Fibonacci sequence,generated by summing the preceding two terms,is a classical sequence renowned for its elegant properties.In this paper,leveraging properties of generalized Fibonacci sequences and formulas for conse...Fibonacci sequence,generated by summing the preceding two terms,is a classical sequence renowned for its elegant properties.In this paper,leveraging properties of generalized Fibonacci sequences and formulas for consecutive sums of equidistant sub-sequences,we investigate the ratio of the sum of numbers along main-diagonal and sub-diagonal of odd-order grids containing generalized Fibonacci sequences.We show that this ratio is solely dependent on the order of the grid,providing a concise and splendid identity.展开更多
USA scientists found </span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">a </span>&l...USA scientists found </span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">a </span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">22-year cycle in Parkfield earthquake sequence, and they predicted that the next quake would come in 1988 ± 5 with 95% possibility, while the quake happened in 2004, which is 11 years later than the prediction. Here, we reanalyze the Parkfield earthquake sequence, and find 11-year cycle;multiple 11-year cycle and Fibonacci sequence existed for earthquake. With these methods, the 2004 earthquake can be predicted well. We also predict that the next earthquake may occur in 2031-2032.展开更多
In this paper, we obtain the period of generalized Fibonacci sequence in finite rings with identity of order p2 by using equality recursively defined by Fn＋2 = A1Fn＋1 ＋ A0Fn, for n ≥ 0, where F0 = 0 （ the zero of...In this paper, we obtain the period of generalized Fibonacci sequence in finite rings with identity of order p2 by using equality recursively defined by Fn＋2 = A1Fn＋1 ＋ A0Fn, for n ≥ 0, where F0 = 0 （ the zero of the ring）, F1 = 1 （the identity of the ring） and A0 , A1 are generators elements of finite rings with identity of order p2. Also, we get some results between the period of generalized Fibonacci sequence in the finite rings oforderp2 and characteristic of these rings.展开更多
The integrable hungry Lotka-Volterra （hLV） system stands for a prey-predator model in mathematical biology. The discrete-time hLV （dhLV） system is derived from a time discretization of the hLV system. The solution...The integrable hungry Lotka-Volterra （hLV） system stands for a prey-predator model in mathematical biology. The discrete-time hLV （dhLV） system is derived from a time discretization of the hLV system. The solution to the dhLV system is known to be represented by using the Casorati determinant. In this paper, we show that if the entries of the Casorati determinant become an extended Fibonacci sequence at the initial discrete time, then those are also an extended Fibonacci sequence at any discrete time. In other words, the extended Fibonacci sequence always appears in the entries of the Casorati determinant under the time evolution of the dhLV system with suitable initial setting. We also show that one of the dhLV variables converges to the ratio of two successive extended Fibonacci numbers as the discrete time goes to infinity.展开更多
The transmission properties in Fibonacci quasi-periodic photonic crystals containing negative-zero-positive index metamaterials(NZPIM)are investigated systematically.Because of linear dispersion,an asymmetric band gap...The transmission properties in Fibonacci quasi-periodic photonic crystals containing negative-zero-positive index metamaterials(NZPIM)are investigated systematically.Because of linear dispersion,an asymmetric band gap appears near the optical Dirac point(DP)only at an oblique incidence.Zero-n gaps and Bragg gaps exist when the refractive indexes of NZPIM are negative and positive,respectively.We also obtain gaps induced by total reflection on both sides of the DP gap.The corresponding Goos-Hänchen(GH)shifts and phase variations are also demonstrated in detail.It is shown that the characteristics of the band gaps,GH shifts,and phase variations depend crucially on the incident angle,number of periods,and layer thickness.展开更多
The purpose of this paper is to analyze and visualize the exact invariant solution of the nonlinear simplified version of the shallow water equations which are being used to simulate equatorial atmospheric waves of pl...The purpose of this paper is to analyze and visualize the exact invariant solution of the nonlinear simplified version of the shallow water equations which are being used to simulate equatorial atmospheric waves of planetary scales. The method of obtaining the exact solution is based on the Lie group invariance principle. It is shown that the obtained invariant solution has a Fibonacci spiral-like form and has two parameters k and t<sub>0</sub>. We have defined a new model hypermarameter Δ<sub>k</sub>t = t – t<sub>0</sub>, where t is time. The question of particular interest is: can we tune the hypermarameter in order to match the exact solution to the actual Fibonacci spiral? It was discovered that the physically relevant part of the solution matches exactly the Fibonacci spiral.展开更多
In this paper, we introduce a new counting function a(m) related to the Lucas number, then use conjecture and induction methods to give an exact formula Ar(N)=α(n), (r=1,2,3) and prove them.
