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The electrostatic terms are notoriously difficult to calculate well because they do not fall off rapidly with distance, and long-range electrostatic interactions are often important features of the system under study (especially for proteins). The basic functional form is the Coulomb potential, which only falls off as r −1. A variety of ...
The closest pair of points problem or closest pair problem is a problem of computational geometry: given points in metric space, find a pair of points with the smallest distance between them. The closest pair problem for points in the Euclidean plane [ 1 ] was among the first geometric problems that were treated at the origins of the systematic ...
Note that since the particles in the ideal gas are non-interacting, the probability of finding a particle at a certain distance from another particle is the same as the probability of finding a particle at the same distance from any other point; we shall use the center of the sphere.
In other words, it is the expected Euclidean distance between two random points, where each point in the shape is equally likely to be chosen. Even for simple shapes such as a square or a triangle, solving for the exact value of their mean line segment lengths can be difficult because their closed-form expressions can get quite complicated.
In simplest terms it is a measure of the probability of finding a particle at a distance of away from a given reference particle, relative to that for an ideal gas. The general algorithm involves determining how many particles are within a distance of r {\displaystyle r} and r + d r {\displaystyle r+dr} away from a particle.
Two cases arise: The first case is theoretical: when you know all the coefficients then you take certain limits and find the precise radius of convergence.; The second case is practical: when you construct a power series solution of a difficult problem you typically will only know a finite number of terms in a power series, anywhere from a couple of terms to a hundred terms.
The distance (or perpendicular distance) from a point to a line is the shortest distance from a fixed point to any point on a fixed infinite line in Euclidean geometry. It is the length of the line segment which joins the point to the line and is perpendicular to the line. The formula for calculating it can be derived and expressed in several ways.
The distance from a point to a plane in three-dimensional Euclidean space [7] The distance between two lines in three-dimensional Euclidean space [8] The distance from a point to a curve can be used to define its parallel curve, another curve all of whose points have the same distance to the given curve. [9]