Solitonic asymptotics for the Korteweg-de Vries equation in the small dispersion limit

We study the small dispersion limit for the Korteweg-de Vries (KdV) equation $u_t+6uu_x+\epsilon^{2}u_{xxx}=0$ in a critical scaling regime where $x$ approaches the trailing edge of the region where the KdV solution shows oscillatory behavior. Using the Riemann-Hilbert approach, we obtain an asymptotic expansion for the KdV solution in a double scaling limit, which shows that the oscillations degenerate to sharp pulses near the trailing edge. Locally those pulses resemble soliton solutions of the KdV equation.Comment: 25 pages, 4 figure

The existence of a real pole-free solution of the fourth order analogue of the Painleve I equation

We establish the existence of a real solution y(x,T) with no poles on the real line of the following fourth order analogue of the Painleve I equation, x=Ty-({1/6}y^3+{1/24}(y_x^2+2yy_{xx})+{1/240}y_{xxxx}). This proves the existence part of a conjecture posed by Dubrovin. We obtain our result by proving the solvability of an associated Riemann-Hilbert problem through the approach of a vanishing lemma. In addition, by applying the Deift/Zhou steepest-descent method to this Riemann-Hilbert problem, we obtain the asymptotics for y(x,T) as x\to\pm\infty.Comment: 27 pages, 5 figure

In vitro formation of Ca-oxalates and the mineral glushinskite by fungal interaction with carbonate substrates and seawater

This study investigates the in vitro formation of Ca-oxalates and glushinskite through fungal interaction with carbonate substrates and seawater as a process of biologically induced metal recycling and neo-mineral formation. The study also emphasizes the role of the substrates as metal donors. In the first experiment, thin sections prepared from dolomitic rock samples of Terwagne Formation (Carboniferous, Viséan, northern France) served as substrates. The thin sections placed in Petri dishes were exposed to fungi grown from naturally existing airborne spores. In the second experiment, fungal growth and mineral formation was monitored using only standard seawater (SSW) as a substrate. Fungal growth media consisted of a high protein/carbohydrates and sugar diet with demineralized water for irrigation. Fungal growth process reached completion under uncontrolled laboratory conditions. The newly formed minerals and textural changes caused by fungal attack on the carbonate substrates were investigated using light and scanning electron microscopy (SEM-EDX), x-ray diffraction (XRD) and Raman spectroscopy. The fungal interaction and attack on the dolomitic and seawater substrates resulted in the formation of Ca-oxalates (weddellite CaC2O4·2(H2O), whewellite (CaC2O4·(H2O)) and glushinskite MgC2O4·2(H2O) associated with the destruction of the original hard substrates and their replacement by the new minerals. Both of Ca and Mg were mobilized from the experimental substrates by fungi. This metal mobilization involved a recycling of substrate metals into newly formed minerals. The biochemical and diagenetic results of the interaction strongly marked the attacked substrates with a biological fingerprint. Such fingerprints are biomarkers of primitive life. The formation of glushinskite is of specific importance that is related, besides its importance as a biomineral bearing a recycled Mg, to the possibility of its transformation through diagenetic pathway into an Mg carbonate. This work is the first report on the in vitro formation of the mineral glushinskite through fungal interaction with carbonate and seawater substrates. Besides recording the detailed Raman signature of various crystal habits of Mg- and Ca-oxalates, the Raman spectroscopy proved two new crystal habits for glushinskite. The results of this work document the role of microorganisms as metal recyclers in biomineralization, neo-mineral formation, sediment diagenesis, bioweathering and in the production of mineral and diagenetic biomarkers. They also reveal the capacity of living fungi to interact with liquid substrates and precipitate new minerals

Universality of a double scaling limit near singular edge points in random matrix models

We consider unitary random matrix ensembles Z_{n,s,t}^{-1}e^{-n tr V_{s,t}(M)}dM on the space of Hermitian n x n matrices M, where the confining potential V_{s,t} is such that the limiting mean density of eigenvalues (as n\to\infty and s,t\to 0) vanishes like a power 5/2 at a (singular) endpoint of its support. The main purpose of this paper is to prove universality of the eigenvalue correlation kernel in a double scaling limit. The limiting kernel is built out of functions associated with a special solution of the P_I^2 equation, which is a fourth order analogue of the Painleve I equation. In order to prove our result, we use the well-known connection between the eigenvalue correlation kernel and the Riemann-Hilbert (RH) problem for orthogonal polynomials, together with the Deift/Zhou steepest descent method to analyze the RH problem asymptotically. The key step in the asymptotic analysis will be the construction of a parametrix near the singular endpoint, for which we use the model RH problem for the special solution of the P_I^2 equation. In addition, the RH method allows us to determine the asymptotics (in a double scaling limit) of the recurrence coefficients of the orthogonal polynomials with respect to the varying weights e^{-nV_{s,t}} on \mathbb{R}. The special solution of the P_I^2 equation pops up in the n^{-2/7}-term of the asymptotics.Comment: 32 pages, 3 figure

