It is well known that there is a close connection between tame kernels and ideal class groups of number fields. However, the latter is a very difficult subject in number theory. In this paper, we prove some results co...It is well known that there is a close connection between tame kernels and ideal class groups of number fields. However, the latter is a very difficult subject in number theory. In this paper, we prove some results connecting the p^n-rank of the tame kernel of a cyclic cubic field F with the p^n-rank of the coinvariants of μp^n×CI(δE,T) under the action of the Galois group, where E = F(ζp^n ) and T is the finite set of primes of E consisting of the infinite primes and the finite primes dividing p. In particular, if F is a cyclic cubic field with only one ramified prime and p = 3, n = 2, we apply the results of the tame kernels to prove some results of the ideal class groups of E, the maximal real subfield of E and F(ζ3).展开更多
For F = Q(√εpq), ε∈ {±1,±2}, primes -p ≡ q ≡ 1 mod 4, we give the necessary and sufficient conditions for 8-ranks of narrow class groups of F equal to 1 or 2 such that we can calculate their densitie...For F = Q(√εpq), ε∈ {±1,±2}, primes -p ≡ q ≡ 1 mod 4, we give the necessary and sufficient conditions for 8-ranks of narrow class groups of F equal to 1 or 2 such that we can calculate their densities. All results are stated in terms of congruence relations of p, q modulo 2^n, the quartic residue symbol (1/q)4 and binary quadratic forms such as q^h(-2p)/^4 = x^2 + 2py^2 where h(-2p) is the class number of Q(√-2p). The results are very useful for numerical computations.展开更多
In this paper, we study the real quadratic function fields K=k(D), given a necessary and sufficient condition for the ideal class group H(K) of any real quadratic function field K to have a cyclic subgroup of order n,...In this paper, we study the real quadratic function fields K=k(D), given a necessary and sufficient condition for the ideal class group H(K) of any real quadratic function field K to have a cyclic subgroup of order n, and obtained eight series of such fields. The ideal class numbers h(O K) of K in the series all have a factor n.[展开更多
Let F = Q(√-p1p2) be an imaginary quadratic field with distinct primes p1 = p2 = 1 mod 8 and the Legendre symbol (p1/p2) = 1. Then the 8-rank of the class group of F is equal to 2 if and only Pl if the followi...Let F = Q(√-p1p2) be an imaginary quadratic field with distinct primes p1 = p2 = 1 mod 8 and the Legendre symbol (p1/p2) = 1. Then the 8-rank of the class group of F is equal to 2 if and only Pl if the following conditions hold: (1) The quartic residue symbols (p1/p2)4 = (p2/p1)4 = 1; (2) Either both p1 and p2 are represented by the form a^2 + 32b^2 over Z and p^h2+(2p1)/4=x^2-2p1y^2,x,y∈Z,or both p1 and p2 are not represented by the form a^2 + 32b^2 over Z and p^h2+(2p1)/4=ε(2x^2-p1y^2),x,y∈Z,ε∈{±1},where h+(2p1) is the narrow class number of Q(√2p1),Moreover, we also generalize these results.展开更多
Imaginary cyclic fields of degree p-1 which have two distinct unramified cyclic extensions of degree p are produced using elementary properties of the Lucas sequences. An infinite family of imaginary cyclic fields of ...Imaginary cyclic fields of degree p-1 which have two distinct unramified cyclic extensions of degree p are produced using elementary properties of the Lucas sequences. An infinite family of imaginary cyclic fields of degree p-1 are then given with the p-rank of the ideal class groups of at least two.展开更多
Theoretical foundations of a new algorithm for determining the p-capitulation type ù(K) of a number field K with p-class rank ?=2 are presented. Since ù(K) alone is insufficient for identifying the seco...Theoretical foundations of a new algorithm for determining the p-capitulation type ù(K) of a number field K with p-class rank ?=2 are presented. Since ù(K) alone is insufficient for identifying the second p-class group G=Gal(F<sub>p</sub><sup>2</sup>K∣K) of K, complementary techniques are deve- loped for finding the nilpotency class and coclass of . An implementation of the complete algorithm in the computational algebra system Magma is employed for calculating the Artin pattern AP(K)=(τ (K),ù(K)) of all 34631 real quadratic fields K=Q(√d) with discriminants 0d<10<sup>8</sup> and 3-class group of type (3, 3). The results admit extensive statistics of the second 3-class groups G=Gal(F<sub>3</sub><sup>2</sup>K∣K) and the 3-class field tower groups G=Gal(F<sub>3</sub><sup>∞</sup>K∣K).展开更多
