The Ellis-Numakura lemma is a result in the field of dynamical systems, particularly in the study of topological dynamics and the behavior of semigroups. It is named after mathematicians John Ellis and Kōji Numakura. The lemma deals with the connection between a compact space and the continuous semigroups acting on it, providing conditions under which certain properties hold for the invariant measures of these semigroups.

The commutation theorem for traces is a result in linear algebra and functional analysis, particularly within the context of operator theory. It deals with the properties of the trace operator, which is a map that takes a square matrix (or, more generally, a bounded operator on a Hilbert space) and sums its diagonal elements. The commutation theorem states that if two operators \( A \) and \( B \) commute (i.e.

Axiom A is a concept in dynamical systems introduced by mathematician Stephen Smale in the 1960s. It describes a class of systems that have certain hyperbolic properties, which means they exhibit chaotic behavior yet retain a structured dynamic. More specifically, Axiom A refers to a dynamical system where: 1. The system's phase space can be decomposed into an unstable manifold and stable manifold, making it possible to analyze orbits and their behavior under iteration.

Annual Premium Equivalent (APE) is a financial metric commonly used in the insurance and financial services industry, particularly in the context of measuring and comparing the performance of life insurance products and sales. APE allows companies to evaluate the value of both regular premium and single premium life insurance policies on a standardized basis.

Equivalent units are a concept used in cost accounting, particularly in process costing, to measure the work done during a period in terms of fully completed units. Since production processes often involve a mix of complete and incomplete units at the end of an accounting period, equivalent units allow businesses to assign costs more accurately.

Washburn's equation describes the capillary action of liquids in porous media or thin tubes. It quantifies the rate at which a liquid will diffuse into a porous material due to capillary forces. The equation is often used in the context of materials science, fluid mechanics, and petroleum engineering, among other fields.

The thin-film equation describes the evolution of a thin liquid film, typically on a solid substrate. This equation is important in fluid dynamics and materials science and is often used in contexts such as coatings, wetting, and thin-film flow dynamics. The thin-film equation can be derived from the Navier-Stokes equations under certain assumptions, specifically when considering a thin film with small thickness compared to its other dimensions.

The Taylor–von Neumann–Sedov (TNNS) blast wave is a theoretical model describing the propagation of a shock wave resulting from an explosion in a homogeneous medium. It is named after three scientists who contributed to the understanding of this phenomenon: G.I. Taylor, J. von Neumann, and L.I. Sedov. The TNNS blast wave model provides a framework for understanding the dynamics of the shock wave and the resulting flow fields in the vicinity of the explosion.

The Taylor–Goldstein equation is a fundamental equation in fluid dynamics and hydrodynamic stability theory, particularly in the study of parallel flows and stability analyses of shear flows. It derives from the linear stability analysis of a basic state in a fluid that is affected by small disturbances.

Stream thrust averaging is a method used in fluid dynamics and aerodynamics to analyze and predict the performance of airfoils, wings, or propellers by averaging the thrust output over a certain stream-wise length or area. This technique is particularly useful in assessing the overall efficiency and behavior of a propulsion system, such as jet engines or helicopters, as it helps to understand how thrust is distributed and how it varies with different operating conditions.

Stokes flow refers to the flow of an incompressible viscous fluid at low Reynolds numbers, where inertial forces can be neglected in comparison to viscous forces. This type of flow is governed by the Stokes equations, which are simplified forms of the Navier-Stokes equations. These equations assume that the fluid is Newtonian, meaning its viscosity is constant and the stress is linearly proportional to the rate of strain.

Stokes' paradox refers to a phenomenon in fluid dynamics that highlights an apparent inconsistency in the flow of a viscous fluid around an object. The paradox is named after the British mathematician and physicist George Gabriel Stokes who analyzed the flow of a viscous (incompressible) fluid around a cylinder. The paradox arises when considering a two-dimensional flow of a viscous fluid past an infinitely long, solid cylinder.

The Shallow Water Equations (SWE) are a set of hyperbolic partial differential equations that describe the flow of a thin layer of fluid, such as water in rivers, lakes, and coastal areas. These equations are particularly useful in hydraulic and environmental engineering for modeling phenomena like flooding, tsunami propagation, and sediment transport. The SWE are derived under the assumption that the horizontal length scale of the fluid flow is much larger than the vertical scale of the fluid depth.

The relativistic Euler equations are a set of equations that describe the dynamics of perfect fluids in the context of relativistic physics. They extend the classical Euler equations, which govern the flow of inviscid (non-viscous), incompressible fluids, to situations where the speeds involved approach the speed of light, or in contexts where relativistic effects are significant, such as in astrophysics or cosmology.

The Rayleigh–Plesset equation is a fundamental equation in the field of fluid dynamics, particularly in the study of cavitation—a phenomenon where vapor bubbles form and collapse in a liquid. The equation describes the dynamics of a spherical gas bubble in an incompressible liquid, accounting for the effects of pressure, surface tension, and viscous forces.

Rayleigh's equation in fluid dynamics refers to a fundamental principle that describes the stability of a fluid flow. It is often associated with the stability analysis of boundary layers and the onset of turbulence and instabilities in various fluid flow situations. One common context in which Rayleigh's equation is discussed is in the study of stability of various flow regimes, particularly in relation to the growth of instabilities in a shear flow. The equation is typically derived from the Navier-Stokes equations under specific assumptions and conditions.

The Rankine–Hugoniot conditions are a set of mathematical conditions used in fluid dynamics and gas dynamics to describe the behavior of shock waves and discontinuities in a medium. These conditions relate the values of physical quantities (such as pressure, density, and velocity) on either side of a discontinuity, which can be a shock wave or a contact discontinuity.

The Oseen equations are a set of equations that describe the steady-state flow of a viscous fluid. They can be seen as a linearization of the Navier-Stokes equations, which govern the motion of fluid substances. The Oseen equations are particularly useful in the study of low Reynolds number flows, where inertial forces are negligible compared to viscous forces.