Punctual Hilbert Schemes and Certified Approximate Singularities

In this paper we provide a new method to certify that a nearby polynomial system has a singular isolated root with a prescribed multiplicity structure. More precisely, given a polynomial system f $=(f\_1, \ldots, f\_N)\in C[x\_1, \ldots, x\_n]^N$, we present a Newton iteration on an extended deflated system that locally converges, under regularity conditions, to a small deformation of $f$ such that this deformed system has an exact singular root. The iteration simultaneously converges to the coordinates of the singular root and the coefficients of the so called inverse system that describes the multiplicity structure at the root. We use $$\alpha$$-theory test to certify the quadratic convergence, and togive bounds on the size of the deformation and on the approximation error. The approach relies on an analysis of the punctual Hilbert scheme, for which we provide a new description. We show in particular that some of its strata can be rationally parametrized and exploit these parametrizations in the certification. We show in numerical experimentation how the approximate inverse system can be computed as a starting point of the Newton iterations and the fast numerical convergence to the singular root with its multiplicity structure, certified by our criteria.Comment: International Symposium on Symbolic and Algebraic Computation, Jul 2020, Kalamata, Franc

Ethanol application at veraison decreases acidity in Cabernet Sauvignon grapes

Research NoteSpraying ethanol (5 % v/v in water) onto grape clusters at mid-veraison led to a 30 % drop in the malic acid concentration at harvest. As a consequence, titratable acidity also dropped by 10 %. The concentration of tartaric acid did not change significantly. The mode of action of ethanol on malic acid metabolism is discussed.

An update on the Hirsch conjecture

The Hirsch conjecture was posed in 1957 in a letter from Warren M. Hirsch to George Dantzig. It states that the graph of a d-dimensional polytope with n facets cannot have diameter greater than n - d. Despite being one of the most fundamental, basic and old problems in polytope theory, what we know is quite scarce. Most notably, no polynomial upper bound is known for the diameters that are conjectured to be linear. In contrast, very few polytopes are known where the bound $n-d$ is attained. This paper collects known results and remarks both on the positive and on the negative side of the conjecture. Some proofs are included, but only those that we hope are accessible to a general mathematical audience without introducing too many technicalities.Comment: 28 pages, 6 figures. Many proofs have been taken out from version 2 and put into the appendix arXiv:0912.423

From error bounds to the complexity of first-order descent methods for convex functions

This paper shows that error bounds can be used as effective tools for deriving complexity results for first-order descent methods in convex minimization. In a first stage, this objective led us to revisit the interplay between error bounds and the Kurdyka-\L ojasiewicz (KL) inequality. One can show the equivalence between the two concepts for convex functions having a moderately flat profile near the set of minimizers (as those of functions with H\"olderian growth). A counterexample shows that the equivalence is no longer true for extremely flat functions. This fact reveals the relevance of an approach based on KL inequality. In a second stage, we show how KL inequalities can in turn be employed to compute new complexity bounds for a wealth of descent methods for convex problems. Our approach is completely original and makes use of a one-dimensional worst-case proximal sequence in the spirit of the famous majorant method of Kantorovich. Our result applies to a very simple abstract scheme that covers a wide class of descent methods. As a byproduct of our study, we also provide new results for the globalization of KL inequalities in the convex framework. Our main results inaugurate a simple methodology: derive an error bound, compute the desingularizing function whenever possible, identify essential constants in the descent method and finally compute the complexity using the one-dimensional worst case proximal sequence. Our method is illustrated through projection methods for feasibility problems, and through the famous iterative shrinkage thresholding algorithm (ISTA), for which we show that the complexity bound is of the form $O(q^{k})$ where the constituents of the bound only depend on error bound constants obtained for an arbitrary least squares objective with $\ell^1$ regularization

Artificial reefs: from ecological processes to fishing enhancement tools

info:eu-repo/semantics/publishedVersio

LRP-1 Promotes Cancer Cell Invasion by Supporting ERK and Inhibiting JNK Signaling Pathways

Background: The low-density lipoprotein receptor-related protein-1 (LRP-1) is an endocytic receptor mediating the clearance of various extracellular molecules involved in the dissemination of cancer cells. LRP-1 thus appeared as an attractive receptor for targeting the invasive behavior of malignant cells. However, recent results suggest that LRP-1 may facilitate the development and growth of cancer metastases in vivo, but the precise contribution of the receptor during cancer progression remains to be elucidated. The lack of mechanistic insights into the intracellular signaling networks downstream of LRP-1 has prevented the understanding of its contribution towards cancer. Methodology/Principal Findings: Through a short-hairpin RNA-mediated silencing approach, we identified LRP-1 as a main regulator of ERK and JNK signaling in a tumor cell context. Co-immunoprecipitation experiments revealed that LRP-1 constitutes an intracellular docking site for MAPK containing complexes. By using pharmacological agents, constitutively active and dominant-negative kinases, we demonstrated that LRP-1 maintains malignant cells in an adhesive state that is favorable for invasion by activating ERK and inhibiting JNK. We further demonstrated that the LRP-1-dependent regulation of MAPK signaling organizes the cytoskeletal architecture and mediates adhesive complex turnover in cancer cells. Moreover, we found that LRP-1 is tethered to the actin network and to focal adhesion sites and controls ERK and JNK targeting to talin-rich structures. Conclusions: We identified ERK and JNK as the main molecular relays by which LRP-1 regulates focal adhesion disassembly of malignant cells to support invasion