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1.
LetK be a number field. Denote byV
3 a split Del Pezzo surface of degree six overK and by ω its canonical divisor. Denote byW
3 the open complement of the exceptional lines inV
3. LetN
W
s(−ω, X) be the number ofK-rational points onW
3 whose anticanonical heightH
−ω is bounded byX. Manin has conjectured that asymptoticallyN
W
3(−ω, X) tends tocX(logX)3, wherec is a constant depending only on the number field and on the normalization of the height. Our goal is to prove the following
theorem: For each number fieldK there exists a constantc
K such thatN
W
3(−ω, X)≤cKX(logX)3+2r
, wherer is the rank of the group of units ofO
K. The constantc
K is far from being optimal. However, ifK is a purely imaginary quadratic field, this proves an upper bound with a correct power of logX. The proof of Manin's conjecture for arbitrary number fields and a precise treatment of the constants would require a more
sophisticated setting, like the one used by [Peyre] to prove Manin's conjecture and to compute the correct asymptotic constant
(in some normalization) in the caseK=ℚ. Up to now the best result for arbitraryK goes back, as far as we know, to [Manin-Tschinkel], who gives an upper boundN
W
3(−ω,X)≤cXl+ε.
The author would like to express his gratitude to Daniel Coray and Per Salberger for their generous and indispensable support. 相似文献
2.
In this paper, we study 2-(v, k, 1) designs with automorphisms of prime orderp, having the maximum possible number of fixed points. We prove an upper bound on the number of fixed points, and we study
the structure of designs in which this bound is met with equality (such a design is called ap-MFP(v, k)). Several characterizations and asymptotic existence results forp-MFP(v, k) are obtained. For (p, k)=(3,3), (5,5), (2,3) and (3,4), necessary and sufficient conditions onv are obtained for the existence of ap-MFP(v, k). Further, for 3≤k≤5 and for any primep≡1 modk(k−1), we establish necessary and sufficient conditions onv for the existence of ap-MFP(v, k). 相似文献
3.
Let p be an odd prime, c be an integer with (c, p) = 1, and let N be a positive integer with N ≤ p − 1. Denote by r(N, c; p) the number of integers a satisfying 1 ≤ a ≤ N and 2 ∤ a + ā, where ā is an integer with 1 ≤ ā ≤ p − 1, aā ≡ c (mod p). It is well known that r(N, c; p) = 1/2N + O(p
1/2log2
p). The main purpose of this paper is to give an asymptotic formula for Σ
c=1
p−1(r(N, c; p) − 1/2N)2. 相似文献
4.
Let P(G, λ) be the chromatic polynomial of a graph G. A graph G is chromatically unique if for any graph H, P(H, λ) = P(G, λ) implies H is isomorphic to G. Liu et al. [Liu, R. Y., Zhao, H. X., Ye, C. F.: A complete solution to a conjecture on chromatic uniqueness of complete
tripartite graphs. Discrete Math., 289, 175–179 (2004)], and Lau and Peng [Lau, G. C., Peng, Y. H.: Chromatic uniqueness of certain complete t-partite graphs. Ars Comb., 92, 353–376 (2009)] show that K(p − k, p − i, p) for i = 0, 1 are chromatically unique if p ≥ k + 2 ≥ 4. In this paper, we show that if 2 ≤ i ≤ 4, the complete tripartite graph K(p − k, p − i, p) is chromatically unique for integers k ≥ i and p ≥ k
2/4 + i + 1. 相似文献
5.
Let p be an odd prime and a be an integer coprime with p. Denote by N(a, p) the number of pairs of integers b, c with bc ≡ a(mod p), 1 ≤ b, c < p and with b, c having different parity. The main purpose of this paper is to study the mean square value problem of (N(a, p) − 1/2 (p−1)) over interval (N, N + M] with M, N positive integers by using the analytic methods, and finally by obtaining a sharp asymptotic formula. 相似文献
6.
Zhou Songping 《分析论及其应用》1989,5(1):11-14
In 1980, M. Hasson raised a conjecture as follows: Let N≥1, then there exists a function f0(x)∈C
[−1,1]
2N
, for N+1≤k≤2N, such that p
n
(k)
(f0,1)→f
0
(k)
(1), n→∞, where pn(f,x) is the algebraic polynomial of best approximation of degree ≤n to f(x). In this paper, a, positive answer to this conjecture
is given. 相似文献
7.
We obtain asymptotic representations as t ↑ ω, ω ≤ + ∞, for all possible types of P
ω(Y
0, λ
0)-solutions (where Y
0 is zero or ±∞ and −∞ ≤ λ0 ≤ +∞) of nonlinear differential equations y
(n) = α
0
p(t)φ(y), where α
0 ∈ {−1, 1}, p: [a, ω[→]0,+∞[ is a continuous function, and φ is a continuous regularly varying function in a one-sided neighborhood of Y
0. 相似文献
8.
