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In the following equations, denotes the sagitta (the depth or height of the arc), equals the radius of the circle, and the length of the chord spanning the base of the arc. As 1 2 l {\displaystyle {\tfrac {1}{2}}l} and r − s {\displaystyle r-s} are two sides of a right triangle with r {\displaystyle r} as the hypotenuse , the Pythagorean ...
For a fixed length n, the Hamming distance is a metric on the set of the words of length n (also known as a Hamming space), as it fulfills the conditions of non-negativity, symmetry, the Hamming distance of two words is 0 if and only if the two words are identical, and it satisfies the triangle inequality as well: [2] Indeed, if we fix three words a, b and c, then whenever there is a ...
is the distance over which the initial time is achieved D 2 {\displaystyle D_{2}} is the distance for which the time is to be predicted Riegel expanded on his thesis in a 1981 article for American Scientist , stating that the formula t = a x b {\displaystyle t=ax^{b}} concerns activities in the "endurance range", namely lasting between 3.5 and ...
Also, the velocities in the directions perpendicular to the frame changes are affected, as shown above. This is due to time dilation, as encapsulated in the dt/dt′ transformation. The V′ y and V′ z equations were both derived by dividing the appropriate space differential (e.g. dy′ or dz′) by the time differential.
Given the coordinates of the two points (Φ 1, L 1) and (Φ 2, L 2), the inverse problem finds the azimuths α 1, α 2 and the ellipsoidal distance s. Calculate U 1, U 2 and L, and set initial value of λ = L. Then iteratively evaluate the following equations until λ converges:
If is the radius of the incircle of the triangle, then the triangle can be broken into three triangles of equal altitude and bases , , and . Their combined area is A = 1 2 a r + 1 2 b r + 1 2 c r = r s , {\displaystyle A={\tfrac {1}{2}}ar+{\tfrac {1}{2}}br+{\tfrac {1}{2}}cr=rs,} where s = 1 2 ( a + b + c ...
By carefully writing the above equations as matrix equations, we obtain its dual problem: [15] {, () + () + (,) and by the duality theorem of linear programming, since the primal problem is feasible and bounded, so is the dual problem, and the minimum in the first problem equals the maximum in the second problem.
In the context of general relativity, it means the problem of finding solutions to Einstein's field equations — a system of hyperbolic partial differential equations — given some initial data on a hypersurface. Studying the Cauchy problem allows one to formulate the concept of causality in general relativity, as well as 'parametrising ...