Objective:In this research,we tried to explore how short-term mindfulness(STM)intervention affects adoles-cents’anxiety,depression,and negative and positive emotion during the COVID-19 pandemic.Design:10 classes were...Objective:In this research,we tried to explore how short-term mindfulness(STM)intervention affects adoles-cents’anxiety,depression,and negative and positive emotion during the COVID-19 pandemic.Design:10 classes were divided into experiment groups(5 classes;n=238)and control(5 classes;n=244)randomly.Hospital Anxi-ety and Depression Scale(HADS)and Positive and Negative Affect Schedule(PANAS)were used to measure par-ticipants’dependent variables.In the experiment group,we conducted STM practice interventions every morning in theirfirst class from March to November 2020.No interventions were conducted in the control group.Methods:Paired-sample t-tests were used to identify if a significant difference exists between every time point of the experimental and control groups.Repeated ANOVA and Growth Mixture Model(GMM)were used to analyze the tendency of positive and negative emotions,anxiety,and depression in the experimental group.Results and Conclusions:(1)With the intervention of STM,there was a significant decrease in negative emotions and an increase in positive emotions in the experimental group,whereas there were non-significant differences in the control group.(2)To explore the heterogeneity trajectories of dependent variables,we built a GMM and found there were two latent growth classes in the trajectories.(3)The results of the models showed their trajec-tories were downward,which meant that the levels of anxiety,depression,and negative emotions of participants decreased during the STM training period.Nonetheless,the score of positive affect showed upward in three loops of intervention,which indicated that the level of the participants’positive affect increased through the STM inter-vention.(4)This research indicated that STM should be given increasing consideration to enhance mental health during the worldwide outbreak of COVID-19.展开更多
Based on the action of the mapping class group on the space of measured foliations,we construct a new boundary of the mapping class group and study the structure of this boundary.As an application,for any point in Tei...Based on the action of the mapping class group on the space of measured foliations,we construct a new boundary of the mapping class group and study the structure of this boundary.As an application,for any point in Teichmüller space,we consider the orbit of this point under the action of the mapping class group and describe the closure of this orbit in the Thurston compactification and the Gardiner-Masur compactification of Teichmüller space.We also construct some new points in the Gardiner-Masur boundary of Teichmüller space.展开更多
The authors introduce a notion of a weak graph map homotopy(they call it M-homotopy),discuss its properties and applications.They prove that the weak graph map homotopy equivalence between graphs coincides with the gr...The authors introduce a notion of a weak graph map homotopy(they call it M-homotopy),discuss its properties and applications.They prove that the weak graph map homotopy equivalence between graphs coincides with the graph homotopy equivalence defined by Yau et al in 2001.The difference between them is that the weak graph map homotopy transformation is defined in terms of maps,while the graph homotopy transformation is defined by means of combinatorial operations.They discuss its advantages over the graph homotopy transformation.As its applications,they investigate the mapping class group of a graph and the 1-order M P-homotopy group of a pointed simple graph.Moreover,they show that the 1-order M P-homotopy group of a pointed simple graph is invariant up to the weak graph map homotopy equivalence.展开更多
Let E / K be an elliptic curve with K-rational p-torsion points. The p-Selmer group of E is described by the image of a map uK and hence an upper bound of its order is given in terms of the class numbers of the S-idea...Let E / K be an elliptic curve with K-rational p-torsion points. The p-Selmer group of E is described by the image of a map uK and hence an upper bound of its order is given in terms of the class numbers of the S-ideal class group of K and the p-division field of E.展开更多
