共查询到20条相似文献,搜索用时 117 毫秒
1.
We study the geometry of a class of group extensions, containing permutational wreath products, which we call “permutational extensions”. We construct for all k∈ℕ a torsion group K k with growth function vKk(n) ~ exp(n1-(1-a)k), 23-3/a+22-2/a+21-1/a=2,v_{K_k}(n)\sim\exp(n^{1-(1-\alpha)^k}),\qquad 2^{3-3/\alpha}+2^{2-2/\alpha}+2^{1-1/\alpha}=2, 相似文献
2.
We consider a class of unconstrained nonsmooth convex optimization problems, in which the objective function is the sum of
a convex smooth function on an open subset of matrices and a separable convex function on a set of matrices. This problem
includes the covariance selection problem that can be expressed as an ℓ
1-penalized maximum likelihood estimation problem. In this paper, we propose a block coordinate gradient descent method (abbreviated
as BCGD) for solving this class of nonsmooth separable problems with the coordinate block chosen by a Gauss-Seidel rule. The
method is simple, highly parallelizable, and suited for large-scale problems. We establish global convergence and, under a
local Lipschizian error bound assumption, linear rate of convergence for this method. For the covariance selection problem,
the method can terminate in O(n3/e){O(n^3/\epsilon)} iterations with an e{\epsilon}-optimal solution. We compare the performance of the BCGD method with the first-order methods proposed by Lu (SIAM J Optim
19:1807–1827, 2009; SIAM J Matrix Anal Appl 31:2000–2016, 2010) for solving the covariance selection problem on randomly generated instances. Our numerical experience suggests that the
BCGD method can be efficient for large-scale covariance selection problems with constraints. 相似文献
3.
Shiyan Hu 《Journal of Global Optimization》2010,46(1):63-73
As a global optimization problem, planar minimum weight triangulation problem has attracted extensive research attention.
In this paper, a new asymmetric graph called one-sided β-skeleton is introduced. We show that the one-sided circle-disconnected (?2b){(\sqrt{2}\beta)} -skeleton is a subgraph of a minimum weight triangulation. An algorithm for identifying subgraph of minimum weight triangulation
using the one-sided (?2b){(\sqrt{2}\beta)} -skeleton is proposed and it runs in O(n4/3+e+min{klogn, n2logn}){O(n^{4/3+\epsilon}+\min\{\kappa \log n, n^2\log n\})} time, where κ is the number of intersected segmented between the complete graph and the greedy triangulation of the point
set. 相似文献
4.
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
5.
Tatiana Shingel 《Constructive Approximation》2010,32(3):597-618
This paper extends previous work on approximation of loops to the case of special orthogonal groups SO(N), N≥3. We prove that the best approximation of an SO(N) loop Q(t) belonging to a Hölder class Lip α , α>1, by a polynomial SO(N) loop of degree ≤n is of order $\mathcal{O}(n^{-\alpha+\epsilon})This paper extends previous work on approximation of loops to the case of special orthogonal groups SO(N), N≥3. We prove that the best approximation of an SO(N) loop Q(t) belonging to a H?lder class Lip
α
, α>1, by a polynomial SO(N) loop of degree ≤n is of order O(n-a+e)\mathcal{O}(n^{-\alpha+\epsilon}) for n≥k, where k=k(Q) is determined by topological properties of the loop and ε>0 is arbitrarily small. The convergence rate is therefore ε-close to the optimal achievable rate of approximation. The construction of polynomial loops involves higher-order splitting
methods for the matrix exponential. A novelty in this work is the factorization technique for SO(N) loops which incorporates the loops’ topological aspects. 相似文献
6.
Sergey Bereg Prosenjit Bose Adrian Dumitrescu Ferran Hurtado Pavel Valtr 《Discrete and Computational Geometry》2009,41(4):513-532
Given a finite set of points S in ℝ
d
, consider visiting the points in S with a polygonal path which makes a minimum number of turns, or equivalently, has the minimum number of segments (links).
