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1.
Giorgio Follo 《Ricerche di matematica》2007,56(1):89-105
Abstract We study entropy numbers of sets and generalize some results shown by B. Carl and I. Stephani in [2]. It is well known that
if Ω is the closed unit ball in the Euclidean space
, then for every
,
We extend estimates of this type to more general subsets of metric spaces. Moreover we study relations between the asymptotic
behaviour of the entropy numbers of bounded subsets of
and their box dimension.
Mathematics Subject Classification (2000): 28A80, 28A78
Keywords: Fractals, Entropy numbers 相似文献
2.
Horst Alzer 《Mediterranean Journal of Mathematics》2008,5(4):395-413
We present several sharp inequalities for the volume of the unit ball in ,
. One of our theorems states that the double-inequality
holds for all n ≥ 2 with the best possible constants
This refines and complements a result of Klain and Rota.
相似文献
3.
Cubic elliptic functions 总被引：1，自引：1，他引：0
Shaun Cooper 《The Ramanujan Journal》2006,11(3):355-397
The function
occurs in one of Ramanujan’s inversion formulas for elliptic integrals. In this article, a common generalization of the cubic
elliptic functions
is given. The function g1 is the derivative of Ramanujan’s function Φ (after rescaling), and χ3(n) = 0, 1 or −1 according as n≡ 0, 1 or 2 (mod 3), respectively, and |q| < 1. Many properties of the common generalization, as well as the functions g1 and g2, are proved.
2000 Mathematics Subject Classification Primary—33E05; Secondary—11F11, 11F27 相似文献
4.
P. Ding 《Journal of Mathematical Sciences》2006,137(2):4645-4653
Let p be a prime number, n be a positive integer, and ƒ(x) = axk + bx. We put
where e(t) = exp(2πit). This special exponential sum has been widely studied in connection with Waring’s problem. We write n in the form n
= Qk + r, where 0 ≤ r ≤ k − 1 and Q ≥ 0. Let α = ord
p(k), β = ord
p(k − 1), and θ = ord
p(b). We define
and J = [ζ]. Moreover, we denote V = min(Q, J). Improving the preceding result, we establish the theorem. Theorem. Let k ≥ 2 and n ≥ 2. If p > 2, then
. An example showing that this result is best possible is given. Bibliography: 15 titles.
Published in Zapiski Nauchnykh Seminarov POMI, Vol. 322, 2005, pp. 63–75. 相似文献
5.
Anastasios D. Simalarides 《Periodica Mathematica Hungarica》2000,40(1):71-75
Let be a primitive character mod k, k > 2. In [1], the following elementary estimate
was given, where
by definition. In the present note we sharpen this estimate by a factor 3/4 in the case of an even primitive character , by improving upon the proof given in [1] in a way which does not alter the elementary character of the method. 相似文献
6.
In the article [2] Ennio De Giorgi conjectured that any compact n-dimensional regular submanifold M of ℝ
n+m
,moving by the gradient of the functional
where ηM is the square of the distance function from the submanifold M and Hn is the n-dimensional Hausdorff measure in ℝ n+m, does not develop singularities in finite time provided k is large enough, depending on the dimension n.
We prove this conjecture by means of the analysis of the geometric properties of the high derivatives of the distance function
from a submanifold of the Euclidean space. In particular, we show some relations with the second fundamental form and its
covariant derivatives of independent interest. 相似文献
7.
M. M. Kabardov 《Vestnik St. Petersburg University: Mathematics》2008,41(4):355-359
A method for inverting the Laplace transform based on expanding the original in Laguerre polynomials as $ f(t) = \sum\limits_{k = 0}^\infty {a_k L_k (bt)} $ is suggested. The representation of the Laguerre series by a linear-fractional mapping is reduced to a power series of the form $ \sum\nolimits_{k = 0}^\infty {a_k z^k } A method for inverting the Laplace transform based on expanding the original in Laguerre polynomials as
is suggested. The representation of the Laguerre series by a linear-fractional mapping is reduced to a power series of the
form , which is summed by the well-known Euler-Knopp method. The summation parameter is chosen in the complex plane so that the
new expansion
of the original corresponding to the Euler-Knopp transformation converge at a maximum rate. On the basis of geometric representations,
the influence of the requirement that the Euler-Knopp transformation must be regular on the choice of the summation parameter
is discussed. Numerical experiments are performed, which demonstrate the high efficiency of the method of choosing a complex
parameter suggested in this paper.
