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division algorithm and Kronecker's approximation theorem

JOURNAL OF ANALYSIS 2021年第1期29卷 207-216页

作者： Aggarwal, Gaurav Singh, Jitender Guru Nanak Dev Univ Dept Math Amritsar Punjab India

Using division algorithm and basic notions of convergence of sequences in real-line, we prove that a real number theta is irrational if and only if there is an eventually nonconstant sequence {pn theta +qn} converging to 0, where pn and qn are integers for each natural number n. This approach leads to alternative proofs of weaker versions of the classical Dirichlet and Kronecker approximation theorems in number theory.

关键词： division algorithm Dirichlet Theorem Kronecker's Theorem Diophantine approximation 11J68 11J20 11J72 11D72

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Modeling, design, and optimization of a cost-effective and reliable hybrid renewable energy system integrated with desalination using the division algorithm

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INTERNATIONAL JOURNAL OF ENERGY RESEARCH 2021年第1期45卷 429-452页

作者： Kiehbadroudinezhad, Mohammadali Rajabipour, Ali Cada, Michael Khanali, Majid Univ Tehran Fac Engn & Technol Dept Mech Engn Agr Machinery Tehran Iran Univ Dalhousie Dept Elect & Comp Engn Halifax NS Canada

People in the Middle East are facing the problem of freshwater shortages. This problem is more intense for a remote region, which has no access to the power grid. The use of seawater desalination technology integrated with the generated energy unit by renewable energy sources could help overcome this problem. In this study, we refer a seawater reverse osmosis desalination (SWROD) plant with a capacity of 1.5 m(3)/h used on Larak Island, Iran. Moreover, for producing fresh water and meet the load demand of the SWROD plant, three different stand-alone hybrid renewable energy systems (SAHRES), namely wind turbine (WT)/photovoltaic (PV)/battery bank storage (BBS), PV/BBS, and WT/BBS are modeled and investigated. The optimization problem was coded in MATLAB software. Furthermore, the optimized results were obtained by the division algorithm (DA). The DA has been developed to solve the sizing problem of three SAHRES configurations by considering the object function's constraints. These results show that this improved algorithm has been simpler, more precise, faster, and more flexible than a genetic algorithm (GA) in solving problems. Moreover, the minimum total life cycle cost (TLCC = 243 763$), with minimum loss of power supply probability (LPSP = 0%) and maximum reliability, was related to the WT/PV/BBS configuration. WT/PV/BBS is also the best configuration to use less battery as a backup unit (69 units). The batteries in this configuration have a longer life cycle (maximum average of annual battery charge level) than two other configurations (93.86%). Moreover, the optimized results have shown that utilizing the configuration of WT/PV/BBS could lead to attaining a cost-effective and green (without environmental pollution) SAHRES, with high reliability for remote areas, with appropriate potential of wind and solar irradiance.

关键词： division algorithm hybrid system life cycle cost optimization sizing reliability renewable energy

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Optimization of wind/solar energy microgrid by division algorithm considering human health and environmental impacts for power-water cogeneration

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ENERGY CONVERSION AND MANAGEMENT 2022年 252卷 115064-115064页

作者： Kiehbadroudinezhad, Mohammadali Merabet, Adel Rajabipour, Ali Cada, Michael Kiehbadroudinezhad, Shahideh Khanali, Majid Hosseinzadeh-Bandbafha, Homa St Marys Univ Div Engn Halifax NS B3H 3C3 Canada Dalhousie Univ Dept Elect & Comp Engn Halifax NS Canada Univ Tehran Fac Agr Engn & Technol Dept Mech Engn Agr Machinery Tehran Iran Dalhousie Univ Dept Math & Stat Halifax NS Canada