Motivated by the work of Birman about the relationship between mapping class groups and braid groups,the authors discuss the relationship between the orbit braid group and the equivariant mapping class group on the cl...Motivated by the work of Birman about the relationship between mapping class groups and braid groups,the authors discuss the relationship between the orbit braid group and the equivariant mapping class group on the closed surface M with a free and proper group action in this paper.Their construction is based on the exact sequence given by the fibration F_(0)^(G) M→F(M/G,n).The conclusion is closely connected with the braid group of the quotient space.Comparing with the situation without the group action,there is a big difference when the quotient space is T^(2).展开更多
Let F = Q(x/P), where p = 8t + 1 is a prime. In this paper, we prove that a speclm case of Qin's conjecture on the possible structure of the 2-primary part of K2OF up to 8-rank is a consequence of a conjecture of ...Let F = Q(x/P), where p = 8t + 1 is a prime. In this paper, we prove that a speclm case of Qin's conjecture on the possible structure of the 2-primary part of K2OF up to 8-rank is a consequence of a conjecture of Cohen and Lagarias on the existence of governing fields. We also characterize the 16-rank of K2OF, which is either 0 or 1, in terms of a certain equation between 2-adic Hilbert symbols being satisfied or not.展开更多
In this paper, we study the p-rank of the tame kernels of pure cubic fields. In particular, we prove that for a fixed positive integer m, there exist infinitely many pure cubic fields whose 3-rank of the tame kernel e...In this paper, we study the p-rank of the tame kernels of pure cubic fields. In particular, we prove that for a fixed positive integer m, there exist infinitely many pure cubic fields whose 3-rank of the tame kernel equal to m. As an application, we determine the 3-rank of their tame kernels for some special pure cubic fields.展开更多
Let K_(0)=Q(√δ)beaquadraticfield.Forthose K_(0) withoddclassnumber,much work has been done on the explicit construction of the Hilbert genus field of a biquadratic extension K=Q(√δ,√d)over Q.Whenδ=2 or p with p...Let K_(0)=Q(√δ)beaquadraticfield.Forthose K_(0) withoddclassnumber,much work has been done on the explicit construction of the Hilbert genus field of a biquadratic extension K=Q(√δ,√d)over Q.Whenδ=2 or p with p≡1 mod 4 a prime and K is real,it was described in Yue(Ramanujan J 21:17–25,2010)and Bae and Yue(Ramanujan J 24:161–181,2011).In this paper,we describe the Hilbert genus field of K explicitly when K_(0) is real and K is imaginary.In fact,we give the explicit construction of the Hilbert genus field of any imaginary biquadratic field which contains a real quadratic subfield of odd class number.展开更多
文摘It is well known that there is a close connection between tame kernels and ideal class groups of number fields. However, the latter is a very difficult subject in number theory. In this paper, we prove some results connecting the p^n-rank of the tame kernel of a cyclic cubic field F with the p^n-rank of the coinvariants of μp^n×CI(δE,T) under the action of the Galois group, where E = F(ζp^n ) and T is the finite set of primes of E consisting of the infinite primes and the finite primes dividing p. In particular, if F is a cyclic cubic field with only one ramified prime and p = 3, n = 2, we apply the results of the tame kernels to prove some results of the ideal class groups of E, the maximal real subfield of E and F(ζ3).
基金Supported by National Natural Science Foundation of China (Grant Nos. 10771100, 10971250)
文摘For F = Q(√εpq), ε∈ {±1,±2}, primes -p ≡ q ≡ 1 mod 4, we give the necessary and sufficient conditions for 8-ranks of narrow class groups of F equal to 1 or 2 such that we can calculate their densities. All results are stated in terms of congruence relations of p, q modulo 2^n, the quartic residue symbol (1/q)4 and binary quadratic forms such as q^h(-2p)/^4 = x^2 + 2py^2 where h(-2p) is the class number of Q(√-2p). The results are very useful for numerical computations.
