.
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Roy, Sharadindu. "Genetic algorithm based approach to solve travelling salesman problem with one point crossover operator:." INTERNATIONAL JOURNAL OF COMPUTERS & TECHNOLOGY 10, no. 3 (2013): 1393–400. http://dx.doi.org/10.24297/ijct.v10i3.3269.
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In this paper, the travelling salesman problem using genetic algorithm has been attempted. In this practical paper solution is easy and we can easily apply genetic operator in this type of problem. Complexity is both in time and space, provided size of the problem an as integer (count is infinite). The solution of the traveling salesman problem is global optimum. There are cities and given distances between them. Traveling salesman has to visit all of them. TSP main objective is to find traveling sequence of cities to minimize the traveling distance.* traverse one time*initially we select parent1 & parent2 by Roulette wheel concept. Apply one point crossover operator on parents and produce the offspring. Again we apply the mutation operator on offspring and created child. But the no. of bits (cities) will be inverted by the mutation operator, that is depended on mutation probability (pm). So one generation contain 6 individual. Then count fitness of the individuals in each generation. For the next generation (for parent1 & parent2) two individuals will be selected whose fitness is best in generation. Here we see crossover between two good solution may not always yield a better or as good a solution. Since parents are good, so the probability of the child being good is high. Every time we have to do, identity the good solution in the population and make multiple copies of the good solution.
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Duan, Huiming, Guang Rong Lei, and Kailiang Shao. "Forecasting Crude Oil Consumption in China Using a Grey Prediction Model with an Optimal Fractional-Order Accumulating Operator." Complexity 2018 (August 1, 2018): 1–12. http://dx.doi.org/10.1155/2018/3869619.
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Crude oil, which is an important part of energy consumption, can drive or hinder economic development based on its production and consumption. Reasonable predictions of crude oil consumption in China are meaningful. In this paper, we study the grey-extended SIGM model, which is directly estimated with differential equations. This model has high simulation and prediction accuracies and is one of the important models in grey theory. However, to achieve the desired modeling effect, the raw data must conform to a class ratio check. Unfortunately, the characteristics of the Chinese crude oil consumption data are not suitable for SIGM modeling. Therefore, in this paper, we use a least squares estimation to study the parametric operation properties of the SIGM model, and the gamma function is used to extend the integer order accumulation sequence to the fractional-order accumulation generation sequence. The first-order SIGM model is extended to the fractional-order FSIGM model. According to the particle swarm optimization (PSO) mechanism and the properties of the gamma function of the fractional-order cumulative generation operator, the optimal fractional-order particle swarm optimization algorithm of the FSIGM model is obtained. Finally, the data concerning China’s crude oil consumption from 2002 to 2014 are used as experimental data. The results are better than those of the classical grey GM, DGM, and NDGM models as well as those of the grey-extended SIGM model. At the same time, according to the FSIGM model, this paper predicts China’s crude oil consumption for 2015–2020.
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Kozub, Pavlo, Svitlana Kozub, Victoria Lukianova, Ganna Reznichenko, Yuvita Koloshko, and Nataly Saidoglu. "METHOD FOR DETERMINING THE SEQUENCE OF REACTIONS TO PREDICT THE CONSEQUENCES OF TECHNOGENIC POLLUTION." Technogenic and Ecological Safety, no. 11(1/2022) (April 7, 2022): 34–39. http://dx.doi.org/10.52363/2522-1892.2022.1.5.
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To predict the negative consequences of man-made pollution with harmful substances, a method for determining possible ways of chemical reactions taking into account the chemical and thermodynamic properties of a given set of components, the theoretical foundations of chemical stoichiometry and thermodynamics, patterns of influence of thermodynamic parameters on the direction of chemical reactions. In this regard, chemical objects were presented in the form of multidimensional vectors, taking into account thermodynamic parameters, for the calculation of which used the methods of chemical thermodynamics. The generation of chemical reactions was carried out using the methods of mathematical statistics and combinatorics, and to determine the sequences of chemical reactions, the methods of graph theory. It is proposed to write a chemical compound as the sum of basis vectors, which allowed to represent compounds and interactions between them in n-dimensional space and to determine possible chemical interactions with the help of the main principles of work with integer vector space. Three conditions for the possibility of chemical interaction by this method are clarified and the values of enthalpy and Gibbs energy of each reaction are introduced, which are determined by the values of the corresponding potentials for each of the compounds. For the convenience of the image and further calculations, the element-enthalpy and element-gibs spaces were used separately. An algorithm for this method is created and tested, features for different boundary conditions of its use are studied, its efficiency on real chemical reaction systems is checked. An algorithm for determining the possibilities of chemical reactions in multicomponent systems to determine their man-made load on the environment is proposed.
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Xuan, Hua, Huixian Zhang, and Bing Li. "An Improved Discrete Artificial Bee Colony Algorithm for Flexible Flowshop Scheduling with Step Deteriorating Jobs and Sequence-Dependent Setup Times." Mathematical Problems in Engineering 2019 (December 12, 2019): 1–13. http://dx.doi.org/10.1155/2019/8520503.
