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For this problem, a rational number p/q is a "good" approximation of a real number α if the absolute value of the difference between p/q and α may not decrease if p/q is replaced by another rational number with a smaller denominator. This problem was solved during the 18th century by means of simple continued fractions.
Stiff problems are ubiquitous in chemical kinetics, control theory, solid mechanics, weather forecasting, biology, plasma physics, and electronics. One way to overcome stiffness is to extend the notion of differential equation to that of differential inclusion, which allows for and models non-smoothness. [24] [25]
Excel maintains 15 figures in its numbers, but they are not always accurate; mathematically, the bottom line should be the same as the top line, in 'fp-math' the step '1 + 1/9000' leads to a rounding up as the first bit of the 14 bit tail '10111000110010' of the mantissa falling off the table when adding 1 is a '1', this up-rounding is not undone when subtracting the 1 again, since there is no ...
A Fermi problem (or Fermi question, Fermi quiz), also known as an order-of-magnitude problem, is an estimation problem in physics or engineering education, designed to teach dimensional analysis or approximation of extreme scientific calculations. Fermi problems are usually back-of-the-envelope calculations.
Borel showed in 1909 that the exceptional set of real pairs (α,β) violating the statement of the conjecture is of Lebesgue measure zero. [2] Manfred Einsiedler, Anatole Katok and Elon Lindenstrauss have shown [3] that it must have Hausdorff dimension zero; [4] and in fact is a union of countably many compact sets of box-counting dimension zero.
There exist inputs to the travelling salesman problem that cause the Christofides algorithm to find a solution whose approximation ratio is arbitrarily close to 3/2. One such class of inputs are formed by a path of n vertices, with the path edges having weight 1, together with a set of edges connecting vertices two steps apart in the path with ...
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