Marcin Dziubiński 《International Journal of Game Theory》2011,40(2):231-262
Two players are endowed with resources for setting up N locations on K line segments of identical length, with N > K ≥ 1. The players alternately choose these locations (possibly in batches of more than one in each round) in order to secure
the area closer to their locations than that of their rival’s. The player with the highest secured area wins the game and
otherwise the game ends in a tie. Earlier research has shown that, if an analogous game is played on disjoint circles, the
second mover advantage is in place only if K = 1, while for K > 1 both players have a tying strategy. It was also shown that these results hold for line segments of identical length when
rules of the game additionally require players to take exactly one location in the first round. In this paper we show that
the second mover advantage is still in place for K ≥ 1 and 2K − 1 ≤ N, even if the additional restriction is dropped, while K ≤ N < 2K − 1 results in the first mover advantage. Our results allow us to draw conclusions about a natural variant of the game, where
the resource mobility constraint is more stringent so that in each round each player chooses a single location and we show
that the second mover advantage re-appears for K ≤ N < 2K − 1 if K is an even number. In all the cases the losing player has a strategy guaranteeing him arbitrarily small loss. 相似文献
9.
S. Staněk 《Ukrainian Mathematical Journal》2008,60(2):277-298
We present existence principles for the nonlocal boundary-value problem (φ(u(p−1)))′=g(t,u,...,u(p−1), αk(u)=0, 1≤k≤p−1, where p ≥ 2, π: ℝ → ℝ is an increasing and odd homeomorphism, g is a Carathéodory function that is either regular or has singularities in its space variables, and α
k: C
p−1[0, T] → ℝ is a continuous functional. An application of the existence principles to singular Sturm-Liouville problems (−1)n(φ(u(2n−)))′=f(t,u,...,u(2n−1)), u(2k)(0)=0, αku(2k)(T)+bku(2k=1)(T)=0, 0≤k≤n−1, is given.
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 2, pp. 240–259, February, 2008. 相似文献
10.
József Beck 《Periodica Mathematica Hungarica》2010,60(2):137-242
We prove that the “quadratic irrational rotation” exhibits a central limit theorem. More precisely, let α be an arbitrary real root of a quadratic equation with integer coefficients; say, α = $
\sqrt 2
$
\sqrt 2
. Given any rational number 0 < x < 1 (say, x = 1/2) and any positive integer n, we count the number of elements of the sequence α, 2α, 3α, …, nα modulo 1 that fall into the subinterval [0, x]. We prove that this counting number satisfies a central limit theorem in the following sense. First, we subtract the “expected
number” nx from the counting number, and study the typical fluctuation of this difference as n runs in a long interval 1 ≤ n ≤ N. Depending on α and x, we may need an extra additive correction of constant times logarithm of N; furthermore, what we always need is a multiplicative correction: division by (another) constant times square root of logarithm
of N. If N is large, the distribution of this renormalized counting number, as n runs in 1 ≤ n ≤ N, is very close to the standard normal distribution (bell shaped curve), and the corresponding error term tends to zero as
N tends to infinity. This is the main result of the paper (see Theorem 1.1). The proof is rather complicated and long; it has
many interesting detours and byproducts. For example, the exact determination of the key constant factors (in the additive
and multiplicative norming), which depend on α and x, requires surprisingly deep algebraic tools such as Dedeking sums, the class number of quadratic fields, and generalized
class number formulas. The crucial property of a quadratic irrational is the periodicity of its continued fraction. Periodicity
means self-similarity, which leads us to Markov chains: our basic probabilistic tool to prove the central limit theorem. We
also use a lot of Fourier analysis. Finally, I just mention one byproduct of this research: we solve an old problem of Hardy
and Littlewood on diophantine sums. 相似文献
11.
E. J. Gómez Ayala 《manuscripta mathematica》1994,83(1):199-213
LetK be an imaginary quadratic field with discriminantd
K
<−4,d
K
≡2, 3 mod 4, andp a prime number,p≡1 mod 8,p split inK; let Ω
p
be the ring class field overK with conductorp andK(p) the ray class field overK with conductorp. An explicit normal basis is constructed for the ring of integers of the unique quadratic extension of Ω
p
contained inK(p) over the ring of integers of Ω
p
. This uses certain classical modular units considered by Deuring and Hecke.
相似文献
12.
A partition of an integer n is a representation n=a
1+a
2+⋅⋅⋅+a
k
, with integer parts 1≤a
1≤a
2≤…≤a
k
. For any fixed positive integer p, a p-succession in a partition is defined to be a pair of adjacent parts such that a
i+1−a
i
=p. We find generating functions for the number of partitions of n with no p-successions, as well as for the total number of such successions taken over all partitions of n. In the process, various interesting partition identities are derived. In addition, the Hardy-Ramanujan asymptotic formula
for the number of partitions is used to obtain an asymptotic estimate for the average number of p-successions in the partitions of n.
This material is based upon work supported by the National Research Foundation under grant number 2053740. 相似文献
13.