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This paper studies a flexible flowshop scheduling problem with step-deteriorating jobs and sequence-dependent setup times (FFSP-SDJ&SDST) where there are multiple unrelated parallel machines at each stage. The actual processing time of each job is modeled as a step function of its starting time. An integer programming model is first formulated with the objective of minimizing the total weighted completion time. Since this problem is NP-complete, it becomes an interesting and challenging topic to develop effective approximation algorithms for solving it. The artificial bee colony (ABC) algorithm has been successfully applied to solve both continuous and combinatorial optimization problems with the advantages of fewer control parameters and ease of implementation. So, an improved discrete artificial bee colony algorithm is proposed. In this algorithm, a dynamic generation mechanism of initial solutions is designed based on job permutation encoding. A genetic algorithm and a modified variable neighborhood search are introduced, respectively, to obtain new solutions for the employed and onlooker bees. A greedy heuristic is proposed to generate the solutions of the scout bees. Finally, to verify the performance of the proposed algorithm, an orthogonal test is performed to optimize the parameter settings. Simulation results on different scale problems demonstrate that the proposed algorithm is more effective compared against several presented algorithms from the existing literatures.
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Wang, Yantao, Yinhan Zhang, Xuesong Qi, Meiqi Wang, and Xinyue Wang. "A Bi-Level Optimization Model for Virtual Power Plant Membership Selection Considering Load Time Series." Sustainability 15, no. 3 (2023): 2138. http://dx.doi.org/10.3390/su15032138.
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In order to improve the level of new energy consumption and reduce the dependence of the power system on traditional fossil energy, this paper proposed a bi-level optimization model for virtual power plant member selection by means of coordination and complementarity among different power sources, aiming at optimizing system economy and clean energy consumption capacity and combining it with the time sequence of load power consumption. The method comprises the following steps: (1) The processing load, wind power, and photovoltaic data by using ordered clustering to reflect the time sequence correlation between new energy and load and (2) uses a double-layer optimization model, wherein the upper layer calculates the capacity configuration of thermal power and energy storage units in a virtual power plant and selects the new energy units to participate in dispatching by considering the utility coefficient of the new energy units and the environmental benefit of the thermal power units. The Latin hypercube sampling (LHS) method was used to generate a large number of subsequences and the mixed integer linear programming (MILP) algorithm was used to calculate the optimal operation scheme of the system. The simulation results showed that by reducing the combination of subsequences between units and establishing a reasonable unit capacity allocation model, the average daily VPP revenue increased by RMB 12,806 and the proportion of new energy generation increased by 1.8% on average, which verified the correctness of the proposed method.
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Costa, Eudes A., Paula M. M. C. Catarino, Francival S. Monteiro, Vitor M. A. Souza, and Douglas C. Santos. "Tricomplex Fibonacci Numbers: A New Family of Fibonacci-Type Sequences." Mathematics 12, no. 23 (2024): 3723. http://dx.doi.org/10.3390/math12233723.
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In this paper, we define a novel family of arithmetic sequences associated with the Fibonacci numbers. Consider the ordinary Fibonacci sequence {fn}n∈N0 having initial terms f0=0, and f1=1 and recurrence relation fn=fn−1+fn−2(n≥2). In many studies, authors worked on the generalizations of integer sequences in different ways, some by preserving the initial terms, others by preserving the recurrence relation, and some for numeric sets other than positive integers. Here, we will follow the third path. So, in this article, we study a new extension tfn∗, with initial terms tf0∗=(f0∗,f1∗,f2∗) and tf1∗=(f1∗,f2∗,f3∗), which is generated by the recurrence relation tfn∗=tfn−1∗+tfn−2∗(n≥2), the Fibonacci-type sequence. The aim of this paper is to define Tricomplex Fibonacci numbers as an extension of the Fibonacci sequence and to examine some of their properties such as the recurrence relation, summation formula and generating function, and some classical identities.
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Alrasheedi, Adel F., Ahmad M. Alshamrani, and Khalid A. Alnowibet. "Investing in Wind Energy Using Bi-Level Linear Fractional Programming." Energies 16, no. 13 (2023): 4952. http://dx.doi.org/10.3390/en16134952.
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Investing in wind energy is a tool to reduce greenhouse gas emissions without negatively impacting the environment to accelerate progress towards global net zero. The objective of this study is to present a methodology for efficiently solving the wind energy investment problem, which aims to identify an optimal wind farm placement and capacity based on fractional programming (FP). This study adopts a bi-level approach whereby a private price-taker investor seeks to maximize its profit at the upper level. Given the optimal placement and capacity of the wind farm, the lower level aims to optimize a fractional objective function defined as the ratio of total generation cost to total wind power output. To solve this problem, the Charnes-Cooper transformation is applied to reformulate the initial bi-level problem with a fractional objective function in the lower-level problem as a bi-level problem with a fractional objective function in the upper-level problem. Afterward, using the primal-dual formulation, a single-level linear FP model is created, which can be solved via a sequence of mixed-integer linear programming (MILP). The presented technique is implemented on the IEEE 118-bus power system, where the results show the model can achieve the best performance in terms of wind power output.
