Francis

@Francis16833887

Oxford, England
Vrijeme pridruživanja: veljača 2019.
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  1. 20. lis 2019.

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  2. 17. lis 2019.

    It turns out that the tropical numbers are multiplicatively closed, once we've properly defined multiplicity, and accounted for the fact that roots may end up at minus infinity! Can you prove it? 16/n, n=16.

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  3. 17. lis 2019.

    Here's an example of the graph of tropical x^2 +x, which doesn't have a constant term. One of its roots has gone off to minus infinity! Thankfully, this point is included in the tropical numbers, so we still have 2 roots. 15/n

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  4. 17. lis 2019.

    Notice that the multiplicity of a root depends on the amount by which the slope of the graph changes! Notice also that a quadratic polynomial always has exactly 2 roots, counted with multiplicity. That is, as long as we have a constant term! 14/n

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  5. 17. lis 2019.

    And once we've made this definition, we can see that the tropical world behaves much more nicely than the real one! For example, the quadratic equation takes a very simple form! 13/n

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  6. 17. lis 2019.

    The roots of a tropical polynomial are defined to be the points where it's graph is discontinuous, or equivalently, where the maximum of it's linear terms is achieved more than once. This may seem like a weird definition, but the story I told gives the motivation. 12/n

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  7. 17. lis 2019.

    And below is it's graph. The discontinuities occur at -3 and 3, where the maximum of the linear terms is achieved twice. 11/n

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  8. 17. lis 2019.

    The tropical version of our polynomial is given by the following. 10/n

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  9. 17. lis 2019.

    The polynomial can only vanish when the two dominant terms are comparable. If we take x to be approximately +/- 10^y, then this occurs precisely when y=-3 and y=3. Below we see that this indeed gives a good approximation to the roots. 9/n

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  10. 17. lis 2019.

    Tropical numbers are useful when we care about solving problems approximately. For example, suppose we want to solve the following polynomial, but we only care about the order of magnitude of the solutions. 8/n

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  11. 17. lis 2019.

    By the way, some mathematicians let t = exp(1/T), and call T the temperature. Then the tropical addition occurs in the limit of zero temperature. There's also an interpretation of this limit as a dequantization. I might get to this later. 7/n

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  12. 17. lis 2019.

    However, this map does not preserve the addition. Instead we get an entire family of t-dependent additions on the set of tropical numbers. And if we let t go to positive infinity, we recover the tropical addition! 6/n

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  13. 17. lis 2019.

    Using the logarithm base t, for a positive number t, we can identify the semi-field of non-negative reals with the tropical numbers. 5/n

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  14. 17. lis 2019.

    Where do they come from? The non-negative real numbers, with the normal operations, also give a semi-field. It's clear why there are no additive inverses in this case: we've thrown away all the negative numbers! 4/n

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  15. 17. lis 2019.

    Tropical addition is also idempotent, meaning that a + a = a, for each tropical number a. 3/n

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  16. 17. lis 2019.

    The tropical numbers give an example of a semi-field. This means that they satisfy all the field axioms except for existence of additive inverses. The tropical 0 is minus infinity. Can you see why there are no inverses? 2/n

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  17. 17. lis 2019.

    The tropical numbers are defined as the real numbers with minus infinity added in, along with the operations of tropical addition and multiplication. 1/n

    2 proslijeđena tweeta 5 korisnika označava da im se sviđa
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  18. proslijedio/la je Tweet
    10. lis 2019.
    Odgovor korisniku/ci

    It's so funny, I have this weird like self-deprecation/hyper-integrity thing where I feel the need to say "So this next part is really cool, but it was actually ________'s idea."

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  19. proslijedio/la je Tweet
    23. kol 2019.

    We Need an International Center for Climate Modeling The science community must join forces to provide the most accurate long-term predictions and make their results publicly accessible

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  20. 9. svi 2019.

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