The study of the chlorite coatings always attracts scholars in China and other countries because the chlorite coatings play an important role in the preservation of residual primary pores in sandstone reservoirs.At pr...The study of the chlorite coatings always attracts scholars in China and other countries because the chlorite coatings play an important role in the preservation of residual primary pores in sandstone reservoirs.At present,the study of the origin and the controlling factors is relatively few.The occurrence,time of formation,genesis,controlling factors,and the mechanism of chlorite coatings inhibiting quartz overgrowths were studied in detail with thin section and SEM analysis.Samples were from the sandstone reservoirs of the T3x Group in the Baojie area,the transitional zone from the middle to the south of Sichuan Basin.The results indicate that the chlorite coatings on the walls of the pore spaces are oriented perpendicular to grain surfaces in the form of isopachous(even-thickness) grain-coating,while the chlorite coatings at the contacts between adjacent detrital grains are arranged with a preferred orientation tangential to the surface of detrital grains.The chlorite coatings were formed in the eogenetic stage.They were formed by recrystallization of Fe-rich clay films during the syndepositional period,and chlorite cements would be recrystallized after the coatings’ formation.The formation of chlorite coatings was mainly controlled by the depositional environment,provenance conditions,and diagenetic environment.The presence of chlorite coatings could result in the preservation of primary pores in deeply buried sandstone reservoirs by effectively inhibiting quartz overgrowths and the development of compaction and pressure solution.展开更多
The 3x + 1 problem, is a math problem that has baffled mathematicians for over 50 years. It’s easy to explain: take any positive number, if it’s even, divide it by 2;if it’s odd, multiply it by 3 and add 1. Repeat ...The 3x + 1 problem, is a math problem that has baffled mathematicians for over 50 years. It’s easy to explain: take any positive number, if it’s even, divide it by 2;if it’s odd, multiply it by 3 and add 1. Repeat this process with the resulting number, and the conjecture says that you will eventually reach 1. Despite testing all starting values up to an enormous number, no one has proved the conjecture is true for all possible starting values. The problem’s importance lies in its simplicity and difficulty, inspiring new ideas in mathematics and advancing fields like number theory, dynamical systems, and computer science. Proving or disproving the conjecture would revolutionize our understanding of math. The presence of infinite sequences is a matter of question. To investigate and solve this conjecture, we are utilizing a novel approach involving the fields of number theory and computer science.展开更多
The aim of this paper is to study the 3x + 1 problem based on the Collatz iterative formula. It can be seen from the iterative formula that the necessary condition for the Collatz iteration convergence is that its slo...The aim of this paper is to study the 3x + 1 problem based on the Collatz iterative formula. It can be seen from the iterative formula that the necessary condition for the Collatz iteration convergence is that its slope being less than 1. An odd number N that satisfies the condition of a slope less than 1 after n<sup>th</sup> Collatz iterations is defined as an n-step odd number. Through statistical analysis, it is found that after n<sup>th</sup> Collatz iterations, the iterative value of any n-step odd number N that is greater than 1 is less than N, which proves that the slope less than 1 is a sufficient and necessary condition for Collatz iteration convergence.展开更多
In this paper, we use two new effective tools and ingenious methods to prove the 3X + 1 conjecture. By using the recursive method, we firstly prove that any positive integer can be turned into an element of fourth col...In this paper, we use two new effective tools and ingenious methods to prove the 3X + 1 conjecture. By using the recursive method, we firstly prove that any positive integer can be turned into an element of fourth column of the infinite-row-six-column-matrix after a finite times operation, thus we convert “the 3X + 1 conjecture” into an equivalent conjecture, which is: Any positive integer n must become 1 after finite operations under formation of <span style="white-space:nowrap;">σ(<em>n</em>)</span> , where <img src="Edit_dad9267d-3c54-455b-b30e-63819c207e54.png" width="300" height="117" alt="" /> Then, with the help of the infinite-row-four-column-matrix, we continue to use the recursive method to prove this conjecture strictly.展开更多
This article introduces a change rule of 3x + 1 problem (Collatz conjecture), it’s named LiKe’s Rule. It’s a map of 3x + 1 problem, and details the path of each step of the change: For any positive integer, change ...This article introduces a change rule of 3x + 1 problem (Collatz conjecture), it’s named LiKe’s Rule. It’s a map of 3x + 1 problem, and details the path of each step of the change: For any positive integer, change by the Collatz conjecture. 1) This positive integer will change to an odd number;2) The odd number must change to a number of LiKe’s second sequence {3<sup>n</sup> – 1| n ∈ Z<sup>+</sup>};3) Then this 3<sup>n</sup> - 1 will change to a smaller 3<sup>n </sup>– 1 and gradually decrease to 8 (that is 3<sup>2</sup> - 1) then back to 1 in the end. If we can determine each step, the Collatz conjecture will be true. This is certainly more valuable than 2<sup>n</sup> (it might even explain 2<sup>n</sup>). And to illustrate the importance of this rule, introduced some important funny corollaries related to it.展开更多
