共查询到20条相似文献,搜索用时 46 毫秒
1.
Friedrich Roesler 《Archiv der Mathematik》1999,73(3):193-198
Abstract. For natural numbers n we inspect all factorizations n = ab of n with a £ ba \le b in \Bbb N\Bbb N and denote by n=an bnn=a_n b_n the most quadratic one, i.e. such that bn - anb_n - a_n is minimal. Then the quotient k(n) : = an/bn\kappa (n) := a_n/b_n is a measure for the quadraticity of n. The best general estimate for k(n)\kappa (n) is of course very poor: 1/n £ k(n) £ 11/n \le \kappa (n)\le 1. But a Theorem of Hall and Tenenbaum [1, p. 29], implies(logn)-d-e £ k(n) £ (logn)-d(\log n)^{-\delta -\varepsilon } \le \kappa (n) \le (\log n)^{-\delta } on average, with d = 1 - (1+log2 2)/log2=0,08607 ?\delta = 1 - (1+\log _2 \,2)/\log 2=0,08607 \ldots and for every e > 0\varepsilon >0. Hence the natural numbers are fairly quadratic.¶k(n)\kappa (n) characterizes a specific optimal factorization of n. A quadraticity measure, which is more global with respect to the prime factorization of n, is k*(n): = ?1 £ a £ b, ab=n a/b\kappa ^*(n):= \textstyle\sum\limits \limits _{1\le a \le b, ab=n} a/b. We show k*(n) ~ \frac 12\kappa ^*(n) \sim \frac {1}{2} on average, and k*(n)=W(2\frac 12(1-e) log n/log 2n)\kappa ^*(n)=\Omega (2^{\frac {1}{2}(1-\varepsilon ) {\log}\, n/{\log} _2n})for every e > 0\varepsilon>0. 相似文献
2.
H. Crauel 《Archiv der Mathematik》2000,75(6):472-480
Let x1,..., xn be points in the d-dimensional Euclidean space Ed with || xi-xj|| £ 1\| x_{i}-x_{j}\| \le 1 for all 1 \leqq i,j \leqq n1 \leqq i,j \leqq n, where || .||\| .\| denotes the Euclidean norm. We ask for the maximum M(d,n) of \mathop?i, j=1n|| xi-xj|| 2\textstyle\mathop\sum\limits _{i,\,j=1}^{n}\| x_{i}-x_{j}\| ^{2} (see [4]). This paper deals with the case d = 2. We calculate M(2, n) and show that the value M(2, n) is attained if and only if the points are distributed as evenly as possible among the vertices of a regular triangle of edge-length 1. Moreover we give an upper bound for the value \mathop?i, j=1n|| xi-xj|| \textstyle\mathop\sum\limits _{i,\,j=1}^{n}\| x_{i}-x_{j}\| , where the points x1,...,xn are chosen under the same constraints as above. 相似文献
3.
A. I. Aptekarev J. S. Dehesa A. Martínez-Finkelshtein R. Yáñez 《Constructive Approximation》2009,30(1):93-119
Given a nontrivial Borel measure on ℝ, let p
n
be the corresponding orthonormal polynomial of degree n whose zeros are λ
j
(n), j=1,…,n. Then for each j=1,…,n,
with
defines a discrete probability distribution. The Shannon entropy of the sequence {p
n
} is consequently defined as
In the case of Chebyshev polynomials of the first and second kinds, an explicit and closed formula for
is obtained, revealing interesting connections with number theory. In addition, several results of numerical computations
exemplifying the behavior of
for other families are presented.
相似文献
4.
Let (t n ) be a sequence of nonnegative real numbers tending to ∞, such that 1≤t n+1?t n ≤α for all natural numbers n and some positive α. We prove that a strongly continuous semigroup {T(t)} t≥0, acting on a Hilbert space H, is uniformly exponentially stable if $$\sum_{n=0}^\infty\varphi\bigl(\bigl|\bigl\langle T(t_n)x, y\bigr\rangle\bigr|\bigr)<\infty, $$ for all unit vectors x, y in H. We obtain the same conclusion under the assumption that the inequality $$\sum_{n=0}^\infty\varphi\bigl(\bigl|\bigl\langle T(t_n)x, x^\ast\bigr\rangle\bigr|\bigr)<\infty, $$ is fulfilled for all unit vectors x∈X and x ?∈X ?, X being a reflexive Banach space. These results are stated for functions φ belonging to a special class of functions, such as defined in the second section of this paper. We conclude our paper with a Rolewicz’s type result in the continuous case on Hilbert spaces. 相似文献
5.