Although freshwater is necessary for the well-being of humankind, increasing population growth and limited resources lead to a serious crisis to supply freshwater. Since the Earth is surrounded by seawater, desalination based on electrical power is introduced as a promising technology to provide freshwater. It is well documented that the connection of remote areas that usually do not have access to freshwater into the electricity grid is not affordable and eco-friendly. Hence, the efforts to design and construct high reliability, cost-effective, and ecofriendly stand-alone hybrid renewable energy system in remote areas. In line with this, this paper describes a novel energy management system for the optimized operation of a stand-alone hybrid energy system based on photovoltaic panels, wind turbines, batteries, and diesel generator. For this purpose, a multi-objective optimization problem is formulated by combining three objective functions, i.e., minimum the total life cycle cost as well as environmental impacts on human health and ecosystems and the maximum system reliability that can conflict with each. To solve the multi-objective optimization problem, a division algorithm is proposed that is more flexible and faster compared with conventional algorithms such as genetic algorithm. In order to show the proposed framework, a real case study in Larak Island, Iran, with appropriate solar and wind is considered. The effectiveness of the applied approach compared with optimization results of genetic algorithm and the artificial bee swarm optimization algorithm that was previously used successfully to solve optimization problems related to desalination integrated with the renewable energy system. The optimization is performed based on different diesel fuel price amounts (0.2, 0.5, and 1 $/liter). It is seen that at fuel price set to 0.2 and 0.5 $/liter, the seawater reverses osmosis desalination/photovoltaic/diesel generator/battery is the most cost-effective energy sy

关键词： Optimization Renewable energy Life cycle cost Environmental impacts division algorithm Desalination

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Optimization of Wind Energy Battery Storage Microgrid by division algorithm Considering Cumulative Exergy Demand for Power-Water Cogeneration

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ENERGIES 2021年第13期14卷 3777页

作者： Kiehbadroudinezhad, Mohammadali Merabet, Adel Hosseinzadeh-Bandbafha, Homa St Marys Univ Div Engn Halifax NS B3H 3C3 Canada Univ Tehran Fac Agr Engn & Technol Dept Mech Engn Agr Machinery Tehran *** Iran

This study investigates the use of division algorithms to optimize the size of a desalination system integrated with a microgrid based on a wind turbine plant and the battery storage to supply freshwater based on cost, reliability, and energy losses. Cumulative exergy demand is used to identify and minimize the energy losses in the optimized system. division algorithms are used to overcome the drawback of low convergence speed encountered by the well-known method genetic algorithm. The findings indicated that there is a positive relationship between cost, cumulative exergy, and reliability. More specifically, when the loss of power supply probability is 10%, compared to when it is 0%, the total cumulative exergy demand and total life cycle cost are reduced by 34.76% when the battery is full and 45.44% when the battery is empty and there is a 44.43% decrease in total life cycle cost, respectively. However, the more reliable system, the less exergy is lost during the production of 1 m(3) freshwater by desalination integrated into wind turbine plant.

关键词： wind energy cumulative exergy demand reliability optimization division algorithm desalination

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The division algorithm in Sextic Truncated Moment Problems

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INTEGRAL EQUATIONS AND OPERATOR THEORY 2017年第4期87卷 515-528页

作者： Curto, Raul E. Yoo, Seonguk Univ Iowa Dept Math Iowa City IA 52242 USA Sungkyunkwan Univ Dept Math Suwon South Korea

