We determined the linear complexity of a family of p2-periodic binary threshold sequences and a family of p2-periodic binary sequences constructed using the Legendre symbol,both of which are derived from Fermat quotie...We determined the linear complexity of a family of p2-periodic binary threshold sequences and a family of p2-periodic binary sequences constructed using the Legendre symbol,both of which are derived from Fermat quotients modulo an odd prime p.If 2 is a primitive element modulo p2,the linear complexity equals to p2-p or p2-1,which is very close to the period and it is large enough for cryptographic purpose.展开更多
Let p be an odd prime and let a,m ∈ Z with a 】 0 and p ︱ m.In this paper we determinep ∑k=0 pa-1(2k k=d)/mk mod p2 for d=0,1;for example,where(-) is the Jacobi symbol and {un}n≥0 is the Lucas sequence given by u0...Let p be an odd prime and let a,m ∈ Z with a 】 0 and p ︱ m.In this paper we determinep ∑k=0 pa-1(2k k=d)/mk mod p2 for d=0,1;for example,where(-) is the Jacobi symbol and {un}n≥0 is the Lucas sequence given by u0 = 0,u1 = 1 and un+1 =(m-2)un-un-1(n = 1,2,3,...).As an application,we determine ∑0【k【pa,k≡r(mod p-1) Ck modulo p2 for any integer r,where Ck denotes the Catalan number 2kk /(k + 1).We also pose some related conjectures.展开更多
The unequal meshsteps are unavoidable in general for scientific and engineering computations especially in large Scale computations. The analysis of difference schemes with nonuniform meshes is very rare even by use o...The unequal meshsteps are unavoidable in general for scientific and engineering computations especially in large Scale computations. The analysis of difference schemes with nonuniform meshes is very rare even by use of fully heuristic methods. For the purpose of the systematic and theoretical study of the finite difference method with nonuniform meshes for the problems of partial differential equations, the general interpolation formulas for the spaces of discrete functions of one index with unequal meshsteps are established in the present work. These formulas give the connected relationships among the norms of various types, such as' the sum of powers of discrete values, the discrete maximum modulo, the discrete Holder and Lipschitz coefficients.展开更多
Diophantine equations have always fascinated mathematicians about existence, finitude, and the calculation of possible solutions. Among these equations, one of them will be the object of our research. This is the Pyth...Diophantine equations have always fascinated mathematicians about existence, finitude, and the calculation of possible solutions. Among these equations, one of them will be the object of our research. This is the Pythagoras’- Fermat’s equation defined as follows. (1) when , it is well known that this equation has an infinity of solutions but has none (non-trivial) when . We also know that the last result, named Fermat-Wiles theorem (or FLT) was obtained at great expense and its understanding remains out of reach even for a good fringe of professional mathematicians. The aim of this research is to set up new simple but effective tools in the treatment of Diophantine equations and that of Pythagoras-Fermat. The tools put forward in this research are the properties of the quotients and the Diophantine remainders which we define as follows. Let a non-trivial triplet () solution of Equation (1) such that . and are called the Diophantine quotients and remainders of solution . We compute the remainder and the quotient of b and c by a using the division algorithm. Hence, we have: and et with . We prove the following important results. if and only if and if and only if . Also, we deduce that or for any hypothetical solution . We illustrate these results by effectively computing the Diophantine quotients and remainders in the case of Pythagorean triplets using a Python program. In the end, we apply the previous properties to directly prove a partial result of FLT. .展开更多
We generalize the congruences of Friedmann-Tamarkine (1909), Lehmer (1938), and Ernvall-Metsänkyla (1991) on the sums of powers of integers weighted by powers of the Fermat quotients to the next Fermat quotient p...We generalize the congruences of Friedmann-Tamarkine (1909), Lehmer (1938), and Ernvall-Metsänkyla (1991) on the sums of powers of integers weighted by powers of the Fermat quotients to the next Fermat quotient power, namely to the third power of the Fermat quotient. Using this result and the Gessel identity (2005) combined with our past work (2021), we are able to relate residues of some truncated convolutions of Bernoulli numbers with some Ernvall-Metsänkyla residues to residues of some full convolutions of the same kind. We also establish some congruences concerning other related weighted sums of powers of integers when these sums are weighted by some analogs of the Teichmüller characters.展开更多
