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1.
Let nsym2fn_{\mathrm{sym}^{2}f} be the greatest integer such that lsym2f(n) 3 0\lambda_{\mathrm{sym}^{2}f}(n)\ge0 for all n < nsym2fnn,N)=1, where lsym2f(n)\lambda_{\mathrm{sym}^{2}f}(n) is the nth coefficient of the Dirichlet series representation of the symmetric square L-function L(s,sym2
f) associated to a primitive form f of level N and of weight k. In this paper, we establish the subconvexity bound: nsym2f << (k2N2)40/113n_{\mathrm{sym}^{2}f}\ll(k^{2}N^{2})^{40/113} where the implied constant is absolute. 相似文献
2.
3.
O. M. Fomenko 《Journal of Mathematical Sciences》2007,143(3):3174-3181
Let f(z) be a holomorphic Hecke eigencuspform of even weight k with respect to SL(2, Z) and let L(s, sym
2 f) = ∑
n=1
∞
cnn−s, Re s > 1, be the symmetric square L-function associated with f.
Represent the Riesz mean (ρ ≥ 0)
as the sum of the “residue function” Γ(ρ+1)−1 Ł(0, sym2f)xρ and the “error term”
.
Using the Voronoi formula for Δρ(x;sym
2f), obtained earlier (see Zap. Nauchn. Semin. POMI. 314, 247–256 (2004)), the integral
is estimated. In this way, an asymptotics for 0 < ρ ≤ 1 and an upper bound for ρ = 0 are obtained. Also the existence of
a limiting distribution for the function
, and, as a corollary, for the function
, is established. Bibliography: 12 titles.
Dedicated to the 100th anniversary of G. M. Goluzin’s birthday
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 337, 2006, pp. 274–286. 相似文献
4.
Let ℂ[−1,1] be the space of continuous functions on [−,1], and denote by Δ2 the set of convex functions f ∈ ℂ[−,1]. Also, let E
n
(f) and E
n
(2) (f) denote the degrees of best unconstrained and convex approximation of f ∈ Δ2 by algebraic polynomials of degree < n, respectively. Clearly, En (f) ≦ E
n
(2) (f), and Lorentz and Zeller proved that the inverse inequality E
n
(2) (f) ≦ cE
n
(f) is invalid even with the constant c = c(f) which depends on the function f ∈ Δ2.
In this paper we prove, for every α > 0 and function f ∈ Δ2, that
where c(α) is a constant depending only on α. Validity of similar results for the class of piecewise convex functions having s convexity changes inside (−1,1) is also investigated. It turns out that there are substantial differences between the cases
s≦ 1 and s ≧ 2.
Dedicated to Jóska Szabados on his 70th birthday 相似文献
5.
We prove inequalities about the quermassintegralsV
k
(K) of a convex bodyK in ℝ
n
(here,V
k
(K) is the mixed volumeV((K, k), (B
n
,n − k)) whereB
n
is the Euclidean unit ball). (i) The inequality
holds for every pair of convex bodiesK andL in ℝ
n
if and only ifk=2 ork=1. (ii) Let 0≤k≤p≤n. Then, for everyp-dimensional subspaceE of ℝ
n
,
whereP
E
K denotes the orthogonal projection ofK ontoE. The proof is based on a sharp upper estimate for the volume ratio |K|/|L| in terms ofV
n−k
(K)/V
n−k
(L), wheneverL andK are two convex bodies in ℝ
n
such thatK ⊆L. 相似文献
6.
Michael Bildhauer 《manuscripta mathematica》2003,110(3):325-342
Suppose that f: ℝ
nN
→ℝ is a strictly convex energy density of linear growth, f(Z)=g(|Z|2) if N>1. If f satisfies an ellipticity condition of the form
then, following [Bi3], there exists a unique (up to a constant) solution of the variational problem
provided that the given boundary data u
0
W
1
1
(ω;ℝ
N
) are additionally assumed to be of class L
∞(ω;ℝ
N
). Moreover, if μ<3, then the boundedness of u
0 yields local C
1,α-regularity (and uniqueness up to a constant) of generalized minimizers of the problem
In our paper we show that the restriction u
0L
∞(ω;ℝ
N
) is superfluous in the two dimensional case n=2, hence we may prescribe boundary values from the energy class W
1
1
(ω;ℝ
N
) and still obtain the above results.
Received: 12 February 2002 / Revised version: 7 October 2002 Published online: 14 February 2003
Mathematics Subject Classification (2000): 49N60, 49N15, 49M29, 35J 相似文献
7.
Leonid Gurvits 《Discrete and Computational Geometry》2009,41(4):533-555
Let K=(K
1,…,K
n
) be an n-tuple of convex compact subsets in the Euclidean space R
n
, and let V(⋅) be the Euclidean volume in R
n
. The Minkowski polynomial V
K
is defined as V
K
(λ
1,…,λ
n
)=V(λ
1
K
1+⋅⋅⋅+λ
n
K
n
) and the mixed volume V(K
1,…,K
n
) as
Our main result is a poly-time algorithm which approximates V(K
1,…,K
n
) with multiplicative error e
n
and with better rates if the affine dimensions of most of the sets K
i
are small. Our approach is based on a particular approximation of log (V(K
1,…,K
n
)) by a solution of some convex minimization problem. We prove the mixed volume analogues of the Van der Waerden and Schrijver–Valiant
conjectures on the permanent. These results, interesting on their own, allow us to justify the abovementioned approximation
by a convex minimization, which is solved using the ellipsoid method and a randomized poly-time time algorithm for the approximation
of the volume of a convex set. 相似文献
8.
O. M. Fomenko 《Journal of Mathematical Sciences》2006,133(6):1733-1748
Let Sk(Γ) be the space of holomorphic Γ-cusp forms f(z) of even weight k ≥ 12 for Γ = SL(2, ℤ), and let Sk(Γ)+ be the set of all Hecke eigenforms from this space with the first Fourier coefficient af(1) = 1. For f ∈ Sk(Γ)+, consider the Hecke L-function L(s, f). Let
It is proved that for large K,
where ε > 0 is arbitrary. For f ∈ Sk(Γ)+, let L(s, sym
2 f) denote the symmetric square L-function. It is proved that as k → ∞ the frequence
converges to a distribution function G(x) at every point of continuity of the latter, and for the corresponding characteristic
function an explicit expression is obtained. Bibliography: 17 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 314, 2004, pp. 221–246. 相似文献
9.
Let K, D be centrally symmetric convex bodies in
Let k < n and let dk(K, D) be the smallest Banach–Mazur distance between k-dimensional sections of K and D. Define
where the supremum is taken over all n-dimensional convex symmetric bodies K, D. We prove that, for any k < n,
where
means that
for some absolute constants C, a > 0. 相似文献
10.
Javier Cilleruelo Florian Luca Adolfo Quirós Igor E. Shparlinski 《Monatshefte für Mathematik》2010,91(5):215-223
Let
f(X) ? \mathbb Z[X]{f(X) \in \mathbb {Z}[X]} be an irreducible polynomial of degree D ≥ 2 and let N be a sufficiently large positive integer. We estimate the number of positive integers n ≤ N such that the product
|
F(n) = ?k = 1n f(k)F(n) = \prod\limits_{k =1}^n f(k) 相似文献
11.
In Part I of the paper, we have proved that, for every α > 0 and a continuous function f, which is either convex (s = 0) or changes convexity at a finite collection Y
s
= {y
i
}
s
i=1 of points y
i
∈ (-1, 1),
|