Suppose thatλ_(1),λ_(2),λ_(3),λ_(4),λ_(5)are nonzero real numbers,not all of the same sign,andλ_(1)/λ_(2)is irrational and algebraic.Let V be a well-spaced sequence,δ>0.In this paper,it is proved that,for ...Suppose thatλ_(1),λ_(2),λ_(3),λ_(4),λ_(5)are nonzero real numbers,not all of the same sign,andλ_(1)/λ_(2)is irrational and algebraic.Let V be a well-spaced sequence,δ>0.In this paper,it is proved that,for anyε>0,the number of v∈V with v≤N such that the following inequality|λ_(1)p_(1)~2+λ_(2)p_(2)~2+λ_(3)p_(3)~4+λ_(4)p_(4)~4+λ_5p_5~4-v|<v^(-δ)has no solution in prime variables p_(1),p_(2),p_(3),p_(4),p_(5)does not exceed O(N^(29/32+2δ+ε)).展开更多
In this paper,a robust and consistent COVID-19 emergency decision-making approach is proposed based on q-rung linear diophantine fuzzy set(q-RLDFS),differential evolutionary(DE)optimization principles,and evidential r...In this paper,a robust and consistent COVID-19 emergency decision-making approach is proposed based on q-rung linear diophantine fuzzy set(q-RLDFS),differential evolutionary(DE)optimization principles,and evidential reasoning(ER)methodology.The proposed approach uses q-RLDFS in order to represent the evaluating values of the alternatives corresponding to the attributes.DE optimization is used to obtain the optimal weights of the attributes,and ER methodology is used to compute the aggregated q-rung linear diophantine fuzzy values(q-RLDFVs)of each alternative.Then the score values of alternatives are computed based on the aggregated q-RLDFVs.An alternative with the maximum score value is selected as a better one.The applicability of the proposed approach has been illustrated in COVID-19 emergency decision-making system and sustainable energy planning management.Moreover,we have validated the proposed approach with a numerical example.Finally,a comparative study is provided with the existing models,where the proposed approach is found to be robust to perform better and consistent in uncertain environments.展开更多
Suppose that c,d,α,β,are real numbers satisfying 1<d<c<14=13,1<α<β<6^(1−d=c).In this paper,we prove that there exist numbers N_(1)^((0))and N_(2)^((0)),depending on c,d,α,β,such that for all re...Suppose that c,d,α,β,are real numbers satisfying 1<d<c<14=13,1<α<β<6^(1−d=c).In this paper,we prove that there exist numbers N_(1)^((0))and N_(2)^((0)),depending on c,d,α,β,such that for all real numbers N1_(1)and N_(2)satisfying N_(1)>N_(1)^((0)),N_(2)>N_(2)^((0))andα≤N_(2)/N_(1)^(d=c)≤β,the system of two Diophantine inequalities|p_^(1)+…+p_(6)^(c)-N_(1)|<N_(1)^(−(1=c)(14=13−c))logN_(1),|p_(1)^(d)+…+p_(6)^(d)|N_(2)^(−(1=d)(14=13−d))logN_(2)has solutions in prime variables p_(1)…,p6.展开更多
The existing concepts of picture fuzzy sets(PFS),spherical fuzzy sets(SFSs),T-spherical fuzzy sets(T-SFSs)and neutrosophic sets(NSs)have numerous applications in decision-making problems,but they have various strict l...The existing concepts of picture fuzzy sets(PFS),spherical fuzzy sets(SFSs),T-spherical fuzzy sets(T-SFSs)and neutrosophic sets(NSs)have numerous applications in decision-making problems,but they have various strict limitations for their satisfaction,dissatisfaction,abstain or refusal grades.To relax these strict constraints,we introduce the concept of spherical linearDiophantine fuzzy sets(SLDFSs)with the inclusion of reference or control parameters.A SLDFSwith parameterizations process is very helpful formodeling uncertainties in themulti-criteria decisionmaking(MCDM)process.SLDFSs can classify a physical systemwith the help of reference parameters.We discuss various real-life applications of SLDFSs towards digital image processing,network systems,vote casting,electrical engineering,medication,and selection of optimal choice.We show some drawbacks of operations of picture fuzzy sets and their corresponding aggregation operators.Some new operations on picture fuzzy sets are also introduced.Some fundamental operations on SLDFSs and different types of score functions of spherical linear Diophantine fuzzy numbers(SLDFNs)are proposed.New aggregation operators named spherical linear Diophantine fuzzy weighted geometric aggregation(SLDFWGA)and spherical linear Diophantine fuzzy weighted average aggregation(SLDFWAA)operators are developed for a robust MCDM approach.An application of the proposed methodology with SLDF information is illustrated.The comparison analysis of the final ranking is also given to demonstrate the validity,feasibility,and efficiency of the proposed MCDM approach.展开更多
设D=multiply from i=1 to s p_i(s≥2),p_i=1(mod 6)(1≤i≤s)为不同的奇素数.关于Diophantine方程x^3-1=Dy^2的初等解法至今仍未解决.主要利用同余式、平方剩余、Pell方程的解的性质、递归序列,证明了q≡7(mod 24)为奇素数.(q/13)=-1...设D=multiply from i=1 to s p_i(s≥2),p_i=1(mod 6)(1≤i≤s)为不同的奇素数.关于Diophantine方程x^3-1=Dy^2的初等解法至今仍未解决.主要利用同余式、平方剩余、Pell方程的解的性质、递归序列,证明了q≡7(mod 24)为奇素数.(q/13)=-1时,Diophantine方程x^3-1=13qy^2仅有整数解(x,y)=(1,0).展开更多
基金Supported by NSFC(Nos.12301006,12471009,12071238,11901566,12001047,11971476)Beijing Natural Science Foundation(No.1242003)。
文摘Suppose thatλ_(1),λ_(2),λ_(3),λ_(4),λ_(5)are nonzero real numbers,not all of the same sign,andλ_(1)/λ_(2)is irrational and algebraic.Let V be a well-spaced sequence,δ>0.In this paper,it is proved that,for anyε>0,the number of v∈V with v≤N such that the following inequality|λ_(1)p_(1)~2+λ_(2)p_(2)~2+λ_(3)p_(3)~4+λ_(4)p_(4)~4+λ_5p_5~4-v|<v^(-δ)has no solution in prime variables p_(1),p_(2),p_(3),p_(4),p_(5)does not exceed O(N^(29/32+2δ+ε)).