We call this minimization problem the minimum link spanning path problem. This natural problem has appeared several times in the literature under different variants. The simplest one is
that in which the allowed paths are axis-aligned. Let L(S) be the minimum number of links of an axis-aligned path for S, and let G
n
d
be an n×…×n grid in ℤ
d
. Kranakis et al. (Ars Comb. 38:177–192, 1994) showed that L(G
n
2)=2n−1 and
and conjectured that, for all d≥3,
We prove the conjecture for d=3 by showing the lower bound for L(G
n
3). For d=4, we prove that
For general d, we give new estimates on L(G
n
d
) that are very close to the conjectured value. The new lower bound of
improves previous result by Collins and Moret (Inf. Process. Lett. 68:317–319, 1998), while the new upper bound of
differs from the conjectured value only in the lower order terms.
For arbitrary point sets, we include an exact bound on the minimum number of links needed in an axis-aligned path traversing
any planar n-point set. We obtain similar tight estimates (within 1) in any number of dimensions d. For the general problem of traversing an arbitrary set of points in ℝ
d
with an axis-aligned spanning path having a minimum number of links, we present a constant ratio (depending on the dimension d) approximation algorithm.
Work by A. Dumitrescu was partially supported by NSF CAREER grant CCF-0444188.
Work by F. Hurtado was partially supported by projects MECMTM2006-01267 and Gen. Cat. 2005SGR00692.
Work by P. Valtr was partially supported by the project 1M0545 of the Ministry of Education of the Czech Republic. 相似文献
7.
Fernando Schwartz 《Annales Henri Poincare》2011,12(1):67-76
We consider asymptotically flat Riemannian manifolds with non-negative scalar curvature that are conformal to
\mathbbRn\ W, n 3 3{\mathbb{R}^{n}{\setminus} \Omega, n\ge 3}, and so that their boundary is a minimal hypersurface. (Here,
W ì \mathbbRn{\Omega\subset \mathbb{R}^{n}} is open bounded with smooth mean-convex boundary.) We prove that the ADM mass of any such manifold is bounded below by
\frac12(V/bn)(n-2)/n{\frac{1}{2}\left(V/\beta_{n}\right)^{(n-2)/n}}, where V is the Euclidean volume of Ω and β
n
is the volume of the Euclidean unit n-ball. This gives a partial proof to a conjecture of Bray and Iga (Commun. Anal. Geom. 10:999–1016, 2002). Surprisingly, we do not require the boundary to be outermost. 相似文献
8.
András Kroó 《Constructive Approximation》2012,35(2):181-200
We consider the problem of the approximation of regular convex bodies in ℝ
d
by level surfaces of convex algebraic polynomials. Hammer (in Mathematika 10, 67–71, 1963) verified that any convex body in ℝ
d
can be approximated by a level surface of a convex algebraic polynomial. In Jaen J. Approx. 1, 97–109, 2009 and subsequently in J. Approx. Theory 162, 628–637, 2010 a quantitative version of Hammer’s approximation theorem was given by showing that the order of approximation of convex bodies
by convex algebraic level surfaces of degree n is
\frac1n\frac{1}{n}. Moreover, it was also shown that whenever the convex body is not
regular (that is, there exists a point on its boundary at which the convex body possesses two distinct supporting hyperplanes), then
\frac1n\frac{1}{n} is essentially the sharp rate of approximation. This leads to the natural question whether this rate of approximation can
be improved further when the convex body is regular. In this paper we shall give an affirmative answer to this question. It
turns out that for regular convex bodies a o(1/n) rate of convergence holds. In addition, if the body satisfies the condition of C
2-smoothness the rate of approximation is
O(\frac1n2)O(\frac{1}{n^{2}}). 相似文献
9.
Christopher S. Withers Saralees Nadarajah 《Annals of the Institute of Statistical Mathematics》2010,62(6):1113-1142
We obtain the Edgeworth expansion for P(n1/2([^(q)]-q) < x){P(n^{1/2}(\hat{\theta}-\theta) < x)} and its derivatives, and the tilted Edgeworth (or saddlepoint or small sample) expansion for P([^(q)] < x){P(\hat{\theta} < x)} and its derivatives where [^(q)]{\hat{\theta}} is any vector estimate having the standard cumulant expansions in powers of n-1{n^{-1}} . 相似文献
10.