J. B. Lasserre 《TOP》2012,20(1):119-129
We consider the semi-infinite optimization problem:
f*:=minx ? X {f(x):g(x,y) £ 0, "y ? Yx},f^*:=\min_{\mathbf{x}\in\mathbf{X}} \bigl\{f(\mathbf{x}):g(\mathbf{x},\mathbf{y}) \leq 0, \forall\mathbf{y}\in\mathbf {Y}_\mathbf{x}\bigr\}, 相似文献
6.
Б. С. кашин 《Analysis Mathematica》1976,2(4):249-266
?n = 1¥ cn jn (x)\sum\limits_{n = 1}^\infty {c_n \varphi _n } (x) 相似文献
7.
Zhenlai Han Tongxing Li Shurong Sun Weisong Chen 《Journal of Applied Mathematics and Computing》2012,40(1-2):143-152
In this paper, by employing Riccati transformation technique, some new sufficient conditions for the oscillation criteria are given for the second order quasilinear neutral delay differential equations with delayed argument in the form $$\bigl(r(t)\bigl|z'(t)\bigr|^{\alpha-1}z'(t)\bigr)'+q(t)f\bigl(x\bigl(\sigma(t)\bigr)\bigr)=0,\quad t\geq t_0,$$ where z(t)=x(t)?p(t)x(??(t)), 0??p(t)??p<1, lim t???? p(t)=p 1<1, q(t)>0, ??>0. Two examples are considered to illustrate the main results. 相似文献
8.
D. Wolke 《Archiv der Mathematik》2000,74(4):276-281
On the assumption of the truth of the Riemann hypothesis for the Riemann zeta function we construct a class of modified von-Mangoldt functions with slightly better mean value properties than the well known function L\Lambda . For every e ? (0,1/2)\varepsilon \in (0,1/2) there is a [(L)\tilde] : \Bbb N ? \Bbb C\tilde {\Lambda} : \Bbb N \to \Bbb C such that¶ i) [(L)\tilde] (n) = L (n) (1 + O(n-1/4 logn))\tilde {\Lambda} (n) = \Lambda (n) (1 + O(n^{-1/4\,} \log n)) and¶ii) ?n \leqq x [(L)\tilde] (n) (1- [(n)/(x)]) = [(x)/2] + O(x1/4+e) (x \geqq 2).\sum \limits_{n \leqq x} \tilde {\Lambda} (n) \left(1- {{n}\over{x}}\right) = {{x}\over{2}} + O(x^{1/4+\varepsilon }) (x \geqq 2).¶Unfortunately, this does not lead to an improved error term estimation for the unweighted sum ?n \leqq x [(L)\tilde] (n)\sum \limits_{n \leqq x} \tilde {\Lambda} (n), which would be of importance for the distance between consecutive primes. 相似文献
9.
This paper is concerned with the oscillatory properties of even order advanced type dynamic equation with mixed nonlinearities of the form $$\bigl[r(t)\varPhi_\alpha\bigl(x^{\Delta^{n-1}}(t) \bigr) \bigr]^\Delta+ p(t)\varPhi_\alpha\bigl(x\bigl(\delta(t)\bigr) \bigr) +\sum_{i=1}^kp_i(t) \varPhi_{\alpha_i} \bigl(x\bigl(\delta(t)\bigr) \bigr)=0 $$ on an arbitrary time scale $\mathbb{T}$ , where Φ ?(u)=|u|??1 u. We present some new oscillation criteria for the equation by introducing parameter functions, establishing a new lemma, using a Hardy-Littlewood-Pólya inequality and an arithmetic-geometric mean inequality and developing a generalized Riccati technique. Our results extend and supplement some known results in the literature. Several examples are given to illustrate our main results. 相似文献
10.
We study the asymptotic behavior as n→∞ of the sequence
|