Characterizations of Lojasiewicz inequalities and applications

The classical Lojasiewicz inequality and its extensions for partial differential equation problems (Simon) and to o-minimal structures (Kurdyka) have a considerable impact on the analysis of gradient-like methods and related problems: minimization methods, complexity theory, asymptotic analysis of dissipative partial differential equations, tame geometry. This paper provides alternative characterizations of this type of inequalities for nonsmooth lower semicontinuous functions defined on a metric or a real Hilbert space. In a metric context, we show that a generalized form of the Lojasiewicz inequality (hereby called the Kurdyka- Lojasiewicz inequality) relates to metric regularity and to the Lipschitz continuity of the sublevel mapping, yielding applications to discrete methods (strong convergence of the proximal algorithm). In a Hilbert setting we further establish that asymptotic properties of the semiflow generated by -∂f are strongly linked to this inequality. This is done by introducing the notion of a piecewise subgradient curve: such curves have uniformly bounded lengths if and only if the Kurdyka- Lojasiewicz inequality is satisfied. Further characterizations in terms of talweg lines -a concept linked to the location of the less steepest points at the level sets of f- and integrability conditions are given. In the convex case these results are significantly reinforced, allowing in particular to establish the asymptotic equivalence of discrete gradient methods and continuous gradient curves. On the other hand, a counterexample of a convex C2 function in R2 is constructed to illustrate the fact that, contrary to our intuition, and unless a specific growth condition is satisfied, convex functions may fail to fulfill the Kurdyka- Lojasiewicz inequality.

On the stability of the optimal value and the optimal set in optimization problems

The paper develops a stability theory for the optimal value and the optimal set mapping of optimization problems posed in a Banach space. The problems considered in this paper have an arbitrary number of inequality constraints involving lower semicontinuous (not necessarily convex) functions and one closed abstract constraint set. The considered perturbations lead to problems of the same type as the nominal one (with the same space of variables and the same number of constraints), where the abstract constraint set can also be perturbed. The spaces of functions involved in the problems (objective and constraints) are equipped with the metric of the uniform convergence on the bounded sets, meanwhile in the space of closed sets we consider, coherently, the Attouch-Wets topology. The paper examines, in a unified way, the lower and upper semicontinuity of the optimal value function, and the closedness, lower and upper semicontinuity (in the sense of Berge) of the optimal set mapping. This paper can be seen as a second part of the stability theory presented in [17], where we studied the stability of the feasible set mapping (completed here with the analysis of the Lipschitz-like property).

Condensation of polyhedric structures onto soap films

We study the existence of solutions to general measure-minimization problems over topological classes that are stable under localized Lipschitz homotopy, including the standard Plateau problem without the need for restrictive assumptions such as orientability or even rectifiability of surfaces. In case of problems over an open and bounded domain we establish the existence of a “minimal candidate”, obtained as the limit for the local Hausdorff convergence of a minimizing sequence for which the measure is lower-semicontinuous. Although we do not give a way to control the topological constraint when taking limit yet— except for some examples of topological classes preserving local separation or for periodic two-dimensional sets — we prove that this candidate is an Almgren-minimal set. Thus, using regularity results such as Jean Taylor’s theorem, this could be a way to find solutions to the above minimization problems under a generic setup in arbitrary dimension and codimension.

Recovering the Elliott invariant from the Cuntz semigroup

Let A be a simple, separable C*-algebra of stable rank one. We prove that the Cuntz semigroup of C (T, A) is determined by its Murray-von Neumann semigroup of projections and a certain semigroup of lower semicontinuous functions (with values in the Cuntz semigroup of A). This result has two consequences. First, specializing to the case that A is simple, finite, separable and Z-stable, this yields a description of the Cuntz semigroup of C (T, A) in terms of the Elliott invariant of A. Second, suitably interpreted, it shows that the Elliott functor and the functor defined by the Cuntz semigroup of the tensor product with the algebra of continuous functions on the circle are naturally equivalent.

A lower bound in Nehari's theorem on the polydisc

By theorems of Ferguson and Lacey ($d=2$) and Lacey and Terwilleger ($d>2$), Nehari's theorem is known to hold on the polydisc $\D^d$ for $d>1$, i.e., if $H_\psi$ is a bounded Hankel form on $H^2(\D^d)$ with analytic symbol $\psi$, then there is a function $\varphi$ in $L^\infty(\T^d)$ such that $\psi$ is the Riesz projection of $\varphi$. A method proposed in Helson's last paper is used to show that the constant $C_d$ in the estimate $\|\varphi\|_\infty\le C_d \|H_\psi\|$ grows at least exponentially with $d$; it follows that there is no analogue of Nehari's theorem on the infinite-dimensional polydisc.

Sensibility and taste alterations after impacted lower third molar extractions. A prospective cohort study

Objectives: To determine the incidence, severity and duration of lingual tactile and gustatory function impairments after lower third molar removal. Study Design: Prospective cohort study with intra-subject measures of 16 patients undergoing lower third molar extractions. Sensibility and gustatory functions were evaluated in each subject preoperatively, one week and one month after the extraction, using Semmes-Weinstein monofilaments and 5 different concentrations of NaCl, respectively. Additionally, all patients filled a questionnaire to assess subjective perceptions. Results: Although patients did not perceive any sensibility impairments, a statistically significant decrease was detected when Semmes-Weinstein monofilaments. This alteration was present at one week after the surgical procedure and fully recovered one month after the extraction. There were no variations regarding the gustatory function. Conclusions: Lower third molar removal under local anesthesia may cause light lingual sensibility impairment. Most of these alterations remain undetected to patients. These lingual nerve injuries are present one week after the extraction and recover one month after surgery. The taste seems to remain unaffected after these procedures.

Pancreatic Cancer Cell Glycosylation Regulates Cell Adhesion and Invasion through the Modulation of a2b1 Integrin and E-Cadherin Function

In our previous studies we have described that ST3Gal III transfected pancreatic adenocarcinoma Capan-1 and MDAPanc-28 cells show increased membrane expression levels of sialyl-Lewis x (SLex) along with a concomitant decrease in α2,6-sialic acid compared to control cells. Here we have addressed the role of this glycosylation pattern in the functional properties of two glycoproteins involved in the processes of cancer cell invasion and migration, α2β1 integrin, the main receptor for type 1 collagen, and E-cadherin, responsible for cell-cell contacts and whose deregulation determines cell invasive capabilities. Our results demonstrate that ST3Gal III transfectants showed reduced cell-cell aggregation and increased invasive capacities. ST3Gal III transfected Capan-1 cells exhibited higher SLex and lower α2,6-sialic acid content on the glycans of their α2β1 integrin molecules. As a consequence, higher phosphorylation of focal adhesion kinase tyrosine 397, which is recognized as one of the first steps of integrin-derived signaling pathways, was observed in these cells upon adhesion to type 1 collagen. This molecular mechanism underlies the increased migration through collagen of these cells. In addition, the pancreatic adenocarcinoma cell lines as well as human pancreatic tumor tissues showed colocalization of SLex and E-cadherin, which was higher in the ST3Gal III transfectants. In conclusion, changes in the sialylation pattern of α2β1 integrin and E-cadherin appear to influence the functional role of these two glycoproteins supporting the role of these glycans as an underlying mechanism regulating pancreatic cancer cell adhesion and invasion