Definable E0 classes at arbitrary projective levels

Using a modification of the invariant Jensen forcing of [11], we define a model of ZFC, in which, for a given $n\ge 3$, there exists a lightface ${\Pi }_n^1$-set of reals, which is a ${\mathsf{E}}_0$-equivalence class, hence a countable set, and which does not contain any OD element, while every non-empty countable ${\Sigma }_n^1$-set of reals is constructible, hence contains only OD reals.     

Countable OD sets of reals belong to the ground model

It is true in the Cohen, Solovay-random, dominaning, and Sacks generic extension, that every countable ordinal-definable set of reals belongs to the ground universe. It is true in the Solovay collapse model that every non-empty OD countable set of sets of reals consists of \(\text {OD}\) elements.     

Mirror symmetry formulae for the elliptic genus of complete intersections

In this paper, we calculate the elliptic genus of certain completeintersections in products of projective spaces. We show thatit is equal to the elliptic genus of the Landau–Ginzburgmodels that are, according to Hori and Vafa, mirror partnersof these complete intersections. This provides additional evidenceof the validity of their construction. Received July 6, 2007.     

Virtual Knots and Infinite-Dimensional Lie Algebras

In the present work, we construct an invariant of virtual knots valued in (infinite-dimensional) Lie Algebras and establish some properties of it. This leads to some heuristic ideas how to construct quandles and extract (virtual) link invariants.     

BRST Operator for Quantum Lie Algebras: Relation to the Bar Complex

Quantum Lie algebras (an important class of quadratic algebras arising in the Woronowicz calculus on quantum groups) are generalizations of Lie (super)algebras. Many notions from the theory of Lie (super)algebras admit quantum generalizations. In particular, there is a BRST operator Q (Q ²=0) that generates the differential in the Woronowicz theory and gives information about (co)homologies of quantum Lie algebras. In our previous papers, we gave and solved a recursive relation for the operator Q for quantum Lie algebras. Here, we consider the bar complex for q-Lie algebras and its subcomplex of q-antisymmetric chains. We establish a chain map (which is an isomorphism) of the standard complex for a q-Lie algebra to the subcomplex of the antisymmetric chains. The construction requires a set of nontrivial identities in the group algebra of the braid group. We also discuss a generalization of the standard complex to the case where a q-Lie algebra is equipped with a grading operator.     

Estimation of the atmospheric greenhouse gas spatial distribution in the Arctic using a back trajectory model

We present a method and results of the retrieval of average effective fields of atmospheric impurity concentration using a passive wind sensing (remote sensing) numeric technology referred to as Fluid Location of the Atmosphere (FLA). The quasi-two-dimensional problem of reconstruction of the spatial distribution of the greenhouse gases assuming the diffusion Peclet number infinitely large was solved numerically. The study is based on in-situ measurements of the atmospheric methane and carbon dioxide during the expeditions to Belyy Island in the Kara Sea in July and August of 2016 to 2017. The differences and common features of CH₄ and CO₂ spatial distribution in this region of the Arctic during specified periods were analyzed. CH₄ concentrations tended to decrease with moving from the continent to the remote sea areas. For CO₂ on the contrary, lower values were observed over the continent, and they increased with a distance from the coastline. For both greenhouse gases, average atmospheric concentrations increased in 2017 relative to 2016.     

The cohomology of the Morava stabilizer group at the prime 3

We compute the cohomology of the Morava stabilizer group at the prime by resolving it by a free product and analyzing the ``relation module.'

     

“Tepid” Geysers above salt cavernsGeysers tièdes au-dessus des cavités salines

The formation of a brine geyser erupting from the wellhead of a large underground salt cavern is described. In most cases, the brine outflow from an opened cavern is slow; it results from the cavern creep closure and the thermal expansion of the cavern brine. These two processes are smooth; however, the brine outflow often is bumpy, as it is modulated by atmospheric pressure variations that generate an elastic increase (or decrease) of both cavern and brine volumes. In addition, when the flow is fast enough, the brine thermodynamic behavior in the wellbore is adiabatic. The cold brine expelled from the cavern wellhead is substituted with warm brine entering the borehole bottom, resulting in a lighter brine column. The brine outflow increases. In some cases, the flow becomes so fast that inertia terms must be taken into account. A geyser forms, coming to an end when the pressure in the cavern has dropped sufficiently. A better picture is obtained when head losses are considered. A closed-form solution can be reached. This proves that two cases must be distinguished, depending on whether the cold brine initially contained in the wellbore is expelled fully or not. It can also be shown that geyser formation is a rare event, as it requires both that the wellbore be narrow and that the cavern be very compressible. This study stemmed from an actual example in which a geyser was observed. However, scarce information is available, making any definite interpretation difficult.