Clifford Fourier Transformation and Uncertainty Principle for the Clifford Geometric Algebra Cl 3,0

First, the basic concept of the vector derivative in geometric algebra is introduced. Second, beginning with the Fourier transform on a scalar function we generalize to a real Fourier transform on Clifford multivector-valued functions Third, we show a set of important properties of the Clifford Fourier transform on Cl_3,0 such as differentiation properties, and the Plancherel theorem. Finally, we apply the Clifford Fourier transform properties for proving an uncertainty principle for Cl_3,0 multivector functions.     

Water adsorption and desorption on microporous solids at elevated temperature

Summary An accurate gravimetric method was used to explore water adsorption/desorption isotherms between 105 and to 250°C for a number of synthetic and natural porous solids including controlled pore glass, activated carbon fiber monoliths, natural zeolites, pillared clay, and geothermal reservoir rocks. The main goal of this work was to evaluate water adsorption results, in particular temperature dependence of hysteresis, for relatively uniform, nano-structured solids, in the context of other state-of-the-art experimental and modeling methods including nitrogen adsorption, spectroscopy, neutron scattering, and molecular simulation. Since no single method is able to provide a complete characterization of porous materials, a combination of approaches is needed to achieve progress in understanding the fluid-solid interactions on the way to developing a predictive capability.     

Minimum dissipative relaxed states in toroidal plasmas

Relaxation of toroidal discharges is described by the principle of minimum energy dissipation together with the constraint of conserved global helicity. The resulting Euler-Lagrange equation is solved in toroidal coordinates for an axisymmetric torus by expressing the solutions in terms of Chandrasekhar-Kendall (C-K) eigenfunctions analytically continued in the complex domain. The C-K eigenfunctions are obtained as hypergeometric functions that are solutions of scalar Helmholtz equation in toroidal coordinates in the large aspect-ratio approximation. Equilibria are constructed by assuming the current to vanish at the edge of plasma. For the m=0, n=0 (m and n are the poloidal and toroidal mode numbers respectively) relaxed states, the magnetic field, current, q (safety factor) and pressure profiles are calculated for a given value of aspect-ratio of the torus and for different values of the eigenvalue λ _{r 0}. The new feature of the present model is that solutions allow for both tokamak as well as RFP-like behaviour with increase in the values of λ _{r 0}, which is related directly to volt-sec in the experiment.     

Phase Behavior of Aqueous Na–K–Mg–Ca–Cl–NO<Subscript>3</Subscript> Mixtures: Isopiestic Measurements and Thermodynamic Modeling

A comprehensive model has been established for calculating thermodynamic properties of multicomponent aqueous systems containing the Na⁺, K⁺, Mg²⁺, Ca²⁺, Cl⁻ and NO₃⁻ ions. The thermodynamic framework is based on a previously developed model for mixed-solvent electrolyte solutions. The framework has been designed to reproduce the properties of salt solutions at temperatures ranging from the freezing point to 300 °C and concentrations ranging from infinite dilution to the fused salt limit. The model has been parameterized using a combination of an extensive literature database and new isopiestic measurements for thirteen salt mixtures at 140 °C. The measurements have been performed using Oak Ridge National Laboratory’s (ORNL) previously designed gravimetric isopiestic apparatus, which can also detect solid phase precipitation. In addition to various Na–K–Mg–Ca–Cl–NO₃ systems, results are reported for LiCl solutions. Water activities are reported for mixtures with a fixed ratio of salts as a function of the total apparent salt mole fraction. The isopiestic measurements reported here simultaneously reflect two fundamental properties of the system, i.e., the activity of water as a function of solution concentration and the occurrence of solid–liquid transitions. The thermodynamic model accurately reproduces the new isopiestic data as well as literature data for binary, ternary and higher-order subsystems. Because of its high accuracy in calculating vapor–liquid and solid–liquid equilibria, the model is suitable for studying deliquescence behavior of multicomponent salt systems.