For each real number x∈(0,1),let[a_(1)(x),a_(2)(x),…,a_n(x),…]denote its continued fraction expansion.We study the convergence exponent defined byτ(x)=inf{s≥0:∞∑n=1(a_(n)(x)a_(n+1)(x))^(-s)<∞},which reflect...For each real number x∈(0,1),let[a_(1)(x),a_(2)(x),…,a_n(x),…]denote its continued fraction expansion.We study the convergence exponent defined byτ(x)=inf{s≥0:∞∑n=1(a_(n)(x)a_(n+1)(x))^(-s)<∞},which reflects the growth rate of the product of two consecutive partial quotients.As a main result,the Hausdorff dimensions of the level sets ofτ(x)are determined.展开更多
Recreational Ecosystem Services(RES)are among the most frequently evaluated ecosystem services.RES are seen as a major opportunity for sustainable development in areas of high ecological value resulting from the prese...Recreational Ecosystem Services(RES)are among the most frequently evaluated ecosystem services.RES are seen as a major opportunity for sustainable development in areas of high ecological value resulting from the presence of emblematic species,habitats or scenery,often integrated in conservation areas,in particular in mountain areas affected by depopulation,rising environmental risks and poverty.Economically,the assessment of RES has been based on Contingent Valuation and Choice Experiments,methods with limitations related to their subjectivity.Alternatively,Input-Output(IO)models provide a very accurate and effective way of valuation of RES with regional information regarding interindustry transactions.Whilst data on nature tourist expenditure may be only available annually,tourism inflows have a strong seasonal behavior,which impacts the profitability of several local economic activities.In this paper,we firstly constructed a novel regionalized IO-model based on Cross-Industry Location Quotients using employment data and,secondly,based on Monte Carlo simulation,we estimated nature tourist expenditure monthly using data on nature tourism inflows.This method allowed for a more precise RES evaluation,estimating direct,indirect and induced monthly economic impacts of nature-based tourism.The method was applied to the Ordesa and Monte Perdido(OMP)National Park in the Spanish Pyrenees.To estimate the average tourist expenditure,we collected and analyzed spending on commerce,hospitality and restaurants,leisure,and transport of 385 visitors.Results suggest that using a regionalized IO model leads to a considerable reduction of over-estimation of the economic impact of tourist expenditure estimated by conventional methods.Taking into consideration the direct,indirect and induced effects of tourist expenditure in 2016-2018,the total annual output effect in the OMP National Park amounted to€208.8 million,while the total income effect amounted to€86.29 million.Also,a total of 2,429 jobs were created which indicates that展开更多
Soil microbes play critical roles in soil biogeochemistry, soil biological health and crop productivity. The current study evaluated the effects of tillage and residue management on changes in soil biochemical indicat...Soil microbes play critical roles in soil biogeochemistry, soil biological health and crop productivity. The current study evaluated the effects of tillage and residue management on changes in soil biochemical indicators at different growth stages of wheat after 5 years of rice-wheat system. Nine treatment combinations of tillage, crop establishment and crop residue management included three main plot treatments applied to rice:(1) conventional till direct dry seeded rice(CTDSR),(2) zero till direct dry seeded rice(ZTDSR), and(3) conventional puddled manual transplanted rice(CTPTR) and three subplot treatments in subsequent wheat:(1) conventional tillage with rice residue removed(CTW-R),(2) zero tillage with rice residue removed(ZTW-R) and(3) zero tillage with rice residue retained as surface mulch(ZTW+R). Irrespective of rice treatments, ZTW+R treatment had higher soil biochemical indicators compared with ZTW-R and CTW-R at all the growth stages of wheat. Generally, all the biochemical indicators were the highest at the flowering stage of wheat. Residual effect of rice treatments was also significant on biochemical quotients in wheat, which were the highest under ZTDSR followed by CTDSR and CTPTR. The present study provided three sensitive and reliable biochemical indicators(microbial biomass, basal soil respiration and microbial quotient) which respond rapidly to change in tillage and residue management practices in RWS of South Asia.展开更多
In this paper we develop a theory of localization for bounded commutative BCK-algebras. We try to extend some results from the case of commutative Hilbert algebras (see [1]) to the case of commutative BCK-alge- bras.