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Shimko, Tyler C., Polly M. Fordyce, and Yaron Orenstein. "DeCoDe: degenerate codon design for complete protein-coding DNA libraries." Bioinformatics 36, no. 11 (2020): 3357–64. http://dx.doi.org/10.1093/bioinformatics/btaa162.
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Abstract Motivation High-throughput protein screening is a critical technique for dissecting and designing protein function. Libraries for these assays can be created through a number of means, including targeted or random mutagenesis of a template protein sequence or direct DNA synthesis. However, mutagenic library construction methods often yield vastly more nonfunctional than functional variants and, despite advances in large-scale DNA synthesis, individual synthesis of each desired DNA template is often prohibitively expensive. Consequently, many protein-screening libraries rely on the use of degenerate codons (DCs), mixtures of DNA bases incorporated at specific positions during DNA synthesis, to generate highly diverse protein-variant pools from only a few low-cost synthesis reactions. However, selecting DCs for sets of sequences that covary at multiple positions dramatically increases the difficulty of designing a DC library and leads to the creation of many undesired variants that can quickly outstrip screening capacity. Results We introduce a novel algorithm for total DC library optimization, degenerate codon design (DeCoDe), based on integer linear programming. DeCoDe significantly outperforms state-of-the-art DC optimization algorithms and scales well to more than a hundred proteins sharing complex patterns of covariation (e.g. the lab-derived avGFP lineage). Moreover, DeCoDe is, to our knowledge, the first DC design algorithm with the capability to encode mixed-length protein libraries. We anticipate DeCoDe to be broadly useful for a variety of library generation problems, ranging from protein engineering attempts that leverage mutual information to the reconstruction of ancestral protein states. Availability and implementation github.com/OrensteinLab/DeCoDe. Contact yaronore@bgu.ac.il Supplementary information Supplementary data are available at Bioinformatics online.
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Chen, Lou, and Guo. "A Robust Formulation Model for Multi-Period Failure Restoration Problems in Integrated Energy Systems." Energies 12, no. 19 (2019): 3673. http://dx.doi.org/10.3390/en12193673.
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The risks faced by modern energy systems are increasing, primarily caused by natural disasters. As a new form of multi-level energy complimentary utilization, integrated energy systems are attracting more and more attention for their high-efficiency and low-cost. However, due to the deep coupling relationship between systems, they are more susceptible to natural disasters, resulting in a cascading failure. To enhance the resilience of the integrated electricity-gas system, this paper proposes a failure restoration strategy after a natural disaster occurs. First, the temporal constraints of the dispatching model are considered, and the failure restoration problem is molded into a multi-period mixed-integer linear programme, aiming to recover the interrupted loads as much as possible. Second, since the uncertain output of distributed generation sources (DGs) such as wind turbines and photovoltaic systems will threat the reliability of restoration results, the robust formulation model is incorporated to cope with this problem. Third, we propose a new modeling method for radial topology constraints towards failure restoration. Moreover, the Column and Constraints Generation (C&CG) decomposition method is utilized to solve the robust model. Then, the piecewise linearization technique and the linear DistFlow equations are utilized to eliminate the nonlinear terms, providing a model that could be easily solved by an off-shelf commercial solver. The obtained results include the sequence of line/pipeline switchgear actions, the time-series dispatching results of electricity storage system, gas storage system, and the coupling devices including the gas-fired turbine, power to gas equipment. Finally, the effectiveness of the proposed restoration strategy is verified by numerical simulation on a 13-6 node integrated energy system.
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Birbrair, Lev, Marlon Gomes, and Wanderley Pereira. "Resonance sequences and recoverability." International Journal of Number Theory 11, no. 02 (2015): 495–506. http://dx.doi.org/10.1142/s1793042115500256.
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Let a = {a1,…, ak} be a finite multiset of positive integers. Consider the sequence of all positive integer multiples of all ais, and note the multiplicity of each term in this sequence. This sequence of multiplicities is called the resonance sequence generated by {a1,…, ak}. Two multisets are called combinatorially equivalent if they generate the same resonance sequence. Sometimes the initial set can be recovered from the resonance sequence, if we assume some additional information. These sets are called recoverable. The paper gives some criteria of recoverability for a rather generic class of generating sets.
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Costa, Eudes A., Paula M. M. C. Catarino, and Douglas C. Santos. "A Study of the Symmetry of the Tricomplex Repunit Sequence with Repunit Sequence." Symmetry 17, no. 1 (2024): 28. https://doi.org/10.3390/sym17010028.
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In this paper, we introduce a new family of sequences related to Horadam-type sequences. Specifically, we consider the repunit sequence {rn}n≥0, which is defined by the initial terms r0=0 and r1=1 and follows the Horadam recurrence relation given by rn=11rn−1−10rn−2 for n≥2. Many studies have explored generalizations of integer sequences in different directions: some by preserving the initial terms, some by preserving the recurrence relation, and some by considering different numerical sets beyond positive integers. In this article, we take the third approach. Specifically, we study these sequences in the context of the tricomplex ring T. We define the Tricomplex Repunit sequence {trn}n≥0, with initial terms tr0=(0,1,11) and tr1=(1,11,111), and governed by the recurrence relation trn=11trn−1−10trn−2, for n≥2. This sequence is also a Horadam-type sequence but defined in the tricomplex ring T. In this paper, we establish the properties of the Tricomplex Repunit sequence and establish several new as well as well-known identities associated with it, including Binet’s formula, Tagiuri–Vajda’s identity, d’Ocagne’s identity, and Catalan’s identity. We also derive the generating function for this sequence. Furthermore, we study various additional properties of these generalized sequences and establish results concerning the summation of terms related to the Tricomplex Repunit sequence, and one of our main goals is to determine analogous or symmetrical properties for the Tricomplex Repunit sequence to those we know for the ordinary repunit sequence.