This paper studies the property of the recursive sequences in the 3x ＋ 1 conjecture. The authors introduce the concept of μ function, with which the 3x ＋ 1 conjecture can be transformed into two other conjectures：...This paper studies the property of the recursive sequences in the 3x ＋ 1 conjecture. The authors introduce the concept of μ function, with which the 3x ＋ 1 conjecture can be transformed into two other conjectures： one is eventually periodic conjecture of the μ function and the other is periodic point conjecture. The authors prove that the 3x ＋ 1 conjecture is equivalent to the two conjectures above. In 2007, J. L. Simons proved the non-existence of nontrivial 2-cycle for the T function. In this paper, the authors prove that the μ function has nol-periodic points for 2 ≤ 1 ≤12. In 2005, J. L. Simons and B. M. M de Weger proved that there is no nontrivial/-cycle for the T function for 1 ≤68, and in this paper, the authors prove that there is no nontrivial l-cycle for the μ function for 2 ≤ 1≤ 102.展开更多
Tropospheric ozone(O_(3))is a phytotoxic air pollutant and the O_(3)-induced visible foliar injury(O_(3)VFI)is a biomarker.A recently developed Free-air O_(3)eXposure(FO_(3)X)is a promising facility to verify field-ob...Tropospheric ozone(O_(3))is a phytotoxic air pollutant and the O_(3)-induced visible foliar injury(O_(3)VFI)is a biomarker.A recently developed Free-air O_(3)eXposure(FO_(3)X)is a promising facility to verify field-observed“O_(3)-like”VFIs and to establish a flux-based threshold for the O_(3)VFI onset.The present study compared O_(3)-like VFI registered in the southern European forest sites with actual O_(3)VFI observed in a FO_(3)X experiment.The O_(3)-like VFIs were evaluated by eye in forests and thus it was subjective.According to the imaging analysis,we firstly demonstrated that major parts of the colors were similar in the field and the FO_(3)X.The color pallets for O_(3)VFI was species-specific and considered a advanced tool for the O_(3)VFI diagnosis.In addition,we calculated a flux-based threshold for the O_(3)VFI onset at the FO_(3)X based on a Phytotoxic Ozone Dose(POD_(1)),which ranged from 4.9 to 18.1 mmol m^(-2)POD1.This FO_(3)X-derived threshold partly explained but did not necessarily match with the observation for several tree species in actual forests.The multivariate analysis showed that O_(3)VFI was decreased by the presence of various species and suggested the importance of continuous monitoring activities in the field for the further analysis.展开更多
基金supported by the Natural Science Key Project of Education Board in Sichuan province,China (No.07ZA139)
文摘The study of the chlorite coatings always attracts scholars in China and other countries because the chlorite coatings play an important role in the preservation of residual primary pores in sandstone reservoirs.At present,the study of the origin and the controlling factors is relatively few.The occurrence,time of formation,genesis,controlling factors,and the mechanism of chlorite coatings inhibiting quartz overgrowths were studied in detail with thin section and SEM analysis.Samples were from the sandstone reservoirs of the T3x Group in the Baojie area,the transitional zone from the middle to the south of Sichuan Basin.The results indicate that the chlorite coatings on the walls of the pore spaces are oriented perpendicular to grain surfaces in the form of isopachous(even-thickness) grain-coating,while the chlorite coatings at the contacts between adjacent detrital grains are arranged with a preferred orientation tangential to the surface of detrital grains.The chlorite coatings were formed in the eogenetic stage.They were formed by recrystallization of Fe-rich clay films during the syndepositional period,and chlorite cements would be recrystallized after the coatings’ formation.The formation of chlorite coatings was mainly controlled by the depositional environment,provenance conditions,and diagenetic environment.The presence of chlorite coatings could result in the preservation of primary pores in deeply buried sandstone reservoirs by effectively inhibiting quartz overgrowths and the development of compaction and pressure solution.
文摘The 3x + 1 problem, is a math problem that has baffled mathematicians for over 50 years. It’s easy to explain: take any positive number, if it’s even, divide it by 2;if it’s odd, multiply it by 3 and add 1. Repeat this process with the resulting number, and the conjecture says that you will eventually reach 1. Despite testing all starting values up to an enormous number, no one has proved the conjecture is true for all possible starting values. The problem’s importance lies in its simplicity and difficulty, inspiring new ideas in mathematics and advancing fields like number theory, dynamical systems, and computer science. Proving or disproving the conjecture would revolutionize our understanding of math. The presence of infinite sequences is a matter of question. To investigate and solve this conjecture, we are utilizing a novel approach involving the fields of number theory and computer science.
