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A New Proof for Congruent Number’s Problem via Pythagorician Divisors

Advances in Pure Mathematics 2024年第4期14卷 283-302页

作者： Léopold Dèkpassi Keuméan François emmanuel Tanoé UFR Mathematics and Computer science F&#233lix Houphou&#235t Boigny University Abidjan Cote d&#8217Ivoire

Considering Pythagorician divisors theory which leads to a new parameterization, for Pythagorician triplets ( a,b,c )∈ ℕ 3∗ , we give a new proof of the well-known problem of these particular squareless numbers n∈ ℕ ∗ , called congruent numbers, characterized by the fact that there exists a right-angled triangle with rational sides: ( A α ) 2 + ( B β ) 2 = ( C γ ) 2 , such that its area Δ= 1 2 A α B β =n;or in an equivalent way, to that of the existence of numbers U 2 , V 2 , W 2 ∈ ℚ 2∗ that are in an arithmetic progression of reason n;Problem equivalent to the existence of: ( a,b,c )∈ ℕ 3∗ prime in pairs, and f∈ ℕ ∗ , such that: ( a−b 2f ) 2 , ( c 2f ) 2 , ( a+b 2f ) 2 are in an arithmetic progression of reason n;And this problem is also equivalent to that of the existence of a non-trivial primitive integer right-angled triangle: a 2 + b 2 = c 2 , such that its area Δ= 1 2 ab=n f 2 , where f∈ ℕ ∗ , and this last equation can be written as follows, when using Pythagorician divisors: (1) Δ= 1 2 ab= 2 s−1 d e ¯ ( d+ 2 s−1 e ¯ )( d+ 2 s e ¯ )=n f 2;Where ( d, e ¯ )∈ ( 2ℕ+1 ) 2 such that gcd( d, e ¯ )=1 and s∈ ℕ ∗ , where 2 s−1 , d, e ¯ , d+ 2 s−1 e ¯ , d+ 2 s e ¯ , are pairwise prime quantities (these parameters are coming from Pythagorician divisors). When n=1 , it is the case of the famous impossible problem of the integer right-angled triangle area to be a square, solved by Fermat at his time, by his famous method of infinite descent. We propose in this article a new direct proof for the numbers n=1 (resp. n=2 ) to be non-congruent numbers, based on an particular induction method of resolution of Equation (1) (note that this method is efficient too for general case of prime numbers n=p≡a ( ( mod8 ) , gcd( a,8 )=1 ). To prove it, we use a classical proof by induction on k , that shows the non-solvability property of any of the following systems ( t=0 , corresponding to case n=1 (resp. t=1 , corresponding to case n=2 )): ( Ξ t,k ){ X 2 + 2 t ( 2 k Y ) 2 = Z 2 X 2 + 2

关键词： Prime Numbers-Diophantine Equations of Degree 2 & 4 Factorization Greater Common Divisor Pythagoras Equation Pythagorician Triplets Congruent Numbers Inductive Demonstration Method Infinite Descent BsD Conjecture

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Fermat and Pythagoras Divisors for a New Explicit Proof of Fermat’s Theorem:a<sup>4sup> + b<sup>4sup> = c<sup>4sup>. Part I

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Advances in Pure Mathematics 2024年第4期14卷 303-319页

作者： Prosper Kouadio Kimou François emmanuel Tanoé Kouassi Vincent Kouakou Laboratoire d’ Informatique et de Math&#233matiques appliqu&#233es Institut Polytechnique F&#233lix Houphou&#235t BOIGNY Yamoussoukro Cote d&#8217Ivoire UFR Mathé matiques et Informatique Universit&#233 F&#233lix Houphouet BOIGNY Abidjan Cote d&#8217Ivoire UFR sciences Fondamentales Appliqué es Universit&#233 NANGUI ABROGOUA Abidjan Cote d&#8217Ivoire

