关于一些二次数域的Tame核的2-Sylow子群

本文主要研究二次数域Ｆ＝Ｑ（ｄ）的 ２－Ｓｙｌｏｗ于群,其中ｄ只有两个不同奇素数和２ ∈ＮＦ／Ｑ（Ｆ）．建立了 Ｅ＝Ｑ（－ｄ）的类群和具有平凡的４－秩或８－秩的Ｋ２ＯＦ两者之间等价关系．     

巧用函数单调性解赛题

函数ｆ（ｘ）在区间［ａ,ｂ］上单调增加（或单调减少）,又ｃ、ｄ∈［ａ,ｂ］上,若ｆ（ｃ）＝ｆ（ａ）,则有ｃ＝ｄ．１　求代数式的值例１　已知ｘ、ｙ∈［－π４,π４］,ａ∈Ｒ,且　ｘ３＋ｓｉｎｘ－２ａ＝０４ｙ３＋ｓｉｎｙｃｏｓｙ＋ａ＝０则ｃｏｓ（ｘ＋２ｙ）＝　　．（１９９４年全国高中数学竞赛题）解　由已知条件,可得　　ｘ３＋ｓｉｎｘ＝２ａ（－２ｙ）３＋ｓｉｎ（－２ｙ）＝２ａ故可设函数ｆ（ｔ）＝ｔ３＋ｓｉｎｔ,则有ｆ（ｘ）＝ｆ（－２ｙ）＝２ａ．由于函数ｆ（ｔ）＝ｔ３＋ｓｉｎｔ,在［－π２,π２］上是单…     

Burgers与组合KdV混合型方程的精确解

该文求出了组合ＫｄＶ方程的渐近值不为零的钟状孤波解和扭状孤波解；求出了Ｂｕｒｇｅｒｓ与组合ＫｄＶ混合型方程ｕｔ＋ａｕｕｘ＋ｂｕ２ｕｘ＋ｒｕ（ｘｘ）＋ｕ（ｘｘｘ）＝０的二类扭状孤波解．作为推论，还求出了波方程ｕ（ｔｔ）－ｋｕ（ｘｘ）＋ｐｕ十ｑｕ２＋ｓｕ３＝０的钟状和扭状孤波解．     

有限域上一类方程的解数公式

本文给出有限域Ｆｑ上一类方程ａ１ｘ１ｄ１１…ｘｎｄ１ｎ＋ａ２ｘ１ｄ２１…ｘｎｄ２ｎ＋…＋ａｓｘ１ｄｓ１…ｘｎｄｓｎ＝ｂ的解数公式，这里ｄｉｊ＞０，ａｉ∈Ｆｑ，ｉ＝１，…，ｓ，ｊ＝１，…，ｎ．特别当ｓ＝ｎ，ｇｃｄ（｜ｄｉｊ｜，ｑ－１）＝１时，得到了简明的解数公式．     

ON THE EXISTENCE OF ALMOST GLOBAL WEAK SOLUTION TO MULTIDIMENSIONAL VLASOV-POISSON EQUATION

§１．ＩｎｔｒｏｄｕｃｔｉｏｎＣｏｎｓｉｄｅｒｔｈｅｆｏｌｏｗｉｎｇｄｄｉｍｅｎｓｉｏｎａｌＶｌａｓｏｖＰｏｉｓｏｎｓｙｓｔｅｍ，ｄ＝２，３，ｔｆ＋ｖ·ｘｆ－Ｅ·ｖｆ＝０，ｆ（０，ｘ，ｖ）＝ｆ０（ｘ，ｖ），Ｅ（ｔ，ｘ）＝ｃ（ｄ）∫ｘ－ｙ｜ｘ...     

THE VARIATION OF INVARIANT LINES AND LIMIT CYCLES FOR A QUADRATIC DIFFERENTIAL SYSTEM

１ＩｎｔｒｏｄｕｃｔｉｏｎＣｏｎｓｉｄｅｒｔｈｅｒｅａｌｑｕａｄｒａｔｉｃｄｉｆｅｒｅｎｔｉａｌｓｙｓｔｅｍｘ＝－ｙ＋δｘ－４ｘ２＋３ｘｙ＋１３ｙ２ｙ＝ｘ（１－１３ｘ－ｙ）．（１）Ｆｉｒｓｔ，ｆｏｒｃｏｎｖｅｎｉｅｎｃｅ，ｗｅｄｅｎｏｔｅｔｈｅｔｗｏ...     

