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The sine and tangent small-angle approximations are used in relation to the double-slit experiment or a diffraction grating to develop simplified equations like the following, where y is the distance of a fringe from the center of maximum light intensity, m is the order of the fringe, D is the distance between the slits and projection screen ...
where is some approximation of the Jacobian matrix of (i.e. Hessian of the objective function) which satisfies the secant equation =. Barzilai and Borwein simplify B {\displaystyle B} with a scalar 1 / α {\displaystyle 1/\alpha } , which usually cannot exactly satisfy the secant equation, but approximate it as 1 α Δ x ≈ Δ g {\displaystyle ...
Small-angle scattering (SAS) is a scattering technique based on deflection of collimated radiation away from the straight trajectory after it interacts with structures that are much larger than the wavelength of the radiation. The deflection is small (0.1-10°) hence the name small-angle. SAS techniques can give information about the size ...
In numerical analysis, the secant method is a root-finding algorithm that uses a succession of roots of secant lines to better approximate a root of a function f. The secant method can be thought of as a finite-difference approximation of Newton's method, so it is considered a quasi-Newton method.
A circular segment (in green) is enclosed between a secant/chord (the dashed line) and the arc whose endpoints equal the chord's (the arc shown above the green area). In geometry, a circular segment or disk segment (symbol: ⌓) is a region of a disk [1] which is "cut off" from the rest of the disk by a straight line.
When the zenith angle is small to moderate, a good approximation is given by assuming a homogeneous plane-parallel atmosphere (i.e., one in which density is constant and Earth's curvature is ignored). The air mass then is simply the secant of the zenith angle : =.
For amplitudes beyond the small angle approximation, one can compute the exact period by first inverting the equation for the angular velocity obtained from the energy method , = and then integrating over one complete cycle, = (), or twice the half-cycle = (), or four times the quarter-cycle = (), which leads to = .
The equation must satisfy the condition that ′ = (no penetration on the solid surface) and also must correspond to conditions behind the shock wave at =, where is the half-angle of shock cone, which must be determined as part of the solution for a given incoming flow Mach number and . The Taylor–Maccoll equation has no known explicit ...