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Sebesta, V. "Data sequences generation by integer mapping." Radioelectronics and Communications Systems 51, no. 3 (2008): 169–72. http://dx.doi.org/10.3103/s0735272708030102.
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Lee, Chong-Dao, and Shao-Hua Hong. "Generation of Long Perfect Gaussian Integer Sequences." IEEE Signal Processing Letters 24, no. 4 (2017): 515–19. http://dx.doi.org/10.1109/lsp.2017.2674972.
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Deveci, Ömür, Yeşim Aküzüm, and Muhammad Eshaq Rashedi. "The Hadamard-type k-step Pell sequences." Notes on Number Theory and Discrete Mathematics 28, no. 2 (2022): 339–49. http://dx.doi.org/10.7546/nntdm.2022.28.2.339-349.
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In this paper, we define the Hadamard-type k-step Pell sequence by using the Hadamard-type product of characteristic polynomials of the Pell sequence and the k-step Pell sequence. Also, we derive the generating matrices for these sequences, and then we obtain relationships between the Hadamard-type k-step Pell sequences and these generating matrices. Furthermore, we produce the Binet formula for the Hadamard-type k-step Pell numbers for the case that k is odd integers and k ≥ 3. Finally, we derive some properties of the Hadamard-type k-step Pell sequences such as the combinatorial representation, the generating function, and the exponential representation by using its generating matrix.
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Kannavou, Zampara, and Capros. "Modelling the EU Internal Electricity Market: The PRIMES-IEM Model." Energies 12, no. 15 (2019): 2887. http://dx.doi.org/10.3390/en12152887.
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The paper presents a newly built model used to simulate the European Union (EU) internal electricity market and assess market reform policies. The model performs an hourly simulation of all stages of the wholesale markets at a Pan-European scale, covering the sequence of day-ahead, intra-day, and balancing/reserve auctions. The model includes market coupling in all market stages, estimates scarcity bidding by generators endogenously, and determines electricity trade as a flow-based allocation of interconnections via the market auctions implicitly. The model solves a unit-commitment program, formulated as a mixed-integer optimisation problem, under demand and generation constraints, interconnection possibilities, technical restrictions of the cyclic operation of power plants, and the provision of ancillary services. The novelty of this approach is the inclusion of distortions in all stages of the markets to evaluate the impacts of their removal, and the operation of the markets in a segmented versus an integrated manner in the EU. The model calculates revenues and costs per power plant in the EU on a country basis and the value of cross-border flows. The model evaluated market reform measures, including the abolishment of priority dispatch of renewable energy plants, the establishment of flow-based allocation of interconnectors without NTC limitations, the activation of demand response, and the market coupling in intra-day markets. The model application has been in the context of the electricity market design initiative included in the “Clean Energy for all Europeans” policy package proposed by the European Commission in 2016.
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Xia, Chaoyu, Yi Wen, Minghua Hu, Hanbing Yan, Changbo Hou, and Weidong Liu. "Microscopic-Level Collaborative Optimization Framework for Integrated Arrival-Departure and Surface Operations: Integrated Runway and Taxiway Aircraft Sequencing and Scheduling." Aerospace 11, no. 12 (2024): 1042. https://doi.org/10.3390/aerospace11121042.
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Integrated arrival–departure and surface scheduling (IADS) is a critical research task in next-generation air traffic management that aims to harmonize the complex and interrelated processes of airspace and airport operations in the Metroplex. This paper investigates the microscopic-level collaborative optimization framework for IADS operations, i.e., the problem of coordinating aircraft scheduling on runways and taxiways. It also describes the mixed-integer linear programming (MILP) bi-layer decision support for solving this problem. In runway scheduling, a combined arrival–departure scheduling method is introduced based on our previous research, which can identify the optimal sequence of arrival and departure streams to minimize runway delays. For taxiway scheduling, the Multi-Route Airport Surface Scheduling Method (MASM) is proposed, aiming to determine the routes and taxi metering for each aircraft while minimizing the gap compared with the runway scheduling solution. Furthermore, this paper develops a feedback mechanism to further close the runway and taxiway schedule deviation. To demonstrate the universality and validity of the proposed bi-layer decision support method, two hub airports, Chengdu Shuangliu International Airport (ICAO code: ZUUU) and Chengdu Tianfu International Airport (ICAO code: ZUTF), within the Cheng-Yu Metroplex, were selected for validation. The obtained results show that the proposed method could achieve closed-loop decision making for runway scheduling and taxiway scheduling and reduce runway delay and taxi time. The key anticipated mechanisms of benefits from this research include improving the efficiency and predictability of operations on the airport surface and maintaining situational awareness and coordination between the stand and the tower.
