We present a network flow model to compute transport, through a pore network, of a compositional fluid consisting of water with a dissolved hydrocarbon gas. The model captures single-phase flow (below local bubble point conditions) as well as the genesis and migration of the gas phase when bubble point conditions are achieved locally. Constant temperature computational tests were run on simulated 2D and 3D micro-networks near bubble point pressure conditions. In the 2D simulations which employed a homogeneous network, negligible capillary pressure, and linear relative permeability relations, the observed concentration of CO2 dissolved in the liquid phase throughout the medium was linearly related to the liquid pressure. In the case of no gravity, the saturation of the gas phase throughout the medium was also linearly related to the liquid pressure; under gravity, the relationship became nonlinear in regions where buoyancy forces were significant. The 3Dheterogeneous network model had non-negligible capillary pressure and nonlinear relative permeability functions. While 100 % of the CO2 entered the 3D network dissolved in the liquid phase, 25 % of the void space was occupied by gas phase and 47 % of the CO2 exiting the outlet face did so via the gaseous phase after 500 s of simulation time.

We give a complete classification of (co)torsion pairs in finite $2$-Calabi-Yau triangulated categories with maximal rigid objects which are not cluster tilting. These finite $2$-Calabi-Yau triangulated categories are divided into two main classes: one denoted by $\A_{n,t}$ called of type $A$, and the other denoted by $D_{n,t}$ called of type $D$. By using the geometric model of cluster categories of type $A, $ or type $D$, we give a geometric description of (co)torsion pairs in $\A_{n,t}$ or $D_{n,t}$ respectively, via defining the periodic Ptolemy diagrams. This allows to count the number of (co)torsion pairs in these categories. Finally, we determine the hearts of (co)torsion pairs in all finite $2$-Calabi-Yau triangulated categories with maximal rigid objects which are not cluster tilting via quivers and relations.

We apply our model of quantum gravity to an AdS black hole resulting in a wave equation in a quantum spacetime which has a sequence of solutions that can be expressed as a product of stationary and temporal eigenfunctions. The stationary eigenfunctions can be interpreted as radiation and the temporal as gravitational waves. The event horizon corresponds in the quantum model to a Cauchy hypersurface that can be crossed by causal curves in both directions such that the information paradox does not occur.

In a recent paper we quantized the interaction of gravity with a Yang-Mills and Higgs field and obtained as a result a gravitational wave equation in a globally hyperbolic spacetime. Assuming that the Cauchy hypersurfaces are compact we proved a spectral resolution for the wave equation by applying the method of separation of variables. In this paper we extend the results to the case when the Cauchy hypersurfaces are non-compact by considering a Gelfand triplet and applying the nuclear spectral theorem.

We quantize the interaction of gravity with a Yang-Mills and Higgs field using canonical quantization. Similar to the approach in a previous paper we discard the Wheeler-DeWitt equation and express the Hamilton constraint by the evolution equation of the mean curvature of the hypersurfaces in the foliation defined by the Hamiltonian setting. Expressing the time derivative of the mean curvature with the help of the Poisson brackets the canonical quantization of this equation leads to a wave equation in $Q=(0,\infty)\times \so$, where $\so$ is one of the Cauchy hypersurfaces in the Hamiltonian setting. The wave equation describes the interaction of an arbitrary Riemannian metric in $\so$ and a given Yang-Mills and Higgs field. If the metric is complete $Q$ is globally hyperbolic. In case $\so$ is compact we also prove a spectral resolution of the wave equation and establish sufficient conditions guaranteeing a mass gap.