The analogy between eigenvalues and singular values has many faces. The current review brings together several examples of this analogy. One example regards the similarity between Symmetric Rayleigh Quotients and Rect...The analogy between eigenvalues and singular values has many faces. The current review brings together several examples of this analogy. One example regards the similarity between Symmetric Rayleigh Quotients and Rectangular Rayleigh Quotients. Many useful properties of eigenvalues stem are from the Courant-Fischer minimax theorem, from Weyl’s theorem, and their corollaries. Another aspect regards “rectangular” versions of these theorems. Comparing the properties of Rayleigh Quotient matrices with those of Orthogonal Quotient matrices illuminates the subject in a new light. The Orthogonal Quotients Equality is a recent result that converts Eckart-Young’s minimum norm problem into an equivalent maximum norm problem. This exposes a surprising link between the Eckart-Young theorem and Ky Fan’s maximum principle. We see that the two theorems reflect two sides of the same coin: there exists a more general maximum principle from which both theorems are easily derived. Ky Fan has used his extremum principle (on traces of matrices) to derive analog results on determinants of positive definite Rayleigh Quotients matrices. The new extremum principle extends these results to Rectangular Quotients matrices. Bringing all these topics under one roof provides new insight into the fascinating relations between eigenvalues and singular values.展开更多
In this paper, a necessary and sufficient condition is given for a commutative Artinian local ring whose annihilating-ideal graph is a star graph. Also, a complete char- acterization is established for a finite local ...In this paper, a necessary and sufficient condition is given for a commutative Artinian local ring whose annihilating-ideal graph is a star graph. Also, a complete char- acterization is established for a finite local ring whose annihilating-ideal graph is a star graph.展开更多
In this paper the existence of discrete vector solutions with bounded second order quotientsfor the difference systems of nonlinear parabolic system is established by the fixed point technique,and then the absolute an...In this paper the existence of discrete vector solutions with bounded second order quotientsfor the difference systems of nonlinear parabolic system is established by the fixed point technique,and then the absolute and relative stability for the general difference schemes is justified in thesense of continuous dependence of the discrete vector solution of the difference schemes on thediscrete data of the original problems.展开更多
It is already known that finitely-generated Fuchsian groups are profinitely rigid among all lattices of connected Lie groups by the result of Bridson,Conder and Reid.Hence the triangle groups are distinguished among t...It is already known that finitely-generated Fuchsian groups are profinitely rigid among all lattices of connected Lie groups by the result of Bridson,Conder and Reid.Hence the triangle groups are distinguished among themselves by their finite quotients.We focus on the question about quantifying the size of a quotient which separates two triangle groups and give an explicit upper bound.展开更多
Let G be a finite group,and let V be a completely reducible faithful finite G-module(i.e.,G≤GL(V),where V is a finite vector space which is a direct sum of irreducible G-submodules).It has been known for a long time ...Let G be a finite group,and let V be a completely reducible faithful finite G-module(i.e.,G≤GL(V),where V is a finite vector space which is a direct sum of irreducible G-submodules).It has been known for a long time that if G is abelian,then G has a regular orbit on V.In this paper we show that G has an orbit of size at least|G/G′|on V.This generalizes earlier work of the authors,where the same bound was proved under the additional hypothesis that G is solvable.For completely reducible modules it also strengthens the 1989 result|G/G′|<|V|by Aschbacher and Guralnick.展开更多
