The authors exhibit some new families of cyclotomic fields which have non-trivial plus parts of their class numbers.They also prove the 3-divisibility of the plus part of the class number of another family consisting ...The authors exhibit some new families of cyclotomic fields which have non-trivial plus parts of their class numbers.They also prove the 3-divisibility of the plus part of the class number of another family consisting of infinitely many cyclotomic fields.At the end,they provide some numerical examples supporting our results.展开更多
A parametrization of quadratic function fields whose divisor class numbers are divisible by 3 is obtained by using free parameters when the characteristics of the fields are not 3.
Let K = $ k(\sqrt \theta ) $ be a real cyclic quartic field, k be its quadratic subfield and $ \tilde K = k(\sqrt { - \theta } ) $ be the corresponding imaginary quartic field. Denote the class numbers of K, k and $ \...Let K = $ k(\sqrt \theta ) $ be a real cyclic quartic field, k be its quadratic subfield and $ \tilde K = k(\sqrt { - \theta } ) $ be the corresponding imaginary quartic field. Denote the class numbers of K, k and $ \tilde K $ by h K , h k and {417-3} respectively. Here congruences modulo powers of 2 for h ? = h K /h K and $ \tilde h^ - = h_{\tilde K} /h_k $ are obtained via studying the p-adic L-functions of the fields.展开更多
Let q 5 be a prime number. Let k d=() be a quadratic number field, where d =--gqq(1)2(1) ---qqqquwuq12((1)+). Then the class number of k is divisible by q for certain integers u,w. Conversely, assume W / k is an unra...Let q 5 be a prime number. Let k d=() be a quadratic number field, where d =--gqq(1)2(1) ---qqqquwuq12((1)+). Then the class number of k is divisible by q for certain integers u,w. Conversely, assume W / k is an unramified cyclic extension of degree q (which implies the class number of k is divisible by q), and W is the splitting field of some irreducible trinomial f(X) = XqaXb with integer coefficients, k Df=(())with D(f) the discriminant of f(X). Then f(X) must be of the form f(X) = Xquq2wXuq1 in a cer-tain sense where u,w are certain integers. Therefore, k d=() with d =-----qqqqqquwuq(1)122(1)((1)+). Moreover, the above two results are both generalized for certain kinds of general polynomials.展开更多
Let K 6 be a real cyclic sextic number fields,and K 2,K 3 be its quadratic and cubic subfields.Let h(L)denote the ideal class number of field L.Seven congruences for h-=h(K 6)/h(K 2)h(K 3)are obtained.In particular,wh...Let K 6 be a real cyclic sextic number fields,and K 2,K 3 be its quadratic and cubic subfields.Let h(L)denote the ideal class number of field L.Seven congruences for h-=h(K 6)/h(K 2)h(K 3)are obtained.In particular,when conductor f\-6 of K 6 is prime p,then Ch-≡B p-16B 5(p-1)6(mod p),where C is an explicitly given constant,and B n is the Bernoulli number.These results for real cyclic sextic fields are an extension of results for quadratic and cyclic quartic fields obtained by Ankeny_Artin_Chowla,Kiselev,Carlitz,Lu Hongwen,Zhang Xianke from 1948 to 1988.展开更多
In this paper, the theory of continued fractions of algebraic functions will be used to give a general theorem on lower bounds for class numbers of real quadratic function fields K=k(D). The bounds are given more expl...In this paper, the theory of continued fractions of algebraic functions will be used to give a general theorem on lower bounds for class numbers of real quadratic function fields K=k(D). The bounds are given more explicitly for six types of real quadratic function fields. As a consequence, six classes of real quadratic function fields with ideal class number greater than one are given.[展开更多
The theory of continued fractions of functions is used to give a lower bound for class numbers h(D) of general real quadratic function fields over k = F q (T). For five series of real quadratic function fields K, the...The theory of continued fractions of functions is used to give a lower bound for class numbers h(D) of general real quadratic function fields over k = F q (T). For five series of real quadratic function fields K, the bounds of h(D) are given more explicitly, e. g., if D = F 2 + c, then h(D) ≥ degF/degP; if D = (SG)2 + cS, then h(D) ≥ degS/degP; if D = (A m + a)2 + A, then h(D) ≥ degA/degP, where P is an irreducible polynomial splitting in K, c ∈ F q . In addition, three types of quadratic function fields K are found to have ideal class numbers bigger than one.展开更多
