Rational degeneration of M-curves, totally positive Grassmannians and KP2-solitons

We establish a new connection between the theory of totally positive Grassmannians and the theory of $\mathtt M$-curves using the finite--gap theory for solitons of the KP equation. Here and in the following KP equation denotes the Kadomtsev-Petviashvili 2 equation, which is the first flow from the KP hierarchy. We also assume that all KP times are real. We associate to any point of the real totally positive Grassmannian $Gr^{TP} (N,M)$ a reducible curve which is a rational degeneration of an $\mathtt M$--curve of minimal genus $g=N(M-N)$, and we reconstruct the real algebraic-geometric data \'a la Krichever for the underlying real bounded multiline KP soliton solutions. From this construction it follows that these multiline solitons can be explicitly obtained by degenerating regular real finite-gap solutions corresponding to smooth $M$-curves. In our approach we rule the addition of each new rational component to the spectral curve via an elementary Darboux transformation which corresponds to a section of a specific projection $Gr^{TP} (r+1,M-N+r+1)\mapsto Gr^{TP} (r,M-N+r)$.Comment: 49 pages, 10 figures. Minor revision

A new data assimilation procedure to develop a debris flow run-out model

Abstract Parameter calibration is one of the most problematic phases of numerical modeling since the choice of parameters affects the model\u2019s reliability as far as the physical problems being studied are concerned. In some cases, laboratory tests or physical models evaluating model parameters cannot be completed and other strategies must be adopted; numerical models reproducing debris flow propagation are one of these. Since scale problems affect the reproduction of real debris flows in the laboratory or specific tests used to determine rheological parameters, calibration is usually carried out by comparing in a subjective way only a few parameters, such as the heights of soil deposits calculated for some sections of the debris flows or the distance traveled by the debris flows using the values detected in situ after an event has occurred. Since no automatic or objective procedure has as yet been produced, this paper presents a numerical procedure based on the application of a statistical algorithm, which makes it possible to define, without ambiguities, the best parameter set. The procedure has been applied to a study case for which digital elevation models of both before and after an important event exist, implicating that a good database for applying the method was available. Its application has uncovered insights to better understand debris flows and related phenomena

Elongation factor 2-diphthamide is critical for translation of two IRES-dependent protein targets, XIAP and FGF2, under oxidative stress conditions

Elongation factor-2 (eEF2) catalyzes the movement of the ribosome along the mRNA. A single histidine residue in eEF2 (H715) is modified to form diphthamide. A role for eEF2 in cellular stress responses is highlighted by the fact that eEF2 is sensitive to oxidative stress and that it must be active in order to drive the synthesis of proteins that help cells to mitigate the adverse effects of oxidative stress. Many of the latter proteins are encoded by mRNAs containing a sequence called an “internal ribosomal entry site” (IRES). Under high oxidative stress conditions diphthamide-deficient cells were significantly more sensitive to cell death. These results suggest that diphthamide may play a role in protection against the degradation of eEF2. Its protection is especially important under those situations where it is necessary for the re-programming of translation from global to IRES synthesis. Indeed, we found that the expression of X-linked inhibitor of apoptosis (XIAP) and fibroblast growth factor 2 (FGF2), two proteins synthesized from mRNAs with IRES that promote cell survival are deregulated in diphthamide-deficient cells. Our findings therefore suggest that eEF2/diphthamide controls the selective translation of IRES-dependent protein targets XIAP and FGF2, critical for cell survival under conditions of oxidative stress.España, Ministerio de Ciencia e Innovación BFU 2010-20882

Periodic billiard orbits on nn--dimensional ellipsoids with impacts on confocal quadrics and isoperiodic deformations

In our paper we study periodic geodesic motion on multidimensional ellipsoids with elastic impacts along confocal quadrics. We show that the method of isoperiodic deformation is applicable.Comment: Latex, 28 pages, 3 figure

Real regular KP divisors on M-curves and totally non-negative Grassmannians

In this paper, we construct an explicit map from planar bicolored (plabic) trivalent graphs representing a given irreducible positroid cell STNN M in the totally non-negative Grassmannian GrTNN(k, n) to the spectral data for the relevant class of real regu lar Kadomtsev–Petviashvili II (KP-II) solutions, thus completing the search of real algebraic-geometric data for the KP-II equation started in Abenda and Grinevich (Commun Math Phys 361(3):1029–1081, 2018; Sel Math New Ser 25(3):43, 2019). The spectral curve is modeled on the Krichever construction for degenerate finite-gap solutions and is a rationally degenerate M-curve, , dual to the graph. The divisors are real regular KP-II divisors in the ovals of , i.e. they fulfill the conditions for selecting real regular finite-gap KP-II solutions in Dubrovin and Natanzon (Izv Akad Nauk SSSR Ser Mat 52:267–286, 1988). Since the soliton data are described by points in STNN M , we establish a bridge between real regular finite-gap KP-II solutions (Dubrovin and Natanzon, 1988) and real regular multi-line KP-II solitons which are known to be parameterized by points in GrTNN(k, n) (Chakravarty and Kodama in Stud Appl Math 123:83–151, 2009; Kodama and Williams in Invent Math 198:637–699, 2014). We use the geometric characterization of spaces of relations on plabic networks intro duced in Abenda and Grinevich (Adv Math 406:108523, 2022; Int Math Res Not 2022:rnac162, 2022. https://doi.org/10.1093/imrn/rnac162) to prove the invariance of this construction with respect to the many gauge freedoms on the network. Such systems of relations were proposed in Lam (in: Current developments in mathematics, International Press, Somerville, 2014) for the computation of scattering amplitudes for on-shell diagrams N = 4 SYM (Arkani-Hamed et al. in Grassmannian geometry of scattering amplitudes, Cambridge University Press, Cambridge, 2016) and govern the totally non-negative amalgamation of the little positive Grassmannians, GrTP(1, 3) and GrTP(2, 3), into any given positroid cell STNN M ⊂ GrTNN(k, n). In our set ting they control the reality and regularity properties of the KP-II divisor. Finally, we explain the transformation of both the curve and the divisor both under Postnikov’s moves and reductions and under amalgamation of positroid cells, and apply our con struction to some examples