文摘The aim of this paper is to study the 3x + 1 problem based on the Collatz iterative formula. It can be seen from the iterative formula that the necessary condition for the Collatz iteration convergence is that its slope being less than 1. An odd number N that satisfies the condition of a slope less than 1 after n<sup>th</sup> Collatz iterations is defined as an n-step odd number. Through statistical analysis, it is found that after n<sup>th</sup> Collatz iterations, the iterative value of any n-step odd number N that is greater than 1 is less than N, which proves that the slope less than 1 is a sufficient and necessary condition for Collatz iteration convergence.
文摘In this paper, we use two new effective tools and ingenious methods to prove the 3X + 1 conjecture. By using the recursive method, we firstly prove that any positive integer can be turned into an element of fourth column of the infinite-row-six-column-matrix after a finite times operation, thus we convert “the 3X + 1 conjecture” into an equivalent conjecture, which is: Any positive integer n must become 1 after finite operations under formation of <span style="white-space:nowrap;">σ(<em>n</em>)</span> , where <img src="Edit_dad9267d-3c54-455b-b30e-63819c207e54.png" width="300" height="117" alt="" /> Then, with the help of the infinite-row-four-column-matrix, we continue to use the recursive method to prove this conjecture strictly.
文摘This article introduces a change rule of 3x + 1 problem (Collatz conjecture), it’s named LiKe’s Rule. It’s a map of 3x + 1 problem, and details the path of each step of the change: For any positive integer, change by the Collatz conjecture. 1) This positive integer will change to an odd number;2) The odd number must change to a number of LiKe’s second sequence {3<sup>n</sup> – 1| n ∈ Z<sup>+</sup>};3) Then this 3<sup>n</sup> - 1 will change to a smaller 3<sup>n </sup>– 1 and gradually decrease to 8 (that is 3<sup>2</sup> - 1) then back to 1 in the end. If we can determine each step, the Collatz conjecture will be true. This is certainly more valuable than 2<sup>n</sup> (it might even explain 2<sup>n</sup>). And to illustrate the importance of this rule, introduced some important funny corollaries related to it.
基金supported by Natural Science Foundation of China under Grant Nos.60833008 and 60902024
文摘This paper studies the property of the recursive sequences in the 3x ＋ 1 conjecture. The authors introduce the concept of μ function, with which the 3x ＋ 1 conjecture can be transformed into two other conjectures： one is eventually periodic conjecture of the μ function and the other is periodic point conjecture. The authors prove that the 3x ＋ 1 conjecture is equivalent to the two conjectures above. In 2007, J. L. Simons proved the non-existence of nontrivial 2-cycle for the T function. In this paper, the authors prove that the μ function has nol-periodic points for 2 ≤ 1 ≤12. In 2005, J. L. Simons and B. M. M de Weger proved that there is no nontrivial/-cycle for the T function for 1 ≤68, and in this paper, the authors prove that there is no nontrivial l-cycle for the μ function for 2 ≤ 1≤ 102.
基金The work was supported by the Consiglio Nazionale delle Ricerche[4ClimAir(SAC.AD002.173.019),OzonPlant(DTA.AD002.640)]European Commission[MODERn(NEC)(LIFE20 GIE/IT/000091),MOTTLES(LIFE15 ENV/IT/000183)]Fondazione Cassa di Risparmio di Firenze[2013/7956]。
文摘Tropospheric ozone(O_(3))is a phytotoxic air pollutant and the O_(3)-induced visible foliar injury(O_(3)VFI)is a biomarker.A recently developed Free-air O_(3)eXposure(FO_(3)X)is a promising facility to verify field-observed“O_(3)-like”VFIs and to establish a flux-based threshold for the O_(3)VFI onset.The present study compared O_(3)-like VFI registered in the southern European forest sites with actual O_(3)VFI observed in a FO_(3)X experiment.The O_(3)-like VFIs were evaluated by eye in forests and thus it was subjective.According to the imaging analysis,we firstly demonstrated that major parts of the colors were similar in the field and the FO_(3)X.The color pallets for O_(3)VFI was species-specific and considered a advanced tool for the O_(3)VFI diagnosis.In addition,we calculated a flux-based threshold for the O_(3)VFI onset at the FO_(3)X based on a Phytotoxic Ozone Dose(POD_(1)),which ranged from 4.9 to 18.1 mmol m^(-2)POD1.This FO_(3)X-derived threshold partly explained but did not necessarily match with the observation for several tree species in actual forests.The multivariate analysis showed that O_(3)VFI was decreased by the presence of various species and suggested the importance of continuous monitoring activities in the field for the further analysis.