Since the time of the ancient Greeks, humans have been aware of this mathematical idea. Golden ratio is an irrational number that is symbolized by the Greek numeral phi (φ). One can find this ratio everywhere. It is ...Since the time of the ancient Greeks, humans have been aware of this mathematical idea. Golden ratio is an irrational number that is symbolized by the Greek numeral phi (φ). One can find this ratio everywhere. It is in nature, art, architecture, human body, etc. But this symbolism can result in a strong connection with mathematical nature. In this paper we will be discussing the connection between Fibonacci sequence (a series of numbers where every number is equal to the sum of two numbers before it) and Golden ratio. Secondly, how this mathematical idea shows up in a nature, such as sunflower and human DNA.展开更多
Criticism is an intellectual process that primarily searches for beauty aspects in the works of art,including architecture.This article explores the mathematical and philosophical principles of classical architectural...Criticism is an intellectual process that primarily searches for beauty aspects in the works of art,including architecture.This article explores the mathematical and philosophical principles of classical architectural criticism.It is hypothesized that design criteria during the Classic period were clear and specific.The research presents theories of classical art that focus on the process of beauty interpretation.It also assesses the mathematical evaluation of architectural art and beauty through“The Golden Ratio”and“The Fibonacci Sequence.”Classical philosophy,and its perception of beauty,is discussed as an essential basis in any artistic critical activity.The research asserts that the science of aesthetics is both objective and subjective,which explains the difference in aesthetic evaluation across eras.Objectivity stems from conditions of proportionality that must be met for an architectural art to be aesthetically judged as beautiful.Subjectivity lies in the time and place of the architectural work,whereby tendencies,tastes,and needs related to the human and geographical environment can affect the standards of beauty.This makes the evaluation of beauty in classical architecture a delicate and complex process in which many aspects must be considered to have an objective,fair,and correct judgment.展开更多
Associating geometric arrangements of 9 Loshu numbers modulo 5, investigating property of golden rectangles and characteristics of Fibonacci sequence modulo 10 as well as the two subsequences of its modular sequence b...Associating geometric arrangements of 9 Loshu numbers modulo 5, investigating property of golden rectangles and characteristics of Fibonacci sequence modulo 10 as well as the two subsequences of its modular sequence by modulo 5, the Loshu-Fibonacci Diagram is created based on strict logical deduction in this paper, which can disclose inherent relationship among Taiji sign, Loshu and Fibonacci sequence modulo 10 perfectly and unite such key ideas of holism, symmetry, holographic thought and yin-yang balance pursuit from Chinese medicine as a whole. Based on further analysis and reasoning, the authors discover that taking the golden ratio and Loshu-Fibonacci Diagram as a link, there is profound and universal association existing between researches of Chinese medicine and modern biology.展开更多
Compared with periodic structures,quasi-periodic structures have superior band gap properties and topological interface states.In this paper,a one-dimensional quasi-periodic Fibonacci water wave metamaterial model tha...Compared with periodic structures,quasi-periodic structures have superior band gap properties and topological interface states.In this paper,a one-dimensional quasi-periodic Fibonacci water wave metamaterial model that can be used to apply quasi-periodic structures to shallow-water wave systems is presented.The fluctuation characteristics of periodic and quasi-periodic structures are examined using finite element numerical calculations based on the shallow-water wave equation.The research results show that the band characteristics of quasi-periodic structures are complex,enabling flexible control of the propagation of shallow-water waves.Furthermore,the mirror-symmetrical design of Fibonacci quasi-periodic water wave metamaterials was created to engineer the topological interface states in shallow-water wave systems,ultimately achieving successful localization of wave energy.This research will greatly enrich our understanding of topology,expand the potential applications of quasi-periodic structures,and provide new insights for manipulating water waves and harvesting energy.展开更多
Sunflower petal is one of the parts of the sunflower which has drawn attention and has several applications these days.These applications justify getting information about physical properties,mechanical properties,dry...Sunflower petal is one of the parts of the sunflower which has drawn attention and has several applications these days.These applications justify getting information about physical properties,mechanical properties,drying trends,etc.in order to design new machines and use new methods to harvest or dry the sunflower petals.For three varieties of sunflower,picking force of petals was measured;number of petals of each head was counted;unit mass and 1000-unit mass of fresh petals were measured and length,width,and projected area of fresh petals were calculated based on image processing technique;frequency distributions of these parameters were modeled using statistical distribution models namely Gamma,Generalized Extreme Value(G.E.V),Lognormal,and Weibull.Results of picking force showed that with increasing number of days after appearing the first petal on each head from 5 to 14 and decreasing loading rate from 150 gmin^-1 to 50 g min^-1 values of picking force were decreased for three varieties,but diameter of sunflower head had different effects on picking force for each variety.Length,width,and number of petals of Dorsefid variety ranged from 38.52 to 95.44 mm,3.80 to 9.28mm and 29 to 89,respectively.The corresponding values ranged from 34.19 to 88.18 mm,4.28 to 10.60 mm and 21 to 89,respectively for Shamshiri variety and ranged from 44.47 to 114.63 mm,7.03 to 20.31 mm and 29 to 89 for Sirena variety.Results of frequency distribution modeling indicated that in most cases,G.E.V and Weibull distributions had better performance than other distributions.展开更多
基金Supported by the National Natural Science Foundation of China(Grant No.12471298)the Shaanxi Fundamental Science Research Project for Mathematics and Physics(Grant No.23JSQ031)the Shaanxi Province College Student Innovation and Entrepreneurship Training Program(Grant Nos.S202210699481 and S202310699324X).