文摘In this paper,a robust and consistent COVID-19 emergency decision-making approach is proposed based on q-rung linear diophantine fuzzy set(q-RLDFS),differential evolutionary(DE)optimization principles,and evidential reasoning(ER)methodology.The proposed approach uses q-RLDFS in order to represent the evaluating values of the alternatives corresponding to the attributes.DE optimization is used to obtain the optimal weights of the attributes,and ER methodology is used to compute the aggregated q-rung linear diophantine fuzzy values(q-RLDFVs)of each alternative.Then the score values of alternatives are computed based on the aggregated q-RLDFVs.An alternative with the maximum score value is selected as a better one.The applicability of the proposed approach has been illustrated in COVID-19 emergency decision-making system and sustainable energy planning management.Moreover,we have validated the proposed approach with a numerical example.Finally,a comparative study is provided with the existing models,where the proposed approach is found to be robust to perform better and consistent in uncertain environments.
文摘Suppose that c,d,α,β,are real numbers satisfying 1<d<c<14=13,1<α<β<6^(1−d=c).In this paper,we prove that there exist numbers N_(1)^((0))and N_(2)^((0)),depending on c,d,α,β,such that for all real numbers N1_(1)and N_(2)satisfying N_(1)>N_(1)^((0)),N_(2)>N_(2)^((0))andα≤N_(2)/N_(1)^(d=c)≤β,the system of two Diophantine inequalities|p_^(1)+…+p_(6)^(c)-N_(1)|<N_(1)^(−(1=c)(14=13−c))logN_(1),|p_(1)^(d)+…+p_(6)^(d)|N_(2)^(−(1=d)(14=13−d))logN_(2)has solutions in prime variables p_(1)…,p6.
文摘The existing concepts of picture fuzzy sets(PFS),spherical fuzzy sets(SFSs),T-spherical fuzzy sets(T-SFSs)and neutrosophic sets(NSs)have numerous applications in decision-making problems,but they have various strict limitations for their satisfaction,dissatisfaction,abstain or refusal grades.To relax these strict constraints,we introduce the concept of spherical linearDiophantine fuzzy sets(SLDFSs)with the inclusion of reference or control parameters.A SLDFSwith parameterizations process is very helpful formodeling uncertainties in themulti-criteria decisionmaking(MCDM)process.SLDFSs can classify a physical systemwith the help of reference parameters.We discuss various real-life applications of SLDFSs towards digital image processing,network systems,vote casting,electrical engineering,medication,and selection of optimal choice.We show some drawbacks of operations of picture fuzzy sets and their corresponding aggregation operators.Some new operations on picture fuzzy sets are also introduced.Some fundamental operations on SLDFSs and different types of score functions of spherical linear Diophantine fuzzy numbers(SLDFNs)are proposed.New aggregation operators named spherical linear Diophantine fuzzy weighted geometric aggregation(SLDFWGA)and spherical linear Diophantine fuzzy weighted average aggregation(SLDFWAA)operators are developed for a robust MCDM approach.An application of the proposed methodology with SLDF information is illustrated.The comparison analysis of the final ranking is also given to demonstrate the validity,feasibility,and efficiency of the proposed MCDM approach.
基金Supported by the NSF of China(11126173)Anhui Province Natural Science Foundation(1208085QA02)+1 种基金the NSF of China(10901002)the NSF of Anhui Province Education Committee(KJ2011Z151)
文摘设D=multiply from i=1 to s p_i(s≥2),p_i=1(mod 6)(1≤i≤s)为不同的奇素数.关于Diophantine方程x^3-1=Dy^2的初等解法至今仍未解决.主要利用同余式、平方剩余、Pell方程的解的性质、递归序列,证明了q≡7(mod 24)为奇素数.(q/13)=-1时,Diophantine方程x^3-1=13qy^2仅有整数解(x,y)=(1,0).