In this paper we prove in a new way, the well known result, that Fermat’s equation a4 + b4 = c4, is not solvable in ℕ , when abc≠0 . To show this result, it suffices to prove that: ( F 0 ): a 1 4 + ( 2 s b 1 ) 4 = c... 详细信息

In this paper we prove in a new way, the well known result, that Fermat’s equation asup>4sup> + bsup>4sup> = csup>4sup>, is not solvable in ℕ , when abc≠0 . To show this result, it suffices to prove that: ( F 0 ): a 1 4 + ( 2 s b 1 ) 4 = c 1 4 , is not solvable in ℕ , (where a 1 , b 1 , c 1 ∈2ℕ+1 , pairwise primes, with necessarly 2≤s∈ℕ ). The key idea of our proof is to show that if (Fsub>0sub>) holds, then there exist α 2 , β 2 , γ 2 ∈2ℕ+1 , such that ( F 1 ): α 2 4 + ( 2 s−1 β 2 ) 4 = γ 2 4 , holds too. From where, one conclude that it is not possible, because if we choose the quantity 2 ≤ s, as minimal in value among all the solutions of ( F 0 ) , then ( α 2 ,2 s−1 β 2 , γ 2 ) is also a solution of Fermat’s type, but with 2≤s−1s , witch is absurd. To reach such a result, we suppose first that (Fsub>0sub>) is solvable in ( a 1 ,2 s b 1 , c 1 ) , s ≥ 2 like above;afterwards, proceeding with “Pythagorician divisors”, we creat the notions of “Fermat’s b-absolute divisors”: ( d b , d ′ b ) which it uses hereafter. Then to conclude our proof, we establish the following main theorem: there is an equivalence between (i) and (ii): (i) (Fsub>0sub>): a 1 4 + ( 2 s b 1 ) 4 = c 1 4 , is solvable in ℕ , with 2≤s∈ℕ , ( a 1 , b 1 , c 1 )∈ ( 2ℕ+1 ) 3 , coprime in pairs. (ii) ∃( a 1 , b 1 , c 1 )∈ ( 2ℕ+1 ) 3 , coprime in pairs, for wich: ∃( b ′ 2 , b 2 , b ″ 2 )∈ ( 2ℕ+1 ) 3 coprime in pairs, and 2≤s∈ℕ , checking b 1 = b ′ 2 b 2 b ″ 2 , and such that for notations: s=s−λ( s−1 ) , with λ∈{ 0,1 } defined by c 1 − a 1 2 ≡λ( mod2 ) , d b =gcd( 2 s b 1 , c 1 − a 1 )= 2 s b 2 and d ′ b = 2 s−s b ′ 2 = 2 s B 2 d b , where ( 2 s B 2 ) 2 =gcd( b 1 2 , c 1 2 − a 1 2 ) , the following system is checked: { c 1 − a 1 = d b 4 2 2+λ = 2 2−λ ( 2 s−1 b 2 ) 4 c 1 + a 1 = 2 1+λ d ′ b 4 = 2 1+λ ( 2 s−s b ′ 2 ) 4 c 1 2 + a 1 2 =2 b ″ 2 4;and this system implies: ( b 1−λ,2 4 ) 2 + ( 2 4s−3 b λ,2 4 ) 2 = ( b ″ 2 2 ) 2;where: ( b 1−λ,2 , b λ,2 , b ″ 2 )={ ( b ′ 2 , b 2 , b ″ 2 ) if λ=

关键词： Factorisation in ℤ Greatest Common Divisor Pythagoras Equation Pythagorician Triplets Fermat's Equations Pythagorician Divisors Fermat's Divisors Diophantine Equations of Degree 2 4-Integral Closure of ℤ in ℚ

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source of metals related to gold occurrences in Cameroon, Central African Belt

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Acta Geochimica 2024年第2期43卷 308-324页

