Computing Puiseux series : a fast divide and conquer algorithm

Let $F\in \mathbb{K}[X, Y ]$ be a polynomial of total degree $D$ defined over a perfect field $\mathbb{K}$ of characteristic zero or greater than $D$. Assuming $F$ separable with respect to $Y$ , we provide an algorithm that computes the singular parts of all Puiseux series of $F$ above $X = 0$ in less than $\tilde{\mathcal{O}}(D\delta)$ operations in $\mathbb{K}$, where $\delta$ is the valuation of the resultant of $F$ and its partial derivative with respect to $Y$. To this aim, we use a divide and conquer strategy and replace univariate factorization by dynamic evaluation. As a first main corollary, we compute the irreducible factors of $F$ in $\mathbb{K}[[X]][Y ]$ up to an arbitrary precision $X^N$ with $\tilde{\mathcal{O}}(D(\delta + N ))$ arithmetic operations. As a second main corollary, we compute the genus of the plane curve defined by $F$ with $\tilde{\mathcal{O}}(D^3)$ arithmetic operations and, if $\mathbb{K} = \mathbb{Q}$, with $\tilde{\mathcal{O}}((h+1)D^3)$ bit operations using a probabilistic algorithm, where $h$ is the logarithmic heigth of $F$.Comment: 27 pages, 2 figure

Short relaxation times but long transient times in both simple and complex reaction networks

When relaxation towards an equilibrium or steady state is exponential at large times, one usually considers that the associated relaxation time $\tau$, i.e., the inverse of that decay rate, is the longest characteristic time in the system. However that need not be true, and in particular other times such as the lifetime of an infinitesimal perturbation can be much longer. In the present work we demonstrate that this paradoxical property can arise even in quite simple systems such as a chain of reactions obeying mass action kinetics. By mathematical analysis of simple reaction networks, we pin-point the reason why the standard relaxation time does not provide relevant information on the potentially long transient times of typical infinitesimal perturbations. Overall, we consider four characteristic times and study their behavior in both simple chains and in more complex reaction networks taken from the publicly available database "Biomodels." In all these systems involving mass action rates, Michaelis-Menten reversible kinetics, or phenomenological laws for reaction rates, we find that the characteristic times corresponding to lifetimes of tracers and of concentration perturbations can be much longer than $\tau$

Using approximate roots for irreducibility and equi-singularity issues in K[[x]][y]

We provide an irreducibility test in the ring K[[x]][y] whose complexity is quasi-linear with respect to the valuation of the discriminant, assuming the input polynomial F square-free and K a perfect field of characteristic zero or greater than deg(F). The algorithm uses the theory of approximate roots and may be seen as a generalization of Abhyankhar's irreducibility criterion to the case of non algebraically closed residue fields. More generally, we show that we can test within the same complexity if a polynomial is pseudo-irreducible, a larger class of polynomials containing irreducible ones. If $F$ is pseudo-irreducible, the algorithm computes also the valuation of the discriminant and the equisingularity types of the germs of plane curve defined by F along the fiber x=0.Comment: 51 pages. Title modified. Slight modifications in Definition 5 and Proposition 1

Computing the equisingularity type of a pseudo-irreducible polynomial

Germs of plane curve singularities can be classified accordingly to their equisingularity type. For singularities over C, this important data coincides with the topological class. In this paper, we characterise a family of singularities, containing irreducible ones, whose equisingularity type can be computed in quasi-linear time with respect to the discriminant valuation of a Weierstrass equation

Large second harmonic generation enhancement in SiN waveguides by all-optically induced quasi phase matching

Integrated waveguides exhibiting efficient second-order nonlinearities are crucial to obtain compact and low power optical signal processing devices. Silicon nitride (SiN) has shown second harmonic generation (SHG) capabilities in resonant structures and single-pass devices leveraging intermodal phase matching, which is defined by waveguide design. Lithium niobate allows compensating for the phase mismatch using periodically poled waveguides, however the latter are not reconfigurable and remain difficult to integrate with SiN or silicon (Si) circuits. Here we show the all-optical enhancement of SHG in SiN waveguides by more than 30 dB. We demonstrate that a Watt-level laser causes a periodic modification of the waveguide second-order susceptibility. The resulting second order nonlinear grating has a periodicity allowing for quasi phase matching (QPM) between the pump and SH mode. Moreover, changing the pump wavelength or polarization updates the period, relaxing phase matching constraints imposed by the waveguide geometry. We show that the grating is long term inscribed in the waveguides, and we estimate a second order nonlinearity of the order of 0.3 pm/V, while a maximum conversion efficiency (CE) of 1.8x10-6 W-1 cm-2 is reached

Human platelets and their capacity of binding viruses: Meaning and challenges?

