In this paper, we prove that if a, b and c are pairwise coprime positive integers such that a^2＋b^2=c^r,a〉b,a≡3 （mod4）,b≡2 （mod4） and c-1 is not a square, thena a^x＋b^y=c^z has only the positive integer solut...In this paper, we prove that if a, b and c are pairwise coprime positive integers such that a^2＋b^2=c^r,a〉b,a≡3 （mod4）,b≡2 （mod4） and c-1 is not a square, thena a^x＋b^y=c^z has only the positive integer solution （x, y, z） = （2, 2, r）. Let m and r be positive integers with 2｜m and 2 r, define the integers Ur, Vr by （m ＋√-1）^r=Vr＋Ur√-1. If a = ｜Ur｜,b=｜Vr｜,c = m^2＋1 with m ≡ 2 （mod 4）,a ≡ 3 （mod 4）, and if r 〈 m/√1.5log3（m^2＋1）-1, then a^x ＋ b^y = c^z has only the positive integer solution （x,y, z） = （2, 2, r）. The argument here is elementary.展开更多
Imaginary cyclic fields of degree p-1 which have two distinct unramified cyclic extensions of degree p are produced using elementary properties of the Lucas sequences. An infinite family of imaginary cyclic fields of ...Imaginary cyclic fields of degree p-1 which have two distinct unramified cyclic extensions of degree p are produced using elementary properties of the Lucas sequences. An infinite family of imaginary cyclic fields of degree p-1 are then given with the p-rank of the ideal class groups of at least two.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NSF of China (No.10571180)the Guangdong Provincial Natural Science Foundation (No.8151027501000114)
文摘In this paper, we prove that if a, b and c are pairwise coprime positive integers such that a^2＋b^2=c^r,a〉b,a≡3 （mod4）,b≡2 （mod4） and c-1 is not a square, thena a^x＋b^y=c^z has only the positive integer solution （x, y, z） = （2, 2, r）. Let m and r be positive integers with 2｜m and 2 r, define the integers Ur, Vr by （m ＋√-1）^r=Vr＋Ur√-1. If a = ｜Ur｜,b=｜Vr｜,c = m^2＋1 with m ≡ 2 （mod 4）,a ≡ 3 （mod 4）, and if r 〈 m/√1.5log3（m^2＋1）-1, then a^x ＋ b^y = c^z has only the positive integer solution （x,y, z） = （2, 2, r）. The argument here is elementary.
基金Supported by the Japan Society for the Promotion of Science (JSPS) (No. 14540030) the JSPS Research Fellowships for Young Scientists
文摘Imaginary cyclic fields of degree p-1 which have two distinct unramified cyclic extensions of degree p are produced using elementary properties of the Lucas sequences. An infinite family of imaginary cyclic fields of degree p-1 are then given with the p-rank of the ideal class groups of at least two.
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.
