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Linear Complexity of Some Binary Sequences Derived from Fermat Quotients 被引量:8
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作者 Chen Zhixiong Hu Lei, Du Xiaoni 《China Communications》 SCIE CSCD 2012年第2期105-108,共4页
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
关键词 CRYPTOGRAPHY pseudorandom binary sequences fermat quotients finite fields linear complexity
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Binomial coefficients,Catalan numbers and Lucas quotients 被引量:5
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作者 SUN ZhiWei Department of Mathematics,Nanjing University,Nanjing 210093,China 《Science China Mathematics》 SCIE 2010年第9期2473-2488,共16页
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
关键词 CONGRUENCES BINOMIAL COEFFICIENTS CATALAN NUMBERS Lucas quotients
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GENERAL INTERPOLATION FORMULAS FOR SPACES OF DISCRETE FUNCTIONS WITH NONUNIFORM MESHES 被引量:5
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作者 Zhou Yu-lin(Institute of Applied Physics and Computational Mathematics, Beijing, China ) 《Journal of Computational Mathematics》 SCIE CSCD 1995年第1期70-93,共24页
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
关键词 Discrete functions INTERPOLATION Mathematical constants Partial differential equations Difference quotients Real lines Finite difference methods Mathematical theorems Ratios Engineering
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Diophantine Quotients and Remainders with Applications to Fermat and Pythagorean Equations
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作者 Prosper Kouadio Kimou François Emmanuel Tanoé 《American Journal of Computational Mathematics》 2023年第1期199-210,共12页
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
关键词 Diophantine Equation Modular Arithmetic Fermat-Wiles Theorem Pythagorean Triplets Division Theorem Division Algorithm Python Program Diophantine quotients Diophantine Remainders
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Some Implications of the Gessel Identity
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作者 Claire Levaillant 《Applied Mathematics》 2023年第9期545-579,共35页
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
关键词 Convolutions Involving Bernoulli Numbers Truncated Convolutions Involving Bernoulli Numbers CONGRUENCES Binomial and Multinomial Convolutions of Divided Bernoulli Numbers Multiple Harmonic Sums Generalized Harmonic Numbers Miki Identity Gessel Identity Sums of Powers of Integers Weighted by Powers of the Fermat quotients Generalization of Kummer’s Congruences Generalizations of Friedmann-Tamarkine Lehmer Ernvall-Metsänkyla’s Congruences p-Adic Numbers Weighted Sums of Powers of Integers
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MULTIFRACTAL ANALYSIS OF CONVERGENCE EXPONENTS FOR PRODUCTS OF CONSECUTIVE PARTIAL QUOTIENTS IN CONTINUED FRACTIONS
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作者 Lulu FANG Jihua MA +1 位作者 Kunkun SONG Xin YANG 《Acta Mathematica Scientia》 SCIE CSCD 2024年第4期1594-1608,共15页
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
关键词 continued fractions product of partial quotients Hausdorff dimension
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模糊结合代数的商及同态定理
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作者 杨晓曼 周鑫 《长春师范大学学报》 2024年第4期10-13,27,共5页
由模糊结合代数的定义出发,利用模糊理想的概念,描述了模糊结合代数的商结构,进而结合模糊同态等概念,给出了相应的模糊同态定理,为讨论模糊结合代数的分类问题进行了有益的探索.
关键词 模糊结合代数 模糊同态 模糊理想 同态定理
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完备布尔代数商格的注(英文) 被引量:3
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作者 王习娟 贺伟 《数学杂志》 CSCD 北大核心 2009年第2期133-138,共6页
本文研究了完备布尔代数L的商格Q(L),建立L上q-集构成的集合Q′(L)与Q(L)之间的同构,得出了完备布尔代数的商格仍是完备布尔代数,并给出了完备布尔代数范畴的某些性质.