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Catarino, P., and Almeida de. "On a quaternionic sequence with Vietoris’ numbers." Filomat 35, no. 4 (2021): 1065–86. http://dx.doi.org/10.2298/fil2104065c.
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Special integers sequences have been the center of attention for many researchers, as well as the sequences of quaternions where its components are the elements of these sequences. Motivated by a rational sequence, we consider the quaternions with components Vietoris? numbers and investigate some of its properties. For this sequence a two and three term recurrence relation is established, as well as a Binet?s type formula. Moreover the generating function for this sequence is introduced and also the determinant of some tridiagonal matrices are used in order to find elements of this sequence.
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Gopalan, M., S. Vidhyalakshmi, and N. Thiruniraiselvi. "CONSTRUCTION OF IRRATIONAL GAUSSIAN DIOPHANTINE QUADRUPLES." International Journal of Engineering Technologies and Management Research 1, no. 1 (2020): 1–7. http://dx.doi.org/10.29121/ijetmr.v1.i1.2015.20.
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Given any two non-zero distinct irrational Gaussian integers such that their product added with either 1 or 4 is a perfect square, an irrational Gaussian Diophantine quadruple ( , ) a0 a1, a2, a3 such that the product of any two members of the set added with either 1 or 4 is a perfect square by employing the non-zero distinct integer solutions of the system of double Diophantine equations. The repetition of the above process leads to the generation of sequences of irrational Gaussian Diophantine quadruples with the given property.
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Schmidt, Maxie D. "Zeta series generating function transformations related to polylogarithm functions and the k-order harmonic numbers." Online Journal of Analytic Combinatorics, no. 12 (December 31, 2017): 1–22. https://doi.org/10.61091/ojac-1202.
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We define a new class of generating function transformations related to polylogarithm functions, Dirichlet series, and Euler sums. These transformations are given by an infinite sum over the jth derivatives of a sequence generating function and sets of generalized coefficients satisfying a non-triangular recurrence relation in two variables. The generalized transformation coefficients share a number of analogous properties with the Stirling numbers of the second kind and the known harmonic number expansions of the unsigned Stirling numbers of the first kind. We prove a number of properties of the generalized coefficients which lead to new recurrence relations and summation identities for the k-order harmonic number sequences. Other applications of the generating function transformations we define in the article include new series expansions for the polylogarithm function, the alternating zeta function, and the Fourier series for the periodic Bernoulli polynomials. We conclude the article with a discussion of several specific new “almost” linear recurrence relations between the integer-order harmonic numbers and the generalized transformation coefficients, which provide new applications to studying the limiting behavior of the zeta function constants, ζ(k), at integers k ≥ 2.
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Cevikcan, Emre. "An optimization methodology for multi model walking-worker assembly systems: an application from busbar energy distribution systems." Assembly Automation 36, no. 4 (2016): 439–59. http://dx.doi.org/10.1108/aa-09-2015-072.
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Purpose Walking-worker assembly lines can be regarded as an effective method to achieve the above-mentioned characteristics. In such systems, workers, following each other, travel workstations in sequence by performing all of the required tasks of their own product. As the eventual stage of assembly line design, efforts should be made for capacity adjustments to meet the demand in terms of allocating tasks to workers via assembly line balancing. In this context, the purpose of this study is to address the balancing problem for multi-model walking-worker assembly systems, with the aim of improving planning capability for such systems by means of developing an optimization methodology. Design/methodology/approach Two linear integer programming models are proposed to balance a multi-model walking-worker assembly line optimally in a sequential manner. The first mathematical programming model attempts to determine number of workers in each segment (i.e. rabbit chase loop) for each model. The second model generates stations in each segment to smooth workflow. What is more, heuristic algorithms are provided due to computational burden of mathematical programming models. Two segment generation heuristic algorithms and a station generation heuristic algorithm are provided for the addressed problem. Findings The application of the mathematical programming approach improved the performance of a tap-off box assembly line in terms of number of workers (9.1 per cent) and non-value-added time ratio (between 27.9 and 26.1 per cent for different models) when compared to a classical assembly system design. In addition, the proposed approach (i.e. segmented walking-worker assembly line) provided a more convenient working environment (28.1 and 40.8 per cent shorter walking distance for different models) in contrast with the overall walking-worker assembly line. Meanwhile, segment generation heuristics yielded reduction in labour requirement for a considerable number (43.7 and 49.1 per cent) of test problems. Finally, gaps between the objective values and the lower bounds have been observed as 8.3 per cent (Segment Generation Heuristic 1) and 6.1 (Segment Generation Heuristic 2). Practical implications The proposed study presents a decision support for walking-worker line balancing with high level of solution quality and computational performance for even large-sized assembly systems. That being the case, it contributes to the management of real-life assembly systems in terms of labour planning and ergonomics. Owing to the fact that the methodology has the potential of reducing labour requirement, it will present the opportunity of utilizing freed-up capacity for new lines in the start-up period or other bottleneck processes. In addition, this study offers a working environment where skill of the workers can be improved within reasonable walking distances. Originality/value To the best knowledge of the author, workload balancing on multi-model walking-worker assembly lines with rabbit chase loop(s) has not yet been handled. Addressing this research gap, this paper presents a methodology including mathematical programming models and heuristic algorithms to solve the multi-model walking-worker assembly line balancing problem for the first time.
