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Each optical element (surface, interface, mirror, or beam travel) is described by a 2 × 2 ray transfer matrix which operates on a vector describing an incoming light ray to calculate the outgoing ray. Multiplication of the successive matrices thus yields a concise ray transfer matrix describing the entire optical system.
The study of hydrodynamic stability aims to find out if a given flow is stable or unstable, and if so, how these instabilities will cause the development of turbulence. [1] The foundations of hydrodynamic stability, both theoretical and experimental, were laid most notably by Helmholtz, Kelvin, Rayleigh and Reynolds during the nineteenth ...
The paradigmatic case is the stability of the origin under the linear autonomous differential equation ˙ = where = [] and is a 2-by-2 matrix. We would sometimes perform change-of-basis by X ′ = C X {\displaystyle X'=CX} for some invertible matrix C {\displaystyle C} , which gives X ˙ ′ = C − 1 A C X ′ {\displaystyle {\dot {X}}'=C^{-1 ...
An exponentially stable LTI system is one that will not "blow up" (i.e., give an unbounded output) when given a finite input or non-zero initial condition. Moreover, if the system is given a fixed, finite input (i.e., a step ), then any resulting oscillations in the output will decay at an exponential rate , and the output will tend ...
The average photon numbers of the three states from top to bottom are n =4.2, 25.2, 924.5 [5] Figure 2: The oscillating wave packet corresponding to the second coherent state depicted in Figure 1. At each phase of the light field, the distribution is a Gaussian of constant width. Figure 3: Wigner function of
The first solution to this problem was provided by Freeman Dyson and Andrew Lenard in 1967–1968, [1] [2] but a shorter and more conceptual proof was found later by Elliott Lieb and Walter Thirring in 1975 using the Lieb–Thirring inequality. [3] The stability of matter is partly due to the uncertainty principle and the Pauli exclusion ...
Scientists have long searched for long-lived heavy isotopes outside of the valley of stability, [6] [7] [8] hypothesized by Glenn T. Seaborg in the late 1960s. [9] [10] These relatively stable nuclides are expected to have particular configurations of "magic" atomic and neutron numbers, and form a so-called island of stability.
For i = 1, 2, 3, … v = Ap i−1; α = ρ i−1 /(r̂ 0, v) h = x i−1 + αp i−1; s = r i−1 − αv; If h is accurate enough, i.e., if s is small enough, then set x i = h and quit; t = As; ω = (t, s)/(t, t) x i = h + ωs; r i = s − ωt; If x i is accurate enough, i.e., if r i is small enough, then quit; ρ i = (r̂ 0, r i) β = (ρ i ...