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1.
Let k0 be a finite extension field of the rational numbers, and assume k0 has at least two Zl-extensions. Assume that at least one Zl-extension has Iwasawa invariant μ = 0, and let L be the composite of K and some other Zl-extension of k0. In this paper we find an upper bound for the number of Zl-extensions of k0 contained in L with nonzero μ. 相似文献
2.
Chen’s Conjecture and Its Generalization 总被引:1,自引:0,他引:1
Let l1, l2, ..., lg be even integers and x be a sufficiently large number. In this paper, the authors prove that the number of positive odd integers k ≤ x such that (k +l1)^2, (k +l2)^2, ..., (k +lg)^2 can not be expressed as 2^n+p^α is at least c(g)x, where p is an odd prime and the constant c(g) depends only on g. 相似文献
3.
Manu Basavaraju 《Discrete Mathematics》2009,309(13):4646-649
An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic (2-colored) cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a′(G). Let Δ=Δ(G) denote the maximum degree of a vertex in a graph G. A complete bipartite graph with n vertices on each side is denoted by Kn,n. Alon, McDiarmid and Reed observed that a′(Kp−1,p−1)=p for every prime p. In this paper we prove that a′(Kp,p)≤p+2=Δ+2 when p is prime. Basavaraju, Chandran and Kummini proved that a′(Kn,n)≥n+2=Δ+2 when n is odd, which combined with our result implies that a′(Kp,p)=p+2=Δ+2 when p is an odd prime. Moreover we show that if we remove any edge from Kp,p, the resulting graph is acyclically Δ+1=p+1-edge-colorable. 相似文献
4.
G.F Clements 《Journal of Combinatorial Theory, Series A》1977,22(3):368-371
Let k1, k2,…, kn be given integers, 1 ? k1 ? k2 ? … ? kn, and let S be the set of vectors x = (x1,…, xn) with integral coefficients satisfying 0 ? xi ? ki, i = 1, 2, 3,…, n. A subset H of S is an antichain (or Sperner family or clutter) if and only if for each pair of distinct vectors x and y in H the inequalities xi ? yi, i = 1, 2,…, n, do not all hold. Let |H| denote the number of vectors in H, let K = k1 + k2 + … + kn and for 0 ? l ? K let (l)H denote the subset of H consisting of vectors h = (h1, h2,…, hn) which satisfy h1 + h2 + … + hn = l. In this paper we show that if H is an antichain in S, then there exists an antichain H′ in S for which |(l)H′| = 0 if , if K is even and |(l)H′| = |(l)H| + |(K ? l)H| if . 相似文献
5.
Daniel J Madden 《Journal of Number Theory》1981,13(4):499-514
The primitive elements of a finite field are those elements of the field that generate the multiplicative group of k. If f(x) is a polynomial over k of small degree compared to the size of k, then f(x) represents at least one primitive element of k. Also f(x) represents an lth power at a primitive element of k, if l is also small. As a consequence of this, the following results holds.Theorem. Let g(x) be a square-free polynomial with integer coefficients. For all but finitely many prime numbers p, there is an integer a such that g(a) is equivalent to a primitive element modulo p.Theorem. Let l be a fixed prime number and f(x) be a square-free polynomial with integer coefficients with a non-zero constant term. For all but finitely many primes p, there exist integers a and b such that a is a primitive element and f(a) ≡ b1 modulo p. 相似文献
6.
7.
Werner Georg Nowak 《manuscripta mathematica》1990,69(1):153-172
In this article we study the arithmetic functiond a,b (l,k;n) which is defined as the number of representationsn=v a w b withw lying in the residue classl modulok (a,b andl,k fixed positive integers). For the remainder term in the asymptotic formula for Σn≤xda,b(l,k,;n) we obtain an Ω± (under a certain restriction onl andk) which is sharper than the known results for the corresponding “unrestricted” problem. 相似文献
8.
