The matrix sign function is a matrix-valued function that generalizes the scalar sign function to matrices. For a square matrix \( A \), the matrix sign function, denoted as \( \text{sign}(A) \), is defined in terms of the eigenvalues of the matrix.

The Moffat distribution is a statistical distribution used primarily in the fields of astrophysics and image processing. It is often employed to model the point spread function (PSF) of optical systems, especially in the context of astronomical observations. The Moffat function is characterized by its ability to describe the spread of light from a point source, allowing for a profile that has more pronounced "wings" compared to Gaussian functions, which decay more rapidly.

The Sigma-D relation, also known as the \(\Sigma-D\) relation or the \(\Sigma-D\) correlation, is a concept in astrophysics and cosmology that describes a relationship between the surface density of galaxies (or their stellar components) and their dynamical properties, particularly their rotational velocity or other measures of mass distribution.

Velocity dispersion is a measure of the range of velocities within a group of objects, such as stars in a galaxy or galaxies in a cluster. It quantifies how much the velocities of the objects deviate from the average velocity of the group. In a more technical sense, it is defined as the standard deviation of the velocities of the objects in the sample. In astrophysics, velocity dispersion is an important metric because it provides insights into the dynamics and mass distribution of celestial bodies.

The Allen-Cahn equation is a partial differential equation that describes the evolution of phase interfaces in materials science and represents the dynamics of gas-liquid phase transitions typically in the context of, but not limited to, crystallization processes. It is an example of a conserved order parameter system and is derived from the principles of thermodynamics and variational calculus.

The Basset–Boussinesq–Oseen (BBO) equation is a mathematical model that describes the motion of small particles suspended in a viscous fluid. This equation accounts for the effects of inertial and viscous forces acting on the particles, along with the interaction between the particles and the surrounding fluid. It is particularly important in the fields of fluid mechanics and particle dynamics, especially in scenarios where the Reynolds number is low.

The Borda-Carnot equation describes the relationship between the temperature, pressure, and specific properties of a fluid in a thermodynamic context, particularly for a fluid undergoing adiabatic (no heat transfer) expansion or compression. It is commonly associated with the performance of turbines and compressors. The equation itself typically relates how the enthalpy, pressure, and temperature of the fluid change during these processes.

The Buckley–Leverett equation is a fundamental equation in petroleum engineering and reservoir engineering that describes the movement of two-phase fluids (typically oil and water) in porous media. It models the flow behavior of immiscible fluids in a reservoir when one fluid displaces another, commonly used to analyze waterflooding operations during oil recovery. The equation is derived from the conservation of mass principle and reflects the dynamics of the interfaces between the two fluids.

The Darcy–Weisbach equation is used in fluid mechanics to calculate the pressure loss (or head loss) due to friction in a pipeline or duct. It is an essential equation for engineers and designers working with fluid flow systems to assess the efficiency and performance of piping and ductwork.

The Davey-Stewartson equation is a nonlinear partial differential equation that arises in the study of wave phenomena, particularly in the context of two-dimensional surface water waves. It is a generalization of the nonlinear Schrödinger equation and describes the evolution of complex wave packets in a two-dimensional setting.

The Navier-Stokes equations describe the motion of fluid substances and are fundamental in fluid mechanics. They are derived from the principles of conservation of mass, momentum, and energy. Here, I'll summarize how these equations are derived step-by-step. ### 1. Conservation of Mass (Continuity Equation) The principle of mass conservation states that the mass of a fluid in a control volume must remain constant over time if no mass enters or leaves the volume.