MetaTalk posts tagged with pvsnp
http://metatalk.metafilter.com/tags/pvsnp
Posts tagged with 'pvsnp' at MetaTalk.Tue, 01 Dec 2015 16:13:38 -0800Tue, 01 Dec 2015 16:13:38 -0800en-ushttp://blogs.law.harvard.edu/tech/rss60Rate of change pedantry
http://metatalk.metafilter.com/23942/Rate%2Dof%2Dchange%2Dpedantry
I don't want to single out a specific recent post, because my pedantic complaint is so not relevant and would harmfully distract from the discussion, but I have a request, my fellow Mefites: "Increasing at an exponential rate" or "increasing exponentially" means that the number of incidences during this time period are some multiple of the number during the previous time period, which were some multiple of the number during the time period before that, etc. If you're comfortable with formulas and algebra, this means
N(t) = r*N(t-1)
N(t-1) = r*N(t-2)
etc.
If M was the original number at some time that we are taking as the initial time zero or baseline, then
N(k) = M*r*r*...*r
for whole numbers k, where the "..." indicates that k copies of r are being multiplied together. Multiplying r by itself k times is what exponential notation means, so a more compact way of writing this formula is with an exponent:
N(k) = M*r^k
In general, if t is any (not necessarily whole number) time past our baseline (initial) time that we are calling time zero, an exponential function with t in the exponent can be defined (kind of by interpolation), and so we still have
N(t) = M*r^t
This is as opposed to a rate of change following a different function. For example, the rate of growth of carbon dioxide in the atmosphere is quadratic - it follows a function that looks like:
Amount of CO_2 t years after we started counting = (some constant)*(t^2) + (a second constant)*t + (a third constant).
In math and computer science, we talk about different "complexity classes" to indicate whether the function that describes a rate of growth is an exponential function, a polynomial function of some power, a logarithm, or whatever. (We don't worry much about the constant coefficients because when t gets large, their influence is negligible.) We care about different complexity classes because it makes a real difference.
Example 1: computer encryption and everything we use the internet for works because the amount of time it takes, using our best algorithms, to factor a number grows at a faster-than-polynomial rate as the size (number of digits) of the number to be factored increases. If you've ever heard of P versus NP, for example, P refers to the class of computations where the time to solve the problem grows at only a polynomial rate with the size of input. Factoring is general believed to be a harder-than-P problem. If anyone ever discovers a polynomial-time algorithm for factoring, encryption and basically the entire internet will break. All of your financial information, private email and facebook conversations, personal photos on your cell phone, and any networked medical or criminal records will be easily accessible to anyone with a reasonably current computer, mediocre technical knowledge, and a grudge or curiosity to motivate them.
Example 2: if the amount of carbon dioxide in the atmosphere were growing at an exponential rate rather than a quadratic rate, then Earth's atmosphere would already be like Venus and probably none of us would be around to care about misuse of the phrase "exponentially increasing". (In fact, if the amount of carbon dioxide were increasing at a higher degree polynomial rate, like even a fourth degree polynomial rate, we'd already be experiencing much more severe consequences worldwide.)
Example 3: if rates of gun violence in the US were increasing exponentially, enough people would be dead by now that we might actually see some gun policy change and the NRA might be unpopular enough that they might disband.
None of this discussion should in any way be misconstrued as arguing against the seriousness of current rates of growth in atmospheric carbon dioxide or gun violence in the US, and this pedantry does not belong in threads on those or similar topics. Just, where possible, could you do a mathematician a favor and try to avoid misusing the term "increasing exponentially"? Thanks! tag:metatalk.metafilter.com,2015:site.23942Tue, 01 Dec 2015 16:13:38 -0800calculuscomplexityexponentialMathpedantrypolynomialpvsnprateofchangerateofgrowtheviemath