Radon Transport from Soil to Air and Monte-Carlo Simulation

The exhalation of geochemical entities from soil to air is significant to understand Lithosphere-Atmospheric relationships. Some of these geochemical entities are capable of modifying the lower atmosphere, and they are employed in various studies. Radon is one of the geochemical gasses widely recognized as a dominant ionization source in near ground regions of the troposphere. The steady state Rn transport equation is considered in many cases for estimating Rn migration from soil to air on the condition that the time evolution is ignored. A method is proposed for estimating radon space-time transport from soil to air. This is achieved by solving the radon transport equation in soil with special boundary conditions. Similar results are obtained with some experimented models, as well as reported radon values in literature for some set of parameter combinations. Strengths and limitations of the method are discussed. The model is useable to study Lithosphere-Atmosphere relationships. It can also be significant in other studies like the Global Electric Circuit or Seismo-Ionospheric studies.

CDT Quantum Toroidal Spacetimes: An Overview

Lattice formulations of gravity can be used to study non-perturbative aspects of quantum gravity. Causal Dynamical Triangulations (CDT) is a lattice model of gravity that has been used in this way. It has a built-in time foliation but is coordinate-independent in the spatial directions. The higher-order phase transitions observed in the model may be used to define a continuum limit of the lattice theory. Some aspects of the transitions are better studied when the topology of space is toroidal rather than spherical. In addition, a toroidal spatial topology allows us to understand more easily the nature of typical quantum fluctuations of the geometry. In particular, this topology makes it possible to use massless scalar fields that are solutions to Laplace’s equation with special boundary conditions as coordinates that capture the fractal structure of the quantum geometry. When such scalar fields are included as dynamical fields in the path integral, they can have a dramatic effect on the geometry.

Discrete-to-continuum limits of planar disclinations

In materials science, wedge disclinations are defects caused by angular mismatches in the crystallographic lattice. To describe such disclinations, we introduce an atomistic model in planar domains. This model is given by a nearest-neighbor-type energy for the atomic bonds with an additional term to penalize change in volume. We enforce the appearance of disclinations by means of a special boundary condition. Our main result is the discrete-to-continuum limit of this energy as the lattice size tends to zero. Our proof relies on energy relaxation methods. The main mathematical novelty of our proof is a density theorem for the special boundary condition. In addition to our limit theorem, we construct examples of planar disclinations as solutions to numerical minimization of the model and show that classical results for wedge disclinations are recovered by our analysis.

K3 surfaces from configurations of six lines in $\mathbb{P}^{2}$ and mirror symmetry II—$\lambda _{K3}$-functions

We continue our study on the hypergeometric system $E(3,6)$ that describes period integrals of the double cover family of K3 surfaces. Near certain special boundary points in the moduli space of the K3 surfaces, we construct the local solutions and determine the so-called mirror maps expressing them in terms of genus 2 theta functions. These mirror maps are the K3 analogues of the elliptic $\lambda $-function. We find that there are two nonisomorphic definitions of the lambda functions corresponding to a flip in the moduli space. We also discuss mirror symmetry for the double cover K3 surfaces and their higher dimensional generalizations. A follow-up paper will describe more details of the latter.