LetG = SLn(C)Cn be the (special) affine group. In this paper we study the representation theory of G and in particular the question of rationality for V/G, where V is a generically free G-representation. We show that ...LetG = SLn(C)Cn be the (special) affine group. In this paper we study the representation theory of G and in particular the question of rationality for V/G, where V is a generically free G-representation. We show that the answer to this question is positive (Theorem 6.1) if the dimension of V is sufficiently large and V is indecomposable. We explicitly characterize two-step extensions 0 → S → V → Q → 0, with completely reducible S and Q, whose rationality cannot be obtained by the methods presented here (Theorem 5.3).展开更多
ymmetry of the world trade network provides a novel perspective to understand the world-wide trading system. However, symmetry in the world trade network (WTN) has been rarely studied so far. In this paper, the auth...ymmetry of the world trade network provides a novel perspective to understand the world-wide trading system. However, symmetry in the world trade network (WTN) has been rarely studied so far. In this paper, the authors systematically explore the symmetry in WTN. The authors construct WTN in 2005 and explore the size and structure of its automorphism group, through which the authors find that WTN is symmetric, particularly, locally symmetric to a certain degree. Furthermore, the authors work out the symmetric motifs of WTN and investigate the structure and function of the symmetric motifs, coming to the conclusion that local symmetry will have great effect on the stability of the WTN and that continuous symmetry-breakings will generate complexity and diversity of the trade network. Finally, utilizing the local symmetry of the network, the authors work out the quotient of WTN, which is the structural skeleton dominating stability and evolution of WTN.展开更多
Recently, Williams [1] and then Yao, Xia and Jin [2] discovered explicit formulas for the coefficients of the Fourier series expansions of a class of eta quotients. Williams expressed all coefficients of 126 eta quoti...Recently, Williams [1] and then Yao, Xia and Jin [2] discovered explicit formulas for the coefficients of the Fourier series expansions of a class of eta quotients. Williams expressed all coefficients of 126 eta quotients in terms of and and Yao, Xia and Jin, following the method of proof of Williams, expressed only even coefficients of 104 eta quotients in terms of and . Here, by using the method of proof of Williams, we will express the even Fourier coefficients of 360 eta quotients i.e., the Fourier coefficients of the sum, f(q) + f(?q), of 360 eta quotients in terms of and .展开更多
基金the National Natural Science Foundation of China,the Open Funds of State Key Laboratory of Information Security (Chinese Academy of Sciences),the Program for New Century Excellent Talents in Fujian Province University
文摘We determined the linear complexity of a family of p2-periodic binary threshold sequences and a family of p2-periodic binary sequences constructed using the Legendre symbol,both of which are derived from Fermat quotients modulo an odd prime p.If 2 is a primitive element modulo p2,the linear complexity equals to p2-p or p2-1,which is very close to the period and it is large enough for cryptographic purpose.
基金supported by National Natural Science Foundation of China (Grant No.10871087)the Overseas Cooperation Fund of China (Grant No.10928101)
文摘Let p be an odd prime and let a,m ∈ Z with a 】 0 and p ︱ m.In this paper we determinep ∑k=0 pa-1(2k k=d)/mk mod p2 for d=0,1;for example,where(-) is the Jacobi symbol and {un}n≥0 is the Lucas sequence given by u0 = 0,u1 = 1 and un+1 =(m-2)un-un-1(n = 1,2,3,...).As an application,we determine ∑0【k【pa,k≡r(mod p-1) Ck modulo p2 for any integer r,where Ck denotes the Catalan number 2kk /(k + 1).We also pose some related conjectures.