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Bertsimas, Dimitris, Allison Chang, Velibor V. Mišić, and Nishanth Mundru. "The Airlift Planning Problem." Transportation Science 53, no. 3 (2019): 773–95. http://dx.doi.org/10.1287/trsc.2018.0847.
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The U.S. Transportation Command (USTRANSCOM) is responsible for planning and executing the transportation of U.S. military personnel and cargo by air, land, and sea. The airlift planning problem faced by the air component of USTRANSCOM is to decide how requirements (passengers and cargo) will be assigned to the available aircraft fleet and the sequence of pickups and drop-offs that each aircraft will perform to ensure that the requirements are delivered with minimal delay and with maximum utilization of the available aircraft. This problem is of significant interest to USTRANSCOM because of the highly time-sensitive nature of the requirements that are typically designated for delivery by airlift, as well as the very high cost of airlift operations. At the same time, the airlift planning problem is extremely difficult to solve because of the combinatorial nature of the problem and the numerous constraints present in the problem (such as weight restrictions and crew rest requirements). In this paper, we propose an approach for solving the airlift planning problem faced by USTRANSCOM based on modern, large-scale optimization. Our approach relies on solving a large-scale mixed-integer programming model that disentangles the assignment decision (which aircraft will pickup and deliver which requirement) from the sequencing decision (in what order the aircraft will pickup and deliver its assigned requirements), using a combination of heuristics and column generation. Through computational experiments with both a simulated data set and a planning data set provided by USTRANSCOM, we show that our approach leads to high-quality solutions for realistic instances (e.g., 100 aircraft and 100 requirements) within operationally feasible time frames. Compared with a baseline approach that emulates current practice at USTRANSCOM, our approach leads to reductions in total delay and aircraft time of 8%–12% in simulated data instances and 16%–40% in USTRANSCOM’s planning instances.
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Fang, Hanzhang. "Three-dimensional tilings of blocks and bracelets and related {2,1,2} sequences." Applied and Computational Engineering 19, no. 1 (2023): 90–102. http://dx.doi.org/10.54254/2755-2721/19/20231014.
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This paper discusses the combinatorial interpretation of the H_n numbers, where each Hn denotes the number of ways to tile a 2 2 n block with 2 2 1 plates and 6-block L shapes. It then investigates a closely related tiling sequence, which is tiling a 22n bracelet with the same two building blocks, and discusses its relation with Hn. The recursive equation for both integer sequences are found using one to one correspondence, induction and Newtons Sum. Additionally, in the case of H_n numbers, its related Lucas Style sequence P_n is found. The relationship between the P_n numbers, the Bn numbers and the Hn numbers is similar to that of the Lucas numbers and the Fibonacci numbers. Finally, identities concerning H_ns generating function and its relations with other existing {2,1,2} sequences are discussed, and a theorem that generalizes the generating function of all tiling sequences is proposed and proved.
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Statsenko, I. V. "APPLICATION OF MULTIHARMONIC NUMBERS FOR THE SYNTHESIS OF CLOSED FORMS OF PARAMETRICALLY MODIFIED FACTORIAL GENERATING SEQUENCES." Prikladnaya Diskretnaya Matematika, no. 55 (2022): 5–13. http://dx.doi.org/10.17223/20710410/55/1.
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A toolkit and a method for reducing sequences of integers belonging to the class of factorial-generating recursions to a closed form are presented. The signs and properties of the modified factorial-generating recursion of one and two variables are determined. The best-known factorial-generating recursion of two variables is the sequence of Stirling numbers of the first kind. Modified hyperharmonic numbers are used to synthesize an analytical recursion model. The advantages of these numbers for constructing closed forms of factorial-generating recursions are revealed. An incomplete closed form of the sequence of Stirling numbers of the first kind is synthesized.
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Bahig, Hazem, and Yasser Kotb. "An Efficient Multicore Algorithm for Minimal Length Addition Chains." Computers 8, no. 1 (2019): 23. http://dx.doi.org/10.3390/computers8010023.
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A minimal length addition chain for a positive integer m is a finite sequence of positive integers such that (1) the first and last elements in the sequence are 1 and m, respectively, (2) any element greater than 1 in the sequence is the addition of two earlier elements (not necessarily distinct), and (3) the length of the sequence is minimal. Generating the minimal length addition chain for m is challenging due to the running time, which increases with the size of m and particularly with the number of 1s in the binary representation of m. In this paper, we introduce a new parallel algorithm to find the minimal length addition chain for m. The experimental studies on multicore systems show that the running time of the proposed algorithm is faster than the sequential algorithm. Moreover, the maximum speedup obtained by the proposed algorithm is 2.5 times the best known sequential algorithm.
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Han, Lin, Guangyan Zhu, and Zongbing Lin. "On the rationality of generating functions of certain hypersurfaces over finite fields." AIMS Mathematics 8, no. 6 (2023): 13898–906. http://dx.doi.org/10.3934/math.2023711.
