This study introduces the representation of natural number sets as row vectors and pretends to offer a new perspective on the strong Goldbach conjecture. The natural numbers are restructured and expanded with the incl...This study introduces the representation of natural number sets as row vectors and pretends to offer a new perspective on the strong Goldbach conjecture. The natural numbers are restructured and expanded with the inclusion of the zero element as the source of a strong Goldbach conjecture reformulation. A prime Boolean vector is defined, pinpointing the positions of prime numbers within the odd number sequence. The natural unit primality is discussed in this context and transformed into a source of quantum-like indetermination. This approach allows for rephrasing the strong Goldbach conjecture, framed within a Boolean scalar product between the prime Boolean vector and its reverse. Throughout the discussion, other intriguing topics emerge and are thoroughly analyzed. A final description of two empirical algorithms is provided to prove the strong Goldbach conjecture.展开更多
This study describes how one can construct sets of composite natural numbers as tensorial products of the vectors created with the natural powers of prime numbers.
A simple recursive algorithm to generate the set of natural numbers, based on Mersenne numbers: M<sub>N</sub> = 2<sup>N</sup> – 1, is used to count the number of prime numbers within the preci...A simple recursive algorithm to generate the set of natural numbers, based on Mersenne numbers: M<sub>N</sub> = 2<sup>N</sup> – 1, is used to count the number of prime numbers within the precise Mersenne natural number intervals: [0;M<sub>N</sub>]. This permits the formulation of an extended twin prime conjecture. Moreover, it is found that the prime numbers subsets contained in Mersenne intervals have cardinalities strongly correlated with the corresponding Mersenne numbers.展开更多
A simple method presenting linear single strain DNA （LssDNA） sequence with natural numbers is introduced in this paper. The method presents LssDNA correspondingly with the numerals 1, 2, 3 and 4. After calculation, ...A simple method presenting linear single strain DNA （LssDNA） sequence with natural numbers is introduced in this paper. The method presents LssDNA correspondingly with the numerals 1, 2, 3 and 4. After calculation, the sequence can be coded in natural numbers which can also be decoded into the DNA sequence. Thus, an LssDNA sequence can be expressed in a natural number and a dot at coordinate axes. In the future, a new LssDNA sequences database termed ＂DotBank＂ would be realized in which each LssDNA sequence is determined as a dot.展开更多
The definition of Collatz Operator, the mathematical avatar of the Collatz Algorithm, permits the transformation of the Collatz conjecture, which is delineated over the whole natural number set, into an equivalent inf...The definition of Collatz Operator, the mathematical avatar of the Collatz Algorithm, permits the transformation of the Collatz conjecture, which is delineated over the whole natural number set, into an equivalent inference restricted to the odd prime number set only. Based on this redefinition, one can describe an empirical-heuristic proof of the Collatz conjecture.展开更多
Pascal Triangle is more of a number construction （body） then an array of the binomial coefficients. It is a mathematical body, like the digital code feeds for computer but with 2 dimensions. And there should be bodi...Pascal Triangle is more of a number construction （body） then an array of the binomial coefficients. It is a mathematical body, like the digital code feeds for computer but with 2 dimensions. And there should be bodies with x-dimensions and even abnormal or irregular appearances.展开更多
This work illustrates the steady state, two dimensional natural convective flow and heat transfer features in square enclosure containing heated hexagonal block maintained either at constant wall temperature(CWT) or u...This work illustrates the steady state, two dimensional natural convective flow and heat transfer features in square enclosure containing heated hexagonal block maintained either at constant wall temperature(CWT) or uniform heat flux(UHF) thermal conditions. Governing equations(mass, momentum and energy) are solved by using finite volume method(FVM) with 3rd order accurate QUICK discretization scheme and SIMPLE algorithm for range of field pertinent parameters such as, Grashof number(10~3≤ Gr ≤ 10~6), Prandtl number(1 ≤ Pr ≤ 100) and power law index(0.5 ≤ n ≤ 1.5). The analysis of momentum and heat transfer characteristics are delineated by evolution of streamlines, isotherms, variation of average Nusselt number value and Colburn factor for natural convection(j_(nH)). A remarkable change is observed on fluid flow and thermal distribution pattern in cavity for both thermal conditions. Nusselt number shows linear variation with Grashof and Prandtl numbers; while rate of heat transfer by convection decreases for power law index value. Higher heat transfer rate can be achieved by using uniform heat flux condition. A Nusselt number correlation is developed for possible utilization in engineering/scientific design purpose.展开更多