Sinan Ünver 《Mathematische Annalen》2010,348(4):833-858
Let ${k[\varepsilon]_{2}:=k[\varepsilon]/(\varepsilon^{2})}Let k[e]2:=k[e]/(e2){k[\varepsilon]_{2}:=k[\varepsilon]/(\varepsilon^{2})} . The single valued real analytic n-polylogarithm
Ln: \mathbbC ? \mathbbR{\mathcal{L}_{n}: \mathbb{C} \to \mathbb{R}} is fundamental in the study of weight n motivic cohomology over a field k, of characteristic 0. In this paper, we extend the construction in ünver (Algebra Number Theory 3:1–34, 2009) to define additive
n-polylogarithms lin:k[e]2? k{li_{n}:k[\varepsilon]_{2}\to k} and prove that they satisfy functional equations analogous to those of Ln{\mathcal{L}_{n}}. Under a mild hypothesis, we show that these functions descend to an analog of the nth Bloch group Bn¢(k[e]2){B_{n}' (k[\varepsilon]_{2})} defined by Goncharov (Adv Math 114:197–318, 1995). We hope that these functions will be useful in the study of weight n motivic cohomology over k[ε]2. 相似文献
11.
Under study is the problem of finding a ball of minimal radius enclosing at least k points of a given finite set in a Euclidean space. In the case of a fixed dimension of the space this problem is polynomially solvable, but in general its complexity has not been previously determined. We prove that the problem is NP-hard in the strong sense and obtain a polynomial-time approximation scheme (PTAS) that enables us to solve the problem with an arbitrary relative error ? in time \(O(n^{1/\varepsilon ^2 + 1} d)\), where n is the cardinality of the original set and d is the space dimension. 相似文献
12.
In this paper, we consider the critical quasilinear Schr?dinger equations of the form
-e2Du+V(x)u-e2[D(u2)]u=|u|2(2*)-2u+g(u), x ? \mathbbRN, -\varepsilon^2\Delta u+V(x)u-\varepsilon^2[\Delta(u^2)]u=|u|^{2(2^*)-2}u+g(u),\quad x\in \mathbb{R}^N, 相似文献
13.
Let S be a set of n points in ℝ3, no three collinear and not all coplanar. If at most n−k are coplanar and n is sufficiently large, the total number of planes determined is at least
1+k\binomn-k2-\binomk2(\fracn-k2)1+k\binom{n-k}{2}-\binom{k}{2}(\frac{n-k}{2}). For similar conditions and sufficiently large n, (inspired by the work of P.D.T.A. Elliott in Acta Math. Sci. Hung. 18:181–188, 1967) we also show that the number of spheres determined by n points is at least
1+\binomn-13-t3orchard(n-1)1+\binom{n-1}{3}-t_{3}^{\mathrm{orchard}}(n-1), and this bound is best possible under its hypothesis. (By t3orchard(n)t_{3}^{\mathrm{orchard}}(n), we are denoting the maximum number of three-point lines attainable by a configuration of n points, no four collinear, in the plane, i.e., the classic Orchard Problem.) New lower bounds are also given for both lines
and circles. 相似文献
14.
Let Λ(n) be the von Mangoldt function, x real and y small compared with x. This paper gives a non-trivial estimate on the exponential sum over primes in short intervals
S2(x,y;a)=?x < n £ x+yL(n)e(n2 a)S_2(x,y;{\alpha})=\sum_{x < n \le x+y}\Lambda(n)e(n^2 {\alpha})
for all α ∈ [0,1] whenever
x\frac23+e £ y £ xx^{\frac{2}{3}+{\varepsilon}}\le y \le x
. This result is as good as what was previously derived from the Generalized Riemann Hypothesis. 相似文献
15.
Silviu Radu 《The Ramanujan Journal》2009,20(2):215-251
In this paper we present an algorithm that takes as input a generating function of the form $\prod_{\delta|M}\prod_{n=1}^{\infty}(1-q^{\delta n})^{r_{\delta}}=\sum_{n=0}^{\infty}a(n)q^{n}
16.
D. R. Heath-Brown 《Journal of Mathematical Sciences》2010,171(6):813-823
Subject to the abc-conjecture, we improve the standard Weyl estimate for cubic exponential sums in which the argument is a
quadratic irrational. Specifically. we show that
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