文摘The unequal meshsteps are unavoidable in general for scientific and engineering computations especially in large Scale computations. The analysis of difference schemes with nonuniform meshes is very rare even by use of fully heuristic methods. For the purpose of the systematic and theoretical study of the finite difference method with nonuniform meshes for the problems of partial differential equations, the general interpolation formulas for the spaces of discrete functions of one index with unequal meshsteps are established in the present work. These formulas give the connected relationships among the norms of various types, such as' the sum of powers of discrete values, the discrete maximum modulo, the discrete Holder and Lipschitz coefficients.
文摘Diophantine equations have always fascinated mathematicians about existence, finitude, and the calculation of possible solutions. Among these equations, one of them will be the object of our research. This is the Pythagoras’- Fermat’s equation defined as follows. (1) when , it is well known that this equation has an infinity of solutions but has none (non-trivial) when . We also know that the last result, named Fermat-Wiles theorem (or FLT) was obtained at great expense and its understanding remains out of reach even for a good fringe of professional mathematicians. The aim of this research is to set up new simple but effective tools in the treatment of Diophantine equations and that of Pythagoras-Fermat. The tools put forward in this research are the properties of the quotients and the Diophantine remainders which we define as follows. Let a non-trivial triplet () solution of Equation (1) such that . and are called the Diophantine quotients and remainders of solution . We compute the remainder and the quotient of b and c by a using the division algorithm. Hence, we have: and et with . We prove the following important results. if and only if and if and only if . Also, we deduce that or for any hypothetical solution . We illustrate these results by effectively computing the Diophantine quotients and remainders in the case of Pythagorean triplets using a Python program. In the end, we apply the previous properties to directly prove a partial result of FLT. .
文摘We generalize the congruences of Friedmann-Tamarkine (1909), Lehmer (1938), and Ernvall-Metsänkyla (1991) on the sums of powers of integers weighted by powers of the Fermat quotients to the next Fermat quotient power, namely to the third power of the Fermat quotient. Using this result and the Gessel identity (2005) combined with our past work (2021), we are able to relate residues of some truncated convolutions of Bernoulli numbers with some Ernvall-Metsänkyla residues to residues of some full convolutions of the same kind. We also establish some congruences concerning other related weighted sums of powers of integers when these sums are weighted by some analogs of the Teichmüller characters.
基金supported by the Scientific Research Fund of Hunan Provincial Education Department(21B0070)the Natural Science Foundation of Jiangsu Province(BK20231452)+1 种基金the Fundamental Research Funds for the Central Universities(30922010809)the National Natural Science Foundation of China(11801591,11971195,12071171,12171107,12201207,12371072)。
文摘For each real number x∈(0,1),let[a_(1)(x),a_(2)(x),…,a_n(x),…]denote its continued fraction expansion.We study the convergence exponent defined byτ(x)=inf{s≥0:∞∑n=1(a_(n)(x)a_(n+1)(x))^(-s)<∞},which reflects the growth rate of the product of two consecutive partial quotients.As a main result,the Hausdorff dimensions of the level sets ofτ(x)are determined.