This scientific paper is a comparative analysis of two mathematical conjectures. The newly proposed -3(-n) - 1 Remer conjecture and how it is related to and a proof of the more well known 3n + 1 Collatz conjecture. An...This scientific paper is a comparative analysis of two mathematical conjectures. The newly proposed -3(-n) - 1 Remer conjecture and how it is related to and a proof of the more well known 3n + 1 Collatz conjecture. An overview of both conjectures and their respective iterative processes will be presented. Showcasing their unique properties and behavior to each other. Through a detailed comparison, we highlight the similarities and differences between these two conjectures and discuss their significance in the field of mathematics. And how they prove each other to be true.展开更多
This work is devoted to the theory of prime numbers. Firstly it introduced the concept of matrix primes, which can help to generate a sequence of prime numbers. Then it proposed a number of theorems, which together wi...This work is devoted to the theory of prime numbers. Firstly it introduced the concept of matrix primes, which can help to generate a sequence of prime numbers. Then it proposed a number of theorems, which together with theorem of Dirichlet, Siegel and Euler allow to prove the infinity of twin primes.展开更多
In Chinese, Mathematics is “数学”, i.e. Numbers （数） Knowledge （学）. Human identifies us ourselves from the whole one of the Universe by means of numbers, and numbers are equal object status identities, and numb...In Chinese, Mathematics is “数学”, i.e. Numbers （数） Knowledge （学）. Human identifies us ourselves from the whole one of the Universe by means of numbers, and numbers are equal object status identities, and numbers counting means repetition, i.e. the process or time dimension. Given a series of n numbers, a1 , a2 , … , an , n∈N, i.e. the n objects of status, and a general formula a（i） for ordering all of the number terms by i （i=1 to n） can be generated, which shows a process of counting, and from sequencing to progression, some properties are extracted.展开更多
The probability of crane living in reedy wetlands can reach 100%, at the same time, the area of reed, the water level and adjacent water area are main factors which control the crane'...The probability of crane living in reedy wetlands can reach 100%, at the same time, the area of reed, the water level and adjacent water area are main factors which control the crane's habitat selection. We all know that all these factors are spatially heterogeneous. For the Xianghai wetland safety and to protect the Xianghai wetland habitat of crane, this paper has mainly identified a solution to these problems. The wetland information is extracted from the TM images, which reflect the whole wetland landscape and are very important for both quantitative analysis of remote sensing observation of the earth system and positioning analysis in GIS database that is automatically extracted from DEM. The DEM for Xianghai characteristics of topography is created. On the basis of the GRID SUBMODULE, applying the GIS spatial overlay analysis, the relationship between the water level and the reed area below the water level and the rating distribution maps of reed area above water level is established. When the water level reaches the altitude of 165 m, the reed area, 981.2 ha is maximum, i.e., the water level of 165 m is the optimal.展开更多
A new set of steps for dividing process, the Reversed Dividing Process, is introduced, in which the position based number representation digits for the value of l/X are reversely output one by one compared with digit ...A new set of steps for dividing process, the Reversed Dividing Process, is introduced, in which the position based number representation digits for the value of l/X are reversely output one by one compared with digit output order of the normal dividing process.展开更多
Rural Ethiopia has significant untapped potential for hydro and solar energy generation systems.However,challenges arise from seasonal variations and unfavourable topographic positions of flowing rivers,hindering the ...Rural Ethiopia has significant untapped potential for hydro and solar energy generation systems.However,challenges arise from seasonal variations and unfavourable topographic positions of flowing rivers,hindering the efficient exploitation of these resources.Despite the country’s abundance in hydro and solar energy resources,>75%of the population still lack access to electricity from the national grid.This work deals with energy resource potential assessment and techno-economic analysis of micro hydro-photovol-taic(PV)hybrid systems,considered in the case study of Goda Warke village,located in the Yaya Gulele district.A novel framework is proposed that utilizes the Natural Resource Soil Conservation Service curve number method to assess the energy potential of micro-hydro energy in ungauged basins,specifically at the exit point of the Girar River basin catchment.The average monthly flow rate in the basin is 0.975 m3/s,while the area exhibits a solar radiation potential of 5.39 kWh/m^(2)/day.Energy policy promotes expanding ac-cess to modern energy sources and utilization of indigenous energy resources.Simulation results indicate that the hydro/PV/diesel generator(DG)/battery and hydro/PV/battery systems are the most optimal choices based on net present cost,with the inclusion of a DG for economic comparison.Micro-hydro energy covers most of the electric load in the area,achieving a capacity factor of 47.5%.The cost of energy and net present cost were found to be sensitive to variables such as the price of diesel fuel,pipe head loss,and the growth of the village load.The optimized system demonstrated a hydro energy potential of 1405.37 MWh/year and a PV energy output of 274.04 MWh/year,resulting in a levelized cost of energy of 0.0057 and 0.049$/kWh for the hydro and PV components,respectively.展开更多