Burgers－KdV 方程的初边值问题

该文借助适当的逼近，用散逸算子理论，差分和估计方法，证明了［０，１］×［０，Ｔ］上Ｂｕｒｇｅｒｓ－ＫｄＶ方程ｕｔ＋ｕ（ｘｘｘ）－Ｕ（ｘｘ）＋ｕｕｘ＝ｆ（ｘ，ｔ）的一类初边值问题存在唯一的解ｕ∈Ｌ∞（０，Ｔ；Ｈ３（０，１））∩Ｃ（０，Ｔ；Ｈ２（０，１））∩Ｗ（１，∞），（Ｏ，Ｔ；Ｌ２（０，１））．     

虚二次域上不可分的正定Hermite型的构作

本文给出了构作虚二次域ＩＱ（－ｍ）上不可分的正定整Ｈｅｒｍｉｔｅ型的方法．对任意给定的自然数ｎ，ｄ和无平方因子的ｍ，当ｍ３（ｍｏｄ４），除了ｍ＝１时ｎ＝２，ｄ＝１；ｎ＝３，ｄ＝１，３；ｎ＝５，ｄ＝１和ｍ＝２时ｎ＝３，ｄ＝１这５个例外，证明了存在ＩＱ（－ｍ）上不可分的正定整Ｈｅｒｍｉｔｅ格，其秩为ｎ且判别式为ｄ，并给出它们的明确结构．在上述５个例外情形下，不存在具有上述性质的格．     

图的最大亏格与2-因子

图Ｇ的一个２因子Ｆ就是Ｇ的这样一个支撑子图，使其任何节点ｖ∈Ｖ的次ｄＦ（ｖ）＝２．易见，Ｇ的每个２因子均为无公共节点的圈之并．若Ｆ的每个圈的长均为３（或４），则称Ｇ含有一个三角形（或四边形）２因子．Ｍ．ｋ∨ｏｖｉｅｒａ［５］得到了含有三角形２因子的３－正则图的最大亏格．本文在３－正则图上，引进了扩张运算和讨论了与最大亏格和Ｂｅｔｉ亏数之间的关系．利用这些运算，得到了所有含四边形２因子的连通３－正则图是上可嵌入的，即γＭ（Ｇ）＝ｎ４（ｎ为Ｇ的节点数ｎ＝｜Ｖ（Ｇ）｜）．然后，基于此证明了含四边形２因子且所有节点ｖ∈Ｖ的次ｄＧ（ｖ）＝３（ｍｏｄ４）的图Ｇ均为上可嵌入的     

On Theorem of the “Mountain Impasse” Type

ＬｅｔＦ（ｕ），Ｇ（ｕ）ｂｅＣ１－ｆｕｎｃｔｉｏｎａｌｓｏｎａＨｉｌｂｅｒｔｓｐａｃｅＨａｎｄｓｕｐｕ∈ＨＦ（ｕ）＝∞．ＬｅｔＫｂｅａｃｏｍｐａｃｔｍｅｔｒｉｃｓｐａｃｅａｎｄｌｅｔＫｂｅａｎｏｎ－ｅｍｐｔｙｃｌｏｓｅｄｓｕｂｓｅｔ≠Ｋ，ｐ∈Ｃ（Ｋ；Ｈ）．ＤｅｎｏｔｅＦＲ＝｛ｕ∈Ｈ｜Ｆ（ｕ）Ｒ｝；　　　　　　ΦＲ＝｛ｐ∈Ｃ（Ｋ；ＦＲ）｜ｐ＝ｐｏｎＫ｝；ｃ（Ｒ）＝ｉｎｆｐ∈ΦＲｍａｘξ∈ＫＧ（ｐ（ξ））；ｃ０＝ｍａｘξ∈ＫＧ（ｐ（ξ））；Φ∞＝｛ｐ∈Ｃ（Ｋ；Ｈ）｜ｐ＝ｐｏｎＫ｝；…     

Hilbert genus fields of biquadratic fields

The Hilbert genus field of the real biquadratic field K=Q(δ~(1/2),d~(1/2)is described by Yue(2010)and Bae and Yue(2011)explicitly in the case=2 or p with p=1 mod 4 a prime and d a squarefree positive integer.In this article,we describe explicitly the Hilbert genus field of the imaginary biquadratic field K=Q(δ~(1/2),d~(1/2)),whereδ=-1,-2 or-p with p=3 mod 4 a prime and d any squarefree integer.This completes the explicit construction of the Hilbert genus field of any biquadratic field which contains an imaginary quadratic subfield of odd class number.     

Neither G9（Q） nor G11（Q） Is a Subgroup of K2（Q）

It is proved that neither G9(Q) nor G11(Q) is a subgroup of K2(Q) which confirms two special cases of a conjecture proposed by Browkin, J. (Lecture Notes in Math., 966, Springer-Verlag, New York, Heidelberg, Berlin, 1982, 1-6).     