文摘In this paper, we study the real quadratic function fields K=k(D), given a necessary and sufficient condition for the ideal class group H(K) of any real quadratic function field K to have a cyclic subgroup of order n, and obtained eight series of such fields. The ideal class numbers h(O K) of K in the series all have a factor n.[
基金Project supported by the National Natural Science Foundation of China(No.10371054)
文摘Let F = Q(√-p1p2) be an imaginary quadratic field with distinct primes p1 = p2 = 1 mod 8 and the Legendre symbol (p1/p2) = 1. Then the 8-rank of the class group of F is equal to 2 if and only Pl if the following conditions hold: (1) The quartic residue symbols (p1/p2)4 = (p2/p1)4 = 1; (2) Either both p1 and p2 are represented by the form a^2 + 32b^2 over Z and p^h2+(2p1)/4=x^2-2p1y^2,x,y∈Z,or both p1 and p2 are not represented by the form a^2 + 32b^2 over Z and p^h2+(2p1)/4=ε(2x^2-p1y^2),x,y∈Z,ε∈{±1},where h+(2p1) is the narrow class number of Q(√2p1),Moreover, we also generalize these results.
基金Supported by the Japan Society for the Promotion of Science (JSPS) (No. 14540030) the JSPS Research Fellowships for Young Scientists
文摘Imaginary cyclic fields of degree p-1 which have two distinct unramified cyclic extensions of degree p are produced using elementary properties of the Lucas sequences. An infinite family of imaginary cyclic fields of degree p-1 are then given with the p-rank of the ideal class groups of at least two.
文摘Theoretical foundations of a new algorithm for determining the p-capitulation type ù(K) of a number field K with p-class rank ?=2 are presented. Since ù(K) alone is insufficient for identifying the second p-class group G=Gal(F<sub>p</sub><sup>2</sup>K∣K) of K, complementary techniques are deve- loped for finding the nilpotency class and coclass of . An implementation of the complete algorithm in the computational algebra system Magma is employed for calculating the Artin pattern AP(K)=(τ (K),ù(K)) of all 34631 real quadratic fields K=Q(√d) with discriminants 0d<10<sup>8</sup> and 3-class group of type (3, 3). The results admit extensive statistics of the second 3-class groups G=Gal(F<sub>3</sub><sup>2</sup>K∣K) and the 3-class field tower groups G=Gal(F<sub>3</sub><sup>∞</sup>K∣K).
基金Regional Science Fund Project of Northwest Normal University,Grant No.31660281.
文摘Objective:In this research,we tried to explore how short-term mindfulness(STM)intervention affects adoles-cents’anxiety,depression,and negative and positive emotion during the COVID-19 pandemic.Design:10 classes were divided into experiment groups(5 classes;n=238)and control(5 classes;n=244)randomly.Hospital Anxi-ety and Depression Scale(HADS)and Positive and Negative Affect Schedule(PANAS)were used to measure par-ticipants’dependent variables.In the experiment group,we conducted STM practice interventions every morning in theirfirst class from March to November 2020.No interventions were conducted in the control group.Methods:Paired-sample t-tests were used to identify if a significant difference exists between every time point of the experimental and control groups.Repeated ANOVA and Growth Mixture Model(GMM)were used to analyze the tendency of positive and negative emotions,anxiety,and depression in the experimental group.Results and Conclusions:(1)With the intervention of STM,there was a significant decrease in negative emotions and an increase in positive emotions in the experimental group,whereas there were non-significant differences in the control group.(2)To explore the heterogeneity trajectories of dependent variables,we built a GMM and found there were two latent growth classes in the trajectories.(3)The results of the models showed their trajec-tories were downward,which meant that the levels of anxiety,depression,and negative emotions of participants decreased during the STM training period.Nonetheless,the score of positive affect showed upward in three loops of intervention,which indicated that the level of the participants’positive affect increased through the STM inter-vention.(4)This research indicated that STM should be given increasing consideration to enhance mental health during the worldwide outbreak of COVID-19.
