177,564 research outputs found
Structure of level sets and Sard-type properties of Lipschitz maps
No full textWe consider certain properties of maps of class C2 from Rd to Rd−1 that are strictly related to Sard’s theorem, and we show that some of them can be extended to Lipschitz maps, while others require some additional regularity. We also give examples showing that, in terms of regularity, our results are optimal
Il Teorema di Morse-Sard
No full textQuesta tesi riguarda il Teorema di Morse-Sard nella sua versione generale. Tale teorema afferma che l'immagine dei punti critici di una funzione di classe C^k da un aperto di R^m a R^n è un insieme di misura di Lebesgue nulla se k >= m-n+1 (se m >= n) o se k >= 1 (se m<n).
Di tale teorema diamo una dimostrazione, tratta dall’articolo di Moreira-Ruas (2009). Dimostriamo inoltre il Teorema di Varberg (1966) riguardante il caso delle funzioni differenziabili con m=n. Si fornisce poi un’applicazione del Teorema di Morse-Sard: la formula di Coarea per funzioni a valori reali
Sard theorems for Lipschitz functions and applications in optimization
No full textWe establish a "preparatory Sard theorem" for smooth functions with a partially affine structure. By means of this result, we improve a previous result of Rifford [17, 19] concerning the generalized (Clarke) critical values of Lipschitz functions defined as minima of smooth functions. We also establish a nonsmooth Sard theorem for the class of Lipschitz functions from R (d) to R (p) that can be expressed as finite selections of C (k) functions (more generally, continuous selections over a compact countable set). This recovers readily the classical Sard theorem and extends a previous result of Barbet-Daniilidis-Dambrine [1] to the case p > 1. Applications in semi-infinite and Pareto optimization are given
The Morse–Sard theorem in Wn,n(Ω): A simple proof
No full textAbstractWe give a simpler and more self-contained proof of the Morse–Sard theorem in the setting of Sobolev space Wn,n(Rn,R) with n⩾2, we already proved in a previous paper [R. van der Putten, The Morse–Sard theorem for Sobolev spaces in a borderline case, Bull. Sci. Math. 136 (4) (2012) 463–475, http://dx.doi.org/10.1016/j.bulsci.2010.02.001]
Recommended from our members
Spectrum Access R&D (SARD) Program: An Update on the Conformal C-Band/Multi-band Antenna Project
No full textIn September 2016, work was initiated on the subprojects which comprise the Conformal C-Band/Multiband Antenna project: SARD #1: Broadband Conformal C-Band Missile Wraparound Antennas; SARD #2: Beam Switching Array Antennas; SARD #3: Multiband Conformal Antennas for Aircraft Applications; SARD #4: High Altitude Coronal Efforts on Antenna Performance; and SARD #5: Small, Medium Gain Multiband Receive Antennas. A brief status of each of them will be discussed and detail the technology areas being developed by each.International Foundation for TelemeteringProceedings from the International Telemetering Conference are made available by the International Foundation for Telemetering and the University of Arizona Libraries. Visit http://www.telemetry.org/index.php/contact-us if you have questions about items in this collection
The Morse-Sard theorem in Sobolev spaces,
No full textLet Ω be an open subset of R n . Consider a differentiable map u : Ω → R m . For many application in differential topology, dynamical systems, and degree theory, it is important to study the “size” of the set of critical values of u . Usually the word “size” we just used is intended in the sense of some measure (e.g. Hausdorff measure, Lebesgue measure, entropy measure). The Morse-Sard Theorem is concerned exactly about the size of such set. To be precise, we will indicate by C u the set of the critical points of u (i.e., the set of points x ∈ Ω such that ∇ u ( x ) is not of maximum rank), and by V u the set u ( C u ) which is by definition the set of the critical values of u . In this paper we will prove that, if u ∈ W loc k , p ( Ω , R m ) for k = n − m + 1 , n < p , then the set of the critical value of u has m -measure zero.
As we are dealing with a very classical theorem, we find it suitable to give an account with discussed bibliography of what is already known about the finite dimensional Morse-Sard theorem. Along the paper we will make the suitable comparisons
The Morse-Sard theorem in W^[n,n](\Omega) : a simple proof.
No full textWe give a simpler and more self-contained proof of the Morse-Sard theorem in the setting of Sobolev space Wn,n(Rn,R) with n≥ 2, we already proved in a previous pape
The curve selection lemma and the Morse-Sard theorem
No full textWe use an inequality due to Bochnak and Lojasiewicz, which follows from the Curve Selection Lemma of real algebraic geometry in order to prove that, given a C(r) function f : U subset of R(m) -> R, we have lim(y -> xy is an element of crit(f)) vertical bar f(y) - f(x)vertical bar/vertical bar y - x vertical bar(r) = 0, for all x is an element of crit(f)` boolean AND U, where crit( f) = {x is an element of U vertical bar df ( x) = 0}. This shows that the so-called Morse decomposition of the critical set, used in the classical proof of the Morse-Sard theorem, is not necessary: the conclusion of the Morse decomposition lemma holds for the whole critical set. We use this result to give a simple proof of the classical Morse-Sard theorem ( with sharp differentiability assumptions).CNPqConselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)FAPESPFundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP
Appropriate Similarity Measures for Author Cocitation Analysis
No full textWe provide a number of new insights into the methodological discussion about author cocitation analysis. We first argue that the use of the Pearson correlation for measuring the similarity between authors’ cocitation profiles is not very satisfactory. We then discuss what kind of similarity measures may be used as an alternative to the Pearson correlation. We consider three similarity measures in particular. One is the well-known cosine. The other two similarity measures have not been used before in the bibliometric literature. Finally, we show by means of an example that our findings have a high practical relevance.information science;Pearson correlation;cosine;similarity measure;author cocitation analysis
The Morse–Sard theorem for Sobolev spaces in a borderline case
No full textAbstractWe extend the Morse–Sard theorem to mappings u belonging to the Sobolev class Wn,n(Rn,R) with n⩾2 under mild regularity assumptions on the critical set of u
- …