Molecular Control of the Amount, Subcellular Location and Activity State of Translation Elongation Factor 2 (eEF-2) in Neurons Experiencing Stress

Eukaryotic elongation factor 2 (eEF-2) is an important regulator of the protein translation machinery wherein it controls the movement of the ribosome along the mRNA. The activity of eEF-2 is regulated by changes in cellular energy status and nutrient availability, and posttranslational modifications such as phosphorylation and mono-ADP-ribosylation. However, the mechanisms regulating protein translation under conditions of cellular stress in neurons are unknown. Here we show that when rat hippocampal neurons experience oxidative stress (lipid peroxidation induced by exposure to cumene hydroperoxide; CH), eEF-2 is hyperphosphorylated and ribosylated resulting in reduced translational activity. The degradation of eEF-2 requires calpain proteolytic activity and is accompanied by accumulation of eEF-2 in the nuclear compartment. The subcellular localization of both native and phosphorylated forms of eEF-2 is influenced by CRM1 and 14.3.3, respectively. In hippocampal neurons p53 interacts with non-phosphorylated (active) eEF-2, but not with its phosphorylated form. The p53 – eEF-2 complexes are present in cytoplasm and nucleus, and their abundance increases when neurons experience oxidative stress. The nuclear localization of active eEF-2 depends upon its interaction with p53, as cells lacking p53 contain less active eEF-2 in the nuclear compartment. Overexpression of eEF-2 in hippocampal neurons results in increased nuclear levels of eEF-2, and decreased cell death following exposure to CH. Our results reveal novel molecular mechanisms controlling the differential subcellular localization and activity state of eEF-2 that may influence the survival status of neurons during periods of elevated oxidative stress.España, Ministerio de Ciencia e Innovación BFU2010-20882.España, Ministerio de Educación, Cultura y Deporte postdoctoral fellowship (EX2009-0918

Reducible M-curves for Le-networks in the totally-nonnegative Grassmannian and KP-II multiline solitons

We associate real and regular algebraic--geometric data to each multi--line soliton solution of Kadomtsev-Petviashvili II (KP) equation. These solutions are known to be parametrized by points of the totally non--negative part of real Grassmannians $Gr^{TNN}(k,n)$. In Ref.[3] we were able to construct real algebraic-geometric data for soliton data in the main cell $Gr^{TP} (k,n)$ only. Here we do not just extend that construction to all points in $Gr^{TNN}(k,n)$, but we also considerably simplify it, since both the reducible rational $M$-curve $\Gamma$ and the real regular KP divisor on $\Gamma$ are directly related to the parametrization of positroid cells in $Gr^{TNN}(k,n)$ via the Le-networks introduced by A. Postnikov in Ref [62]. In particular, the direct relation of our construction to the Le--networks guarantees that the genus of the underlying smooth $M$-curve is minimal and it coincides with the dimension of the positroid cell in $Gr^{TNN}(k,n)$ to which the soliton data belong to. Finally, we apply our construction to soliton data in $Gr^{TP}(2,4)$ and we compare it with that in Ref [3].Comment: 72 pages; several figures. We have decided to split our paper in Arxiv:1801.00208v1 into two parts. This preprint is the fully revised version of the first part of it. In the next version Arxiv:1801.00208 this part will be removed V2: Minor modifications, proof of Theorem 3.1 improve

Potential for ill-posedness in several 2nd-order formulations of the Einstein equations

Second-order formulations of the 3+1 Einstein equations obtained by eliminating the extrinsic curvature in terms of the time derivative of the metric are examined with the aim of establishing whether they are well posed, in cases of somewhat wide interest, such as ADM, BSSN and generalized Einstein-Christoffel. The criterion for well-posedness of second-order systems employed is due to Kreiss and Ortiz. By this criterion, none of the three cases are strongly hyperbolic, but some of them are weakly hyperbolic, which means that they may yet be well posed but only under very restrictive conditions for the terms of order lower than second in the equations (which are not studied here). As a result, intuitive transferences of the property of well-posedness from first-order reductions of the Einstein equations to their originating second-order versions are unwarranted if not false.Comment: v1:6 pages; v2:7 pages, discussion extended, to appear in Phys. Rev. D; v3: typos corrected, published versio