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Infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of matter , space , time , money , or abstract mathematical objects such as the continuum .
This presents Zeno's problem not with finding the sum, but rather with finishing a task with an infinite number of steps: how can one ever get from A to B, if an infinite number of (non-instantaneous) events can be identified that need to precede the arrival at B, and one cannot reach even the beginning of a "last event"? [10] [11] [12] [13]
The concept of infinite divisibility of probability distributions was introduced in 1929 by Bruno de Finetti. This type of decomposition of a distribution is used in probability and statistics to find families of probability distributions that might be natural choices for certain models or applications. Infinitely divisible distributions play ...
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Using a mathematical approach, specifically geometric models, Gottfried Leibniz and Descartes discussed the infinite divisibility of extension. Actual divisibility may be limited due to unavailability of cutting instruments, but its possibility of breaking into smaller pieces is infinite.
Barndorff-Nielsen and Halgreen proved that the GIG distribution is infinitely divisible and since the GH distribution can be obtained as a normal variance-mean mixture where the mixing distribution is the generalized inverse Gaussian distribution, Barndorff-Nielsen and Halgreen showed the GH distribution is infinitely divisible as well.
A graph that shows the number of balls in and out of the vase for the first ten iterations of the problem. The Ross–Littlewood paradox (also known as the balls and vase problem or the ping pong ball problem) is a hypothetical problem in abstract mathematics and logic designed to illustrate the paradoxical, or at least non-intuitive, nature of infinity.
The solution (g, h, k) is another solution to the original equation, but smaller (0 < g < d < x). Applying the same procedure to (g, h, k) would produce another solution, still smaller, and so on. But this is impossible, since natural numbers cannot be shrunk indefinitely. Therefore, the original solution (x, y, z) was impossible.
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