Original Russian Text ? M.M. Kabardov, 2008, published in Vestnik Sankt-Peterburgskogo Universiteta. Seriya 1. Matematika,
Mekhanika, Astronomiya, 2008, No. 4, pp. 84–89. 相似文献
8.
M. I. Gil’ 《Acta Appl Math》2007,99(2):117-159
Let f be an entire function. Denote by z
1(f),z
2(f),… the zeros of f with their multiplicities. In the paper, estimates for the sums
and for the counting function of the zeros of f are established. If f is of finite order ρ(f), we derive bounds for the series
as well as relations between the series
and the traces of certain matrices. The contents of the paper is closely connected with the following well-known results:
the Hadamard theorem on the convergence exponent of the zeros of an entire function, the Jensen inequality for the counting
function, the Cauchy theorem on the comparison of the zeros of polynomials, Ostrowski’s inequalities for the real and imaginary
parts of the zeros of polynomials and the Cartwright–Levinson theorem. The suggested approach is based on the recent development
of the spectral theory of linear operators.
This research was supported by the Kamea fund of the Israel. 相似文献
9.
E. G. Kwon 《Integral Equations and Operator Theory》2009,64(2):251-260
We characterize the composition operators mapping Blochs boundedly into the weighted Bergman spaces of logarithmic weight.
For 0 < p < ∞, 1 < α < ∞, let Ap, log α denote the space of holomorphic functions F in the unit disc D for which
and let Ap, log ασ denote the class of holomorphic self maps f of D for which
Then for the Bloch pullback operator Cf, the following are equivalent:
This work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion
Fund) (KRF-2007-313-C00026). 相似文献
| (1) | Cf maps Bloch space boundedly into A2p, log α |
| (2) | |
| (3) | . |
10.
A. S. Zhuk 《Journal of Mathematical Sciences》2008,150(3):2034-2044
Let M be either the space of 2π-periodic functions Lp, where 1 ≤ p < ∞, or C; let ωr(f, h) be the continuity modulus of order r of the function f, and let
, where
, be the generalized Jackson-Vallée-Poussin integral. Denote
. The paper studies the quantity Km(f − Dn,r,l(f)). The general results obtained are applicable to other approximation methods. Bibliography: 11 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 350, 2007, pp. 52–69. 相似文献
11.
O. M. Fomenko 《Journal of Mathematical Sciences》2003,118(1):4910-4917
Let
be the Hecke eigenbasis of the space
of
-cusp forms of weight 2. Let p be a prime. Let
be the Hecke L-series of form
. The following statements are proved:
and
We also give a correct proof of a previous author's theorem on automorphic L-functions. Bibliography: 12 titles. 相似文献
12.
We determine exact values for the k-error linear complexity L
k
over the finite field
of the Legendre sequence
of period p and the Sidelnikov sequence
of period p
m
− 1. The results are
for 1 ≤ k ≤ (p
m
− 3)/2 and
for k≥ (p
m
− 1)/2. In particular, we prove
相似文献
13.
On sums of a prime and four prime squares in short intervals 总被引：1，自引：1，他引：0
In this paper, we prove that each sufficiently large integer N ≠1（mod 3） can be written as N=p＋p1^2＋p2^2＋p3^2＋p4^2, with
｜p-N/5｜≤U,｜pj-√N/5｜≤U,j=1,2,3,4,
where U=N^2/20＋c and p,pj are primes. 相似文献
｜p-N/5｜≤U,｜pj-√N/5｜≤U,j=1,2,3,4,
where U=N^2/20＋c and p,pj are primes. 相似文献
14.