作者： Kevin Igor Azeuda Ndonfack Jinlin Zhang Yuling Xie Nsamiratou Yaya emmanuel Archelaus Afanga Basua JCHX Mining Management Co. LtdBeijing 101500People’s Republic of China Institute of Geology Chinese Academy of Geological SciencesBeijing 100037People’s Republic of China school of Civil and Environmental Engineering University of Science and Technology BeijingBeijing 100083People’s Republic of China Department of Earth science University of BamendaBamendaRepublic of Cameroon

suites of granitic and metamorphic rocks form a perfect geological environment in which to investigate the source of metals related to gold *** the East Province of Cameroon,the source of metals-related gold(grade~4.6 g/t)that occurs within metamorphic and granitic rocks remains *** host rocks were subjected to XRF and ICP-Ms whole-rock geochemistry and BsE/EDs mineral analyses to investigate the source(s)of metals related to gold in the ***,chalcopyrite-pyrite and magnetite-ilmenite are the main sulfi des and oxides of granites,*** low-grade metamorphic rocks are sulfi de-rich consisting of arsenopyrite,chalcopyrite,pyrite,and sphalerite,and oxides include magnetite,ilmenite,hematite,and ***,Ni,Co,sb,sc,and Zn concentrations depleted in high-grade metamorphic rocks and granites compared to low-grade metamorphic *** well,Bi,Cd,Co,Cu,Ni,sb,V,W,and Zn concentrations decrease with increasing temperatures from low-grade rocks to high-grade rocks and *** suite of depleted elements in high-grade rocks and granites is almost identical to those enriched in gold *** conclude that metals(Bi,Cd,Co,Cu,Ni,sb,and Zn)related to gold mineralization in the East Province of Cameroon likely originated from metal-rich low-grade metamorphic rocks during prograde metamorphic processes prior to partial melting.

关键词： East Province of Cameroon Metamorphic rocks Granites Oxide–sulﬁdes Metal and gold Geochemistry

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Comparative Analysis of Machine Learning Models for Customer Churn Prediction in the U.s. Banking and Financial services: Economic Impact and Industry-specific Insights

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Journal of Data Analysis and Information Processing 2024年第3期12卷 388-418页

作者： Omoshola s. Owolabi Prince C. Uche Nathaniel T. Adeniken Oghenekome Efijemue samuel Attakorah Oluwabukola G. Emi-Johnson emmanuel Hinneh Department of Data science Carolina University Winston-Salem NC USA Department of Computer science Austin Peay State University Clarkesville TN USA Department of statistical sciences Wake Forest University Winston-Salem NC USA

Customer churn poses a significant challenge for the banking and finance industry in the United states, directly affecting profitability and market share. This study conducts a comprehensive comparative analysis of machine learning models for customer churn prediction, focusing on the U.s. context. The research evaluates the performance of logistic regression, random forest, and neural networks using industry-specific datasets, considering the economic impact and practical implications of the findings. The exploratory data analysis reveals unique patterns and trends in the U.s. banking and finance industry, such as the age distribution of customers and the prevalence of dormant accounts. The study incorporates macroeconomic factors to capture the potential influence of external conditions on customer churn behavior. The findings highlight the importance of leveraging advanced machine learning techniques and comprehensive customer data to develop effective churn prevention strategies in the U.s. context. By accurately predicting customer churn, financial institutions can proactively identify at-risk customers, implement targeted retention strategies, and optimize resource allocation. The study discusses the limitations and potential future improvements, serving as a roadmap for researchers and practitioners to further advance the field of customer churn prediction in the evolving landscape of the U.s. banking and finance industry.