Asymptotics for a special solution to the second member of the Painleve I hierarchy

We study the asymptotic behavior of a special smooth solution y(x,t) to the second member of the Painleve I hierarchy. This solution arises in random matrix theory and in the study of Hamiltonian perturbations of hyperbolic equations. The asymptotic behavior of y(x,t) if x\to \pm\infty (for fixed t) is known and relatively simple, but it turns out to be more subtle when x and t tend to infinity simultaneously. We distinguish a region of algebraic asymptotic behavior and a region of elliptic asymptotic behavior, and we obtain rigorous asymptotics in both regions. We also discuss two critical transitional asymptotic regimes.Comment: 19 page

Stable isotopic composition of bivalve shell organic matrix: <i>Mytilus edulis</i> collected along the Scheldt estuary

Bivalve shells are biostructures composed of a mineral and an organic phase. For paleoclimatology applications, the mineral part (carbonates) is most widely studied. In contrast, understanding of the composition and the proxyfunction of the organic matrix is much less developed. The quantity of organic matrix in shells is relatively small compared to the mineral phase (a few wt %) and the biochemical composition is quite complex, consisting mainly of sugars and proteins. Lipids, which represent a small fraction of the organic matrix, are rather poorly known. We studied the potential of stable isotope composition (C, N, H) of bulk organic matrix and specific lipid compounds of Mytilis edulis shells, as environmental and climatic proxies, with special focus on the effects due to changing salinity. Mytilus specimens were collected along the salinity gradient of the Scheldt estuary (The Netherlands) and we analysed the isotopic composition of the organic matrix and associated specific lipid compounds and related these to averaged physico-chemical characteristics of the water, in particular salinity. We discuss these relationships in the light of their usefulness as proxies for reconstructing past environmental conditions

Moments of Moments of the Characteristic Polynomials of Random Orthogonal and Symplectic Matrices

Using asymptotics of Toeplitz+Hankel determinants, we establish formulae for the asymptotics of the moments of the moments of the characteristic polynomials of random orthogonal and symplectic matrices, as the matrix-size tends to infinity. Our results are analogous to those that Fahs obtained for random unitary matrices in [14]. A key feature of the formulae we derive is that the phase transitions in the moments of moments are seen to depend on the symmetry group in question in a significant way

Intra- and intertaxon stable O and C isotope variability of fossil fish otoliths: an early Eocene test case

Knowledge of basic data variability is essential for the interpretation of any proxy-based paleotemperature record. To evaluate this for d18O stable isotope paleothermometry based on early Paleogene fish otoliths from marginal marine environments, an intra- and interspecific stable O and C isotope study was performed at a single locality in the southern North Sea Basin (Ampe Quarry, Egem, Belgium), where shallow marine sands and silts are exposed. The age of the deposits is early late Ypresian (ca. 50.9 Ma) and falls within the early Eocene climatic optimum (EECO) interval. In each of four fossiliferous levels sampled, the same three otolith species were analyzed (Platycephalus janeti, Paraconger papointi and “genus Neobythitinorum” subregularis). Intrataxon stable isotope spread amounts on average 2.50-3.00‰ for all taxa and is present in all levels. This implies that each sample level comprises substantial variability, which can be attributed to a combination of temporal and taphonomic effects. More importantly, intertaxon offsets of 4.60‰ in d13C and 2.20‰ in d18O between the mean values of the three otolith species are found, with “N.” subregularis representing more positive values relative to the other species. We hypothesize that freshwater influence of coastal waters is the most likely cause for these discrepancies. Similar analyses on two coastal bivalve species (Venericardia sulcata and Callista laevigata) corroborate this hypothesis. Accordingly, d18O values measured on “N.” subregularis otoliths probably represent a more open oceanic signal, and therefore seem well-suited for d18O stable isotope paleothermometry. This study highlights the importance of investigating data variability of a biogenic carbonate paleotemperature proxy at the species level, before applying paleotemperature equations and interpreting the outcome