In this paper, the author applies adjacent lattice method and Siegel mass formula to determine the classes of positive definite unimodular lattices of rank 4 over Z , and obtains that the class number of unit genus ...In this paper, the author applies adjacent lattice method and Siegel mass formula to determine the classes of positive definite unimodular lattices of rank 4 over Z , and obtains that the class number of unit genus gen( I 4 ) is nine and the class number of even unimodular lattices is three, and also gives the representative lattices of each class.展开更多
In this paper, the author applies adjacent lattice method and Siegel mass formula to determine the classes of positive definite unimodular lattices of rank 4 over Z , and obtains that the class number of unit genus ...In this paper, the author applies adjacent lattice method and Siegel mass formula to determine the classes of positive definite unimodular lattices of rank 4 over Z , and obtains that the class number of unit genus gen( I 4 ) is nine and the class number of even unimodular lattices is three, and also gives the representative lattices of each class.展开更多
We investigate the asymptotic behavior of the entropy numbers of Besov classes BBΩp,θ（Sd 1）of generalized smoothness on the sphere inL q（Sd 1）for 1≤p,q,θ≤∞,and get their asymptotic orders.We also obtain the ...We investigate the asymptotic behavior of the entropy numbers of Besov classes BBΩp,θ（Sd 1）of generalized smoothness on the sphere inL q（Sd 1）for 1≤p,q,θ≤∞,and get their asymptotic orders.We also obtain the exact orders of entropy numbers of Sobolev classesBWr p（Sd 1）inL q（Sd 1）whenpand/orqis equal to 1 or∞.This provides the last piece as far as exact orders of entropy numbers ofBWr p（Sd 1）inL q（Sd 1）are concerned.展开更多
A necessary condition is presented for the ideal class group of an imaginary quadratic function field K=k(D)(k=F-q(x),2q)to be of exponent≤2.The condition is proved to be sufficient in some cases.An analogue of Loubo...A necessary condition is presented for the ideal class group of an imaginary quadratic function field K=k(D)(k=F-q(x),2q)to be of exponent≤2.The condition is proved to be sufficient in some cases.An analogue of Louboutin’s result in function field case is particularly presented.展开更多
We investigate arithmetic properties of certain subsets of square-free positive integers and obtain in this way some results concerning the class number h（d） of the real quadratic field Q（√d）. In particular, we g...We investigate arithmetic properties of certain subsets of square-free positive integers and obtain in this way some results concerning the class number h（d） of the real quadratic field Q（√d）. In particular, we give a new proof of the result of Hasse, asserting that in this case h（d） = 1 is possible only if d is of the form p, 2q or qr. where p.q. r are primes and q≡r≡3（mod 4）.展开更多
Ideal class groups H(K) of algebraic quadratic function fields K are studied. Necessaryand sufficient condition is given for the class group H(K) to contain a cyclic subgroup of anyorder n, which holds true for both r...Ideal class groups H(K) of algebraic quadratic function fields K are studied. Necessaryand sufficient condition is given for the class group H(K) to contain a cyclic subgroup of anyorder n, which holds true for both real and imaginary fields K. Then several series of functionfields K, including real, inertia imaginary, and ramified imaginary quadratic function fields, aregiven, for which the class groups H(K) are proved to contain cyclic subgroups of order n.展开更多
Let k=Q((D 2+md)(D 2+nd)(D 2+rd)), this paper proves firstly that the fundamental unit of k is ε=((D 2+md)(D 2+nd)+D 2(D 2+rd)) 2/(|mn|d 2), where D,d,m,n, and r are rational integers satisfying certain cond...Let k=Q((D 2+md)(D 2+nd)(D 2+rd)), this paper proves firstly that the fundamental unit of k is ε=((D 2+md)(D 2+nd)+D 2(D 2+rd)) 2/(|mn|d 2), where D,d,m,n, and r are rational integers satisfying certain conditions. Consequently, we describe the fundamental unit system of K=Q(D 2+md,D 2+nd,D 2+rd) explicitly by the fundamental unit of all the quadratic subfields and the class number h K explicitly by the class numbers of all the quadratic subfields. We also provide the fundamental unit system of some fields of (2,2) type.展开更多
In this paper,the authors show that there exists infinitely many family of pairs of quadratic fields Q(√D)and Q((√D+n)(1/2))with D,n∈Z whose class numbers are both divisible by 3.
