共查询到20条相似文献,搜索用时 15 毫秒
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We show an effective method to compute the Łojasiewicz exponent of an arbitrary sheaf of ideals of OX{\mathcal{O}_X} , where X is a non-singular scheme. This method is based on the algorithm of resolution of singularities. 相似文献
3.
We prove a sharp regular separation condition for arbitrary projective algebraic sets. As a corollary an extension of Kollár's
results on the Łojasiewicz exponent of polynomial mappings is obtained.
Oblatum 9-II-1998 & 3-VI-1998 / Published online: 14 January 1999 相似文献
4.
Carles Bivià-Ausina 《Mathematische Zeitschrift》2009,262(2):389-409
Let be a finite analytic map. We give an expression for the local Łojasiewicz exponent and for the multiplicity of g when the component functions of g satisfy certain condition with respect to a set of n monomial ideals I
1,..., I
n
. We give an effective method to compute Łojasiewicz exponents based on the computation of mixed multiplicities. As a consequence
of our study, we give a numerical characterization of a class of functions that includes semi-weighted homogenous functions
and Newton non-degenerate functions.
Work supported by DGICYT Grant MTM2006-06027. 相似文献
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We propose different types of Łojasiewicz inequality at infinity for polynomials in two real variables. The formulas for the Łojasiewicz exponents are given. 相似文献
7.
Carles Bivià-Ausina 《Mathematische Zeitschrift》2007,257(4):745-767
In this paper we give a lower bound for the Łojasiewicz exponent at infinity of a special class of polynomial maps , s ≥ 1. As a consequence, we detect a class of polynomial maps that are global diffeomorphisms if their Jacobian determinant never vanishes.
Work supported by DGICYT Grant BFM2003–02037/MATE. 相似文献
8.
Rafał Pierzchała 《Potential Analysis》2014,40(1):41-56
The aim of this paper is to address a problem raised originally by L. Gendre, later by W. Ple?niak and recently by L. Bia?as–Cie? and M. Kosek. This problem concerns the pluricomplex Green function and consists in finding new examples of sets with so–called ?ojasiewicz–Siciak ((?S) for short) property. So far, the known examples of such sets are rather of particular nature. We prove that each compact subset of ? N , treated as a subset of ? N , satisfies the ?ojasiewicz–Siciak condition. We also give a sufficient geometric criterion for a semialgebraic set in ?2, but treated as a subset of ?, to satisfy this condition. This criterion applies more generally to a set in ? definable in a polynomially bounded o–minimal structure. 相似文献
9.
Dominikus Noll 《Journal of Optimization Theory and Applications》2014,160(2):553-572
We investigate the convergence of subgradient-oriented descent methods in non-smooth non-convex optimization. We prove convergence in the sense of subsequences for functions with a strict standard model, and we show that convergence to a single critical point may be guaranteed if the Kurdyka–?ojasiewicz inequality is satisfied. We show, by way of an example, that the Kurdyka–?ojasiewicz inequality alone is not sufficient to prove the convergence to critical points. 相似文献
10.
In this paper, we study the Kurdyka–?ojasiewicz (KL) exponent, an important quantity for analyzing the convergence rate of first-order methods. Specifically, we develop various calculus rules to deduce the KL exponent of new (possibly nonconvex and nonsmooth) functions formed from functions with known KL exponents. In addition, we show that the well-studied Luo–Tseng error bound together with a mild assumption on the separation of stationary values implies that the KL exponent is \(\frac{1}{2}\). The Luo–Tseng error bound is known to hold for a large class of concrete structured optimization problems, and thus we deduce the KL exponent of a large class of functions whose exponents were previously unknown. Building upon this and the calculus rules, we are then able to show that for many convex or nonconvex optimization models for applications such as sparse recovery, their objective function’s KL exponent is \(\frac{1}{2}\). This includes the least squares problem with smoothly clipped absolute deviation regularization or minimax concave penalty regularization and the logistic regression problem with \(\ell _1\) regularization. Since many existing local convergence rate analysis for first-order methods in the nonconvex scenario relies on the KL exponent, our results enable us to obtain explicit convergence rate for various first-order methods when they are applied to a large variety of practical optimization models. Finally, we further illustrate how our results can be applied to establishing local linear convergence of the proximal gradient algorithm and the inertial proximal algorithm with constant step sizes for some specific models that arise in sparse recovery. 相似文献
11.
A compact set
K ì \mathbbCN{K \subset \mathbb{C}}^{N} satisfies (ŁS) if it is polynomially convex and there exist constants B,β > 0 such that
VK(z) 3 B(dist(z,K))b if dist(z,K) £ 1, \labelLS V_K(z)\geq B(\rm{dist}(z,K))^\beta\qquad \rm{ if}\quad \rm{ dist}(z,K)\leq 1, \label{LS} 相似文献
12.