基金financial support from the Organismo Autónomo de Parques Nacionales(Project V09-Cátedra de Parques Nacionales)。
文摘Recreational Ecosystem Services(RES)are among the most frequently evaluated ecosystem services.RES are seen as a major opportunity for sustainable development in areas of high ecological value resulting from the presence of emblematic species,habitats or scenery,often integrated in conservation areas,in particular in mountain areas affected by depopulation,rising environmental risks and poverty.Economically,the assessment of RES has been based on Contingent Valuation and Choice Experiments,methods with limitations related to their subjectivity.Alternatively,Input-Output(IO)models provide a very accurate and effective way of valuation of RES with regional information regarding interindustry transactions.Whilst data on nature tourist expenditure may be only available annually,tourism inflows have a strong seasonal behavior,which impacts the profitability of several local economic activities.In this paper,we firstly constructed a novel regionalized IO-model based on Cross-Industry Location Quotients using employment data and,secondly,based on Monte Carlo simulation,we estimated nature tourist expenditure monthly using data on nature tourism inflows.This method allowed for a more precise RES evaluation,estimating direct,indirect and induced monthly economic impacts of nature-based tourism.The method was applied to the Ordesa and Monte Perdido(OMP)National Park in the Spanish Pyrenees.To estimate the average tourist expenditure,we collected and analyzed spending on commerce,hospitality and restaurants,leisure,and transport of 385 visitors.Results suggest that using a regionalized IO model leads to a considerable reduction of over-estimation of the economic impact of tourist expenditure estimated by conventional methods.Taking into consideration the direct,indirect and induced effects of tourist expenditure in 2016-2018,the total annual output effect in the OMP National Park amounted to€208.8 million,while the total income effect amounted to€86.29 million.Also,a total of 2,429 jobs were created which indicates that
基金financial as well technical support of International Maize and Wheat Improvement Center (CIMMYT) through USAID and BMGF funded Cereal System Initiative for South Asia (CSISA) ProjectConsultative Group of International Agriculture Research (CGIAR) Program on wheat (CRP 3.1)
文摘Soil microbes play critical roles in soil biogeochemistry, soil biological health and crop productivity. The current study evaluated the effects of tillage and residue management on changes in soil biochemical indicators at different growth stages of wheat after 5 years of rice-wheat system. Nine treatment combinations of tillage, crop establishment and crop residue management included three main plot treatments applied to rice:(1) conventional till direct dry seeded rice(CTDSR),(2) zero till direct dry seeded rice(ZTDSR), and(3) conventional puddled manual transplanted rice(CTPTR) and three subplot treatments in subsequent wheat:(1) conventional tillage with rice residue removed(CTW-R),(2) zero tillage with rice residue removed(ZTW-R) and(3) zero tillage with rice residue retained as surface mulch(ZTW+R). Irrespective of rice treatments, ZTW+R treatment had higher soil biochemical indicators compared with ZTW-R and CTW-R at all the growth stages of wheat. Generally, all the biochemical indicators were the highest at the flowering stage of wheat. Residual effect of rice treatments was also significant on biochemical quotients in wheat, which were the highest under ZTDSR followed by CTDSR and CTPTR. The present study provided three sensitive and reliable biochemical indicators(microbial biomass, basal soil respiration and microbial quotient) which respond rapidly to change in tillage and residue management practices in RWS of South Asia.
文摘In this paper we develop a theory of localization for bounded commutative BCK-algebras. We try to extend some results from the case of commutative Hilbert algebras (see [1]) to the case of commutative BCK-alge- bras.
文摘The analogy between eigenvalues and singular values has many faces. The current review brings together several examples of this analogy. One example regards the similarity between Symmetric Rayleigh Quotients and Rectangular Rayleigh Quotients. Many useful properties of eigenvalues stem are from the Courant-Fischer minimax theorem, from Weyl’s theorem, and their corollaries. Another aspect regards “rectangular” versions of these theorems. Comparing the properties of Rayleigh Quotient matrices with those of Orthogonal Quotient matrices illuminates the subject in a new light. The Orthogonal Quotients Equality is a recent result that converts Eckart-Young’s minimum norm problem into an equivalent maximum norm problem. This exposes a surprising link between the Eckart-Young theorem and Ky Fan’s maximum principle. We see that the two theorems reflect two sides of the same coin: there exists a more general maximum principle from which both theorems are easily derived. Ky Fan has used his extremum principle (on traces of matrices) to derive analog results on determinants of positive definite Rayleigh Quotients matrices. The new extremum principle extends these results to Rectangular Quotients matrices. Bringing all these topics under one roof provides new insight into the fascinating relations between eigenvalues and singular values.
基金The first author is supported by Fundamental Research Funds for the Central Universi- ties (No. XDJK2013C060), Chongqing Research Program of Application Foundation and Advanced Technology (No. cstc2014jcyjA00028) and Scientific Research Foundation for Doctors of Southwest University (No. SWUl12054). The second author is supported by National Natural Science Foundation of China (No. 11271250).