The classical chi-squared goodness of fit test assumes the number of classes is fixed,meanwhile the test statistic has a limiting chi-square distribution under the null hypothesis.It is well known that the number of c...The classical chi-squared goodness of fit test assumes the number of classes is fixed,meanwhile the test statistic has a limiting chi-square distribution under the null hypothesis.It is well known that the number of classes varying with sample size in the test has attached more and more attention.However,in this situation,there is not theoretical results for the asymptotic property of such chi-squared test statistic.This paper proves the consistency of chi-squared test with varying number of classes under some conditions.Meanwhile,the authors also give a convergence rate of KolmogorovSimirnov distance between the test statistic and corresponding chi-square distributed random variable.In addition,a real example and simulation results validate the reasonability of theoretical result and the superiority of chi-squared test with varying number of classes.展开更多
With the help of continued fractions, we plan to list all the elements of the set Q△ = {aX2 ＋ bXY ＋ cY2 ： a,b, c ∈Z, b2 - 4ac = △ with 0 ≤ b 〈 √△}of quasi-reduced quadratic forms of fundamental discriminant ...With the help of continued fractions, we plan to list all the elements of the set Q△ = {aX2 ＋ bXY ＋ cY2 ： a,b, c ∈Z, b2 - 4ac = △ with 0 ≤ b 〈 √△}of quasi-reduced quadratic forms of fundamental discriminant △. As a matter of fact, we show that for each reduced quadratic form f = aX2 ＋ bXY ＋ cY2 = （a, b, c） of discriminant △〉0（and of sign σ（f） equal to the sign of a）, the quadratic forms associated with f and defined by {〈a＋bu＋cu2,b＋2cu.c〉},with 1≤σ（f）u≤b/2｜c｜ （whenever they exist）, 〈c,-b-2cu,a＋bu＋cu2〉 with b/2｜c｜≤σ（f）u≤[w（f）]=[b＋√△/2｜c｜], are all different from one another and build a set I（f） whose cardinality is #I（f）={1＋[ω（f）],when（2c）｜b,[ω（f）],when （2c）｜b. If f and g are two different reduced quadratic forms, we show that I（f） ∩ I（g） = Ф. Our main result is that the set Q△ is given by the disjoint union of all I（f） with f running through the set of reduced quadratic forms of discriminant △〉0. This allows us to deduce a formula for #（Q△） involving sums of partial quotients of certain continued fractions.展开更多
基金Supported by NNSF of China and SF of Chinese Education Committee ,and has been done when the author visited the Department of Mathematics of Purduc Unuversity in 1993
文摘It is a survey of the problem on class numbers of quadratic number fields.
基金supported by the SERB MATRICS Grant(No.MTR/2021/00762)the ANRF(SERB)Govt.of India Grant(No.CRG/2023/007323).
文摘The authors exhibit some new families of cyclotomic fields which have non-trivial plus parts of their class numbers.They also prove the 3-divisibility of the plus part of the class number of another family consisting of infinitely many cyclotomic fields.At the end,they provide some numerical examples supporting our results.
基金Supported by National Natural Science Foundation of China （Grant No. 10131010）
文摘A parametrization of quadratic function fields whose divisor class numbers are divisible by 3 is obtained by using free parameters when the characteristics of the fields are not 3.
基金supported by National Natural Science Foundation of China (Grant No. 10771111)
文摘Let K = $ k(\sqrt \theta ) $ be a real cyclic quartic field, k be its quadratic subfield and $ \tilde K = k(\sqrt { - \theta } ) $ be the corresponding imaginary quartic field. Denote the class numbers of K, k and $ \tilde K $ by h K , h k and {417-3} respectively. Here congruences modulo powers of 2 for h ? = h K /h K and $ \tilde h^ - = h_{\tilde K} /h_k $ are obtained via studying the p-adic L-functions of the fields.