Zhong Gen Su 《数学学报(英文版)》2011,27(8):1573-1580
Let π be a minimal Erdös-Szekeres permutation of 1, 2, ..., n 2, and let l n,k be the length of the longest increasing subsequence in the segment (π(1), ..., π(k)). Under uniform measure we establish an exponentially decaying bound of the upper tail probability for l n,k , and as a consequence we obtain a complete convergence, which is an improvement of Romik’s recent result. We also give a precise lower exponential tail for l n,k . 相似文献
9.
Zhang Xianke 《Journal of Number Theory》1984,18(3):350-355
Let k = (√u) (u ≠ 1 squarefree), K any possible cyclic quartic field containing k. A close relation is established between K and the genus group of k. In particular: (1) Each K can be written uniquely as K = (√vwη), where η is fixed in k and satisfies η ? 1, (η) = 2√u, |2| = |(√u)|, (v, u) = 1, v ∈ is squarefree, w|u, 0 < w < √u. Thus if u ≠ a2 + b2, there is no K ? k. If u = a2 + b2 then for each fixed v there are 2g ? 1K ? k, where g is the number of prime divisors of u. (2) has a relative integral basis (RIB) (i.e., OK is free over Ok) iff N(ε0) = ?1 and w = 1, where ε0 is the fundamental unit of k, (or, equivalently, iff K = (√vε0√u), (v, u) = 1). (3) A RIB is constructed explicitly whenever it exists. (4) disc(K) is given. In particular, the following results are special cases of (2): (i) Narkiewicz showed in 1974 that has a RIB if u is a prime; (ii) Edgar and Peterson (J. Number Theory12 (1980), 77–83) showed that for u composite there is at least one K ? k having no RIB. Besides, it follows from (4) that the classification and integral basis of K given by Albert (Ann. of Math.31 (1930), 381–418) are wrong. 相似文献
10.
Asymptotic Upper Bounds for Ramsey Functions 总被引:5,自引:0,他引:5
We show that for any graph G with N vertices and average degree d, if the average degree of any neighborhood induced subgraph is at most a, then the independence number of G is at least Nf
a
+1(d), where f
a
+1(d)=∫0
1(((1−t)1/(
a
+1))/(a+1+(d−a−1)t))dt. Based on this result, we prove that for any fixed k and l, there holds r(K
k
+
l
,K
n
)≤ (l+o(1))n
k
/(logn)
k
−1. In particular, r(K
k
, K
n
)≤(1+o(1))n
k
−1/(log n)
k
−2.
Received: May 11, 1998 Final version received: March 24, 1999 相似文献
11.
Let f be a cusp form of weight k + 1/2 and at most quadratic nebentype character whose Fourier coefficients a(n) are all real. We study an equidistribution conjecture of Bruinier and Kohnen for the signs of a(n). We prove this conjecture for certain subfamilies of coefficients that are accessible via the Shimura lift by using the Sato–Tate equidistribution theorem for integral weight modular forms. Firstly, an unconditional proof is given for the family {a(tp 2)} p , where t is a squarefree number and p runs through the primes. In this case, the result is in terms of natural density. To prove it for the family {a(tn 2)} n where t is a squarefree number and n runs through all natural numbers, we assume the existence of a suitable error term for the convergence of the Sato–Tate distribution, which is weaker than one conjectured by Akiyama and Tanigawa. In this case, the results are in terms of Dedekind–Dirichlet density. 相似文献
12.
Aaron Clark 《代数通讯》2013,41(11):4097-4104
Let d be an odd integer, and let k be a field which contains a primitive dth root of unity. Let l 1 and l 2 be cyclic field extensions of k of degree d with norms n l 1/k and n l 2/k . Minà?'s approach which showed that quadratic Pfister forms are strongly multiplicative is applied to the form n l 1/k ? n l 2/k of degree d. Let K = k(X 1,…, X d 2 ). We compute polynomials which are similarity factors of a form of the kind N ? (n l 2/k ? k K) over K, where N is the norm of a certain field extension of K of degree d. These polynomials arise by specializing certain indeterminates of the homogeneous polynomial representing the form n l 1/k ? n l 2/k to be zero. Similar results are obtained for the tensor product of the norm of a cubic division algebra and a cubic norm n l 1/k . 相似文献
13.