We provide a reasonably optimal answer to the natural question of the conditions under which an analytic function on an infinite-dimensional Hilbert space satisfies the ?ojasiewicz gradient inequality. 相似文献
13.
J. Gwoździewicz 《Commentarii Mathematici Helvetici》1999,74(3):364-375
Let f be a real analytic function defined in a neighborhood of
0 ? \Bbb Rn 0 \in {\Bbb R}^n such that f-1(0)={0} f^{-1}(0)=\{0\} . We describe the smallest possible exponents !, #, / for which we have the following estimates: |f(x)| 3 c|x|a |f(x)|\geq c|x|^{\alpha} , |grad f(x)| 3 c|x|b |{\rm grad}\,f(x)|\geq c|x|^{\beta} , |grad f(x)| 3 c|f(x)|q |{\rm grad}\,f(x)|\geq c|f(x)|^{\theta} for x near zero with c > 0 c > 0 . We prove that a = b+1 \alpha=\beta+1, q = b/a\theta=\beta/\alpha . Moreover b = N+a/b \beta=N+a/b where $ 0 h a < b h N^{n-1} $ 0 h a < b h N^{n-1} . If f is a polynomial then |f(x)| 3 c|x|(degf-1)n+1 |f(x)|\geq c|x|^{(\deg f-1)^n+1} in a small neighborhood of zero. 相似文献
14.
We provide a sharp quantitative version of the Gaussian concentration inequality: for every \(r>0\), the difference between the measure of the r-enlargement of a given set and the r-enlargement of a half-space controls the square of the measure of the symmetric difference between the set and a suitable half-space. We also prove a similar estimate in the Euclidean setting for the enlargement with a general convex set. This is equivalent to the stability of the Brunn–Minkowski inequality for the Minkowski sum between a convex set and a generic one. 相似文献
15.
Journal of Optimization Theory and Applications - A correction to this paper has been published: https://doi.org/10.1007/s10957-020-01779-7. 相似文献
16.
Beata Osińska 《Bulletin des Sciences Mathématiques》2011,(2):215-229
Let F:V→Cm be a regular mapping, where V⊂Cn is an algebraic set of positive dimension and m?n?2, and let L∞(F) be the ?ojasiewicz exponent at infinity of F. We prove that F has a polynomial extension G:Cn→Cm such L∞(G)=L∞(F). Moreover, we give an estimate of the degree of the extension G. Additionally, we prove that if then for any β∈Q, β?L∞(F), the mapping F has a polynomial extension G with L∞(G)=β. We also give an estimate of the degree of this extension. 相似文献
17.
A sub-Riemannian curvature-dimension inequality,volume doubling property and the Poincaré inequality
Let $\mathbb M $ be a smooth connected manifold endowed with a smooth measure $\mu $ and a smooth locally subelliptic diffusion operator $L$ satisfying $L1=0$ , and which is symmetric with respect to $\mu $ . We show that if $L$ satisfies, with a non negative curvature parameter, the generalized curvature inequality introduced by the first and third named authors in http://arxiv.org/abs/1101.3590, then the following properties hold:
18.
The aim of this paper is to analyze the heat semigroup ${(\mathcal{N}_{t})_{t >0 } = \{e^{t \Delta}\}_{t >0 }}$ generated by the usual Laplacian operator Δ on ${\mathbb{R}^{d}}$ equipped with the d-dimensional Lebesgue measure. We obtain and study, via a method involving some semigroup techniques, a large family of functional inequalities that does not exist in the literature and with the local Poincaré and reverse local Poincaré inequalities as particular cases. As a consequence, we establish in parallel a new functional and integral inequality related to the Ornstein–Uhlenbeck semigroup. 相似文献
19.
In this work, we first prove a generalized version of a parameterized Jordan-type inequality. We then use it to prove the generalized versions of Janous’s inequality and Tsintsifas’s inequality which reduce to two inequalities conjectured by Janous and Tsintsifas as special cases. 相似文献
20.
Rafał Pierzchała 《Journal d'Analyse Mathématique》2016,129(1):285-307
The paper contains a full geometric characterization of compact semialgebraic sets in C satisfying the ?ojasiewicz-Siciak condition. The ?ojasiewicz-Siciak condition is a certain estimate for the Siciak extremal function. In a previous paper, we gave a sufficient criterion for a compact, connected, and semialgebraic set in C to satisfy this condition. In the present paper, we remove completely the connectedness assumption and prove that the aforementioned sufficient condition is also necessary. Moreover, we obtain some new results concerning the ?ojasiewicz-Siciak condition in CN. For example, we prove that if K1,...,Kp are compact, nonpluripolar, and pairwise disjoint subsets of CN, each satisfying the ?ojasiewicz-Siciak condition, and K:= K1?· · ·?Kp is polynomially convex, then K satisfies this condition as well. 相似文献
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