关键词： Churn Prediction Machine Learning Economic Impact Industry-specific Insights Logistic Regression Random Forest Neural Networks

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Methane cracking in molten tin for hydrogen and carbon production—a comparison with homogeneous gas phase process

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Frontiers of Chemical science and Engineering 2024年第7期18卷 155-166页

作者： emmanuel Busillo Benedetta de Caprariis Maria Paola Bracciale Vittoria Cosentino Martina Damizia Gaetano Iaquaniello Emma Palo Paolo De Filippis Chemical Engineering Department Sapienza University of RomeRome 00184Italy NextChem s.p.A. Rome 00156Italy

Methane cracking is considered a bridge technology between gray and green hydrogen production *** this work an experimental study of methane cracking in molten tin is *** tests were conducted in a quartz reactor(i.d.=1.5 cm,L=20 cm)with capillary injection,varying temperature(950–1070℃),inlet methane flow rate(30–60 mL·min^(-1))and tin height(0–20 cm).The influence of the residence time in the tin and in the headspace on methane conversion and on carbon morphology was *** conversions obtained in tin and in the empty reactor were measured and compared with results of detailed kinetic simulations(CRECK).Furthermore,an expression of a global kinetic constant for methane conversion in tin was also *** highest conversion(65%at Q_(0)=30 mL·min^(-1) and t=1070℃)is obtained for homogeneous gas phase reaction due to the long residence time(70 s),the presence of tin leads to a sharp decrease of residence time(1 s),obtaining a conversion of 35%at 1070℃,thus meaning that tin owns a role in the *** characterization(scanning electron microscopy,Raman)reported a change in carbon toward sheet-like structures and an increase of the carbon structural order in the presence of molten tin media.

关键词： methane cracking molten media H2 production carbon morphology

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Detection and clearance in Alzheimer’s disease: leading with illusive chemical, structural and morphological features of the targets

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Neural Regeneration Research 2024年第3期19卷 497-498页

作者： Luís Gales i3s−Instituto de Investigação e Inovação em saúde Rua Alfredo AllenPortoPortugal(Gales L)IBMC−Instituto de Biologia Molecular e CelularUniversidade do PortoRua Alfredo AllenPortoPortugal ICBAs−Instituto de Ciências Biomédicas Abel salazar Rua de Jorge Viterbo FerreiraPortoPortugal

Highly specific interactions between biomolecules,such as antigen-antibody,protein-ligand,or nucleic acid base pair complementary are on the basis of the organization of complex *** same principles may be tentatively used in molecular medicine for diagnosis and therapeutics.A molecule can be designed to selectively bind a protease and thereby inhibit the production of a peptide that forms toxic aggregates in the brain or an antibody may be produced to bind specifically to that peptide for detection or clearance ***,interference in biological systems is not that *** a start there is the inhibition of the physiological role of the protease;moreover,several cleavage fragments may be produced,which may continue to diverge due to putative post-translational modification and self-assembly processes,hiding the toxic target in a“soup”of peptide species varying in size,structure and chemical composition.A perspective of the current status and challenges in targeting peptide species for diagnosis and treatment in the context of Alzheimer’s disease is given.

关键词： Alzheimer diagnosis thereby

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Data-driven analysis of chemicals,proteins and pathways associated with peanut allergy:from molecular networking to biological interpretation

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Food science and Human Wellness 2024年第3期13卷 1322-1335页

作者： emmanuel Kemmler Julian Braun Florent Fauchère sabine Dölle-Bierke Kirsten Beyer Robert Preissner Margitta Worm Priyanka Banerjee Institute for Physiology Charité-University Medicine BerlinBerlin 10115Germany Member of the KFO339 FOOD@Berlin 10115Germany si-M/“Der simulierte Mensch”a science framework of Technische Universität Berlin and Charité-Universitätsmedizin Berlin Berlin 10115Germany Charité-Universitätsmedizin Berlin corporate member of Freie Universität BerlinHumboldt-Universität zu Berlinand Berlin Institute of HealthBerlin 10115Germany Division of Allergy and Immunology Department of DermatologyVenerology and AllergyCharité-Universitätsmedizin Berlincorporate member of Freie Universität Berlin and Humboldt-Universität zu BerlinBerlin 10115Germany Pediatrics Campus VirchowCharité-University Medicine BerlinBerlin 10115Germany

