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A polytropic process is a thermodynamic process that obeys the relation: = where p is the pressure , V is volume , n is the polytropic index , and C is a constant. The polytropic process equation describes expansion and compression processes which include heat transfer.
The normalized density as a function of scale length for a wide range of polytropic indices. In astrophysics, a polytrope refers to a solution of the Lane–Emden equation in which the pressure depends upon the density in the form = (+) / = + /, where P is pressure, ρ is density and K is a constant of proportionality. [1]
The particular choice of a polytropic gas as given above makes the mathematical statement of the problem particularly succinct and leads to the Lane–Emden equation. The equation is a useful approximation for self-gravitating spheres of plasma such as stars, but typically it is a rather limiting assumption.
In a nozzle or other constriction, the discharge coefficient (also known as coefficient of discharge or efflux coefficient) is the ratio of the actual discharge to the ideal discharge, [1] i.e., the ratio of the mass flow rate at the discharge end of the nozzle to that of an ideal nozzle which expands an identical working fluid from the same initial conditions to the same exit pressures.
The strain hardening exponent (also called the strain hardening index), usually denoted , is a measured parameter that quantifies the ability of a material to become stronger due to strain hardening. Strain hardening (work hardening) is the process by which a material's load-bearing capacity increases during plastic (permanent) strain , or ...
It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential gives the exponential map between a matrix Lie algebra and the corresponding Lie group. Let X be an n×n real or complex matrix. The exponential of X, denoted by e X or exp(X), is the n×n matrix given by the power series = =!
Toyesh Prakash Sharma, Etisha Sharma, "Putting Forward Another Generalization Of The Class Of Exponential Integrals And Their Applications.," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.10, Issue.2, pp.1-8, 2023.
To introduce Lyapunov exponent consider a fundamental matrix () (e.g., for linearization along a stationary solution in a continuous system), the fundamental matrix is (() |) consisting of the linearly-independent solutions of the first-order approximation of the system.