Wen Jiwei 《高校应用数学学报(英文版)》2001,16(4):423-429
Let {Y(t);t=(t 1,t2)≥0}={Xk(t1,t2);t1≥0,t2≥0} k=1 ∞ , be a sequence of two-parameter Ornstein-Uhlenbeck processes (OUP2) with coefficient a k>0,ßk>0.. A Fernique type inequality is established and the sufficient condition for a. s. l 2 continuity of Y(?) is studied by means of the inequality. 相似文献
14.
Hiroshi Yamashita 《manuscripta mathematica》1993,79(1):1-5
We shall show two sufficient conditions under which the Iwasawa invariants λ
k
and μ
k
of a totally real fieldk vanish for an odd primel, based on the results obtained in [1], [3] and [4]. LetK
n be the composite ofk and thel
n-th cyclotomic extension of the fieldQ of rational numbers. LetC
n be the factor group of thel-class group ofK
n by a subgroup generated by ideals whose prime factors divide the principal ideal (l). Let ϕ1 be an idempotent of the group ringZ
l[Gal(K
1/k)] defined in the below. We shall prove λ
k
= μ
k
=0 if there is a natural numbern such that ε1
C
n
vanishes, under additional conditions concerning ramifications inK
n/k. 相似文献
15.
A graph is said to be claw-free if it does not contain an induced subgraph isomorphic to K 1,3. Let K 4 ? be the graph obtained by removing exactly one edge from K 4 and let k be an integer with k ? 2. We prove that if G is a claw-free graph of order at least 13k ? 12 and with minimum degree at least five, then G contains k vertex-disjoint copies of K 4 ? . The requirement of number five is necessary. 相似文献
16.
M. Bhaskaran 《Journal of Number Theory》1979,11(4):488-497
Let k be a number field of finite degree. The narrow genus field K of k (genus field of k in the sense of Fröhlich) is defined as the maximal extension of k which is unramified at all finite primes of k of the form kk1, where k1 is an Abelian number field. In this article, K is determined and some applications are given. The results indicate a possibility that many class field theoretic properties of normal number fields could be extended to nonnormal number fields. 相似文献
17.
Let N′(k) denote the number of coprime integral solutions x, y of y2 = x3 + k. It is shown that lim supk→∞N′(k) ≥ 12. 相似文献
18.
Attila Pethö 《Journal of Number Theory》1982,14(2):232-236
Let {ai} with a1 ≥ 2 be an infinite bounded sequence of positive integers, and d1 = 1, di = ±1 for i = 2, 3,…. Let {Qi} be another sequence defined by the recursion Q1 = 1, Qi = ai?1Qi?1k for i = 2, 3,…, where k ≥ 2 an integer. Put Ck(a) = Σi = 1∞diQi?1. In this paper we shall determine the simple continued fraction expansion for the real numbers Ck(a). 相似文献
19.
Yunhe Sheng 《代数通讯》2013,41(5):1929-1953
Let Y be an integral projective curve whose singularities are of type Ak, i.e. with only tacnodes and planar (perhaps non-ordinary) cusps. Set g:= pa(Y). Here we study the Brill - Noether theory of spanned line bundles on Y. If the singularities are bad enough, we show the existence of spanned degree d line bundles, L, with h0(Y, L) ≥ r + 1 even if the Brill - Noether number ρ(g, d, r) < 0. We apply this result to prove that genus g curves with certain singularities cannot be hyperplane section of a simple K3 surface S ? P g. 相似文献
20.
Let A be an n × n matrix with real eigenvalues λ1 ? … ? λn, and let 1 ? k < l ? n. Bounds involving trA and trA2 are introduced for λk/λl, (λk ? λl)/(λk + λl), and {kλk + (n ? l + 1)λl}2/{kλ2k + (n ? l + 1)λ2l}. Also included are conditions for λl >; 0 and for λk + λl > 0. 相似文献