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In nuclear physics, the Bateman equation is a mathematical model describing abundances and activities in a decay chain as a function of time, based on the decay rates and initial abundances. The model was formulated by Ernest Rutherford in 1905 [1] and the analytical solution was provided by Harry Bateman in 1910. [2]
One of the most popular techniques to accomplish this is the use of key-rate durations (KRDs), introduced by Thomas Ho (1992). Ho defines a number of maturities on the yield curve as being the key rate durations, with typical values of 3 months, 1, 2, 3, 5, 7, 10, 15, 20, 25 and 30 years.
A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation , where N is the quantity and λ ( lambda ) is a positive rate called the exponential decay constant , disintegration constant , [ 1 ] rate constant , [ 2 ] or ...
Some investment products earn interest that works similarly to a variable rate. For example, floating-rate notes (FRNs) have rates based on the 13-week Treasury bill, plus a spread — similar to ...
The formula above can be used for more than calculating the doubling time. If one wants to know the tripling time, for example, replace the constant 2 in the numerator with 3. As another example, if one wants to know the number of periods it takes for the initial value to rise by 50%, replace the constant 2 with 1.5.
In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the distance between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate; the distance parameter could be any meaningful mono-dimensional measure of the process, such as time ...
In this case, E is the center-of-mass energy that produces the resonance, M is the mass of the resonance, and Γ is the resonance width (or decay width), related to its mean lifetime according to τ = 1 / Γ. (With units included, the formula is τ = ħ / Γ.)
A formula that is accurate to within a few percent can be found by noting that for typical U.S. note rates (< % and terms =10–30 years), the monthly note rate is small compared to 1. r << 1 {\displaystyle r<<1} so that the ln ( 1 + r ) ≈ r {\displaystyle \ln(1+r)\approx r} which yields the simplification: