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In mathematics, summation by parts transforms the summation of products of sequences into other summations, often simplifying the computation or (especially) estimation of certain types of sums. It is also called Abel's lemma or Abel transformation , named after Niels Henrik Abel who introduced it in 1826.
Seth Lloyd proposed an alternative approach to time travel with closed timelike curves (CTCs), based on "post-selection" and path integrals. [21] Path integrals are a powerful tool in quantum mechanics that involve summing probabilities over all possible ways a system could evolve, including paths that do not strictly follow a single timeline ...
Another simple activity is predicting the position of a vehicle from its initial position, direction and speed of travel, using the equation that distance traveled is the product of time and speed. This is known as dead reckoning when used more formally. Mathematical modeling in this way does not necessarily require formal mathematics; animals ...
Outer Wilds (2019): A video game involving time travel which does not follow the principle, causing a game over if the player experiments to test it. All time travel in the Hallmark Channel original series The Way Home follows the Novikov self-consistency principle. Two of the main characters can travel backwards in time by jumping into a pond ...
The S-matrix in quantum field theory is an example of a time-ordered product. The S-matrix, transforming the state at t = −∞ to a state at t = +∞, can also be thought of as a kind of "holonomy", analogous to the Wilson loop. We obtain a time-ordered expression because of the following reason: We start with this simple formula for the ...
For example, given a = f(x) = a 0 x 0 + a 1 x 1 + ··· and b = g(x) = b 0 x 0 + b 1 x 1 + ···, the product ab is a specific value of W(x) = f(x)g(x). One may easily find points along W(x) at small values of x, and interpolation based on those points will yield the terms of W(x) and the specific product ab. As fomulated in Karatsuba ...
Thus, it is possible that the worst-case running time for any algorithm for the TSP increases superpolynomially (but no more than exponentially) with the number of cities. The problem was first formulated in 1930 and is one of the most intensively studied problems in optimization. It is used as a benchmark for many optimization methods.
It runs in polynomial time on inputs that are in SUBSET-SUM if and only if P = NP: // Algorithm that accepts the NP-complete language SUBSET-SUM. // // this is a polynomial-time algorithm if and only if P = NP. // // "Polynomial-time" means it returns "yes" in polynomial time when // the answer should be "yes", and runs forever when it is "no".