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The no-three-in-line problem in discrete geometry asks how many points can be placed in the grid so that no three points lie on the same line. The problem concerns lines of all slopes, not only those aligned with the grid. It was introduced by Henry Dudeney in 1900. Brass, Moser, and Pach call it "one of the oldest and most extensively studied ...
For example, it was used in the analysis of the Markov equation back in 1879 and in the 1953 paper of Mills. [ 1 ] In 1988, the method came to the attention to mathematical olympiad problems in the light of the first olympiad problem to use it in a solution that was proposed for the International Mathematics Olympiad and assumed to be the most ...
The jump can be iterated into transfinite ordinals: there are jump operators for sets of natural numbers when is an ordinal that has a code in Kleene's (regardless of code, the resulting jumps are the same by a theorem of Spector), [2] in particular the sets 0 (α) for α < ω 1 CK, where ω 1 CK is the Church–Kleene ordinal, are closely ...
The term ‖ ‖ = # {: +} penalizes the number of jumps and the term ‖ ‖ = = | | measures fidelity to data x. The parameter γ > 0 controls the tradeoff between regularity and data fidelity . Since the minimizer u ∗ {\displaystyle u^{*}} is piecewise constant the steps are given by the non-zero locations of the gradient ∇ u ∗ ...
The algorithm can be modified by performing multiple levels of jump search on the sublists, before finally performing the linear search. For a k-level jump search the optimum block size m l for the l th level (counting from 1) is n (k-l)/k. The modified algorithm will perform k backward jumps and runs in O(kn 1/(k+1)) time.
In mathematics, a jumping line or exceptional line of a vector bundle over projective space is a projective line in projective space where the vector bundle has exceptional behavior, in other words the structure of its restriction to the line "jumps". Jumping lines were introduced by R. L. E. Schwarzenberger in 1961. [1] [2] The jumping lines ...
For example, the infinite sequence (,, …) of the natural numbers increases infinitively and has no upper bound in the real number system (a potential infinity); in the extended real number line, the sequence has + as its least upper bound and as its limit (an actual infinity).
The Heaviside step function is an often-used step function.. A constant function is a trivial example of a step function. Then there is only one interval, =. The sign function sgn(x), which is −1 for negative numbers and +1 for positive numbers, and is the simplest non-constant step function.