Rolf Kultze 《manuscripta mathematica》1990,69(1):31-42
LetX be a connected, locally finite spectrum and letk(n) (n>-1) denote the (−1)-connected cover of then-th MoravaK-Theory associated to the primep.k(n) is aBP-module spectrum with π*(k(n)) ≅ ℤ
p
[υ
n
] where |v
n
| = 2(p
n
-1). We prove the following splitting theorem: Thek(n)
*-torsion ofk(n)
* (X) is already annihilated byv
n
e
(e≥1) if and only ifk(n)ΛX is homotopy equivalent to a wedge of spectrak(n) and
r
k(n) (0≤r≤e-1) where
r
k(n) denotes ther-th Postnikov factor ofk(n). Moreover we investigate splitting conditions for
r
k(n)ΛX. 相似文献
14.
Harish Seshadri 《Proceedings Mathematical Sciences》2009,119(2):197-201
Using elementary comparison geometry, we prove: Let (M, g) be a simply-connected complete Riemannian manifold of dimension ≥ 3. Suppose that the sectional curvature K satisfies −1 − s(r) ≤ K ≤ −1, where r denotes distance to a fixed point in M. If lim
r → ∞ e2r
s(r) = 0, then (M, g) has to be isometric to ℍ
n
.
The same proof also yields that if K satisfies −s(r) ≤ K ≤ 0 where lim
r → ∞
r
2
s(r) = 0, then (M, g) is isometric to ℝ
n
, a result due to Greene and Wu.
Our second result is a local one: Let (M, g) be any Riemannian manifold. For a ∈ ℝ, if K ≤ a on a geodesic ball B
p
(R) in M and K = a on ∂B
p
(R), then K = a on B
p
(R). 相似文献
15.
L. M. Camacho E. M. Cañete J. R. Gómez B. A. Omirov 《Siberian Mathematical Journal》2011,52(5):840-853
The n-dimensional p-filiform Leibniz algebras of maximum length have already been studied with 0 ≤ p ≤ 2. For Lie algebras whose nilindex is equal to n−2 there is only one characteristic sequence, (n − 2, 1, 1), while in Leibniz theory we obtain the two possibilities: (n − 2, 1, 1) and (n − 2, 2). The first case (the 2-filiform case) is already known. The present paper deals with the second case, i.e., quasi-filiform
non-Lie-Leibniz algebras of maximum length. Therefore this work completes the study of the maximum length of the Leibniz algebras
with nilindex n − p with 0 ≤ p ≤ 2. 相似文献
16.
A Dirichlet series with multiplicative coefficients has an Euler product representation. In this paper we consider the special
case where these coefficients are derived from the numbers of representations of an integer by an integral quadratic form.
At first we suppose this quadratic form to be positive definite. In general the representation numbers are not multiplicative.
Instead we consider the average number of representations over all classes in the genus of the quadratic form. And we consider
only representations of integers of the form tk
2 with t square-free. If we divide the average representation number for these integers by a suitable factor, we do get a multiplicative
function. Using results from Siegel (Ann. Math. 36:527–606, 1935), we derive a uniform expression for the Euler product expansion of the corresponding Dirichlet series. As a special case,
we consider the standard quadratic form in n variables corresponding to the identity matrix. Here we use results from Shimura (Am. J. Math. 124:1059–1081, 2002). For 2≤n≤8, the genus of this particular quadratic form contains only one class, and this leads to a rather simple expression for
the Dirichlet series, where the coefficients are just the number of representations of a square as the sum of n squares. Finally we consider the indefinite case, where we can get results similar to the definite case. 相似文献
17.
A. M. Samoilenko 《Ukrainian Mathematical Journal》1998,50(10):1559-1585
We describe an asymptotic method for the integration of m-frequency oscillation systems of order 2n, analyze averaged equations in nonresonance and resonance cases, prove a theorem
on the preservation of smooth p-dimensionai invariant tori under perturbation for any 0 ≤ p ≤ n, and indicate forms of decomposable m-frequency oscillation systems.
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 10, pp. 1366–1387, October, 1998. 相似文献
18.
This paper continues the search, started in [10], for relatives of the ham sandwich theorem. We prove among other results,
the following implications {fx21-1} whereK(n, k) is an important instance of the Knaster’s conjecture so thatK(n, n − 1) reduces to the Borsuk-Ulam theorem,B(n, k) is a R. Rado type statement about (k + 1) measures inR
n whereB(n, n − 1) turns out to be the ham sandwich theorem andC(n, k) is a topological statement, established in this paper in the caseC(n, n − 2),n = 3 orn ≥ 5. 相似文献
19.
The asymptotics for the number of representations ofN asN→∞ is expressed as the sum of a number havingk prime divisors and a product of two natural numbers. The asymptotics is found fork≤(2−ε) ln lnN and (2+ε) ln lnN≤k≤b ln lnN, whereε>0. The results obtained are uniform with respect tok.
Translated fromMatematicheskie Zametki, Vol. 59, No. 4, pp. 585–602, April, 1996.
This research was partially supported by the Russian Foundation for Basic Research under grant No. 93-01-00260. 相似文献
20.
Hyunsuk Moon 《The Ramanujan Journal》2008,16(1):73-81
Let F(z)=∑
n=1∞
A(n)q
n
denote the unique weight 6 normalized cuspidal eigenform on Γ0(4). We prove that A(p)≡0,2,−1(mod 11) when p≠11 is a prime. We then use this congruence to give an application to the number of representations of an integer by quadratic
form of level 4.
相似文献