J.-P. Allouche 《The Ramanujan Journal》2007,14(1):39-42
We answer a question of Berndt and Bowman, asking whether it is possible to deduce the value of the Ramanujan integral I from the value of the Ramanujan integral J, where
and
We also show that the second integral can be deduced from a classical expression of the ψ function due to Dirichlet and from
the classical equality
which is a simple consequence of Frullani-Cauchy’s theorem.
2000 Mathematics Subject ClassificationPrimary—33B15
Partially supported by MENESR, ACI NIM 154 Numération. 相似文献
15.
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
16.
V. M. Babich 《Journal of Mathematical Sciences》2006,132(1):2-10
The paper is devoted to a detailed consideration of an ansatz known from the seventies:
where
Here the Dp are parabolic-cylinder functions. Analytic expressions in the first approximation for the wave field in the penumbra of the
wave reflected by an impedance or transparent cone are obtained. Bibliography: 11 titles.
Dedicated to P. V. Krauklis on the occasion of his seventieth birthday
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 308, 2004, pp. 9–22. 相似文献
17.
V. A. Bykovskii 《Journal of Mathematical Sciences》1998,89(1):915-932
A trace formula expressing the mean values of the form (k=2,3,...)
via certain arithmetic means on the group Г0(N1) is proved. Here the sum is taken over a normalized orthogonal basis in the space of holomorphic cusp forms of weight 2k
with respect to Г0(N1). By H
f
(x)
(s) we denote the Hecke series of the form f, twisted with the primitive character χ (mod N2), and λf(d), (d, N1N2)=1, are the eigenvalues of the Hecke operators
. The trace formula is used for obtaining the estimate
for the newform f for all ε>0, l=0,1,2,.... This improves the known result (Duke-Friedlander-Iwaniec, 1993) with upper bound
(1+|t|)2N
2
1/2−1/22+ε
on the right-hand side. As a corollary, we obtain the estimate
for the Fourier coefficients of holomorphic cusp forms of weight k+1/2, which improves Iwaniec' result (1987) with exponent
1/4–1/28+ε. Bibliography: 25 titles.
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 14–36. 相似文献
18.
Florian Luca 《Czechoslovak Mathematical Journal》2008,58(1):51-59
For a positive integer n we write φ(n) for the Euler function of n. In this note, we show that if b > 1 is a fixed positive integer, then the equation
has only finitely many positive integer solutions (x, y, m, n).
Dedicated to William D. Banks on his birthday. 相似文献
19.
O. M. Fomenko 《Journal of Mathematical Sciences》2004,122(6):3685-3698
Let h(d) be the class number of properly equivalent primitive binary quadratic forms ax2+bxy+cy2 with discriminant d=b2-4ac. The behavior of h(5p2), where p runs over primes, is studied. It is easy to show that there are few discriminants of the form 5p2 with large class numbers. In fact, one has the estimate
where is an arbitrary constant number in (0;1/2). Assume that (x) is a positive function monotonically increasing for x and (x). If
, then (assuming the validity of the extended Riemann hypothesis for certain Dedekind zeta-functions) it is proved that
It is also proved that for an infinite set of p with
one has the inequality
where log
k
p is the k-fold iterated logarithm (k is an arbitrary integer, k3). Results on mean values of h(5p
2
) are also obtained. Similar facts are true for the residual indices of an integer a2 modulo p:
where o(a,p) is the order of a modulo p. Bibliography: 13 titles. 相似文献
| x^{1 - \delta } \} \ll x^{2\delta } ,$$ " align="middle" vspace="20%" border="0"> |
| \alpha (x)} \right\} \asymp \frac{{\pi (x)}}{{\alpha (x)}}.$$ " align="middle" vspace="20%" border="0"> |
20.
Vincent Borrelli Fabiano Brito Olga Gil-Medrano 《Annals of Global Analysis and Geometry》2003,23(2):129-140
We construct a one-parameter family of unit smooth vector fieldsglobally defined on the sphere
2k+1 for k 2, with energyconverging to the energy of the unit radial vector field, which isdefined on the complementary of two antipodal points. So we prove thatthe infimum of the energy of globally defined unit smooth vector fieldsis
相似文献