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<abstract><p>Let $ a, n $ be positive integers and let $ p $ be a prime number. Let $ \mathbb F_q $ be the finite field with $ q = p^a $ elements. Let $ \{a_i\}_{i = 1}^\infty $ be an arbitrary given infinite sequence of elements in $ \mathbb F_q $ and $ a_1\neq 0 $. For each positive integer $ i $, let $ \{d_{i+j, i}\}_{j = 0}^\infty $ be an arbitrary given sequence of positive integers with $ d_{ii} $ coprime to $ q-1 $. For each integer $ n\ge 1 $, let $ N_n $, $ \bar N_n $ and $ \widetilde{N}_n $ denote the number of $ \mathbb F_q $-rational points of the hypersurfaces defined by the following three equations:</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ a_1x_1+\cdots+a_nx_n = b, $\end{document} </tex-math></disp-formula></p> <p><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ x_1^2+\cdots+x_n^2 = b $\end{document} </tex-math></disp-formula></p> <p>and</p> <p><disp-formula> <label/> <tex-math id="FE3"> \begin{document}$ a_1 x_1^{d_{11}}+a_2 x_1^{d_{21}}x_2^{d_{22}}+ \cdots+a_n x_1^{d_{n1}}x_2^{d_{n2}} \cdots x_n^{d_{nn}} = b, $\end{document} </tex-math></disp-formula></p> <p>respectively. In this paper, we show that the generating function $ \sum_{n = 1}^{\infty}N_nt^n $ is a rational function in $ t $. Moreover, we show that if $ p $ is an odd prime, then the generating functions $ \sum_{n = 1}^{\infty}\bar N_nt^n $ and $ \sum_{n = 1}^{\infty}\widetilde{N}_nt^n $ are both rational functions in $ t $. Moreover, we present the explicit rational expressions of $ \sum_{n = 1}^{\infty}N_nt^n $, $ \sum_{n = 1}^{\infty}\bar N_nt^n $ and $ \sum_{n = 1}^{\infty}\widetilde{N}_nt^n $, respectively.</p></abstract>
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Akala, Omobolaji Oyekunle, Ronak Shah, Brian R. Untch, et al. "Widespread somatic loss of heterozygosity as a possible marker of disease aggressiveness in metastatic well differentiated pancreatic neuroendocrine tumors." Journal of Clinical Oncology 36, no. 4_suppl (2018): 275. http://dx.doi.org/10.1200/jco.2018.36.4_suppl.275.
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275 Background: To assess the utility of molecular profiling to predict tumor biology, we performed next-generation sequencing (NGS) of metastatic well differentiated (WD) pancreatic neuroendocrine tumors (panNETs) in the routine practice setting. Our institutional NGS platform allows for evaluation of genetic alterations that contribute to tumorigenesis (sequence variants, copy number alterations, insertions/deletions, select rearrangements) as well as somatic loss of heterozygosity (LOH). Methods: NGS was performed using MSK-IMPACT, a matched tumor-normal sequencing platform that interrogates 468 cancer-related genes. LOH for all cases was determined by analysis of total, allele-specific, and integer DNA copy number genome-wide using the FACETS algorithm. Results: Since 2014, NGS has been performed in 96 tumor samples from 80 patients. The most commonly altered genes were MEN1 (56%), DAXX (40%), ATRX (25%), and TSC2 (25%). LOH was highly prevalent, identified in 51/96 samples (53%). A significant association (q-value < 0.05) was noted between LOH and the presence of altered MEN1, DAXX, PTEN, or TSC2. LOH was recurrently observed in chromosomes 1, 2, 3, 6, 8, 10, 11, 15, 16, 21, and 22. Since MEN1 alterations predict improved outcome, comparison was made between patients with LOH present/MEN1 wild type (WT) tumors (n = 16) and LOH absent/MEN1 altered tumors (n = 21); inferior overall survival (OS) was noted with LOH present/MEN1 WT status (p < 0.01). High grade pathology was observed in 6/16 (38%) LOH present/MEN1 WT tumors and in 1/21 (5%) LOH absent/MEN1 altered tumors (p < 0.0001). The mean Ki-67 index of LOH present/MEN1 WT tumors was 22% and that of LOH absent/MEN1 altered tumors was 9%. Conclusions: This is the first study to report widespread somatic LOH in metastatic WD panNETs. PanNETs appear to exhibit more LOH overall than virtually any tumor type studied. In this cohort, LOH was a prognostic marker of inferior OS and was associated with more aggressive pathologic features. LOH warrants further investigation as it may help better characterize these clinically heterogeneous tumors.
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Falcon, Sergio. "Rational Generating Functions of Numerical Sequences." Journal of Advances in Mathematics and Computer Science 39, no. 9 (2024): 1–13. http://dx.doi.org/10.9734/jamcs/2024/v39i91923.
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If division is performed on a rational (non-integer) function, an infinite series is obtained that is relatively easy to find. But the inverse problem can also be solved and, given an infinite numerical sequence, the rational function that that can generate it can be found. In this article, different cases are studied in which this generating function can be found in a more or less simple way.
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Lennerstad, Håkan. "The n-dimensional Stern–Brocot tree." International Journal of Number Theory 15, no. 06 (2019): 1219–36. http://dx.doi.org/10.1142/s1793042119500672.
