Euclidean space Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called “Euclidean geometry“, which is the study of the relationships between angles and distances in space. Euclid first developed “plane geometry” which dealt with the geometry of two-dimensional objects on a flat surface. He then went on to develop “solid geometry” which analyzed the geometry of three-dimensional objects. All of the axioms of Euclid have been encoded into an abstract mathematical space known as a two- or three-dimensional Euclidean space. These mathematical spaces may be extended to apply to any dimension, and such a space is called an n-dimensional Euclidean space or an n-space. Euclidean structure Euclidean space is more than just a real coordinate space. In order to apply Euclidean geometry one needs to be able to talk about the distances between points and the angles between lines or

In situ Air Sparging (IAS) and Soil Vapor Extraction (SVE) systems in the Subsurface Introduction Groundwater and soil contamination Non-Aqueous Phase Liquids (NAPLs) Heterogeneous porous media (subsurface system): Significant heterogeneities in terms of physicochemical and geological transport properties In situ Air Sparging (IAS) and Soil Vapor Extraction (SVE) Study objectives Three-dimensional models for groundwater flow and contaminant transport in subsurface system Optimization model for integrated IAS and SVE technology User-friendly window environment for input data generation Numerical simulation Finite Element Method (FEM) Monte Carlo Technique Optimization & Modified Genetic Algorithm High Performance Computing (HPC) system for computation-oriented procedure Program language : C/C++ & MS Visual C++ Application and Uses 3D numerical simulation in combined saturated and vadose zones Optimized remediation strategies for integrated IAS and SVE system Graphical environments for end users Practial remediation system setup and its application to contaminated sites

Concentration Evolution of Gas Species within a Collapsing Bubble in a Liquid Medium Wonyong Jang and Mustafa M. Aral Multimedia Environmental Simulation Laboratory School of Civil and Environmental Engineering Georgia Institute of Technology, Atlanta, GA, USA Abstract: In this study numerical methods are used to investigate the relationship between chemical concentration of gas species within a cavitating bubble, equilibrium radius of the gas bubble and pressure variations in the ambient liquid. For this purpose, governing equations are developed to describe the dynamic equilibrium of a bubble in a flowing fluid and mass transfer between gas and liquid phases, where it was assumed that gases undergo isothermal compression, obey the ideal gas law, Henry law. It is further assumed that the concentration of each phase within the bubble is uniform. The resulting nonlinear equations are solved using implicit Trapezoidal method with Newton iteration. Four gas species are modeled under various initial