In this paper we prove in a new way, the well known result, that Fermat’s equation a<sup>4</sup> + b<sup>4</sup> = c<sup>4</sup>, is not solvable in ℕ , when abc≠0 . To show this ...In this paper we prove in a new way, the well known result, that Fermat’s equation a<sup>4</sup> + b<sup>4</sup> = c<sup>4</sup>, is not solvable in ℕ , when abc≠0 . To show this result, it suffices to prove that: ( F 0 ): a 1 4 + ( 2 s b 1 ) 4 = c 1 4 , is not solvable in ℕ , (where a 1 , b 1 , c 1 ∈2ℕ+1 , pairwise primes, with necessarly 2≤s∈ℕ ). The key idea of our proof is to show that if (F<sub>0</sub>) holds, then there exist α 2 , β 2 , γ 2 ∈2ℕ+1 , such that ( F 1 ): α 2 4 + ( 2 s−1 β 2 ) 4 = γ 2 4 , holds too. From where, one conclude that it is not possible, because if we choose the quantity 2 ≤ s, as minimal in value among all the solutions of ( F 0 ) , then ( α 2 ,2 s−1 β 2 , γ 2 ) is also a solution of Fermat’s type, but with 2≤s−1<s , witch is absurd. To reach such a result, we suppose first that (F<sub>0</sub>) is solvable in ( a 1 ,2 s b 1 , c 1 ) , s ≥ 2 like above;afterwards, proceeding with “Pythagorician divisors”, we creat the notions of “Fermat’s b-absolute divisors”: ( d b , d ′ b ) which it uses hereafter. Then to conclude our proof, we establish the following main theorem: there is an equivalence between (i) and (ii): (i) (F<sub>0</sub>): a 1 4 + ( 2 s b 1 ) 4 = c 1 4 , is solvable in ℕ , with 2≤s∈ℕ , ( a 1 , b 1 , c 1 )∈ ( 2ℕ+1 ) 3 , coprime in pairs. (ii) ∃( a 1 , b 1 , c 1 )∈ ( 2ℕ+1 ) 3 , coprime in pairs, for wich: ∃( b ′ 2 , b 2 , b ″ 2 )∈ ( 2ℕ+1 ) 3 coprime in pairs, and 2≤s∈ℕ , checking b 1 = b ′ 2 b 2 b ″ 2 , and such that for notations: S=s−λ( s−1 ) , with λ∈{ 0,1 } defined by c 1 − a 1 2 ≡λ( mod2 ) , d b =gcd( 2 s b 1 , c 1 − a 1 )= 2 S b 2 and d ′ b = 2 s−S b ′ 2 = 2 s B 2 d b , where ( 2 s B 2 ) 2 =gcd( b 1 2 , c 1 2 − a 1 2 ) , the following system is checked: { c 1 − a 1 = d b 4 2 2+λ = 2 2−λ ( 2 S−1 b 2 ) 4 c 1 + a 1 = 2 1+λ d ′ b 4 = 2 1+λ ( 2 s−S b ′ 2 ) 4 c 1 2 + a 1 2 =2 b ″ 2 4;and this system implies: ( b 1−λ,2 4 ) 2 + ( 2 4s−3 b λ,2 4 ) 2 = ( b ″ 2 2 ) 2;where: ( b 1−λ,2 , b λ,2 , b ″ 2 )={ ( b ′ 2 , b 2 , b ″ 2 ) if λ=0 ( b 2 , b ′ 2 , b ″ 2 ) if λ=1;From where, it is quite easy to conclude, following the method explained above, and which thus closes, part I, of this article. .展开更多
The J_2-integral induced from the interface of bimaterial solids(J_2^(interface))is stud- ied by numerical method.First,the effect on the J_2-integral induced from the interface is very significant in bimaterial solid...The J_2-integral induced from the interface of bimaterial solids(J_2^(interface))is stud- ied by numerical method.First,the effect on the J_2-integral induced from the interface is very significant in bimaterial solids,which is inherently related to that induced from the subinterface cracks.Moreover,it can be concluded that either the first or the second component of the J_k- vector is always equal to zero when the contour encloses both the cracks and the whole interface in bimaterial solids.Secondly,it can also be concluded that the interface does produce significant effect on the J_2-integral induced from the subinterface cracks(J_2^(sub))in bimaterial solids.This effect depends on the geometry of the crack arrangement,which is corresponding to the different interaction effect among the cracks and the interface.Moreover,the interface effect on the J_2^(sub) can be neglected when the distance from the crack center to the interface is large enough,which reveals that the bimaterial solids can be regarded as homogenous solids in fracture analysis when the subinterface crack is far enough from the interface.Three examples are given in this paper.展开更多
One existence integral condition was obtained for the adapted solution of the general backward stochastic differential equations(BSDEs). Then by solving the integral constraint condition, and using a limit procedure, ...One existence integral condition was obtained for the adapted solution of the general backward stochastic differential equations(BSDEs). Then by solving the integral constraint condition, and using a limit procedure, a new approach method is proposed and the existence of the solution was proved for the BSDEs if the diffusion coefficients satisfy the locally Lipschitz condition. In the special case the solution was a Brownian bridge. The uniqueness is also considered in the meaning of "F0-integrable equivalent class" . The new approach method would give us an efficient way to control the main object instead of the "noise".展开更多
