Small Modular Solutions to Fermat’s Last Theorem

The proof by Andrew Wiles of Fermat’s Last Theorem in 1995 resolved the existence question for non-trivial solutions in integers x,y,zto the equation xn+yn=znfor n>2. There are none. Surprisingly, there are infinitely many solutions if the problem is recast in terms of modular arithmetic. Over a hundred years ago Issai Schur was able to show that for any n there is always a sufficiently large prime p0such that for all primes p≥p0the congruence xn+yn≡zn(modp)has a non-trivial solution. Schur’s argument wasnon-constructive, and there is no systematic method available at present to construct specific examples for small primes. We offer a simple method for constructing all possible solutions to a large class of congruences of this type.

Whole Perfect Vectors and Fermat’s Last Theorem

A naïve discussion of Fermat’s last theorem conundrum is described. The present theorem’s proof is grounded on the well-known properties of sums of powers of the sine and cosine functions, the Minkowski norm definition, and some vector-specific structures.

Revisiting East Asian monsoon change during the Last Glacial Maximum using PMIP4 simulations

利用PMIP4多模式试验数据,本文重新检查了末次冰盛期(距今约21000年)东亚季风变化.结果表明:相对于工业革命前期,所有5个模式一致模拟显示末次冰盛期东亚季风减弱,冬季和夏季减幅分别为1%-18%和2-32%;不同模式中东亚季风环流变化的空间分布存在差异,这主要源于该时期大尺度变冷和海平面气压梯度变化的空间分布不同;由于模式之间的差异和重建记录之间的不确定性,未来有待开展更多模拟和重建工作以更好地理解冰期东亚季风变化.

Situational Analysis of Malaria Control Following the 2021 Distribution of Long Lasting Insecticide-Treated Bed Nets in the Tiko Health District, South West Region, Cameroon

Malaria is still a major public health concern in Cameroon as more than 20 million people are at risk of the disease. Despite the government’s effort to scale up the distribution of long lasting insecticide-treated bed nets (LLINs), full ownership and optimal use is still limited. In 2021, over a million LLINs were made available for distribution in the South West Region. However, the current situation of malaria has not yet been ascertained following the 2021 mass distribution campaign (MDC). Therefore, the aim of this study was to determine LLIN ownership, coverage, usage and the prevalence of malaria following the 2021 MDC in the Tiko Health District (THD), South West Region of Cameroon. A cross-sectional study was carried from May 1 to 27, 2022 in which a multi-stage sampling technique was used to enroll eligible participants. Data were collected from household heads using questionnaires and capillary blood samples collected from consented household members were used to test for malaria. Data were analyzed using SPSS version 25 at 0.05 significance level. From the 150 households surveyed, 96 (64.0%) owned a mosquito net before the recent MDC. There was an overall significant increase (p p = 0.644) in LLINs ownership between areas where door-to-door (79.5%) and fixed “Hit and Run” (75.8%) strategies were used in the distribution of LLINs in THD. The prevalence of malaria was 10.2%. The regular use of LLINs was significantly associated (p = 0.007) with low malaria prevalence. The 2021 MDC had an impact on malaria prevalence in those who properly used them.

Products of Odd Numbers or Prime Number Can Generate the Three Members’ Families of Fermat Last Theorem and the Theorem Is Valid for Summation of Squares of More Than Two Natural Numbers

Fermat’s last theorem, had the statement that there are no natural numbers A, B, and C such that An + Bn = Cn, in which n is a natural number greater than 2. We have shown that any product of two odd numbers can generate Fermat or Pythagoras triple (A, B, C) following n = 2 and also it is applicable A2 + B2 + C2 + D2 + so on =An2 where all are natural numbers.

On Fermat Last Theorem: The New Efficient Expression of a Hypothetical Solution as a Function of Its Fermat Divisors

Denote by a non-trivial primitive solution of Fermat’s equation (p prime).We introduce, for the first time, what we call Fermat principal divisors of the triple defined as follows. , and . We show that it is possible to express a,b and c as function of the Fermat principal divisors. Denote by the set of possible non-trivial solutions of the Diophantine equation . And, let (p prime). We prove that, in the first case of Fermat’s theorem, one has . In the second case of Fermat’s theorem, we show that , ,. Furthermore, we have implemented a python program to calculate the Fermat divisors of Pythagoreans triples. The results of this program, confirm the model used. We now have an effective tool to directly process Diophantine equations and that of Fermat. .