For a degree 2n finite sequence of real numbers beta beta((2n)) = {beta(00,) beta 10, beta 01,..., beta 2n, 0, beta 2n-1,1,.., beta 1,2n-1, beta 0,2n}to have a representing measure mu, it is necessary for the associated moment matrix M(n) to be positive semidefinite, and for the algebraic variety associated to beta, nu beta nu(M(n)),, to satisfy rank M(n) <= card nu(beta) as well as the following consistency condition: if a polynomial p(x,y) equivalent to Sigma(ij) a(ij) x(4) y(i) of degree at most 2n vanishes on V beta, then the Riesz functional Lambda(p) equivalent to p(beta) := Sigma(ij) a(ij) beta(ij)=0. Positive semidefiniteness, recursiveness, and the variety condition of a moment matrix are necessary and sufficient conditions to solve the quadratic (n=1) and quartic (n=2) moment problems. Also, positive semidefiniteness, combined with consistency, is a sufficient condition in the case of extremal moment problems, i.e., when the rank of the moment matrix (denoted by r) and the cardinality of the associated algebraic variety (denoted by v) are equal. For extremal sextic moment problems, verifying consistency amounts to having good representation theorems for sextic polynomials in two variables vanishing on the algebraic variety of the moment sequence. We obtain such representation theorems using the division algorithm from algebraic geometry. As a consequence, we are able to complete the analysis of extremal sextic moment problems.

关键词： Extremal moment problems Algebraic variety division algorithm Riesz functional Consistency

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Efficient multiple-precision integer division algorithm

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INFORMATION PROCESSING LETTERS 2014年第3期114卷 152-157页

作者： Mukhopadhyay, Debapriyay Nandy, Subhas C. IXIA Technol Pvt Ltd Kolkata 700091 India Indian Stat Inst Kolkata 700108 India

Design and implementation of division algorithm is one of the most complicated problems in multi-precision arithmetic. Huang et al. [1] proposed an efficient multi-precision integer division algorithm, and experimentally showed that it is about three times faster than the most popular algorithms proposed by Knuth [2] and Smith [3]. This paper reports a bug in the algorithm of Huang et al. [1], and suggests the necessary corrections. The theoretical correctness proof of the proposed algorithm is also given. The resulting algorithm remains as fast as that of [1]. (C) 2013 Elsevier BM. All rights reserved.

关键词： division algorithm Normalization Computational arithmetic Cryptography

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An optimised solar-based microgrid integrated with desalination to enhance exergy sustainability

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INTERNATIONAL JOURNAL OF EXERGY 2023年第2期41卷 111-130页

作者： Kiehbadroudinezhad, Mohammadali Merabet, Adel Hosseinzadeh-Bandbafha, Homa St Marys Univ Div Engn Halifax NS B3H 3C3 Canada Univ Tehran Coll Agr & Nat Resources Fac Agr Engn & Technol Dept Mech Engn Agr Machinery Karaj Iran

The size of a solar microgrid system should be designed based on aspects of sustainability, such as cost, exergy, etc. This study by the division algorithm finds the optimal size with maximum reliability vs. the cumulative exergy demand. Larak Island, Iran, is used as an example of how the concept could be applied to the real world. According to the introduced optimal size, 1 m(3) of freshwater generated by the solar energy system leads to an average cumulative exergy demand of 17 MJ. While integrating a diesel generator into system means, exergy demand reaches 33 MJ/m(3) of freshwater.

关键词： solar energy cumulative exergy demand desalination optimisation reliability division algorithm

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Environmental assessment of optimized renewable energy-based microgrids integrated desalination plant: considering human health, ecosystem quality, climate change, and resources

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ENVIRONMENTAL SCIENCE AND POLLUTION RESEARCH 2023年第11期30卷 29888-29908页

作者： Kiehbadroudinezhad, Mohammadali Merabet, Adel Hosseinzadeh-Bandbafha, Homa Ghenai, Chaouki St Marys Univ Div Engn Halifax NS B3H 3C3 Canada Univ Tehran Fac Agr Engn & Technol Dept Mech Engn Agr Machinery Coll Agr & Nat Resources Karaj Iran Univ Sharjah Sustainable & Renewable Energy Engn Dept Coll Engn Sharjah U Arab Emirates