Let D be an integral domain with quotient field K,D be the integral closure of D in K,and D^[w] be the ω-integral closure of D in K;so D ■ D^[w],and equality holds when D is Noetherian or dim(D)=1.The Mori-Nagata th...Let D be an integral domain with quotient field K,D be the integral closure of D in K,and D^[w] be the ω-integral closure of D in K;so D ■ D^[w],and equality holds when D is Noetherian or dim(D)=1.The Mori-Nagata theorem states that if D is Noetherian,then D is a Krull domain;it has also been investigated when D is a Dedekind domain.We study integral domains D such that D^[w] is a Krull domain.We also provide an example of an integral domain D such that D ■ D ■ D^[w],t-dim(D)=1,D is a Priifer multiplication domain with v-dim(D)=2,and D^[w] is a UFD.展开更多
In this paper, we establish some Rosenthal type inequalities for maximum partial sums of asymptotically almost negatively associated random variables, which extend the corresponding results for negatively associated r...In this paper, we establish some Rosenthal type inequalities for maximum partial sums of asymptotically almost negatively associated random variables, which extend the corresponding results for negatively associated random variables. As applications of these inequalities, by employing the notions of residual Cesàro α-integrability and strong residual Cesàro α-integrability, we derive some results on Lp convergence where 1 < p < 2 and complete convergence. In addition, we estimate the rate of convergence in Marcinkiewicz-Zygmund strong law for partial sums of identically distributed random variables.展开更多
文摘In this paper we prove in a new way, the well known result, that Fermat’s equation a<sup>4</sup> + b<sup>4</sup> = c<sup>4</sup>, is not solvable in ℕ , when abc≠0 . To show this result, it suffices to prove that: ( F 0 ): a 1 4 + ( 2 s b 1 ) 4 = c 1 4 , is not solvable in ℕ , (where a 1 , b 1 , c 1 ∈2ℕ+1 , pairwise primes, with necessarly 2≤s∈ℕ ). The key idea of our proof is to show that if (F<sub>0</sub>) holds, then there exist α 2 , β 2 , γ 2 ∈2ℕ+1 , such that ( F 1 ): α 2 4 + ( 2 s−1 β 2 ) 4 = γ 2 4 , holds too. From where, one conclude that it is not possible, because if we choose the quantity 2 ≤ s, as minimal in value among all the solutions of ( F 0 ) , then ( α 2 ,2 s−1 β 2 , γ 2 ) is also a solution of Fermat’s type, but with 2≤s−1<s , witch is absurd. To reach such a result, we suppose first that (F<sub>0</sub>) is solvable in ( a 1 ,2 s b 1 , c 1 ) , s ≥ 2 like above;afterwards, proceeding with “Pythagorician divisors”, we creat the notions of “Fermat’s b-absolute divisors”: ( d b , d ′ b ) which it uses hereafter. Then to conclude our proof, we establish the following main theorem: there is an equivalence between (i) and (ii): (i) (F<sub>0</sub>): a 1 4 + ( 2 s b 1 ) 4 = c 1 4 , is solvable in ℕ , with 2≤s∈ℕ , ( a 1 , b 1 , c 1 )∈ ( 2ℕ+1 ) 3 , coprime in pairs. (ii) ∃( a 1 , b 1 , c 1 )∈ ( 2ℕ+1 ) 3 , coprime in pairs, for wich: ∃( b ′ 2 , b 2 , b ″ 2 )∈ ( 2ℕ+1 ) 3 coprime in pairs, and 2≤s∈ℕ , checking b 1 = b ′ 2 b 2 b ″ 2 , and such that for notations: S=s−λ( s−1 ) , with λ∈{ 0,1 } defined by c 1 − a 1 2 ≡λ( mod2 ) , d b =gcd( 2 s b 1 , c 1 − a 1 )= 2 S b 2 and d ′ b = 2 s−S b ′ 2 = 2 s B 2 d b , where ( 2 s B 2 ) 2 =gcd( b 1 2 , c 1 2 − a 1 2 ) , the following system is checked: { c 1 − a 1 = d b 4 2 2+λ = 2 2−λ ( 2 S−1 b 2 ) 4 c 1 + a 1 = 2 1+λ d ′ b 4 = 2 1+λ ( 2 s−S b ′ 2 ) 4 c 1 2 + a 1 2 =2 b ″ 2 4;and this system implies: ( b 1−λ,2 4 ) 2 + ( 2 4s−3 b λ,2 4 ) 2 = ( b ″ 2 2 ) 2;where: ( b 1−λ,2 , b λ,2 , b ″ 2 )={ ( b ′ 2 , b 2 , b ″ 2 ) if λ=0 ( b 2 , b ′ 2 , b ″ 2 ) if λ=1;From where, it is quite easy to conclude, following the method explained above, and which thus closes, part I, of this article. .