Unfavorable environmental conditions for tropical cyclone genesis over the western North Pacific during the Last Interglacial based on PMIP4 simulations

研究过去暖期西北太平洋热带气旋的变化有助于理解未来气候变暖情景下热带气旋的可能变化.本研究基于PMIP4多模式输出结果,分析了末次间冰期西北太平洋热带气旋大尺度生成因子的变化及相关机制.结果表明,末次间冰期西北太平洋风暴季潜在强度降低,湿熵亏损升高,绝对涡度减弱,中部垂直风切变增强,西南部减弱.进一步,基于生成潜势指数,指出末次间冰期西北太平洋生成潜势降低,这表明环境条件不利于热带气旋生成。

Strength of the North African monsoon in the Last Interglacial and under future warming

与人为强迫引起的全球变暖相比,末次间冰期是轨道强迫引起的过去80万年来最暖的一个间冰期,但鲜有人研究末次间冰期中北非季风的响应.因此,本文基于CMIP6多模式模拟结果对比研究了末次间冰期和SSP5-8.5情景下北非季风的响应,发现末次间冰期下北非季风平均降水及其降水变率均远大于SSP5-8.5情景下的结果.轨道强迫导致的北大西洋暖于南大西洋增加了北非季风环流和平均降水,同时,南北大西洋海温梯度变化通过增强热带北大西洋的海气相互作用增大了海温梯度和撒哈拉低压的变率,从而增强了北非季风降水变率.

Can a global mean sea-level rise reduce the Last Interglacial model-data mismatch in East Asia?

末次间冰期有着丰富的重建和模拟资料,为研究未来温暖气候提供了一个理想的参考.然而,关于末次间冰期的东亚气候,模拟与重建的结果间长期存在着不匹配的情况,模拟结果普遍较重建结果更为冷干。本研究利用挪威地球系统模式(NorESM1-F),探讨了在末次间冰期模拟试验中纳入全球平均海平面上升能否减少模式-数据的不匹配.该试验结果表明,海平面上升情况下东亚地区会产生一定的增温增湿效应,但不足以消除模式-数据不匹配.基于这些结果,作者探讨了其它可能造成不匹配的因素以供进一步研究.

ACE、CICARE及LAST护理服务理念在脑梗死患者康复护理中的应用效果

目的:观察ACE、CICARE及LAST护理服务在脑梗死患者康复护理中的应用效果。方法:将脑梗死恢复期患者60例随机分为观察组和对照组各30例。对照组行常规护理,观察组在对照组基础上行ACE、CICARE及LAST护理服务,比较2组患者护理满意度、Barthel指数、护理依从性、回归社会生活信心等心理状况。结果:观察组护理满意度、Barthel指数、护理治疗依从性、对回归社会生活的信心及心理状况优于对照组,差异具有统计学意义(P<0.05)。结论:ACE、CICARE及LAST护理服务有利于提高脑梗死患者临床康复护理满意度,提高患者慢病自我管理能力,增加护理治疗依从性及患者回归社会生活的信心,减少焦虑及抑郁情绪等心理状况的发生。

LAST理论与外语教学

本文简要介绍了最新的语言加工模式后句法指派理论 ,并探讨了这一理论对外语教学的启示。L AST认为 ,语言理解是联想与符号计算两种过程的组合 ,人在理解过程中首先根据输入的表层线索形成意义 /形式假设 ,即假句法 ,然后根据句法规则生成完整的结构 ,并与输入序列进行对比 ,如果相符 ,则将结果存储到概念记忆中 ,进入下一句的理解。反之 ,则重复上述过程。句子是理解过程中联想信息和符号信息融合的自然层次。大量的研究证据无疑都支持上述理论 ,而且用这种理论可以更好地解释以前难以解释的一些心理语言学现象。同时 ,该理论对我们重新审视语言习得理论有重要意义 :人不同于其他动物 ,具有语言学习的能力 ,但外部刺激对语言发展起重要的作用 ,出现频率高的SVO结构先得到发展。这就为外语教学中传统教学法的应用提供了理论依据。

基于Hadoop Streaming的Last比对软件并行化的研究与实现

随着下一代测序技术的到来,单机版Last比对软件已经不能满足海量数据的处理需求。使用Hadoop Streaming技术将Last比对软件快速部署到云计算环境中,解决当前单机版Last比对软件处理大数据能力差的问题。通过自定义的基于NFS文件系统的数据集切分方法和基于Partitioner的任务分配方式能够实现均衡高效的数据切分,并保证并行化粒度可控。实验结果表明,在保证与单机运行结果一致的情况下,这种方法能有效缩减软件运行时间,具有较高的加速比。

书信中语篇人际意义的实现——从对“A Soldier’s Last Letter”语言的分析谈起

结合功能语言学理论,以"A Soldier’s Last Letter"中的部分段落为例进行分析,可以展示如何通过语气系统和情态系统来进行选择形成语篇,从而突显语言选择对语篇人际意义实现的重要性以及功能语言学在分析文学作品时的可行性。