CONVERGENCE RATE OF MULTIVARIATE K-NEAREST NEIGHBOR DENSITY ESTIMATES

Let be the collection of m-times continuously differentiable probability densities fon R~d such that 丨D~af(x_1)-D~af(x_2)丨≤M‖x_1-x_2‖~β for x_1,x_2∈R~d,[a]=m,where D~adenotes the differential operator defined by D~a=([a])/(x_1~a…x_d~a_d).Under rather weak conditionson K(x),the necessary and sufficient conditions for sup丨_n(x)-f(x)丨=0(((logn/n)~λ/(d+3λ),λ=m+β,f∈ are that ∫x~aK(xi)dx=0 for 0<[a]≤m.Finally the convergenco rate at apoint is given.     

An elementary proof of the law of quadratic reciprocity over function fields

Let and be relatively prime monic irreducible polynomials in (). In this paper, we give an elementary proof for the following law of quadratic reciprocity in :

where is the Legendre symbol.

     

伪压缩族公共不动点的复合隐格式迭代逼近问题

设H是一实Hillber空间,K是H之一非空间凸子集,设{Ti}Ni=1是N个Lipschitz伪压缩映象使得F=∩Ni=1F(Ti)≠0,其中F(Ti)={x∈K:Tix=x}并且{αn}n∞=1,{βn}∞n=1[0,1]是满足如下条件的实序列(i)∑∞n=1(1-αn)2= ∞;(ii)limn→∞(1-αn)=0;(iii)∑∞n=1(1-βn)< ∞;(iv)(1-αn)L2<1,n1;(v)αn(1-βn)2 αn[βn L(1-βn)]2<1,其中L1是{Ti}iN=1的公共Lipschitz常数,对于x0∈K,设{xn}n∞=1是由下列定义的复合隐格式迭代xn=αnxn-1 (1-αn)Tnyn,yn=βnxn (1-βn)Tnxn,其中Tn=TnmodN,则(i)limn→∞‖xn-p‖存在,对于所有的p∈F;(ii)limn→∞d(xn,f)存在,其中d(xn,F)=infp∈F‖xn-p‖;(iii)liminfn→∞‖xn-Tnxn‖=0.本文的结果推广并且改进H-K.Xu和R.G.Ori在2001年的结果和Osilike在2004年的结果,并且在这篇文章中,主要的证明方法也不同与H-K.Xu和Osilike的方法.     

关于Q(4k~(2n)+1)的理想类群的循环子群

设ｄ,ａ,ｋ,ｎ是适合４ｋ２ｎ＋１＝ｄａ２,ｋ＞１,ｎ＞２,ｄ无平方因子的正整数;又设Ｃ（Ｋ）和ｈ（Ｋ）分别是实二次域Ｋ的理想类群和类数．本文证明了：当ａ＜０．５ｋ０．５６ｎ时,则ｈ（ｋ）＝０（ｍｏｄｎ）和Ｃ（Ｋ）必有ｎ阶循环子群．     

Differential geometry on almost tangent manifolds

Summary We start from a tensor field Q of type (1, 1) defined in a2n-dimensional manifold M which satisfies Q ²⁼⁰ and has rank n. The tensor field Q defines an almost tangent structure in M. We then introduce another tensor field P of the same type and having properties similar to those of Q. We then define and study the tensors H=PQ, V=QP, J=P−Q, K=P+Q, L=PQ−QP, (J, K, L) defining an almost quaternion structure of the second kind on M. We study the differential geometry on almost tangent manifolds in terms of these tensors. To ProfessorBeniamino Segre on his seventieth birthday Entrata in Redazione il 7 giugno 1973.     

SYMMETRIC AND ASYMMETRIC DIOPHANTINE APPROXIMATION

§1. Introduction Let ξbe an irrational number with simple continued fraction expansion ξ= [a0;a1,···,ai,···]and pi be its ith convergent. In [1], the present author considered the well-known inequality q     

On Normal Integral Bases

in this paper, let K = Q ( ) be an abelian number field,where mi's are distinct square-free integers and Gal (K/Q ) (Z/2Z )n+1, and let k Q(M). When k is imaginary or k is real and has a unit of norm -1, we give a necessaryand sufficient condition for K/k to have a normal integral basis.     

Dedekind zeta-functions and Dedekind sums

In this paper we use Dedekind zeta functions of two real quadratic number fields at -1 to denote Dedekind sums of high rank. Our formula is different from that of Siegel’s. As an application, we get a polynomial representation of ζK(-1): ζK(-1) = 1/45(26n³ -41n± 9),n = ±2(mod 5), where K = Q(√5q), prime q = 4n² + 1, and the class number of quadratic number field K2 = Q(vq) is 1.