Using hybrid renewable energy technology is an efficient method for greenhouse gas mitigation caused by fossil fuel combustion. However, these renewable microgrids are not free from environmental damages, especially during the lifetime of hybrid renewable energy systems (HRES). The main objective of this study is to assess the environmental impacts of three optimized HRES for the Sea Water Reverse Osmosis Desalination (SWROD) plant. An objective optimization was developed using the division algorithm, and the environmental impacts of the optimized HRES were investigated by the life cycle assessment approach. The results showed that producing 1 m(3) freshwater by an optimal size SWROD integrated with wind turbine/battery is responsible for 3.56E - 07 disability-adjusted life year (DALY). It is significantly less than 1 m(3) freshwater production by an optimal size SWROD integrated with solar PV/battery (5.88E - 07 DALY) and solar PV/wind turbine/battery (5.13E - 07 DALY) energy systems. Moreover, 1 m(3) freshwater by a SWROD integrated with proposed microgrids in this study led to a damage of 0.089 to 0.193 potentially disappeared fraction of species (PDF)*m2*yr to ecosystem quality. It also results in an emission of 0.143 to 0.339 kg CO2 eq per 1 m(3) freshwater. Furthermore, resources for 1 m(3) freshwater production by a SWROD are calculated at 2.77 to 4.806 MJ primary. Freshwater production by an optimal size SWROD integrated with solar wind/battery compared with solar PV/battery and solar PV/wind turbine/battery had less damage to ecosystem quality, climate, and resources. The results showed reductions of 91.23% in human health, 73.51% in an ecosystem quality, 92.43% in climate change, and 90.08% in resources for producing 1 m(3) of freshwater using SWROD integrated with wind turbine/battery bank compared to fossil-based desalination. Finally, the result showed that solving the optimization problem using the division algorithm compared to other algorithms leads

关键词： Environmental impacts Life cycle assessment Objective optimization division algorithm Desalination

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Research on Euclidean algorithm and Reection on Its Teaching

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应用数学 2025年第1期38卷 308-310页

作者： ZHANG Shaohua School of Mathematics and Statistics Yangtze Normal UniversityChongqing 408102China

In this paper,we prove that Euclid's algorithm,Bezout's equation and Divi-sion algorithm are equivalent to each *** result shows that Euclid has preliminarily established the theory of divisibility and the gre... 详细信息

关键词： Euclid's algorithm division algorithm Bezout's equation

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Diophantine Quotients and Remainders with Applications to Fermat and Pythagorean Equations

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American Journal of Computational Mathematics 2023年第1期13卷 199-210页

作者： Prosper Kouadio Kimou François Emmanuel Tanoé Department of Mathematics and Computer Science Félix Houphouet-Boigny National Polytechnic Institute Yamoussoukro Ivory Coast UFR Mathematics of Computer Science Université Félix Houphouet-Boigny Abidjan Ivory Coast

Diophantine equations have always fascinated mathematicians about existence, finitude, and the calculation of possible solutions. Among these equations, one of them will be the object of our research. This is the Pythagoras’- Fermat’s equation defined as follows. (1) when , it is well known that this equation has an infinity of solutions but has none (non-trivial) when . We also know that the last result, named Fermat-Wiles theorem (or FLT) was obtained at great expense and its understanding remains out of reach even for a good fringe of professional mathematicians. The aim of this research is to set up new simple but effective tools in the treatment of Diophantine equations and that of Pythagoras-Fermat. The tools put forward in this research are the properties of the quotients and the Diophantine remainders which we define as follows. Let a non-trivial triplet () solution of Equation (1) such that . and are called the Diophantine quotients and remainders of solution . We compute the remainder and the quotient of b and c by a using the division algorithm. Hence, we have: and et with . We prove the following important results. if and only if and if and only if . Also, we deduce that or for any hypothetical solution . We illustrate these results by effectively computing the Diophantine quotients and remainders in the case of Pythagorean triplets using a Python program. In the end, we apply the previous properties to directly prove a partial result of FLT. .

关键词： Diophantine Equation Modular Arithmetic Fermat-Wiles Theorem Pythagorean Triplets division Theorem division algorithm Python Program Diophantine Quotients Diophantine Remainders

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