Blood platelets are first aimed at ensuring primary hemostasis. Beyond this role, they have been acknowledged as having functions in the maintenance of the vascular arborescence and, more recently, as being also innate immune cells, devoted notably to the detection of danger signals, of which infectious ones. Platelets express pathogen recognition receptors that can sense bacterial and viral moieties. Besides, several molecules that bind epithelial or sub-endothelial molecules and, so forth, are involved in hemostasis, happen to be able to ligate viral determinants, making platelets capable of either binding viruses or even to be infected by some of them. Further, as platelets express both Fc-receptors for Ig and complement receptors, they also bind occasionally virus-Ig or virus-Ig-complement immune complexes. Interplays of viruses with platelets are very complex and viral infections often interfere with platelet number and functions. Through a few instances of viral infections, the present review aims at presenting some of the most important interactions from pathophysiological and clinical points of view, which are observed between human viruses and platelets.Fil: Chabert, Adrien. Universite Lyon 2; FranciaFil: Hamzeh Cognasse, Hind. Universite Lyon 2; FranciaFil: Pozzetto, Bruno. Universite Lyon 2; FranciaFil: Cognasse, Fabrice. Universite Lyon 2; FranciaFil: Schattner, Mirta Ana. Consejo Nacional de Investigaciones Científicas y Técnicas. Instituto de Medicina Experimental. Academia Nacional de Medicina de Buenos Aires. Instituto de Medicina Experimental; ArgentinaFil: Gomez, Ricardo Martin. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata. Instituto de Biotecnología y Biología Molecular. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Instituto de Biotecnología y Biología Molecular; ArgentinaFil: Garraud, Olivier. Universite Lyon 2; Franci

The control of short-term ice mélange weakening episodes on calving activity at major Greenland outlet glaciers

The dense mixture of iceberg of various sizes and sea ice observed in many of Greenland's fjords, called ice mélange (sikussak in Greenlandic), has been shown to have a significant impact on the dynamics of several Greenland tidewater glaciers, mainly through the seasonal support it provides to the glacier terminus in winter. However, a clear understanding of shorter-term ice mélange dynamics is still lacking, mainly due to the high complexity and variability of the processes at play at the ice–ocean boundary. In this study, we use a combination of Sentinel-1 radar and Sentinel-2 optical satellite imagery to investigate in detail intra-seasonal ice mélange dynamics and its link to calving activity at three major outlet glaciers: Kangerdlugssuaq Glacier, Helheim Glacier and Sermeq Kujalleq in Kangia (Jakobshavn Isbræ). In those fjords, we identified recurrent ice mélange weakening (IMW) episodes consisting of the up-fjord propagation of a discontinuity between jam-packed and weaker ice mélange towards the glacier terminus. At a late stage, i.e., when the IMW front approaches the glacier terminus, these episodes were often correlated with the occurrence of large-scale calving events. The IMW process is particularly visible at the front of Kangerdlugssuaq Glacier and presents a cyclic behavior, such that we further analyzed IMW dynamics during the June–November period from 2018 to 2021 at this location. Throughout this period, we detected 30 IMW episodes with a recurrence time of 24 d, propagating over a median distance of 5.9 km and for 17 d, resulting in a median propagation speed of 400 m d−1. We found that 87 % of the IMW episodes occurred prior to a calving event visible in spaceborne observations and that ∼75 % of all detected calving events were preceded by an IMW episode. These results therefore present the IMW process as a clear control on the calving activity of Kangerdlugssuaq Glacier. Finally, using a simple numerical model for ice mélange motion, we showed that a slightly biased random motion of ice floes without fluctuating external forcing can reproduce IMW events and their cyclic influence and explain observed propagation speeds. These results further support our observations in characterizing the IMW process as self-sustained through the existence of an IMW–calving feedback. This study therefore highlights the importance of short-term ice mélange dynamics in the longer-term evolution of Greenland outlet glaciers