文摘Fibonacci sequence,generated by summing the preceding two terms,is a classical sequence renowned for its elegant properties.In this paper,leveraging properties of generalized Fibonacci sequences and formulas for consecutive sums of equidistant sub-sequences,we investigate the ratio of the sum of numbers along main-diagonal and sub-diagonal of odd-order grids containing generalized Fibonacci sequences.We show that this ratio is solely dependent on the order of the grid,providing a concise and splendid identity.
文摘USA scientists found </span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">a </span></span></span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">22-year cycle in Parkfield earthquake sequence, and they predicted that the next quake would come in 1988 ± 5 with 95% possibility, while the quake happened in 2004, which is 11 years later than the prediction. Here, we reanalyze the Parkfield earthquake sequence, and find 11-year cycle;multiple 11-year cycle and Fibonacci sequence existed for earthquake. With these methods, the 2004 earthquake can be predicted well. We also predict that the next earthquake may occur in 2031-2032.
文摘In this paper, we obtain the period of generalized Fibonacci sequence in finite rings with identity of order p2 by using equality recursively defined by Fn＋2 = A1Fn＋1 ＋ A0Fn, for n ≥ 0, where F0 = 0 （ the zero of the ring）, F1 = 1 （the identity of the ring） and A0 , A1 are generators elements of finite rings with identity of order p2. Also, we get some results between the period of generalized Fibonacci sequence in the finite rings oforderp2 and characteristic of these rings.
文摘The integrable hungry Lotka-Volterra （hLV） system stands for a prey-predator model in mathematical biology. The discrete-time hLV （dhLV） system is derived from a time discretization of the hLV system. The solution to the dhLV system is known to be represented by using the Casorati determinant. In this paper, we show that if the entries of the Casorati determinant become an extended Fibonacci sequence at the initial discrete time, then those are also an extended Fibonacci sequence at any discrete time. In other words, the extended Fibonacci sequence always appears in the entries of the Casorati determinant under the time evolution of the dhLV system with suitable initial setting. We also show that one of the dhLV variables converges to the ratio of two successive extended Fibonacci numbers as the discrete time goes to infinity.
基金supported by the National Natural Science Foundation of China(No.61975109)the Science and Technology Commission of Shanghai Municipal(No.19ZR1417900)。
文摘The transmission properties in Fibonacci quasi-periodic photonic crystals containing negative-zero-positive index metamaterials(NZPIM)are investigated systematically.Because of linear dispersion,an asymmetric band gap appears near the optical Dirac point(DP)only at an oblique incidence.Zero-n gaps and Bragg gaps exist when the refractive indexes of NZPIM are negative and positive,respectively.We also obtain gaps induced by total reflection on both sides of the DP gap.The corresponding Goos-Hänchen(GH)shifts and phase variations are also demonstrated in detail.It is shown that the characteristics of the band gaps,GH shifts,and phase variations depend crucially on the incident angle,number of periods,and layer thickness.
文摘The purpose of this paper is to analyze and visualize the exact invariant solution of the nonlinear simplified version of the shallow water equations which are being used to simulate equatorial atmospheric waves of planetary scales. The method of obtaining the exact solution is based on the Lie group invariance principle. It is shown that the obtained invariant solution has a Fibonacci spiral-like form and has two parameters k and t<sub>0</sub>. We have defined a new model hypermarameter Δ<sub>k</sub>t = t – t<sub>0</sub>, where t is time. The question of particular interest is: can we tune the hypermarameter in order to match the exact solution to the actual Fibonacci spiral? It was discovered that the physically relevant part of the solution matches exactly the Fibonacci spiral.
基金Supported by the Education Department Foundation of Shaanxi Province(03JK213) Supported by the Weinan Teacher's College Foundation(03YKF001)
文摘In this paper, we introduce a new counting function a(m) related to the Lucas number, then use conjecture and induction methods to give an exact formula Ar(N)=α(n), (r=1,2,3) and prove them.
