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Often the independent variable is time. Described as a function, a quantity undergoing exponential growth is an exponential function of time, that is, the variable representing time is the exponent (in contrast to other types of growth, such as quadratic growth). Exponential growth is the inverse of logarithmic growth.
RGR is a concept relevant in cases where the increase in a state variable over time is proportional to the value of that state variable at the beginning of a time period. In terms of differential equations, if is the current size, and its growth rate, then relative growth rate is
First order LTI systems are characterized by the differential equation + = where τ represents the exponential decay constant and V is a function of time t = (). The right-hand side is the forcing function f(t) describing an external driving function of time, which can be regarded as the system input, to which V(t) is the response, or system output.
Upon comparing the ratio on the left hand side to the equation on the right hand side, we conclude that the ratio between the final and initial concentrations follows an exponential function, of which e is the base. As mentioned above, the units for k are inverse time. If we were to take the reciprocal of this, we would be left with units of time.
The stretched exponential is also the characteristic function, basically the Fourier transform, of the Lévy symmetric alpha-stable distribution. In physics, the stretched exponential function is often used as a phenomenological description of relaxation in disordered systems.
In mathematics, the exponential function can be characterized in many ways. This article presents some common characterizations, discusses why each makes sense, and proves that they are all equivalent. The exponential function occurs naturally in many branches of mathematics. Walter Rudin called it "the most important function in mathematics". [1]
Exponential polynomials—see also Touchard polynomials (combinatorics) Exponential response formula; Exponential sheaf sequence; Exponential smoothing; Exponential stability; Exponential sum; Exponential time. Sub-exponential time; Exponential tree; Exponential type; Exponentially equivalent measures; Exponentiating by squaring; Exponentiation ...
There are no inherent limitations on the number of variables, parameters etc. Lyap which includes source code written in Fortran, can also calculate the Lyapunov direction vectors and can characterize the singularity of the attractor, which is the main reason for difficulties in calculating the more negative exponents from time series data.