文摘Based on the action of the mapping class group on the space of measured foliations,we construct a new boundary of the mapping class group and study the structure of this boundary.As an application,for any point in Teichmüller space,we consider the orbit of this point under the action of the mapping class group and describe the closure of this orbit in the Thurston compactification and the Gardiner-Masur compactification of Teichmüller space.We also construct some new points in the Gardiner-Masur boundary of Teichmüller space.
基金supported by the National Natural Science Foundation of China(No.11771116)。
文摘The authors introduce a notion of a weak graph map homotopy(they call it M-homotopy),discuss its properties and applications.They prove that the weak graph map homotopy equivalence between graphs coincides with the graph homotopy equivalence defined by Yau et al in 2001.The difference between them is that the weak graph map homotopy transformation is defined in terms of maps,while the graph homotopy transformation is defined by means of combinatorial operations.They discuss its advantages over the graph homotopy transformation.As its applications,they investigate the mapping class group of a graph and the 1-order M P-homotopy group of a pointed simple graph.Moreover,they show that the 1-order M P-homotopy group of a pointed simple graph is invariant up to the weak graph map homotopy equivalence.
文摘Let E / K be an elliptic curve with K-rational p-torsion points. The p-Selmer group of E is described by the image of a map uK and hence an upper bound of its order is given in terms of the class numbers of the S-ideal class group of K and the p-division field of E.
基金supported by the National Natural Science Foundation of China(No.11971112)。
文摘Motivated by the work of Birman about the relationship between mapping class groups and braid groups,the authors discuss the relationship between the orbit braid group and the equivariant mapping class group on the closed surface M with a free and proper group action in this paper.Their construction is based on the exact sequence given by the fibration F_(0)^(G) M→F(M/G,n).The conclusion is closely connected with the braid group of the quotient space.Comparing with the situation without the group action,there is a big difference when the quotient space is T^(2).
基金Supported by NSFC(Grant Nos.11201225,11271177 and 11171141)
文摘Let F = Q(x/P), where p = 8t + 1 is a prime. In this paper, we prove that a speclm case of Qin's conjecture on the possible structure of the 2-primary part of K2OF up to 8-rank is a consequence of a conjecture of Cohen and Lagarias on the existence of governing fields. We also characterize the 16-rank of K2OF, which is either 0 or 1, in terms of a certain equation between 2-adic Hilbert symbols being satisfied or not.
基金Supported by National Natural Science Foundation of China(Grant Nos.10971091 and 10871088)Specialized Research Fund for the Doctoral Program of Higher Education(Grant Nos.200802840003 and 200802841042)
文摘In this paper, we study the p-rank of the tame kernels of pure cubic fields. In particular, we prove that for a fixed positive integer m, there exist infinitely many pure cubic fields whose 3-rank of the tame kernel equal to m. As an application, we determine the 3-rank of their tame kernels for some special pure cubic fields.
基金partially supported by National Key Basic Research Program of China(Grant No.2013CB834202)National Natural Science Foundation of China(Nos.11501429,11171150 and 11171317)Fundamental Research Funds for the Central Universities(Grant No.JB150706).
文摘Let K_(0)=Q(√δ)beaquadraticfield.Forthose K_(0) withoddclassnumber,much work has been done on the explicit construction of the Hilbert genus field of a biquadratic extension K=Q(√δ,√d)over Q.Whenδ=2 or p with p≡1 mod 4 a prime and K is real,it was described in Yue(Ramanujan J 21:17–25,2010)and Bae and Yue(Ramanujan J 24:161–181,2011).In this paper,we describe the Hilbert genus field of K explicitly when K_(0) is real and K is imaginary.In fact,we give the explicit construction of the Hilbert genus field of any imaginary biquadratic field which contains a real quadratic subfield of odd class number.