基金Project supported by the National Natural Science Foundation of China(No.19472053)the Doctorate Foundation of Xi'an Jiaotong University(No.DFXJU2000-15).
文摘The J_2-integral induced from the interface of bimaterial solids(J_2^(interface))is stud- ied by numerical method.First,the effect on the J_2-integral induced from the interface is very significant in bimaterial solids,which is inherently related to that induced from the subinterface cracks.Moreover,it can be concluded that either the first or the second component of the J_k- vector is always equal to zero when the contour encloses both the cracks and the whole interface in bimaterial solids.Secondly,it can also be concluded that the interface does produce significant effect on the J_2-integral induced from the subinterface cracks(J_2^(sub))in bimaterial solids.This effect depends on the geometry of the crack arrangement,which is corresponding to the different interaction effect among the cracks and the interface.Moreover,the interface effect on the J_2^(sub) can be neglected when the distance from the crack center to the interface is large enough,which reveals that the bimaterial solids can be regarded as homogenous solids in fracture analysis when the subinterface crack is far enough from the interface.Three examples are given in this paper.
基金National Natural Science Foundation of China ( No. 11171062 ) Natural Science Foundation for the Youth,China ( No.11101077) Innovation Program of Shanghai Municipal Education Commission,China ( No. 12ZZ063)
文摘One existence integral condition was obtained for the adapted solution of the general backward stochastic differential equations(BSDEs). Then by solving the integral constraint condition, and using a limit procedure, a new approach method is proposed and the existence of the solution was proved for the BSDEs if the diffusion coefficients satisfy the locally Lipschitz condition. In the special case the solution was a Brownian bridge. The uniqueness is also considered in the meaning of "F0-integrable equivalent class" . The new approach method would give us an efficient way to control the main object instead of the "noise".
基金supported by the Academic Research Fund of Hoseo University in 2017(no.2017-0047).
文摘Let D be an integral domain with quotient field K,D be the integral closure of D in K,and D^[w] be the ω-integral closure of D in K;so D ■ D^[w],and equality holds when D is Noetherian or dim(D)=1.The Mori-Nagata theorem states that if D is Noetherian,then D is a Krull domain;it has also been investigated when D is a Dedekind domain.We study integral domains D such that D^[w] is a Krull domain.We also provide an example of an integral domain D such that D ■ D ■ D^[w],t-dim(D)=1,D is a Priifer multiplication domain with v-dim(D)=2,and D^[w] is a UFD.
基金supported by National Natural Science Foundation of China (Grant No.10871217) the SCR of Chongqing Municipal Education Commission (Grant No.KJ090703)
文摘In this paper, we establish some Rosenthal type inequalities for maximum partial sums of asymptotically almost negatively associated random variables, which extend the corresponding results for negatively associated random variables. As applications of these inequalities, by employing the notions of residual Cesàro α-integrability and strong residual Cesàro α-integrability, we derive some results on Lp convergence where 1 < p < 2 and complete convergence. In addition, we estimate the rate of convergence in Marcinkiewicz-Zygmund strong law for partial sums of identically distributed random variables.
基金J.Koolen was partially supported by the National Natural Science Foundation of China(Grants No.11471009 and No.11671376)by the fund Anhui Initiative in Quantum Information Technologies(No.AHY150000)Q.Q.Yang was partially supported by the China Scholarship Council(No.201806340049)when she was studying at Tohoku University as a joint Ph.D.student.
文摘In this note we give several problems and conjectures on graphs with fixed smallest eigenvalue.