Very Original Proofs of Two Famous Problems: “Are There Any Odd Perfect Numbers?” (Unsolved until to Date) and “Fermat’s Last Theorem: A New Proof of Theorem (Less than One and a Half Pages) and Its Generalization”

This article presents very original and relatively brief or very brief proofs about of two famous problems: 1) Are there any odd perfect numbers? and 2) “Fermat’s last theorem: A new proof of theorem and its generalization”. They are achieved with elementary mathematics. This is why these proofs can be easily understood by any mathematician or anyone who knows basic mathematics. Note that, in both problems, proof by contradiction was used as a method of proof. The first of the two problems to date has not been resolved. Its proof is completely original and was not based on the work of other researchers. On the contrary, it was based on a simple observation that all natural divisors of a positive integer appear in pairs. The aim of the first work is to solve one of the unsolved, for many years, problems of the mathematics which belong to the field of number theory. I believe that if the present proof is recognized by the mathematical community, it may signal a different way of solving unsolved problems. For the second problem, it is very important the fact that it is generalized to an arbitrarily large number of variables. This generalization is essentially a new theorem in the field of the number theory. To the classical problem, two solutions are given, which are presented in the chronological order in which they were achieved. Note that the second solution is very short and does not exceed one and a half pages. This leads me to believe that Fermat, as a great mathematician was not lying and that he had probably solved the problem, as he stated in his historic its letter, with a correspondingly brief solution. To win the bet on the question of whether Fermat was telling truth or lying, go immediately to the end of this article before the General Conclusions.

China's English Learners' Verbal Phrase Acquisition Research Based on the Corpus——“at last”and“in the end”as an Example

This study is a comparative research based on corpus. This article compares China's English learners with English na?tive speakers about the use of "at last" and "in the end" from three aspects: frequency of use, location, semantic prosody. The study shows that the use of the two words are greatly different between China's English learners and English native speakers. In addition, this study discusses about causes of obvious differences and teaching enlightenments.

A Post-colonial Approach to The Last of the Mohicans

The Last of the Mohicans is one of the five frontier novels written by famous American novelist James Fenimore Cooper in the early 19thcentury.Although it has been criticized for its narrative flaws,its length and formal prose style,it was one of the most popular in its time and is of a high literary value nowadays.With the coming of the white European colonialists,the Indians underwent various changes.Some Indians submitted themselves to the white while some Indians rebelled against the invaders and sought their revenge.With a post-colonial approach,the changes of the Indians will be better understood,and people nowadays will also have a better understanding of the Indian history and culture in early American history.

A Multi-cycle Climatic Fluctuation Record of the Last Interglacial Period:Typical Stratigraphic Section in the Salawusu River Valley on the Ordos Plateau,China

The Last Interglacial Period strata in the Milanggouwan section in the Salawusu River valley on the Ordos Plateau, China, have 8.5 sedimentary cycles composed alternately of eolian dune sands, fluvio-lacustrine facies and paleosols. Based on comprehensive analyses on the distribution of magnetic susceptibility and CaCO3 and paleo-ecology indicated by fossils in the region, it is considered that the sedimentation cycles resulted from dry-cold and warm-humid climate fluctuations. Magnetic susceptibility values and CaCO3 contents in stratigraphic sectors I, III, V and II, IV basically respectively present peaks and low vales, and the former three can in time correlate with MIS5a, MIS5c and MIS5e successively and the latter two with MIS5b and MIS5d. In addition, some horizons of eolian dune sands and the low vales of their magnetic susceptibility and CaCO3 are also correlated with 6 periods of cooling events indicated by the higher content of foraminifer Neogloboquadrina pachyderma (S.) documented in the V29—191 drill in the North Atlantic and the cold events recorded by δ 18O in the ice cores in GRIP, especially with 9 periods of dust events in Chinese Loess Plateau.

Glacier Relics of the Last Glacial and Holocene Periods Discovered in the Middle-low Mountains of Eastern China: Sedimentary Sequences and Environmental Evolution of Mengshan Mountain in Shandong Province since 80 ka