文摘Since the time of the ancient Greeks, humans have been aware of this mathematical idea. Golden ratio is an irrational number that is symbolized by the Greek numeral phi (φ). One can find this ratio everywhere. It is in nature, art, architecture, human body, etc. But this symbolism can result in a strong connection with mathematical nature. In this paper we will be discussing the connection between Fibonacci sequence (a series of numbers where every number is equal to the sum of two numbers before it) and Golden ratio. Secondly, how this mathematical idea shows up in a nature, such as sunflower and human DNA.
文摘Criticism is an intellectual process that primarily searches for beauty aspects in the works of art,including architecture.This article explores the mathematical and philosophical principles of classical architectural criticism.It is hypothesized that design criteria during the Classic period were clear and specific.The research presents theories of classical art that focus on the process of beauty interpretation.It also assesses the mathematical evaluation of architectural art and beauty through“The Golden Ratio”and“The Fibonacci Sequence.”Classical philosophy,and its perception of beauty,is discussed as an essential basis in any artistic critical activity.The research asserts that the science of aesthetics is both objective and subjective,which explains the difference in aesthetic evaluation across eras.Objectivity stems from conditions of proportionality that must be met for an architectural art to be aesthetically judged as beautiful.Subjectivity lies in the time and place of the architectural work,whereby tendencies,tastes,and needs related to the human and geographical environment can affect the standards of beauty.This makes the evaluation of beauty in classical architecture a delicate and complex process in which many aspects must be considered to have an objective,fair,and correct judgment.
文摘Associating geometric arrangements of 9 Loshu numbers modulo 5, investigating property of golden rectangles and characteristics of Fibonacci sequence modulo 10 as well as the two subsequences of its modular sequence by modulo 5, the Loshu-Fibonacci Diagram is created based on strict logical deduction in this paper, which can disclose inherent relationship among Taiji sign, Loshu and Fibonacci sequence modulo 10 perfectly and unite such key ideas of holism, symmetry, holographic thought and yin-yang balance pursuit from Chinese medicine as a whole. Based on further analysis and reasoning, the authors discover that taking the golden ratio and Loshu-Fibonacci Diagram as a link, there is profound and universal association existing between researches of Chinese medicine and modern biology.
基金supported by the National Natural Science Foundation of China(Grant No.11972034)the Youth Innovation Promotion Association of the Chinese Academy of Science(Grant No.2020018).
文摘Compared with periodic structures,quasi-periodic structures have superior band gap properties and topological interface states.In this paper,a one-dimensional quasi-periodic Fibonacci water wave metamaterial model that can be used to apply quasi-periodic structures to shallow-water wave systems is presented.The fluctuation characteristics of periodic and quasi-periodic structures are examined using finite element numerical calculations based on the shallow-water wave equation.The research results show that the band characteristics of quasi-periodic structures are complex,enabling flexible control of the propagation of shallow-water waves.Furthermore,the mirror-symmetrical design of Fibonacci quasi-periodic water wave metamaterials was created to engineer the topological interface states in shallow-water wave systems,ultimately achieving successful localization of wave energy.This research will greatly enrich our understanding of topology,expand the potential applications of quasi-periodic structures,and provide new insights for manipulating water waves and harvesting energy.
文摘Sunflower petal is one of the parts of the sunflower which has drawn attention and has several applications these days.These applications justify getting information about physical properties,mechanical properties,drying trends,etc.in order to design new machines and use new methods to harvest or dry the sunflower petals.For three varieties of sunflower,picking force of petals was measured;number of petals of each head was counted;unit mass and 1000-unit mass of fresh petals were measured and length,width,and projected area of fresh petals were calculated based on image processing technique;frequency distributions of these parameters were modeled using statistical distribution models namely Gamma,Generalized Extreme Value(G.E.V),Lognormal,and Weibull.Results of picking force showed that with increasing number of days after appearing the first petal on each head from 5 to 14 and decreasing loading rate from 150 gmin^-1 to 50 g min^-1 values of picking force were decreased for three varieties,but diameter of sunflower head had different effects on picking force for each variety.Length,width,and number of petals of Dorsefid variety ranged from 38.52 to 95.44 mm,3.80 to 9.28mm and 29 to 89,respectively.The corresponding values ranged from 34.19 to 88.18 mm,4.28 to 10.60 mm and 21 to 89,respectively for Shamshiri variety and ranged from 44.47 to 114.63 mm,7.03 to 20.31 mm and 29 to 89 for Sirena variety.Results of frequency distribution modeling indicated that in most cases,G.E.V and Weibull distributions had better performance than other distributions.