Peanut allergy is majorly related to severe food induced allergic *** food including cow's milk,hen's eggs,soy,wheat,peanuts,tree nuts(walnuts,hazelnuts,almonds,cashews,pecans and pistachios),fish and shellfish are responsible for more than 90%of food ***,we provide promising insights using a large-scale data-driven analysis,comparing the mechanistic feature and biological relevance of different ingredients presents in peanuts,tree nuts(walnuts,almonds,cashews,pecans and pistachios)and ***,we have analysed the chemical compositions of peanuts in different processed form raw,boiled and *** the data-driven approach we are able to generate new hypotheses to explain why nuclear receptors like the peroxisome proliferator-activated receptors(PPARs)and its isoform and their interaction with dietary lipids may have significant effect on allergic *** results obtained from this study will direct future experimeantal and clinical studies to understand the role of dietary lipids and PPARisoforms to exert pro-inflammatory or anti-inflammatory functions on cells of the innate immunity and influence antigen presentation to the cells of the adaptive immunity.

关键词： Allergy informatics Knowledge-graph Data analysis Food allergy Peroxisome proliferator-activated receptors Fatty acids

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A cross-sectional analysis of the effectiveness of a nutritional support programme for people with tuberculosis in southern Madagascar using secondary data from a non-governmental organisation

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Infectious Diseases of Poverty 2024年第1期13卷 101-102页

作者： Mara Anna Franke Julius Valentin Emmrich Fierenantsoa Ranjaharinony Onja GbrilleRavololohanitra Harizaka emmanuel Andriamasy samuel Knauss Nadine Muller Global Digital Health Lab at Charité Center for Global Health Charité – Universitätsmedizin Berlin Berlin Germany London school of Hygiene and Tropical Medicine London UK Ärzte Für Madagaskar E.V. Leipzig Germany Heidelberg Institute of Global Health Heidelberg University Heidelberg Germany Berlin Institute of Health at Charité - Universitätsmedizin Berlin Berlin Germany Doctors for Madagascar Antananarivo Madagascar speciality Network: Infectious Diseases and Respiratory Medicine Charité - Universitätsmedizin Berlin Berlin Germany

Background There is a strong,bi-directional link between tuberculosis(TB)and undernutrition:TB often causes undernutrition,and undernourished people are more likely to contract TB and experience worse ***,several TB nutritional support programmes exist;however,evidence on their effectiveness is limited and *** study evaluates the effect of a nutritional support programme implemented for people with TB in the Atsimo-Andrefana region,Madagascar in *** this programme,undernourished people with TB[with a body mass index(BMI)ofsoy-wheat blend per month throughout their TB *** We analysed secondary non-governmental organisation data collected between January and Novem-ber 2022 in the Atsimo-Andrefana region,southern Madagascar,including information on an individual's medical conditions(e.g.,type of TB,treatment outcomes)and nutritional status measured prior to,during,and after comple-tion of treatment(e.g.,height,weight,mid-upper arm circumference).We conducted descriptive analyses of patient baseline characteristics and outcomes to assess the impact of the provided nutritional support on the BMI of people With *** A total of 1310 people with TB were included in the study[9.9%(130)children under the age of 5,32.1%(420)children between 5 and 18 years,58.0%(760)adults].55.4% of children under 5,28.1% of children between ages 5 and 18,and 81.3%of adults were undernourished at treatment initiation.42.3%(55/130)of children under 5 experi-enced severe acute malnutrition at treatment *** the average BMl of adults with TB receiving food support increased over time,from 17.1 kg/m^(2)(interquartile range:15.8-18.3,range:10.3-22.5)to 17.9 kg/m^(2)(interquartile range:16.6-19.1,range:11.9-24.1),most adults remained undernourished even after completing TB *** The current TB nutritional support programme falls short of sufficiently increasing the BM

关键词： Tuberculosis sub-saharan Africa Undernutrition Nutritional support

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Optimization of stealthwatch Network security system for the Detection and Mitigation of Distributed Denial of service (DDos) Attack: Application to smart Grid system