文摘In this paper, a necessary and sufficient condition is given for a commutative Artinian local ring whose annihilating-ideal graph is a star graph. Also, a complete char- acterization is established for a finite local ring whose annihilating-ideal graph is a star graph.
文摘In this paper the existence of discrete vector solutions with bounded second order quotientsfor the difference systems of nonlinear parabolic system is established by the fixed point technique,and then the absolute and relative stability for the general difference schemes is justified in thesense of continuous dependence of the discrete vector solution of the difference schemes on thediscrete data of the original problems.
文摘It is already known that finitely-generated Fuchsian groups are profinitely rigid among all lattices of connected Lie groups by the result of Bridson,Conder and Reid.Hence the triangle groups are distinguished among themselves by their finite quotients.We focus on the question about quantifying the size of a quotient which separates two triangle groups and give an explicit upper bound.
基金supported by National Natural Science Foundation of China(Grant No.11671063)a grant from the Simons Foundation(Grant No.280770 to Thomas M.Keller)a grant from the Simons Foundation(Grant No.499532 to Yong Yang)。
文摘Let G be a finite group,and let V be a completely reducible faithful finite G-module(i.e.,G≤GL(V),where V is a finite vector space which is a direct sum of irreducible G-submodules).It has been known for a long time that if G is abelian,then G has a regular orbit on V.In this paper we show that G has an orbit of size at least|G/G′|on V.This generalizes earlier work of the authors,where the same bound was proved under the additional hypothesis that G is solvable.For completely reducible modules it also strengthens the 1989 result|G/G′|<|V|by Aschbacher and Guralnick.
基金supported by the Natural Science Foundation of USA (Grant No. DMS 0701578)supported by the German Research Foundation (Deutsche Forschungsgemeinschaft (DFG)) through the Institutional Strategy of the University of Gttingen
文摘LetG = SLn(C)Cn be the (special) affine group. In this paper we study the representation theory of G and in particular the question of rationality for V/G, where V is a generically free G-representation. We show that the answer to this question is positive (Theorem 6.1) if the dimension of V is sufficiently large and V is indecomposable. We explicitly characterize two-step extensions 0 → S → V → Q → 0, with completely reducible S and Q, whose rationality cannot be obtained by the methods presented here (Theorem 5.3).
基金supported by the National Natural Science Foundation of China under Grant No. 70371070 Shanghai Leading Academic Discipline Project under Grant No. S30504Key Project for Fundamental Research of STCSM under Grant No. 06JC14057
文摘ymmetry of the world trade network provides a novel perspective to understand the world-wide trading system. However, symmetry in the world trade network (WTN) has been rarely studied so far. In this paper, the authors systematically explore the symmetry in WTN. The authors construct WTN in 2005 and explore the size and structure of its automorphism group, through which the authors find that WTN is symmetric, particularly, locally symmetric to a certain degree. Furthermore, the authors work out the symmetric motifs of WTN and investigate the structure and function of the symmetric motifs, coming to the conclusion that local symmetry will have great effect on the stability of the WTN and that continuous symmetry-breakings will generate complexity and diversity of the trade network. Finally, utilizing the local symmetry of the network, the authors work out the quotient of WTN, which is the structural skeleton dominating stability and evolution of WTN.
文摘Recently, Williams [1] and then Yao, Xia and Jin [2] discovered explicit formulas for the coefficients of the Fourier series expansions of a class of eta quotients. Williams expressed all coefficients of 126 eta quotients in terms of and and Yao, Xia and Jin, following the method of proof of Williams, expressed only even coefficients of 104 eta quotients in terms of and . Here, by using the method of proof of Williams, we will express the even Fourier coefficients of 360 eta quotients i.e., the Fourier coefficients of the sum, f(q) + f(?q), of 360 eta quotients in terms of and .