The action of N on l^2(N) is studied in association with the multiplicative structure of N. Then the maximal ideal space of the Banach algebra generated by N is homeomorphic to the product of closed unit disks indexed...The action of N on l^2(N) is studied in association with the multiplicative structure of N. Then the maximal ideal space of the Banach algebra generated by N is homeomorphic to the product of closed unit disks indexed by primes, which reflects the fundamental theorem of arithmetic. The C*-algebra generated by N does not contain any non-zero projection of finite rank. This assertion is equivalent to the existence of infinitely many primes. The von Neumann algebra generated by N is B(l^2(N)), the set of all bounded operators on l^2(N).Moreover, the differential operator on l^2(N,1/n(n+1)) defined by ▽f = μ * f is considered, where μ is the Mbius function. It is shown that the spectrum σ(▽) contains the closure of {ζ(s)-1: Re(s) > 1}. Interesting problems concerning are discussed.展开更多
文摘This study introduces the representation of natural number sets as row vectors and pretends to offer a new perspective on the strong Goldbach conjecture. The natural numbers are restructured and expanded with the inclusion of the zero element as the source of a strong Goldbach conjecture reformulation. A prime Boolean vector is defined, pinpointing the positions of prime numbers within the odd number sequence. The natural unit primality is discussed in this context and transformed into a source of quantum-like indetermination. This approach allows for rephrasing the strong Goldbach conjecture, framed within a Boolean scalar product between the prime Boolean vector and its reverse. Throughout the discussion, other intriguing topics emerge and are thoroughly analyzed. A final description of two empirical algorithms is provided to prove the strong Goldbach conjecture.
文摘This study describes how one can construct sets of composite natural numbers as tensorial products of the vectors created with the natural powers of prime numbers.
文摘A simple recursive algorithm to generate the set of natural numbers, based on Mersenne numbers: M<sub>N</sub> = 2<sup>N</sup> – 1, is used to count the number of prime numbers within the precise Mersenne natural number intervals: [0;M<sub>N</sub>]. This permits the formulation of an extended twin prime conjecture. Moreover, it is found that the prime numbers subsets contained in Mersenne intervals have cardinalities strongly correlated with the corresponding Mersenne numbers.
基金Supported by the National High Technology Research and Development Program of China (863 Program) (No.2003AA622030)
文摘A simple method presenting linear single strain DNA （LssDNA） sequence with natural numbers is introduced in this paper. The method presents LssDNA correspondingly with the numerals 1, 2, 3 and 4. After calculation, the sequence can be coded in natural numbers which can also be decoded into the DNA sequence. Thus, an LssDNA sequence can be expressed in a natural number and a dot at coordinate axes. In the future, a new LssDNA sequences database termed ＂DotBank＂ would be realized in which each LssDNA sequence is determined as a dot.
文摘The definition of Collatz Operator, the mathematical avatar of the Collatz Algorithm, permits the transformation of the Collatz conjecture, which is delineated over the whole natural number set, into an equivalent inference restricted to the odd prime number set only. Based on this redefinition, one can describe an empirical-heuristic proof of the Collatz conjecture.
文摘Pascal Triangle is more of a number construction （body） then an array of the binomial coefficients. It is a mathematical body, like the digital code feeds for computer but with 2 dimensions. And there should be bodies with x-dimensions and even abnormal or irregular appearances.
文摘This work illustrates the steady state, two dimensional natural convective flow and heat transfer features in square enclosure containing heated hexagonal block maintained either at constant wall temperature(CWT) or uniform heat flux(UHF) thermal conditions. Governing equations(mass, momentum and energy) are solved by using finite volume method(FVM) with 3rd order accurate QUICK discretization scheme and SIMPLE algorithm for range of field pertinent parameters such as, Grashof number(10~3≤ Gr ≤ 10~6), Prandtl number(1 ≤ Pr ≤ 100) and power law index(0.5 ≤ n ≤ 1.5). The analysis of momentum and heat transfer characteristics are delineated by evolution of streamlines, isotherms, variation of average Nusselt number value and Colburn factor for natural convection(j_(nH)). A remarkable change is observed on fluid flow and thermal distribution pattern in cavity for both thermal conditions. Nusselt number shows linear variation with Grashof and Prandtl numbers; while rate of heat transfer by convection decreases for power law index value. Higher heat transfer rate can be achieved by using uniform heat flux condition. A Nusselt number correlation is developed for possible utilization in engineering/scientific design purpose.