The goal of this research study is to describe academic issues which have been debated in the field of Chinese geosciences for a century. In 1922, Jonquei S. Lee(Li Siguang) discovered Quaternary glacial relics at Taihang Mountainin eastern China. In 1947, he published his research findings in the magazine Mount Lushan in Glacial Age. The research results had established three Ice Ages: Poyang(Gonzi), Dagu(Minde), and Lushan(Lisi). However, at that time, no Wurm glacial relics of the last Ice Age had been found in Lushan Mountain. Since then, the research team represented by Shi Yafeng, who is considered to be "the father of glaciers in China", questioned Jonquei S. Lee’s research results and concluded that "Professor Jonquei S. Lee’s Quaternary glacier research in Lushan Mountain having misread the debris flow". In 2005, the "middle-low mountains" in eastern China were finally defined as follows: "We clearly and unambiguously believe that there were no glacial activities in the middle-low mountainous areas of eastern China(east of 102° to 104°E;below 3,000 and 2,500 m) during the Quaternary Period". Currently, the long-standing academic debate appears to have come to a conclusion. As of 2015, the author and others began to investigate and study the Quaternary glacial relics in Mengshan Mountain(1,156 m above sea level), Shandong Province, one of the "middle-low mountains" of eastern China. The relics have been observed to posses the systematic features of glacial erosion, trough and valley striations, and moraine deposits. The applied dating method shave confirmed that there were not only glacial relics of the last Ice Age(Wurm), but also Holocene glacial relics in the Mengshan Mountain area. Therefore, in order to further establish the corresponding relationship between the glacier, loess, stream sediment series, and MIS in the Mengshan Mountain area, a large number of chronological studies have been carried out regarding the various types of sediments in the area, and 24 dating datahave been obtained using OSL, CRN, and 14 Cmethods.On this basis, the corresponding relationship between the sedimentary sequences and the MIS was established for the first time in eastern China, which in dicates the environmental changes which had occurred in eastern China since 80 ka. These discoveries s and chronological study results confirm the existence of the Last Ice Age, as well as Holocene glacial relics at Mengshan Mountain, there by confirming that Quaternary glaciation had occurred in the middle-low mountain areas of eastern China.

Regional-scale Surface Air Temperature and East Asian Summer Monsoon Changes during the Last Millennium Simulated by the FGOALS-gl Climate System Model

The spatial patterns and regional-scale surface air temperature （SAT） changes during the last millennium,as well as the variability of the East Asian summer monsoon （EASM） were simulated with a low-resolution version of Flexible Global Ocean-Atmosphere-Land-Sea-ice （FGOALS-gl） model.The model was driven by both natural and anthropogenic forcing agents.Major features of the simulated past millennial Northern Hemisphere （NH） mean SAT variations,including the Medieval Climate Anomaly （MCA）,the Little Ice Age （LIA） and the 20th Century Warming （20CW）,were generally consistent with the reconstructions.The simulated MCA showed a global cooling pattern with reference to the 1961-90 mean conditions,indicating the 20CW to be unprecedented over the last millennium in the simulation.The LIA was characterized by pronounced coldness over the continental extratropical NH in both the reconstruction and the simulation.The simulated global mean SAT difference between the MCA and LIA was 0.14°C,with enhanced warming over high-latitude NH continental regions.Consistencies between the simulation and the reconstruction on regional scales were lower than those on hemispheric scales.The major features agreed well between the simulated and reconstructed SAT variations over the Chinese domain,despite some inconsistency in details among different reconstructions.The EASM circulation during the MCA was stronger than that during the LIA The corresponding rainfall anomalies exhibited excessive rainfall in the north but deficient rainfall in the south.Both the zonal and meridional thermal contrast were enhanced during the MCA.This temperature anomaly pattern favored a stronger monsoon circulation.

Variation of Surface Temperature during the Last Millennium in a Simulation with the FGOALS-gl Climate System Model

A reasonable past millennial climate simulation relies heavily on the specified external forcings, including both natural and anthropogenic forcing agents. In this paper, we examine the surface temperature responses to specified external forcing agents in a millennium-scale transient climate simulation with the fast version of LASG IAP Flexible Global Ocean-Atmosphere-Land System model （FGOALS-gl） developed in the State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics, Institute of Atmospheric Physics （LASG/IAP）. The model presents a reasonable performance in comparison with reconstructions of surface temperature. Differentiated from significant changes in the 20th century at the global scale, changes during the natural-forcing-dominant period are mainly manifested in the Northern Hemisphere. Seasonally, modeled significant changes are more pronounced during the wintertime at higher latitudes. This may be a manifestation of polar amplification associated with sea-ice-temperature positive feedback. The climate responses to total external forcings can explain about half of the climate variance during the whole millennium period, especially at decadal timescales. Surface temperature in the Antarctic shows heterogeneous and insignificant changes during the preindustrial period and the climate response to external forcings is undetectable due to the strong internal variability. The model response to specified external forcings is modulated by cloud radiative forcing （CRF）. The CRF acts against the fluctuations of external forcings. Effects of clouds are manifested in shortwave radiation by changes in cloud water during the natural-forcing-dominant period, but mainly in longwave radiation by a decrease in cloud amount in the ant hropogenic- forcing-dominant period.