基金the National Natural Science Foundation of China (No.10071041)
文摘Let q 5 be a prime number. Let k d=() be a quadratic number field, where d =--gqq(1)2(1) ---qqqquwuq12((1)+). Then the class number of k is divisible by q for certain integers u,w. Conversely, assume W / k is an unramified cyclic extension of degree q (which implies the class number of k is divisible by q), and W is the splitting field of some irreducible trinomial f(X) = XqaXb with integer coefficients, k Df=(())with D(f) the discriminant of f(X). Then f(X) must be of the form f(X) = Xquq2wXuq1 in a cer-tain sense where u,w are certain integers. Therefore, k d=() with d =-----qqqqqquwuq(1)122(1)((1)+). Moreover, the above two results are both generalized for certain kinds of general polynomials.
基金supported the by the National Natural Science Foundation of China(Grant No.19771052)
文摘Let K 6 be a real cyclic sextic number fields,and K 2,K 3 be its quadratic and cubic subfields.Let h(L)denote the ideal class number of field L.Seven congruences for h-=h(K 6)/h(K 2)h(K 3)are obtained.In particular,when conductor f\-6 of K 6 is prime p,then Ch-≡B p-16B 5(p-1)6(mod p),where C is an explicitly given constant,and B n is the Bernoulli number.These results for real cyclic sextic fields are an extension of results for quadratic and cyclic quartic fields obtained by Ankeny_Artin_Chowla,Kiselev,Carlitz,Lu Hongwen,Zhang Xianke from 1948 to 1988.
文摘In this paper, the theory of continued fractions of algebraic functions will be used to give a general theorem on lower bounds for class numbers of real quadratic function fields K=k(D). The bounds are given more explicitly for six types of real quadratic function fields. As a consequence, six classes of real quadratic function fields with ideal class number greater than one are given.[
文摘The theory of continued fractions of functions is used to give a lower bound for class numbers h(D) of general real quadratic function fields over k = F q (T). For five series of real quadratic function fields K, the bounds of h(D) are given more explicitly, e. g., if D = F 2 + c, then h(D) ≥ degF/degP; if D = (SG)2 + cS, then h(D) ≥ degS/degP; if D = (A m + a)2 + A, then h(D) ≥ degA/degP, where P is an irreducible polynomial splitting in K, c ∈ F q . In addition, three types of quadratic function fields K are found to have ideal class numbers bigger than one.
文摘In this paper, the author applies adjacent lattice method and Siegel mass formula to determine the classes of positive definite unimodular lattices of rank 4 over Z , and obtains that the class number of unit genus gen( I 4 ) is nine and the class number of even unimodular lattices is three, and also gives the representative lattices of each class.
文摘In this paper, the author applies adjacent lattice method and Siegel mass formula to determine the classes of positive definite unimodular lattices of rank 4 over Z , and obtains that the class number of unit genus gen( I 4 ) is nine and the class number of even unimodular lattices is three, and also gives the representative lattices of each class.
基金supported by the NSFC(1080110530871444)+2 种基金the Key Project of Natural Science of Anhui Provincial Department of Education(KJ2009A44)the Doctoral Special Fund of Hefei University of Technology(GDBJ2010-012)the Key Project of Science and Technology of Anhui Province(08010302070)
文摘In this paper, we give a lower bound exp（2.2 × 10~8 ） for those discriminants of real quadratic fields Q（√ d） with d= N^2-4 and h（d）=1.
基金Supported by National Natural Science Foundation of China(Grant Nos.10871132,11271263)Beijing Natural Science Foundation(Grant Nos.1102011,1132001)Specialized Research Fund for the Doctoral Program of Higher Education(Grant No.20091108110004)
文摘We investigate the asymptotic behavior of the entropy numbers of Besov classes BBΩp,θ（Sd 1）of generalized smoothness on the sphere inL q（Sd 1）for 1≤p,q,θ≤∞,and get their asymptotic orders.We also obtain the exact orders of entropy numbers of Sobolev classesBWr p（Sd 1）inL q（Sd 1）whenpand/orqis equal to 1 or∞.This provides the last piece as far as exact orders of entropy numbers ofBWr p（Sd 1）inL q（Sd 1）are concerned.