文摘This scientific paper is a comparative analysis of two mathematical conjectures. The newly proposed -3(-n) - 1 Remer conjecture and how it is related to and a proof of the more well known 3n + 1 Collatz conjecture. An overview of both conjectures and their respective iterative processes will be presented. Showcasing their unique properties and behavior to each other. Through a detailed comparison, we highlight the similarities and differences between these two conjectures and discuss their significance in the field of mathematics. And how they prove each other to be true.
文摘This work is devoted to the theory of prime numbers. Firstly it introduced the concept of matrix primes, which can help to generate a sequence of prime numbers. Then it proposed a number of theorems, which together with theorem of Dirichlet, Siegel and Euler allow to prove the infinity of twin primes.
文摘In Chinese, Mathematics is “数学”, i.e. Numbers （数） Knowledge （学）. Human identifies us ourselves from the whole one of the Universe by means of numbers, and numbers are equal object status identities, and numbers counting means repetition, i.e. the process or time dimension. Given a series of n numbers, a1 , a2 , … , an , n∈N, i.e. the n objects of status, and a general formula a（i） for ordering all of the number terms by i （i=1 to n） can be generated, which shows a process of counting, and from sequencing to progression, some properties are extracted.
基金Knowledge Innovation Project of CAS No.KZCX2-SW-320-1+1 种基金 National Key Project for Science and Technology No.2001-BA608B-03-02
文摘The probability of crane living in reedy wetlands can reach 100%, at the same time, the area of reed, the water level and adjacent water area are main factors which control the crane's habitat selection. We all know that all these factors are spatially heterogeneous. For the Xianghai wetland safety and to protect the Xianghai wetland habitat of crane, this paper has mainly identified a solution to these problems. The wetland information is extracted from the TM images, which reflect the whole wetland landscape and are very important for both quantitative analysis of remote sensing observation of the earth system and positioning analysis in GIS database that is automatically extracted from DEM. The DEM for Xianghai characteristics of topography is created. On the basis of the GRID SUBMODULE, applying the GIS spatial overlay analysis, the relationship between the water level and the reed area below the water level and the rating distribution maps of reed area above water level is established. When the water level reaches the altitude of 165 m, the reed area, 981.2 ha is maximum, i.e., the water level of 165 m is the optimal.
文摘A new set of steps for dividing process, the Reversed Dividing Process, is introduced, in which the position based number representation digits for the value of l/X are reversely output one by one compared with digit output order of the normal dividing process.
文摘Rural Ethiopia has significant untapped potential for hydro and solar energy generation systems.However,challenges arise from seasonal variations and unfavourable topographic positions of flowing rivers,hindering the efficient exploitation of these resources.Despite the country’s abundance in hydro and solar energy resources,>75%of the population still lack access to electricity from the national grid.This work deals with energy resource potential assessment and techno-economic analysis of micro hydro-photovol-taic(PV)hybrid systems,considered in the case study of Goda Warke village,located in the Yaya Gulele district.A novel framework is proposed that utilizes the Natural Resource Soil Conservation Service curve number method to assess the energy potential of micro-hydro energy in ungauged basins,specifically at the exit point of the Girar River basin catchment.The average monthly flow rate in the basin is 0.975 m3/s,while the area exhibits a solar radiation potential of 5.39 kWh/m^(2)/day.Energy policy promotes expanding ac-cess to modern energy sources and utilization of indigenous energy resources.Simulation results indicate that the hydro/PV/diesel generator(DG)/battery and hydro/PV/battery systems are the most optimal choices based on net present cost,with the inclusion of a DG for economic comparison.Micro-hydro energy covers most of the electric load in the area,achieving a capacity factor of 47.5%.The cost of energy and net present cost were found to be sensitive to variables such as the price of diesel fuel,pipe head loss,and the growth of the village load.The optimized system demonstrated a hydro energy potential of 1405.37 MWh/year and a PV energy output of 274.04 MWh/year,resulting in a levelized cost of energy of 0.0057 and 0.049$/kWh for the hydro and PV components,respectively.
基金supported by National Natural Science Foundation of China (Grant Nos. 11371290 and 11701549)Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2017JM1045)
文摘The action of N on l^2(N) is studied in association with the multiplicative structure of N. Then the maximal ideal space of the Banach algebra generated by N is homeomorphic to the product of closed unit disks indexed by primes, which reflects the fundamental theorem of arithmetic. The C*-algebra generated by N does not contain any non-zero projection of finite rank. This assertion is equivalent to the existence of infinitely many primes. The von Neumann algebra generated by N is B(l^2(N)), the set of all bounded operators on l^2(N).Moreover, the differential operator on l^2(N,1/n(n+1)) defined by ▽f = μ * f is considered, where μ is the Mbius function. It is shown that the spectrum σ(▽) contains the closure of {ζ(s)-1: Re(s) > 1}. Interesting problems concerning are discussed.