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Communications and Network 2024年第3期16卷 108-134页

作者： emmanuel s. Kolawole Penrose s. Cofie John H. Fuller Cajetan M. Akujuobi emmanuel A. Dada Justin F. Foreman Pamela H. Obiomon Electrical and Computer Engineering Department Prairie View A&M University Prairie View TX USA

The smart Grid is an enhancement of the traditional grid system and employs new technologies and sophisticated communication techniques for electrical power transmission and distribution. The smart Grid’s communication network shares information about status of its several integrated IEDs (Intelligent Electronic Devices). However, the IEDs connected throughout the smart Grid, open opportunities for attackers to interfere with the communications and utilities resources or take clients’ private data. This development has introduced new cyber-security challenges for the smart Grid and is a very concerning issue because of emerging cyber-threats and security incidents that have occurred recently all over the world. The purpose of this research is to detect and mitigate Distributed Denial of service [DDos] with application to the Electrical smart Grid system by deploying an optimized stealthwatch secure Network analytics tool. In this paper, the DDos attack in the smart Grid communication networks was modeled using stealthwatch tool. The simulated network consisted of secure Network Analytic tools virtual machines (VMs), electrical Grid network communication topology, attackers and Target VMs. Finally, the experiments and simulations were performed, and the research results showed that stealthwatch analytic tool is very effective in detecting and mitigating DDos attacks in the smart Grid system without causing any blackout or shutdown of any internal systems as compared to other tools such as GNs3, Nessi2, NIssT Framework, OMNeT++, INET Framework, ReasE, Ns2, Ns3, M5 simulator, OPNET, PLC & TIA Portal management software which do not have the capability to do so. Also, using stealthwatch tool to create a security baseline for smart Grid environment, contributes to risk mitigation and sound security hygiene.

关键词： smart Grid system Distributed Denial of service (DDos) Attack Intrusion Detection and Prevention systems Detection Mitigation and stealthwatch

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Assessment of Axial Power Peaking Factors in GHARR-1 LEU Core: A Decadal simulation Analysis

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World Journal of Nuclear science and Technology 2024年第1期14卷 72-85页

作者： emmanuel Kwame Ahiave emmanuel Ampomah-Amoako Rex Gyeabour Abrefah Mathew Asamoah Department of Mathematics and statistics Ho Technical University Ho Ghana Department of Nuclear Engineering Graduate School of Nuclear and Allied Sciences University of Ghana Accra Ghana Nuclear Regulatory Authority Accra Ghana National Nuclear Research Institute Accra Ghana

This study aims to thoroughly investigate the axial power peaking factors (PPF) within the low-enriched uranium (LEU) core of the Ghana Research Reactor-1 (GHARR-1). This study uses advanced simulation tools, like the MCNPX code for analysing neutron behavior and the PARET/ANL code for understanding power variations, to get a clearer picture of the reactor’s performance. The analysis covers the initial six years of GHARR-1’s operation and includes projections for its whole 60-year lifespan. We closely observed the patterns of both the highest and average PPFs at 21 axial nodes, with measurements taken every ten years. The findings of this study reveal important patterns in power distribution within the core, which are essential for improving the safety regulations and fuel management techniques of the reactor. We provide a meticulous approach, extensive data, and an analysis of the findings, highlighting the significance of continuous monitoring and analysis for proactive management of nuclear reactors. The findings of this study not only enhance our comprehension of nuclear reactor safety but also carry significant ramifications for sustainable energy progress in Ghana and the wider global context. Nuclear engineering is essential in tackling global concerns, such as the demand for clean and dependable energy sources. Research on optimising nuclear reactors, particularly in terms of safety and efficiency, is crucial for the ongoing advancement and acceptance of nuclear energy.

关键词： GHARR-1 Power Peaking Factor Nuclear Reactor safety Low Enriched Uranium Core Operational Longevity Thermal Hydraulics

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