文摘A necessary condition is presented for the ideal class group of an imaginary quadratic function field K=k(D)(k=F-q(x),2q)to be of exponent≤2.The condition is proved to be sufficient in some cases.An analogue of Louboutin’s result in function field case is particularly presented.
文摘We investigate arithmetic properties of certain subsets of square-free positive integers and obtain in this way some results concerning the class number h（d） of the real quadratic field Q（√d）. In particular, we give a new proof of the result of Hasse, asserting that in this case h（d） = 1 is possible only if d is of the form p, 2q or qr. where p.q. r are primes and q≡r≡3（mod 4）.
文摘Ideal class groups H(K) of algebraic quadratic function fields K are studied. Necessaryand sufficient condition is given for the class group H(K) to contain a cyclic subgroup of anyorder n, which holds true for both real and imaginary fields K. Then several series of functionfields K, including real, inertia imaginary, and ramified imaginary quadratic function fields, aregiven, for which the class groups H(K) are proved to contain cyclic subgroups of order n.
文摘Let k=Q((D 2+md)(D 2+nd)(D 2+rd)), this paper proves firstly that the fundamental unit of k is ε=((D 2+md)(D 2+nd)+D 2(D 2+rd)) 2/(|mn|d 2), where D,d,m,n, and r are rational integers satisfying certain conditions. Consequently, we describe the fundamental unit system of K=Q(D 2+md,D 2+nd,D 2+rd) explicitly by the fundamental unit of all the quadratic subfields and the class number h K explicitly by the class numbers of all the quadratic subfields. We also provide the fundamental unit system of some fields of (2,2) type.
基金supported by Anhui Initiative in Quantum Information Technologies(No.AHY150200)
文摘In this paper,the authors show that there exists infinitely many family of pairs of quadratic fields Q(√D)and Q((√D+n)(1/2))with D,n∈Z whose class numbers are both divisible by 3.
基金supported by the Natural Science Foundation of China under Grant Nos.11071022,11028103,11231010,11471223,BCMIISthe Beijing Municipal Educational Commission Foundation under Grant Nos.KZ201410028030,KM201210028005Jishou University Subject in 2014(No:14JD035)
文摘The classical chi-squared goodness of fit test assumes the number of classes is fixed,meanwhile the test statistic has a limiting chi-square distribution under the null hypothesis.It is well known that the number of classes varying with sample size in the test has attached more and more attention.However,in this situation,there is not theoretical results for the asymptotic property of such chi-squared test statistic.This paper proves the consistency of chi-squared test with varying number of classes under some conditions.Meanwhile,the authors also give a convergence rate of KolmogorovSimirnov distance between the test statistic and corresponding chi-square distributed random variable.In addition,a real example and simulation results validate the reasonability of theoretical result and the superiority of chi-squared test with varying number of classes.
文摘With the help of continued fractions, we plan to list all the elements of the set Q△ = {aX2 ＋ bXY ＋ cY2 ： a,b, c ∈Z, b2 - 4ac = △ with 0 ≤ b 〈 √△}of quasi-reduced quadratic forms of fundamental discriminant △. As a matter of fact, we show that for each reduced quadratic form f = aX2 ＋ bXY ＋ cY2 = （a, b, c） of discriminant △〉0（and of sign σ（f） equal to the sign of a）, the quadratic forms associated with f and defined by {〈a＋bu＋cu2,b＋2cu.c〉},with 1≤σ（f）u≤b/2｜c｜ （whenever they exist）, 〈c,-b-2cu,a＋bu＋cu2〉 with b/2｜c｜≤σ（f）u≤[w（f）]=[b＋√△/2｜c｜], are all different from one another and build a set I（f） whose cardinality is #I（f）={1＋[ω（f）],when（2c）｜b,[ω（f）],when （2c）｜b. If f and g are two different reduced quadratic forms, we show that I（f） ∩ I（g） = Ф. Our main result is that the set Q△ is given by the disjoint union of all I（f） with f running through the set of reduced quadratic forms of discriminant △〉0. This allows us to deduce a formula for #（Q△） involving sums of partial quotients of certain continued fractions.