关键词 完备布尔代数 核算子
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A regionalized IO-model to value seasonal recreational ecosystem services in a mountain National Park in Spain 被引量:2
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作者 Víctor MARTIN BARROSO Mónica de CASTRO-PARDO +1 位作者 Pascual FERNÁNDEZ MARTÍNEZ João C.AZEVEDO 《Journal of Mountain Science》 SCIE CSCD 2022年第1期211-227,共17页
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
关键词 INPUT-OUTPUT Monte Carlo Simulation Location quotients Recreational Ecosystem Services Nature tourism Sustainable mountain development
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Changes in soil biochemical indicators at different wheat growth stages under conservation-based sustainable intensification of rice-wheat system
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作者 Tanushree Bera Sandeep Sharma +3 位作者 H.S.Thind Yadvinder-Singh H.S.Sidhu M.L.Jat 《Journal of Integrative Agriculture》 SCIE CAS CSCD 2018年第8期1871-1880,共10页
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
关键词 microbial metabolic quotients tillage practices residue retention cropping system
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The Localization of Commutative Bounded BCK-Algebras
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作者 Dana Piciu Dan Dorin Tascau 《Advances in Pure Mathematics》 2011年第6期367-377,共11页
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.
关键词 BCK-ALGEBRA COMMUTATIVE BCK-ALGEBRA ALGEBRA of FRACTIONS Maximal ALGEBRA of quotients ∨-Closed System Topology LOCALIZATION ALGEBRA
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From Eigenvalues to Singular Values: A Review
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作者 Achiya Dax 《Advances in Pure Mathematics》 2013年第9期8-24,共17页
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
关键词 EIGENVALUES Singular Values RAYLEIGH quotient ORTHOGONAL quotient Matrices The ORTHOGONAL quotients EQUALITY Eckart-Young Theorem Ky Fan’s Extremum Principles
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On Standard Invariants of Bi-polymatroidal Ideals
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作者 Monica La Barbiera 《Algebra Colloquium》 SCIE CSCD 2014年第2期267-274,共8页
The standard invariants of the bi-polymatroidal ideals of Veronese type are computed.
关键词 bi-polymatroidal ideals linear quotients
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Artinian Local Rings Whose Annihilating-ideal Graphs Are Star Graphs
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作者 Houyi Yu Tongsuo Wu Weiping Gu 《Algebra Colloquium》 SCIE CSCD 2015年第1期73-82,共10页
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
关键词 Artinian rings local rings quotients of polynomial rings annihilating-ideals star graphs
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STABILITY OF DIFFERENCE SCHEMES FOR NONLINEAR PARABOLIC SYSTEMS WITH THREE DIMENSIONS
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作者 袁光伟 周毓麟 《Acta Mathematicae Applicatae Sinica》 SCIE CSCD 1998年第3期284-299,共16页
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
关键词 Difference scheme bounded quotients of second order stability nonlinear parabolic systems with two and three space dimensions
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Efficiency of Profinite Rigidity of Triangle Groups
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作者 Ji Ming MA Zi Xi WANG 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2022年第10期1939-1948,共10页
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
关键词 Triangle groups profinite completion finite quotients
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Abelian quotients and orbit sizes of linear groups
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作者 Thomas Michael Keller Yong Yang 《Science China Mathematics》 SCIE CSCD 2020年第8期1523-1534,共12页
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
关键词 abelian quotients orbits of group actions linear groups
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Rationality of quotients by linear actions of affine groups
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作者 BOGOMOLOV Fedor BHNING Christian GRAF VON BOTHMER Hans-Christian 《Science China Mathematics》 SCIE 2011年第8期1521-1532,共12页
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
关键词 RATIONALITY linear group quotients affine groups
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SYMMETRY IN WORLD TRADE NETWORK
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作者 Hui WANG Guangle YAN Yanghua XIAO 《Journal of Systems Science & Complexity》 SCIE EI CSCD 2009年第2期280-290,共11页
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
关键词 Automorphism group network quotients SYMMETRY world trade network.
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Fourier Coefficients of a Class of Eta Quotients of Weight 16 with Level 12
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作者 Baris Kendirli 《Applied Mathematics》 2015年第8期1426-1493,共68页
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
关键词 Dedekind Eta Function Eta quotients Fourier Series
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