Ax-Schanuel condition in arbitrary characteristic

We prove a positive characteristic version of Ax's theorem on the intersection of an algebraic subvariety and an analytic subgroup of an algebraic group. Our result is stated in a more general context of a formal map between an algebraic variety and an algebraic group. We derive transcendence results of Ax-Schanuel type

The origin of peculiar molecular bands in cool DQ white dwarfs

The DQ white dwarfs are stars whose atmosphere is enriched with carbon, which for cool stars ($T_{\rm eff}<8000\rm \, K$) is indicated by the Swan bands of $\rm C_2$ in the optical part of their spectra. With decreasing effective temperature these molecular bands undergo a significant blueshift ($\sim 100-300 \AA$). The origin of this phenomenon has been disputed over the last two decades and has remained unknown. We attempt to address this problem by investigating the impact of dense helium on the spectroscopic properties of molecular carbon under the physical conditions encountered inside helium-rich, fluid-like atmospheres of cool DQ white dwarfs. We found that the electronic transition energy $T_e$ increases monotonically with the helium density ($\Delta T_{\rm e}\rm\, (eV)\sim1.6 \, \it \rho \rm \, (g/cm^3)$). This causes the Swan absorption to occur at shorter wavelengths compared with unperturbed $\rm C_2$. On the other hand the pressure-induced increase in the vibrational frequency is insufficient to account for the observed Swan bands shifts. This is consistent with the observations and indicates that the observed Swan-like molecular bands are most likely the pressure-shifted bands of $\rm C_2$.Comment: 4 pages, 5 figures, accepted for publication in A&A letter

On the Dissociation Equilibrium of H2 in Very Cool, Helium-Rich White Dwarf Atmospheres

We investigate the dissociation equilibrium of $\rm H_2$ in very cool, helium-rich white dwarf atmospheres. We present the solution of the non-ideal chemical equilibrium for the dissociation of molecular hydrogen in a medium of dense helium. We find that at the photosphere of cool white dwarfs of $T_{\rm eff}\rm=4000 K$, the non-ideality results in an increase of the mole fraction of molecular hydrogen by up to a factor of $\sim 10$, compared to the equilibrium value for the ideal gas. This increases the $\rm H_{2}-He$ CIA opacity by an order of magnitude and will affect the determination of the abundance of hydrogen in very cool, helium-rich white dwarfs.Comment: 9 pages, 5 figures, 1 table; Accepted for publication in The Astrophysical Journa

Integrating Hasse-Schmidt derivations

We study integrating (that is expanding to a Hasse-Schmidt derivation) derivations, and more generally truncated Hasse-Schmidt derivations, satisfying iterativity conditions given by formal group laws. Our results concern the cases of the additive and the multiplicative group laws. We generalize a theorem of Matsumura about integrating nilpotent derivations (such a generalization is implicit in work of Ziegler) and we also generalize a theorem of Tyc about integrating idempotent derivations

Existentially closed fields with G-derivations

We prove that the theories of fields with Hasse-Schmidt derivations corresponding to actions of formal groups admit model companions. We also give geometric axiomatizations of these model companions.Comment: In version 2: new proof of (the current) Proposition 3.3

Ab initio Stellar Astrophysics: Reliable Modeling of Cool White Dwarf Atmospheres

Over the last decade {\it ab initio} modeling of material properties has become widespread in diverse fields of research. It has proved to be a powerful tool for predicting various properties of matter under extreme conditions. We apply modern computational chemistry and materials science methods, including density functional theory (DFT), to solve lingering problems in the modeling of the dense atmospheres of cool white dwarfs ($T_{\rm eff}\rm <7000 \, K$). Our work on the revision and improvements of the absorption mechanisms in the hydrogen and helium dominated atmospheres resulted in a new set of atmosphere models. By inclusion of the Ly-$\rm \alpha$ red wing opacity we successfully fitted the entire spectral energy distributions of known cool DA stars. In the subsequent work we fitted the majority of the coolest stars with hydrogen-rich models. This finding challenges our understanding of the spectral evolution of cool white dwarfs. We discuss a few examples, including the cool companion to the pulsar PSR J0437-4715. The two problems important for the understanding of cool white dwarfs are the behavior of negative hydrogen ion and molecular carbon in a fluid-like, helium dominated medium. Using {\it ab initio} methods we investigate the stability and opacity of these two species in dense helium. Our investigation of $\rm C_2$ indicates that the absorption features observed in the ``peculiar'' DQp white dwarfs resemble the absorption of perturbed $\rm C_2$ in dense helium.Comment: 6 pages, 4 figures, submitted to proceedings of 17th European White Dwarf Workshop, Tuebingen, Germany 201

Distribution of minor metallic elements within waste incineration bottom ashes defined by WDX/EDX spectrometry

A number of metallic elements are inherited from waste during thermal treatment and concentrated in the incineration residues. Because the major part of the incineration residue mass are bottom ashes (BAs), their study is of great importance from the point of view of their environmental impact or resource potential. The general focus of this study was on the minor metallic elements present in BAs. They co-occurred with main phases and often determined the inherited potential of the material. The analysed residues were produced from municipal and industrial waste. The BAs were studied using spectroscopic methods of chemical microanalysis: energy dispersive X-ray spectrometry (EDX) and wavelength dispersive X-ray spectrometry (WDX). Both the main and minor metallic elements were concentrated in metallic components. They were typically present as separate grains and metallic inclusions (commonly in the glass matrix of the grains) ranging in size from several to hundreds of micrometres. Despite Fe-, Al- and Cu-rich occurrences, metallic elements rarely occurred in fragments composed of a single element. Their main forms of occurrence were alloy grains, admixtures in polymetallic occurrences and micro-inclusions in glassy matrix. The content of particular elements in those forms was investigated and described in greater detail. Even though two types of bottom ash were formed from different types of waste and differences in used technologies were present, the obtained materials contained metallic components having similar attributes. Elevated concentrations of not only Fe and Al, but also Ti, Cu and Zn, allow us to consider bottom ash as a promising material from the point of view of metallic elements' recovery (e.g. by the physical concentration of elements through gravity or magnetic methods)