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This paper generalizes the Stern–Brocot tree to a tree that consists of all sequences of [Formula: see text] coprime positive integers. As for [Formula: see text] each sequence [Formula: see text] is the sum of a specific set of other coprime sequences, its Stern–Brocot set [Formula: see text], where [Formula: see text] is the degree of [Formula: see text] With an orthonormal base as the root, the tree defines a fast iterative structure on the set of distinct directions in [Formula: see text] and a multiresolution partition of [Formula: see text]. Basic proofs rely on a matrix representation of each coprime sequence, where the Stern–Brocot set forms the matrix columns. This induces a finitely generated submonoid [Formula: see text] of [Formula: see text], and a unimodular multidimensional continued fraction algorithm, also generalizing [Formula: see text]. It turns out that the [Formula: see text]-dimensional subtree starting with a sequence [Formula: see text] is isomorphic to the entire [Formula: see text]-dimensional tree. This allows basic combinatorial properties to be established. It turns out that also in this multidimensional version, Fibonacci-type sequences have maximal sequence sum in each generation.
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Yarmolik, V. N., N. A. Shevchenko, V. А. Levantsevich, and D. V. Demenkovets. "Generation of address sequences with specified switching activity and address repeatability." Informatics 19, no. 3 (2022): 7–24. http://dx.doi.org/10.37661/1816-0301-2022-19-3-7-24.
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Objectives. The problem of developing a methodology for generating address sequences with a given switching activity and repeatability of addresses widely used in testing modern computing systems is being solved. The relevance of this problem lies in the fact that the main characteristic of the difference and their effectiveness for address sequences is the switching activity of both individual address bits and their sequences.Methods. Presented results are based on a universal method for generating quasi-random Sobol sequences, which are effectively used to generate targeted test sequences. As an initial mathematical model, a modification of the indicated generation method proposed by Antonov and Saleev is used. The main idea of proposed approach is based on the use of rectangular (m + k) × m generating matrices V of arbitrary rank r to generate address sequences.Results. The main properties of sequences generated in accordance with the new mathematical model are determined. A number of statements are given that substantiate the requirements for generator matrices to ensure the maximum period of generated sequences and the multiplicity of repetition of used addresses. The problem of synthesizing the sequences with given values of switching activity F(A) and F(ai) is solved. It is shown that in order to find a generating matrix for generating such sequences, it is necessary to solve the problem of decomposing an integer into terms. This decomposition represents the value of switching activity in the (m + k)-ary mixed number system, in which the weights of the digits are represented as powers of two from 20 to 2m+k-1, and the values of the digits w(vi) lie in the range from 0 to m+k-1. On the basis of proposed restrictions, the notion of an integer decomposition diagram similar to the Young diagram is introduced, and the operation of its modification is defined.Conclusion. The proposed mathematical model expands the possibilities of generating test address sequences with the required values of switching activity of both test sets and their individual bits. The use of generating matrices of non-maximal rank makes it possible to formalize the method of generating address sequences with even repetition of addresses.
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Ojera, Dariel. "Unveiling the Properties and Relationship of Yellowstone Permutation Sequence." Psychology and Education: A Multidisciplinary Journal 27, no. 2 (2024): 173–84. https://doi.org/10.5281/zenodo.13993059.
Full textAbstract:
This paper explores a mathematical sequence known as the Yellowstone permutation, introduced by Zumkeller (2004). This sequence, characterized by alternating even and odd integers with prime and composite number patterns, is studied for its unique properties and connections to mathematical structures like Pythagorean triples and quadruples. The research employs descriptive and expository methods to explore the sequence’s nature, establishing it as infinite, containing infinitely many primes, and ensuring that all integers appear at least once. The paper also delves into how the Yellowstone permutation sequence can generate both primitive and non-primitive Pythagorean triples and quadruples. It demonstrates that the expressions derived from terms in this sequence consistently yield these triples and quadruples through a combination of algebraic properties and geometric interpretations. Additionally, the study formulates propositions to clarify the relationships between the terms of the Yellowstone permutation sequence and their behavior, particularly in generating Pythagorean constructs. The findings underscore the sequence's intriguing mathematical characteristics, offering insights into number theory and its potential applications. This work highlights the role of such sequences in exploring deeper mathematical relationships and fostering curiosity in combinatorial number theory.
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Goenawan, Stephanus Ivan. "Fractional Generating Function from The Square Root of Two with A-B Goen Numbers." Engineering, MAthematics and Computer Science (EMACS) Journal 4, no. 1 (2022): 11–14. http://dx.doi.org/10.21512/emacsjournal.v4i1.8073.
Full textAbstract:
The square root of two is an irrational number that cannot be written as a fraction of the numerator and denominator. By using the generating function of A-B Goen, from the resulting set of numbers, the sequence of A-B Goen numbers can be obtained by selecting integer numbers. The A Goen numbers are generated from the generator function which have integer numbers, while the B Goen numbers are obtained from the sequence numbers. In this study, through the generating function of A-B Goen, it can be proven that the division between the A Goen number and the B Goen number in an infinite sequence will result the value of the square root of two.